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Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends

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UNIVERSITY OF CALIFORNIA,
IRVINE
Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Electrical and Computer Engineering
by
Javier Rodriguez De Luis
Dissertation Committee:
Professor Franco De Flaviis, Chair
Associate Professor Pai H. Chou
Assistant Professor Filippo Capolino
2011
UMI Number: 3456974
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent on the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 3456974
Copyright 2011 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106 - 1346
Chapter 2 © 2010 IEEE
Portion of Chapter 3 © 2009 IEEE.
Portion of Chapters 4 to 7 © 2011 IEEE
All other materials © 2011 Javier R. De Luis
Dedication
To
my parents
ii
TABLE OF CONTENTS
Page
LIST OF FIGURES
vii
LIST OF TABLES
xvii
ACKNOWLEDGMENTS
xix
CURRICULUM VITAE
xx
ABSTRACT OF THE DISSERTATION
xxi
1. INTRODUCTION
1
1.1
Interest in Reconfigurable antennas and microwave circuits.
1
1.2
Document organization
3
2. A FREQUENCY AGILE SWTICHED BEAM ANTENNA ARRAY SYSTEM
6
2.1
Introduction
6
2.2
Design of the single antenna element
9
2.2.1 Gap Size Effect on the resonant frequencies.
11
2.2.2 Optimum number of diodes analysis.
12
2.2.3 Radiation Efficiency analysis
16
The dual frequency phase shifter
17
2.3.1 The Dual Frequency phase delay section.
20
2.3.2 The Dual Frequency DC Bias.
21
2.4
Simulations and measurement results
22
2.5
Conclusions
28
2.3
References
28
3. BANDWIDTH ENHANCEMENT OF HIGH ISOLATION ISO-FREQUENCY REPEATERS USING
MEMS-RECONFIGURABLE LOADED PARASITICS
32
3.1
Introduction
32
3.2
Description of the repeater system
34
3.3
Design Of In-plane (IPPs) and Out-of-plane (OUPPs) parasitics
36
iii
3.4 Combined isolation and matching optimization using the genetic algorithm
optimizer.
37
3.5
Isolation and impedance bandwidth enhancement.
40
3.6
Additional Information.
41
43
References
4. AN ANALYTICAL ALGORITHM FOR PI-NETWORK IMPEDANCE TUNERS.
45
4.1
Introduction
45
4.2
Matching parameters and tunable pi-network configuration
47
4.3
Explanation of composite components frequently used in the algorithm
49
4.3.1 Equivalent Series Inductance & Its Characteristics
50
4.3.2 Composite Load and Source Impedance
51
Formulas and algorithm for perfect match
53
4.4.1 Perfect Match Solutions
53
4.4.2 Algorithm for Perfect Match
54
Formulas and algorithm for best match
55
4.5.1 Formulas for Best Match
55
4.5.2 Algorithm for Best Match
59
4.5.3 Algorithm for Branch 3
60
4.6
Comparison of algorithm and optimizer results
62
4.7
Algorithm extension for different Pi-network topologies.
67
4.4
4.5
4.7.1 Shunt Tunable Capacitor Having a Fixed Inductor Connected in Parallel. 68
4.8
4.9
4.7.2 Series portion of a pi-network consisting of series LC combination
68
Application examples.
69
4.8.1 Replacing Optimizer to Analyze Matching Performance
69
4.8.2 Predetermining the Match Tuning Setting for Known Antenna Load
Impedance
70
4.8.3 Control Algorithm for Closed loop Impedance Matching
71
Conclusions.
73
Appendix A. Derivation of formulas (4.14)-(4.17)
iv
73
Appendix B. Derivation of input reflection coefficient.
76
Appendix C. Parameters of Pi-Network tuner.
77
References
78
5. TUNABLE MEMS NOTCH FILTER AND ITS FREQUENCY CONTROL LOOP FOR WIRELESS
TRANSCEIVERS
81
5.1
Introduction
81
5.2
Tunable Notch filter design
83
5.3
Notch filter frequency control loop based on reflected reference signal
85
5.3.1 System Description.
85
5.3.2 Mathematical Model and Formulation
89
5.3.3 Calculations and Simulations of Notch Filter Frequency Control Loop
Performance
92
Implementations and measurements
96
5.4.1 Standalone Tunable Filter
96
5.4.2 Notch Filter Control Loop
100
Conclusions
104
5.4
5.5
Appendix A. Derivation of the transfer function of the notch filter.
104
Appendix B. Derivation of the frequency control loop differential equation.
105
References
107
6. APPROXIMATE CLOSE FORM DESIGN EXPRESSIONS FOR CAPACITIVELY LOADED PLANAR
INVERTED-F ANTENNA
109
6.1
Introduction
109
6.2
Closed form expressions for capacitively loaded PIFA design
111
6.2.1 Transmission line model for a capacitively loaded PIFA
111
6.2.2 Input impedance.
113
6.2.3 Determining the patch length for a given operation frequency and loading
capacitance.
115
6.2.4 Required loading capacitance for a given length and operating
frequency.
118
6.2.5 Radiation efficiency and quality factor.
119
v
6.2.6 Impedance bandwidth.
126
6.2.7 Design Guidelines.
128
6.3
Measurement results
130
6.4
Conclusions
133
134
References
7. TUNABLE DUPLEXING ANTENNA SYSTEM FOR WIRELESS TRANSCEIVERS
137
7.1
Introduction
137
7.2
Duplexing Antenna System
139
7.2.1 Advantages from a system perspective
139
7.2.2 Single Tunable Element Design
141
7.2.3 Antenna Pair Design
148
7.2.4 Single Antenna Radiated Test
152
Conclusions
153
7.3
156
References
vi
LIST OF FIGURES
Page
Fig. 1.1 Conventional non tunable cell phone system architecture from Wispry Inc
(www.wispry.com). ........................................................................................................... 2
Fig. 1.2 Novel tunable cell phone system architecture from Wispry Inc (www.wispry.com). ...... 2
Fig. 2.1 (a) Ominidirectional antenna system with single frequency operation showing
interference between aircrafts. (b) switched beam dual frequency antenna system
proposed in this chapter. .................................................................................................... 7
Fig. 2.2 Complete system diagram formed by a 4x1 dual frequency phased array and two
different types of switched line phase shifters (Type 1, Type 2) ...................................... 8
Fig. 2.3 The antenna element consists of a dual microstrip patch topology connected through PIN
diodes, printed over RT duroid 5880 (εr=2.2,tanδ=0.0009, 35µm copper thickness ,
1.575mm dielectric thickness) with dimensions: Lhf=11.1mm, Whf=12.15mm,
Llf=19.5mm, Wlf=21mm, g=0.5mm, gd=0.3mm, iip=2mm, iop=0.34mm,
Wdc=0.2mm. Diode length=0.7mm, width=0.3mm and pad gap=0.3mm ....................... 9
Fig. 2.4 Diode circuit model for the ‘On’ and ‘Off’ states. Rs=1.52Ω, Ls=0.25nH and Coff=47fF,
Rp=10KΩ. Maximum insertion loss @4-8GHz 0.4dB. Minimum isolation @4-8GHz
11dB. ............................................................................................................................... 10
Fig. 2.5 Scalar current distribution over the antenna at 4.7GHz for the cases: (a) Ideal ‘On’ with
metallic connections (b) Ideal ‘Off’ with no connections, (c) Real ‘On’ with diode
connections and (d) Real ‘Off’ with diode connections. Center of figure, standard
rectangular patch resonating at 4.7GHz. ......................................................................... 14
vii
Fig. 2.6 Resonant frequency change with increasing gap between IP and OP for cases: (a) Ideal
‘On’: Metal connection between Patches (low frequency) (b) Ideal ‘Off’: No connection
between patches (high frequency), (c) Real ‘On’: Diodes between patches in ‘On’ state
(low frequency) and (d) Real ‘Off’: Diodes between patches in ‘Off’ state (high
frequency). ....................................................................................................................... 14
Fig. 2.7 Scalar current distribution over antenna different number of diodes in ‘On state: 1 diode
(a) first resonance, (b) second resonance; 2 diodes (c) first resonance, (d) second
resonance; 3 diodes (e) first resonance, (f) second resonance and 5 diodes (j) first
resonance, (k) second resonance ..................................................................................... 15
Fig. 2.8 Imaginary part of the antenna edge impedance versus frequency showing first and
second resonances for the cases of (a) 1 diode ‘On’, (b) 3 diodes ‘On’ and (c) 3 diodes
‘On’ ................................................................................................................................. 15
Fig. 2.9 Normalized radiation pattern (dB) corresponding to the second resonance of cases (a) 1
diode ‘On’ (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’ ........................................................ 15
Fig. 2.10 Radiation Efficiency as a function of IP and OP slit lengths. The dynamic range
between worst and best results is found to be 35%. ........................................................ 17
Fig. 2.11 Complete Type 1 Phase Shifter with phase delays from top to bottom branch of
{0,45,90,135} degrees ..................................................................................................... 19
Fig. 2.12 Zoom into the phase shifter input region that contains the DF matching network, input
SP4T distribution section and equal length launches. The Output SP4T distribution
section is also included in the figure to show the direction of the diodes. ...................... 19
Fig. 2.13 Dual frequency phase delay section with dual stub configuration and showing
optimization variables. .................................................................................................... 20
viii
Fig. 2.14 Simulation results from a zero degree phase delay section.Input/Output magnitude of
the reflection coefficient in dB (left axis) and Input/Output phase delay in degrees (right
axis). Both frequencies remain matched while providing the desired zero degrees phase
delays. .............................................................................................................................. 21
Fig. 2.15 Dual frequency DC bias formed by two quarter wavelength stubs, a dual frequency
quarterwave section and a DC input pad to bias the diodes. ........................................... 22
Fig. 2.16 Fabricated Single element antenna and Zoom into the diode region. The devices are
attached using silver epoxy to avoid overheating ........................................................... 24
Fig. 2.17 Simulated versus measured antenna magnitude of the reflection coefficient. For
simplicity, the cases ‘On’ and ‘Off’ corresponding to operation at 4.7GHz and 7.5GHz
are superimposed in the same graph. ............................................................................... 25
Fig. 2.18 Measured radiation patterns for the single element antenna operating in the (a-b) ‘On’
and (c-d) ‘Off’ cases. (a) and (c) show the E-plane while (b) and (d) show the H-plane.
Solid and dashed line represents co-pol and cross-pol components respectively. The
measured efficiency is 79% and 68% with associated gains of 5.4dB and 5dB at 7.5GHz
and 4.7GHz, respectively. ............................................................................................... 25
Fig. 2.19 Fabricated 4x1 switched beam antenna array and phase shifters. ................................. 26
Fig. 2.20 Simulated with IE3D infinite ground (solid) and measured (dashed) normalized E-total
radiation patterns for 4.7GHz (lf) and 7.5GHz (hf) corresponding to different
progressive phases between antennas (a) lf,α=-135 sim-gain=10.8dB (b)lf, α=-45, simgain=12.5dB (c) lf,α=+135,sim-gain=10.5dB (d) lf,α=+45 sim-gain=13.7dB; (e) hf, α=135 sim- gain=14.8dB (f) hf, α=-45 sim-gain=14.7dB (g) hf, α=+135 sim-gain=12.7dB
(h) hf, α=+45 sim-gain=15.2dB. ..................................................................................... 27
ix
Fig. 3.1 Block Diagram of an Iso-frequency repeater. ................................................................. 33
Fig. 3.2 Proposed Iso-frequency repeater using RF-MEMS loaded parasitics. ............................ 35
Fig. 3.3 UCI’s RF MEMS switch picture. This specific device was built over a quartz substrate
and is intended to be used as depicted in Fig. 3.2 ........................................................... 35
Fig. 3.4 A) OUPPs detail with parameters wo =0.5mm,= so =5mm,= do =20mm No =8 (each side)
B) IPPs details. wi=0.5mm,= si =0.75mm Ni=46 (each side). ......................................... 37
Fig. 3.5 Network cascaded connection representing the loading conditions for each of the
parasitic ports. The loading block is represented by a set of reflection coefficients being
+1 for the open circuit and -1 for the short circuit conditions. ........................................ 39
Fig. 3.6 Results after optimization. (a) case 1, no obstacle. (b) case 2, obstacle at 1λx0.5λ. ....... 42
Fig. 3.7 Results after optimization of the isolation bandwidth. Case 2 and different fitness
functions. ......................................................................................................................... 42
Fig. 3.8 a) Radiation patterns after optimization for case 1 and 2. b) Aperture coupled microstrip
antenna with optimized impedance bandwidth and high isolation.................................. 42
Fig. 4.1 (a) ideal tunable low-pass pi-network and (b) equivalent circuit of practical pi-network
tuner with fixed inductor and tunable capacitors. ........................................................... 48
Fig. 4.2 Circuit topology of pi-network tuner with driving source and load to be matched......... 49
Fig. 4.3 Equivalent or composite inductance Le as a function of frequency for different values of
C3 ranging (a) 0.5 pF to 1.5 pF and (b) 2 pF to 5 pF. ...................................................... 51
Fig. 4.4 Tuning algorithm flow chart for Branch 1 and 2 ............................................................. 58
Fig. 4.5 Algorithm flow chart for Branch 3 .................................................................................. 61
Fig. 4.6 Input VSWR contour plots for 0.05 ≤ |ΓL| ≤ 0.95 obtained with (a) the algorithm and (b)
the optimizer simulation at 700 MHz ............................................................................. 62
x
Fig. 4.7 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤
180o (a) algorithm and (b) optimizer at 2170 MHz for the case of lossless network. ... 64
Fig. 4.8 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤
180o (a) algorithm and (b) optimizer at 700 MHz. ......................................................... 67
Fig. 4.9 Pi-network shunt C2 having a fixed inductor Ls connected in parallel........................... 68
Fig. 4.10 Pi-network with a series LC circuit in its series path. ................................................... 69
Fig. 4.11 VSWR versus frequency of a given antenna impedance with and without using the Pinetwork tuner. The tuner has components L= 6.8 nH, C1,min = C2,min = 0.8 pF,
C1,max = C2, max = 5 pF, C3,min = 0.25 pF and C3,max = 4 pF and the values of the
capacitor settings have been found using the presented tuning algorithm. ..................... 71
Fig. 4.12 Conceptual block diagram of closed loop impedance matching control system. The
proposed algorithm is used here to adjust the tunable capacitor settings once the antenna
complex impedance is detected. ...................................................................................... 72
Fig. 4.13 Equivalent circuit for (4.14) and (4.15) derivations. ..................................................... 74
Fig. 4.14 Equivalent circuit for (4.16) and (4.17) derivations. ..................................................... 75
Fig. 4.15 Equivalent circuit for Γin derivation ............................................................................. 76
Fig. 4.16 Equivalent circuit of pi-network tuner........................................................................... 77
Fig. 5.1 Block Diagram of an Iso-frequency repeater. ................................................................. 82
Fig. 5.2 Circuit schematic of the SPSZ tunable filter design. ....................................................... 84
Fig. 5.3 S-parameter response of notch filter without the nearby co-exisisting pass band........... 86
Fig. 5.4 Block diagram of tunable notch filter automatic frequency control loop utilizing
reflected reference signal................................................................................................. 87
xi
Fig. 5.5 Mathematical model of notch filter frequency automatic control loop utilizing the
reflection signal. .............................................................................................................. 88
Fig. 5.6 Comparison of Closed form expressions and ADS simulations for the second order filter
response with In this experiment, 20, ∆
15 1000 and
0 10 ............................................................................................................. 93
Fig. 5.7 Simulated filter frequency control loop transient response for different values with
=20KHz, A=0.12 and ∆
=15MHz, τ=0us. .............................................................. 94
Fig. 5.8 Simulated filter frequency control loop transient response for different values with
=80, A=0.12 and ∆
=15MHz, =0us. ..................................................................... 95
Fig. 5.9 Simulated filter frequency control loop transient response for different A values with
=20KHz, =80 and ∆
=15MHz, =0us ............................................................... 95
Fig. 5.10 Simulated filter frequency control loop transient response for different values with
=80, =20KHz and ∆
=15MHz , =0.03............................................................. 96
Fig. 5.11 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four
capacitance bits of a cell; b) 3D image of capacitance bits ............................................. 97
Fig. 5.12 Detailed TDC layout. Red cells and green cells are used for the resonator and insertion
loss blocks, respectively .................................................................................................. 98
Fig. 5.13 Fabricated SPSZ tunable filter prototype board and filter structure detail. ................... 98
Fig. 5.14 Measurement results for the transmission and return loss characteristics of the SPSZ
tunable filter when tuning resonator block capacitor 2................................................. 99
Fig. 5.15 Measurement results for the transmission and return loss characteristics of the SPSZ
tunable filter when tuning insertion loss block capacitors 3 and 4. ......................... 100
Fig. 5.16 Discrete component implementation of the proposed notch filter control loop. ......... 101
xii
Fig. 5.17 Measured filter frequency control loop transient response for different A=0.3 and
∆
=49MHz. ................................................................................................................ 102
Fig. 5.18 Measured filter frequency control loop transient response for different A=3 and
∆
=50MHz. ................................................................................................................ 103
Fig. 5.19 Simple notch filter circuit topology ............................................................................. 105
Fig. 5.20 Low pass loop filter and integrator .............................................................................. 106
Fig. 6.1 Capacitively loaded tunable PIFA ................................................................................. 111
Fig. 6.2 Transmission line equivalent model for the capacitively loaded PIFA ......................... 111
Fig. 6.3 (a) Equivalent transmission line model of a capacitively loaded PIFA, (b) the capacitor
has been replaced by an open circuited section of transmission line ............................ 113
Fig. 6.4 Magnitude of the reflection coefficient using the transmission line model and method of
moments simulations (IE3D) for a PIFA over air with 2, 4, ! 34.5, ! 1. Error refers to the absolute value of the difference in frequency
between simulations and closed form expressions. ....................................................... 116
Fig. 6.5 Magnitude of the reflection coefficient using the transmission line model and method of
moments simulations (IE3D) for a PIFA over air with 4, 4, ! 33.5, ! 1. Error refers to the absolute value of the difference in frequency
between simulations and closed form expressions. ....................................................... 116
Fig. 6.6 Simulations and closed form (CF) expressions results for the total patch length (LT)
required to make the PIFA resonating at 2GHz versus capacitve loading for different
height values and 4 ...................................................................................... 118
xiii
Fig. 6.7 Simulations and closed form (CF) expressions results for the resonance frequency versus
capacitive loading for different height values. The patch total length was chosen to
resonate at #$ 2% in absence of capacitive loading ( 0&'........................... 119
Fig. 6.8 (a) Surface current distribution in the fundamental mode opeation for a regular half
wavelength microstrip antenna, (b) current distribution in a capacitively loaded PIFA
antenna. The shadowed region indicates the current integration area. .......................... 122
Fig. 6.9 (a) Radiation efficiency versus capacitive loading obtained by MoM simulations (Zeland
IE3D) and obtained closed form (CF) expressions. The antenna has parameters 4, #$ 2% ()* 0&', +$ 1, ,1, - 6/ (b) Radiation
efficiency versus resonant frequency with ,0&, - 2&/. ................................... 123
Fig. 6.10 Efficiency reduction versus capacitive loading for different values of with 2. ............................................................................................................................. 125
Fig. 6.11 Efficiency reduction versus capacitive loading for different values of with 4, ............................................................................................................................. 125
Fig. 6.12 a) Impedance bandwidth reduction versus capacitive loading for different values of with 0 4. (b) Impedance bandwidth reduction versus frequency when loading
capacitance is changed from C=0pF to 2pF. The patch total length was chosen to
resonate at #$ 2% in absence of capacitive loading ( 0&' for each case. .... 127
Fig. 6.13 Relative impedance bandwidth reduction versus capacitive loading for different values
of with 2 ................................................................................................... 128
Fig. 6.14 Relative impedance bandwidth reduction versus capacitive loading for different values
of with 4. .................................................................................................. 129
xiv
Fig. 6.15 Fabricated PIFA prototype over large ground plane. The ceramic capacitor is loaded at
the antenna radiating edge. ............................................................................................ 131
Fig. 6.16 Measured magnitude of the PIFA reflection coefficient for different capacitive
loadings. ........................................................................................................................ 132
Fig. 6.17 Fabricated Wheeler Cap covering the PIFA antenna for radiation efficiency
measurement. ................................................................................................................. 132
Fig. 7.1 Duplexing antenna tunable front end concept depicting tunable narrowband antennas in
combination with tunable notch filter, (b) Conventional non tunable front end
comprising of a multiband broadband antenna with diplexer and external multiple
duplexer modules. ......................................................................................................... 140
Fig. 7.2 Tunable PIFA antenna concept with shunt capacitor loading on antenna open edge. .. 141
Fig. 7.3 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four
capacitance bits of a cell; b) 3D image of capacitance bits. .......................................... 142
Fig. 7.4 Tunable PIFA with two shunt tunable capacitors loading the antenna radiating edge and
a series external chip capacitor in series with the tunable capacitors. .......................... 144
Fig. 7.5 Designed tunable single element and PCB dimensions................................................. 144
Fig. 7.6 Top and bottom views of the fabricated prototype. The top view shows the duplexing
antenna structure, tunable capacitor dies and RF ground plane. The bottom side contains
the control circuitry and interfacing multi-pin connector.. ........................................... 145
Fig. 7.7 Picture of the duplexing antenna built prototype........................................................... 146
Fig. 7.8 Simulated (HFSS) and Measured reflection coefficient of the single element antenna for
different values of capacitive loading. .......................................................................... 147
xv
Fig. 7.9 Measured reflection coefficient of the single element antenna for different values of
capacitive loading. ......................................................................................................... 147
Fig. 7.10 Different configurations considered in the duplexing antenna isolation study. The
distance ‘d’ is varied in discrete intervals and the S21 parameter is recorded for each
variation at high and low frequency pair situations. ..................................................... 149
Fig. 7.11 Simulated S-parameters resulting from configuration of Fig. 7.10(a) when the two
antennas are separated a distance (a)11.8mm and (b)36.8mm. Antenna with port 1 and
port 2 are operating at 1850MHz (transmit antenna) 1930MHz (receive antenna),
respectively. ................................................................................................................... 151
Fig. 7.12 Simulated parametric study of isolation (S21) between the two antennas operating at
high (1980MHz, 2170MHz) and low (1850MHz, 1930MHz) frequency pairs for the
configurations depicted in Fig. 7.10. ............................................................................. 154
Fig. 7.13 Measured reflection coefficients and S21(dB) between the two antennas operating at
(a) high (Tx-Rx offset=190MHz) and (b) low (Tx-Rx offset=80MHz) frequency pairs.
....................................................................................................................................... 154
Fig. 7.14 Simulated (blue solid) and measured (red dashed) normalized radiation patterns for
three orthogonal planes at two different loading conditions: (a) C=Cmax and (b)
C=Cmin.. ....................................................................................................................... 155
xvi
LIST OF TABLES
Page
Table 2.1 Phase delays for each antenna phase shifter ................................................................. 19
Table 2.2 Simulated and measured antenna center frequencies ................................................... 26
Table 2.3 Simulated and measured phase delays at 4.7GHz ........................................................ 26
Table 2.4 Simulated and measured phase delays at 7.5GHz ........................................................ 26
Table 3.1 Reflection coefficient (S11) for several combinations of OUPPs and optimized
reflection coefficient using best configuration of IPPs. Bit ‘0’ meaning open circuited
parasitic and bit ‘1’ meaning short circuited ................................................................ 37
Table 4.1 comparison of average VSWR resulting from optimizer and algorithm for lossless
network. 0. 5<|ΓL|<0.9 .................................................................................................. 63
Table 4.2 comparison of average ∆GT resulting from optimizer and algorithm for lossless
network. 0. 5<|ΓL|<0.9 .................................................................................................. 64
Table 4.3 Comparison of ∆GT resulting from algorithm and optimizer at different frequencies for
low loss network case. .................................................................................................. 67
Table 4.4 Matching performance and tunable capacitor settings considering matching network
with continuous capacitance. ........................................................................................ 71
Table 4.5 Matching performance and tunable capacitor settings considering matching network
with discrete capacitance steps. .................................................................................... 71
Table 5.1 Final Frequency Error versus Value ........................................................................ 94
Table 5.2 Summary of Suppression and insertion losses within 5MHz Bandwidth ................... 100
Table 6.1 antenna resonant frequency versus capacitive loading .............................................. 131
Table 6.2 antenna impedance bandwidth versus capacitive loading .......................................... 132
xvii
Table 6.3 antenna radiation efficiency versus capacitive loading .............................................. 133
Table 7.1 Wireless standards frequency specifications .............................................................. 138
Table 7.2 Requirement for cell phone antenna bandwidths ........................................................ 138
Table 7.3 antenna radiation efficiency versus capacitive loading ............................................. 153
xviii
ACKNOWLEDGMENTS
I would like to begin this dissertation expressing my most sincere appreciation to Prof. Franco De Flaviis
whose valuable advice, guidance and trust have been the light of this journey.
I would also like to thank Prof. Lluis Jofre, Prof. Roger Rangel and Mr. Pete Balsells for giving me the
opportunity of becoming part of this family and enabling the dreams of others. I would not be at this point
without them.
Special thanks to the members of my committee Prof. Filippo Capolino and Prof. Pai Chou for their
willingness, commitment and encouragement.
To my friends and colleagues, Dr. Alfred Grau, Salvatore Campione, Anna Papio, Andrea Massenz, Nick
Chopra and Ali Hosseini for their friendship and enjoyable presence during these years. Special thanks to
Dr. Qizheng Gu and Dr. Art Morris for giving me the opportunity to be part of what they do and helping
me to become a better engineer.
To my friends back at my second home, Jose A. Parras, Miquel Cardona, Pere Salvatella, David
Navarrete, Roger Piqueras, Raul Macule and Mika Pedros for making me laugh and sharing the best
experiences of our life together.
To the most loving people of my life, my parents. For making their unconditional support the role model
of my life. I will never find enough words to express my gratitude. To my brother Roberto and my sister
Carolina, for the incredible gift of growing up with them.
To Evonne, for making my days happier, transforming me into a better person and always being with me
no matter what.
This work has been funded by the California Catalonia Innovation Program (2007-2008 and 2010-2011)
and the Balsells fellowship for graduate studies (2006-2007 and 2009-2010). My deepest appreciation to
everyone involved on these programs.
xix
CURRICULUM VITAE
Javier R. De Luis
2000-06
B.S. in Telecommunications Engineering
Universitat Politecnica Catalunya, Barcelona, Spain
2006-08
M.S in Electrical Engineering and Computer Science
University of California, Irvine, USA
2008-09
Teaching Assistant, Henry Samueli School of Engineering
University of California, Irvine
2006-11
Ph.D in Electrical and Computer Engineering
University of California, Irvine, USA
2009-11
Intern Antenna/RF Engineer
Wispry Inc, Irvine, USA
FIELD OF STUDY
Reconfigurable Antennas and microwave circuits, Electrical and Computer Engineering.
SELECTED PUBLICATIONS
De Luis, J.R.; de Flaviis, F.; , "Frequency Agile Switched Beam Antenna Array System,"
Antennas and propagation, IEEE Transactions on , vol.58, no.10, pp.3196-3204, Oct. 2010
De Luis, J. R.; Morris III, A.; Gu, Q.; De Flaviis, F.;,” A Tunable Asymmetric Notch Filter using
RFMEMS” Microwaves theory and Techniques Society International Symposium, 2010. MTT-S
2010
De Luis, J.R.; Gu, Q.; Morris III, A.;De Flaviis, F, “Tunable MEMS Notch Filter and its
Frequency Control Loop for Wireless Transceivers”, Microwave Theory and Techniques, IEEE
Transactions on, (Submitted, under review).
Vallechi, A.; De Luis, J.R; Capolino, F.; De Flaviis, F., “A Low Profile Folded Dipole
Antenna on a Reactive High Impedance Substrate”, Antennas and propagation, IEEE
Transactions on. (submitted, under review).
De Luis, J.R.; Capdevila, S. Gu, Q.; Morris III, A.;De Flaviis, F, “Closed Form Expressions for
Capacitively Loaded Planar Inverted-F Antenna Design”. Antennas and propagation, IEEE
Transactions on. (submitted, under review)
U.S Patent provisional app. “Tuning methods for tunable matching networks”. Q. Gu and J.R De
Luis
U.S Patent provisional app. “MEMS tunable notch filter frequency automatic control loop” Q.
Gu and J.R De Luis.
xx
ABSTRACT OF THE DISSERTATION
Tunable Antennas and Microwave Circuits for Next Generation Reconfigurable Front Ends
By
Javier Rodriguez De Luis
Doctor of Philosophy in Electrical and Computer Engineering
University of California, Irvine, 2011
Professor Franco De Flaviis, Chair
Reconfigurable antennas and microwave circuits have attracted much attention in recent
years due to their advantages compared to conventional designs. It is recognized that systems
exploiting a certain extent of tunability potentially benefit from multi-functionality and
performance enhancement, which ultimately translates into lower component count, size and
overall cost of the transceiver. In this work, a collection of tunable/reconfigurable designs
comprising of antennas and passive microwave circuits that spans to different application areas is
presented in a comprehensive manner. In chapter 2 a dual frequency reconfigurable antenna
array using PIN diodes and its associated phase shifter are presented. The system is capable of
operation at two independent frequencies (4.7GHz and 7.5GHz) while switching between four
different radiation patterns types for each frequency. Chapter 3 explores the feasibility of a novel
of iso-frequency repeater system based on reconfigurable parasitic elements to maximize the
isolation between the transmitting and receiving antennas.
In chapter 4, an analytical tuning algorithm for a reconfigurable impedance matching pi-network
based on tunable capacitors is presented. The algorithm is able to determine all tunable network
component values for matching any given load impedance.
xxi
A single pole single zero notch filter using RF MEMS tunable capacitors and its associated
frequency control loop system for automatic frequency tracking are presented in chapter 5. The
high Q value of the tunable capacitors enables this filter to achieve 22dB rejection with less than
0.8dB insertion loss over a 5MHz bandwidth in the International Mobile Telecommunications
band (IMT, 2.1 GHz).
In chapter 6, a set of approximate closed form expressions for the input impedance, resonant
frequency, radiation efficiency, quality factor and impedance bandwidth of capacitively-tunable
loaded planar inverted-F antennas (PIFAs) used for cell phone applications are presented.
Finally, chapter 7 presents a tunable antenna pair for wireless transceivers using RF MEMS
tunable capacitors. This design utilizes antenna frequency agility to cover a frequency tuning
range from 1850MHz to 2170MHz (16%) corresponding to the cellular wireless bands IMT-I
and PCS and is designed to provide built-in filtering between antennas.
xxii
1. INTRODUCTION
1.1 Interest in reconfigurable antennas and microwave circuits.
Reconfigurable antennas and microwave circuits have received much attention in the past years
playing an important role in the design of smart and adaptive systems. The evolution of such
systems has been enormously benefited from recent improvements in the performance of the
existing switching/tunable solid state or RF MEMS technologies.
In contrast to conventional microwave designs, reconfigurable systems are able to adapt one or
more of its operational characteristics in order to enhance performance or provide additional
functionalities to the system. This multi-functional advantage most often enables a reduction in
the number of required components compared to non tunable solutions, which ultimately impacts
the overall cost of the front end.
As an illustrative example of the potential advantages allowed by tunability let’s consider a
classic cell phone system architecture as the one shown in Fig. 1.1. The conventional approach
uses multiple multiband antennas followed by arrays of switches or duplexer modules in order to
select the desired transmission branch specific of each standard. Additionally, each transmission
path contains its own filtering and amplifiers stages. Recognizing the fact that the cell phone
market evolves towards the increase in number of frequencies and standards, the classic problem
approach leads to a complex front end and elevated number of components, which profoundly
affects the design cycle time and cost of the transceiver.
A solution based on a novel tunable front end is shown as an alternative in Fig. 1.2 where the
different antennas have been substituted by a multiband antenna with a tunable matching
network (or a tunable antenna), followed by a lower number of switching sections, tunable
1
filters, amplifiers and duplexers. The comparison of Fig. 1.1 and Fig. 1.2 makes apparent the
flexibility that allows the reduction on the transceiver component number and provides a sensible
argument for the need of research in the reconfigurable RF field.
Fig. 1.1 Conventional non tunable cell phone system architecture from Wispry Inc (www.wispry.com).
Fig. 1.2 Novel tunable cell phone system architecture from Wispry Inc (www.wispry.com).
2
Although a wide debate and extensive comparison can be found in literature regarding the most
suited tunable device technology, the reality is that RF MEMS and solid state components offer
advantages and disadvantages that must be clear prior to the design stage. The objective of this
thesis is not to discuss the several technology options but to present different design concepts
based on reconfigurable antennas and microwave circuits that cover different application areas
from telemetry to cellular communications and use both RF MEMS and PIN diode devices.
1.2 Document organization
In chapter 2 a dual frequency reconfigurable antenna array and its associated phase shifter are
presented. The system is capable of operation at two independent frequencies (4.7GHz and
7.5GHz) while switching between four different radiation patterns pointing at different spatial
locations for each frequency. The reconfigurability is achieved by using high performance PIN
diodes acting as microwave switches. Chapter 3 explores the capacity and feasibility of a new
concept of iso-frequency repeater system based on reconfigurable parasitic elements that is able
to maximize the isolation between the transmitting and receiving antennas. This is done with the
objective to allow a higher the power amplifier gain which results in an increased signal
coverage area.
In chapter 4, an analytical tuning algorithm for a reconfigurable pi-network impedance matching
network is presented. The pi-network consists of tunable capacitors with finite tuning range, and
a fixed value inductor. The algorithm is able to determine all tunable network component values
for matching any given load impedance. The resulting matching performance measured either in
terms of the input VSWR or transducer gain, is equivalent to that obtained from a commercial
iterative optimization methods, but the algorithm runs significantly faster than the optimizer
simulation. The proposed algorithm can be extended to a network tuner topology comprised of
3
four or less tunable components as long as it can be transformed into an equivalent pi-network
topology. It is suitable for the design of fixed as well as tunable matching networks.
A single pole single zero notch filter using RF MEMS tunable capacitors and its associated
frequency control loop system are presented in chapter 5. The high Q value of the tunable
capacitors enables this filter to achieve 22dB rejection with less than 0.8dB insertion loss over a
5MHz bandwidth in the International Mobile Telecommunications band (IMT, 2.1 GHz). The
proposed design allows for independent tuning of the rejection and pass bands. A filter frequency
control loop based on sensing the reflection phase change of the notch filter is developed to
achieve automatic channel frequency tracking. The filter frequency control loop uses a reference
signal (in this case, the transmitter carrier) to keep the tunable filter tracking the transmitter
operation frequency.
In chapter 6, a set of approximate closed form expressions for the input impedance, resonant
frequency, radiation efficiency, quality factor and impedance bandwidth of capacitively-tunable
loaded planar inverted-F antennas (PIFAs) used for cell phone applications are presented. These
expressions yield insight into the effects of the most important design parameters over antenna
performance and are useful to establish fundamental tradeoffs in early stages of the design.
Experimental results by means of measurements and full wave simulations are presented to
support the validity of the proposed closed form expressions showing good agreement between
them.
Finally, chapter 7 presents a tunable antenna pair for wireless transceivers using RF MEMS
tunable capacitors. This design utilizes antenna frequency agility to cover a frequency tuning
range from 1850MHz to 2170MHz (16%) corresponding to the cellular wireless bands IMT-I
and PCS, although the same concept can be extended to additional bands. Each antenna in the
4
pair is designed to operate at the transmit and receive channels of the band, respectively, while
the narrow impedance bandwidth characteristics of the single element provides built-in filtering
between antennas. The level of isolation between elements is studied for several antenna
arrangements depending on their relative orientation within the handset and can be exploited
from a system perspective towards relaxing the duplexer specifications or allowing its
substitution by narrowband RF MEMS tunable notch filters. A fabricated prototype is presented
and measured to proof the concept and validate the design performance.
Chapters 2 and 3: Personal use of this material is permitted. However, permission to
reprint/republish this material for advertising or promotional purposes or for creating new
collective works for resale or redistribution to servers or lists or to reuse any copyrighted
component of this work in other works must be obtained from the IEEE
Chapters 4 to 7: This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be accessible.
5
2. A FREQUENCY AGILE SWTICHED BEAM ANTENNA ARRAY
SYSTEM
2.1 Introduction
Increasing the angular resolution of antenna systems enhances the performance of the wireless
mobile communication link. A phased array comprising of several elements and a control
algorithm provides virtually unlimited control over azimuth and elevation scan angles at high
speeds. This capability, however, comes with a high complexity and cost associated with a large
number of elements, each having a digital phasing network with switching semiconductor or
electromechanical elements. The high complexity and cost become more severe if a wideband
(or multiple band) phased array is needed. Thus, as of today the use of phased array antennas is
nearly limited to sophisticated military and space systems. However, many applications do not
require a full scan capability involving the complexity of a phased array. In these cases, a simpler
system, such as a reconfigurable switched beam antenna array, can be used. Reconfigurable
antennas have received much attention in the past years playing an important role in the design
of smart and adaptive systems [1]. Recent improvements in the performance of switching
technologies such as RFMEMS and solid state switches, integrated in antennas and microwave
circuits, have proved to be useful for a wide range of applications [2]-[10], including switched
beam arrays. 3D structures, based on circular array configurations having a single active antenna
surrounded by several loaded parasitics have been proposed to provide endfire beam switching
for single [11] and dual frequency applications [12]. A multi-layered three beam system using
triangular microstrip antennas has been presented in [13]. Other multi-beam approaches using
6
Butler Matrix [14] or phase switched solutions [15] can be found in the literature. However, the
systems in [13]-[15] are limited to operation at a single frequency band.
In this chapter, a novel dual frequency reconfigurable microstrip antenna array for next
generation telemetry application with beam switching capability at 4.7GHz and 7.5GHz is
presented. In contrast to [12], only one of the two operating bands remains active at each time,
rejecting the non-active frequency without the need of an external diplexer. Furthermore, the
system is a true planar implementation printed in a two layer PCB, reducing the complexity,
space and total cost of the overall system.
The dual frequency switching capability will provide a better usage of the spectrum
resources and will also allow several nearby systems to communicate at high speed with the
ground station simultaneously. A system comprising of ominidirectional antennas and single
frequency operation as shown in Fig. 2.1(a) may create harmful interferences between aircraft.
Fig. 2.1 (a) Ominidirectional antenna system with single frequency operation showing interference between
aircrafts. (b) switched beam dual frequency antenna system proposed in this chapter.
However, a system deployed as shown in Fig. 2.1(b), with two different frequencies and switched
beam capability, minimizes the jamming risk and enables a fully functional system. Based on
7
this concept, the system proposed in this chapter is composed of a dual frequency (DF)
reconfigurable antenna and a single feed dual frequency reconfigurable switched line phase
shifter. Two different features, frequency and beam maximum position, can be selected by
introducing PIN diodes in the system. Progressive phases between antennas of {+45,+135,-45,135} can be chosen to generate a switched beam with beam maxima pointing to {15,45,-15,45}
and {13,30,-13,-30} degrees at 4.7GHz and 7.5GHz respectively.
The proposed design is shown in Fig. 2.2 and can be divided in two blocks: a DF reconfigurable
microstrip antenna and DF switched line phase shifter explained in sections 2.2 and 2.3,
respectively. Section 2.4 presents system simulations and measurement results. The input and
output of each block is matched to a system impedance of 50Ω. The modular design approach of
each sub-block allows for a simple design process flow that can be reproduced for any desired
frequency ratio.
Fig. 2.2 Complete system diagram formed by a 4x1 dual frequency phased array and two different types of switched
line phase shifters (Type 1, Type 2)
8
Fig. 2.3 The antenna element consists of a dual microstrip patch topology connected through PIN diodes, printed
over RT duroid 5880 (εr=2.2,tanδ=0.0009, 35µm copper thickness , 1.575mm dielectric thickness) with dimensions:
Lhf=11.1mm, Whf=12.15mm, Llf=19.5mm, Wlf=21mm,
g=0.5mm, gd=0.3mm, iip=2mm, iop=0.34mm,
Wdc=0.2mm. Diode length=0.7mm, width=0.3mm and pad gap=0.3mm
2.2 Design of the single antenna element
The antenna element used in the 4x1 linear array of Fig. 2.2 is shown in detail in Fig. 2.3. The
design consists of a printed patch over a 1.575mm thick Duroid RT-5880 substrate with εr=2.2,
tanδ=0.0009 at f=10GHz and 34µm copper thickness, backed with a metallic ground plane. A
small inner rectangular patch (IP) is partially surrounded by a larger U-shaped outer patch (OP).
Both patches are connected through three high frequency GaAs PIN diodes (model
MA4AGCFCP910, vendor M/A-COM) acting as microwave switches with equivalent circuit
model shown in Fig. 2.4. The element values of the equivalent lumped circuit model were found
in order to match a single diode measurement mounted in a microstrip line test fixture over the
frequency range from 4GHz to 8GHz. The device was characterized and the test fixture was deembedded from the measurement, using a TRL calibration procedure.
The antenna is fed by a microstrip line connected to the middle point of the IP left edge. In the
fundamental mode, the left and right edges of the antenna are responsible for the radiation
9
phenomena and the RF current resonates along the x-axis (Fig. 2.3). The diodes are placed on the
right edge of the inner patch to allow the currents to flow to the OP when required.
Fig. 2.4 Diode circuit model for the ‘On’ and ‘Off’ states. Rs=1.52Ω, Ls=0.25nH and Coff=47fF, Rp=10KΩ.
Maximum insertion loss @4-8GHz 0.4dB. Minimum isolation @4-8GHz 11dB.
The patches are separated by a distance of 0.5mm, which is sufficient to leave space for the
diode placement and to provide physical isolation. When the diodes are in the 'Off’ state (or
reverse bias), the RF signal flows mainly in the inner patch while the outer patch acts as a
parasitic element and the antenna operates in the high frequency mode (7.5GHz). Similarly,
when the diodes are switched to the 'On' state (forward bias), the current flows in both patches
resulting in an increase of the effective area of the antenna and the system operates in the low
frequency mode (4.7GHz). The sizes of both patches have been optimized simultaneously for
resonance at both frequency bands, as explained in the next subsections.
The DC current path returning to ground necessary for the diode biasing, is designed using a
high impedance 4.7GHz quarter wave microstrip line ending in a pad connected to the antenna
ground plane. The short circuit at the pad is transformed into an open circuit seen from the OP
edge, making the ground path section transparent to the RF signal. It was observed that the effect
of the quarterwave biasing line on the 7.5GHz resonance, when the diodes are in the ‘Off’ state,
was negligible due to the small RF current flowing in the OP as shown in Fig. 2.5. When the
diodes are in the ‘On’ state, the effect of the biasing line on the radiation pattern at 4.7GHz
produced a decrease of 0.4dB on the antenna gain.
10
Due to the different topology for both operation states, the edge resonance resistance seen from
the feeding point has different values depending on the operative frequency band. Therefore, a
dual frequency matching network based on a dual stub configuration was designed.
2.2.1 Gap Size Effect on the resonant frequencies.
As a first step in the design of any dual-frequency dual-patch antenna, it is important to
understand the effects of the gap size between the IP and OP on the high and low resonant
frequencies. In order to analyze this effect, two stand alone rectangular patches were designed to
resonate at 4.7GHz (L=20mm 0.46λeff at 4.7GHz, W=22mm) and 7.5GHz (L=12.2mm 0.45λeff at
7.5GHz, W=13.25mm) respectively according to [16]-[18]. Both independent patches were then
combined together into the same space leaving a 0.5mm gap between them as shown in Fig. 2.5(ad). Depending on the interconnection and operating state, four different cases of the current
distribution in the antenna were plotted. The antenna current distribution gives a qualitative idea
of the level of IP/OP isolation in the diode ‘Off’ state and the current ability to flow from IP to
OP in the ‘On’ state. Fig. 2.5 shows the low frequency operation mode with three metallic
connections (ideal ‘On’ state) and their absence (ideal ‘Off’ state), respectively. On the other
hand, Fig. 2.5 shows the high frequency operation using the forward biased diode S-parameters
(real ‘On’ state) and reverse biased diode parameters (real ‘Off’ state) respectively. It is observed
that the current distributions of the antenna at low frequency for the ‘ideal’ and ‘real’ ‘On’ states
are very similar to the standard reference patch in the center of the figure. Fig. 2.5(b) gives an idea
on how much current is induced in the OP only due to the proximity coupling effect when both
patches are physically isolated. In addition to this coupling, Fig. 2.5(d) produces an additional
current leakage near the diode connections due to the imperfect isolation of the diode in ‘Off’
state.
11
Intuitively, it is expected that the biggest deviation in frequency value from the stand alone patch
case would be produced for the case of Fig. 2.5(d) because the IP current distribution presents
more differences with respect to the stand alone patch. To verify this statement and to study the
gap size effect on the antenna resonant frequency, a numerical experiment using full wave
analysis was performed. The gap size was increased from 0.05mm to 2mm in 0.05mm steps,
while the associated frequency change was monitored as shown in Fig. 2.6. The difference,
between ‘ideal’ (a) and ‘real’ (c) ‘On’ states curves for low frequency is clearly less significant
than the difference between (b) and (d) ‘Off’ state (high frequency) cases, which matches with
the current distribution comparison.
The average low frequency in Fig. 2.6 is 4.6GHz which is only 1% lower than the stand alone
patch case. This decrease is due to the slightly longer current path forced by the connections
between both patches. When the gap distance gets larger, curve (b) shows no difference in
frequency with respect to the 7.5GHz stand alone patch due to the negligible proximity coupling
between IP and OP. However, this effect is not seen on curve (d) due to existing signal leakage
produced by the diode in the ‘Off’ state. Similarly, when the gap distance is small, both curves
(b) and (d) show a decrease in the resonant frequency produced by the effective increase on
antenna size created by the IP/OP mutual coupling.
2.2.2 Optimum number of diodes analysis.
In this study we found that the number of diodes integrated in the antenna affects the radiation
efficiency and system losses. In addition, a higher number of devices will require more DC
power consumption and will have higher noise generation. Therefore, a minimum number of
diodes for good operation at both frequencies must be found. One to five diodes were integrated
in the antenna as shown in Fig. 2.7 and considered in the following analysis. Fig. 2.8 shows the
12
behavior of the imaginary part of the input impedance with frequency when one, two and three
diodes are switched ‘On’ forcing the antenna to operate at low frequency.
The fundamental resonant modes for these are observed at 4.48GHz, 4.59GHz and 4.6GHz
respectively. When a smaller number of diodes is used, frequency is lowered due to the increased
current path enforced by the topology. Even more important is the higher frequency resonances
observed at 8.5GHz, 8.73GHz and 9.2GHz for the one, two and three diode cases respectively. In
order to study all the mentioned resonances, the average current distribution over the patch
operating in ‘On’ state is shown in Fig. 2.7. The low frequency fundamental modes and higher
frequency resonances are shown in the top and bottom rows, respectively. An additional case of
5 diodes is also introduced (Fig. 2.7(j-k)) to show that further increase in diode number does not
create a significant difference in the patch current distribution. This is because the two diodes
placed horizontally in the non radiating edges do not contribute to the vertical current flow in the
patch. The fundamental mode current flow and the value of the associated resonant frequency are
clearly closer to the stand alone patch case of Fig. 2.5 when three diodes are used.
The radiation patterns corresponding to the higher frequency resonances for all considered cases
are shown in Fig. 2.9. The pattern for the second resonance (at 8.5GHz), corresponding to 1-diode
case has a radiation behavior typical of a fundamental mode. This is due to the fact that the
current is very confined to the diode region when crossing to the OP, thus the IP remains
resonant resulting in a dual frequency antenna (instead of a switchable one). One can actually
perceive that behavior on the current plot of Fig. 2.7(b). However, when three diodes are used, the
second resonance appears considerably higher at 9.2GHz (around 2flow) and the radiation pattern
shows a null in the normal direction typical of a higher order mode of the radiating OP (instead
13
of a fundamental one). In this case, the IP is not contributing to radiation and switching becomes
effective. The two diode pattern case of Fig. 2.9(b) is in between the two previous cases.
If only one diode was chosen for the design, its 8.5Ghz resonance could create an undesirable
spurious radiating mode close to the target high frequency of 7.5GHz. This could potentially
harm the receiver if filtering circuitry is not implemented. Therefore, three diodes will be used
for the design to avoid requiring an external diplexer while guaranteeing a switchable behavior
with good operation at 4.7GHz.
Fig. 2.5 Scalar current distribution over the antenna at 4.7GHz for the cases: (a) Ideal ‘On’ with metallic connections
(b) Ideal ‘Off’ with no connections, (c) Real ‘On’ with diode connections and (d) Real ‘Off’ with diode connections.
Center of figure, standard rectangular patch resonating at 4.7GHz.
7.75
7.50
7.25
7.00
Frequency (GHz)
6.75
(a) Metal Connection between patches
(b) No connection between patches
(c) Diodes between pacthes in ON state
(d) Diodes between patches in OFF state
6.50
6.25
6.00
5.75
Gap=2mm
Gap=0.05mm
5.50
5.25
5.00
4.75
4.50
4.25
0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 1.95
Gap (mm)
Fig. 2.6 Resonant frequency change with increasing gap between IP and OP for cases: (a) Ideal ‘On’: Metal
connection between Patches (low frequency) (b) Ideal ‘Off’: No connection between patches (high frequency), (c)
Real ‘On’: Diodes between patches in ‘On’ state (low frequency) and (d) Real ‘Off’: Diodes between patches in
‘Off’ state (high frequency).
14
Fig. 2.7 Scalar current distribution over antenna different number of diodes in ‘On state: 1 diode (a) first resonance,
(b) second resonance; 2 diodes (c) first resonance, (d) second resonance; 3 diodes (e) first resonance, (f) second
resonance and 5 diodes (j) first resonance, (k) second resonance
200
(a) 1 Diode
(b) 2 Diodes
(c) 3 Diodes
150
100
Im(Zin)
50
0
-50
-100
-150
-200
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Frequency(GHz)
Fig. 2.8 Imaginary part of the antenna edge impedance versus frequency showing first and second resonances for the
cases of (a) 1 diode ‘On’, (b) 3 diodes ‘On’ and (c) 3 diodes ‘On’
Fig. 2.9 Normalized radiation pattern (dB) corresponding to the second resonance of cases (a) 1 diode ‘On’ (b) 3
diodes ‘On’ and (c) 3 diodes ‘On’
15
2.2.3 Radiation Efficiency analysis
Each diode can be considered like a passive device that introduces losses into the system. The
loss sources are mainly ohmic loss (insertion loss) due to the device itself and mismatch loss due
to the diode placement within the antenna. The transducer gain expression [19] takes into
account both loss sources and can be expressed as:
2
GT =
1 − Γs
Pload
=
Pavailable 1 − Γs Γin
2
S 21
2
1 − ΓL
2
1 − S 22 ΓL
2
(2.1)
Where Pload and Pavailable in this cases, refer the power delivered to the outer patch and inner patch
respectively. Γs is the reflection coefficient looking towards the IP from the diode positive
terminal. Γin is the reflection coefficient looking to the positive terminal of the diode from the IP
. ΓL is the reflection coefficient looking to the OP from the diode negative terminal, S22 is the
reflection coefficient looking towards the IP from the negative terminal of the diode and S21
represents the transmission coefficient in 50Ω environment. Dealing with passive devices, the
maximum achievable transducer gain is 0dB. Assuming ohmic losses, the optimum transducer
gain for a specific device can be achieved satisfying simultaneous matching conditions (SMC)
imposed by Γs = Γin* and ΓL = S22*, which minimize the mismatch loss. This is an important
condition as long and it can be related to the radiation efficiency of the antenna. The input
impedance seen at the edge of the IP and OP where the terminals of the diode are connected can
be considerably different. In order to find impedances that satisfy the SMC condition, slits are
introduced in the IP and OP sections as shown in Fig. 2.3. The length of both slits is changed
independently from 0mm (no slit) to 5.3mm (half of the IP length). The associated radiation
efficiency of the antenna is evaluated using Full Wave (Zeland IE3D MoM base software)
simulations for each different pair of slit length for a total of 2500 combinations. The radiation
16
efficiency versus slit length information is then extracted from the simulation data and plotted in
Fig. 2.10.
Efficiency varies from 40% to 74% between the two extreme cases. The efficiency
improvement due to the proper slit length has an associated gain increase from 5.88dB to
6.48dB. This result demonstrates the importance of finding SMC for any diode placed within the
antenna. In a second study, the slits for the center and corner diodes were optimized
independently, but no improvement above the mentioned 74% efficiency was observed by using
this approach.
Fig. 2.10 Radiation Efficiency as a function of IP and OP slit lengths. The dynamic range between worst and best
results is found to be 35%.
2.3 The dual frequency phase shifter
A microwave phase shifter is a two port network that provides a specific phase delay to a signal
travelling from its input to output ports at a given frequency. Different design approaches and
topologies are widely found in the literature such us loaded line phase shifters [20], quadrature
reflection phase shifters [21] or Schiffman phase shifter [22]. However, when identical phase
delays are required at two different frequencies, a solution based on a non-tunable passive circuit
17
is not available or its complexity increases [23] allowing only a very specific combination of
phase delays. In general, switchable solutions are required for dual frequency operation, but the
complexity and fabrication challenges may play an important role as in some reported solutions
[24]. In this chapter, a novel reconfigurable switched line phase shifter using commercial PIN
diodes is presented following a simple design procedure.
In a switched line phase shifter, the RF input signal can travel between different transmission
paths before arriving to the output resulting in different phase delays. In this system, four paths
or branches can be selected as shown in Fig. 2.11. The phase shifters (PS) for antennas #1 and #4
(Type 1 PS) shown in Fig. 2.2 are identical, and can provide a switchable phase delay of
{0,45,90,135} degrees at 4.7GHz and 7.5GHz simultaneously. Similarly, the PS connected to
antennas #2 and #3 (Type 2 PS) can provide a delay of {270,45,90,135} degrees at both
frequencies. The different combinations of states in PS #1 to #4 shown in Table I can produce
four different progressive phases between antennas of {-135,-45,+45,+135}, that have associated
radiation beam maxima at {+15,+45,-15,-45} degrees for 4.7GHz and {+13,+30,-13,-30}
degrees for 7.5GHz. The design consists of four passive sub-circuits that are designed separately
and cascaded to form the complete system. Each of these will be explained in the following
sections: the input/output matching networks, the SP4T distribution blocks, the dual frequency
phase delay section and the dual frequency DC bias network. The design shown in Fig. 2.11 is
fully planar using microstrip transmission lines printed on a RT Duroid 6006 substrate (εr=6.15,
tanδ=0.0027 at f=10GHz, 35µm copper thickness and 0.635mm board thickness). The same PIN
diode model used for the antenna element was also used for the phase shifter design.
18
Fig. 2.11 Complete Type 1 Phase Shifter with phase delays from top to bottom branch of {0,45,90,135} degrees
Desired
Progressive
Phase (deg)
-135
-45
45
135
Table 2.1 Phase delays for each antenna phase shifter
Phase
Phase
Phase
PS # 1
PS # 2
PS # 3
(deg)
(deg)
(deg)
45
270
135
135
90
45
0
45
90
0
135
270
Phase
PS # 4
(deg)
0
0
135
45
Fig. 2.12 Zoom into the phase shifter input region that contains the DF matching network, input SP4T distribution
section and equal length launches. The Output SP4T distribution section is also included in the figure to show the
direction of the diodes.
In reality, due to the non perfect isolation of the diode, the non active ‘Off’ branches always
contribute with some degree of shunt parasitic reactance, deteriorating return loss conditions in
the input and output ports of the phase shifter. Therefore, dual frequency input and output
19
matching networks based on stepped impedance transmission lines are connected to the input
and output SP4T ports (Fig. 2.12), to provide good reflection coefficient characteristics
(VSWR<2) for all eight possible combinations (four phases and two frequencies). Both,
matching networks and distribution sections must be identical for both types of phase shifter to
avoid any phase perturbation.
2.3.1 The Dual Frequency phase delay section.
Fig. 2.13 Dual frequency phase delay section with dual stub configuration and showing optimization variables.
The DF phase delay section of Fig. 2.13 is the fundamental building block of the phase shifter. It
consists of a transmission line loaded with two shunt opened stubs that are able to provide the
same phase delay at two different frequencies simultaneously. In [25], a dual frequency delay
network is presented and the closed form expressions are obtained for the particular case of 90
degrees phase delay. However, phases different from 90 degrees make the mathematical
approach unfeasible so an optimization tool is required. The length (Lstub), width (Wstub) and
separation (L_line) between stubs are set as optimization variables in a microwave circuit
simulator (Applied Wave Research, AWR) to achieve each specific phase delay goal. IE3D was
used afterwards to perform full wave analysis and for fine tuning of the phase shifter purposes.
The simulation result for the case of zero degree phase delay at 4.7GHz and 7.5 GHz with
20
magnitude of the reflection coefficient below -20dB at both frequencies is shown in Fig. 2.14. The
same approach is used for the rest of the targeted phases with similar results.
0
200
100
50
-20
0
-30
-50
Phase (Deg)
Mag. Reflection Coefficient (dB)
150
-10
-100
-40
(a) Input/Output Return Loss
(b) Input/Output Phase Difference
-150
-50
-200
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Frequency (GHz)
Fig. 2.14 Simulation results from a zero degree phase delay section.Input/Output magnitude of the reflection
coefficient in dB (left axis) and Input/Output phase delay in degrees (right axis). Both frequencies remain matched
while providing the desired zero degrees phase delays.
2.3.2 The Dual Frequency DC Bias.
Biasing current is required to control the diode states for switching the circuit ‘On’ and ‘Off’.
Surface mount components may be used for this purpose. However, when operating at high
frequencies, catalog rated values may change and issues may arise with component SRF. In these
cases a printed approach is preferred such as the DC Bias circuit shown in Fig. 2.15, which does
not require surface mount components. Two quarter wavelength radial open stubs are connected
in shunt to a microstrip line. Each of them transforms the open circuit in point 'A' into a short
circuit seen from point 'B' for each frequency. The network after point 'B' is a dual frequency
quarterwave transformer that transforms the short circuit in ‘B’ to open circuit in ‘C’ for 4.7GHz
and 7.5GHz. Thus, the network seen from point ‘C’ becomes transparent to the RF signal flow.
Point ‘D’ is used as a DC-current input pad that allows the diodes to be biased in the circuit.
Anything surrounding the DC pad should not affect RF performance if both radial stubs are
21
properly designed. Each diode requires a current of 10mA in forward mode (‘On’), while zero
volts are used in reverse mode (‘Off’). If the RF signal needs to travel through the upper branch
of Fig. 2.11, 0V are applied to the pads DC#1,3,4,5,6 while a voltage that provides 10mA would
be applied to DC pad #2.
Fig. 2.15 Dual frequency DC bias formed by two quarter wavelength stubs, a dual frequency quarterwave section
and a DC input pad to bias the diodes.
2.4 Simulations and measurement results
The single element antenna, with optimized slit lengths and patch sizes was fabricated as shown
in Fig. 2.16. The three PIN diodes were placed by hand using conductive epoxy to avoid
overheating the device. For testing purposes, the bias current was introduced through an external
coaxial bias T. However, a printed dual frequency bias could be designed for the same purpose.
The magnitude of the reflection coefficient comparison between simulated (IE3D) and measured
results is in very good agreement as shown in Fig. 2.17, where both frequency bands present good
matching (<-15dB). For convenience, the ‘On’ and ‘Off’ cases are superimposed in the same
graph. Table II shows the target versus simulated and measured frequencies. In all cases, return
loss below 10dB was achieved for the target frequency.
22
The measured radiation patterns in both principal planes for the single antenna element at 4.7GHz
and 7.5GHz are shown in Fig. 2.18(a-b) and Fig. 2.18(c-d), respectively. Peak gains of 5dB and
5.4dB are obtained for 4.7GHz and 7.5GHz respectively. Efficiency values of 68% for the low
band and 79% for the high band were obtained.
Four identical antenna elements were used to form the 4x1 linear array. Deciding the distance
between elements in a dual frequency array is not a simple task and requires a tradeoff analysis of
array gain and side lobe level (e.g. in order to obtain such gain at 4.7GHz, an increase in SLL
must be accepted at 7.5GHz). For this specific application a high system gain greater than 10dB
was required. Therefore a separation distance between antennas of 30mm (0.47λ at 4.7GHz,
0.75λ at 7.5GHz) was chosen. With this distance, the simulated array gains are greater than 10dB
at both frequencies, despite of an increased side lobe level at 7.5GHz. In addition, the distance
avoids input impedance distortion due to mutual coupling, keeping good return loss conditions in
the antennas.
Phase shifter Types 1 and 2 were fabricated and arranged resembling Fig. 2.2, and all the possible
phase combinations were measured to evaluate their performance. Tables III and IV show the
simulated and measured progressive phase values between antennas compared to the ideal target
values for 4.7GHz and 7.5GHz respectively. All measured values remain close to the simulations
and target phases with an average phase error with respect to target values of 4.43 degrees at
4.7GHz and 4.63 degrees at 7.5GHz. Magnitude of the reflection coefficient conditions below 10dB were measured for the input and output phase shifter ports when all eight paths were
activated at both frequencies. Average insertion loss of 2.8dB including connectors and diode
losses was measured.
23
The final system is shown in Fig. 2.19. The four input ports of the phase shifters shown in Fig. 2.19
were connected to a 4-way SMA power divider to converge to a single input port. The DC
control lines for the PIN diodes were externally controlled by a Labjack U3 device [26]
connected through a USB interface to a PC, were custom software was designed. The system
radiation patterns were measured in the UCI far field anechoic chamber. The simulated (solid
line) and measured (dashed line) normalized radiation patterns for the array system with each
one of the progressive phases between antennas provided by the phase shifter is shown in Fig.
2.20.
A total of four switchable beams at each frequency can be obtained by using the proposed
phase shifter, with the measured patterns resulting reasonably close to the simulated ones. The
simulated patterns were obtained using Zeland IE3D with infinite ground plane, which explains
the absence of the back radiation that is present in measurements. On the other hand, in order to
simulate all different patterns the antenna array without phase shifter was fed directly with the
theoretical phases. Therefore, the simulated absolute peak gain is on average 3.1dB higher than
the measured one, which corresponds to the phase shifter and inter-stage connector average
losses.
Fig. 2.16 Fabricated Single element antenna and Zoom into the diode region. The devices are attached using silver
epoxy to avoid overheating
24
0
Mag. Reflection Coefficient (dB)
-5
-10
-15
-20
Sim. 4.7GHz (Diode ON)
Sim. 7.5GHz (Diode OFF)
Meas. 7.5GHz (Diode OFF)
Meas. 4.7GHz (Diode ON)
-25
-30
-35
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Frequency (GHz)
Fig. 2.17 Simulated versus measured antenna magnitude of the reflection coefficient. For simplicity, the cases ‘On’
and ‘Off’ corresponding to operation at 4.7GHz and 7.5GHz are superimposed in the same graph.
Fig. 2.18 Measured radiation patterns for the single element antenna operating in the (a-b) ‘On’ and (c-d) ‘Off’
cases. (a) and (c) show the E-plane while (b) and (d) show the H-plane. Solid and dashed line represents co-pol and
cross-pol components respectively. The measured efficiency is 79% and 68% with associated gains of 5.4dB and
5dB at 7.5GHz and 4.7GHz, respectively.
25
Target Frequencies
(GHz)
4.7
7.5
Target Values
At 4.7GHz
-135
-45
45
135
Target Values
At 7.5GHz
-135
-45
45
135
Table 2.2 Simulated and measured antenna center frequencies
Simulated
Measured
Return Loss<10dB at
Frequency
Frequency
target frequency
(GHz)
(GHz)
4.72
4.7
YES
7.53
7.55
YES
Table 2.3 Simulated and measured phase delays at 4.7GHz
Sim./Meas.
Sim. /Meas.
Sim. /Meas.
Ant#1~Ant#2
Ant#2~Ant#3
Ant#3~Ant#4
(deg)
(deg)
(deg)
-131.5/-128.73
-137.5/-136.37
-134.5/-139.76
-48.50/-50.00
-45.20/-44.55
-45.50/-49.31
45.50/45.77
45.20/42.37
48.50/50.00
134.50/139.76
137.50/136.37
131.50/131.77
Table 2.4 Simulated and measured phase delays at 7.5GHz
Sim./Meas.
Sim. /Meas.
Sim. /Meas.
Ant#1~Ant#2
Ant#2~Ant#3
Ant#3~Ant#4
(deg)
(deg)
(deg)
-134.50/-130.79
-131.00/-138.96
-138.50/-130.14
-43.50/-40.60
-49.00/-49.05
-46.00/-44.53
46.00/46.09
49.00/50.00
43.50/40.60
138.5/130.14
131.00/138.96
134.50/134.47
Fig. 2.19 Fabricated 4x1 switched beam antenna array and phase shifters.
26
120°
135°
105° 90° 75°
120°
135°
60°
45°
150°
0°
-30°
105° 90° 75°
30°
165°
0 -10
-20 -30
± 180°
0°
-30°
120°
135°
-30°
0°
-15°
45°
30°
15°
0°
-165°
-15°
-30°
-135°
-45°
-120°
-60°
-105° -90° -75°
-135°
-45°
-120°
-60°
-105° -90° -75°
60°
120°
135°
45°
30°
105° 90° 75°
150°
15°
60°
45°
30°
165°
0 -10
-20 -30
± 180°
0°
-165°
60°
-150°
-30°
165°
0 -10
-20 -30
± 180°
105° 90° 75°
165°
0 -10
-20 -30
± 180°
15°
-165°
-15°
-150°
0°
150°
30°
150°
15°
-150°
45°
105° 90° 75°
30°
-135°
-45°
-120°
-60°
-105° -90° -75°
165°
0 -10
-20 -30
± 180°
120°
135°
45°
-15°
60°
-150°
60°
-165°
-135°
-45°
-120°
-60°
-105° -90° -75°
150°
105° 90° 75°
165°
0 -10
-20 -30
± 180°
-15°
105° 90° 75°
-30°
150°
15°
-165°
120°
135°
-15°
120°
135°
45°
-150°
0°
-135°
-45°
-120°
-60°
-105° -90° -75°
60°
150°
15°
-150°
-135°
-45°
-120°
-60°
-105° -90° -75°
120°
135°
30°
-165°
-15°
-150°
45°
165°
0 -10
-20 -30
± 180°
15°
-165°
60°
150°
30°
165°
0 -10
-20 -30
± 180°
105° 90° 75°
-30°
0°
-165°
-15°
-150°
-135°
-45°
-120°
-60°
-105° -90° -75°
15°
-30°
-135°
-45°
-120°
-60°
-105° -90° -75°
Fig. 2.20 Simulated with IE3D infinite ground (solid) and measured (dashed) normalized E-total radiation patterns
for 4.7GHz (lf) and 7.5GHz (hf) corresponding to different progressive phases between antennas (a) lf,α=-135 simgain=10.8dB (b)lf, α=-45, sim-gain=12.5dB (c) lf,α=+135,sim-gain=10.5dB (d) lf,α=+45 sim-gain=13.7dB; (e) hf,
α=-135 sim- gain=14.8dB (f) hf, α=-45 sim-gain=14.7dB (g) hf, α=+135 sim-gain=12.7dB (h) hf, α=+45 simgain=15.2dB.
27
2.5 Conclusions
In this chapter, a dual frequency reconfigurable switched beam antenna array with phase shifter
using PIN diodes for telemetry applications was presented. A dual patch approach was used to
design the single antenna element, while a switched line topology was chosen for the phase
shifter. The system is capable of switching the beam to four selectable space positions at two
different frequencies with 1.6:1 ratio.
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31
3. BANDWIDTH ENHANCEMENT OF HIGH ISOLATION ISOFREQUENCY REPEATERS USING MEMS-RECONFIGURABLE
LOADED PARASITICS
3.1 Introduction
One of the main problems in wireless communications systems is the lack of signal coverage in
some specific areas shadowed by obstacles. The use of signal repeaters becomes a cost-effective
solution to extend the coverage to areas of low signal strength. To avoid frequency conversions,
an iso-frequency repeater (gap-filler) can be used. This device acts as a directional amplifier that
receives signal from a donor base station and retransmits it with the same frequency after an
amplification stage. A block diagram of a general repeater is shown in Fig. 3.1. On the other hand,
compact gap-filler solutions result in designs with a small physical separation between transmit
and receive antennas and its radio-frequency sections. As a consequence a strong undesired
electromagnetic coupling between the transmitting and receiving stages may arise, thus bringing
unacceptable isolation levels. This lack of isolation between transmitter and receiver may
severely degrade the performance of the system in the following ways:
•
Magnitude and phase errors in the repeated signal.
•
Magnification of high undesired spurious signals
•
Instability and oscillation in the amplifier
•
Mismatch between the antennas and the amplifier.
In addition, the nature of this coupling do not only depends on the system architecture but also
on the characteristics of the time-variant surrounding environment in which the gap-filler is
working. As a consequence of poor isolation, the active gain of the gap-filler must be reduced,
32
and the maximum coverage range reduces drastically. To improve isolation in iso-frequency
repeaters, a sheltering panel between the transmitter and the receiver can be used, or the physical
separation between the transmit and receive antennas can be increased [1]. However, those are
not practical solutions for compact systems. Some studies have focused in the design of an
adaptive feedback cancellation system [2] or store-and forward techniques [3, 4], but the
computational complexity and power consumptions in these solutions increase their cost
drastically.
In [5], we presented a preliminary concept of a narrowband back-to-back iso-frequency repeater
antenna system which used MEMS-reconfigurable parasitics to improve signal isolation. In this
work, we present a bandwidth enhanced solution based on a different and more flexible repeater
architecture which instead uses MEMS-reconfigurable loaded parasitics and an aperture coupling
feeding mechanism. The proposed gap-filler can be smartly reconfigured in real-time to optimize
its performance (matching and isolation level and bandwidth) in the presence of a changing
surrounding environment.
Fig. 3.1 Block Diagram of an Iso-frequency repeater.
33
3.2 Description of the repeater system
The proposed MEMS-reconfigurable iso-frequency repeater shown in Fig. 3.2 is composed of two
active microstrip patch antennas placed back-to-back over a 2x2 in quartz substrate (εr=3.78 and
δ≈0) with a thickness of 1.587 mm. The feeding mechanism is done through coaxial probe in an
optimum antenna location to ensure impedance matching at the operating frequency of 5.2 GHz.
The ground planes of the microstrip patches (1.5λx1.5λ) also act as sheltering panels between the
transmitting and receiving side. At a distance of 0.13λ, two sets of out-of-plane parasitics
(OUPPs) are placed symmetrically with respect to the sheltering planes. These parasitic elements
are loaded with MEMS switches or PIN diodes, which allow them to be reconfigured into a
short- or open-circuit state. These parasitics act as directors of the active patches when they are
short-circuited while they will become transparent to the system in the open–circuit state. The
OUPPs are used to improve the isolation capabilities of the gap-filler over a certain frequency
band. On the other hand, to recover from mismatches of the antenna due to a specific
environment configuration (such as close obstacles, etc.), two sets of in-plane parasitics (IPPs)
are placed next to each microstrip antenna and loaded as well with MEMS. Therefore, these IPPs
can also be set into an open- or short-circuit state. The capacitive couplings produced from the
main patch to these elements in combination with the capability to select the state of their
respective switches, allows one to effectively elongate the antenna width and thus to incorporate
an additional degree of freedom to reconfigurable the impedance matching over a certain
frequency band. Also, the IPPs can contribute to minimize the coupling between the antennas.
Both sets of parasitics (92 in total) can be driven to dynamically enhance the isolation and
matching conditions and thus to optimize the performance of the system under a wide variety of
working environment.
34
Fig. 3.2 Proposed Iso-frequency repeater using RF-MEMS loaded parasitics.
Fig. 3.3 UCI’s RF MEMS switch picture. This specific device was built over a quartz substrate and is intended to be
used as depicted in Fig. 3.2
35
3.3 Design Of In-plane (IPPs) and Out-of-plane (OUPPs) parasitics
The width (wo), separation (so), distance (do) and Number (No) of OUPPs (Fig. 3.4) was optimized
first in order to obtain the maximum single tone isolation level for all cases with obstacles up to
1.5x1.5 λ distance and 45 degrees tilt. This study concluded with an optimum number of eight
OUPPs that can be switched in two different states, open circuit (bit ‘0’ or non active) and short
circuited (bit ‘1’ or active).
When the antenna is placed near a metallic object, the OUPPs will activate or de-activate to find
the optimum condition that satisfies maximum isolation. However, the input impedance of the
antenna will change, resulting on reflection coefficient increase and thus mismatch with the
amplifier stage. When this happens, the amplified signal may not be radiated efficiently into the
space. So this is one major problem that needs to be addressed.
It is known that the input impedance and bandwidth of a microstrip patch antenna can be
modified by controlling its width. The idea behind the In-plane parasitic concept is being able to
dynamically change this dimension to recover a good antenna reflection coefficient condition (<10dB) in all case scenarios.
In plane parasitics (IPPs) are capacitively coupled to the microstrip patch, increasing its effective
width. These parasitics will also change the radiation pattern characteristics, contributing to
maximize the isolation between radiating elements.
The width (wi), separation (si) and Number (Ni) of IPPs (Fig. 3.4) was optimized in order to
recover the minimum antenna reflection coefficient conditions for six standard loading cases for
the OUPPs.
36
Table 3.1,
shows the antenna S11 for several combinations of OUPPs without the effect of the
IPPs and how an optimal S11 condition can be achieved with and optimal configuration of the
IPPs.
Table 3.1 Reflection coefficient (S11) for several combinations of OUPPs and optimized reflection coefficient using
best configuration of IPPs. Bit ‘0’ meaning open circuited parasitic and bit ‘1’ meaning short circuited
Binary Combination for OUPPs
S11(dB)
S11 optimized (dB)
[1 0 1 0 0 1 0 1]
[1 1 1 1 1 1 1 1]
[1 1 1 0 1 1 1 1]
[1 0 1 0 1 0 1 0]
-4.2966
-14.1311
-11.3433
-4.6566
-18.212 (+14)
-16.9427 (+2)
-58.8151 (+47)
-40.1912 (+36)
Fig. 3.4 A) OUPPs detail with parameters wo =0.5mm,= so =5mm,= do =20mm No =8 (each side) B) IPPs details.
wi=0.5mm,= si =0.75mm Ni=46 (each side).
3.4 Combined isolation and matching optimization using the genetic algorithm
optimizer.
In order to optimize the matching and isolation in the proposed reconfigurable gap-filler, we use
a genetic optimization algorithm (GOA) to select the optimal state of the MEMS switches for
each of the parasitic elements. This evolutionary technique has been successfully used in several
electromagnetic problems in the last years [6].Using the GOA, a binary chromosome (array)
composed of 92 bits is evaluated as a potential solution candidate. This binary array contains the
37
state information of each MEMS switch. MEMS in the OFF state (open-circuit) are assigned a
bit value of ‘1’, while MEMS in the ON state (short-circuit) are assigned a bit value of ‘0’. Each
chromosome candidate receives a score according to a specific fitness function. Greater scores
are assigned to solutions that maximize the isolation between active antennas (S12) at 5.2 GHz
while maintaining a minimum reflection coefficient level (S11,S22) of 15 dB. The GOA
maximizes the following fitness function:
abs( S11 ) if abs( S11 ) < 15dB
fitness = abs( S12 @ 5.2 GHz ) − f ( S11@ 5.2GHz ) where f ( S11 ) = 
0 otherwise

(3.1)
By using this approach, only one full wave simulation need to be performed which makes the
optimization process feasible. In this case the structure is drawn in HFSS and each of the
radiating antenna elements and parasitics are assigned a lumped port. The structure is then EM
simulated resulting in a S-parameter matrix of N by M dimensions SNxM, where N is the number
of radiating antennas (in this case N=2) and M is the number of parasitic elements (in this case
M=92). This 184 element matrix can be divided into four different sub-matrices named:
•
S11NxN: A NxN square matrix containing the reflection coefficient information from the
radiating elements
•
S21MxN,S12NxM: A MxN or NxM rectangular matrix containing the transmission coefficient
information between the radiating and parasitic elements and viceversa.
•
S22MxM: A MxM square matrix containing the reflection coefficient information from the
parasitic elements
Once these sub-matrices have been defined, we can mathematically load the M parasitic
elements with ideal open or short circuits as depicted in Fig. 3.5, where the cascaded ΓR vector
represents the reflection coefficients from the open and short circuiting loads taking values of +1
38
or -1, respectively. The resulting network after the parasitic loading now consists of a NxN
matrix containing the S-parameter information (isolation and matching) of just the radiating
elements. With proper manipulation we can find this matrix (Sin) by using expression (3.2).
Fig. 3.5 Network cascaded connection representing the loading conditions for each of the parasitic ports. The
loading block is represented by a set of reflection coefficients being +1 for the open circuit and -1 for the short
circuit conditions.
(
NxN
NxM MxM
MxM
S in = S11
+ S12
I
− ΓR S 22
(
NxN
NxM MxM
MxM
S in = S11
− S12
I
+ S 22
)
−1
)
−1




MxN
ΓR S 21
→  ΓR( i ,i )
= ±1
short _ circuit ( − )
 open

_ circuit ( + )
(3.2)
MxN
S 21
After a unique full wave multiport simulation is performed, the parasitic elements are loaded
using a combination decided by the genetic algorithm (assigning bit ‘1’ to ΓR,i=1 and bit ‘0’ to
ΓR,i=-1). The genetic algorithm is then in charge of evaluating the fitness of each solution
candidate and conclude with a global optimum genome.
In order to verify the reconfigurable capabilities of the proposed gap-filler, we consider for now
a single-tone analysis (at 5.2 GHz) for the following three cases. Case 1 corresponds to a
standalone reconfigurable gap-filler without any scatterers surrounding the antennas. In case 2, a
1λx1λ metallic plate is placed around the proposed repeater at a distance of 1λx0.5λ and forming
39
a 45º angle with respect to the antennas ground plane. Finally, in order to compare the results to
those of a conventional (non-reconfigurable) repeater, a reference case (Case ref.) is also
considered which basically consists of two back-to-back microstrip patches. Fig. 3.6(a-b) shows
the simulation results after optimization for cases 1 and 2, respectively, using the proposed gapfiller system. A maximum level of 85 dB of isolation is achieved for case 1 and 125 dB for the
case 2, at the operating frequency 5.2 GHz. The dashed line shows the reference case
corresponding to the isolation of the conventional repeater, which is approximately 30 dB at 5.2
GHz. Therefore, a 55 dB and 95 dB isolation improvement is achieved in cases 1 and 2,
respectively. The impedance bandwidth at -10dB is around 40 MHz (0.77%), while the isolation
bandwidth at -50 dB is around 0.6%, in both cases. Fig. 3.8(a) shows the resultant 3D radiation
patterns after optimization for case 1 and 2. Peak gains of approximately 8 dBi are achieved in
both cases. Notice that the reconfigurable repeater, in the adaptive process, tries to place a
radiation null in the direction of the metallic obstacle to avoid the undesired back-scattered
radiation.
3.5 Isolation and impedance bandwidth enhancement.
We now investigate the capabilities of the proposed reconfigurable gap-filler to improve its
bandwidth for a particular isolation level (-50 dB) and matching level (-10 dB). In this occasion,
for the configuration shown in case 2 of the previous section, we let the GOA maximize any of
following two fitness functions:
5
fitness1 = ∑ abs ( S12@ f k GHz ) − f ( S11@ 5.2 GHz ),
k =1
fitness2 = min(abs ( S12@ f k GHz )) − f ( S11@ 5.2 GHz )
(3.3)
where fk corresponds to frequencies points ([5.18, 5.19, 5.2, 5.21, 5.22] GHz) within the
considered band. The results of the optimization are shown in Fig. 3.7(a) and Fig. 3.7(b), when
40
using the fitness function 1 and 2, respectively. Values of 3.3% and 2.5% isolations bandwidths
at -50 dB are achieved using the fitness functions 1 and 2, respectively. This represents a
significant improvement in the isolation bandwidth with respect to the 0.6% bandwidth achieved
with the single-tone analysis (case 2 in . The impedance bandwidth is 0.77% and 1.15% for the
fitness functions 1 and 2, respectively. On the other hand, in order to further improve the
inherent narrowband frequency response of the used microstrip antennas, we now use a thicker
substrate (4.762 mm) in combination with an aperture coupling feeding technique [7]. Applying
this idea, the proposed repeater can achieve higher impedance and isolation bandwidths. Results
after optimization with this approach are shown in Fig. 3.8, where one can observe that the
impedance bandwidth has now been increased to 17% while the isolation bandwidth is 1.5%.
3.6 Additional Information.
For additional information on this topic the reader is referred to a recent publication where a
prototype of a version of this repeater using PIN diodes has been built and successfully tested [8,
9]. Additionally, such repeater system includes some extent of beam steering capabilities besides
isolation enhancement and good matching capabilities.
41
Fig. 3.6 Results after optimization. (a) case 1, no obstacle. (b) case 2, obstacle at 1λx0.5λ.
Fig. 3.7 Results after optimization of the isolation bandwidth. Case 2 and different fitness functions.
Fig. 3.8 a) Radiation patterns after optimization for case 1 and 2. b) Aperture coupled microstrip antenna with
optimized impedance bandwidth and high isolation.
42
References
[1]
W. T. Slingsby and J. P. McGeehan, "Antenna isolation measurements for on-frequency
radio repeaters," in Ninth International Conference on Antennas and Propagation (Conf. Publ.
No.407), Eindhoven, Netherlands, 1995, pp. 239-43 vol.1.
[2]
S. J. Kim, et al., "Adaptive feedback interference cancellation system (AF-ICS)," in
Microwave Symposium Digest, 2003 IEEE MTT-S International, 2003, pp. 627-630 vol.1.
[3]
J. J. Nehez, "Multiple Channel Same Frequency Repeater flight test," in Proceedings of
the IEEE 1974 National Aerospace and Electronics Conference, Dayton, OH,, 1974, pp. 333340.
[4]
T. Gluszczak, et al., "Wideband Digital UHF Same Frequency Repeater," MAL-TR-73-
22, 1973.
[5]
A. Grau, et al., "Back-to-back high-isolation iso-frequency repeater antenna using
MEMS-Reconfigurable-Parasitics," in Antennas and Propagation Society International
Symposium, 2007 IEEE, 2007, pp. 497-500.
[6]
Y. Rahmat-Samii, "Genetic algorithm (GA) and particle swarm optimization (PSO) in
engineering electromagnetics," in Conference Proceedings ICECom 2003. 17th International
Conference on Applied Electromagnetics and Communications, Dubrovnik, Croatia, 2003, pp. 15.
[7]
P. B. R.Garg, I. Bahl, A. Ittipiboon, Microstrip Antenna Design Handbook: Artech House
antennas and propagation, 2001.
[8]
E. Diaz, et al., "Pixeled-Dipole Based Isofrequency Reconfigurable RF Repeater,"
presented at the European Conference on Antennas and Propagation, EUCAP, Rome, 2011.
43
[9]
E. Diaz, et al., "Isofrequency Reconfigurable 8-bit RF Repeater," IEEE, Transactions on
Antennas and Propagation, (Submitted) 2011.
44
4. AN ANALYTICAL ALGORITHM FOR PI-NETWORK IMPEDANCE
TUNERS.
4.1 Introduction
Tunable matching networks are expected to play an important role in the realization of adaptive
and reconfigurable radio front-end architectures. One particular example is the compensation of
handset antenna impedance mismatch loss caused by user proximity effects using tunable
antenna impedance matching networks [1-5].
Different matching network topologies have been reported in the literature. Basic L-type
networks are able to achieve conjugate matching over a limited Smith chart region [6-8]. On the
other hand, pi-networks [1, 3, 5, 9, 10] similar to (Fig. 4.1(a)) provide an extra degree of freedom
that enables conjugate matching over a substantially wider impedance range. In the ideal case
where the component values range is unbounded [0,∞] pi-networks can provide complete Smith
Chart coverage and the component values for perfect conjugation matching can be calculated by
using the approaches given in [10-12].
The finite nature of the component tunable range is due to practical implementation limitations
such as parasitic influences and component properties. For a matching network with finite
component tuning ranges the perfect conjugate match can be achieved only if the load impedance
lays within the matching domain [11, 13]. In reality, the network component available range may
be predetermined and the unknown load impedance may often be located outside of the matching
domain. Therefore, in practical impedance tuners with finite component tuning range, where a
perfect conjugation match may not exist, optimization techniques have been commonly used to
minimize the reflected signal (minimize VSWR) [5, 14-18]. Different optimization approaches,
45
such as simplex and single step [5], genetic algorithm [15, 16, 18], or simulated annealing [17,
18] have been used to minimize the network input reflection coefficient as much as possible or
at least down to an acceptable level. These optimization methods search for the right component
tuning setting through an iterative process, consuming a considerable amount of time to reach the
tuning goal. In addition, depending on the optimizer choice and its initial settings, there is a risk
of converging into local minima.
Thus it is desirable to develop a deterministic approach to directly compute the final component
tuning setting for the impedance match in order to reduce the tuning time and avoid the
intermediate tuning states. This chapter presents an analytical algorithm to determine the
required component values for a tunable pi-network having finite tuning range components
matching any load impedance on the Smith chart at a given frequency. The analytical tuning
algorithm is based on closed-form formulas and a direct calculation procedure. For a given load
impedance, the algorithm first attempts to find a perfect conjugation match solution, and in case
it does not exist, the algorithm then calculates the setting providing the maximum power transfer
to the load.
It is acknowledged that in the case of a matching network without loss, tuning for achieving
conjugation match or minimizing the reflection coefficient means maximizing the power transfer
to the load. However in reality, the matching network has a certain amount of loss and the above
statements are no longer equivalent. Thus any impedance matching approach or algorithm based
on (or partially based on) minimizing the input reflection coefficient, only has good accuracy for
lossless and low loss matching networks or tuners where the final matching goal is maximizing
the power delivered to the load. The scope of application of this algorithm is not exclusive to
46
antenna impedance tuning control; it can be used for tunable or non-tunable matching network
design or performance analysis to replace an optimization tool.
Sections 4.4 to 4.5 of this chapter present the algorithm for perfect and optimal matching
conditions based on VSWR or maximum power transfer criteria. Section 4.6 compares the
experimental results from this algorithm with those obtained by means of the robust and simplex
optimization methods [19]. The results are shown to be in very good agreement, but the
algorithm using significantly less time than the iterative optimizer to achieve the same results.
The application of the algorithm is not constrained to a specific pi-network topology, but may
also be extended to other pi-network topologies as described in Section 4.7. Section 4.8 provides
additional application examples to illustrate use of the algorithm.
4.2 Matching parameters and tunable pi-network configuration
In this chapter, we evaluate the performance of a pi-network tuner algorithm based on the input
voltage standing wave ratio (VSWR) and transducer gain and/or relative transducer gain. The
VSWR at the input port of the pi-network tuner is related to its input reflection coefficient ( Γin )
as
VSWRin =
1 + Γin
1 − Γin
(4.1)
The impedance match is achieved by minimizing the magnitude of reflection coefficient or
equivalently the VSWR. In the case of perfect match, the magnitude of the reflection coefficient
is zero and VSWR is equal to 1.
On the other hand, the transducer gain (GT) of the pi-network tuner is defined as the ratio of the
power delivered to the load to the available power from the source, and can be expressed as [20]
47
2
GT =
(
2
S 21 1 − ΓL
1 − S 22 ΓL
2
)
(4.2)
where |S21|2 and S22 are the insertion loss and output reflection coefficient, respectively, of the
pi-network tuner. ΓL is the load reflection coefficient. The relative transducer gain is the ratio of
the GT to the transmission loss (1-|ΓL|2) caused by the mismatched load impedance. The relative
transducer gain (∆GT) is the GT improvement achieved by introducing the pi-network tuner and it
can be expressed as:
∆ GT =
S 21
2
1 − S 22 ΓL
2
(4.3)
Maximizing the GT or ∆GT for a given load impedance, i.e. delivering maximum power to the
load, is often a better merit than minimizing the VSWR since GT and ∆GT include the effect of
the tuner’s internal loss.
Fig. 4.1 (a) ideal tunable low-pass pi-network and (b) equivalent circuit of practical pi-network tuner with fixed
inductor and tunable capacitors.
An ideal tunable low-pass pi-network topology is shown in Fig. 4.1(a) using a tunable inductor
and two tunable capacitors. A practical pi-network tuner consisting of tunable capacitors may be
implemented as shown in Fig. 4.1(b). In this case, since there is no practical tunable inductor
available, the series equivalent inductance tuning is achieved by adjusting the tunable capacitor
48
C3 connected in parallel to a fixed inductor L [21]. As a result, the series equivalent inductance
Le value is determined by
Le =
L
1 − ω 2 LC3
(4.4)
where ω equal to 2πf is the angular frequency in (rad/s). In Fig. 4.1(b), C1 to C3 represent high-Q
tunable capacitors having a finite tuning range
Ck,min ≤ Ck ≤ Ck,max,
(4.5)
where k={1,2, or 3}.
4.3 Explanation of composite components frequently used in the algorithm
The tuning algorithm for the pi-network tuner is developed based on the circuit topology,
depicted in Fig. 4.2, which consists of a load impedance, ZL = RL + jXL, a source with a internal
impedance Ro and the pi-network tuner. The pi-network tuner input port (source port) is defined
between terminals 1-1’ while the terminals 2-2’ define the output port where the load (e.g.
antenna) is to be connected. The equivalent inductance Le comprised by L and C3 and composite
source/load impedances, which are frequently used in the tuning algorithm development, will be
discussed in this section.
Fig. 4.2 Circuit topology of pi-network tuner with driving source and load to be matched
.
49
4.3.1 Equivalent Series Inductance & Its Characteristics
In Fig. 4.2, Le represents the equivalent inductor of the parallel combination of C3 and L with the
inductance value determined by (4.4). To obtain a desired equivalent inductance Le for a fixed L
at a given frequency ω, the required C3 can be derived from (4.4) as
C3 =
Le − L
(4.6)
ω 2 Le L
The series equivalent inductor Le presents a resonance at the frequency given by
fr =
1
2π LC 3
(4.7)
The equivalent inductance Le for L = 3 nH varies with operating frequency as shown in Fig. 4.3
when C3 takes values from 0.5 pF to 5 pF. Fig. 4.3(b) shows that Le becomes negative when the
operating frequency is beyond fr. Also, for frequencies near or beyond the resonance frequency fr
the effective Q of the series element will be significantly reduced. To keep Le positive, C3 should
be small enough to meet the following inequality
C3 <
1
(4.8)
ω 2L
20
C3 Range
Effective Inductance Le (nH)
18
0.500pF
0.625pF
0.750pF
0.875pF
1.000pF
1.125pF
1.250pF
1.375pF
1.500pF
16
14
12
10
8
6
4
2
800
1000
1200
1400
1600
Frequency (MHz)
(a)
50
1800
2000
2200
4.3.2 Composite Load and Source Impedance
As described in [6], any impedance in the Smith chart can be perfectly matched by properly
choosing the appropriate L-network topology and component values. The L-network topology
choice roughly depends upon RL ≤ Ro or RL > Ro (formally, they should be the ‘Yin and Yan’
areas in Smith chart [6]). Two main topologies of the L-network are used: the L-network with
the shunt capacitor on the right side of the series inductor or the L-network having the shunt
capacitor on the left side of the series inductor. A tunable low-pass pi-network is turned into
either of the two L-networks by adjusting the shunt capacitor, C1 or C2, to zero. However, in
practice this is not possible since a realizable capacitor always has parasitic capacitance making
its minimum capacitance, Cmin, greater than zero.
100
Effective Inductance Le (nH)
C3 Range
2pF
3pF
4pF
5pF
50
0
-50
-100
800
1000
1200
1400
1600
1800
2000
2200
Frequency (MHz)
(b)
Fig. 4.3 Equivalent or composite inductance Le as a function of frequency for different values of C3 ranging (a) 0.5
pF to 1.5 pF and (b) 2 pF to 5 pF.
Therefore, depending on the load impedance, we either absorb C1,min into the source impedance
Ro to form a composite source impedance ZS or absorb C2,min into the load impedance ZL to form
a composite load impedance Ze. Thus, we can adopt the matching approaches similar to the ones
used for the aforementioned L-networks.
51
From Fig. 4.2, the composite source impedance ZS is comprised of Ro and C1,min and can be
expressed as
ZS =
R o B C 1, min
Ro
1
1
=
=
+ j
2
2
YS
G o + jB C 1, min 1 + R o B C 1, min
1 + R o2 B C2 1, min
(4.9)
In (4.9), G o = 1 Ro , and
B C 1 , min = 2 π f ⋅ C 1 , min = ω ⋅ C 1 , min
(4.10)
The composite load impedance Ze consists of ZL and C2,min and its expression is given in (4.11)
Z e = Re + jX e =
Re =
Xe =
1
G L + j (BL + BC 2,min )
GL
G + (B L + BC 2,min )
2
2
L
− (B L + B C 2 , min
)
G + (B L + B C 2 , min
2
L
(4.11)
)
2
where GL and BL are given in (4.12)
YL =
R
X
1
= G L + jB L = 2 L 2 − j 2 L 2
ZL
RL + X L
RL + X L
(4.12)
And
BC 2 ,min = 2πf ⋅ C 2 ,min = ω ⋅ C 2 , min
(4.13)
The portion of the Smith chart that can be effectively perfect matched is reduced after using the
composite source or composite load impedance. This means that not every load impedance in the
Smith chart can be perfectly matched by using a realizable pi-network tuner. If a perfectly
52
matched solution does not exist, the tuning algorithm will provide the best matching based on
maximizing ∆GT or alternatively minimizing VSWR.
4.4 Formulas and algorithm for perfect match
4.4.1 Perfect Match Solutions
If the perfect match solution exists, there are two sets of formulas to find pi-network component
values. Either set of formulas is chosen depending on the load impedance characteristics. For a
given load impedance ZL = RL + jXL, and a specific operating frequency f, the perfect match
solutions for RL ≤ Ro and RL > Ro are as follows:
In the case of RL ≤ Ro, setting C2 = C2,min and BC2 = ωC2,min, we have solutions for C1 and Le of
(see Appendix A)
C1 =
1
2π f
G o (G e − G o )
(
R e ( Ro − R e ) − X e
(4.14)
and
Le =
1
2πf
)
(4.15)
where Ge = 1/Re. Capacitance C3 can be obtained by substituting (4.15) into (4.16), where L is a
fixed given inductance.
In the case of RL > Ro (accurately, the condition should be 1/GL > Ro and in the algorithm
calculation this restricted condition has been considered at step 11.), setting C1 = C1,min and BC1 =
ωC1,min, the solutions for C2 and Le to achieve perfect match are (see Appendix A)
C2 =
1
2πf
 GL


1 + Ro2 BC21,min − GL2 − BL 
 R

o


(
)
and
53
(4.16)
 GL



1 + Ro2 BC21, min − GL2
2
Ro BC1, min 
1  Ro
Le =
+


GL
2πf 
1 + Ro2 BC21, min 
1 + Ro2 BC21, min


Ro


(
)
(
)
(4.17)
Substituting (4.13) into (4.6), we can obtain the required C3 capacitance value for this match
solution.
4.4.2 Algorithm for Perfect Match
For a better understanding of the algorithm the reader is advised to refer to the flow chart
provided in Fig. 4.4 when reading the following steps.
1. Check the real part of the load impedance, RL, first if RL ≤ Ro or RL > Ro.
a. Algorithm for RL ≤ Ro (Branch1):
2. Set C2 = C2,min and let BC2 = ωC2,min.
3. Calculate GL, BL from (4.12) and use the resulting values to calculate the equivalent load
impedance Ze from (4.11)
4. Utilize (4.14) and (4.15) to obtain C1 and Le values.
5. If real or real part of Le < 0 or C1 < C1,min, then go to branch ‘Algorithm for RL > Ro
(Branch2)’ and follow that branch to complete the calculations. Otherwise proceed with
step 6.
6. If C1 ≤ C1,max, compute C3 by using (4.6). Otherwise, set C1 = C1,max, re-compute Le by
using (19) and calculate C3 with (4.6).
7. If C3 < C3,min, set C3 = C3,min and follow the algorithm presented in Section 4.5.
Otherwise, we obtain solutions C1,max, C2,min, and C3 if C3 ≤ C3,max and C1 ≥ C1,max.
54
8. If C3 > C3,max, set C3 = C3,max or set C3 = C3,min only if C3 > 1/(ω2L), and then follow the
algorithm given in Section 4.5. Otherwise, we obtain solutions C1, C2,min, and C3 if C3 ≤
C3,max and C1 ≤ C1,max.
b. Algorithm for RL>Ro (Branch 2):
9. Set C1 = C1,min and let BC1 = ωC1,min.
10. Calculate C2 and Le values by using (4.16) and (4.17).
11. If C2 < C2,min, go to ‘Algorithm for RL ≤ Ro (Branch1)’ and follow the algorithm given in
that branch.
12. If C2 < C2,max, use (4.6) to calculate C3 from the Le derived from (4.17). Otherwise, set C2 =
C2,max and recompute Le by utilizing (4.19); then calculate C3 with (4.6).
13. If C3 < C3,min, set C3 = C3,min and follow the algorithm presented in Section 4.5.
Otherwise, we obtain solutions C1,min, C2,max, and C3 if C3 ≤ C3,max and C2 ≥ C2,max.
14. If C3 > C3,max, set C3 = C3,max or set C3 = C3,min only if C3 > 1/(ω2L), and then follow the
algorithm given in Section 4.5. Otherwise, we obtain solutions C1,min, C2 and C3 if C3 ≤
C3,max and C2 ≤ C2,max.
4.5 Formulas and algorithm for best match
4.5.1 Formulas for Best Match
From the previous section, we have seen that perfect match solutions do not always exist for all
the load impedances in the Smith chart. This is due to the limitations imposed by the finite tuning
range of the tunable capacitors, Ck,min ≤ Ck ≤ Ck,max (k = 1,2 & 3), and the fixed value of the
55
series inductor. In this case, we can employ the best match solutions, which maximize GT or ∆GT
and/or minimize input VSWR (or reflection coefficient). The expressions of the GT and ∆GT are
given in (4.2) and (4.3), respectively, and the VSWR at the input port of the pi-network tuner, is
defined by (4.1), where Γin has an expression (see Appendix B)
Γin =
Go [1− X Le (BL + BC2x )] − GL (1− X LeBC1x ) + j[GoGL X Le − BC1x − (BC2x + BL )(1− X LeBC1x )]
Go [1− X Le (BL + BC2x )] + GL (1− X LeBC1x ) + j[GoGL X Le + BC1x + (BC2x + BL )(1 − X LeBC1x )]
(4.18)
where X Le = ωLe
In Subsection 4.4.2, steps 6 and 12, we may be forced to take C1 = C1,max and C2 = C2,min, or to
employ C1 = C1,min and C2 = C2,max, respectively. In either case, we need to find an equivalent
series inductor Le, which minimize the input VSWR of the pi-network tuner. The Le providing
the best match resulting from
Le =
∂VSWR in ∂L e = 0 is
presented in (4.19),
[
1 BC 2 x + BL + BC1x Ro2 (BC 2 x + BC1x + BL )(BC 2 x + BL ) + G L2
⋅
2πf
(BC 2 x + BL )2 + GL2 ⋅ BC21x Ro2 + 1
[
](
)
]
(4.19)
where BCkx = ω ⋅ C k ,min or Ck ,max (k = 1 or 2)
In the case of C3 < C3,min or C3 > C3,max as occurs in steps 7 – 8 and 13 – 14 of Subsection 4.4.2,
we need to set C3 = C3,min or C3 = C3,max. If the C2 value has been previously assigned to C2,min or
C2,max , the C1 value should be chosen to minimize the input VSWR of the pi-network tuner from
∂VSWR in ∂C 1 = 0 ,
and the C1 for the best match is
C1 =
1
2πf
⋅
[
]
X Le (B L + BC 2 x ) + G L2 − (B L + BC 2 x )
2
[1 − X Le (B L + BC 2 x )]2 + X Le2 G L2
56
(4.20)
where X Le = ωLe =
ωL
2
1 − ω LC3, min
or
ωL
1 − ω 2 LC3,max
On the other hand, if the C1 value has been defined
as equal to C1,min or C1,max , the C2 value that minimizes the input VSWR of the pi-network tuner
derived from
∂VSWR in ∂C 2 = 0
C2 =
is
[
(
]
)
1 X Le 1− X Le BL 1+ Ro2 BC21x + BC1x Ro2 (2BL + BC1x ) − Ro2 (BL + BC1x )
⋅
2
2
2πf
X Le
+ Ro2 (1− X Le BC1x )
(4.21)
The corresponding formulas of Le, C1, and C2 resulting from maximizing ∆GT can also be
derived, but such a derivation is outside the scope of this chapter. On the other hand, we found
that the final results of using the formulas derived either from maximizing ∆GT or from
minimizing input VSWR differ insignificantly for low loss networks, i.e., the former set of
formulas may provide only a few hundredths of a dB better average ∆GT than the latter ones.
57
C3 > 1
C3 > 1
Lω 2
Fig. 4.4 Tuning algorithm flow chart for Branch 1 and 2
58
Lω 2
4.5.2 Algorithm for Best Match
1. From step 6 in Section 4.4.2, we have C1 = C1,max and C2 = C2,min or BC1x = ωC1,max and
BC2x = ωC2,min,. Use (4.19) to recalculate Le to obtain the best match. Then, follow the
steps presented in Section 4.4.2.
2. From step 7, we have C1 = C1,max and C3 = C3,min if C3 ≤ C3,min. Employ (4.21) to compute
C2 for the best match.
3. From step 8, we obtain C2 = C2,min and C3 = C3,max if C3 > C3,max and C3 < 1/(ω2L).
Employ (4.20) to calculate C1 for the best match. We shall take C2 = C2,min and C3 = C3,min
even if C3 > C3,max but C3> 1/(ω2L). Then use (4.20) to calculate C1.
4. From step 12, we have C1 = C1,min and C2 = C2,max, or BC1x = ωC1,min and BC2x = ωC2,max.
Use (4.19) to recalculate Le to obtain the best match. Then, follow the steps presented in
Section 4.4.2.
5. From step 13, we have C2 = C2,max and C3 = C3,min if C3 ≤ C3,min. Employ (4.20) to compute
C1 for the best match.
6. From step 8, we obtain C1 = C1,min and C3 = C3,max if C3 > C3,max and C3 < 1/(ω2L). Adopt
(4.21)
to calculate C2 for the best match. We shall take C1 = C1,min and C3 = C3,min even if
C3 > C3,max but C3 > 1/(ω2L). Then use (4.21) to calculate C2.
The C1 and C2 values calculated from (4.20) and (4.21), respectively, in steps 16 – 17 and 19 – 20
may not be within their boundaries. Therefore, in the implementation of the best match
algorithm, we use the following Branch 3 algorithm when the calculated C3 value for the perfect
match is out of its boundaries, i.e.,
C 3 ≤ C 3, min
or
C 3 ≥ C 3, max .
59
4.5.3 Algorithm for Branch 3
The reader is again advised to refer to the flow chart given in Fig. 4.5 when reading the algorithm
for this branch. The goal of the algorithm in Branch 3 is to search for the maximized GT or ∆GT
based on using formulas (4.20) and (4.21). In the ∆GT calculations, S parameters of the pi-network
tuner need to be used. The formulas for the S parameters of the pi-network can be found in
Appendix C.
a. We shall first set C3 = C3,max if C3 ≥ C3,max, or set C3 = C3,min if C3 ≤ C3,min or C3 ≥ C3,max
while meeting inequality (4.8).
b. Set C1 = C1,min and use (4.21) to compute C2. If C2 is within the range of C2,min ≤ C2 ≤
C2,max, we utilize (4.3) to calculate
∆ GT = ∆ GT 1 ,
or otherwise setting C2 = C2,min if C2 ≤
C2,min or C2 = C2,max if C2 ≥ C2,max, we then compute
∆ GT = ∆ GT 1
by using (4.3) ( ∆GTx
represents a calculated results to identify from the calculated result ∆GTx).
c. Set C1 = C1,max and use (4.21) to compute C2. If C2 is within the range of C2,min ≤ C2 ≤
C2,max, we utilize (4.3) to calculate
∆ GT = ∆GT 2 ,
or otherwise setting C2 = C2,min if C2 ≤
C2,min or C2 = C2,max if C2 ≥ C2,max, we then compute
∆GT = ∆GT 2
by using (4.3).
d. Set C2 = C2,min and use (4.20) to compute C1. If C1 is within the range of C1,min ≤ C1 ≤
C1,max, we utilize (4.3) to calculate
∆GT = ∆GT 3 ,
or otherwise setting C1 = C1,min if C1 ≤
C1,min or C1 = C1,max if C1 ≥ C1,max, we then compute
∆GT = ∆GT 3
by using (3).
e. Set C2 = C2,max and use (4.20) to compute C1. If C1 is within the range of C1,min ≤ C1 ≤
C1,max, we utilize (4.3) to calculate
∆ GT = ∆GT 4 ,
or otherwise setting C1 = C1,min if C1 ≤
C1,min or C1 = C1,max if C1 ≥ C1,max, we then compute ∆GT = ∆GT 4 by using (4.3).
60
f. Comparing ∆GT 1 / ∆GT 1 , ∆GT 2 / ∆GT 2 , ∆GT 3 / ∆GT 3 , and ∆GT 4 / ∆GT 4 , we choose the solutions or
the setting of the tunable capacitors corresponding to the largest ∆GTx or
∆GTx
among
these four ∆GT values calculated from steps b to e. The solutions can be one of the
following permutations sets: ( C1,min / C1,max , C2, &
C 3, min / C 3, max
) or ( C1,min / C1,max ,
C3,min / C3,max )
C 2, min / C 2, max ,& C3,min / C3,max ),
or (C1, C2,min / C2,max ,&
where we should read
Cx or Cy. These are the best match solutions in the terms of maximizing ∆GT.
∆GT 2
∆GT 3
∆GT 1−4
∆GT 1− 4
∆GT 1
∆GT 4
Fig. 4.5 Algorithm flow chart for Branch 3
61
Cx / C y
as
4.6 Comparison of algorithm and optimizer results
In this section, different ‘experimental’ matching results calculated by using the algorithm are
compared to those obtained by using a commercial and validated optimizer [19]. We first
compare of the results obtained from a lossless pi-network turner and then those resulting from
the same tuner but using components with finite Q factor.
As an example, it is assumed that the pi-network tuner used here has the following parameters:
tunable shunt capacitors C1 and C2 have a tuning range from 0.8 pF to 5 pF, C3 has a tuning
range from 0.25 pF to 4 pF, and the fixed series inductor L has a value of either 6.8 nH or 2.3 nH
for low frequency band (700 to 960 MHz) and high frequency band (1710 to 2170 MHz),
respectively. The input VSWR contours of match tuning the load with reflection |ΓL| varying
from 0.05 to 0.95 at 700 MHz by using the algorithm and utilizing the optimizer are shown in
Fig. 4.6.
The average VSWR over the Smith chart within the region of 0.05≤ |ΓL| ≤ 0.95 and -180o
≤ ∠ΓL ≤ 180o is 2.13 for both the algorithm and the optimizer.
Fig. 4.6 Input VSWR contour plots for 0.05 ≤ |ΓL| ≤ 0.95 obtained with (a) the algorithm and (b) the optimizer
simulation at 700 MHz
62
In the region of |ΓL| < 0.5, the input VSWR is low. In most cellular handset applications a VSWR
< 3:1 is usually required after matching. If the load impedance is located inside the area defined
by |ΓL| < 0.5, no matching task would be required.. Therefore, the most interesting area in the
Smith chart to check tuner performance is within the region of 0.5 ≤ |ΓL| ≤ 0.90 and -180o ≤ ∠ΓL
≤ 180o. From this point on, we will only discuss the matching performance within this region to
highlight the magnitude of improvement where a tuner is most beneficial. A comparison of the
average input VSWR obtained from the optimizer and algorithm at different frequencies are
given in Table 1.
Table 4.1 comparison of average VSWR resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9
Frequency (MHz)
700
960
1710
2170
Optimizer Average
VSWR
3.22
2.31
1.22
1.41
Algorithm Average
VSWR
3.22
2.31
1.23
1.42
63
∆VSWR
0
0
0.01
0.01
Fig. 4.7 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm
and (b) optimizer at 2170 MHz for the case of lossless network.
The plots of the ∆GT versus the load reflection coefficient (0.5 ≤ |ΓL| ≤ 0.9 and -180o ≤
∠ΓL ≤
180o) at 2170 MHz derived from the tuning algorithm and the optimizer simulation are given in
Fig. 4.7
(a) and (b), respectively. The average ∆GT over the above area is found to be 3.19 dB for
the algorithm and 3.20dB for the optimizer. A comparison of the average ∆GT resulting from the
algorithm and the optimizer at different operating frequencies is presented in Table 2. The
difference is equal to or less than 0.01 dB over the frequency from 700 MHz to 2170 MHz. In
the case of the pi-network without loss, the ∆GT derived by means of minimizing input VSWR or
maximizing GT or ∆GT approach is exactly the same.
Table 4.2 comparison of average ∆GT resulting from optimizer and algorithm for lossless network. 0. 5<|ΓL|<0.9
Frequency (MHz)
700
960
1710
2170
Optimizer Average
∆GT (dB)
2.12
2.68
3.31
3.20
Algorithm Average
∆GT (dB)
2.12
2.67
3.30
3.19
64
∆GT Discrepancy (dB)
0
-0.01
-0.01
-0.01
In practice, all of the components of the pi-network have a finite Q factor instead of infinite. We
now compare the algorithm tuning results of a pi-network tuner having loss with those resulting
from the optimizer simulations. Assuming that the tuner utilizes the components with the same
tunable range and the nominal value as defined in the previous example but having a finite Q
factors of QC1,2 = 100 for C1 and C2, QC3 = 150 for C3 and QL = 55 for the fixed value series
inductors. In order to take the finite Q of the components into account, the final ∆GT and/or input
VSWR calculations need to use the following component values
and

j  , k= 1,2, or 3

Cˆ k = C k 1 −
Q
Ck ( f ) 

(4.22)

j 

Lˆ = L  1 −
Q L ( f ) 

(4.23)
In the case of a tuner having loss, the plots of the ∆GT versus the load reflection coefficient (0. 5
≤ |ΓL| ≤ 0.9 and -180o ≤
∠ΓL ≤
180o) at 700 MHz derived from the tuning algorithm and the
optimizer simulation are presented in Fig. 4.8. The average ∆GT over the above area is found to be
1.79dB and 1.78dB for the optimizer and algorithm, respectively, a difference of only 0.01 dB.
A comparison of the average ∆GT resulting from the algorithm and the optimizer at different
operating frequencies is presented in Table 3. The difference is lower than 0.1 dB over the
frequency from 700 MHz to 2170 MHz. In the case of finite but high Q components, the ∆GT
obtained from minimizing input VSWR or maximizing GT or ∆GT approach is slightly different,
and the later approach provides higher average ∆GT than the former one. Therefore, for a
network tuner with loss, achieving the perfect match does not maximize the GT or ∆GT to the
load.
From Table 3, we notice that the average ∆GT resulting from the algorithm for a lossy tuner is
always slightly less than the average ∆GT obtained from the optimizer. The reason is that the
65
algorithm’s main goal is to achieve a perfect match (VSWR = 1) and only in the case of having
no perfect match situation, to then maximize the ∆GT. In addition, the algorithm first determines
the tunable component values for matching based on an assumption of a lossless network tuner
and then computes the final ∆GT and/or input VSWR with finite Q components. Contrarily, the
optimizer takes the losses into account from the beginning of the process. However, it requires
approximately one hour and fifty minutes generating the results shown in Fig. 4.8(b) when using
commercial RF CAD optimizers on a typical PC, while it takes less than five seconds to obtain
the same plot by utilizing the algorithm on the same PC. Thus tuning using this analytical
algorithm is approximately one thousand times faster than the optimizer simulation with
negligible errors for high Q components. This implies that the time saved by using this approach
can significantly reduce the network design cycle duration. The algorithm can be coded for fast
closed loop control of any tunable pi-, L- or certain double pi-network, independent of the tuning
technology.
(a)
66
(b)
Fig. 4.8 ∆GT (dB) results for load reflection coefficient 0.5 ≤ |ΓL| ≤ 0.9 and -180o≤ phase (ΓL)≤ 180o (a) algorithm
and (b) optimizer at 700 MHz.
Table 4.3 Comparison of ∆GT resulting from algorithm and optimizer at different frequencies for low loss network
case.
Frequency (MHz)
700
960
1710
2170
Optimizer Average
∆GT (dB)
1.79
2.10
2.29
2.29
Algorithm Average
∆GT (dB)
1.78
2.05
2.22
2.22
∆GT Discrepancy (dB)
-0.01
-0.05
-0.07
-0.07
4.7 Algorithm extension for different Pi-network topologies.
The present algorithm can be easily extended for network circuit topologies different from that
presented in Fig. 4.1(b) by reformulations. Two different topology examples are introduced in this
section.
67
4.7.1 Shunt Tunable Capacitor Having a Fixed Inductor Connected in
Parallel.
A shunt inductor Ls may be connected in parallel with C2 and/or C1 in the network shown in Fig.
4.1(b) to
make the minimum capacitance of the composite equivalent tunable capacitor formed by
Ls||C2 or C1 lower than the original C1,min or C2,min.. Assuming that the shunt tunable capacitor C2
on the load side of the pi-network has a fixed inductor Ls connected with C2 in parallel as shown
in Fig. 4.9, the equivalent capacitance formed by the LS and C2 at frequency ω can be expressed as

1
C 2e = C 2 1 − 2

ω LS C 2


 , k= 1,2, or 3


(4.24)
The algorithm presented in Sections 4.2 to 4.5 can then be used for the pi-network configuration
shown in Fig. 4.9 as long as the C2 value in the algorithm is replaced by the C2e value defined by
(4.24) while ensuring C2e is
positive, i.e.,
C2 >
1
2
ω LS
(4.25)
Fig. 4.9 Pi-network shunt C2 having a fixed inductor Ls connected in parallel.
4.7.2 Series portion of a pi-network consisting of series LC combination
The series portion of the Pi-network in Fig. 4.1(b) can be changed to a series LC combination as
shown in Fig. 4.10. This is an alternative method achieving an equivalent effective inductance.
The main reason for such a change is to increase the network frequency tuning range. However
68
this series configuration needs to use larger inductors and much larger tunable capacitors than the
original parallel structure for the same operation frequencies.
Fig. 4.10 Pi-network with a series LC circuit in its series path.
In this case, the equivalent series inductance can be expressed as

1 
Le = L1 − 2
 ω LC 
3

(4.26)
Then, the C3 calculation must be done using (4.27) instead of (4.6)
C3 =
1
ω (L − Le )
2
(4.27)
To keep Le positive, C3 should be large enough to meet the inequality
C3 >
1
ω 2L
(4.28)
All other formulas given in the previous sections can then be reused in the tuning algorithm for
the network topology shown in Fig. 4.10.
4.8 Application examples.
4.8.1 Replacing Optimizer to Analyze Matching Performance
As was shown in 4.6, the analytical tuning algorithm presented in this chapter can be used to
analyze the matching performance of the pi-network tuner in terms of transducer gain GT or
relative transducer gain ∆GT. The use of this algorithm requires considerably less time than an
iterative optimizer. More examples will not be introduced here for space limitations.
69
4.8.2 Predetermining the Match Tuning Setting for Known Antenna Load
Impedance
In the case of an antenna impedance match based on open loop control, the antenna impedance
is premeasured under different conditions over different frequencies. The analytic tuning
algorithm can be adopted to determine the component setting of the matching network for the
measured antenna impedance immersed in different scenarios at each frequency.
Fig. 4.11
presents the VSWR versus frequency curves for a mobile phone antenna before and
after using a pi-network tuner. In this example, a pi-network as shown in Fig. 4.1(b) containing a
series inductor L=6.8 nH, and capacitor values C1,min = C2,min = 0.8 pF, C1,max = C2, max = 5 pF,
C3,min = 0.25 pF and C3,max = 4 pF is employed as the impedance matching network and it is
tuned by means of our algorithm. From Fig. 4.11, we can see the input maximum VSWR after
using the tuner is reduced from a VSWR = 5.8:1 reduces to a VSWR of 1.2:1 over the frequency
range from 820 MHz to 970 MHz. The relative transducer gain ∆GT and resulting tunable
capacitor C1 ~ C3 settings at frequencies 850, 880, 925, and 970 MHz, are given in Table 4.
If instead of a continuous capacitance setting, the tunable capacitors are adjusted in discrete steps
of 0.125pF, the matching performance degradation is shown in Table 5. The ∆GT degrades only
0.1 dB or less, and the VSWR increases 0.1. Usually, in this frequency range the matching
performance is not highly sensitive to component value quantization. The capacitance step size
for commercially available surface mount chip capacitors may be larger than this particular
example.
70
7
At Tuner Input
6
At Ant. Input
VSWR
5
4
3
2
1
0
820
850
880
910
940
970
Frequency (MHz)
Fig. 4.11 VSWR versus frequency of a given antenna impedance with and without using the Pi-network tuner. The
tuner has components L= 6.8 nH, C1,min = C2,min = 0.8 pF, C1,max = C2, max = 5 pF, C3,min = 0.25 pF and
C3,max = 4 pF and the values of the capacitor settings have been found using the presented tuning algorithm.
Table 4.4 Matching performance and tunable capacitor settings considering matching network with continuous
capacitance.
Freq (MHz)
850
880
925
970
VSWR
1.04
1.10
1.05
1.06
∆GT (dB)
1.63
1.17
0.55
2.16
C1 (pF)
5
0.8
0.8
5
C2 (pF)
0.80
2.86
2.84
0.80
C3 (pF)
2.45
2.66
1.73
1.45
Table 4.5 Matching performance and tunable capacitor settings considering matching network with discrete
capacitance steps.
Freq (MHz)
850
880
925
970
VSWR
1.11
1.13
1.08
1.16
∆GT (dB)
1.55
1.11
0.54
2.06
C1 (pF)
5
0.875
0.875
5
C2 (pF)
0.875
2.875
2.875
0.875
C3 (pF)
2.5
2.75
1.75
1.5
4.8.3 Control Algorithm for Closed loop Impedance Matching
Since the algorithm deterministic nature provides much faster tuning settings than its optimizer
counterpart, it is a good candidate to be used as a control algorithm for closed loop dynamic
antenna impedance match closed loop control. A conceptual block diagram of the closed loop
impedance matching system utilizing this algorithm is depicted in Fig. 4.12.
71
The objective of this loop system implementation is the determination of the antenna complex
input impedance through the measurement of input and output voltages, Vin and Vout. The
antenna impedance ZAnt can be determined by the initial values Y2 and Y3 in the pi-network and
the voltages Vin and Vout by using the following expression
Z Ant =
1
(Vin / Vout − 1)Y3 − Y2
(4.29)
Vin
Vo ut
ZAnt=
1
(Vin/Vout−1)Y3 −Y2
∠(Vin Vout )
Fig. 4.12 Conceptual block diagram of closed loop impedance matching control system. The proposed algorithm is
used here to adjust the tunable capacitor settings once the antenna complex impedance is detected.
This type of complex impedance determination using both, voltage and current measurements
has been proposed in [7] for an L-type matching network. Because we are dealing with a
different topology (no series component is connected directly to antenna to measure its current),
the impedance determination is based only on two node voltage measurements. The analytical
tuning algorithm can then be applied to calculate the required tunable capacitor values, C1 ~ C3
for the best match as long as the instant antenna impedance is known. This block diagram is
given as an illustrative example of a particular application of the algorithm however specific
implementation details are outside the scope of this chapter.
72
4.9 Conclusions.
This chapter presents a novel tuning algorithm that has been shown to be very effective in
determining the optimum tuning settings for a pi-network tuner with finite tuning range
capacitors. Perfect match solutions are obtained if the values C1~C3 derived from the tuning
algorithm are within the available tuning range. Otherwise, the algorithm always achieves the
best possible match solutions. Formulas for the component values that achieve the perfect or the
best match solutions are presented along with flow chats depicting the operation of the
algorithm. Based on the description of the algorithm presented herein, a software version of this
algorithm can be implemented in any suitable programming language.
Using the algorithm to find the network component values resulting in the best possible match
for any given load impedance is more than a thousand times faster than the time required to find
an equivalent solution by using a commercially available optimizer. Furthermore, it was
demonstrated that the match tuning accuracy derived from the tuning algorithm method is as
good as that resulting from the optimizer simulation approach, thus the algorithm represents a
practical and a useful method for closed feedback control of pi-network tuners.
The proposed algorithm can be possibly extended to a network tuner topology comprised of four
or less tunable components having a limited tuning range if these circuit structures can be
transformed into an equivalent pi-network topology.
Appendix A. Derivation of formulas (4.14)-(4.17)
The derivation of equations of (4.14) – (4.17) needs to be split into two cases, i.e.,
R L < Ro
and
R L > Ro , then the derivation can be done by means of two different configurations of the
equivalent circuit topologies. For (4.14) and (4.15), our derivation is based on the equivalent
73
circuit configuration shown in Fig. 4.13. C2 in the pi-network tuner has been absorbed by Xe and
Re as given in (4.11).
Z left =
1
Go + jBC1
Fig. 4.13 Equivalent circuit for (4.14) and (4.15) derivations.
The impedance on the left side of the dashed line is given by
Z left =
G
B
1
= 2 0 2 − j 2 C1 2
Go + jBC1 Go + BC1
Go + BC1
(4.30)
In the conjugation match condition, we should have (4.31) and (4.32) satisfied.
G0
= Re
G o2 + B C2 1
(4.31)
BC1
= X e + 2πf ⋅ Le
G + BC21
(4.32)
2
o
where BC1 = 2πf ⋅ C1 .From (4.31), we obtain (4.14) and from (4.32) we derive (4.10) as,
C1 =
Le =
1
2πf
1
1
BC1 =
2πf
2πf
Go
1
11 1
 − 
− Go2 =
Go (Ge − Go ) =
Re
2πf
Ro  Re Ro 
 BC1
 1

− Xe  =
 G2 + B2
 2πf
 o C1

 Re Go
 1

− Go2 − Xe  =
G R
 2πf
 o e

(4.33)
( R (R − R ) − X )
e
o
e
e
(4.34)
In the derivation of (4.16) and (4.17), the equivalent circuit as depicted in Fig. 4.14 is used. For
R L > Ro , C1 is set to minimum Cmin. Zleft and Zright in Fig. 4.14 can be expressed respectively as,
74
Z left
Z right
Fig. 4.14 Equivalent circuit for (4.16) and (4.17) derivations.
Z left =
Ro
BC1 min Ro2
1
=
−
j
Go + jBC1 min 1 + Ro2 BC21 min
1 + Ro2 BC21 min
(4.35)
and
Z right =
R L2 (B C 2 + B L )
RL
1
=
−
j
G L + j (B C 2 + B L ) 1 + R L2 (B C 2 + B L )2
1 + R L2 (B C 2 + B L )2
(4.36)
where BC1min = 2πf ⋅ C1,min and BC 2 = 2πf ⋅ C 2
In conjugation match condition, we should have the real part of (4.35) equal to
Ro
RL
=
2 2
2
1 + Ro BC1 min 1 + RL (BC 2 + BL )2
(4.37)
and the imaginary part of (4.35) without the negative sign equal to the following expression
B C 1 min R o2
R L2 (B C 2 + B L )
=
2
π
f
⋅
L
−
e
2
1 + R o2 B C21 min
1 + R L2 (B C 2 + B L )
(4.38)
From (4.37), after manipulating we derive (4.16)
C2 =
1
2πf
 GL

2 2
2


 R 1 + Ro BC1,min − G L − BL 
o


(
)
(4.39)
and from (4.38) and (4.16) we obtain (4.17)
 GL

1 + Ro2 BC21,min − G L2
Ro2 BC1, min
1  Ro
Le =
+

GL
2πf 
1 + Ro2 BC21, min
1 + Ro2 BC21, min

Ro

(
)
(
)
75







(4.40)
Appendix B. Derivation of input reflection coefficient.
Z in
Zx
Fig. 4.15 Equivalent circuit for Γin derivation
The input reflection coefficient Γin is defined as
Γin =
Z in − Ro
Z in + Ro
(4.41)
where Zin is the input impedance as shown in Fig. 4.15, and it can be expressed as
Z in =
where
B C1 = 2πf ⋅ C1 and
BC 2 = 2πf ⋅ C 2
(4.42)
Yx is expressed as
Yx =
where
1
1
=
jBC1 + 1 Z x
jBC1 + Y x
and
X Le = 2πf ⋅ L e .
Z in =
G L + j ( BC 2 + B L )
1
=
Z x 1 − X Le (BC 2 + B L ) + jG L X Le
(4.43)
Substituting (4.43) to (4.42), we obtain Zin to be
1 − X Le (BC 2 + BL ) + jGL X Le
GL (1 − X LeBC1 ) + j[BC1 + (BC 2 + BL )(1 − X Le BC1 )]
(4.44)
Finally, plugging (4.44) into (4.41), we derive the input reflection coefficient Γin (4.18) to be
expressed as,
Γin =
Go [1− XLe(BL + BC2x )] −GL (1− XLeBC1x ) + j[GoGL XLe − BC1x −(BC2x + BL )(1− XLeBC1x )]
Go[1− XLe(BL + BC2x )] +GL (1− XLeBC1x ) + j[GoGL XLe + BC1x +(BC2x + BL )(1− XLeBC1x )]
76
(4.45)
Appendix C. Parameters of Pi-Network tuner.
Fig. 4.16 Equivalent circuit of pi-network tuner
The equivalent circuit of the tunable pi-network tuner is simply depicted in Fig. 4.16, which
consists of an equivalent inductor Le and two tunable capacitors C1 and C2. These components
are better expressed as impedance ZLe and admittance YC1 & YC2 if the loss of the components
needs to be considered. The loss of the components can be characterized by a resistance or
conductance as (4.46) – (4.48)
Z Le = RLe + jX Le
(4.46)
YC 1 = G C1 + jBC1
(4.47)
And
YC 2 = GC 2 + jBC2
(4.48)
or otherwise by the Q factor of the components as


1
Z Le = 2π f ⋅ L e 
+ j 
 Q Le ( f )



1
YC 1 = 2π f ⋅ C 1 
+ j 
 QC1 ( f )

(4.49)
(4.50)
And
 1
YC 2 = 2πf ⋅ C 2 
+
 QC 2 ( f )

j 

The S parameters of the pi-network in Fig. 4.16 are derived as
77
(4.51)
S11 =
(
) [ (
+ Y + (1 + Y
)
]
)⋅ Z
− YC1 + YC 2 + 1 + YC 2 − YC1 − YC1 YC 2 ⋅ Z Le
2 + YC1
S 21 = S12 =
C2
C1
+ YC 2 + YC1 YC 2
2
(
Le
)
2 + YC1 + YC 2 + 1 + YC1 + YC 2 + YC1 YC 2 ⋅ Z Le
(4.52)
(4.53)
And
S 22 =
(
) [ (
+ Y + (1 + Y
)
]
)⋅ Z
− YC1 + YC 2 + 1 − YC 2 − YC1 − YC1 YC 2 ⋅ Z Le
2 + YC1
C2
C1
+ YC 2 + YC1 YC 2
(4.54)
Le
where
Z Le =
Z Le
,
Ro
YC 1 = YC1 ⋅ Ro , and YC 2 = YC 2 ⋅ Ro
(4.55)
References
[1]
H. Song, et al., "Automatic antenna tuning unit for software-defined and cognitive radio,"
in Antennas and Propagation Society International Symposium, 2007 IEEE, 2007, pp. 85-88.
[2]
P. Sjoblom and H. Sjoland, "An adaptive impedance tuning CMOS circuit for ISM 2.4-
GHz band," Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 52, pp. 11151124, 2005.
[3]
J.-S. Fu, et al., "A ferroelectric-based impedance tuner for adaptive matching
applications," in Microwave Symposium Digest, 2008 IEEE MTT-S International, 2008, pp.
955-958.
[4]
K. Brito de Brito and R. Nunes de Lima, "Impedance Network for an Automatic
Impedance Matching System," in Microwave Conference, 2007. APMC 2007. Asia-Pacific,
2007, pp. 1-4.
78
[5]
J. de Mingo, et al., "An RF electronically controlled impedance tuning network design
and its application to an antenna input impedance automatic matching system," Microwave
Theory and Techniques, IEEE Transactions on, vol. 52, pp. 489-497, 2004.
[6]
R. Rea. (2006, The Yin-Yang of Matching: Part 1: Basic Matching Concepts. pp. 16 – 25.
Available: www.highfrequencyelectronics.com
[7]
A. van Bezooijen, et al., "Adaptive Impedance-Matching Techniques for Controlling L
Networks," Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 57, pp. 495-505,
2010.
[8]
P. Smith, Electronic Applications of the Smith Chart, 2nd ed. Raleigh, NC:
SciTech/Noble, 1995.
[9]
J. R. Moritz and Y. Sun, "Frequency agile antenna tuning and matching," in HF Radio
Systems and Techniques, 2000. Eighth International Conference on (IEE Conf. Publ. No. 474),
2000, pp. 169-174.
[10]
Y. Sun and J. K. Fidler, "Component value ranges of tunable impedance matching
networks in RF communications systems," in HF Radio Systems and Techniques, Seventh
International Conference on (Conf. Publ. No. 441), 1997, pp. 185-189.
[11]
M. Thompson and J. K. Fidler, "Determination of the impedance matching domain of
impedance matching networks," Circuits and Systems I: Regular Papers, IEEE Transactions on,
vol. 51, pp. 2098-2106, 2004.
[12]
Y. Sun and J. K. Fidler, "Design method for impedance matching networks," Circuits,
Devices and Systems, IEE Proceedings -, vol. 143, pp. 186-194, 1996.
[13]
Y. Sun and F. J. K., "Determination of the impedance matching domain of passive LC
ladder networks," Journal of the Franklin Institute, vol. Vol. 333(B), p. 15, 1996.
79
[14]
M. Thompson, "Application of Multi Objective Evolutionary Algorithms to Analogue
Filter
Tuning,"
in
Evolutionary
Multi-Criterion
Optimization.
vol.
1993/2001,
ed
Berlin/Heidelberg: Springer, 2001, pp. 546-559.
[15]
Y. Sun and W. K. Lau, "Evolutionary tuning method for automatic impedance matching
in communication systems," in Electronics, Circuits and Systems, 1998 IEEE International
Conference on, 1998, pp. 73-77 vol.3.
[16]
Y. Sun and W. K. Lau, "Antenna impedance matching using genetic algorithms," in
Antennas and Propagation, 1999. IEE National Conference on., 1999, pp. 31-36.
[17]
M. Thompson and J. K. Fidler, "Fast antenna tuning using transputer based simulated
annealing," Electronics Letters, vol. 36, pp. 603-604, 2000.
[18]
M. Thompson and J. K. Fidler, "Application of the genetic algorithm and simulated
annealing to LC filter tuning," Circuits, Devices and Systems, IEE Proceedings -, vol. 148, pp.
177-182, 2001.
[19]
(Feb. 5 2009). AWR: Microwave Office. Available: http://web.awrcorp.com
[20]
G. Gonzalez, "Microwave Transistor Amplifiers Analysis and Design," 2nd ed Upper
Saddle River, NJ: Prentice-Hall, 1997, pp. 183-184.
[21]
A. S. Morris III, "Tunable Matching Network Circuit Topology Selection," US Patent
2009 0267705 A1, 2009.
80
5. TUNABLE MEMS NOTCH FILTER AND ITS FREQUENCY CONTROL
LOOP FOR WIRELESS TRANSCEIVERS
5.1 Introduction
In a frequency division duplex (FDD) wireless transceiver, such as CDMA and WCDMA, the
transmitter and receiver sections of a mobile station must operate simultaneously. A typical RF
FDD front-end single-band block diagram of the wireless transceiver is shown in Fig. 5.1 [1].
The duplexer in the RF front-end is used to separate the transmission and reception signals. The
duplexer specification requirements on suppressing unwanted signal and/or interference are very
high. Typically 55dB or greater of isolation is required to suppress the transmission signal
leaking into the receiver and a minimum 45 dB is required to suppress the transmitter noise in
the receiver frequency band.
Excessive transmission leakage through the duplexer to the
receiver will cause inter-modulation and/or cross-modulation interference desensitizing the
receiver [1]. An external SAW filter with modest rejection level (typically 20dB), is often
placed after the LNA as shown in Fig. 5.1 to relax the mixer linearity and duplexer rejection
requirements. However SAW filters historically have shown resistance to integration and
frequency tunability, thus increasing the size, component count and cost of the overall
transceiver.
In order to address these problems, tunable solutions such as YIG filters have been proposed [2],
which exhibit low loss and broad tuning bandwidth characteristics, but require an externally
applied magneto-static field, suffer from slow tuning times due to hysteresis effects and exhibit
high power consumption.
81
Fig. 5.1 Block Diagram of an Iso-frequency repeater.
Distributed filter designs using coupled sections of resonant printed structures such as loaded
combline filters [3], loaded loop resonators [4]-[5] or interdigitated filters [6], have been reported
in literature. However, the large footprint required becomes the main disadvantage for any
distributed implementation when designed for operation at typical cell phone frequency bands
(700MHz-2.7GHz). In general, either acoustic or tunable lumped element filters must be used to
meet cell phone real estate constraints. In [7], a non tunable notch filter using bond wire
inductors operating at the IMT band (TX: 1.92-1.98GHz, RX: 2.11-2.17GHz) was reported.
However, low suppression level (approx. 12dB) and high insertion losses (3dB) were measured.
In addition, due to the non tunable nature of the design, the suppression level shows high
variation within in the operating frequency band.
In [8], a tunable notch filter based on RF MEMS showed 1.2dB insertion loss in the pass band
and 25.5dB notch rejection. In this chapter, an improved version of the notch filter of [8], having
less than 0.8dB pass band insertion losses is presented. Section 5.2 presents the theoretical
analysis of the filter while Section 5.4.1 focuses on its implementation and measurement results.
In the application of wireless mobile station transceivers, the narrow band notch filter frequency
needs to have frequency accuracy within 100 kHz for 2.5G and 3G mobile stations where the
82
channel spacing is 200 kHz or 50 kHz for the long term evolution (LTE) mobile stations where
channel spacing is 100KHz. Therefore, accurate frequency selection and tracking capabilities are
required. A novel filter frequency control loop based on sensing the reflection phase of the
leaking transmitter signal has been designed for this objective. The filter frequency control loop
uses the transmitter carrier itself as a reference signal to keep the filter tracking the transmitter
operation frequency.
A mathematical model is developed to theoretically analyze the notch filter frequency control
loop. A closed form solution to this frequency control loop equation is derived in section 5.3.
Circuit model simulations in section 5.3.3 are employed to validate the theoretical solution, to
investigate the loop control transient behavior, and to consider design parameters tradeoff.
Finally, section 5.4.2 describes the implementation and performance assessment of the control
loop and its interaction with the tunable filter to conform the overall autonomous tunable filter
system.
5.2 Tunable Notch filter design
In practical notch filter designs, a single bandpass is usually designed to coexist near the notch
frequency. When this is the case, the design is also referred as single-pole single-zero (SPSZ)
filter denoting a unique pole and zero in the filter transfer function. The proposed notch filter
consist of a series-LC resonator (dashed box in Fig. 5.2) providing the signal notch or rejection at
a transmitter frequency, and two shunt capacitors C3 and C4 that combine with the excess
reactance of the series-LC resonator to form a low loss passband at the corresponding receiving
frequency. The proposed tunable notch filter circuit using tunable components is shown as Fig.
5.2.
This topology allows for the notch filter suppression and the bandpass filter insertion loss to
be tuned separately. All the tunable capacitors used in this filter are RF MEMS capacitors [9].
83
This series-LC block comprising !1 , 1 and 2 resonates at the notch frequency.
3 1
4!1 3
(5.1)
where Cr represents the resonator composite capacitance resulting from the series connection of
1 and 2 can be expressed as
3 1 2
1 5 2
(5.2)
Fig. 5.2 Circuit schematic of the SPSZ tunable filter design.
At resonance (#3 ), the series-LC block presents a short circuit to ground reflecting most of the
signal traveling along the transmission line. The values of external components, !1 and 1 must
be chosen in order to obtain a resonance frequency higher than 1.98 GHz when all MEMS
capacitor cells used in 2 are in the Off state (minimum capacitance). By tuning the value of 2
the resonance frequency can be dynamically adjusted to generate a notch at one of the transmitter
operating frequencies. The center frequency of this notch filter should be capable of being tuned
to cover the entire IMT transmission frequency band (from 1.92 to 1.98 GHz).
On the other hand, the bandpass filter comprised of the series-LC in combination with the
tunable capacitors 6 and 7 which are located in a symmetrical fashion on both sides of the
84
resonator. This arrangement minimizes the insertion loss of bandpass filtering and makes
impedance matching identical seen from ports 1 and 2.
The total capacitance || of 6 plus 7 for resonating at the receiver frequency ωrx can be
obtained from (5.3).
9 1
2
3:
!1 ;
1
3
(5.3)
The S-parameters for the notch and bandpass combination filter can be derived and expressed as
;
3 (26 32 !1 1 2 ; 26 2 ; 26 1 ; 1 2 '=>
<11 ?
Where
<12 <21 ;2@(
32 !1 1 2 ; 2 ; 1 '
?
? (2@
32 !1 1 2 5 2@2 5 2@1 5 2
36 => 6 !1 1 2 ; 2
3 => 6 62
; 2
3 => 6 1 ; 3 => 1 2 '
(5.4)
(5.5)
(5.6)
where => is a reference impedance. It is expected that <11 <22 (symmetric matching condition)
only if the capacitors 6 and 7 are identical.
5.3 Notch filter frequency control loop based on reflected reference signal
5.3.1 System Description.
The notch filter in the previous section provides an inherent narrow rejection bandwidth. The
filter center notch frequency needs to be tuned with an accuracy of 100 kHz or less for 2.5G and
3G mobile systems or 50 kHz for the LTE system. To achieve the accurate frequency tuning of
the narrow band notch filter, a frequency automatic control loop is required. This loop utilizes
the transmitter carrier as a reference signal and uses the reflection phase change of the reference
85
signal from the notch filter to tune the filter frequency and to track carrier frequency that the
mobile station transmitter operates on. To the authors knowledge, the frequency control loop for
automatically tuning a narrow bandwidth notch filter has not been reported to date, although the
frequency automatic control loop for the MMIC bandpass filters has been discussed in [10]-[13].
The bandpass filter frequency control loops are usually based upon sensing the transmission
coefficient phase (or <21 phase).
One key difference from the bandpass filters is that the signal passing through the notch filter
should be suppressed to a very weak level. Additionally, the notch filter S21 presents 180o phase
jump at its notch (or center) frequency (Fig. 5.3). As a consequence, the transmission phase
information is difficult to utilize for the frequency control loop to tune the filter frequency.
In the case of notch or other narrowband rejection filters, we can use the phase information of
the reflected reference signal from the filter (i.e., the phase information of the S11 around the
notch frequency). The advantage of sensing S11 is that the magnitude of the reflection coefficient
of the notch filter near its notch frequency is very high. In addition, the S11 reflection phase
behavior versus frequency is continuous across its notch frequency and quasi linear fashion as
shown in Fig. 5.3.
Fig. 5.3 S-parameter response of notch filter without the nearby co-exisisting pass band.
86
The block diagram of a proposed novel tunable notch filter automatic frequency control loop
utilizing the reflected reference signal is shown in Fig. 5.4. This control loop will tune the filter
notch frequency to suppress the transmitter leakage by utilizing the carrier of the transmitter
leakage with a frequency ωr as the reference signal.
A cos(ωot + m(t ) + ϕ o + ∆ϕ )
A cos( ω o t + m ( t ) + ϕ o )
Fig. 5.4 Block diagram of tunable notch filter automatic frequency control loop utilizing reflected reference signal.
The basic operation principle can be described as follows: The reference injected (leakage)
signal having a frequency ωr passes through a directional coupler. Most of the leakage signal is
forwarded to the input port of the tunable filter while a small fraction of the signal is coupled to
the left logarithmic amplifier of Fig. 5.4.
The forward signal is reflected by the notch filter and has a phase change ∆φ different
from 180o if the leakage signal carrier frequency ωr is different from the notch filter originally
tuned frequency ωo. The reflected signal comes back to the directional coupler and it is coupled
to the right logarithmic amplifier of Fig. 5.4.
87
The signal amplified by the left logarithmic amplifier passes through a phase shifter, which is
used to equalize the group delay of the two paths, and arrives at the first input of the phase
detector (PD) (see point “B” of Fig. 5.4).
The reflected signal with an extra phase shift ∆φ after amplification by the right
logarithmic amplifier reaches the second input port of the PD (see point “A” of Fig. 5.4). The
phase shifter is used to adjust the initial phase difference between the two PD inputs to 90o.
The output of the PD contains a low frequency voltage Vc(t) proportional to sin (∆D'
and some high frequency products. The high frequency products are filtered by a low-pass RC
Fig. 5.5 Mathematical model of notch filter frequency automatic control loop utilizing the reflection signal.
loop filter. Only the low frequency output passes through the lowpass filter and it is digitized by
a analog-to-digital converter (ADC). The ADC and next step is not necessary if the tunable filter
comprises of analog tuning devices.
The digital signal is then coded by an encoder that creates a set of proper tuning code
words that tune the MEMS capacitors in the notch filter and make the notch filter frequency
aligning with the leakage carrier frequency ωr.
88
5.3.2 Mathematical Model and Formulation
Under the assumption that the notch filter can be modeled as voltage controlled analog tunable
filter, the notch filter automatic frequency control loop can be fully characterized by the
mathematical model as presented in Fig. 5.5.
The transfer function EFF (@
' of the notch filter reflection versus input reference signal has an
expression (see Appendix A for its derivation)
2 (-'
;1 V
;G3
32
Q
S
EFF (@
' *K @$-O
G
M
U
3
2
2 (-'
L
HI
2 ; 1J 5 G32
P
RT
3
N
Where
G3 =>
2!
3
(5.7)
(5.8)
Note that for the mathematical description in this section a simple notch filter is considered to
simplify the expressions. However the same procedure can be made analogously in the case of
having a nearby passband working from (5.5) at expenses of a considerable increase in the
solution complexity. From the frequency control loop mathematical model of Fig. 5.6 we can
derive the automatic frequency control loop equation of the notch filter as (see Appendix B for
derivation)
1 2 (-' (-'
5
W -
;
GX Y W Z>2
2
2 (- ; '
; 1`
]
G3
32
\
_
sin
G
\
_
3
2
2 ([
^
HI
2; ' ; 1J 5 G32
3
89
(5.9)
where
W
the low pass filter cut off frequency
W 1⁄a , and
τ the MEMS capacitor actuation
delay time.
Equation (5.9) is a second order nonlinear differential equation. From this equation, we can see
that the output signal level from the phase detector (P.D.) is not only dependent on the phase
difference between the reference signal and the reflected signal by the notch filter, but also upon
the magnitude of the reflected signal. However, the magnitude variation near the frequency ωr
within notch filter bandwidth is very small. In addition, the logarithmic amplifier further reduces
the PD output signal amplitude difference.
Considering !1 and 3 in the filter both having a finite Q factor, they can expressed as
!c1 !1 d1 ; @/fF g
(5.10)
h3 3 d1 ; @/W g
i
And
> Substituting (5.12) and
(-' 3 5 ∆
(-'
(5.11)
fF Wi
fF 5 Wi
(5.12)
into (5.9) and considering the magnitude of the PD inputs
being constant, (-' j 3 and > k 1, we obtain a differential equation of ∆
(-' as
1 2 (-' (-'
5
l
W -
GX Y W Z>2
;
sin m
2
2∆
(- ; '
G
; 23
3
>
n
1
G3 ; >
(5.13)
With the above assumptions, the right side of (5.13) can be further linearized as (5.14)
1 2 (-' (-'
2> ∆
(- ; '
G3
5
l ;∆
op: GX q
;
r
W G3 > ; 1
3
2(G3 > ; 1'
90
(5.14)
where
∆
op:
is the maximum frequency control range achieved when |sin( ∆D'| 1, which is
determined by the overall loop gain and the PD output low frequency signal level
∆
op: Y W Z>2
2
(5.15)
The process of solving equation (5.14) by using Laplace transformation is not described here in
detail as it involves lengthy manipulations. During this solution process, in order to find a closed
form solution for the inverse Laplace transform we need to assume that the MEMS capacitor
actuation delay is small (to linearize
exp(;' j 1 5 '.
Equation (5.14) can be approximately
solved by using Laplace transformation and the final time domain solution is
∆
(-' v
Where
∆
op: G3 GX W 1 ,1 ; exp(;2 -'/ ; 2 ,1 ; exp(;1 -'/
·
2(G3 > ; 1'
1 (2 ; 1 '2
exp(;1 -' (1 ; W ' ; exp(;2 -' (2 ; W '
5 x
> q
r
2 ; 1
1,2 ;
y (1 ; 3 GX ' z 4
y2 (1 ; 3 GX ' ; 4
y 3 GX
2
3 2∆
op: >
(G3 > ; 1'
3
(5.16)
(5.17)
(5.18)
and ∆
> is the initial filter frequency error when t = 0. The frequency control error of the filter
control loop depends upon the Qo factor of the notch filter and the value of the fixed inductor !1 .
The final frequency error can be obtained by using (5.16) and letting - j ∞.
∆
(- j ∞' =>
8!1 >
(5.19)
Note that (5.19) can be also interpreted as the final frequency error related to the notch/rejection
bandwidth if we consider
} ~ 1/> .
The gain loop parameters GX , Y , and Z> can be set during
91
the design or selection of the building blocks in the control loop (integrator, phase detector and
amplifier). The parameter y can be derived as follows. The capacitance versus control voltage
Zy (-' equation can be expressed as
(-' Zy (-' 5 >
(5.20)
Using (5.20) the filter instantaneous notch frequency can be written as
(-' 1
4!1 (-'
1
4!1 (Zy (-' 5 > '
1
q
1
4!1 > 41 5 Zy(€' />
r
(5.21)
where > is the capacitance when the control voltage Zy (-' 0 (i.e 3‚ƒ ) and  is a constant
that depends on the MEMS tunable capacitor design that relates capacitance to applied bias
voltage. We can linearize (5.21) using Taylor expansion series as:
(-' l
1
4!>
;
1 
Z (-'
2 4!>6 y
(5.22)
The terms in (5.20) can be associated with those from the frequency voltage tuning equation
(-' 3 5 y Zy (-' to conclude that
y l ;
1 
2 4!1 >6
(5.23)
5.3.3 Calculations and Simulations of Notch Filter Frequency Control Loop
Performance
In this subsection the effect of several parameters in the performance of tunable notch filter and
control loop system (locking time, final error and exponential behavior) are studied in detail.
The notch filter under the study is implemented by using a fix value inductor !1 10.6 and a
3 0.6& which produces a notch frequency at 2GHz.
92
When y 20, > 1000 and 0 ; 10 (close to real MEMS actuation delay), the
transient responses of this frequency control to an initial frequency offset (∆
> 15)
derived from (5.16) and ADS simulations are shown in Fig. 5.6. Good agreement of the results
obtained from both approaches can be observed for both cases of actuation delay, which
validates the theoretical approach solution.
4
2
0
∆ω(t) in MHz
-2
-4
-6
-8
-10
τ=0µs (ADS Sim.)
τ=0µs (Calculation)
τ=10µs(ADS Sim.)
τ=10µs (Calculation)
-12
-14
-16
0
100
200
300
400
500
600
Time (µs)
Fig. 5.6 Comparison of Closed form expressions and ADS simulations for the second order filter response with In
this experiment, 20, ∆
> 15 > 1000 and 0 10 Using the same loop parameters as above, the locking time responses for the filter > =1000, 100,
80 and 40 are shown in Fig. 5.7. All responses have similar exponential behavior but the final
frequency error decreases with a > increase. Table I shows the numerical final frequency error
for all considered > values.
The low pass filter cut off frequency y or the loop bandwidth impacts the filter frequency
locking time. The locking time responses for different y and Qo = 80, A = 0.12, ∆
> 15, and τ = 0 are shown in Fig. 5.8. Small y values create damped oscillations in the
transient responses and therefore, longer frequency locking times. To achieve a shorter frequency
convergence time for this loop, we choose the cut-off frequency y 10~20 kHz in the final
design.
93
2
0
-2
Qo=1000
Qo=100
Qo=80
Qo=40
∆ω(t) in MHz
-4
-6
-8
-10
-12
-14
-16
0
100
200
300
400
500
600
700
800
900
1000
Fig. 5.7 Simulated filter frequency control loop transient response for different > values with y =20KHz, A=0.12
and ∆
> =15MHz, τ=0us.
Time (µs)
>
1000
100
80
40
Table 5.1 Final Frequency Error versus „… Value
Final Frequency Error (KHz)
11
87
148
627
Relative Error (%) respect to ∆
>
0.073
0.58
0.98
4.18
The overall loop gain (controlled by parameter ‘A’) impacts the locking time and oscillatory
behavior. A high loop gain may cause frequency divergence situations. Fig. 5.9 shows the filter
frequency control error transient response for different values of ‘A’. The number of oscillations
and locking time increases with an increasing gain. It was found that a gain value higher than
A>4.1 results in a divergent situation. Due to the loop gain importance, good care must be given
not to exceed the convergence threshold during the control loop design.
The capacitor actuation delay time can also affect the frequency locking time of the filter and
may originate a divergent solution. The filter frequency control error transient response for
different MEMS capacitor actuation delays is shown in Fig. 5.10. The time scale has been
expanded to 3ms in order to observe convergence. For the delay times larger than 150 s, the
filter frequency control loop becomes unstable and the frequency is divergent. It is important to
94
choose the overall loop gain in consonance with the actuation delay time to make the loop
operate in convergent region with the fastest locking time. A low loop gain should be chosen for
large actuation delays. In this case the price of achieving the convergence is a longer locking
time.
15
ωc=20 KHz
10
ωc=10 KHz
ωc=5 KHz
∆ω(t) in MHz
5
ωc=1 KHz
0
-5
-10
-15
-20
0
100
200
300
400
500
600
700
800
900
1000
200
225
250
Fig. 5.8 Simulated filter frequency control loop transient response for different y values with > =80, A=0.12 and
∆
> =15MHz, =0us.
Time (µs)
30
A=0.3
∆ω(t) in MHz
20
A=1
A=3
10
0
-10
-20
0
25
50
75
100
125
150
175
Fig. 5.9 Simulated filter frequency control loop transient response for different A values with y =20KHz, > =80
and ∆
> =15MHz, =0us
Time (µs)
95
Fig. 5.10 Simulated filter frequency control loop transient response for different values with > =80, y =20KHz
and ∆
> =15MHz , =0.03
5.4 Implementations and measurements
In this section, the standalone tunable filter implementation is first discussed in detail.
Measurement results from tuning range, bandwidth, rejection and insertion loss values are
presented. The filter is then integrated with a discrete implementation of the control loop to
conform the overall tunable filter system. The total system performance is evaluated through the
filter transient response and reference signal tracking capability.
5.4.1 Standalone Tunable Filter
An existing high Q tunable digital capacitor array (TDCA) (see Fig. 5.11) flip chip solution from
Wispry Inc [9] was utilized in the design. The TDCa consists of twenty tunable capacitor cells of
nominal value 1pF or 0.875pF depending on the specific cell. The minimum capacitance step
resolution is 0.125pF. The cells in the TDCa can be interconnected on the PCB level in order to
achieve any desired topology.
The Q of the die level capacitors was measured to be greater than 150 at 2GHz, allowing low
insertion loss designs. In addition, the value of capacitance is highly repeatable, which is an
important feature for narrowband tunable filtering circuits. The IP3 level for this device is
96
65dBm. The group delay distortion is below 1ns in the received signal pass band. The CMOS
biasing circuitry is integrated in the same chip and transforms a 3.3V supply voltage to the
required 35V voltage actuation level. The power consumption is 6 µA and 90 µA in the sleep and
the active mode (charge pump on), respectively. A Serial Peripheral Interface (SPI) is used to
control the capacitor banks states. A USB port is used to transmit the tuning commands from PC
control software.
Fig. 5.11 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell;
b) 3D image of capacitance bits
Two identical filters were designed using a single TDCA die. The objective is to involve one of
the filters in the control loop system and replicate any action taken in the remaining filter that is
used for monitoring and reproducibility testing purposes. The final PCB layout distribution can
be seen in Fig. 5.12. The red and green color rectangles represent cells that were used for the notch
and bandpass sections, respectively. Fig. 5.12 also provides information on how the cells are
interconnected underneath the chip showing that half of the TDCa cells where used for each
filter.
The SPSZ filter of section 5.2 was fabricated using a 0.254mm thick Rogers 4003C substrate
(εr=3.55, tanδ=0.0021 at 2.5GHz) backed with 14µm thick copper (Fig. 5.13). The 50 ohms
microstrip line width was calculated to be 0.58mm for this substrate. A bottom layer of thick
FR4 was used for mechanical stability purposes and to allocate the control circuitry and SPI
97
buffers. Side launch SMA connectors were used to connect the structure to a VNA through
3.5mm coaxial cables. The components used for the filter are SMD 0402 Murata high Q
multilayer ceramic capacitors and 0603 chip CoilCraft inductors.
Fig. 5.12 Detailed TDC layout. Red cells and green cells are used for the resonator and insertion loss blocks,
respectively
Fig. 5.13 Fabricated SPSZ tunable filter prototype board and filter structure detail.
98
The measured transmission and return loss of the filter for 39 different tuning states of the
resonator block are shown in Fig. 5.14. The measured suppression in the transmitter band is equal
or higher than 22 dB within a 5MHz bandwidth, and the insertion loss in the receiver band is less
than 0.8dB within the reception signal bandwidth. The return loss is better than 20dB in the pass
band on the receiver side. The suppression level is almost constant over the entire transmitter
operating band as shown in Fig. 5.14 The filter tuning range is 90MHz which covers the
transmitter band (1920MHz-1980MHz) and the receive band (2110MHz-2170MHz). Fig. 5.15
shows the insertion and return loss tuning using capacitor 6 and 7 banks when 2 is in the
minimum capacitance state. As expected, this tuning does not affect the position of the resonator
block resonant frequency. Therefore, this tuning capability can be used to compensate for
changes in the insertion and return loss values as shown in while tuning the notch. Finally, Table
II summarizes the suppression and insertion loss levels achieved by tuning the TDCA chip for
the two limiting frequency pairs 1.92GHz-2.11GHz and 1.98GHz-2.17GHz.
S21/S12
C2 ↑↑
S11/S22
C2 ↑↑
Fig. 5.14 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when
tuning resonator block capacitor 2 .
99
Fig. 5.15 Measurement results for the transmission and return loss characteristics of the SPSZ tunable filter when
tuning insertion loss block capacitors 6 and 7 .
Table 5.2 Summary of Suppression and insertion losses within 5MHz Bandwidth
Tx Leakage
Suppression
RX Insertion Loss
IMT Band Lower Pair
Tx: 1.92GHz, Rx: 2.11GHz
22.1 dB
IMT Band Upper Pair
Tx: 1.98GHz, Rx: 2.17GHz
21.3 dB
0.8 dB
0.7 dB
5.4.2 Notch Filter Control Loop
The notch filter control loop was implemented using discrete components as shown in Fig. 5.16 to
prove the concept (The integrated circuit implementation is reserved as future work). A 20dB
directional coupler (Meca 722S-20-1.950) distributes the input and reflected leakage signals. An
analog adjustable phase shifter (Narda 3752) was used to provide the 90 degrees phase difference
at the phase detector input when the transmit leakage and notch frequency are aligned. The phase
detector board (AD8302) includes the logarithmic amplifiers and provides a single ended output
with dynamic range between 0 and 2 volts. The integrator is implemented using and operational
amplifier (THS3091DDA). The low pass filter is a single stage RC circuit with a cutoff
frequency of ωc=20KHz. The analog to digital convertor (ADC)-(NI USB-6009) is used to
100
convert the output voltage from the integrator into a digital signal. A PC was used here as an
encoder in order to generate the tuning words that actuate the tunable RF MEMS capacitors. The
communication link between the PC and the TDCA die is based on SPI commands sent via USBSPI interface (Total Phase Cheetah). The left tunable filter ports are connected to a vector
network analyzer for monitoring purposes.
Due to the unknown delay associated with the ADC and PC processing speed, it is very difficult
to accurately determine the fast filter locking response time. Therefore, in order to check the
filter convergence the time axis is considered here as number of iterations (or tuning words)
required for the filter to achieve the frequency tracking. The locking time can then be estimated
by multiplying the number of iterations by the MEMS actuation delay (typically 10µs with this
technology). The final estimated value would be reasonably close to the expected locking time in
a future IC implementation.
Fig. 5.16 Discrete component implementation of the proposed notch filter control loop.
101
The loop performance and locking time will be evaluated in high or low loop gain conditions.
Due to the digital nature of the tunable notch filter, only certain discrete frequency states are
possible, which explains the expected step behavior of locking time response curves
Fig. 5.17
shows the filter transient response when the initial frequency offset ∆
> 49 and
the loop gain 0.4. In this case, approximately 35 iterations where needed to achieve
convergence yielding an estimated locking time of 350µs. This matches reasonably well with the
predicted behavior of Fig. 5.9.
Fig. 5.18
shows the transient response when the gain is increased to 3. The system then
presents oscillations that eventually are damped to reach convergence in approximately 125
iterations (1250µs). Gains higher than 3 resulted in non convergent situations.
From Fig. 5.17 and Fig. 5.18 we can conclude that, as we expected from simulation, it is important
to properly set the overall loop gain in order to obtain the fastest locking time, smoothest
exponential behavior and frequency convergence. In addition, high loop gains may potentially
create undesired frequency fluctuations after convergence in case of system noise pick up.
0
∆ω(t) in MHz
-10
-20
-30
-40
-50
-60
5
10
15
20
25
30
35
40
45
50
Number of Iterations
55
60
65
70
75
80
Fig. 5.17 Measured filter frequency control loop transient response for different A=0.3 and ∆
> =49MHz.
102
40
∆ω(t) in MHz
20
0
-20
-40
-60
0
20
40
60
80
100
120
140
Number of Iterations
Fig. 5.18 Measured filter frequency control loop transient response for different A=3 and ∆
> =50MHz.
The estimation of the filter frequency error in this practical discrete implementation is a very
challenging task for several reasons: (a) the tuning resolution of the present TDCA being 0.125
pF, the tunable digital filter can only achieve certain frequencies, (b) the noise pick-up of the
discrete system is not negligible and (c) the analog-to-digital convertor number of bits is limited.
The final frequency error of this experimental filter frequency control loop will not exclusively
depend on the quality factor of the components but will be considerably affected by the before
mentioned factors.
Assuming these limitations we tried to estimate the final frequency for our discrete
implementation by changing the filter external components to !1 27, 1 0.1&, which
reduces the step resolution to a maximum of 500KHz at expenses of reduced notch tuning range
of 5.6MHz. The loop gain was chosen as in Fig. 5.17 to avoid oscillations in the response. An
experiment was conducted choosing different ∆
> 5 MHz. The maximum recorded frequency
error is 277KHz for this particular implementation. The final frequency control error could be
further reduced if the filter frequency tuning resolution and the overall Q factor increase.
103
5.5 Conclusions
A complete tunable filter system comprising of an RF MEMS tunable notch filter and its
associated frequency control loop has been presented. The theoretical analysis and the derived
closed form solutions and formulas have been proved very useful to understand the design and
provide an implementation of the tunable filter system. This tunable filter can be practically used
in the transceivers of wireless mobile stations. It can be possibly integrated into mobile
transceiver RF ICs. The notch filter frequency control loop formulation developed here is not
only applicable to this specific filter topology but can be applied to any narrowband band-stop
tunable filters. Concepts of this tunable filtering system may be also used in the design of a more
complex future tunable duplexer system.
Appendix A. Derivation of the transfer function of the notch filter.
A simplified notch filter circuit topology (excluding the insertion loss block) is shown Fig. 5.19.
The ABCD matrix of this filter can be expressed as:
where
‹
Ž>€y
1
}
I


Œ Ž>€y
Ž>€y
@
3 1 ; 3 2 !
0
J
1
(5.24)
@
3
(-'

3 2 >
1;
(-'2
(5.25)
And
(-' 1
(-'
; > H
!
4!(-'
104
(5.26)
Fig. 5.19 Simple notch filter circuit topology
We can obtain the S-parameters matrix from (5.24) and (5.25) as:
I
<11
<21
1
<12
;
=
J
I Ž>€y >
<22
2
2 5 Ž>€y =>
2
;Ž>€y =3
J
(5.27)
Substituting (5.25) into (5.27) we can derive the transfer function of <11as (5.7)
Appendix B. Derivation of the frequency control loop differential equation.
We derive the differential equation of the notch filter frequency control loop starting with the
low pass loop filter and integrator as shown in Fig. 5.20. From this figure, we can obtain the
following equations:
Z’ (-' ; ZX (-'
“(-'
a
ZX (-' 1 €
” “(-' •–
€
(5.28)
(5.29)
ZW (-' GX ” ZX (-'-
(5.30)
2
“(-' Z (-'
GX - 2 W
(5.31)
•–
105
Ve (t ) =
ki ∫ Vi (t )dt
Vi (t )
Vo2 AK d ⋅ H S 11 ( jω )
⋅ sin (∠H S 11 ( jω ))
2
VC (t )
Fig. 5.20 Low pass loop filter and integrator
Substituting (5.29) and (5.31) into (5.28) and rearranging its terms, we have the second order non
linear differential equation
1 2
ZW (-' 5 ZW (-' GX Z’ (-'
2
W -
where
W 1
a
(5.32)
(5.33)
The frequency voltage tuning equation of the notch filter is
(-' 3 5 W ZW (- ; '
And
ZW (- ; ' (-' ; 3
W
(5.34)
(5.35)
Substituting (5.35) into (5.32), we derive the differential equation (5.9):
1 2 (-' (-' GX Y W Z>2
5
—EFF — sin( ∠ (<11 ''
W - 2
2
(5.36)
Finally, using the transfer function (5.7) of the notch filter reflection coefficient, we finally obtain
the differential equation (5.9).
106
References
[1]
Q. Gu, “RF System Design of Transceivers for Wireless Communications,” Springer, 2005
[2]
W. J. Keane, “YIG Filters aid wide Open Receivers”, Microwave Journal. Vol. 17, no 8, Sept
1980.
[3]
I. C. Hunter, J.D. Rhodes, “Electronically Tunable Microwave Bandpass Filters”. IEEE
Trans. on Microwave Theory and Techniques. Vol. 30, no 9, Sept. 1982.
[4]
S. J. Park, K. Y. Lee and G. M. Rebeiz, “Low-Loss 5.15-5.70GHz RF MEMS Switchable
Filter for Wireless LAN Applications”, IEEE Trans. on Microwave Theory and Techniques.
Vol 54, no 11, Nov. 2006
[5]
B. Jitha, P. C. Bybi, C. K. Aanandan, P. Mohanan, “Microstrip Band Rejection Filter using
Open Loop Resonator”, Microwave and Optical Technology Letters. Vol. 50, no 6, June
2008.
[6]
A. R. Brown, G. M. Rebeiz, “A Varactor Tuned RF Filter”, IEEE Trans. on Microwave
Theory and Techniques. Vol. 48, no 7, Jul. 2000.
[7]
H. Khatri, L. E. Larson, D. Y. C. Lie, “On-chip Monolithic Filters for Receiver Interference
Suppression using Bond-Wire Inductors”, Student paper, Silicon Monolithic Integrated
Circuits in RF Systems (SiRF), Jan 2006.
[8]
J De Luis, J. R.; Morris, A. S.; Gu, Q.; De Flaviis, F.; , "A tunable asymmetric notch filter
using RFMEMS," Microwave Symposium Digest (MTT), 2010 IEEE MTT-S International ,
vol., no., pp.1-1, 23-28 May 2010
[9]
Wispry Inc. Tunable RF solutions. www.wispry.com. Email: rf_mems@wispry.com. 20,
Fairbanks. Suite 198. 92618 Irvine, CA. USA.
107
[10]
P. Katzin; B. Bedard; Y. Ayasli; , "Narrow-band MMIC filters with automatic tuning and Qfactor control," Microwave Symposium Digest, 1993., IEEE MTT-S International , vol., no.,
pp.403-406 vol.1, 1993
[11]
Aparin, V.; Katzin, P.; , "Active GaAs MMIC band-pass filters with automatic frequency
tuning and insertion loss control," Solid-State Circuits, IEEE Journal of , vol.30, no.10,
pp.1068-1073, Oct 1995
[12]
Quintanel, S.; Serhan, H.; Jarry, B.; Billonnet, L.; Guillon, P.; , "Theoretical and
experimental implementation of microwave active filters using automatic frequency control
techniques,"
High
Power
Microwave
Electronics:
Measurements,
Identification,
Applications, 1999. MIA-ME '99. Proceedings of the IEEE-Russia Conference , vol., no.,
pp.IV7-I12, 1999
[13]
L. Billonnet, et al., “Recent Advances in Microwave Active Filter Design, Part 2: Tunable
Structure and Frequency Control Techniques,” Int. Journal RF and Microwave CAE 12, pp.
177 – 189, Wiley Periodicals, Inc., 2002.
108
6. APPROXIMATE CLOSE FORM DESIGN EXPRESSIONS FOR
CAPACITIVELY LOADED PLANAR INVERTED-F ANTENNA
6.1 Introduction
External monopoles used as antennas in earliest handset devices experienced a fast migration
towards the dominant use of internal antennas. These antennas feature a lower cost, lower
profile, simpler fabrication and, to some extent, easier integration with the rest of the RF front
end modules.
Most internal antenna designs consist of multi-resonant printed structures usually referred as
planar inverted F antennas (PIFAs). A PIFA used for cell phone applications as shown in Fig. 6.1
can be viewed as a quarter wavelength narrow patch where the shorting pin/wall serves the
purpose of ‘built-in’ matching network [1],[2].
Efforts have been focused over the past years in increasing the number of operating frequency
bands [3], minimizing the overall antenna size and ground plane dimensions [4] and human body
interaction effects [5]. One common technique in handset antenna design to achieve wider
impedance bandwidth and higher radiation efficiencies is to use air as substrate below the
antenna or to print the patch on top of a very thin layer of low loss and low permittivity dielectric
material providing mechanical support. Multiple antenna arms of different lengths can be
designed to obtain multiple coupled or uncoupled resonances to cover multiple cell phone
frequency bands [6].
109
Various simulation experiments and prototypes have demonstrated that a capacitively loaded
PIFA as the one in Fig. 6.1 can achieve additional size reduction [2] and/or frequency agility
[7],[8]. However, the capacitive loaded PIFA antenna concept has been only studied for very
specific topologies and its fundamental tradeoffs with other important antenna parameters have
not been clearly understood. The material in the following sections will show how this approach
provides the mentioned advantages at expenses of degradation in the antenna impedance
bandwidth and radiation efficiency.
Previous works have focused on the determination of the resonant frequency of conventional
(non loaded) PIFAs, which was approximately predicted using modal theory [9] or transmission
model analysis [10],[11]. However, to our knowledge, an extended set of closed form
expressions to quantitatively predict other important antenna parameters is not available to date.
The objective of this chapter is to present a set of approximate closed form expressions to predict
the most important parameters for capacitively loaded handset PIFAs such as input impedance,
resonant frequency, radiation efficiency, quality factor and impedance bandwidth. Without
losing generality, the same expressions can be applied for conventional PIFA design assuming
absence of loading capacitance. This set of approximate closed form expressions will yield
insight into the effects of the several design parameters over the antenna performance and allow
the assessment of fundamental tradeoffs decisions in the early stages of antenna design. In order
to illustrate the usefulness of these expressions, section 6.2 is concluded with a set of guidelines
for a hypothetical example design of a tunable PIFA under certain specified conditions of
frequency range, radiation efficiency and impedance bandwidth. Finally, section 6.3 presents a
comparison of measurements and full wave simulation results in order to validate the proposed
approximate closed form expressions.
110
6.2 Closed form expressions for capacitively loaded PIFA design
A canonical representation of a capacitively loaded tunable PIFA antenna of patch length !˜ ,
width ™ and height is shown in Fig. 6.1. A tunable (or switchable) shunt capacitor with
capacitance šš is connected between the antenna radiating open edge and its ground plane. The
shorting wall is located in close proximity to the feeding probe and typically has the same width
as the antenna patch.
Fig. 6.1 Capacitively loaded tunable PIFA
6.2.1 Transmission line model for a capacitively loaded PIFA
The antenna in Fig. 6.1 can be modeled using a transmission line equivalent circuit shown in Fig.
6.2.
Because of its simplicity, this model can be easily implemented in any commercial
microwave circuit simulator, and provides practical intuition into the basic antenna behavior.
Fig. 6.2 Transmission line equivalent model for the capacitively loaded PIFA
In Fig. 6.2, Gr is the radiation conductance associated with the power radiated by the antenna open
edge and Bs is the radiation susceptance related to the energy stored in the antenna near field.
111
a›y is the short circuit wall impedance. Expressions for the model components will be given later
in this section.
Note that in the transmission line model, the tunable capacitor is connected in parallel with the
radiation admittance. The total patch length (!˜ ) is the sum of the transmission line length from
the feed point to the antenna open edge (!™ ) and the distance between the feed and short wall
(!› ). => is the characteristic impedance of the antenna patch, which is related to the patch width,
height over the ground plane and permittivity (εr) of the supporting substrate [12].
The edge capacitive load effect can be graphically explained in the simplified transmission line
model of Fig. 6.3. The capacitor in Fig. 6.3(a) is replaced by an equivalent open circuited
transmission line of length !W in Fig. 6.3(b). The value of the equivalent transmission line length
can be can be found using microwave theory as:
!W 1
1
- œ
ž
G
=> 2#3 (6.1)
Where G=2/ Ÿ and Ÿ is the effective wavelength. The new circuit for the antenna in Fig. 6.3(b)
will now resonate when the equivalent length !’ !W 5 !˜ , is approximately quarter
wavelength at the operating frequency. In other words, the capacitive loading is acting as an
effective electrical length extension of the antenna.
This model reveals two important features that are valid for any edge capacitively loaded PIFA:
1. For a constant operating frequency (#3 ), the required physical patch length (!˜ ) in order
to achieve resonance will be less than quarter wavelength due to the capacitive loading
length extension effect. This allows for antenna size miniaturization using a fixed
capacitor loading and can be used in non tunable antenna designs [13].
112
2. For a fixed !˜ , the antenna operating frequency can be decreased by increasing the
amount of capacitive loading. Therefore, tunable antenna designs can make use of a
variable (or switchable) capacitor to achieve frequency agility.
Fig. 6.3 (a) Equivalent transmission line model of a capacitively loaded PIFA, (b) the capacitor has been replaced by
an open circuited section of transmission line
By looking at points 1 and 2, related exclusively with size and operation frequency, a capacitive
loaded antenna is an attractive candidate for reducing the size and/or tuning the frequency of
handset antennas. However, these two advantages will produce a decrease in impedance
bandwidth and radiation efficiency as it will be shown in the following sections.
In the following subsections, different approximate closed forms expressions will be presented to
predict the behavior of input impedance, resonant frequency, radiation efficiency, quality factors
and impedance bandwidth versus different capacitive loading conditions. All expressions can be
applied to standard non loaded PIFAs by considering the particular case of 0.
6.2.2 Input impedance.
An approximate expression for %3 and }› can be found in [14] and [12] respectively. To find %3 ,
a slot of length
’ parallel to the ground plane
was chosen to model the radiation from the open patch
edge as
113
%3 Where ’ 12¡¢
£¤ 4¥¦§§
2
’ 1
2
&
œ
ž
v
120 2 2 Ÿ ’ Ÿ2 90
(6.2)
is the equivalent width that models the spreading of the slot field from the
strip edge. &2 is the integral defined in [14]. The approximation that simplifies (6.2) performed
over &2 integral in is more accurate for ’ << Ÿ. On the other hand, the equivalent electrical
length extension (Δ©' due to the open edge fringing fields is used to model the radiation
susceptance as
}› > tan (GΔ©'
where
Δ© ¬
(6.3)
0.95
0.075(+3 ; 2,45'
;
­
1 5 0.85G
1 5 10G
(6.4)
and where > 1/=> . The short circuit radiation resistance for narrow patches can be modeled
as that of a small element of constant current [15] while the loss resistance is determined by the
material and operation frequency.
where
² and a›yi®¯
# 2
#3 H
80 œ ž ; a›y °>›› >
±
²
2
are the material conductivity and permeability, respectively and
(6.5)
>
is the speed of
light. The short circuit wall impedance of Fig. 6.2 can be considered to be the sum of radiation
and loss resistances a›y a›y 3pY 5 a›y °>›› . The short circuit loss resistance will be in most
cases negligible compared to the short circuit radiation resistance. The latter will increase with
the patch height and operation frequency and will be valid as long as the current can in the short
wall can be considered constant ( ³ Ÿ).
114
The input admittance (XŽ ) of the equivalent model can be found as the sum of admittances 1
and 2 in Fig. 6.2
XŽ > ,f 5 @> tan(´!™ '/ > ,›y 5 @> tan(´!› '/
5
> 5 @f tan(´!™ '
> 5 @›y tan(´!› '
(6.6)
where f %3 5 @}› 5 @2#3 is the capacitively loaded edge admittance, ›y 1/a›y and ´ is
the complex propagation constant of the line.
The reflection coefficient calculated with this model is compared with simulation results
obtained from method of moments (Zeland IE3D) in Fig. 6.4 and Fig. 6.5. Two different heights
2, 4 are considered with lengths !± 34.5 and !± 33.5 chosen to make
the antenna resonate at 2GHz with 0& for each case. From these results, the frequency
decrease provided by the increased capacitance is clearly visible.
It is apparent that the accuracy of the transmission line model is slightly better for lower antenna
heights. The frequency error increases for higher capacitive loading. Despite these differences
and the limitations imposed by the approximations, this model is a good tool to obtain a quick
representation of the loaded PIFA behavior in typical cell phone profile sizes.
6.2.3 Determining the patch length for a given operation frequency and
loading capacitance.
For design purposes, it is desired to determine the physical patch length (!˜ ) required to make
the antenna operate at a certain frequency (#3 ) using a fixed value of capacitive loading (e.g. if
antenna size reduction is intended).
This could be done by solving for !™ in (6.6) after imposing the condition µ¶XŽ · 0.
However, although a solution can be obtained, it results in a very long expression that provides
very little physical insight or practical application. Therefore, a different, more practical
115
approach will be used here. In [16], an approximate expression for the resonant frequency of a
PIFA operating in the fundamental mode and valid when the short wall length is equal to the
antenna width is given as
Fig. 6.4 Magnitude of the reflection coefficient using the transmission line model and method of moments
simulations (IE3D) for a PIFA over air with 2, ± 4, !± 34.5, !› 1. Error refers to the
absolute value of the difference in frequency between simulations and closed form expressions.
Fig. 6.5 Magnitude of the reflection coefficient using the transmission line model and method of moments
simulations (IE3D) for a PIFA over air with 4, ± 4, !± 33.5, !› 1. Error refers to the
absolute value of the difference in frequency between simulations and closed form expressions.
116
#3 >
4(! 5 '√+3
(6.7)
Where ! in [16] was the total metallic patch length of a non loaded PIFA.
In our case,
substituting ! in (6.7) by the equivalent length !’ !W 5 !˜ and using (6.1), the expression for
the new required physical patch length as function of the desired operating frequency and
loading capacitance can be obtained as
1
> - œ
ž ; 2#3 √+3 2=> #3 !˜ 2#3 √+3
Fig. 6.6
(6.8)
shows the required patch length in terms of wavelength versus loading capacitance in
order to operate at a fix
resonant frequency of #3 2%
for the cases of
2, 4, 6 obtained by method of moments (Zeland IE3D) simulations and closed
form expression (6.8). It is observed that the patch length decreases in an exponential manner
with increasing capacitance. As expected, the required patch length for zero capacitive loading is
close to quarter wavelength (particular case of non loaded PIFA antenna) and is shorter for
higher values of due to the increased presence of fringing field effects. In addition, Fig. 6.6
curves obtained from expression (6.8) are in concordance with the intuitive explanation offered
by the transmission line model: the reduction in patch effective length with an increase
capacitive loading at a fix operating frequency is clearly visible. Full wave simulations and
calculations from (6.8) are shown in good agreement.
117
0.25
h=2mm
h=2mm
h=4mm
h=4mm
h=6mm
h=6mm
Patch Lenght LT (λ)
0.20
0.15
Closed Form
Sim. IE3D
Closed Form
Sim. IE3D
Closed form
Sim. IE3D
0.10
0.05
0.00
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Loading capacitance C (pF)
Fig. 6.6 Simulations and closed form (CF) expressions results for the total patch length (LT) required to make the
PIFA resonating at 2GHz versus capacitve loading for different height values and ± 4
6.2.4 Required loading capacitance for a given length and operating
frequency.
An expression for the required capacitance to operate at a certain frequency for a fix patch length
can be of interest for tunable PIFA designs. This expression could be used to find a proper finite
tuning range of the tunable/switchable capacitor/s.
The value of required capacitance for a fix patch length and some desired resonant frequency can
be found by substituting ! in (6.7) by the equivalent length !’ !W 5 !˜ , using (6.1) and
solving for . The resulting capacitance value is then
1
2#3 √+3 (! ˜ 5 '
q- ¬
­ => #3 r
>
2
•1
Fig. 6.7
(6.9)
compares the required capacitance value using versus resonating frequency using (6.9) and
118
method of moments simulations (Zeland IE3D) for a fix patch length (!˜ was chosen to make the
antenna resonate at 2GHz when 0&) and different 1, 2, 4, 6.
This graph shows once again the exponential behavior of the frequency with capacitive loading.
An important conclusion can be extracted from Fig. 6.7; In case the capacitance is to be increased
in discrete steps (i.e. tunable digital capacitor/switched capacitor banks), a finer capacitance step
resolution would be specially desirable for higher frequencies (low capacitive loading) in order
to avoid coarse frequency jumps.
2.0
h=1mm Closed Form
h=1 Sim. IE3D
h=2mm Closed Form
h=2mm Sim. IE3D
h=4mm Closed Form
h=4mm Sim. IE3D
h=6mm Closed Form
h=6mm Sim. IE3D
Resonance frequency, fr (GHz)
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Capacitive loading, C(pF)
Fig. 6.7 Simulations and closed form (CF) expressions results for the resonance frequency versus capacitive loading
for different height values. The patch total length was chosen to resonate at #3 2% in absence of capacitive
loading ( 0&'.
6.2.5 Radiation efficiency and quality factor.
Radiation efficiency is a key antenna parameter and is paid special attention in cell phone
antenna design. The radiation efficiency quantifies how much of the energy accepted by the
antenna is radiated to the space. The radiation efficiency can be expressed in terms of the quality
factors of the antenna as [12]
¹3 >
> 5 3
119
(6.10)
where > is the loss quality factor associated to the conductor and dielectric loss dissipation
mechanisms and can be found as [12]
With
1
1
1
5
> y Y
(6.11)
y 4#3 > ²
(6.12)
Y 1
1;»
œ1 5
ž
-º
»+3
(6.13)
Where ² is the metal conductivity, -º is the dielectric loss tangent and » (0.5 ¼ » ¼ 1) is the
dielectric filling factor representing the fraction of total fields in the dielectric. Due to the fact
that most practical PIFA antennas for cell phone applications are usually built over a relatively
thick air gap using a very thin layer of supporting substrate [6], we will not consider the effect of
surface waves or dielectric losses in this study. 3 in (6.10) is the antenna radiation quality factor
and is a uniquely defined parameter [17] given by
3 2#3 ½›
03
(6.14)
where ½› represents the total energy stored in the antenna and 03 is the total power radiated to the
space. From (6.9) it is clear that in absence of losses, or when > k 3 , the radiation efficiency
approaches one.
In [18], an approximate expression for ½› and 03 in a half wavelength microstrip antenna
was found using cavity mode analysis and the expressions from the radiation of a Hertzian dipole
120
on a grounded layer. Using the same approach and assuming a fundamental mode surface current
distribution flowing in the y-axis direction (see Fig. 6.8) as
¾›¿ (À' 1¡ 2
2À
œ
ž
@
> Ÿ>
Ÿ>
(6.15)
where 1¡ is the amplitude of the dominant mode-field that depends on the patch length to width
aspect ratio and feed position [19],[20]. The power radiated by the half wavelength antenna was
found [18] to be approximately
Áf/2
1
2
2 2
(G
03 v 2 > ' 80 ™ ¬”
¾›¿ (À'À­
Ÿ>
•f/2
2
(6.16)
where ! Ÿ /2 was the total length of the patch that serves as limits for the current integration
as shown in Fig. 6.8(a).
On the other hand, the total energy stored in the half wavelength antenna can be found at
resonance, when the electric (½’ ) and magnetic (½o ) energies are equal, as
Áf/2
1
2
½› 2½o > ™ ”
—¾›¿ — À
2
•f/2
Fig. 6.8(b)
(6.17)
shows the case of the capacitively loaded PIFA which is resonating with a physical
length less than quarter wavelength. In this case, the current integration in (6.16) and (6.17) needs
to be performed over a smaller interval [0,!˜ ], and we will use the !˜ formula that was found in
(6.8).
The final expressions after performing the integration within the new intervals are:
03 320
2
2 1¡
™2  2
Ÿ2> ¹>2 (4=>2  2 #32 2 5 1'
121
(6.18)
½› 2
™ ! ˜
1¡
1
4! ˜
q 25
“ œ
žr
2
Ÿ>
4#3 > Ÿ> 4Ÿ>
Where ¹> is the free-space wave impedance and
 4! ˜ /Ÿ>
is an empirical constant. By
substituting (6.18) and (6.19) into (6.14) the general expression for the capacitively loaded PIFA
radiation quality factor is obtained as
3 1
1
ž
Â2 - œ
2560
2=> #3 1
Ÿ6> ¹>2 (4= 2  2 #32 2 5 1'
5 sin I2 - œ
žJÃ
2=> #3 ™ > > 4! ˜  2
(6.19)
Fig. 6.8 (a) Surface current distribution in the fundamental mode opeation for a regular half wavelength microstrip
antenna, (b) current distribution in a capacitively loaded PIFA antenna. The shadowed region indicates the current
integration area.
When 0, the obtained radiation quality factor corresponds to the one of a regular quarter
wavelength PIFA antenna.
Radiation efficiency can now be calculated combining (6.20), (6.11) and (6.12) into (6.10). A
comparison of the PIFA radiation efficiency using different heights (h = 1mm, 2mm, 4mm, 6mm)
versus capacitive loading from 0pF to 2pF obtained using method of moments simulations
122
(IE3D) and the final closed form expression is shown in Fig. 6.9(a). On the other hand, Fig. 6.9(b)
shows the efficiency versus antenna resonant frequency for the mentioned loading capacitance
range.
100
90
Radiation Efficiency ηr (%)
80
70
60
50
40
h=1mm
h=1mm
h=2mm
h=2mm
h=4mm
h=4mm
h=6mm
h=6 CF
30
20
10
0
0.00
0.25
IE3D
CF
IE3D
CF
IE3D
CF
IE3D
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Capacitive Loading (pF)
(a)
100
Radiation Efficiency (%)
80
60
40
h=1mm
h=1mm
h=2mm
h=2mm
h=4mm
h=4mm
h=6mm
h=6mm
20
IE3D
CF
IE3D
CF
IE3D
CF
IE3D
CF
0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Frequency (GHz)
(b)
Fig. 6.9 (a) Radiation efficiency versus capacitive loading obtained by MoM simulations (Zeland IE3D) and
obtained closed form (CF) expressions. The antenna has parameters ± 4, #3 2% ()* 0&', +3 1, ,1, - 6/ (b) Radiation efficiency versus resonant frequency with ,0&, - 2&/.
123
The simulated and obtained radiation efficiencies differ less than 6% for all cases. An important
observation is that the efficiency decreases considerably with capacitive loading. In [13] it is
stated that the efficiency of a capacitively loaded PIFA would be slightly less than that of a non
loaded PIFA. Here we point out that this will only be true in cases of small capacitive loading. In
order to have reasonably high radiation efficiencies the antenna must be only ‘lightly’ loaded.
However, this approach will in turn reduce the overall antenna frequency tuning range
(according to Fig. 6.7) or restrict the patch length decrease (according to Fig. 6.8). Therefore, this is
a fundamental tradeoff the designer must be aware of.
For design purposes is interesting to observe the effect of the antenna dimensions (™ and ' on
the radiation efficiency versus capacitive loading so we define the efficiency reduction factor as
Δ¹3 1 ;
¹3 ( : '
¹3 ( 0'
(6.20)
where ¹3 ( : ' represents the radiation efficiency when the antenna is loaded with a
capacitance : . Fig. 6.10 shows the effect of changing the patch width (™ ) in the radiation
efficiency for different loading conditions when 2 and #3 changes as indicated in Fig. 6.7.
It is observed than a wider antenna patch provides better radiation efficiencies. Also, the degree
of linear behavior in the efficiency reduction curve is increased for greater ™ . For ™ 31.1 the radiation efficiency reduction factor behavior becomes quasi linear with capacitive
loading.
Fig. 6.11,
shows the effect of varying in the radiation efficiency reduction factor for different
loading conditions when ™ 4, #3 2%. It can be observed that the efficiency
increases considerably with higher patch heights.
124
In summary, from Fig. 6.10 and Fig. 6.11 we can conclude that the patch width and height should
be chosen to be as high as possible always that the space allocated for the antenna is sufficient, in
order to minimize the efficiency reduction due to capacitive loading.
Radiation Efficiency reduction, ∆ηr (%)
100
Wp=31.1 mm
Wp=13.2 mm
Wp=7.5 mm
Wp=4.7 mm
Wp=3.2 mm
Wp=2.1 mm
Wp=1.5 mm
Wp=1.0 mm
Wp=0.7 mm
Wp=0.5 mm
90
80
70
60
50
40
30
20
10
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Fig. 6.10 Efficiency reduction versus capacitive loading for different values of ± with 2.
Capacitive Loading,C(pF)
100
h=1mm
h=2mm
h=4mm
h=6mm
Radiation Efficiency reduction, ∆ηr
90
80
70
60
50
40
30
20
10
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Fig. 6.11 Efficiency reduction versus capacitive loading for different values of with ± 4,
Capacitive Loading,C(pF)
125
6.2.6 Impedance bandwidth.
The impedance bandwidth of the antenna can be estimated from the total quality factor ˜
defined as
1
1
1
5
˜ 3 >
(6.21)
A good approximation for the fractional (or relative) impedance bandwidth for narrowband
matched antennas is given in [21] as
}3 v
Z<a ; 1
√Z<a˜
1.6
√2.6˜
)* Z<a 2.6
(6.22)
where a maximum VSWR of 2.6 (return loss of 7dB) has been selected as typical for cell phone
applications. If we compute the total quality factor using expressions (6.22), (6.20) and (6.11), we
can obtain the relative bandwidth for the capacitive loaded microstrip antenna using (6.23). Fig.
6.12(a)
shows the comparison of calculated and simulated antenna input impedance bandwidth
(when VSWR=2.6) for different values of capacitive loading. Fig. 6.12(b) plots bandwidth versus
resonant frequency for the loading capacitance range 0& - 2& It is observed that the
bandwidth reduces quickly with increased capacitance and more abruptly for higher antenna
heights. This feature ads and additional design tradeoff between capacitive loading and
impedance bandwidth that needs to be paid special attention if minimum operational bandwidth
is required.
126
50
h=2mm IE3D
h=2mm CF
h=4mm IE3D
h=4mm CF
h=6mm IE3D
h=6mm CF
Bandwidth (MHz)
40
30
20
10
0
0.00
0.25
0.50
0.75
1.00
1.25
Capacitive Loading (pF)
(a)
50
h=2mm IE3D
h=2mm CF
h=4mm IE3D
h=4mm CF
h=6mm IE3D
h=6mm CF
Bandwidth (MHz)
40
30
20
10
0
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency (GHz)
(b)
Fig. 6.12 a) Impedance bandwidth reduction versus capacitive loading for different values of with ™ 4.
(b) Impedance bandwidth reduction versus frequency when loading capacitance is changed from C=0pF to 2pF. The
patch total length was chosen to resonate at #3 2% in absence of capacitive loading ( 0&' for each case.
It is interesting to study the relative bandwidth reduction in a similar manner than was done with
radiation efficiency. The relative bandwidth reduction and can be expressed as
Δ}3 1 ;
}3 ( : '
}3 ( 0'
127
(6.23)
Fig. 6.13 shows the relative bandwidth reduction dependence on ™ with different loading
conditions when 2 changes as indicated in Fig. 6.7. Similarly, Fig. 6.14 shows the effect
of varying in the relative bandwidth reduction for different loading conditions when ™ 4 and #3 2%.
From Fig. 6.13 it can be observed that the wider the antenna patch, the less severe the bandwidth
reduction is. On the other hand, from Fig. 6.14 we can see how the bandwidth reduction factor is
lower when the patch height is also lower. However, is important to keep in mind that the
absolute bandwidth when 0 will be also be lower for lower values of as seen in Fig. 6.12.
Impedance bandwidth reduction, ∆ΒW r (%)
100
90
80
70
60
Wp= 31.1 mm
Wp= 13.2 mm
Wp= 7.5 mm
Wp= 4.7 mm
Wp= 3.2 mm
Wp= 2.1 mm
Wp= 1.5 mm
Wp= 1.0 mm
Wp= 0.7 mm
Wp= 0.5 mm
50
40
30
20
10
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Fig. 6.13 Relative impedance bandwidth reduction versus capacitive loading for different values of ± with
2
Capacitive Loading,C(pF)
6.2.7 Design Guidelines.
Subsections 6.2.1 to 6.2.6 summarized the most important parameters usually found in
specifications for cell phone antenna design. The conclusion from this study is that size reduction
and/or frequency agility can be achieved using a capacitively loaded PIFA at expenses of lower
radiation efficiency and impedance bandwidth. As direct consequence, a practical antenna to be
128
used in cell phone applications will have to be only ‘lightly’ loaded. The scope of this chapter is
not focused on the design of a particular antenna (as this is regarded as future work) but to
introduce the different design expressions. In order to illustrate how these tools can be used in
practice, a set of guidelines for a design example of a capacitively
Impedance bandwidth reduction, ∆ΒW r (%)
100
90
80
70
60
50
40
30
h=1mm
h=2mm
h=4mm
h=6mm
20
10
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Fig. 6.14 Relative impedance bandwidth reduction versus capacitive loading for different values of with ± 4.
Capacitive Loading,C(pF)
a. Decide antenna parameters , as high as possible accounting for the antenna allocated
space constraint.
b. Find the required patch length (!˜ ) for an specific minimum capacitance “ and
#K using (6.8). In a realistic implementation, the capacitance “ will be always
greater than zero due to the inherent device shunt parasitics.
c. Substitute the !˜ value found in (6.8) and calculate the maximum capacitance K
required to operate at the minimum frequency (#“) using (6.9). Steps (a) and (b) will
determine the required capacitor tuning range (“~K). The required capacitor
tuning ratio can be also calculated by manipulating (6.9) in terms of a desired frequency
range as
129
K
-((k op: (! ˜ 5 ''
∆ ∆#
“
- (GoXŽ (! ˜ 5 ''
(6.24)
With ∆# #K/#“,k op: 2#K√+3 /> and k oXŽ 2#“√+3 /> .
d. Calculate the radiation efficiency and impedance bandwidth values for the pairs
[“, #K] and [K, #“] using (6.10) and (6.23) respectively. Check if these
values meet the efficiency and bandwidth design goals. If the specifications are not met
iterate to (a), by increasing , or accept a decrease in frequency tuning range.
6.3 Measurement results
To validate the above expressions and full wave simulations, a PIFA antenna prototype was built
as shown in Fig. 6.15. The antenna square ground plane is made of copper having a side
dimension of 230mm (1.5Ÿ> at 2GHz). The PIFA metallic strip has a width of ± 4. The
patch is 2 away from the ground plane and is supported by foam material (+3 v 1'. The
prototype’s height value was chosen to be the most convenient to match the length of the
available surface mount ceramic capacitors, Murata 0805 GQM series, of chip size 1.25mm
(width) by 2mm (length). By choosing the same capacitor length as the patch height we avoid
additional parasitic effects that could be introduce from any interconnecting path distorting the
main purpose of the experimental results.
The patch length is chosen to make the antenna operate at 2GHz for absence of loading
capacitance ( 0&). The measured resulting patch length is !˜ 35 (0.233Ÿ> '. This
result is in good agreement with the calculated patch length of !˜ 35.5 (0.236Ÿ> ' using
(6.8).
Then, the capacitor is soldered between the antenna radiating edge and the ground plane as
indicated in Fig. 6.1 and Fig. 6.15. For this experiment different available capacitor values will be
130
used to compare the closed form expressions presented in the previous section with the
measurements results.
Fig. 6.16
shows the antenna reflection coefficients for different values of capacitor loading. The
bandwidth reduction and frequency decrease with capacitive loading is apparent from this plot.
Tables I and II summarize the values for the resonant frequency and impedance bandwidth
respectively. Good agreement is observed between closed form expressions, IE3D simulations
and measurements. The average error between calculations and measurements is 30MHz and
2.75MHz for the resonant frequency and impedance bandwidth respectively.
Fig. 6.15 Fabricated PIFA prototype over large ground plane. The ceramic capacitor is loaded at the antenna
radiating edge.
Table 6.1 antenna resonant frequency versus capacitive loading
C (pF)
0.2
0.5
1.2
1.6
1.8
Closed Form
Expression (6.8) (GHz)
1.754
1.498
1.161
1.046
1.001
IE3D
Simulation (GHz)
1.757
1.458
1.130
1.016
0.972
131
Measurements
(GHz)
1.727
1.442
1.132
1.027
0.980
Fig. 6.16 Measured magnitude of the PIFA reflection coefficient for different capacitive loadings.
Table 6.2 antenna impedance bandwidth versus capacitive loading
C (pF)
0
0.2
0.5
1.2
1.6
1.8
Closed Form
Expression (6.22)
16.5MHz (0.83%)
10MHz (0.57%)
5.5MHz (0.37%)
2.5MHz (0.21%)
2MHz (0.19%)
1.7MHz (0.17%)
IE3D
Simulation
16MHz (0.80%)
11MHz (0.62%)
6MHz (0.41%)
2.5MHz (0.22%)
2MHz (0.19%)
1.9MHz (0.19%)
Measurements
22MHz (1.1%)
12MHz (0.69%)
8MHz (0.55%)
5MHz (0.44%)
4MHz (0.40%)
3.5MHz (0.36%)
The efficiency was calculated using the generalized Wheeler cap method proposed by Johnston
and McRory [22] and further applied on [23]. For this measurement a cylindrical Wheeler cap
made of aluminum material with internal diameter of 90 mm and height of 150 mm was built.
Fig. 6.17 Fabricated Wheeler Cap covering the PIFA antenna for radiation efficiency measurement.
132
The resulting radiation efficiencies are compared in Table III with those obtained using the
closed form expressions and IE3D simulations. Good agreement is observed between the results
with an average error between measured and calculated radiation efficiencies of 6.7%.
Table 6.3 antenna radiation efficiency versus capacitive loading
C (pF)
0
0.2
0.5
1.2
1.6
1.8
Closed Form
Expression ((6.13,(6.17,(6.18)
90.0%
84.7%
74.4%
46.0%
32.7%
29.1%
IE3D
Simulation
89.1%
81.9%
70.1%
44.1%
31.9%
28.5%
Measurements
86%
77%
66%
43%
27%
24%
6.4 Conclusions
In order to predict the resonant frequency, quality factor, radiation efficiency and impedance
bandwidth for an edge capacitively loaded PIFA, a set of useful approximate closed form
expressions have been presented. The effects of the antenna loading over the resonant frequency
or antenna size and related tradeoffs with impedance bandwidth and radiation efficiency have
been elaborated. In order to meet typical cell phone antenna specifications, it was shown that the
capacitive loading must be relatively light for an end-loaded PIFA, restricting the frequency
tuning range or achievable size reduction.
Some design guidelines have been presented as example of the usefulness of these expressions
in a practical antenna design example. The results from closed form expressions have been
validated using full wave simulations as well as measurement results.
133
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136
7. TUNABLE DUPLEXING ANTENNA SYSTEM FOR WIRELESS
TRANSCEIVERS
7.1 Introduction
With the increase in number of wireless bands and standards, tunable antennas can play an
important role in reducing the overall system size and enhancing the power transfer to/from the
transceiver. Traditionally, non-tunable cell phone antennas have been designed to cover multiple
frequency bands by establishing tradeoffs with other important parameters such as antenna
efficiency or overall size [1]. Designing and developing non-tunable small multiband antennas
with reasonable efficiency becomes a more difficult task with the increase of allocated frequency
bands, especially those at lower frequencies. As an example, Table 7.1 summarizes some of the
current wireless standards frequency specifications ranging from 700MHz up to 2700MHz. In
addition, it is well known that the user interaction with the handset affects the antenna
performance considerably [2, 3]. In particular, the antenna is detuned (mismatch increases) and
its efficiency decreases when the user hand/fingers/head move close to the antenna location
leading to a reduced sensitivity (possible dropped calls) and increasing the power consumption
(shortening the battery life).
Tunability within the radiating element can be achieved by using tunable capacitors [4],[5],[6] or
RF switches [7],[8]. For example, if frequency agility is exploited, the tunable antenna
impedance bandwidth can be relaxed leading to a potential size decrease and a reduction of
design cycle time. In addition, the antenna can be re-tuned in order to mitigate user or other
loading effects, thus optimizing the power transfer.
137
As an enabling technology, RF MEMS tunable capacitors are able to meet the stringent
requirements of handset front ends such as low loss, high linearity, and low power consumption
[9].
While designing tunable handset antennas, two different design approaches can be considered:
•
Design a single antenna with broad enough bandwidth to cover the transmit and receive
channels of each operating band simultaneously.
•
Design an antenna pair with dedicated transmitter and receiver antennas respectively.
The minimum bandwidth of these antennas is the operating channel bandwidth.
Both of these approaches are compared in Table 7.2 in terms of impedance bandwidth
requirements for different wireless standards versus conventional non tunable antennas.
Band
Designation
Band 17 (LTE)
US-CELL
E-CELL
DCS
PCS
IMT
Band 7 (LTE)
Band
Designation
Band 17 (LTE)
US-CELL
E-CELL
DCS
PCS
IMT
Band 7 (LTE)
Table 7.1 Wireless standards frequency specifications
Transmit Uplink
Receive Downlink
Channel
Frequency
Frequency
BW
(MHz)
(MHz)
(MHz)
704~716
734~746
1.4~20
824~849
869~894
0.2~5
880~915
925~960
0.2
1710~1785
1805~1880
0.2
1850~1910
1930~1990
0.2~5
1920~1980
2110~2170
5
2500-2570
2620-2690
1.4 - 20
TX~RX Offset
(MHz)
30
45
45
95
80
190
120
Table 7.2 Requirement for cell phone antenna bandwidths
Conventional
Single Tunable Antenna
Tunable Antenna Pair
Antenna BW
Minimum BW
Minimum BW
(MHz)
(MHz)
(MHz)
42 (5.8%)
32.8~50(4.3~6.9%)
1.4~20 (0.19~2.7%)
70 (8.1%)
45.4~55 (5.3~6.4%)
0.2~5 (0.025~0.58%)
80 (8.7%)
45.4 (4.9%)
0.2 (0.022%)
170 (9.5%)
95.4 (5.3%)
0.2 (0.012%)
140 (7.3%)
80.4~90 (4.2~4.6%)
0.2~5 (0.010~0.26%)
250 (12.2%)
200 (9.8%)
5 (0.24%)
190 (7.3%)
90 (3.4%)
20 (0.7%)
From Table 7.2, the lower instantaneous bandwidth requirements of tunable antennas are apparent
compared to conventional antennas. Further, the tunable antenna pair is shown to have far lower
138
required impedance bandwidth. Note that the tuning resolution using tuned pairs must be high
enough to allow for optimal tuning for each different communications channel (spacing as low as
100KHz). On the other hand, a single tunable antenna with broader bandwidth can be
implemented with fewer RF MEMS switches or lower resolution RF MEMS varactor devices.
Additionally, according to classic fundamental limits for small antennas [10, 11], a narrower
impedance bandwidth antenna can be designed to occupy a smaller size than its conventional
broadband antenna counterpart. In this context, the reduction in bandwidth requirement shown in
Table 7.2
may allow for smaller tunable antenna footprints.
The objective of this chapter is to examine the feasibility of the antenna pair concept by means of
simulations and measurements of a fabricated prototype. Section 7.2.1 and 7.2.2 presents the
proposed antenna pair solution and highlights some its additional advantages (such as the built-in
filtering characteristics) from a system point of view. Section 7.2.3 analyzes the isolation level
between both antennas for different arrangements within the handset. Through section 7.2,
simulations will be supported by measurements on the fabricated prototype. Finally, section
7.2.4 presents the results obtained from the radiated test performed at UCI´s anechoic chamber
over the single antenna element.
7.2 Duplexing Antenna System
7.2.1 Advantages from a system perspective
An additional advantage of using an antenna pair with narrow impedance bandwidth is the ability
to enhance the isolation levels between transmit and receive antennas [5, 12-15].
If the antenna pair configuration is properly designed, the isolation level between both elements
may reach up to 25dB. Due to this inherent ‘filtering’ ability, the antenna pair is given the name
139
of duplexing antenna. If these antennas were combined with the tunable notch filter presented in
a companion paper [16] as shown in Fig. 5.12(a), the overall isolation level between transmit and
received branches would approach nearly 50dB levels. This value of isolation is comparable to a
conventional full duplex front end solution shown in Fig. 5.12(b) comprising of a broadband
multiband tunable antenna, diplexer and external duplexer module based on non integratable
FBAR technology. Contrarily to external duplexers, RF MEMS tunable filters using RFCMOS
process are compatible with on-chip integration. In addition, because of the desired narrowband
characteristics of the duplexing antenna elements, the antenna pair may be made smaller than its
broadband antenna counterpart. These features added to the advantage of multiband operability,
can reduce the number of components, overall size and cost of cell phone front ends. The
following sub-sections will evaluate the concept of tunable duplexing antenna in a realistic
handset platform.
Fig. 7.1 Duplexing antenna tunable front end concept depicting tunable narrowband antennas in combination with
tunable notch filter, (b) Conventional non tunable front end comprising of a multiband broadband antenna with
diplexer and external multiple duplexer modules.
140
7.2.2 Single Tunable Element Design
Planar inverted-F antennas (PIFAs) have been widely used as internal mobile phone antennas
due to their low profile, easiness of fabrication and relatively small footprint. PIFAs are a
particular type of microstrip antenna where the resonating patch length is approximately a
quarter wavelength at the operating frequency. This is accomplished by placing short circuit
plate/pins in one of the antenna edges and placing the feeding probe near the shorting pins in a
location selected for good impedance matching.
One possible option for designing a tunable PIFA is to load a shunt tunable capacitor on its open
end as shown in Fig. 7.2.
Fig. 7.2 Tunable PIFA antenna concept with shunt capacitor loading on antenna open edge.
In this case, when the capacitance is minimum (high impedance) the antenna resonates at the
highest operation frequency. On the other hand, when the capacitance is maximized, the
capacitor acts as a longer equivalent length extension of PIFA arm, making the antenna to appear
electrically longer and resonating at the lowest frequency. This allows for the antenna to be
considerably smaller than quarter wavelength at low frequency.
The antenna size reduction allowed by the loading capacitor is accompanied by a reduction in the
impedance bandwidth and radiation efficiency. Narrow impedance bandwidth (as long as is
141
greater than channel bandwidth) is a desired feature in the duplexing antenna concept to enhance
the transmit-receive isolation level, although not the only factor. On the other hand, radiation
efficiency reduction comes from the fact that the loaded small antenna radiation resistance
decreases and starts to be comparable to the loss resistance. Efficiency values greater than 50%
are usually required in cell phone applications, therefore, care must be given to avoid excessive
efficiency drop by limiting the amount of capacitive loading, which in turns imposes limit on the
overall frequency tuning range.
In this chapter, an existing high Q tunable digital capacitor array (TDCA) (see Fig. 7.3) flip die
solution from Wispry Inc. [9] was utilized in the designs. The TDCA consists of twenty tunable
capacitor cells, each with a nominal tuning range of 1pF or 0.875pF. Only two 1pF cells will be
utilized in this specific antenna application (avoiding excessive capacitive loading).
Nonetheless, for product applications, much smaller custom TDCAs would be utilized. The
minimum capacitance step resolution of each cell is 0.125pF in this chip. The cells in the TDCa
can be interconnected on the PCB level in order to achieve any desired amount of capacitance
and circuit topology.
Fig. 7.3 Tunable CMOS-integrated RF-MEMS digital capacitor cell: a) die photo of four capacitance bits of a cell;
b) 3D image of capacitance bits.
142
The Q of the die level capacitors was measured to be greater than 100 at 2GHz with a value of
capacitance highly repeatable. The IP3 level for this device is greater than +65dBm. The CMOS
biasing circuitry is integrated within the same chip and transforms a 3.3V regulated supply
voltage coming from the USB port to the required 35V actuation voltage. The current
consumption is 6 µA and 90 µA in the sleep and the active mode (charge pump on), respectively.
A Serial Peripheral Interface (SPI) is used to control the capacitor bank states. A USB port is
used to issue the tuning commands from PC control software.
Fig. 7.4
shows the designed shunt loaded PIFA. The antenna length was adjusted to its final value
of Lp=15mm for operation at highest frequency when C=Cmin. The antenna width was chosen to
be Wp=2mm to occupy the smallest area while providing sufficient radiation efficiency.
Placing the tunable capacitor flip chip at the open edge of the antenna patch requires additional
routing for the SPI control lines and via/pins connecting to the ground plane below as shown in
Fig. 7.4.
A total of eight control lines are then routed to the opposite layer of the PCB (made of
FR4 material) where the control circuitry is located. In order to shield the low frequency lines
from the RF signal and serve as ground connection for the tunable capacitors, a row of four
grounding pins are located between the antenna structure and the eight control lines.
A tunable capacitance ratio of C=0.26-1.86pF (Cmin-Cmax) corresponding to two parallel
capacitor cells is used to cover the desired frequency range. The antenna patch is printed on top
of a supporting dielectric material (40mm by 20mm, Rogers 5880 εr=2.2, tanδ=0.009 at 10GHz,
0.381mm thick) as shown in Fig. 7.5. The ground plane has dimensions 40mm by 90mm and is
located at 4 mm distance from the antenna patch. Two pins are used as shorting point separated
0.5 mm from coaxial line fed in order to match the antenna at the operating frequency.
143
Fig. 7.4 Tunable PIFA with two shunt tunable capacitors loading the antenna radiating edge and a series external
chip capacitor in series with the tunable capacitors.
Fig. 7.5 Designed tunable single element and PCB dimensions.
To increase the resolution of the antenna operating frequency steps, an external surface mount
capacitor (Murata SMD 0402) has been added in series with the tunable capacitor die as shown
in the detail of Fig. 7.4. This will produce finer capacitance step size at expenses of reducing the
capacitance tuning range. The design has been simulated using Ansys HFSS finite element
method solver. The lumped components such as series capacitor or tunable cells have been
144
modeled in this tool using horizontal lumped ports. The multiport S-parameter matrix resulting
from full wave simulations was loaded with the proper values of fixed and tunable capacitance
using AWR Microwave Office circuit simulator.
A prototype using the design dimensions was built and assembled as shown in Fig. 7.6 and Fig. 7.7.
In this design, the integrity of the simulated geometry of Fig. 7.5 was preserved with minor
additions mainly from the DC control circuitry in the opposite side of the PCB. The control
circuitry consist of connection pins for the SPI lines, voltage regulator and a few surface mount
components such as capacitors/resistors.
Fig. 7.6 Top and bottom views of the fabricated prototype. The top view shows the duplexing antenna structure,
tunable capacitor dies and RF ground plane. The bottom side contains the control circuitry and interfacing multi-pin
connector..
145
Fig. 7.7 Picture of the duplexing antenna built prototype
The simulation and measured results for different cases of capacitive loading are in good
agreement as shown in Fig. 7.8. These representative cases are interesting to observe the effect of
different capacitive loading on antenna resonance frequency and are listed from higher lo lower
resonance frequency as: (a) absence of surface mount fix capacitor or tunable capacitors with
antenna resonating at 3.32GHz, (b) a 0.5pF surface mount capacitor with absence of tunable
capacitors and resonance at 2890MHz, (c) a 0.5pF surface mount capacitor and tunable
capacitors set to C=Cmin=2x0.13pF with resonance at 2170MHz and (d) a 0.5pF surface mount
capacitor and tunable capacitors set to C=Cmax=2x0.93pF operating at 1850MHz.
From Fig. 7.8, it is also observed that if the tunable capacitors minimum capacitance was reduced
to zero, the antenna tuning range would increase considerably. However, a residual parasitic
capacitance is always present in any realistic device. With the current configuration the antenna
tuning range is 2170MHz-1850MHz (16%).This is sufficient for operation in the IMT-I and PCS
bands (refer to Table 7.1). If the capacitance step resolution was finer than the current 0.125pF (i.e
146
tunable capacitors were customized for this application) and therefore the series SMD capacitor
was not needed, the same capacitance tuning range (2x0.13pF to 2x0.93pF) would allow for a
resonance tuning range of 1990MHz to 990MHz (67%).
Fig. 7.8 Simulated (HFSS) and Measured reflection coefficient of the single element antenna for different values of
capacitive loading.
Fig. 7.9 Measured reflection coefficient of the single element antenna for different values of capacitive loading.
147
Fig. 7.9
shows the measured single antenna reflection coefficient for all the possible tunable
capacitor states. An overall tuning range of 1850MHz to 2170MHz with a frequency step
ranging from 55MHz to 15MHz and a measured impedance bandwidth from 54MHz to 25MHz
(at VSWR=2.6).
7.2.3 Antenna Pair Design
Once the single element was designed, the duplexing antenna system is formed using two
identical tunable antennas. One of the most important features of this system is the isolation
between transmit and received antennas. This parameter is also of high importance in
conventional non tunable antenna designs and depends not only on the far field antenna
characteristics but also in the coupling of currents induced in the PCB ground plane. Several
studies have studied the isolation between different handset antennas by analyzing different
antenna orientations within the PCB [17, 18], using ground plane notches [19, 20], connecting
strips and resonating elements [17, 21]. A generalization is difficult in these cases because the
isolation level ultimately depends on the specific antenna topology, size of the PCB ground
plane, proximity of other cell phone components and user interaction. In general, a specific study
must be performed for each particular device platform. In practice, it is usually challenging to
achieve more than 15dB isolation between broadband elements in a realistic handsets.
148
Fig. 7.10 Different configurations considered in the duplexing antenna isolation study. The distance ‘d’ is varied in
discrete intervals and the S21 parameter is recorded for each variation at high and low frequency pair situations.
Full wave simulations provide some quick understanding of the best antenna geometry for a
specific platform. In this work, several geometries were considered as shown in Fig. 7.10(a-d).
The main goal was to allocate the two antennas within the same region of the handset (top
portion) and decide the separation distance and relative orientation.
Besides the isolation level, the other important factor is the ability to tune both antennas
independently. These two concepts are ultimately related as shown in Fig. 7.11 (a) where the
antenna with port 1 is acting as transmitting antenna at 1850MHz while port 2 corresponds to the
receiving antenna at 1930MHz. The distance between elements is d=11.8mm (referring to Fig.
7.10(a)).
In this case, a second resonance appears in the transmit antenna at the receive frequency
and vice versa, providing a dual frequency behavior. Besides the low isolation level under these
circumstances, a change in the capacitive loading value on any of the two antennas will strongly
affect the opposite antenna resonance frequency. It is obvious that this tuning dependency is not
149
a desired effect due to the fact that the control of both antennas becomes unfeasible. On the other
hand, Fig. 7.11 (b) shows the same frequency pair situation when both antennas are separated a
greater distance of d=36.8mm. In this case, not only the isolation level between elements will be
much higher but the coupled dual resonance behavior does not appear and both antennas can be
tuned independently.
In order to find configurations with higher isolation, a simulation parametric study has been
performed using HFSS with the topologies considered in Fig. 7.10. The separation distance ‘d’ for
each case was varied in incremental steps. The resulting S-parameters matrix was then
capacitively loaded in AWR Microwave office to make the antenna operate at two different
frequency pairs: high frequency pair (1980MHz, 2170MHz) and low frequency pair (1850MHz,
1930MHz). The value of the S21 parameter versus separation distance for cases of Fig. 7.10(a-d) at
these four frequencies are shown in Fig. 7.12(a-d), respectively. For the cases Fig. 7.12(a-b), the
isolation between antennas increases with the separation distance. In particular the higher
isolation level is obtained for the case of Fig. 7.12(a) when d=36.8mm and both antennas are
located close to opposite PCB edges. In that situation 26.6dB and 21.8dB are the minimum
isolation levels achieved for the high and low frequency pairs, respectively.
On the other hand, in Fig. 7.12(c) the antenna the isolation is higher for the cases of d=0mm and
d=36.8mm when the respective feeding point and radiating edge of both antennas are further
away. In the intermediate cases, from approximately d=4mm to d=24mm it was not possible to
tune the antennas at the specific frequencies due to the previously mentioned tuning dependency
between elements. This seems to indicate that when one antenna’s radiating edge is closer to the
other antenna’s feeding point the existing coupling between elements is maximum.
150
(a)
(b)
Fig. 7.11 Simulated S-parameters resulting from configuration of Fig. 7.10(a) when the two antennas are separated a
distance (a)11.8mm and (b)36.8mm. Antenna with port 1 and port 2 are operating at 1850MHz (transmit antenna)
1930MHz (receive antenna), respectively.
151
Finally, as intuitively expected, the case of Fig. 7.12(d) where both antennas are oriented in
orthogonal polarizations exhibits higher isolation level and slower variations with distance.
However, it was not desirable to choose this option for final implementation as the transmit and
receive antennas in the system are desired to operate at the same polarization in the
communication link. For this reason, the configuration with maximum isolation (Fig. 7.10(a)) with
d=36.8mm was chosen for the final implementation.
The duplexing antenna prototype was built as shown in Fig. 7.7. Both antennas were then
connected to the two-port vector network analyzer and tuned to operate at the high and low
frequency pairs while the S21 isolation parameter was measured as shown in Fig. 7.13
For the high (Tx-Rx offset=190MHz, Fig. 7.13(a)) and low (Tx-Rx offset=80MHz, Fig. 7.13(b))
frequency pairs, the worst obtained isolation level was 18.5dB and 22.5dB, respectively. This
value is slightly lower than the simulated isolation level and may be due to additional coupling
created by the DC control circuitry that was not included in simulations and VNA measurement
setup. However, the obtained values of isolation are still particularly high and remain as a valid
proof of concept for the duplexing antenna system.
7.2.4 Single Antenna Radiated Test
The single antenna was measured in UCI’s far field anechoic chamber facility in order to obtain
the radiation pattern and radiation efficiency. The normalized measured radiation pattern
corresponding to the two extreme capacitive cases C=Cmax and C=Cmin for the main plane cuts
are shown in Fig. 7.14(a) and (b), respectively. The results are in reasonably good agreement with
simulations. The obtained peak gain was 2.95dBi and 2.25dBi for the C=Cmax and C=Cmin cases,
respectively.
152
The radiation efficiency was measured in UCI’s far field anechoic chamber at the optimum
matched frequency for each one of the cases shown in Fig. 7.8 and compared to those obtained by
HFSS simulations in Table 7.3. In general the measured efficiency is lower than the predicted by
simulations and this may be due to additional losses introduced due to solder joints and other
fabrication imperfections. In general, the efficiency values are still above the target 50% for all
frequencies within the tuning range which was within the acceptable efficiency threshold
predominantly admitted for cell phone applications.
7.3 Conclusions
A duplexing antenna concept based on a narrowband antenna pair with high isolation between
elements has been presented. The feasibility of this system for the use in handset applications has
been validated with simulations and measurements. This work demonstrated the potential
advantages of using tunable handset antennas for next generation reconfigurable front ends. A
modified topology that allows for an increase in the number of covered bands and decrease in the
frequency resolution steps by using a chip with smaller capacitance increments is regarded as
future work.
Table 7.3 antenna radiation efficiency versus capacitive loading
SMD
Chip Capacitor
(pF)
0
0.5
0.5
0.5
Tunable Capacitor
(pF)
0
0
0.26
1.86
Antenna
operating
frequency
(MHz)
3320
2890
2170
1850
153
Simulated Radiation
Efficiency
Measured
Radiation
Efficiency
92%
91%
74%
68%
89%
83%
68%
55%
(a)
(b)
(c)
(d)
Fig. 7.12 Simulated parametric study of isolation (S21) between the two antennas operating at high (1980MHz, 2170MHz)
and low (1850MHz, 1930MHz) frequency pairs for the configurations depicted in Fig. 7.10.
(a)
(b)
Fig. 7.13 Measured reflection coefficients and S21(dB) between the two antennas operating at (a) high (Tx-Rx
offset=190MHz) and (b) low (Tx-Rx offset=80MHz) frequency pairs.
154
Fig. 7.14 Simulated (blue solid) and measured (red dashed) normalized radiation patterns for three orthogonal planes
at two different loading conditions: (a) C=Cmax and (b) C=Cmin..
155
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