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Equivalent boundary conditions of strip gratings and their application to microwave filters

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EQUIVALENT BOUNDARY CONDITIONS OF STRIP
GRATINGS AND THEIR APPLICATION
TO MICROWAVE FILTERS
by
Tungyi Wu
B.S.C.E., National Chiao-Tung University, 1988
M.S.E.E., University of Massachusetts, 1992
A thesis submitted to the
Faculty o f the Graduate School of the
University of Colorado in partial fulfillment
o f the requirements for the degree of
Doctor of Philosophy
Department of Electrical and Computer Engineering
1996
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This thesis for the Doctor of Philosophy degree by
Tungyi Wu
has been approved for the
Department of
Electrical and Computer Engineering
by
Edward F. Kuester
Zoya Popovic
Date
8 M e u e J L l<
i ’? 6
L
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j
Wu, Tungyi (Ph.D., Electrical Engineering)
Equivalent Boundary Conditions of Strip Gratings and Their Application to
Microwave Filters
Thesis directed by Professor Edward F. Kuester
This thesis develops the equivalent boundary conditions (EBCs) for a
periodic metallic strip grating which is located between two media with uniform
phase shift between adjacent strips. A method of homogenization based on the
technique of multiple scales is employed. The derived EBCs modify previous results
which didn’t take into account the required phase shift and are suitable for two
different media on either side of the grating.
Based on the developed EBCs, the propagation of surface waves along a
periodic strip grating on grounded dielectric slab is then investigated in an efficient
and accurate approach. Propagation along a grating with a uniform phase shift
between strips and at an oblique angle with respect to the strips is assumed. The
determination o f propagation characteristics o f metal gratings on grounded substrates
then becomes analytically simple and computationally fast compared to a numerical
approach.
Finally, dispersion equations o f periodic grating elements formed by
truncating an infinite array of strips, such as microstrip meander lines, comb lines,
and hairpin lines, are investigated and show the filtering properties. Then a class of
compact grating filters is designed. To efficiently compute the responses of grating
\
1
j_ _ _ _ _ _ _ _ _ _ _ _
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filters, an equivalent transmission line (ETL) model based on EBCs is employed.
The behaviors o f filters are also verified by experiment and compare well. For a
meander line loaded with lumped elements, an interesting property of bandwidthtuned exists which can find application in voltage-tunable microwave filters.
iv
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ACKNOWLEDGMENTS
I would first like to thank my advisor, Prof. Edward F. Kuester, for his
supervision and encouragement during the entire course of my thesis research.
Professor Kuester provided the important ingredients that helped me finish this work
during my three years at the University of Colorado. I would also like to thank the
members o f my thesis committee, K. C. Gupta, Zoya Popovic, Melinda Piket-May,
and Arlan Ramsay, for their valuable suggestions about this thesis.
I would also like to pay many thanks to several people outside the
University o f Colorado. Dr. Ming H. Chen, President of Victory Industrial Corp. in
Taiwan, who offered me the interships for several summers and winters. My loving
parents and families supported and encouraged me to make all things possible in my
life. Special thanks to my wife Suming for all the love and infinite patience in the
past years.
v
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CONTENTS
CHAPTER
1 INTRODUCTION
2
1
1.1
Thesis Goals
1
1.2
Thesis Overview
3
EQUIVALENT BOUNDARY CONDITIONS
5
2.1
Introduction
5
2.2
The Geometry and Assumptions
6
2.3
Maxwell’s Equations and Expansion o f the Scattered Fields
10
2.4
Boundary Conditions
13
2.5
The Zeroth Order Fields
16
2.5.1 The Zeroth Order Magnetic Fields
17
2.5.2 The Surface Current Density
25
2.5.3 The Odd Current Density
28
2.5.4 The Even Current Density
29
2.5.5 The Zeroth Order Electric Fields
37
2.5.6
38
2.6
Boundary-layer Grating Voltages and Currents
The First Order Fields
!
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39
2.7 Equivalent Boundary Conditions (EBCs)
3
2.7.1
EBCs - Electric Field
44
2.7.2
EBCs - Magnetic Field
46
2.8 Results and Conclusions
51
SURFACE WAVES ALONG A METALLIC STRIP GRATING
55
3.1 Introduction
55
3.2 Determination of the Propagation Characteristics
57
3.2.1
z-directed Propagation Constant Pvs. Phase Shift 0
57
3.2.2
Propagation Constant k j vs.Propagation Angle (J)
67
3.2.3
Characteristic Impedance Zc
69
3.3 Numerical Results
4
43
73
3.3.1 Comparison of 3 with Weiss’ Model
74
3.3.2 Comparison of k t with Bellamine’s Result
76
3.3.3 Comparison of Zc with Weiss’ Model
76
3.4 Conclusions
79
CHARACTERISTICS OF MICROSTRIP GRATING ELEMENTS
80
4.1 Introduction
80
4.2 Dispersion Equations
81
4.2.1
Meander Lines
81
4.2.2
Hairpin Lines
87
4.2.3 Comb Lines
90
vii
L.
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4.3
4.4
4.5
5
Image Impedance Zim
93
4.3.1
Meander Lines
93
4.3.2
Hairpin Lines
97
4.3.3
Comb Lines
99
Numerical Results and Prediction o f Stopband and Passband
99
4.4.1
Meander Lines
99
4.4.2
Hairpin Lines
108
4.4.3
Comb Lines
112
Conclusions
MICROSTRIP GRATING FILTERS
118
5.1
Introduction
118
5.2
Meander-line Bandreject Filters
119
5.2.1
6
117
Equivalent Transmission Line (ETL) Model:
Normal Modes
119
5.2.2
Dual-mode ETL Model: Complex Wave Modes
122
5.2.3
Design of a Microstrip Meander-line Bandreject Filter
126
5.3
Microstrip Comb-line Bandreject Filters
130
5.4
Microstrip Hairpin-line Bandpass Filters
133
5.5
Tunable Microstrip Meander-line Bandreject Filters
136
5.6
Conclusions
143
SUMMARY AND FUTURE WORK
viii
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146
BIBLIOGRAPHY
148
APPENDIX
A Evaluation o f Equation (2.87)
158
B Evaluation o f Equation (2.125)
161
C Evaluation o f Identities
164
D Grating Voltages and Currents
167
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FIGURES
FIGURE
2.1
Geometry o f a periodic strip grating.
2.2
Cross-section of a strip grating on “fast-variable” coordinate.
3.1
A periodic array of strips loaded by a dielectric slab over a ground
plane: (a) cross section; (b) top view.
3.2
7
24
56
Comparison o f P between quasi-TEM Green’s function method
and homogenization EBCs method for even and odd modes. The
parameters used are: kod = 3.325e - 3 , p / d = 1.632, a /p = 0.5588.
and e r = 6.5. Weiss’ result (— ); present method (—).
3.3
75
Comparison of normalized propagation constant versus propagation
angle. The parameters used are: kod = 2.0944e - 3, p / d = 1.86,
a/p = 0.07, and e r = 9.6. Bellamine’s result ( - •); Crampagne’s
result ( - -); present method (—).
3.4
77
Comparison o f Zc between quasi-TEM Green’s function method
and homogenization EBCs method for even and odd modes. The
parameters used are the same as those in Figure 3.2. Weiss’ result
(— ); present method (—).
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78
4.1
Top view o f a microstrip meander-line structure consisting of an
infinite array o f strips.
4.2
82
Top view o f a microstrip meander-line structure loaeded with
series impedances Zi and Z i .
83
4.3
Hairpin-line structure with a defined unit cell.
88
4.4
(a) A comb-line structure loaded by reactances; (b) a unit cell of (a).
91
4.5
Comparison o f dispersion diagram (backward-wave) of a meander
line between homogenization method ( - - ) and Crampagne’s result
(—) with a = 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm,
and e r —9.6.
4.6
101
Dispersion diagram (backward-wave) o f a meander-line structure
with p = 1.2mm, d = 1.012mm, L - 18.36mm, and e r = 10.2
for ajb = 2.0 (—), 5.0 (• •), 1.0 (— ), and 0.5 ( - •).
4.7
102
Calculated image impedances (forward-wave) of a meander line
with p - 1.2mm, d - 1.012mm, L = 18.36mm, and e r = 10.2
for ajb = 2.0 (—), 5.0 (• •), 1.0 (— ), and 0.5 ( - •)-
4.8
103
Dispersion diagram (backward-wave) of a meander-line structure
with a = 0.8mm, b = 0.4mm, d - 1.016mm, and z r = 10.2 for
4.9
L = 18.9mm (—), 18.36mm (• •), 17.5mm (— ), and 19.5mm ( - •).
105
Example o f a diode-loaded meander-line structure.
106
4.10 Dispersion diagram (backward-wave) o f a loaded meander line
XI
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for Ct = 1 p F (—), 15 p F (• •), 5 p F (— ), and 0.5 p F ( - •)
with a = 0.8mm, b = 0.4m m , L = 19mm, d = 1.016mm, and
e r = 10.2.
107
4.11 Normalized group velocity o f a meander-line structure with a = 0.13
m m , b = 0.8mm, d = 1.0mm, L = 18.36mm, and e r = 9.6.
109
4.12 Dispersion diagram o f a hairpin line for c/ = 1.27mm, L = 18.36mm
, and e r = 10.2in terms o f a and b .
110
4.13 Dispersion diagram o f a hairpin line with a = 0.8mm, b = 0.4mm ,
d = 1.27mm, and e r = 10.2 for L = 18.9mm (— ), 18.36mm (—)
, 18.8mm (• •), and19.2mm ( - •)•
111
4.14 Image Impedance of a hairpin line for p = 1.2mm and d = 1.27mm
with (a) L = 18.66mm and a /p = 2/3; (b) s r = 10.2 and a/p = 2/3.
113
4.15 Dispersion diagram of a comb-line structure for a = 0.8mm,
b = 0.4m m , L = 18.36mm, d = 1.016mm, and e r = 10.2.
Equivalent reactances: Ls = 0.7239/?// and Cs = 0.0112//F
with Cp = 0 (— , C s = 0 ), 0.03p F ( - •), 0.3//F ( - -), and
\.0pF (” ).
115
4.16 Calculated image impedances of a comb-line structure in terms
of a and b with p = 1.2mm, L = 18.36mm, d = 1.016mm,
and e r = 10.2: a/b = 2.0 (—), 1.0 (• •), 0.5 ( - - ) , and 0.2 ( - •)•
xii
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116
5.1
(a) Top view o f a microstrip meander-line filter, (b) Equivalent
transmission line model of (a).
5.2
120
(a) Dual-mode ETL model for a meander-line filter, (b)
Admittance-matrix representation of (a).
5.3
124
Complex propagation constant y of a microstrip meander-line
filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, Z, = 18.36
m m , and e r = 10.2: y in passbands (—) and P5 ( - •) and - a s
(• •) in stopband.
5.4
127
Measured (• •) and computed (—) responses of a microstrip meander
-line bandreject filter with a = 0.8mm, b = 0.4m m , d = 1.016mm,
w = 30mm, L = 18.36mm, and e r = 10.2.
128
5.5
An example o f a microstrip comb-line bandreject filter.
130
5.6
Measured (• •) and calculated (—) responses o f a microstrip combline filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36
5.7
mm, w= 14.4mm, and s r = 10.2.
131
An example o f a microstrip hairpin-line bandpass filter.
133
5.8 Measured insertion loss of a microstrip hairpin-line bandpass filter
for a - 0.8mm, b - 0.4mm, d = 1.27mm, L = 18.36mm, w =
31.2mm, and e r = 10.2.
5.9
134
Calculated responses of a microstrip hairpin-line bandpass filter
with a = 0.15mm, b = 2.0mm, d = 0.25mm, L = 18.36mm,
xm
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w = 43mm, and e r = 18.
5.10 An example of a tunable microstrip meander-line bandreject
filter with loaded capacitances.
5.11 Dispersion diagram (backward-wave) o f a loaded meander line
for C* =1.5 p F (— ), 3.3 p F (• •), 4.8 p F ( - •), 10 p F ( - • • •)
, and short link (—) with a = 3.5mm, b = 2.0mm, L = 100mm
, d = 1.27mm, and e r = 10.2.
5.12 Measured responses o f a loaded meander-line bandreject filter for
G =1-5 p F
3.3 p F ( - ) , 4.8 p F ( - •), 10 p F ( - • • •),
and short link (• •) with a = 3.5mm, b = 2.0mm , L = 100mm,
d = 1.27mm, w = 110m m , and e r = 10.2.
5.13 Calculated ( - -) and measured (—) 1^21| of a loaded meander-line
filter with a = 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm,
w = 1lOmm and e r = 10.2 : (a) short link; (b) C j= 10 p F ; (c)
G = 4 .8 p F \ (d) G =3.3 p F .
5.14 Calculated image impedances of a loaded meander line for Cs
=1.5 p F ( - - ) , 3.3 p F (• •)» 4.8 p F (-•), 10 p F (------ ), and
short link (—) with a = 3.5mm, b - 2.0m m , L = 100mm,
d = 1.27mm, and e r = 10.2.
5.15 Measured transmission loss of a meander-line bandreject filter
xiv
Ii
j
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(— ) and hairpin-line bandpass filter (• •) with the same circuit
dimensions: a = 2.5mm, b = 2m m , L = 50m m , d = 1.27mm
, w = 90m m , and e r = 10.2.
xv
|
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CHAPTER 1
INTRODUCTION
1.1
Thesis Goals
The method of homogenization had been used to derive the equivalent
boundary conditions in analyzing a strip grating antenna [11-12]. The use of
homogenization method simplifies the antenna analysis and requires
less
computational memory compared to the analysis o f numerical method, yet the
method employed in [11-12] is not quite general for most microwave circuits since
the required phase shift between adjacent strips o f a periodic strip grating was
assumed zero.
In this dissertation, the equivalent or averaged boundary conditions
(EBCs or ABCs) of a semi-infinite strip grating located at two different media are
derived by using a modified method of homogenization based on the technique of
multiple scales; the developed EBCs successfully modify the previous results [9, 1112] by considering a general phase shift between adjacent strips and are suitable for
two different media on either side of the grating. To investigate the characteristics of
surface-wave propagation on the grounded dielectric substrate covered by periodic
strips for circuit application, the modified EBCs have been shown to make the
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analysis analytically simple and computationally fast compared to the results using
numerical method [57-59],
By making connections or disconnections between finger ends of periodic
strips on grounded dielectric substrate, several compact planar grating elements are
formed such as microstrip meander lines, hairpin lines, comb lines, and so on.
Periodic strip grating structures can propagate slow-wave whose group velocity is
less than the velocity of light and have pass bands and stop bands. By imposing the
proper boundary conditions o f strip ends of the periodic grating elements, loaded or
unloaded with lumped devices, the dispersion equations of microstrip grating
elements are thus obtained and the calculated Brillouin diagrams show that they exist
filtering properties similar to filters. Using an accurate and efficient model for metal
gratings on grounded substrate based on the obtained EBCs, a new class of
microwave grating filters is designed and their behaviors are also verified by
experiment. Especially, when a meander line loaded with lumped elements, a
promising bandwidth-tuned grating filter is constructed; this achievement enable us
to design a voltage-tuned filter if nonlinear devices such as varactor diodes are
loaded with a meander line. The impact of these grating filters promises to deliver
broadband properties or highly selective behaviors and these techniques are
applicable to more compact filters used in microwave and millimeter-wave
integrated circuits.
2
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1.2
Thesis Overview
This thesis contains six chapters. In chapter 2, the modified method of
homogenization based on the technique of multiple scales is used to analyze a semi­
infinite periodic strip grating in a general manner. Then the equivalent boundary
conditions (EBCs) for the average fields at a periodic array of perfectly conducting
strips with a uniform phase shift between adjacent strips are developed. The firstorder EBCs which take into account the phase shift between strips are obtained
under the assumption that the period length of the grating is much smaller than the
wavelength o f incident wave such that the fields approach the average fields existing
around the grating at a appreciably exceeding the period. Also the derived EBCs
here generalize the previously obtained results.
Chapter 3 deals with the propagation characteristics of surface waves
along a grounded dielectric slab covered by periodic strip grating using the
developed EBCs in chapter 2. Without loss of generality, the fields are assumed to
have a uniform phase shift between adjacent strips and propagate at an oblique angle
with respect to the strips. With the aid of EBCs, the propagation constant and
characteristic impedance o f a strip grating on grounded substrate are obtained
analytically in a simple and accurate manner compared to the results using numerical
method in some other literatures.
Chapter 4 presents the dispersion equations and image (iterative)
impedances o f periodic grating elements loaded or unloaded with lumped devices,
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i
such as microstrip meander lines and hairpin lines, etc, formed by truncating an
infinite array of strips. The calculated Brillouin diagrams predict that these grating
elements have filtering properties and have the potential application in compact filter
design. The calculated filtering characteristics of grating elements loaded with
lumped devices are also presented; they find possible applications in electronically
tunable filters.
Chapter 5 compares the measured and calculated results of microstrip
meander-line bandreject filters, hairpin-line bandpass filters, and comb-line
bandreject filters. There the calculated responses are obtained by using the
equivalent transmission line (ETL) model based on the obtained EBCs by
considering normal-mode and complex-mode propagation along the grating
structure. The measured results o f a microstrip meander-line filter loaded with
lumped devices are also presented for the promising application in bandwidth-tuned
filters.
Finally, Chapter 6 make conclusions from the previous chapters and
points out the suggestions and directions for future work.
4
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CHAPTER 2
EQUIVALENT BOUNDARY CONDITIONS
2.1
Introduction
This chapter develops equivalent boundary conditions (EBCs) of a
periodic metal grating o f strips with uniform phase shift between adjacent strips
using a modified method o f homogenization based on the technique of multiple
scales. The development o f EBCs is useful for efficiently modelling the propagation
characteristics o f fields on periodic slow-wave grating structures which form the
basis of many microwave and millimeter-wave components. Several investigators
have previously formulated equivalent boundary conditions for a strip grating
assuming that there is no phase shift between conductors [1-4]. However, this
assumption will not be valid for many microwave or millimeter-wave slow-wave
structures [5-8, 26, 57-59], Ivanov [9-10] derived equivalent boundary conditions for
an array o f strips or circular conductors excited by a field with a phase shift in free
space using a quasi-static approximation, and these generalized the equivalent
boundary conditions obtained by Vainshtein [4]. Although Ivanov's research was a
pioneer work, it is not directly useful in application to microstrip or other substrate-
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based structures because it cannot describe a grating at the interface between
different media. Such situations will be treated in this chapter.
In this chapter, we will follow a modification o f DeLyser’s approach [1112] on homogenization analysis o f strip gratings with zero phase shift and obtain
EBCs for a strip grating lying at the interface between two different media with
nonzero phase shift. The detailed mathematical treatment o f the homogenization
method has been discussed extensively in [13-20] and will not be repeated in this
dissertation.
2.2
The Geometry and Assumptions
Assume the periodic structure under analysis consisting of infinitely long
and ideally conducting metallic ribbons of zero thickness in the z -direction and
extending to infinity in the y -direction with period p , width b, and spacing a is
located between two different media as shown in Figure 2.1, where the material
properties are denoted by a “ + ” subscript for x > 0 and a “
subscript for x < 0.
An assumption is made that the period length p of the structure is much smaller than
the wavelength of incident waves. If incident waves
exist in x > 0 plane, for
example, they cause scattered fields everywhere. Thus the total fields can be
represented as:
6
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t
£ + = £ ? c + £+
^ H+ = H T + Hs+
x > 0
and
(2.1)
«
E l = EsH i
=
H t
x
< 0
t
A J
M
w
F
mmm
a
Figure 2.1: Geometry o f a periodic strip grating.
where the fields in equation (2.1) include the phase shift 0 occurring from the
incident fields and -7t < 0 < 7t is assumed in the following derivation. The incident
7
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waves in (2.1) can be represented as ~
e!+ c = E inc(r,Q)e~J®y' and owing to the
dispersive homogenization theory [17-23], a more general incident waves can be
handled as £ + c =
][inc(r,Q)e~jQy'dB where the variables r and y ' will be
defined below. For a wave is incident on the metal strip, there exists a phase shift 9
between adjacent strips. Since the grid period p is much smaller than the wavelength
of incident waves, the fields approach the average fields existing around the grating
at a distance appreciably exceeding the period. Therefore a perturbation method
called the technique o f multiple scales is applied [13-16] where the scattered fields
are expanded in powers o f the period p . Assume the scattered fields are functions of
a “fast” variable r ' and a “slow” variable r which are defined as:
r = (x, y, z)
.
,
r = rjp
(2-2)
where x , y , and z are the usual rectangular coordinates. Here we assume the “fast”
variable is independent o f the “slow” variable temporarily [15], Thus the del vector
operator is defined as:
8
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where we assume there is no field dependence on the “fast” variable z ' and this
assumption will be used in the following derivation of EBCs. The MKS system of
units and the time dependence eJat are used. Now expand the scattered fields with a
phase shift in powers of p ,
Es ~ E ° i r X , y ’) e - j Qy' + P E l (r’x ' y ) e ~ JQy’ + 0(p2)
Hs
where
e
~
' e ~ jQy'
(2-4)
H ° ( r , x ' y ) e - J Qy' + P H l ( . r , x ' , y ' ) e - j Qy ' + o ( P 2)
and
e~^y \ i
= 0,1,2 •••) are vector fields containing the non­
boundary layer fields (in captial letters) and boundary layer fields (in lower letters)
which are defined as:
El ~ E‘(r,x^ + e ' i r y y )
H 1~
(2.5)
+ V (r, x ' , y ' )
According to Floquet’s theorem [24],
e
i =
0 , 1,2 ■
‘ and H l in (2-5) will correspond to the
fundamental Floquet-Bloch mode, while e‘ and h l are the sum of all the higher
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order modes. Also £ ' and f f 1 are not function of y ' because they are defined as the
averaged fields of e ‘ and
h
‘ over one period, which are
E !(r,x') = f o E \ r , x ' y W
—’
( 2 .6 )
p l_ :
H l (r ,x ’) = | aH , ( r , x ' , y ' W
i = 0,1,2 -
and equation (2.6) results in
foe '( r , x \ y ') d y ' = 0
and
£ h ‘ (r,x\y')<fy' = 0
i = 0 ,1 ,2 -
(2.7)
which means that the averaged boundary layer fields without e~jQy> dependence
over one period are zero. Equation (2.7) is useful and important in our derivation of
the EBCs and it also implies that e' (or p ) is periodic in y ' with a period of 1.
2.3
Maxwell’s Equations and Expansion of the Scattered Fields
Apply the source-free Maxwell's equations to the scattered fields
V x £ * = -j<a\iHs
(2.8)
V x f j s = y<ae£*
and from equations (2.4) and (2.5), we have the scattered fields expanded in powers
of p
10
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E? - [ E V . x O + e ^ r .x '.j O le - j0/ +
+
* ', / ) ]
+o(p2)
(2.9)
H s ~ [H° (>.*') + A1V , x ' . y ’yie-fiy' + p[H l (r,*') + /»I(r >Jc'.yX k”^ '
+o(p2)
We substitute equation (2.9) into Maxwell’s equations and then equate identical
powers of p on both sides o f (2.8), and get
P~l :
p°:
Vr - x (£ ° + e ° )+J/9 (£ ° +e°) x a y = 0
Vr x ( £ ° + e°) + Vr - x ( £ l + el) + J6( £ l + el) x a ,
=
Pl■
+ h°)
^210^
v r X (E l + e1) + V r- X ( £ 2 + e2) + 7 0( £ 2 + e2) x
= - > H ( ^ l + Al)
Now, take the average of the first equation of (2.10) over one period. W e have
V r>x e° = 70 ay x e°
(211)
where equations (2.6) and (2.7) have been used.
Again, take the average of the second equation of (2.10) and use in
addition:
11
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j lQ(V r x & W = 0
and f^ (v r . x
=0
/ = 0,1,2 —
(2. 12)
Then we get
Vr x £ ° + v r *x £ l + 70 £ l x gy = - j o (XH °
(2.13)
Substituting equation (2.13) into the second equation of (2.10) yields
(Vr - - y'0 a y ) x el = - J o H f ? - V r x e °
(2.14)
Next, applying the operator (Vr«-JQ ay) • to equation (2.14) yields after
some manipulation
(V r ' -./Q o y M n /j0) = 0
(2.15)
Similarly, following the same procedure for the third equation of (2.10), we have
( V r>-jQ ay ) h l = - V r t
(2-16)
Equations (2.11), (2.14), (2.15), and (2.16) are important for determining the
boundary layer fields as will be shown later.
12
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If we start from the second equation o f (2.8), the similar results to
equations (2.11), (2.14), (2.15), and (2.16) can be obtained. Thus, gathering the
results for boundary layer fields, we have:
(Vr »- JQ ay ) x e° = 0
(2.17)
(Vr ' - jQ ay) x /j° = 0
(Vr>—jQ ay) •(e e°) = 0
(2.18)
(Vr - —jQ ay) •(n h°) - 0
( W - jQ ay ) x e l =
- Vr x e°
(2.19)
(Vr <- ; e ay) X A1= jme e° - Vr x h°
' (Vr <-y’0 ay)-el = - V r -e°
( 2 .20)
(V r. - j B a y ) - h l = - V r h0
Note that when the phase shift 0 = 0, they reduce to DeLyser’s equations (2.20) (2.23) [12] as expected.
2.4
Boundary Conditions
The boundary conditions at the plane o f the grating are:
ax x(]E+C+ E+~ E - )|x=x'=0 - 0
; ~E+C= E ‘+e~jQy
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(2.21)
ax -[e+( £ ? C+ £ “1) - £ -
^= x'= 0 = P.
(2.22)
= [Ps(r) + P°s(r,r')\e-JQy' + p[Pls (r ) + P\( r ,r ') \e -j*y' + 0 ( p 2)
; H T ^ lf+ e -fly '
a * - [p + ( t f ? C+ 7 7 * ) ' P _ t f ^ x ' =o = 0
(2.23)
a x x [ t f T + 7 & -7 F - ^x=x'=0 =
(2.24)
* [ j h r ) + f s (r,r')]e-JQy' +pCA(r) + j ls (r,r')]e-j* S + o (p 2)
where P5 and j s are the surface charge density and surface current density which
contain averaged and boundary-layer components have been expanded in powers of
p . Then substitute equations (2.4) and (2.5) for E s and Jjs into equations (2.21) (2.24) and collect the like powers o f p to get the p ° components
_
,_0 -0 \
— r—i —0 —0 ,
ax x (e+ - e - )x=r--o - - ax x [£+ + E + ~ E - \ix=x'=0
(2.25)
— r —
0
—
0l
Clx'\£ + e+ E-e- Jjjc=ar'=0
- - a x - [e+(H!+ + E+) - s _
0
30 ^ o0
)x=x'=o + /*? + ?
(2.26)
- (K
ax x (fj+ h-j^x=x'=0
-
r —zl
— 0
—0
-,o .
-0
1
(2.27)
-0
- ~ ax x \jf+ + H +~ / / - ^x=x'=0 + J s + J 5
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_
r
-0
ax-[V-+h+ ~
-0
]jx=x'=0
(2.28)
= - ax ■[ n + Q /V + h I) ~ H_ H - }x=x'=0
and the p l components
ax x (e+ - e l -\ x=x >=q - - a x * [E?+ - E l- )x=x'=0
I
!
(2.29)
ax •[e + e+ - e _ e - ]|x=x'=0
(2.30)
-
- a x - [ e +£ i-e _ £ L ] |x = x ,=o + / >! + Py
ax * (/i+ - /jl )|x=x'=0
(2.31)
- - a x x ( t f i - H - ) |x=x'=0 + v i
ax ■[M-+h+ ~ t1_ h - ]jx=x'=o
(2.32)
= ~ ax •O + h \ - H_H - ^x=x'=0
The boundary conditions (2.25) - (2.32) and equations (2.17) - (2.20) can be
combined and reduced to the boundary-value problems for the zeroth order and first
order boundary layer fields which will be discussed below.
15
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2.S
The Zeroth O rder Fields
Now, we have to rewrite equations (2.17) - (2.20) for easily solving the
static electric and magnetic boundary-value problems. Let us define new functions e'
and H' such that
(2.33)
i = 0,1,2 -
Then equations (2.17) - (2.20) can be rewritten in terms of e' and h' as
'
Vr 'x e °=0
(2.34)
Vr -x h °=0
' Vr '-ee°=0
(2.35)
Vr .-Hh°=0
Vr ' x el = - y o n h 0 - Vr x e°
(2.36)
Vr »x 5 1 = yoee^ —Vr x ii0
Vr '- e l = - V r e
(2.37)
. V r - E l = - V r -E°
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Equations (2.34) - (2.37) are similar to those obtained in DeLyser's
results [11-12], We can follow the same process and assumptions as done in [11-12]
to obtain the zeroth order fields.
2.5.1
The Zeroth O rd er Magnetic Fields
From equation (2.34), we find as in [ 11] that
e° = h2 = 0
(2.38)
The zeroth order z -directed boundary layer fields are zero which imply from
equation (2.25) that the z -components of the “slow” variable fields are continuous.
Now, we need to solve for the magnetic field h°. From the second
equation of (2.35), we define a vector potential A such that
Uh° = Vr «x(Ae-fly")
(2.39)
where A = a z A z ( x ' , y ' ) is periodic in y ' with period 1. Because A is periodic in y '
with period 1, A can be expanded in a Fourier series as:
a o(x')
A z ( x ' , y ’)
n=»
n=Q0
n = ®
= — - — + £ a a(x ')co s2n7ty, + £ 6 n(x')sin2m ty'
2
n=l
n=l
where
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(2.40)
r I/2
a n = 2 1_^ Az ( x y ') cos 2 nny 'efy'
n = 0,1,2 '
(2.41)
fl/2
bn = 2 J-i/2 A z ( x 'j O s i n 2 n 7 t y '# '
n = 1,2 —
Since /j° has zero average in one period with respect to y ' as mentioned previously,
A also has zero average in one period. This leads to ao(x') = 0. Thus equation
(2.40) becomes the following expression by multiplying e~JQy' on both sides of
equation (2.40)
n=oo
n=ao
A z {x’, y ' ) e ~jQy' = ^ aa(x')e -JQy' cos2mty’ + 5 ^ n ( x ')e"J0'y' sin2n;ry'
n=l
n=l
= 2 Cn(x ') ey(2ll7l~ 0 ) y
n=-oo
a *°
where
Cn = J_V^ ( A z e - j Qy')e-->i2Tm _9)y'cfy'
n*-
(2.43)
If taking the curl o f equation (2.39) with respect to the “fast” variable and
using equation (2.34), we have
V ^ x [ V r f x ( A e -ye/ ) ] = 0
(2.44)
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or by a vector identity
a2
«
a1
{ k z e - f l y ,) + -2- T { \ z e-fly') = Q
dx'2
dy'2
(2.45)
Substituting equation (2.42) into equation (2.45), we have
d2
2
- c n( x ') - ( 2 n7t - 0 ) c n(*') = 0
dx'2
(2.46)
whose solution is
Cn(x ') = A ne(2n7t- 0)jf' + B ne - (2rOT- 0)jf'
n = ±l,±2,----
(2.47)
where An and Bn are constants. Then the potential A z on either side o f the grid is
given by equation (2.42) which is
Az + e - W
(2.48)
= S Bne - (2lut “ 0) * V ( 2im - 0) / + A.a e _(2im +0)*'e--/(2mt + 0).y']
a=l
for x ' > 0, and
i
19
i
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A Z-e -J Qy'
~
^ [ A n e ( 2n7t - 0 )Jr' ey(2nji - 0 ) / + B_n<?(2n* + 9 )* 'e -/(2im +0).y']
n=l
for x ' < 0.
From the continuity o f |dh° at x = 0 , the x -component of (2.39)
together with equations (2.48) and (2.49) give the conditions for the constants A n
and Bn :
(2.50)
where g n and f n are constants yet to be determined. Thus, the potential A z in
equations (2.48) and (2.49) now becomes
A z+ e-W
n=0°
= ^ ] [ - . / g ne “(2im - 0)*'e/(2n* ~ Q)y' + j f ae~(lnn +0)*'e- /( 2iHt +0 ) / ]
n=l
C2 51)
for x ’ > 0, and
P iz-e -W
n=o°
- j S ne(-2lm -Q)x'ej(2m -0 )y + y fQg(2njt +Q)x'e-j(2m + 0)/]
n=l
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(2 52)
for x ' < 0 .
Then from equation (2.39), the zeroth order boundary layer fields without the phase
dependence e ~jQy' can be solved as
1 n
h i+ = ---- £[(2n7C + 0 )fn e _(2im +Q)x'e- j 2rmy' +
n=l
(2n7t - 0 )gne~(2lut - Q)Jc'ej 2imy']
(2.53)
1 n
h%_ = — £ [ ( 2mt + 0 )fn e (2rot +0)^ 'e-y2njry' +
n=l
(2mt ~ 0 )gae(2mt ~Q)x 'e jinny']
and
= — T [/(2n7t + 0 ) f ne -(2nn + 0)x'e -y2njt/ _
y(2njt - 0 )gne _(2nn -Q)x 'ejloxy']
i n=oo
ifi
"V- = — 2 ["-/(2niz + 0 )fn e (2mt +Q)X' e - j 2imy' +
I*- n=l
(2.54)
j ( 2an - 0 )gne(2mt -®)x'e jinny']
which can be used to determine the surface current density on the strips.
From equation (2.27), the ay and a2 components of the surface current
density are
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J % + j% - [hy+ - hy_ + H y + + H y + - Hy_]^x =x'=0
*
(2.55)
J s y + j s y = ~[H z+ + H'z+ ~ H*z- ^x=r'=0
where h? = 0 is used. Let Kh = (H°y+ + H ‘y+ - H°y_ ^x=*'=0, and note that the
transverse current density
+ yO j js identically zero as obtained in [11- 12].
Thus we have
J sz+ J<
L = K h + (hy+ - Ay_)|x=x'=0
(2.56)
(H°z+ + H iz + - H ° Z-] x = x '= o = 0
Substituting (2.54) into equation (2.56), we obtain
fsz
= K h - J % + (— + — ) 5 f[/(2n7t + 0 ) f ne - / 2n n / - j ( 2 n n -Q )g neJ*my']
lt +
j
(2.57)
It- n=l
j
n=oo
= Kh ~ J% + (— + — ) ^ [ Gn cos 2n7ty' + Fnsin 2nny’]
lI + I1- n=l
where
G a = y(2n7t + 0 ) f n - y ( 2n7i - 0 ) g n
Fn = (2n7t + 0 ) f n + (2n7t - 0 )gn
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(2.58)
Note that the right-hand side of equation (2.57) is the Fourier series
representation o f j \^ which is periodic in y ' with period 1, so the Fourier
coefficients are
K h -J L ^ l^ lw w
(2.59)
(— + — )G „ = 2 j ^ n j l , ( y ’) c o s 2 im y ,d y ’
(— + — )F„ = 2
it
M*—
j % ( y ’) s m 2 n x y ’d y '
n = 1,2,3
Using the strip geometry on the “fast-variable” coordinate shown in Figure 2.2, the
first equation o f (2.59) can be rewritten as
(2.60)
since J ^ + 7^ = 0 in the gaps. Then equation (2.60) becomes
M
(2.61)
' w
23
.. i
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Figure 2.2: Cross-section of a strip grating on “fast-variable” coordinate.
The second equation o f (2.59) gives
Gn =
-b /ip
[J°sz + fsz(y')]cos2My'cfy'
2(1 |i_
where u = ---and since
H+ + H_
„
(2.62)
is .y'-independent, the following identity has
been used:
\ ^ P/szcoslvm y'dy' = -
^0
J%cos2mzy'cfy'
Jbf2 p
Similarly, from the third equation of (2.59) we get
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(2.63)
p®/2|> .n
Fn = V-a]_m p Jsz(y ')v n 2 M y 'd y '
(2.64)
To evaluate Kh, Fn and Gn explicitly, the zeroth order z -directed
surface current density
2.5.2
must be evaluated.
The Surface C urrent Density
According to the boundary condition on the strips, the normal component
of the magnetic flux density
b
equals to zero, i.e.
(/»2 + # 2 + / & W = o = 0
on strip
where hx can be found from equations
(2.33) and (2.39)
(2.65)
as
h°x = - i - ^ 7 A z - j Q A z)
|i qy
(2.66)
Thus equation (2.65) becomes
d
( - v , A Z- jQ A2)|x=x'=0 =
°y
+ ^/x)|x=x'=0
1
1
—
~
on strip
_ p0
D xi\x=x"=0
Obviously, the solution of equation (2.67) is
25
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(2.67)
eP
Az,x=x,=o = c iejQy' +
(1 - ej Qy')
where c\ is a constant to be determined and
on strip
(2.68)
can be found from equations
(2.51) and (2.52) as
Aa„ x..o = "ft - j Z n ‘ J l m y + J
n=l
(2.69)
n=oo
= Z t(Sn + f n)sin lx™y’ + A f n “ 8n) cos 2n7^ ' l
n=l
Note that (fn + g ^ and (f n ~gn) can be found in terms of Fa 111(1 G n from equation
(2.58), they are
(f + s ) =
2mt Fa +jQ G n
U n gnJ
(2n7t + 0 )(2n7t - 0 )
(2JQ)
re - a \ = - ^ F n - ^ n T tG n
U n gn'
(2n7t + 0 )(2n7t - 0 )
Substituting F n and G a in equations (2.62) and (2.64) into equation (2.70), then
(2.69) becomes
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dO
^x/V=o
— {cie-fiy' + — -^-(1 - eJQy')}
V-a
jQ
2k
=
0 4
J-*/^
♦a
o o i
i
(n + 5 )(n -8 )
(2.71)
5=i
(n + 8 )(n -5 )
n=oo
y'S cos InKy' sin 2 nTty'
- JJ-*/2p
L V ^ o " ) “s Ly' T ( 7n +«5 v) ( n" -~5 ) : r ■»'
where 5 = 0/2 7t.
To solve equation (2.71), we separate
into the even and odd parts:
(2.72)
Substituting (2.72) into equation (2.71) and after a little algebra, comparing the even
and odd functions on both sides, we get
pU
2k Dxi\x,=a
Va
jQ ~
pO
+ (ci~
^xiV-o
) cosOy'}
fi
5=1
n=oo
- r '/ s o '^ z
(n + 5 )(n -5 )
y'5cos2n7ty' cos 2n7ty "
<fy"
(n + 5 )(n -5 )
27
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and
pO
27C ,
a xi\x*=Q
— { (/C l------r — )sm 0y }
0
=Jo
5=1
V
(n + 5 )(n -5 )
<?■»)
fA/2P 0Of 'y ° n sin 2n?ty' sin 2n7g/" ,
J0 J s z W ) ^
(n + 5 )(n -8 )
^
The even current density fj£ and odd current density fj£ will be found below.
2.5.3
The O dd C urrent Density
Taking the derivative o f equation (2.74) with respect to y ', we have
pO
2rc f
£>x/|x’=o
— KC1------X—) cos03/ }
Va
JQ
Jo
5=1
p -w )
(n + S )(n -5 )
. Cbl2p ;Oo/ ..„xIl^ 0 n2cos2n7^, sin2n7ty"_t „
+ Jn
Jo
Jsz (V ) 2 *
5=i
T?
TiTv
sT
y 5 ( n + 5 ) (n - 5 )
^
and then subtracting equation (2.75) from equation (2.73) gives
^Xi\x'-n
Min o.
n 00
— = - \ bQ2Pfsz O'") £ c o s 2n7iy 'sin 2n;iy ’d y”
6/2 d
= -JL
a
1
n=oo
n
^ °
n=ao
n
=oo
y'SO'")r { £ s m 2 n jiO '+y") - £ s in 2 n x 0 ” - X W
2
n=l
n=l
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(2.76)
Using the formula
j
x
sinnx = —cot—
n=l
2
2
n=oo
0 < x < 7t
(2.77)
and after a little algebraic simplification, equation (2.76) yields
nO
^xi\x'= 0
_ 1 rW p oo „
= IJ o ^ C v " )[c o tn C y ' + / ' ) - c o t 7 t ( y - y w
4
(2.78)
= 1 f ' ^ ( T ) ------------------------ dy"
2 ■'°
cos 2-izy' - cos 2xy "
Note that equation (2.78) is exactly the same as DeLyser's equation (C.43) [12], thus
we have the solution for
y g (V > -—
i
^a
2.5.4
:
(2.79)
Jsin 2——~ s in 2tty'
V
2P
The Even C urrent Density
Substituting (2.79) into equation (2.73), we get an integral equation for
Q J % + jg )
29
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A * £ (? ■ )]■ E
S=i
r e
Jo
pO
(n + 8 )(n -5 )
pO
2;c . ^ x/Ijc’sO ,
"xrV=o.
_ ,,
= — {—
+ (C1--------- ^ - ) c o s e y '}
Va
JQ
_ 1*6
Jo
/
2
(2.80)
sinny"______ r^ ° cos2 nfly'sin2 m t y " „
»a
I ■ z b*
JsurV
2p
surfly
(n + 5 )(n -8 )
^
Use the integral representation of a Legendre polynomial [25]
f6*/2p cos(2n +1)/
7C
Atc,
J
—r = = = = = = d t —~ P n(cos )
• 2 blz
./sin
r
V
2/?
■ 2 ,
sm
*
2
, , on
(2.81)
^
where p n is a Legendre polynomial of order n. Then equation (2.80) becomes
Jo
2i t r
Q,
o
.
[ e ia ^
+^
h=i
(n + 5 )(n -S )
,c pO
l-c o s 9 y '
J °xi\x'=Q n ” cos2nny'
_
_
] _ _ ^ [g _
_
(2-82)
....
s . (A)1
where ^ ( A ) = P n _ i(c o s A ) -/, n (cosA) = / >_n ( c o s A )-/)n(cosA) and A = bit/p.
Letting the right-hand side of equation (2.82) equal to f ( y ' ) 7 an alternative
representation o f (2.82) is
30
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or in integral form
r t 2 / a + ^ c v " ) ] Si i cosf
° s 2-S
" )f v
Jo
n ( n^+ 5 ‘)(n
(2 84)
= - 47t 2J0 Jjj 2f ( y ' \ ) d y ’\ d y '2 + A.\y' + A.2
where Ai and A2 are constants. Since the left-hand side of equation (2.84) is an
even function, the right-hand side must be an even function too. Thus we have the
condition o f Ai = 0.
Now, apply the operator (d2ld y '2 + 4 n 25 2) on both sides o f equation
(2.84), we have
(2.85)
where A 3 is a constant. Substituting fly ') which is the right-hand side o f equation
(2.82) into equation (2.85), after some manipulation we get
31
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271
e ^x»v=0
- f c 1+ —
*
,
B xi\x'=o n^ f c o s 2 n 7 ^ '
2,
> - _ _ _ ( £ _
_
S l(4 )
I B»(. + « X n - « ) S- M ) * A i
2y
(Aa
n^ ° ^ n ( A )
n^ ° 5 n (A )
n=l
n=l
a=lH2 - 5 2
„
„
R§ (A') = P_§ (cos A') + P s (cos A'),
P§
+ X
T~+ S
n
(2 '86)
A
r - c o s 2n7t/ } + A 3
n-4
Now, in Appendix A it is shown that
y ° S n(A) _ k R 8 (A ’)
n^l n 2 ~ 5 2
where
28 sin57c
1
52
is the Legendre function of
noninteger order 5 , A = b n /p , and A' = a n / p . Furthermore, from (A.6) and (A.7)
n=l
nz
n=l
nz
_ lim rV" ^ n ( ^ )
= c -> o l2 *
n=l n 2
2
~
•S'n(^) „ o n_ /-i
2rcos2n 7iy ]
n=l n 2 - ?
= 27t2/ 2
where L’Hospitai’s rule has been used twice. Thus equation (2.86) reduces to
32
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f'* ??\
(2.88)
_2k
\ia Cl
j Dxi\x'=o r7t R8 (A')
u
2sin57c
1
8
3
To determine A 3 in equation (2.89), we compare equation (2.89) with
equation (2.82) by letting 8 -> 0, we find that A3 is equal to zero because
lim
R5 (A ')__ 1_
5-*° 2sin§7E
28
_ lim rric/>_5 (cosA')
1 , , rn P 5 (cos A')
1 „
.........
= ( - - In — ” S-A') + ln - + 005 A ’)
2
2
2
2
=
0
Finally, we have the integral equation for (2 J% + j ^ ) :
0
0
n=l
11
(2 M )
=
\ia
li a
= 2k,
Cl
2sin87c
^ > '= 0 tc/? s(A ')
2rt
2sinS7t
8
1
8
Note that this integral equation is similar to equation (C.35) in [12]. The solution for
equation (2.91) is
33
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2 J%+j°sHy')
- 8 ft
,
ii inrc;«2 A \
Hflln(sm
- ^ y . o rlt jfe(A ')
CI
2k
2sin87C
1„
5
c o s tt/
(2.92)
I . ,A
^ ~
Jsin2 — - s i n 2ny
Next, we need to determine the constant ci in equation (2.92). The
zeroth order surface current density is
Substituting equations (2.79) and (2.92) into (2.93) gives
;0 f . . i \ _
,0
JSZV / ~ ~ J sz
|
2 fP
sinity'
I-=
y]s-sin2ity'
^Xt|jC'=o 7tj?s (A')
Ha lns
2sin5Tt
4JCC1
cos7ty'
;
; =
^ a lns ^Js-sin2Tiy’
(2.94)
cos Tty'
5 J s - s m 2W
where ■J= sin2 (A/2) and J% is constant over y ' as seen in (2.24). Integrating
equation (2.94) with respect to y ' from - b / 2 p to b/2p and using equation (2.61),
then we have
34
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Kh ___ ~4 k ci | 2J ^S.y-o tc /?§ (A')
Ha ln s
f^a ln 5
2 sin57t
i
8
(2.95)
so that
1
- ^ lnV L i y ^ /'*,°0r* /?g (A'>
-4tc
2tc
2sin8it
5
(2.96)
Substituting c\ into equation (2.94), the total zeroth order surface current density in
the z -direction is thus
[J°sz+jsz(y')]e-^y'
=
—1 A?®
»mg '
+^
y j s - sin2 w
_ co s^
]e. j e y
(2.97)
y j s - sin2w
The zeroth order boundary layer magnetic fields are shown in equations
(2.53) and (2.54), where f n and gn can be expressed in terms of Fn and G n by
equation (2.58), they are
c _ Fn VGn
n
2(2n7t + 0 )
=
n
(2.98)
Fq+yGn
2(2nx - 0 )
then equations (2.53) and (2.54) becomes
35
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h i = -5— Y r ^ — iS±ILe +(2rm + 0 )x 'e -y'2n7t/ +
1
H± n=l
2
F n ~f; / G ne y (2n7t - Q ) x ' e j2zmy']
+ , n=o r
p2
/,0 = _£L Y [ ! £ n l Z £ j L e +(2mt + 0 )x 'e -y2n n / +
* H± S=i
P S?)
2
—•/^ rte +(2mt -0)x'gy'2im /]
2
where Fn and G n can be represented in Legendre polynomials by substituting
equations (2.97) and (2.94) into equations (2.62) and (2.64). And the integral
representation of a Legendre polynomial is shown in (2.81). After some
manipulation, we have Fn and Gn as
Fn = -fi2,V , 0S„(A)
(2. 100)
Gn = ^ K n ( A )
where
7?n(A) = / >n _i(cosA) + / >n(cosA) = / >_I1(cosA) + />n(cosA).
Note
that
when substituting (2.100) into equation (2.99), the zeroth order boundary layer
magnetic fields are expressed in terms of Legendre polynomials.
|
36
i
i
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2.5.5
The Zeroth Order Electric Fields
Following the similar procedures, we assume
= -(Vr - - jQay)<& in the
First equation of (2.17), where <t>(x',y') is periodic in y ' with period 1. Then we
have the surface charge density on strip
n 0 . « 0 / . . n _ o , r-0
si*1* /
* s LV /
^ S O vi\x'—Q r '
'J s - s i n 2ny'
47tea c2
*s
1
^n,y
cos7ty'
i ■ -~
'J s - s i n 2 ny'
(2 . 101)
2J £ q £ y i\X'=o
Ins
R$ (A') _
cosny'
2 sin57t
5 J s-sin W
2 sin57t
5
where c i js
4tte a
2tc
Then the total zeroth order surface charge density is
[P? + P ? C v ') ] e '^ '
(2.103)
- n * F°
sin Tty'
cosny'
~ l2 ea^yi\X'=o I
— ----- + Ke I
Je
VJ - s in 2n y'
y /s - s in ny'
where ea = ^ ~ ~
and K e = [s +(£x++ E%+) ~ s -£ ? _ ]x=r'=o • Finally, the zeroth
order boundary layer fields e0± and e0± are
37
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± £ | - K n j ^ y L j i e :p(2njt + Q)X' e - j l i m y ' +
eQ
x* -—
n=l
2
K n ~ =^ ne +(2im -Q )x 'e j2rm y']
2
n=SO
e°
ey±
_
n=l
r
(2.104)
_
—■ 9 g+(2rot +0)x' e-/2imy' +
2
.L n t Z ^ f l e T(2nff - 0 )x'e J2imy']
where
K„ = ^ - ( 2S „)
2s Q
(2.105)
where 0 n and <Pn are defined in (B.2).
2.5.6
Boundary-layer Grating Voltages and Currents
From the zeroth order electric fields in (2.104) and magnetic fields in
(2.99), the voltages and current on the surface of grating due to the zeroth order
boundary layer fields can also be derived. The voltage at a strip center due to the
component of the boundary-layer electric field (which we will call the boundarylayer voltage) is
38
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Vh
^
:
2za
i M
A ”> + J ^
4 sm 87i
1 [± - M
2
8tc
^ 1 ]
( 2 m
2 sin87t
while the current flowing on a strip is
pKh\x'=o
2sin(A 8) , JR^xi\x,=o
lh = — T 1----- [Jt Rs (A )------- ^ _ 2 ] + -------- 1— S5 (A)
2*
5
Ho
(2.107)
The transverse voltage between adjacent strips is given by V&(e~jQ- 1), where
JP sin(A'8 ) ^
Fe' M
l ^ 5 ) £ lv = o
(2.108)
The voltage Vfe and current //, are evaluated at .y' = 0. The detailed mathematical
discussions are shown in Appendix D.
2.6
The First Order Fields
Equations (2.19) and (2.20) and the boundary conditions in section 3 give
the boundary conditions for the first order fields, they are
39
i
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(Vr ' - j Q
ay ) x e l =
-
Vr Xe°
(Vr ' - y ’9 ay) - el = - Vr •e°
ax x (e+ - ei)|x=x,=0 = ~ a x x \Jl + - £i]|x=x'=0
(2.109)
ax ■(e +e+ - e -C-)jx=x'=o
= ~ ax •(e+E+ ~ £ —£^- )|x=x'=0 +
and
(V r' - 7 0 ay) x a 1 = ycos e° - V r x
( V r.-T Q a jd - ^ - V ^ A 0
5x • (H + h \ - H_ h- )|x=x'=0
( 2 . 110)
= —ax'([i-+H+~ M-_//^-)jx=x'=0
Ox x (h+ - h i )jx=x'=0
= ~ a x x [//+ —//_ ]|x=x'=o + «/i + js
Expand the first equation of (2.109) and (2.110) by using d/dz' = 0, e® 0, and
tPz = 0, we have
«*(“dy' 4 -y0 4) - ay^-4+&
(tax'^4 ~^
t4+;e 4)
obc
ay'
= a x ( ~ > p^x + ^ -e j) + a y ( - j ® l i h y
& '
- j-e $ )
3r
+ a* ( ^ - 4 - -|-e$)
ay
ax
and
40
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( 2 . 11 1 )
a x (- z ~ ;h \
ay'
- jQ h \) - a y ^ - j h l + a z { ~ - h ly - ^ - h lx +jQ h \ )
ox'
dx' ' dy
(2.112)
= a x C/o e e%+ JJ- hy ) + a v O e e$ hx) + a z (£ - hx - 4~ A?)
oz
dz
ay
ox
Equating ax and ay components, we have
^^7
4 -jQ 4 =-j®V>h°x +^-eO
dy
dz
(2.113)
^ ~ h \ - jQ h\ = yoe e%+
dy
dz
and
^ —4 = > H A°+ -?-*?
dx
oz
(2.114)
A -lj\ = - j aee0+^-h x
. dx
dz
Note that equation (2.114) gives the easy way to solve for e\ and h . Integrate
(2.114) with respect to x ' from ±oo to x ’, we get
£ (S 7e**)‘fc' = Ja v- + £ h<y+dx' + f
dz
for x' > 0, and
41
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(2.115)
for x' < 0
where g0± and hy^ are shown in (2.104) and (2.99), then equations (2.115) and
(2.116) become
2 e \t = -yea V {5 iLtZZlLe+(2n3t +0)x'e-j2mty' +
ji=I 2n?t ^“0
SLl—Z f l e T(2n)i -Q)x' ej2rmy'\
2n% - 9
'
(2.117)
_ ny ° , j _ KnJ-y'Lgx x(2tOT +Q)X' -y2ntcy' +
dz 2mt +0 '
j - ( K n - y L nx T(2nTC -Q)x'j2n n y’\
dz 2nrc - 0 '
'
Similarly, substitute (2.104) and (2.99) into the second equation of (2.114), we have
2 h \± = ±y<0 e± Y {
1
t£ l
n + j/K llg+(2n7t + 0 )x ' e -j2nny' _
2ntt +0
L n_t .yK n +(2mt-0 )x ' ej2nny'\
2m i -0
'
(2.118)
+ _L y (A(Ln. A 0 * ) g+(2rm + 0 )x 'e -j2wty' +
&
2n7i +0 '
■ £ .(E n + /G nx x(2ik -0 )x ' y2m t/\
az
2nrc - 0
'
Then the first order fields in (2.117) and (2.118) can be employed to obtain the
equivalent boundary conditions (EBCs) for the total z -directed fields.
42
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2.7
Equivalent Boundary Conditions (EBCs)
Evaluate (2.117) and (2.118) at the interface x ' = 0 , we obtain
24
V { Gn + U * .e-j2imy' + G n _./ F n y2imy'}
' o = “>
^
2 n tt+ 0
2n7t - 0
'
(2.119)
_ " y { — ( Kn +J-l?n) e -J2nny' + l . ( E }L~. J ^ ) e j2nny'\
dz
2mz + 0
dz
2n7t - 0
and
(4 +
h \- )|*'=o
= - > e flny { - - jK n-e - j 2™y + k n -V l ^ a gy2rmy'\
2n7t +0
2njt - 0
(2 . 120)
L y 0/j_(Fn~ j
&
_ -n) g-y2rwty' +
+■
/ G n) ;2n ^ ,\
2n7t + 0 '
d z K 2 mz - 0 '
'
where
G „ ± y F „ = • ^ « „ ( A ) + y f l “li.,0S „(4 )
F„±yG „ =
(2. 121)
K „± y
l„
=
L n + yK n =
2ea
R n (A )
± y 4 > '- » s »(A)
T ^ ^ n (A )
2e a
43
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Equations (2.119) and (2.120) will be shown to be the main contribution
to the EBCs for a semi-infinite strip grating.
2.7.1
EBCs - Electric Field
Take the ax component when expanding equation (2.13), we have
dy
where
e\
4t4 ~JQE\= -J°
(2 -122)
dz 7 + dy
is independent of y ' and e® = -E?z, and e\ - -
e
\ on the strip. As we
know e0 = 0, then equation (2. 122) becomes
E°y i - j Q e \ = j(j)B0xi
at x ' = 0
(2.123)
or
J
E%\X'=o ~
ez|x'=o ® Bxi\X'=0
(2.124)
where the incident fields must be considered. And ^ x,=0 can be found in (2.119) by
inserting the constants in (2.121). Appendix B shows that e\\x'=o equals to
44
i
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aj2n8y'
p> jr
ArA^ a + - ( - ^ ) ] f t ( A ,) +
dz en
8sm07t
(2.125)
Substitute (2.125) into (2.124), we have
e1
'r ^ “°
OZ
Ea
4tt5 R$ (A')
(2.126)
—_ r l
—
■c 'Z |x '= 0
or
U<»Kh \ia + -L ( (^X±y
) ] S 5 (A')
-1
_ “ '~ " nr’a ' d z \ a
E m -0 = --------------------- —-----------z|x-°
47t5 R$ (A')
(2.127)
where as before the p$ are the Legendre functions of order 5 for |S| < 1/2.
The average field ( without the phase dependence e~ jQy' ) is given by
[H - 12]
Eav = (l?nC+ £?) + P E l
(2.128)
45
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If we are interested in the equivalent boundary conditions for z -directed fields.
Since the zeroth order fields are continuous, only the first order and higher order
fields contribute the averaged fields of the EBCs.
E «K - o= p £ * "
o
Therefore we
have
and
K e ( y , z ) — [5 -t-(£r+ + £ * . ) —£ -£ ^ c _ ]r = x ’=0
= [e + £ j - z - E ^ ^x=x'=0
(2.129)
K h (y .z ) = [(.Hy+ + H°y+)- H ° y _ ] x=x'=0 = [H $ - H } ^ = x '= 0
Then, the desired EBC for the z -directed electric field is
Ez\x'=0
P
S k (A')
47t5 R $(A')
2.7.2
.
“
1 a
1
^
z a dz
(2.130)
1
EBCs - M agnetic Field
Now, we need to evaluate equation (2.120) in the gap to obtain the
equivalent boundary condition for the z -directed magnetic field. Take the ay
component of equation (2.31), we have
J\y = - j ' s y - m l - H i V -o - i h l - h i
(2.131)
Taking the average value of equation (2.131) yields
46
i
_
__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .._ _ _ _ _ _ _ _ _ _ _ _
___
____
Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
4 y = - ( f l l _ f li - V = 0
(2 l3 2 )
where we use j j\y d y ' = 0 and j^hldy' = 0 which are mentioned in (2.7). Thus
equation (2.131) becomes
-J/ sl y ~~W(hz l+ -A"z-J\x'=o
1 \
<2133>
Substituting (2.120) into (2.133) yields
t ?3 e~J2imy' eJ2imy'
|i [
+
1<p" +
i
-Jsy= > e»
n=«
°
eS
y2iwy'
e -j2 sa q f
2n3C- 0
2n7t +0
n
(2*134)
&
2n7t - 0
a_
n^ ° r e~J2any'
^ dz h
2nn +0
2n7t +0
a
eJ2imy 'ln
2nrt- 0
n
where Q n and (pn are defined in terms of Legendre polynomials in (2.105) or
0 .2).
As we know, j\y + j \ y = 0 in the gap, or
47
. _ L —
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/sy = - ( 4 . - h\_ Jjx’sO = - J sl y = ( H i - H\_ V=o
(2-I35)
And the zeroth order y-directed magnetic field need to be continuous in the gap, we
have
( # y ++ Hy^ + hy+)]jr,=o = (Hy_ 'l' hy_ )|x'=o
(2.136)
Taking the average value of equation (2.136), we obtain that //^ + + 4 . - 4 - = °
which is equivalent to K h ~ 0 and hy+= hy_. From the boundary condition, the
normal component o f electric flux density ~D must be continuous in the gap, this
yields
[s + (■£*■+ + £r+) —S - ■£*_ )x=x'=0 = (s —ex- —£ + er+ )lx=x' =0
1
1
= Ke
in the gap
Making use o f (2.135), equation (2.134) becomes
48
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(2.137)
r-
rO
a
p—jLvm
pjinny'
^ x / | x ' •=l zot e~J2my'
Y [ — ----- + - ---- —I©
2n7i +9
2n7i - 0
n
gjtimy
I\z « ej
2iwy' ^-yziuiy
e -/2iBiy'
- a K e 2 J - ---------------------2nrc+0
n=l 2n7
4 t -0
(2.138)
n
d , B° x i\ * - = o ^ (re - n ™ y , ej2 m y'
> > '= ° '& (~ 7 r )1n4 l' {
CD
Ka
4k
n+?~
n^ °
e /Z iH ty '
/frl
n -5
Tn^-55 ~
]•
e -jin n y'
^ nr +o
^ r ]* " (4)}
From the second equation o f (2.113) when evaluated on both sides of the grating, we
have
^ 7 Aj+ - jQ h\+ = j o e +e§+ + J j / # +
(2.139)
4dy'r f i - ~ JQ h\_ =y©e_e§_+oz
Subtracting the first equation from the second equation in (2.139) yields
h \j- m l- h \j
(2.140)
When evaluated in the gap, we have the following conditions
49
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!
h \+ -h \_ = - ( H zl +- H \_ )
(2-141)
- hy+~ hy_ = 0
Ke —(e - ex_ 6 +e x+ )|x=x'=0
Note that the right-hand side o f the first equation of (2.141) is independent of y ' .
Substitute (2.141) into equation (2.140) yielding
Q K e = -Q (H z1+~
h
\_)
in the gap
(2.142)
Then substitution of (2.142) into (2.138) with the aid of (C.7) and (C.8) yields
( h \+ ~
(2.143)
R°
—r in o IT®
_ ^ f jal,x's^Mr ^5 (^) i
- L/© Q^^/i\x'=0
a ( H
y‘\x - Q dz
7t5 R$ (A)
Thus as stated previously, the EBC for the magnetic field is mainly contributed by
the first order field, i.e.
( H t - H 7 ) |*=x'=o
=
(2.144)
~ f i \ - ) |x=x'=o
50
1
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2.8
Results and Conclusions
From equations (2.130) and (2.144), we then have the EBCs for a semi-
infinite metal strip grating with a phase shift 0 at the grating surface
E^
[W
) ]U a
(2.145)
rr+
Hz
where
r r - _ P r ^ S (A) lr
r- .
5
Hz [ _ - - - ][/© e a ^ v /y = o ~ ~ (
*6 £ s ( A)
dz
S5 (A') = £ -5 (cos A' ) - £5 (cos A'),
2 |i , (!_
A = b n /p , A' = a n / p , u = — ±— ,
a | i + + (i_
11a
)]
£g(A ') = £ _ 5 (cosA') + £ 5 (cosA'),
s +g_
£a = — -— , and £„• and Bxi denote
2
y
total fields including the incident fields. The subscript “ i ” will be omitted in the
following expressions.
The obtained results will be compared by previous approximation [4, 10]
for the strip grating in the air, i.e., |i+ = |i_ = |i0
e + = e_ = eo - From [4], the
equivalent boundary condition can be expressed in terms of our notation as follows:
E+
z + E~ = ./* o /o (5 )(tfJ; - H ~ ) + / q ( 5 ) £ ( £ ^ - E ~ )
(2.146)
for CGS units, or transfer to MKS units as
51
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E z+ + E Z = j t o / 0( S ) J — ( / / t ~ H V ) + /0( 5 ) - |- ( £ + - E ~ )
Veo
'
oz
(2.147)
where ko is the free-space wavenumber and /0(5) is defined in [10] which is
(2I48)
Note that Ez = E z = E z at x = 0 plane, then (2.147) becomes
which agrees with the first equation o f our EBCs in (2.145) with (J.a = fi0 ^
s a = eo- Similarly, [4] also gives
3
+ / l(5) — ( H i + H x )
dz
H l - f G = -jk o li(5 \ + £ ? )
yn0
(2.150)
where /j(5 ) is defined in [10] which is
(2151)
Also Ey = Ey = E y and
= H x for n + = n_ = p Q. Thus we have
52
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715 Pg (A)
(2.152)
dz
which also agrees with the second equation of (2.145) with e a = eo and M-a = |a0.
Therefore, we see that the derived EBCs using homogenization analysis to the first
order are identical to Ivanov's results [10] which consider the fundamental mode
only with quasi-static approximation.
If we let the phase shift 0 equal to 0, i.e., 8 = 0 in (2.145), since
lim P -8 (c o sA p -P g (cosAp
5_>0 8 [P_§ (cos A p + Pg (cos A p]
=
(cos A P - | - P g (COSAP]
(2.153)
= - In cos ^ (^ -)
and
lim Ps (cos A) - P_g (cos A)
5_>0 8 [Pg (cos A) + P_g (cos A)]
=
58
(“ s A> -
(“ s 4 >]
(2.154)
= In cos2 ( y )
where — p z ( u ) = ln -^ -^ and — p s(w) = -ln -^ -^ - have been used [25]. Then
68 6
2
68
6
2
the EBCs with zero phase shift become
53
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and
A
i 3
H t - H I = - In cos2 ( - ) [ > e aEy ------ — H x]
K
2
n a dz
(2.156)
which are identical to DeLyser's results [11-12],
The equivalent boundary conditions (EBCs) to the first order considering
a uniform phase shift for a semi-infinite strip grating lying on different media have
been derived based on the homogenization analysis. Comparison with Ivanov’s
results [ 10] shows that our homogenization method solution is less cumbersome and
more straightforward and our results generalize those of DeLyser [11-12] by taking
into account phase shift between strips. Having obtained the EBCs, we will use them
to investigate the propagation characteristics of the slow-wave grating structures on
grounded substrates for planar microwave filter applications. This will be the
subjects of next chapters.
54
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CHAPTER 3
SURFACE WAVES ALONG A METALLIC STRIP GRATING
3.1
Introduction
Periodic slow-wave structures have been well known for their
transmission-line application such as filters, antennas, and other components at
microwave and millimeterwave frequencies for many years [26-45], To analyze
these periodic structures for circuit applications, the propagation characteristics of
guided waves must be investigated. In this chapter, the propagation of surface waves
on a semi-infinite periodic strip grating lying on a grounded dielectric slab (Figure
3.1) will be determined in an efficient and accurate approach.
Several authors [46-59] have investigated the propagation of surface
waves on the periodic structures using analytical or numerical methods. But most o f
their work had assumed that the direction of propagation is perpendicular to or
parallel to the strips or had additional restriction in practical application. Recently,
Bellamine et al. [60] investigated the surface waves propagating at an oblique angle
with respect to the strips over the periodic structure, as illustrated in Figure 3.1,
where the equivalent boundary conditions (EBCs) based on the homogenization
analysis derived in [11-12] were used to simplify the analysis. Although Bellamine’s
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x=d
x=0
m
Ilf;;
Q ound Plane
(a)
(b)
Figure 3.1: A periodic array o f strips loaded by a dielectric slab over a
ground plane: (a) cross section; (b) top view.
56
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analysis is analytically simple and computationally fast, its application is limited
since the required phase shift over a period length was assumed zero in the
derivation of the EBCs in [11-12]. In general, there exists a phase shift between
adjacent strips in the application of several periodic slow-wave structures.
Without loss o f generality, the fields along a grating are assumed to have
a uniform phase shift 0 between adjacent strips and propagate at an oblique angle
with respect to the strips, as illustrated in Figure 3.1(b). Then the modified EBCs
derived in chapter 2 will be employed to investigate the propagation characteristics
of surface waves along a strip grating. Comparisons will be made and show good
results.
3.2
Determination of the Propagation Characteristics
3.2.1
z -directed Propagation Constant (3 vs. Phase Shift 6
Figure 3.1 shows the geometry of strip grating on a grounded dielectric
slab where the strips extend to infinity in the z -direction; the width o f a slot in the
grating is a and the width o f a strip is b . Based on the derivation of the equivalent
boundary conditions in chapter 2, the grating period p is assumed much smaller in
comparison with the wavelength of radiation, i.e., koP «
1, where ko is the free-
space wavenumber and p is comparable to the substrate height, i.e., p / d will not be
too large. Also, in the following discussion of the propagation of surface waves, we
adopt the hypotheses made in [60] for the periodic structure of Figure 3.1.
57
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The EBCs at a strip grating lying at the interface between two different
media (Figure 3.1) derived in equations (2.145) can be rewritten as ( in MKS units
and the time dependence eJat )
(3.1)
where
u = 2^ i
H2 + *l,
-
Efl =
'
g2 4~g l
2
S8 (A) = F*_5 (cos A) —P 8 (cos A ), A'
= o k / p
Rs (A) = p _5 (cos A) + p 8 (cos A ),
,
A = bit/p, the P 8 are Legendre
functions o f noninteger order 5 which is defined as 5 = Q/2n, and the superscript I
or II denotes the fields in dielectric region ( / ) or half-space region above the slab
(II). Note that E z and H z shown in (3.1) do not contain the term e~jQyfP
according to the derivation in Chapter 2.
Assume the waves propagate in arbitrary direction on the surface o f the
slab as shown in Figure 3.1(b), the fundamental Floquet-mode electric and magnetic
fields can be denoted as
E=E(x)e-jPze~jyy
H=H(x)e~JVze-jyy
(3.2)
; y =9/p
58
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I
where 3 and y are the propagation constants in the z -direction (along the strips)
and y-direction (transverse to the strips) respectively and the field quantities E and
H are functions of x only. The higher-order Floquet modes (boundary-layer fields)
concentrated near the grating surface can be ignored when applying boundary
conditions at the ground plane and do not explicitly enter into the EBCs in (3.1).
Then substitute equation (3.2) into the source-free Maxwell's equations, namely
f V x E = —/cd \iH
[ V xW = j a s E
(3-3)
in both the dielectric region ( / ) and the half-space region (II) in Figure 1. And “V”
in equation (3.3) is the regular del vector operator. Using the following substitution
dE
dy
~jy E
dH
■--jyH
dy
dE
dz
(3.4)
~3E
dH
■ -J0
dz
to expand (3.3), the tangential components of (3.3),
and Hy. can be represented
in terms o f Ex and Hx, we obtain
59
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E r ~ 72"[®Ja(^zY -5yP)H x~y(ayY + a z 3 ) —
kj*
dx
—
1
d H
Hr = -y [-ffle(azy - a y P ) E x - y ( a y Y + a z P )— -]
kf
dx
(3.5)
where k \ = p 2 +Y 2 = p 2 + (^ ) , and Ex as well as Hr satisfy
region I ( 0 < x < d )
(£
+ h2)H/ °
(3.6)
h2 =co2 ^ o e i - ^ r
and region II ( x ^ d )
(3.7)
; Re(q) > 0
where q must have a positive real part for a proper guided mode.
Obviously, the forms of the solutions in region I are standing waves.
Take into account the boundary condition at ground plane x - 0, the solutions for
equation (3.6) are
60
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cos(hx)
1cos(hcf)
x
Hx = IIjSin(hx)
x
sin(luf)
(38)
and the forms o f solutions for equation (3.7) are exponentially decayed waves in xdirection, they are
Ex= E 2e - ^ x- ^
(3.9)
Hx= H 2e - <l(x- </)
where E i, H i, E 2 ,and H2 are constants.
Now, the boundary condition at interface x = d must be matched. By
inserting equations (3.8) and (3.9) into equation (3.5), the tangential electric field
on the slab are
E f l w “ 7j [ a n ( 5zY - a y P ) H i + y ( a y y +az 3 )Eihtan(hcO]
Kt
l
_
= 7rt© ^(5rY -5 y P )H 2 + j ( a y j + 5zP)E2q]
(3l0)
kT
Equate the az and ay components in equation (3.10) by making E ^ a * = E r ^ * - ,
we have
61
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f cony H i+ y P h E ita n (h i/) = Q(iy H 2 + y ‘P q E 2
{ -o (.iP H i+ y h Y E itan(ht/) = -<DjipH 2 + yqY E 2
^ *
Simplifying equation (3.11) results in the following conditions
f Hi = h 2
(312)
| q E 2 = hE[tan(ht/)
Let us now substitute equation (3.2) into the EBC at x = d for the
tangential E -field in equation (3.1), we have
] •O
Va(Hy1e-JV* - Htye-Jfr)
(3.13)
+ — | - ( e 2Ejc7
e a oz
or
(3.14)
0 |i ( H y ~Hy)
where M-a =
= Ho for nonmagnetic materials, s fl = -g l+ -g2, A' = a % f p , and the
parameter I q(5) is defined as
62
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/ (Sx
0
P r? -5 ( c o s A ')-P 5 (cosA')
2tiS P_8(cosA') + Pg(cosA')
- j u
s m
(3,5)
2iz5l R5 (A t)1
From equations (3.5) and (3.12), we have Er and Hy at the interface x =
d as
Ez\x=d
= 4 ( o f i y H i+y‘P qE 2)
kT
= 4-(ffl s 2 p E 2 + yqy H 2)
(3.16)
kT
Hy\*=d- = Tjt® e iP E l-y h Y H ic o tM ]
kf
where Er and Hr in equations (3.8) and (3.9) are used. Then substitute equation
(3.16) into equation (3.14), we have
T [o HY Hi +ypq E 2] =
kj
7
2
P s 2E 2 +yqy H 2 - a p e t E i +
kf
(3.17)
jh y Hicot(hflO]- — ~ — (e2E2 - e {Ei)}
E2 +6!
Use condition (3.12) to simplify equation (3.17), after some arrangement we get
I
I
63
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Aronoy {1 + ^ p [ q + h cot(h^]}Hi
(3.18)
= -y'P Ei tan(hrf){h +
-------- 2—][en cot(hd) - — ]}
S ri+ e r2
q
2
where % = yj\x0/ e 0 is the intrinsic impedance in free space, e t = s r lso> e 2 = e r 2so>
and ko is the free-space wavenumber.
Similarly, substitute equation (3.2) into the equivalent boundary
condition for the tangential H -field in equation (3.1) at x = d, we have
W / e - J f r - Hle-JP* =
ea E y e " ^ + — ^ ( B x e " ^ ) ]
V-adz
iz5 R§ (A)
(3-19)
or
H? -
Hi
= 2 /, (6 )[-y *0.££1L|££2.E>,- yp HJ
at
x=d
(3.20)
where A = b n / p and ^ ( 5 ) is defined as
/j(5 ) = P ^ - 5 (cos A ) - P s (cos A)j
27t5 P _5 (cos A) + P§ (cos A)
P f 58 (A)
27t8
/? s (A )
With the aid o f equations (3.8) and (3.9), equation (3.5) gives |-t as
64
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(3.21)
H zfr=«/+ “ T F H ® Y E 2 E 2 + 7 p q H 2]
kT
(3.22)
= i [ - ® Y e 1Ei -yhP H icot(hJ)]
kT
Substitute equation (3.22) into (3.20) and use (3.5) for Ey and Hx, we obtain
|j~Y E 1tan(iu0[srl cot(hrf) -
- / i ( 8 )(er l + e r2)h]
(3.23)
= -y'P H i {q + h cot(hc0 + / t(5 )[2 k j - Ar§(e rl + e r2)]}
where the conditions in (3.12) are used.
Now, combine equations (3.18) and (3.23) to eliminate the common term
Eitan(h</). For nontrivial solution of Hi, we obtain the dispersion equation for the
surface wave propogation on the periodic strip grating
(2 y 2(1 + M l [ q + h cot(hrf)]} {eMcot(kO 2.
= - p * {q + h cot(lu/) + 1{(5 )[2 kj- {h + ^ [ * 5
2
e r l + s r2
q
- /j<5 ) te ,,+ e r2)h}
(erl + e r2)]}
co«M ) - ^ S ] }
q
For simplicity, let gr2 = 1 and e r i - e r in Figure 3.1, then (3.24) becomes
65
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(3.24)
k2y 2 {1 + / # ) [q + h
2
}{e r cot(hii) _ h _ /i(5 )(1 + g r)h}
q
= - p z {q + h c o t M + /i( 5 ) [ 2 A : f - ^ ( i+ e r )]}
{h + /o ff).
2
(3.25)
2 fPh
- -— —][er cot(hfO - - ] }
1+ E r
q
where some parameters are summarized as
k l = P2 +Y2
,
; Y = 0 /p
h i = kiQz r - k \
q2 = k2T - k l
P p^sCA'),
; A' = an/p
P r^sC^!
(3.26)
; A = bn/p
Dispersion equation (3.25) describes the z-directed propagation constant P versus
the phase shift 0 when surface waves propagate along the infinite array of strips on
a grounded dielectric slab.
Now, let us investigate the limiting case when the phase shift 0 -> 0, i.e.,
5 -> 0, then Iq and l\ in (3.26) are obtained with the aid of equations (2.153) and
(2.154) as
66
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Then equation (3.25) reduces to
q + h cot(hrf) + 1’\[2 k\- - k \ (1 + e r)] = 0
(3-28)
for TE mode, or
i'n
2 Ict
h + ^ r [* 0 -
2
l +er
h
r COt(hrf) ——] = 0
(3.29)
Q
for TM mode which are identical to those obtained in [60], This is quite evident
because the EBCs in (3.1) reduce to DeLyser’s equivalent boundary condition [1112] when 9 - > 0
and the dispersion equation in (3.25) obviously reduces to
Bellamine’s equation [60],
3.2.2
Propagation Constant k t vs* Propagation Angle <(>
Equation (3.25) gives the propagation constant parallel to the strips, 3,
versus phase shift 0 between adjacent strips. As seen in Figure 3.1(b), 3 is related to
the propagation constant k f by an oblique angle 4>, so is y . Thus we introduce
67
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P = ^7'C0S<j)
0
,
. .
(3.30)
Y s — = kj-sm^
P
where <j> is the propagation angle of waves with respect to the strips. From the
second equation o f (3.30), the phase shift 0 (or 5 ) can be reprsented as
With (3.30) and (3.31), equation (3.25) can be rewritten for k j in terms of <j> as
tan2(J){1 +
2
[q + hcot(h^)]}(er cot(bd) - —- / 1(5)(l + gr )h}
q
= -{q + hcot(h£/) + / l ( 5 ) [ 2 4 - ^ ( l + 6r)]}-
(3.32)
{h + ^ - [ k l - i ^ L ] [ 8r cot(hd) - -]}
2
l+er
q
where the parameters are now defined as
h —j-Jkj'~SrkQ
q=
,
1
— ■]
^ 7 sm<t> /?§ (A )
/ o (p ) = ';—
'■ P
^ 7’Sin<j) R § ( A)
R
/JAr7’Sin(j)
------2tc
, 5 = —
; A = b n /p , A’ = m / p
68
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(3.33)
Now, equation (3.32) shows implicitly the propagation constant k r as a function of
propagation angle <j) for the surface-wave propagation on a strip grating with an
arbitrary angle.
3.2.3
Characteristic Impedance Z c
The characteristic impedance Z c of a strip grating structure shown in
Figure 3.1 is defined as
Zc = y
(3.34)
where V is total voltage between grating surface and ground plane and I is total
current flowing on one o f the strips in z -direction, this definition is similar to the
characteristic impedance in parallel plate waveguide [61], The quantities V and / in
(3.34) are both due to the boundary layer and non-boundary layer fields mentioned
in Chapter 2. Then V is equal to
V = V E + Ve
fd
where Ex = E\
„
(3.35)
- e e ~ t i y in dielectric region and Ve is the voltage due to
cos(h a)
the zeroth order boundary-layer electric fields e% which is defined in (2.104). Since
69
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the boundary-layer fields decayed rapidly from the grating,
taken identical to the
V jl
Ve
in (3.35) could be
obtained in (D.4) which depends on the non-boundary
layer fields in equations (3.8) and (3.10), then
p ie 0 e S - E e 'x
Ve
in (3.35) becomes
J & * y /p
? ' = ---------- :---------—
Ss(A'1 +
2ea
4sin57t
.p E * x - d r i
J
2
57t
(3-36)
J^y/p
2sin57c
where e = e r so - For simplicity, take the strip located at >> = 0 in Figure 3.1 as
reference, and omit the factor e~J^z hereafter. Then we have kr£ = - q £ 2/h 2 and
from
Ex
and
Ey
in equation (3.10), we have
K e\x=d = [zoEx - e Ex \ x = d - ^ o E i ~ ^ E \
Ey\x=d =
1
-y[-<d l^o P H i+ /y Eihtan(hrf)]
kT
(3.37)
Substitute (3.37) into (3.36) with the aid of equations (3.12) and (3.18), then we
have the total voltage V on the strip located at y = 0
(3.38)
70
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where
ko Tlo hy {1 + y [ q + h cot(lu/)]}
T=
(3-39)
At 2
P ( h q - f [*o - - J 1 ][q e rC 0 t(h < /) + h]}
2
1 +Br
The total current / on the strip located at y = 0 is determined from
integrating current density due to nonboundary- and boundary-layer magnetic fields
over that strip, then I is ( omit the factor e ~ ^ z )
1= J
!
- H
V
* '* +\ y i „ -
(3-40)
~ lH + Ih
where current
is due to non-boundary layer fields Hy and If, is due to zeroth
order boundary layer magnetic fields hy . In calculating I H, equation (3.16) is used
along with the aid o f equations (3.12) and (3.18), thus
kr
e o n + 3 i - ( ^ a P _ ! 3 .) e a m i ^ s S b M
T
h
T
Y
where T is defined in equation (3.39) and If, is obtained in (D ll) which is
71
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(3-41)
in 27C
ft(A ) - ^ 5 > ] +
5
s 5 (A)
(3-42)
Ho
where
K h\x=d -
(Hy - Hy )]*=</
(3.43)
5*1*=* = (i0 Hxi^rf =
Hx\x=d
Substitute Hr and Hy into (3.42), after some manipulation we have
lb =
2lt
f t (A) -
+ ^ ~ f t ( A)
7
(3-44)
E«P + VT ' (£ Th ® " ' T )cot(W)!
(3.45)
5
where
Therefore the total current on the strip located at y = 0 is
^ i M
i i M & S)+ E m t l1 tR sW . ^
m
]+p j 2 S s W
72
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(3-46)
i
I
where Kh and T are defined in equations (3.45) and (3.39) respectively. Note that
in the expressions of
V
and
I
in equations (3.38) and (3.46), there exists a common
constant E i which will be canceled when calculating characteristic impedance
Zc ■
By observing equations (3.38) and (3.46), computation of characteristic impedance
Zc
depends on the determination o f propagation constant (3 in the transcendental
equation in (3.25).
3.3
Numerical Results
This section presents results of z -directed propagation constant P versus
phase shift 9 , propagation constant k j
characteristic impedance
Zc
versus propagation angle <{>, and
o f strip gratings when solving the transcendental
equations (3.25) and (3.32). Although the dispersion equations (3.25) and (3.32)
show that P (or k f ) ^ d
Zc
are frequency-dependent, once a small value of ko p is
chosen and the assumption for the homogenization analysis o f the strip grating is
satisfied, P (or k r ) and
Zc
can be treated to have a quasi-TEM character which
means that they are frequency-independent in this way.
To increase the speed o f computation and convergence, the Legendre
function />§ in equations (3.25) and (3.32) are represented by a hypergeometric
function [25]
PS (cos£) = F(-8 ,5 +1; l;sin2| )
73
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(3.47)
to simplify computer computation and 5 in (3.47) could be a real or complex
quantity.
3.3.1
Comparison of (3 with Weiss' Model
The z -directed propagation constant P obtained in equation (3.25) is a
function o f the phase shift 0 between adjacent strips. Now, we would like to
compare P versus 0 based on homogenization method with Weiss’ quasi-TEM
model for microstrip meander-line structure. In [57], each unit cell of the meanderline periodic structure is defined to contain two strips and there is a fixed phase shift
cp between successive cells. Within each cell, the potential on the two strips can be
either identical or 180° out of phase, they are called “even” mode for phase
difference between strips o f cp/2 and “odd” mode for cp/2 + 7t phase difference.
When compared to our derivation and notation for phase shift 0 between strips, 0
and 0 + 7t is equivalent to cp/2 in Weiss' definition [57] for even mode and cp/2 + n
for odd mode, respectively. Figure 3.2 shows the normalized z-directed propagation
constant versus phase shift per strip on the infinite array of parallel strips for even
and odd modes when solving the transcendental equation (3.25). Note that Weiss'
effective dielectric constant
[57] is equivalent to our (J3/yfco) ^ t^ie definition
o f the homogenization process.
The parameters used to determine Figure 3.2 are: £o<^=3.325e-3,
p / d =1.632, a j p =0.5588, and e r =6.5. Since k o p is chosen small enough, good
74
i
I
I
I
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5
Even Mode
D
e.
r
3 fj-
z0
.
,
.__________ .
50
,
'CO
__________ _
150
Phase Shift 8 (D ec )
Figure 3.2: Comparison o f (3 between quasi-TEM Green’s function method
and homogenization EBCs method for even and odd modes. The parameters
used are: k o d = 3 .3 2 5 e -3 , p / d = 1.632, a/p = 0.5588, and e r = 6.5.
Weiss’ result ( - -); present method (—).
75
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agreements are achieved between the homogenization EBCs method and Weiss'
quasi-TEM Green's function approximation. Due to the mode symmetry, Figure 3.2
is symmetrical at 0 = 90° as expected.
3.3.2
Comparison of k j with Beilamine’s Result
Bellamine has illustrated the normalized propagation constant hf/lco
versus the propagation angle <j> in comparison with Crampagne’s results [58, 60],
Good agreement was achieved only for the low propagation angle since Beilamine’s
analysis did not take into account the phase shift between adjacent strips (i.e.,
0->O ); in fact, 0 is related to propagation angle <j) by equation (3.30). Now, the
dispersion equation in (3.32) has improved Bellamine’s situation. Figure 3.3 shows
that equation (3.32) does improve Beilamine’s results when propagation angle
greater than 45° and good agreement is achieved in comparison with Crampagne’s
result [58, 60], The parameters used to determine Figure 3.3 are fco^=2.0944e-3,
p / d = 1.86, a j p =0.07, and e r =9.6 where the assumptions of hop «
I and p / d
be finite are fulfilled.
3.3.3
Comparison of Z c with Weiss' Model
The normal-mode characteristic impedance Z c of a grounded dielectric
slab covered by an infinite array of metal strips is obtained by the ratio of voltage in
equation (3.38) to current in equation (3.46). In those formulas, the z -directed
propagation constant P in (3.25) has to be determinted first. Figure 3.4 shows good
76
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T
2 0 r—
i-
0.00
O.’ O
0 .3 0
0.20
0 .4 0
3.50
Figure 3.3: Comparison o f normalized propagation constant versus propagation
angle. The parameters used are: ko d = 2.0944e - 3, p / d = 1.86, a / p - 0.07,
and s r = 9.6. Bellamine’s result ( - •); Crampagne’s result (— ); present method
(-)-
77
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T
Even Mode
20
Odd M ode
*0i
20 -
0
50
'0 0
150
Phase Shift 0 (D eg )
Figure 3.4: Comparison of Zc between quasi-TEM Green’s function method
and homogenization EBCs method for even and odd modes. The parameters
used are the same as those in Figure 3.2. Weiss’ result (— ); present method
(-)•
78
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results of Zc in comparison with Weiss’ model [57] since Figure 3.2 gives good
agreement for p . The parameters used to obtain Figure 3.4 are the same as those in
Figure 3.2.
3.4
Conclusions
The EBCs with a phase shift has been used to study the propagation
characteristics o f surface waves along a strip grating on a grounded dielectric slab
with finite thickness substrate under the assumption of ko p «
1- Unlike previous
analyses, we take into account the phase shift between adjacent strips and assumes
that the direction o f surface-wave propagation is at arbitrary angle with respect to
strips on the grating. With the aid of the developed EBCs in Chapter 2,
determination of eigenvalues of the dispersion equations for the propagation
constants becomes analytically simple and computationally fast compared to the
previous quasi-TEM Green’s function method [57-58], Comparisons have been
made and show good results. The obtained propagation constant and characteristic
impedance o f a semi-infinite strip grating on a grounded substrate are important in
modelling the properties o f metal grating elements, such as meander-line structures,
for compact microwave circuit elements design. This will be illustrated in the
subsequent chapters.
79
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CHAPTER 4
CHARACTERISTICS OF MICROSTRIP GRATING ELEMENTS
4.1
Introduction
Chapter 4 will address the potential application of microstrip grating
elements such as microstrip meander lines, hairpin lines, and comb lines for compact
microwave components using the derived propagation characteristics of surface
waves along a periodic array of strips discussed in chapter 3. Impose the proper
boundary conditions for different grating element structures, the dispersion equations
of microstrip grating elements are derived and show the filtering characteristics
similar to filters. For grating elements loaded with nonlinear lumped devices, tunable
filtering properties exist and show practical application in electronically tuned
microwave filters. Also the image (iterative) impedances o f a semi-infinite periodic
grating elements are defined and derived for efficiently modelling filter responses
which will be discussed in Chapter 5.
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4.2
Dispersion Equations
4.2.1
M eander Lines
Consider a section of length L of an infinite array of strips on a
grounded substrate shown in Figure 3.1, a microstrip meander-line is formed by
connecting alternating ends of strips as illustrated in Figure 4.1. Without loss of
generality, linear or nonlinear lumped elements can be loaded between adjacent
strips, as illustrated in Figure 4.2, where the impedances Zi and Z2 could be
inductive or capacitive. To apply the results o f propagation characteristics of metal
gratings to meander-line structures, the previous assumptions made for periodic
strips in Chapter 3 must be followed. We will show that this meander-line structure
has bandreject properties, which indicate that this circuit element will be a bandstop
filter.
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Figure 4.1: Top view of a microstrip meander-line structure consisting of
an infinite array of strips.
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z
unit cell
Figure 4.2: Top view of a microstrip meander-line structure loaded with
series impedances Zl and Z l ■
As stated in Chapter 3, the propagation of surface waves in Figure 4.2 is
in both z (along the strips) and y (transverse to strips) directions and the phase shift
between adjacent strip is assumed to be 0 . In this case, the voltages and currents on
strips 1 and 2 in Figure 4.2 can be denoted by
83
i
I
i
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Vi —[A+e J $ez + A - e J $ez —B+e j $ o z + B-e JP oz]e-i®/^
V2 ~ \_A+e~i P*2 + A —eJ $ez + B+e~J &oz —B~eJ Poz]e - /®/^
/ l = {— [A+e~jVez - A - e j V e z] + — [ - B +e-JPoz - B - e j V o z]}eJQ/ 2
Zee
(4 I )
Zco
I 2 = {— [A+e-jVez - A - e j V e z] + — [B+e-JPoz + B-eJPoz]}e-JQ/ 2
Zee
where
A±
Zco
and
B±
are the amplitudes of the even- and odd-mode waves,
respectively, for waves propagating in +z and - z directions.
Zee, Z co,
Pe , and
P0 are the even- and odd-mode characteristic impedances and z -directed
propagation constants for a strip grating, respectively. Assume the currents depicted
in Figure 4.2 flow in the +z direction, the boundary conditions of the meander line
are
\ V \ ( L / 2 ) - V i ( L / 2 ) = Z \Ii( L/2)
V2( - L / 2 ) - V 3( -L /2) = Z 2 I 3(-L/2)
/ 1(Z/2) = - / 2(L/2)
(4.2)
I l ( ~ L/2) = - / 3( - L/2)
V 2(z) = V ^ e - j ™
. h ( z ) = I l( z )e - J 2Q
where the last two equations of (4.2) are always true due to the iterative phase
relationship. Applying equation (4.1) to boundary conditions in (4.2) gives a system
of homogeneous equations for mode amplitudes
A±
and
B±:
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-4n
An
An
An
A 13 A 14 A+
A 14 A n A .-
A 31 A 32 A 33 A 34 B+
A n A 42 A 43 An _
=0
(4.3)
where the elements in the 4 x 4 matrix are given by
A n = cos(0/2)e“/P e V 2
A 12 = -co s(0 /2 )e / PeV2
^13 = -/nsin(0/2)e-y'PoV2
^14 = —jr\ sin(0/2) eJ PoV 2
^31 = [-TiieCOs(0/2) +y(2 -n ie)sin (0 /2 )]e^'P ez/ 2
^32 = h i ecos(0/2)+y(2 +Tlle)sin(0/2)]e/Pe V 2
(4.4)
^33 = [Olio - 2)cos(0/2) + y rilosin(0/2)]e-y P0^ /2
A 34 = [(Tlio +2)cos(0/2)+yTilosin(0/2)]e/PoV 2
A n = [Tl2ccos(0/2)+y(2+Ti2e)sin(0/2)]ey Pei / 2
^42 = [_Tl2ecos(0/ 2) + y(2 -r)2e)sin(0/ 2)]e_y P ei/2
-443 = C0l2o + 2)cos(0/2)+yTi2osin(0/2)]eyPoi / 2
-444 =
where
[Cn2o-
2)cos(0/2) + y ri2osin(0/2)]e-/Poi / 2
r\ = Z e e / Z c o ,
*11,2e = Z u /Z c e ,
and
rji 2o =
Z i,2 /Z c o .
For nontrivial
solutions o f (4.3), the determinant o f the 4 x 4 matrix in (4.3) must be equal to
zero. After a lengthy algebraic and trigonometric simplification, we obtain the
dispersion equation for a loaded meander-line (as illustrated in Figure 4.2)
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Cnsin2 (9/2) tan(Pe L/2) - cos2 (9/2) cot(P0 L/2) - y'nio/2] •
fa sin2 (9/2) cot(pe L/2) - cos2 (9/2) tan(p 0 L / 2 ) +jr \2o/2]
-J
“ 0 (^20
2
- , rrisin2(9/2)
rllo)L . fn , x
sm(PeZ,)
(4.5)
cos2(9/2)^
. rn ~ J
sm (p0 I )
Two limiting cases should be noted. When the loaded impedances are
equal, i.e., Zl = Z l = Zs in Figure 4.2, the dispersion equation (4.5) becomes
Zc<?sin2(9/2)cot(Pe Z,/2) - Zcocos2(9/2)tan(P0 I /2 ) = - j Z s/ l
Zcesin2(9/2)tan(PeZ ,/2)-Z cocos2(9/2)cot(P0 Z,/2) = jZ s / 2
which is identical to Dashenkov’s results [62] if only the series elements in his paper
are considered. If Zs = 0 , which means that the parallel strips of the meadner-line
structure are connected as short circuits as shown in Figure 4.1, then equations (4.5)
and (4.6) become
Zee
,
f tan(Pe L/2) tan(P0 L/2)
— tan2(9 /2 )H
Zco
'
{ cot(PeZ./2)cot(P0 Z,/2)
(4-7)
which has been shown in [57-58], with 0 from equation (4.7), to be equivalent to
<j>/2. According to equation (4.6) or (4.7), there exist mathematically two dispersion
equations in a meander-line structure, that is, there are two normal modes
propagating along the structure. The upper and lower equations of (4.6) or (4.7) are
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then referred to as forward-wave and backward-wave branches of the dispersion
diagram, respectively. The forward-wave and backward-wave modes are defined by
their directions of the group velocity (or energy velocity) and phase velocity. For
forward-wave propagation, the group velocity and phase velocity are in the same
direction, and for backward-wave in the opposite direction [87],
Note that, as mentioned in Chapter 3, the even- and odd-mode grating
characteristic impedances and propagation constants shown in equation (4.7) are
functions o f phase shift 0 (or frequency). Obviously, at frequencies for which the
arguments Pe £/2 and P0 L/2 in (4.7) are in different quadrants, the right-hand side
of (4.7) must be negative. This requires that the phase shift 9 on the left-hand side
of (4.7) be imaginary, i.e., the propagation along the meander line is cut off and
stopband properties occur. This stopband phenomenon is applicable to bandstop
filters. If a loaded impedance is considered in the dispersion equation as shown in
equation (4.6), the location o f stopbands will shift, which means that the bandwidth
of the stopbands is tunable by loading devices; this phenomenon is important and
will be discussed numerically later. The existence and prediction of stopbands is
essential and important for filter design purposes. Obviously, the bandwidth o f the
bands will depend on the circuit geometry and impedances loading the strips.
4.2.2
Hairpin Lines
By letting the impedances Zl - » °° and Z l = 0 in Figure 4.2, the
structure then becomes a hairpin line or C-section line illustrated in Figure 4.3. Note
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that there are two strips in a unit cell which is defined from the center of one Csection to center of adjacent C-section line. This definition gives a reasonable way to
find the image impedance of a hairpin line when modelled by transmission line
theory and this will be discussed more in Chapter 5. Taking the limits of Zi
*
and Z l - 0 on both sides of equation (4.5) results in the dispersion equation for a
hairpin line
a
4
b
4-
►
.,1
V,
d
V2
#2
#1
d
V3
#3
unit cell
Figure 4.3: Hairpin-line structure with a defined unit cell.
88
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lim lim {[q sin2(0/2) tan(Pe 1/2) - cos2(0/2) cot(p0 L/2) - j q lo/2] z r « z 2-*°
[Tisin2(0/2)cot(PeZ ./2)-cos2(0/2)tan(PoZ./2)+yrj2o/2]
(4.8)
j /•„
. rrisin2(0/2) cos2(0/2)11 n
~ T v l 2 o ~ n i 0)[ .--------- --------. . q ~7 x'JJ = 0
2
sm(PeL)
sm ^L )
where ri20 = ZilZco and r|j0 = Zi/Zco • After simple arrangement, equation (4.8)
becomes
tan2(0/2) = -
Zco tan(PeZ)
Zee tan(P0 Z.)
(4.9)
which is identical to Crampagne’s dispersion equation for C-section periodic
structure [59] where the phase shift between strips was defined as <p/2 which is
equivalent to 0 in this thesis. Clearly from equation (4.9), at the frequencies for
which arguments p eZ, and P0 L are in different quadrants, the right-hand side of
equation (4.9) is positive and results in real phase shift 0 existing in the propagation
band (passband). This phenomenon implies that this hairpin-line structure could be
designed for a bandpass filter.
In deriving the dispersion equation of (4.9) for a hairpin line (Figure 4.3),
the end-effects o f strips are not considered, i.e., the coupling capacitance between
strip ends is assumed to be small enough so that it can be ignored. On the other hand,
the excess capacitance due to the open end discontinuity can be transformed into an
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equivalent length of small transmission line adding to the physical length L [63].
This approximation is essential and valid for the microwave frequency range.
By looking at the dispersion equation (4.9), since the even- and odd­
mode propagation constant P and characteristic impedance Zc are symmetrical at
9 = 90° as calculated in chapter 3, then dispersion curve o f a hairpin line will also
be symmetrical at 6 = 90° in the passband; this will be verified numerically later.
4.2.3
Comb Lines
Figure 4.4 shows the general representation of a comb-line structure with
reactive loads at the finger ends of the lines. Since only one strip is contained in the
unit cell of a strip grating, the voltage and current on strip #1 of Figure 4.4 can be
denoted as
Vi = Ae~JPz + BeJPz
(4.10)
Zc
where P and Zc are propagation constant in +z direction and characteristic
impedance of strip grating, respectively. The boundary conditions for Figure 4.4(a)
now are
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#0 I
#1
1 1 Io,V0
I V
°
1012 #2 I
fl^V ,
1 1 I*V2
l
Figure 4.4: (a) A comb-line structure loaded by reactances; (b) a unit cell of (a).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Uoi + 1 l( V 2) = I \g + I C12
V0 ( - L / 2 ) - V l ( - L / 2 ) = j X l 0l
(4.11)
V X= V 0 e~JQ
.
I c l 2 = I c O i e ~ jQ
where the last two equations o f (4.11) are true for the iterative phase assumption.
Using KirchhofPs voltage and current laws on strips #1 and #2 of Figure 4.4, (4.11)
can be reduced to the following conditions
h
( i / 2) = - A
Ki ( - Z./2) =
± -
X2
+ -k e-jB - lX«/0 - 0] n ( i / 2 )
X\
(4.12)
jX
h(-L/2)
Substituting equation (4.10) into (4.12) gives a system of homogeneous equations
for wave amplitudes A and B . For nontrival solutions of A and B , it gives the
following dispersion equation for a comb line with nonzero reactance X \
[S i" 2
0 /2 ) + f
1 • [sin2 0 /2 ) + —
4 Z ctan(PZ.)
--7^ — — 1
4ZcXi
(4.13)
sin 2 (6/ 2)
-2Zcsin(2pZ,)
For the limiting case of X \ —> °o and X 2
00 • (4.13) then becomes
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sin2 (0 / 2) = -^-tan(PZ-)
4 Zc
(4.14)
where we see that the reactance X in equation (4.14) can not be zero for proper
solution of phase shift and for the consistency of unit on both sides of (4.14). This
reactance would be the self-inductance due to the transmission line for practical
reason. Equation (4.14) also shows that at frequencies for which tan(PL) is negative,
then the phase shift 0 in the left-hand side of (4.14) is imaginary which cuts off the
propagation and stopband phenomenon occurs.
4.3
Image Impedance Z/m
4.3.1
M eander Lines
The image (iterative) impedance of a loaded meander line shown in
Figure 4.2 is defined as
Zim = ~ ~
IA
(4.15)
where V a and I a are voltage and current at point A o f Figure 4.2. This concept of
image impedance o f (4.15) is derived based on the input impedance of an infinite
array of strip conductors extending in the transverse direction (.y-direction) with
connected alternating ends. However, as long as the periodic structure consists of a
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sufficient large finite number o f strips such that a uniform wave-propagating in the
transverse direction is established, the mentioned input impedance can be considered
equivalent to the image impedance for an infinite array of strips. The image
impedance is sometimes referred to as iterative impedance since it repeats every
period.
Now according to Figure 4.2 with Zl = Z l = Z s , the image impedance
defined in (4.15) can be written as
7- —-----Z d
Lim
IA
_ Zs , Vi(tLI2)
2
_
Iii-L/2)
eJPe^/2 A+ + e~j Pe^/2/f_ — e J B + + e~i
Zs
^
[eJPe^/2A+—e~J
Zee
A^]-------[e/Po^/2 B+ + e~j Po^/2 BJ\
Zco
(4.16)
where V\ and I\ defined in equation (4.1) are used. To find out (4.16), the relation
between wave amplitudes must be evaluated first. Then use the 4 set of equations in
(4.3) to eliminate amplitudes B+ and B- , we obtain
i±
A(1 + e i P' L eJ Vol )[2t\ sin 2 (6/ 2) + 2 cos2(0 / 2)] -% (1 - eJ P«* ei Pql )
(eJ $ e L + eJ
Po^)[2 cos2(0 / 2) - 2r|sin2(0 /2)]+ r\s (eJ PoL - eJ P « L)
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(41?)
where % = Z s {Z c o • The lower equation of dispersion (4.6) for backward-wave
mode can be expanded as
(1 + eJ P* L e i Po L)[2t\ sin2(0/2) + 2 cos2(0/2)] - t\s (1 - eJ P*L ei Po i )
—- ( e i P*^ + e i Po^)[2cos2(0/2) —2 q sin 2(0 /2 )]—^ (e /P o ^ - - g 7 Pe ^)
Equation (4.18) is then used to simplify (4.17), we obtain the following relation for
wave amplitudes A±
A+ = - A -
for backward-wave
(4.19)
Similarly, by substituting the upper dispersion equation in (4.6) for the forwardwave branch in (4.17), we get another relationship
A+ - A -
for forward-wave
(4.20)
Similarly, by eliminating amplitudes A+ and A - from the homogeneous
equations (4.3), we have
B+ _ (1 + e i Pej ei Po^)[2t| sin2 (0/2) + 2 cos 2 (0 /2)] + % (ei P<L e i Pql - 1 )
B~
(ei Pe^ + e i Po^)[2 cos2 (0/2) - 2r) sin2 (0/2)] +r\s (e i^o^ - e i Pe^)
(4.21)
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Again, use the dispersion equation for forward-wave and backward-wave branches
in (4.6) to simplify (4.21), we obtain the following relation for wave amplitudes B ±
for forw ard-w ave
, ,
.
for backward - wave
(4.22)
Employing the relationship o f wave amplitudes in (4.19), (4.20), and (4.22), after a
lengthy algebraic and trigonometric simplification, the image impedance of a loaded
meander line defined in (4.16) becomes
- S‘-" ~ [Z ee tan(J3 e L f 2) + Zco cot(P 0 L /2 )]
backward - wave
(4 23)
Zim —
— —- [ Z e e cot(Pe L / 2 ) + Zco
tan(P0 L /2 )]
forward - wave
For the case in Figure 4.1 where no series elements exist, i.e.,
Zi = Z l = Z s = 0 , we use the dispersion equation in (4.7) for zero series elements to
simplify (4.23), and the image impedance in (4.23) becomes
Zim \y
tan(P0 Z,/2)
=0 = J Z ee Z c o —
=-
I
Zee Z c o
V
cot(p0Z /2)
—-—
cot(Pe£/2 )
„ „
for forw ard-w ave
_ ,
for backw ard-w ave
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which is identical to Weiss’ meander-line image impedance [57].
4.3.2
Hairpin Lines
The voltages and currents on strip #1 and #2 in Figure 4.3 for a hairpin-
line structure can also be represented by equation (4.1). Now in this case, the
boundary conditions o f the structure are
/l( L /2) = 0
I 2{Lf 2) = 0
/ 2( - L / 2 ) + / 3(-L /2 ) = 0
(4.25)
V 2( - L / 2 ) = V3( ' L / 2 )
r 3(z)=Fi(z)e- / 20
h ( z ) = h ( z ) e ~ J 2Q
Substituting the expression of voltages and currents in equation (4.1) into boundary
conditions in (4.25), we obtain a system o f 4 homogeneous equations and
simplifying these 4 equations, the following relationship for wave amplitudes are
obtained
A+ = eJ$eL A A - _ y'sin(Q/2)r|eJ Pq V 2sin(P0Z,)
B- ~
cos(9 /2)ej P«^/2 sin(P e L)
B+ = - e J $ol B~
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(4.26)
Now, the definition o f image impedance shown in (4.15) can be adapted for a
hairpin-line structure. Since a unit cell o f a hairpin-line structure is defined from
center of one C-section to center of adjacent C-section such that the image
impedance of this structure can be well defined. Thus the image impedance of the
hairpin line shown in Figure 4.3 is
_
VA
V x (-L !2)
Zm~TA - W
w )
ei V'L/ 2 A + + e - j P* V 2 A .- - e i P° L/ 2 B+ + e~J
=
— [ei Pe A + —e~i Pe V2
Zee
V2B-
(4-27)
[gj P0 £/2 B++e~i Po^/2 /j_]
Zco
where Fj and /[ are depicted in Figure 4.3 and expressed in equation (4. 1). Use
conditions (4.26) and hairpin-line dispersion equation in (4.9) to simplify equation
(4.27), we then obtain
^ _ Z e e tan(0/2)
£im —------- —— ~~—
tan(PgZ)
—Zco
(4.28)
tan(0/2)tan(PoI )
—Z e e Zco
tan(Pe Z,)tan(P0Z,)
where the third expression is obtain by taking square root of product of the first and
second expressions. As stated previously, if PeL and P 0 L are in different
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quadrants and the phase shift 8 is real in the passband region, then the image
impedance shown in (4.28) is real in the propagation band, which is expected.
4.3.3
Comb Lines
From Figure 4.4, the image impedance o f a comb-tine structure can be
defined as
;
V o i - V \ ( r L [ 2 ) + j — lQ[
'01
(4.29)
2
where Foi and / qi are depicted in Figure 4.4. Using KirchhofFs voltage and
current laws on the unit cell o f Figure 4.4(b), equation (4.29) becomes
Zim = -~cot (9/2)
(4.30)
Note that this image impedance depends on the reactance loaded on the series line
which connects the parallel strips, and reactance X could not be zero for reasonable
image impedance.
4.4
Numerical Results and Prediction of Stopband and Passband
4.4.1
Meander Lines
The dispersion equation of unloaded meander-line shown in (4.7) will be
used to calculate the important 0 - / diagram (Brillouin diagram) and predict the
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filtering characteristics of the structure. Figure 4.5 shows the comparison of
prediction of stopband and dispersion diagram o f backward-wave branch for an
unloaded meander-line between homogenization EBCs method and Crampagne’s
results [58]. The frequency range between the extremes of a 0 - / diagram is called
a passband since the phase shift is real and a frequency range between two passbands
is a stopband. These two results agree well as expected since the characteristic
impedance Zc and z -directed propagation constant (J of the periodic strip grating
were numerically compared well as shown in Chapter 3. Figure 4.6 shows the
dispersion diagram (backward-wave) of an unloaded meander-line structure in terms
of slot width a and strip width 6 with p - 1.2mm, L = 18.36mm, d = 1.016mm,
and g r = 10.2. We see that the bandwidth and the diagram curves do not change
much when a/6 is adjusted. Even though, the image impedance will change
significantly with different a /6 . Figure 4.7 shows the calculated image impedances
of a meander line (forward-wave) where the circuit parameters used are the same as
those in Figure 4.6 and we see that the image impedance goes to either zero or
infinity at the edges of stopband; this is obvious since the propagation approaches to
cut off near the edges o f the stopband. The selection of image impedances is
important to this class of grating filter design since they correspond to the
“equivalent” characteristic impedances of the structure when modelled by
transmisssion line theory and this willl be explained more in the next chapter. The
dispersion diagram in terms of the strip length L with fixed slot and strip width is
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2QG;—
50
x
2
4
F re q u e n c y (GHz)
Figure 4.5: Comparison o f dispersion diagram (backward-wave) of a meander
line between homogenization method ( - - ) and Crampagne’s result (—) with
a - 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm, and e r = 9.6.
101
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200-
X
• 50
50
-L
2
F r e c u e n c y (GHz)
Figure 4.6: Dispersion diagram (backward-wave) of a meander-line structure
with p = 1.2mm, d = 1.012mm, L = 18.36mm, and e r = 10.2 for a[b = 2.0
(—X 5.0 (• •), 1-0 (— ), and 0.5 ( - •).
102
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I
I
(Olitna)
t
Imcnjo
Impedance
r
00
t
0
1
2
3
4
-
5
6
►Vecuency (G h z )
Figure 4.7: Calculated image impedances (forward-wave) o f a meander line
with p = 1.2m m , d - 1.012mm, L = 18.36mm, and e r = 10.2 for a/b =
2.0 (— ), 5.0 (• •), 1.0 ( - -), and 0.5 ( - •)•
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shown in Figure 4.8 for backward-wave branch where we see that the location of
stopband varies with the strip length. It is expected since the resonant frequency of a
single strip varies with the length of strip. For an unloaded meander-line structure,
the physical strip length L is approximately equal to the guided half-wavelength in
microstrip and this will be shown experimentally later.
The loaded series impedances shown in Figure 4.2 can be replaced by
identical nonlinear devices such as varactor diodes, as illustrated in Figure 4.9,
where the DC bias circuit is not shown and the junction capacitance Cj of the
diodes can be tuned by appling DC voltage. Figure 4.10 shows the backward-wave
branch dispersion characteristics in terms of total capacitance Ct of a device-loaded
meander-line structure using equation (4.6), where Ct is the total capacitance loaded
between strips. We see from Figure 4.10 that, for different capacitance Ct near the
stopband, the first edge does not change too much, but the second stopband edge
shifts as the loaded capacitance is changed. Basically, the bandwidth of the stopband
of this device-loaded meamder-line structure is electronically tunable when the
grating element is loaded with nonlinear lumped devices and this phenomenon is
promising for tunable compact filter design.
104
i
j.
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T
2CG—
CO
50
0
1
2
3
F r e q u e n c y (GHz)
4
5
5
Figure 4.8: Dispersion diagram (backward-wave) of a meander-line structure
with a = 0.8mm, b - 0.4mm, d = 1.016mm, and z r = 10.2 for L = 18.9mm
(—), 18.36mm (• •), 17.5mm ( - - ) , and 19.5mm ( - •).
I
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Figure 4.9: Example of a diode-loaded meander-line structure.
To show that a periodic structure such as a meander line is a class of
slow-wave structures, the group velocity vg must be examined. With the aid of the
Brillouin diagram ( 0 - /
diagram) shown in Figure 4.5, group velocity Vg is
defined as
3(9 /p)
da
2np%
(4'31)
where Q/p is the “equivalent” propagation constant along the periodic system in y direction for fundamental mode as mentioned in Chapter 3, / is the frequency, and
df/dQ is the inverse o f slope of the Brillouin diagram shown in Figure 4.5. To
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2C0 —
0
3
2
O
f r e q u e n c y (GHz)
Figure 4.10: Dispersion diagram (backward-wave) o f a loaded meander line
for Ct= 1.0 p F (—), 15 p F (• •), 5 p F
and 0.5 p F (-•) with a =
0.8mm, b = 0.4mm, L = 19mm, d = 1.016mm, and e r = 10.2.
107
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evaluate the slope of the dispersion curve, one can fit the curve using mathematical
functions or polynomials with high accuracy and take the derivative directly from
the fitted curve. The absolute value of normalized group velocity to the light speed
in free space vq versus frequency for the meander-line structure with the same
parameters as those in Figure 4.5 is shown in Figure 11, where the dispersion
diagram o f Figure 4.5 is fitted using 10th degree polynomial. We see that the group
velocity o f this meander-line structure is much smaller than the speed of light in free
space as expected for slow-wave structure. Also near the edges of stopband, group
velocity approaches to zero which indicates that the structure carries no power in
stopband region.
4.4.2
H airpin Lines
The prediction of passband characteristics or 0 - / diagram of a hairpin
line can be calculated by using equation (4.9). Figure 4.12 shows the calculated
dispersion diagram of a hairpin-line structure in terms of slot width a and strip
width b . These dispersion curves are symmetrical at 0 = 90° as predicted by (4.9)
in section 4.2.2. We also see that the passband characteristics do not change too
much for different values of strip and slot width except for the tightly coupled case
of a = 0.2mm and b = 1.8mm. In this case, the dispersion curve seems degenerate
with little bandwidth and the group velocity (corresponding to inverse of the slope of
the curve) approaches to zero resulting in little power flowing through the structure.
Figure 4.13 shows that the location of passband in terms of the length of strip; this
108
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\
|OA/f>A|
0
1
2
2
rreauency
^
5
5
(G H z)
Figure 4.11: Normalized group velocity of a meander-line structure with
a = 0.13mm, b = 0.8mm, d = 1.0mm, L = 18.36mm, and s r = 9.6.
109
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2C 0-
T
a
a
t/
a
a
= 02mm /> = 18mm
- 10mm h - 05mm
= 0 8mm h = 0 4mm
- 06mm h = 0 3mm
= 0 2mm h = 0 4mm
50-
02.0
2.5
3.0
7
F r e q u e n c y (GHz)
Figure 4.12: Dispersion diagram of a hairpin line for d = 1.27mm, L = 18.36mm,
and e r = 10.2 in terms of a and b .
110
i
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\ v
•co -
Phose
Shill (Di.fj )
\
50
or
1.0
1.5
2 .0
2.5
3 .0
3.5
F r e c - e r i c y (GHz)
Figure 4.13: Dispersion diagram of a hairpin line with a = 0.8mm, b = 0.4mm,
d = 1.27mm, and e r = 10.2 for L - 18.9mm
18.36mm (—), 18.8mm (• •)
, and 19.2mm ( - •)•
111
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phenomenon is obvious and expected as mentioned in the case of meander-line
structure.
Figure 4.14 shows the calculated image impedances for various er and
L using equation (4.28). We see that the image impedance becomes sharply large at
the edges of passband for each case and for higher dielectric constant, the image
impedance of passband decreases and the location of passband shifts toward lower
frequency. From Figure 4.14(b) we also see that as strip length L increases, the
bandwidth of passband decreases; however, the fractional bandwidth is the same.
The variation of L doesn’t change the impedance level as expected. Although
Figure 4.14 does not show good circuit dimensions for filter design purpose since
the image impedances are quite high, it provides an important information for a
hairpin-line structure.
4.4.3
Comb Lines
For the comb-line structure illustrated in Figure 4.4, if no external
elements are loaded with the structure, then X j could represent the reactance due to
open-end discontinuity effect which can be represented by an excess capacitance Cp
[63] to ground, X \ could be the reactance due to coupling capacitance Cs between
adjacent strips which can be found in Getsinger’s paper using the concept of
negative inductance in quasi-TEM transmission line [65], and X is the reactance
due to the self-inductance Ls produced by the short transmission line connected
between two parallel strips and can be modelled by
112
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^ ik-.CC
(a)
L
-
18 6 6 m m ■
18 0 m m •
19 5m m ■
21 0mm ■
22 5m m
-
-re c u e rc y (GHz)
| K , ' n __
\
5CG r ~
(b)
;
Er= 102
96
11 9
130
150
o2.0
2.5
" re c u e rc y (GHz)
3.0
Figure 4.14: Image impedance of a hairpin line for p = 1.2mm and d = 1.27mm
with (a) L = 18.66mm and ajp = 2/3; (b) e r - 10.2 and ajp = 2/3.
113
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I
L s — A/
Zo -J&e
(4.32)
CO
where A/ is the length of the short transmission line which is equal to the period of
the structure, Zo and ee represent the characteristic impedance of the short
transmission line A/ and effective dielectric constant of the substrate, respectively,
and co is the speed of light in free space. Then the reactances shown in Figure 4.4
can be represented as
X —co Ls
(4.33)
X i = - l / o Cs
Xi =
- l/ o C P
Now, use the structure parameters o f a = 0.8mm, b = 0.4mm, d = 1.016mm,
L
= 18.36mm, and g r = 10.2, then the mentioned reactances are calculated as:
Cp
= 0.02p F , C s = 0 01 \ 2p F , and L s = 0.1229nH. The dispersion diagram of the
comb line with the above computed equivalent reactances are shown in dash-dot line
in Figure 4.15. Also, if the end-loading capacitance Cp is replaced by an identical
varactor diode whose capacitance is tunable by applied DC voltage, then the location
of stopband will be tuned electronically as shown in Figure 4.15. Thus an
electronically tunable comb-line filter suggested in [66, 67] can be constructed in
microstrip. Figure 4.16 shows the calculated image impedance of a comb-line
114
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2QC-
•
00
—
<
u
m
50-
0
o
7
2
Frequency
(G H z)
Figure 4.15: Dispersion diagram o f a comb-line structure for a = 0.8mm, b =
0.4mm, L = 18.36mm, d = l.016m m , and e r = 10.2. Equivalent reactances:
Ls = 0.7239nH and Cs = 0.0112p F with Cp = 0 (—, Cs = 0), 0.03p F ( - -),
03 p F
and 1.0p F (• •)•
115
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i)
u
1cr>
)
c
50
0
2
3
4
5
“ r e q u e n c y (G h z)
Figure 4.16: Calculated image impedances o f a comb-line structure in terms of
a and b with p = 1.2mm, L = 18.36mm, d = 1.016mm, and s r = 10.2: a/b
= 2 (—), 1.0 (• •), 0.5 (— ), and 0.2 ( - •)■
116
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structure in terms o f a and b using equation (4.30) where the equivalent reactances
are considered. The calculated image impedances approach to either zero or infinity
at the edges o f stopband; this phenomenon is quite obvious or can be understood by
checking the expression of image impedance in (4.30).
4.5
Conclusions
Dispersion equations of microstrip periodic grating elements such as
microstrip meander lines, hairpin lines, and comb lines are derived analytically in
this chapter. With the aid of propagation characteristics o f surface waves along
periodic strip gratings, the filtering properties of the grating elements are
numerically predicted which provide the important information to microstrip grating
filter design. The image impedances o f the grating elements are also defined and
derived. Since image impedance corresponds to the “equivalent” characteristic
impedance o f grating elements when modelled by equivalent transmission line
theory, the selection of circuit parameters must be careful to provide proper
impedance level for impedance-matching purpose. The assumptions made in
previous chapter that the substrate height has to be finite and the grating period p
must be much smaller than the wavelengh of radiation should be obeyed for all the
design. The design of the grating filters will be discussed in next chapter.
117
i
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CHAPTER 5
MICROSTRIP GRATING FILTERS
5.1
Introduction
To efficiently and accurately compute the responses of microstrip grating
elements as microwave filter applications, the grating filters will be modelled by an
equivalent transmission line (ETL) with a defined propagation constant and
characteristic impedance based on the structure. The results of propagation
characteristics of strip gratings on grounded substrates derived from the EBCs will
be used to model the ETLs. Then the performances of unloaded and loaded
microstrip meander-line bandreject filters, hairpin-line bandpass filters, and combline bandreject filters are calculated using the ETL model based on the normal and
complex modes theory. For the grating elements loaded with lumped devices, they
find potential applications in bandwidth-tuned filters. The calculated responses are
also compared with the measured results. The development of this class of grating
filters promises to deliver broadband properties or highly selective behaviors of
compact filters.
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5.2
Meander-line Bandreject Filters
5.2.1
Equivalent Transmission Line (ETL) Model: Normal Modes
Consider a section o f microstrip meander-line structure with equivalent
length o f w in the transverse direction ( y -direction) and the structure is connected
with nominal 50- Cl microstrip lines on both input and output ports as shown in
Figure 5.1, where w is equal to the number of strips times the period length p . To
represent this meander-line structure from the standpoint of circuit, an equivalent
transmission line (ETL), as illustrated in Figure 5.1(b), with propagation constant
y = 9 Ip
in transverse direction as mentioned in Chapter 3 and characteristic
impedance replaced by Zim which is defined in Chapter 4 as an image (iterative)
impedance for semi-infinite periodic structure will be employed. The use of Q/p as
the propagation constant of ETL is quite obvious as mentioned in Chapter 2 and the
characteristic impedance of the ETL can be modelled by Zim without doubt
according to the defintion o f image impedance for symmetrical two-port networks
[61]. Then the transmission or ABCD matrix of the ETL o f Figure 5.1(b) is
n
[ cos(wy) y Z/m sin(HY)
sin(ivy)
cos(vvy)
Zim
and return loss S n and insertion loss S 21
(5.1)
this transmission line can be obtained
by transforming transmission matrix o f (5.1) into scattering matrix, which is
119
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port #1
■ port #2
)
-W
(a)
w
z*
y=Q/p
i ------------------------------------•
(b)
Figure 5.1: (a) Top view of a microstrip meander-line filter, (b) Equivalent
transmission line model of (a).
120
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2 co s(w y ) + j ( Z i m / Z o + Z o/Z /m )sin(w Y )
(5.2)
5 i 1 = j - ^ i Z i m / Z o - Zo/Zim)sin(wy)
where Zo is the characteristic impedance o f microstrip line connected on both ports
of the meander line and is set to 50 £2 normally. As mentioned in the meander-line
dispersion diagram in Chapt 4, y =Q/p in equation (5.2) is a real quantity for
passbands and is imaginary for stopband. If a Bloch wave propagating in +y
direction (see Figure 4.1) is assumed, we can define the equivalent propagation
constant o f ETL in more general way (wave propagation with exp(-jyy)
dependence is assumed)
y =Q/p
passband
(5 .3 )
=Q /p= $s + jas
stopband
where 0 and 0 are real and complex, respectively. The second equation of (5.3)
turns out that (3^ and a s are resultant phase constant (positive) and attenuation
constant (negative) o f ETL at stopband; this complex representation of resultant
propagation constant due to the mode coupling will be discussed more later.
Note that this proposed ETL model only holds for a normal mode
propagation since the interaction or coupling between two normal modes existing in
121
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meander-line structures is not considered. The importance of the coupling between
two different propagation modes will be discussed in next section. The derived
equations of finding S-parameters of ETL shown in (5.2) can also be applied to
hairpin lines and comb lines since only a single normal mode exists in those
structures.
5.2.2
Dual-mode ETL Model: Complex Wave Modes
As derived in equation (4.6) or (4.7), there are two normal modes
existing in meander-line structures. We now must take into account the interaction
between these two modes; the interaction o f normal modes of propagation is referred
as complex wave modes. Complex waves are the modes having complex propagation
constants in spite o f the assumption that the structures are lossless. Complex wave
modes in lossless systems were first observed in a waveguide with anisotropic
impedance walls by Miller [68]. Later, the existence and properties of complex
waves in lossless waveguiding structures, e.g., dielectric-loaded circular waveguides,
shielded microstrip lines, and finlines, has been investigated by several authors [6985] for the analysis o f discontinuity problems o f the structures. Investigations
showed that complex wave modes constitute essential parts o f mode spectrum
(complex propagation constant). In addition, in coupled lossless transmission lines
and periodic structures, complex wave propagation was predicted to occur in a
certain frequency band and carry no power flow [83-85]; this peculiar feature is
characteristic o f wave filters. It has been mentioned in [85] that waves with complex
122
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propagation constants can only occur in structures which exist at least two normal
modes. For our meander-line structure, there exist two normal modes propagation as
derived in dispersion equations of (4.6) and (4.7). Then the coupling o f the
backward waves and forward waves constitutes complex wave modes and gives rise
to the stopbands. As pointed out in [77], ignoring the effects of complex wave
modes in certain frequency band will lead to erroneous solutions. So we will apply
the concept o f complex wave modes to the previous microstrip meander-line
bandreject filter and modify the normal-mode ETL model.
It has been shown that complex waves always exit in pair [72, 76], so
there are a set o f four possible complex waves resulting in standing and evanescent
waves along the meander-line structure. To consider the coupling of these two
modes using a circuit model, the concept of ETL model shown in Figure 5.1(b) still
holds; the difference now is, we use two ETLs connected in parallel representing
forward- and backward-wave mode as shown in Figure 5.2 where y b , Z bm , y A
and Z{m are equivalent complex propagation constants and image impedances for
backward- and forward-wave mode, respectively. This concept was first introduced
by Clarricoats [86] in 1963 by using homogeneous and inhomogeneous waveguides
coupled with forward and backward waves for a frequency-selector application.
According to complex mode theory [72, 76, 85], y b and y f are complex conjugate
with opposite sign in such a way that there is no real power being carried by these
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Forward-wave mode, Z^,
Backward-wave mode,
yf
y1
(a)
[Y*]
•------------C-
---- 1
[Y 6]
(b)
Figure 5.2: (a) Dual-mode ETL model for a meander-line filter, (b) Admittancematrix representation of (a).
124
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two modes. Thus these waves at stopband in the system contain one of the forms in
y -dependence
exp(±y Psy)exp(a.<r>')
exp(±yp^)exp(-a5>0
(5.4)
where Ps = Re(y) and a s = Im(y) < 0. Since these two ETLs are connected in
parallel, we may use admittance-matrix to represent each section of the dual-mode
ETL, as illustrated in Figure 5.2(b), and find the scattering matrix. Then transform
the transmission matrix shown in (5.1) for each section of the ETL of Figure 5.2(a)
into the admittance-matrix, we have
Db’f
Bb' f Cb' f - A b' f D b' f
B b' f
-1
Bb' f
Ab' f
_Bb' f
Bb’f
(5.5)
where A, B, C, and D are shown in equation (5.1) with the superscripts b and /
standing for backward- and forward-wave modes. Then the overall admittancematrix for the dual-mode ETL is
[K ]=[r*]+[r/]
(5.6)
125
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and convert admittance-matrix into scattering-matrix, we have the S-parameters
representation o f the dual-mode ETL
P l = - ( P 1 + I 'o M r 'd H - I'o W
(5.7)
where [/] is the unit diagonal matrix and Y q = I/Z q .
5.2.3
Design of a Microstrip Meander-line Bandreject Filter
Based on the previously developed ETL models for normal and complex
modes, a microstrip meander-line bandreject filter shown in Figure 5.1(a) with
parameters o f a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, w = 30mm,
and e r = 10.2
will be designed. Figure 5.3 shows the calculated complex
propagation constant versus frequency of a microstrip meander-line filter using ETL
models. At stopband region, we see that the phase constant (3 * ) is small and merges
with the edges o f passbands and minus of attenuation constant ( - a s ) is maximum at
the center of stopband and goes to zero at the edges of stopband as expected.
Figure 5.4 shows the measured [Sul and |^2i| of a microstrip meanderline bandreject filter. The calculated results without considering loss factors are
obtained from the developed ETL models; the parameters used are the same as those
in Figure 5.3. We see that the measured and calculated results agree well except the
frequency shift o f about 0.2 GHz; this discrepancy of frequency shift at stopband is
due to the strip width of short transmission line connected between two parallel
126
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I
30
25
2C
\
\
stopband retpon
i
2
3
-
5
5
Frequency (GHz)
Figure 5.3: Complex propagation constant y of a microstrip meander-line filter
with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, and e r = 10.2: y
in passbands (—) and Ps ( - •) and - a * (• •) in stopband.
127
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-2G
-5 0-
0
-
1
2
3
F-ecuency (GHz)
4
-
5
20-
rs
A
-ac-
j
-
100-
0
1
2
3
4
5
Frequency (GHz)
Figure 5.4: Measured (• •) and computed (— ) responses of a microstrip meander
-line bandreject filter with a = 0.8mm, b = 0.4m m , d = 1.016mm, w = 30mm,
L = 18.36mm, and e r = 10.2.
128
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strips. This phenomena was experimentally predicted by Vesslkov et al. [8] in
calculating transmission loss o f a meander-line structure at stopband. Their
experimental curves show that the strip width of the short link not only affects the
transmission loss but also shifts the center frequency of stopband o f a meander-line
structure. This can be explained that the equivalent circuit due to the strip width and
discontinuities of strip ends between two parallel strips varies the transmission
characteristics of the structure. To improve this frequency discrepancy, the
equivalent circuit between two parallel strips should be modelled correctly. In this
filter structure, the strip length L is approximately equal to half guided-wavelength
computed from the center frequency of stopband and the measured fractional
bandwidth is about 25%. Compared to the conventional quarter-wave coupled line
filters, this class of grating filter provides both wider bandwidth and compact design.
The maximum measured insertion loss of this filter at passbands is about
2 dB mainly due to the conductor loss since the transmission loss of the whole
structure is calculated about 1.7 dB at 2 GHz [61], The average return loss is about
10 dB in the first passsband; To improve the performance of the return loss, the
image impedance of the structure (or characteristic impedance o f ETL) should be
matched, i.e., the dimensions o f this meander-line filter must be adjusted such that
the image (iterative) impedance of filter is close to the microstrip characteristic
impedances (50 £2) connected on both ports, as illustrated in Figure 4.7. Some other
loss may be due to the step junction between microstrip line and meander-line
129
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I
structure, this type of discontinuities should be modelled properly. Also from the
measured insertion loss o f stopband in Figure 5.4, we see that the cut-off slopes are
relatively steep, better than 0.07 dB/MHz. Although the meander-line bandreject
filter shown in Figure 5.4 is not optimumly designed, it provides the potential
application in compact microwave filter design with sharp skirts in
i| and with
less complicated computation by using the accurate and efficient ETL models.
5.3
M icrostrip Comb-line Bandreject Filters
Figure 5.5: An example of a microstrip comb-line bandreject filter.
Figure 5.5 shows an example of a comb-line bandreject filter in
microstrip. The measured and calculated return loss and insertion loss of a comb-line
bandreject filter are shown in Figure 5.6. The calculated results are based on the
ETL model mentioned in section 5.2.1 for a single normal mode. The circuit
parameters used to achieve Figure 5.6 are: a = 0.8mm, b = 0.4mm, L = 18.36mm,
130
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- • o
—
-20
A
i
- JG
i
‘I
-5 0
0
Frequency (GHz)
- 30 r
uu
-4
-5 0 —
0
Frequency (GHz)
Figure 5.6: Measured (• •) and calculated (—) responses o f a microstrip comb-line
filter with a = 0.8mm, b = 0.4mm, d = 1.016mm, L = 18.36mm, w = 14.4mm,
and, e r = 10.2.
131
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w - 14.4mm (12strips),
d = 1.016mm, and e r = 10.2. Based on the circuit
geometry of Figure 4.4, the equivalent reactances are calculated in Chapter 4 as:
Cp
= 0.03pF,
Cs
= O.Q\\2pF, and
Ls
= 0.7239nH where
Cp
is the excess
capacitance due to the open-end discontinuity and Cs is the coupling capacitance
between parallel strips. To match the measured results at lower frequency,
Ls
is
chosen as 0.9049n H in obtaining Figure 5.6. As illustrated in Figure 5.6, the
calculated results are quite close to the measurement except at higher frequency
edge. The return loss at the first passband is not quite good since the circuit
parameters were not optimumly chosen such that the image impedance is about one
half o f the characteristic impedance of the microstrip line (50
) connected on both
ports, as predicted numerically in Figure 4.16, and this contributes a severe
impedance mismatch.
For this type o f filter, equivalent reactances due to the side- and endeffects are more sensitive to the reponses since the dispersion characteristics of comb
lines are strongly dependent on the circuit model as shown in (4.13). For practical
reason, an electronically bandwidth-tuned filter can be constructed by loading
varactor diodes at the end o f strips, as suggested in [66, 67]; an example of tunable
dispersion curve o f a comb-line structure is shown in Figure 4.15.
132
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5.4
M icrostrip Hairpin-line Bandpass Filters
Figure 5.7: An example o f a microstrip hairpin-line bandpass filter.
Figure 5.7 shows an example of microstrip hairpin-line bandpass filter.
The measured insertion loss of fabricated hairpin-line filter is shown in Figure 5.8
with circuit parameters o f a = 0.8mm, b = 0.4mm, d = 1.27mm, L - 18.36mm,
w = 31.2mm (26 strips), and e r = 10.2. We see that the average insertion loss at
passband is about 5 dB which is not quite good; this is expected since the calculated
image impedance at passband with the above circuit parameters is quite high as
shown in Figure 4.14 (solid line) where the equivalent length o f about 0.3mm due to
open-end discontinuities [63] is considered. This high image impedance will result in
a severe impedance mismatch at passband. However, the edge frequencies of
passband are well predicted. In addition, this bandpass filter provides sharp
attenuation rate at the edges which is of importance in practical filter design.
133
.1
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REEF" —t>0 . 0 d B
1 0 .0 dB/
START
STO P
1.000000000
4.000000000
GHz
GHz
Figure 5.8: Measured insertion loss of a microstrip hairpin-line bandpass filter
for a = 0.8mm, b = 0.4mm , d = 1.27mm, L = 18.36mm, w = 3 1.2mm, and
e r = 10.2.
134
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• 'G —
:n
£
-20
u
—
- 3 0 -
L
-40 —
2.0
2. 5
F r e q u e n c y (GHe)
Figure 5.9: Calculated responses of a microstrip hairpin-line bandpass filter with
a - 0.15mm, b = 2.0mm, d = 0.25mm, L = 18.36mm, w = 43mm, and e r =
18.
135
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Figure 5.9 shows the computed return loss and insertion loss of ahairpinline bandpass filter using ETL model of (5.2) with a = 0.15mm, b = 2.0mm,
L = 18.36mm, d = 0.25mm, w = 43mm (20 strips), and s r = 18. Since this set of
circuit parameters provides good image impedance at passband region, the calculated
responses show better insertion loss at passband and sharp skirts at the edges.
5.5
Tunable Microstrip Meander-line Bandreject Filters
Cs
H h
Cs
Cs
Cs
Cs
Hh
Hh
HH
HH
H
HH
HH
HH
HH
2Cs
Cs
Cs
Cs
Cs
4
2Cs
Figure 5.10: An example of a tunable microstrip meander-line bandreject filter
with loaded capacitances.
Figure 5.10 shows a meander-line structure loaded with lumped
capacitances in microstrip form. As shown in Figure 4.10, this kind of loaded
meander-line has bandwidth-tuned property. To demonstrate the phenomenon, a
136
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meander line structure loaded with chip capacitances is fabricated with a = 3.5mm,
b = 2.0mm, d = 1.27mm, L = 100mm, w = 110mm (20 strips), and e r - 10.2.
Figure 5.11 shows the calculated periodic dispersion diagram for this structure using
the lower equation o f (4.6). We see that the bandwidth of stopband increases as
frequency goes up; this phenomenon will be verified by experiment. Figure 5.12
shows the measured insertion loss and return loss of the capacitance-loaded
meander-line bandreject filter, where we see that the lower edges of each stopband
stay almost close for all cases and the location of stopband and bandwidth are
tunable as the loaded capacitance is varied; both phenomena are correctly predicted
in the dispersion curves o f Figure 5.11. So basically speaking, this kind of meanderline filter is bandwidth-tuned. This structure can be extended to load nonlinear
devices such as varactor diodes to meander lines, as illustrated in Figure 4.9 with
proper bias circuit, for voltage-tuned filter applications. The calculated and
measured IS21I are shown in Figure 5.13 where the measured insertion losses in
passband increase for lower value of capacitance and the losses are mainly due to the
conductor loss and impedance-mismatched loss as shown Figure 5.14. To correctly
compute the responses, the loss factors and the model of chip capacitances should be
taken into account, especially for higher frequency.
Another promising application suggested by Onoh et al. [64] is loading
PIN diodes into meander lines to control the passband and stopband properties for
multiplexer application. In our meander-line structure shown in Figure 4.2, if PIN
137
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2:0
•
—
r
i
’
\
1
■A
* 52 —
-
V
,
!
\
-
,
;! ■ '
-
I
—
\
-
\ ,
-
\
(
V \
\ V :
w\ •\
v„ ■
v .
; '■
V
-
.
\
/ . ' .
■■
F
:
:
50 —
:
:
■
:
n
i
f
*'
-
f \
/
r
■
•\
V -
0—
-
•
■
F
/;
/'
/..
/' •
/"
i
0. 5
/
'
:
r
’ 0
-'?cue
/
' .5
- c y (GHz)
A .
;
i ‘
-
■i
i
2.3
2:
Figure 5.11: Dispersion diagram (backward-wave) o f a loaded meander line for
Cs =1.5 p F
3.3 p F (• •), 4.8 p F ( - •), 10.0 p F ( - • • •), and short link
(—) with a = 3.5mm, b - 2.0mm, L = 100mm, d = 1.27mm, and e r = 10.2.
138
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Figure 5.12: Measured responses of a loaded meander-line bandreject filter for
Cj=l-5 p F (— ), 3.3 p F (—), 4.8 p F ( - •), 10 p F ( - • • •), and short link
(• •) with a = 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, w = 110mm,
and e r = 10.2 .
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fa)
-20
-
C.
0 .5
' 0
" '■ e c je r’c y (G h z )
140
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
i
t
(c)
"I ".'I | I
1 ;|j
;
y,r
■<
-2C ^ e c - e r c v ''GHz,
i
Or
A
(d)
; _ i'M
i
- i •i. 1-'Jn
f
aw
F
0.0
0.5
^ 'ecuencv (GHz)
Figure 5.13: Calculated (— ) and measured (—) 1^21| of a loaded meander-line
filter with a - 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, w = 110mm,
and e r = 10.2 : (a) short link; (b) Cj=10 p F \ (c) Cy=48 p F \ (d) G =3.3 p F .
141
!
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I.'irnl
( wt . i n: .
V
1
0.5
VC
'.5
-'ecu e-cy
'G Hz;
Figure 5.14: Calculated image impedances of a loaded meander line for Cr=l-5
p F ( - -), 3.3 p F (• •), 4.8 p F (- •), 10 p F ( - • • •), and short link (—) with a
= 3.5mm, b = 2.0mm, L = 100mm, d = 1.27mm, and e r = 10.2.
142
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diodes are loaded on the upper side of structure (replace Z i) and Z i is replaced by
a short link, then this structure becomes a meander-line bandreject filter when the
diode is “on” and becomes a hairpin-line bandpass filter when the diode is “o ff’.
Thus a two-staged filter with the same bandwidth in one circuit can be constructed.
Figure 5.15 demonstrates the measured transmission loss of a meander-line and
hairpin-line filter with the same circuit dimensions. We see that the passband and
stopband of the structures can be switched with the same bandwidth by controling
Z i in Figure 4.2. This idea can be realized by loading PIN diodes into the structure
in future work according to the good measured results of Figure 5.15.
5.6
Conclusions
This chapter develops accurate and efficient models for microstrip
grating elements such as meander lines, hairpin lines, and comb lines for compact
filter applications. The computed results are verified by experiment and compared
well. Especially, a loaded meander-line structure provides application in voltagetuned filters. These grating filters promise to deliver sharp attenuation rate at
stopband or passband edges which is necessary in practical design. The only thing
needed to be improved is the loss factor. To well design the grating filters, the most
important thing is that one should select the optimum circuit parameters to provide
proper image impedance to match the characteristic impedance on both ports such
that the impedance-mismatched loss can be minimized. To be more accurate, the
143
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h,/11
" r e c u e r c y (GHz)
Figure 5.15: Measured transmission loss of a meander-line bandreject filter (—)
and hairpin-line bandpass filter (• •) with the same circuit dimensions: a = 2.5mm
, b = 2.0m m , L = 5 0 m m , d = 1.27mm, w = 9 0 m m , and e r = 10.2.
I
144
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model o f the side edge effect should be considered seriously. The overall
performances of filters presented in this chapter are able to be further improved by
optimum design. All the measured results we show in this chapter are performed on
HP 8510 using full two-port calibration in coaxial connectors. For accurate
measurement in microstrip circuits, the TRL calibration should be used by taking
into account the effects on both ports.
145
i
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CHAPTER 6
SUMMARY AND FUTURE WORK
The main contribution in this dissertation is the development of
equivalent boundary conditions (EBCs) of a periodic strip grating lying at the
interface between two media. The obtained EBCs taking into account the phase shift
between adjacent strips are derived from the modified method of homogenization
based on the technique of multiple scales. This technique is employed to expand the
scattered fields with a phase shift from a strip grating in powers of grating period p
and this process leads to solving the static electric and magnetic boundary-value
problems to obtain EBCs.
The derived EBCs with a phase shift are then used to investigate the
propagation characteristics o f surface waves along a periodic strip grating on a
grounded substrate. By using EBCs, the analysis of the periodic structure becomes
less complicated and more efficient compared to the numerical method used in [5759]. Having the propagation characteristics of metal gratings on grounded substrates,
planar grating elements loaded or unloaded with lumped devices are then
investigated and the results show the stopband or passband properties which are
characteristics o f periodic structures. Then grating filters are designed by using
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equivalent transmission line (ETL) models and verified experimently. The impact of
these compact grating filters is to promise to deliver either broadband properties or
highly selective behaviors. In this analysis, the key limitations are p /d will not be
too large and the period length p of the grating is much smaller than the wavelength
in free space due to the homogenization process. In all the design through this thesis,
the loss factor and the model of side edge effect o f the structure are not considered.
This will be very important in practical design and could be the future subject.
Finally, it should be very possible to extend the grating filters to other
prominent transmission-line structures, such as coplanar waveguide (CPW),
stripline, and multi-layer microstrip. For some other grating elements application for
planar microwave circuits, such as switches and pulse shapers, the developed EBCs
need to be used to efficiently model metal grating on substrates and this will be an
interesting subject.
147
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I
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APPENDIX A
EVALUATION OF EQUATION (2.87)
The goal o f Appendix A is to evaluate the expression shown in equation
(2.87). Ivanov showed that [10]
n=c»
2
—/2nJtv'
/ ’n ( « » 4 )[ 5
-/2 (n + l)n v '
«.
— + i --------- —] = ey 2 S » y '_ _ _ ^ _ s (cosA ')
n+5
n + 1 -5
smott
(A .l)
where 0 < j27ty'| < A, A' = 7t - A, and 6 is not an integer. Taking the real part of
both sides o f (A. 1), we have
cos 2n8y' —- — P_g (cos A')
sinSre
^ /
A\rCOs2n7tv' cos2(n + l)7ty',
= S P n(cos A)[------ f - +
V
i ]
£0
n +5
n + 1 -5
r
A.cos2n7iy'
= £ P n(cos A)
n=0
n+0
IS ° ^
0052(0 + 1)79/'
J P n(cosA)
^
>*
n=0
n + 1 -5
Let n -+ n - 1 in the second term of right-hand side of (A.2), we get
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A-2)
I
cos27t5y'—- — />_§(cos A')
sin 87c
o r
* \ c° s 2 n it/
n
/
A,c o s 2 n7cy'
= X P n (c°sA )c. + Z P n-lC cosA )
- fn=0
11 + 5
n=l
(A.3)
11-5
or
"v n t
2
;
Pa
n=l
...cos2miy'
(COS A ) ---------— i - +
2
..c o s 2n7ty'
/ ^ ( c o s A ) -----------
n+8
=
7rcos27t5y'
n -5
,
. .
(A.4)
I
r - r - ^ - P - s i c o s A ') - sino7t
0
By letting 5 —> -5 in (A.4), we have
V °n r
ANcos2n7ty'
A.c o s 2n7ty'
2 ; P n (cos A)------- ■~—Jr 2 P n _ 1(cos A)
-~—
n=l
n -8
n^i
n+5
(A-5)
7t cos 27c5y' „ .
. 1
= ----- — — i»5(cosA') + r
-sino7t
0
Then subtracting (A. 5) from (A.4) yields
ny ° *^n(A)cos2n7ty' _ tc cos 2tt8y' R§ (A')
n2 - 8 2
25 sin 87c
1
52
Let y ’ = 0 in (A.6), we then obtain the identity shown in equation (2.87)
159
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(A.6)
n=oo
y
S n ( A ) _ 7C/?g(A>)
n=l
n2 -8 2
25 sinSre
1
(A.7)
52
where Sn(A) = P n _ l ( c o s A ) - P a(cosA), /?5 (A') = P_5 (cosA') + P 8 (cosA'),
A = bit/p, and A' = aiz/p.
160
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APPENDIX B
EVALUATION OF EQUATION (2.125)
Appendix B finds the first-order z -component boundary-layer electric
field on the surface o f grating, as shown in equation (2.125) for
Substituting
(2. 121) into equation (2.119), we then have
i
-2 V - 0 = U®
d K „ It ? 3 <?-y2n7iy' „/2imy'
+ a 11— ) 1 Z [ + 2 S T ¥ 19 »
(B .l)
r no
. dr)
" r - e - J ' 2n*y' eJ2imy \
where
S n = T ^ n - l ( cos A ) + F’nCcos A)] = ^ -fln (A )
2
2
cpn = P n( c o s A ) - P n _ 1(cosA) = - 5 n (A)
(B.2)
; & = bn/p
Now, we have to evaluate the summation terms in (B .l) first. Substitute
in (B.2) into (B.l), the summation terms become
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and
ny ° r ^ — — gy2n7ly'n
2nit+0 + 2n7C-0
1 n^ °
g-y'2njty'
{Z
ey2n7tv'
=—
[^n(cos A)------— +Pn- i( cos A)------— ]+
4tt n=i
n+5
n -5
ejinny'
^2°
(B.3)
p-jinny'
Z
t^n(cosA)------— + /3n_ 1(cosA)----- —
“ l
n -5
n+5
]}
and
ejinny'
ny ° - e -y2n7ty'
£i
2n7t +0
+ 2n7t -0
1 I\z ?
=— { Z
2k
n
ey2n7ry'
p-jinny'
I A ( c° s A )------ — +/>n_ l(c o s A )------ — - ] -
n -5
^
n+5
(B.4)
“2^°
e-jinny'
ej2nny'
X [PnCcosA)------— — + P n_ l (c o sA )----- — ]}
n=i
n+5
n—5
where 5 = 9 /2 tc. We have to make use of (A.1) to obtain (B.3) and (B.4). After
regrouping the terms in the summation in (A.1), (A.1) can be rewritten as
n_z?°
e -jin n y'
pjln n y’
n -«
£ P n(cosA)-------— + £ P n _!(cosA )
—
n=i
n+8
n=1
n -5
n ejin n y’
i
= — -------- P _ 5 ( cosA ' ) - sin57t
^
5
Also replace 5 by 5 +1 in (A.1), with a little rearrangement we have
162
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nJZ?
g -jlr a iy '
n=°°
p jltm y '
£ P n _i(cos A)
— + £ Pn (cosA )
—n-,
n+6
n-8
k
1
eJ'2my '
= - = ^ - * - ' (“ s 4 ') + r
Substitute (B.5) and (B.6) into (B.3) and (B.4), we obtain
n“f° p -jinn y'
pj2nvy'
p j’& ny'
and
It2 ° -e-jln xy' eJlnxy'
1 1 trpj^Snty'
S [ - - ‘ J . - + * ----- „]<Pa = i [ i - S ^ - / - « S(A')]
£=l 2n7t +6
2n7t - 0
7t 5 2 sin57t
(B.8)
Then substitution o f (B.7) and (B.8) into (B .l) yields (2.125),
- eer|*'=o
l
pjln 8 y'
K
W
a ^ ) W
I
A
' )+-
n
r)
n
la
(B.9)
1 ireJ'2n8y'
5
2 smote
where identities P _ s _ 1(cosA ') = />5 (cosA ') and
P _ 8 ( c o sA ')
= P s_ i(co sA ') are
used.
163
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APPENDIX C
EVALUATION OF IDENTITIES
Appendix C shows two important identities which are helpful to evalute
equation (2.138). Ivanov [10] gave the following identity
^
n( c o s A ) [ ^ ^ + ^
n=0
^
]
=0
A <| / | <
(C l)
Multiply both sides by e'Jt5 and integrate this equation with respect to t from 2izy'
to 7t, we obtain another identity
a-f°
e -j2n(n+8)y'
£ Pn(cosA)[S
n—o
n"°
gj2n (n + l-5)y'
-------- 1------n+5
n + 1 -5
(C.2)
= n=0
Z ° (- 0 " ^ i.( cosA) [nr+~oT *nr+r ri T
—o^ " - /"6
Use the following formula [ 10]
! _ ] ( _ i ) « = _ J L - f 6 (cosA)
"If />n(c o s A )[-!------------5 -n
n + 1+5
sinSre
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(C.3)
By making 5 —>-5 in (C.3) and substituting into (C.2), then (C.2) becomes
n^ °
,
,
X /»n(cosA)[
i^O
e -/2rc(n+5)y'
ej2 n (n + \-§ )y'
— ----------------- — — ]
n+5
n + I-5
(C.4)
7t e
= - — — /*_5 (cos A)
smS7t
After a little rearrangement, (C.4) yields
ay P n.-l(cos A) ej2my' _ y ° ^ n (cos A) g-y?nny»
n^l
n -5
£i
n+5
T
i
7t
ej^ny'-7t)5
(C.5)
; A < |2jty '|< it
Also replace 5 by 5 +1 in (C.4), we obtain the identity
y 0 /!n-l(£ps A) ^ _y2n^y' _ ny
n=l
n+5
P n(cos A) / 2my*
n -5
- i 7t e j(2ny' ~n ^
= T~ + ------ r 1 -------P 5 (cos A)
5
sm 57t
(C.6)
; A< 2 7 t/< 7 t
where p_g _ ^ co sA ) = PgtcosA ) has been used in obtaining (C.5) and (C.6).
Adding and subtracting o f (C.5) and (C.6) respectively, we obtain the following
important identities
165
i
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e -jinn y'
I
ejinny'
; l _ _ +£
(
~ „j(lny'-n)S
C
.
7
)
n=l
and
n=oo e jinny'
e -jinny'
2
n p lO ^ y '-^
I t
<c -8>
n=l
where S n , S5 , R n , and R$ are shown in Appendix A or B.
166
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APPENDIX D
GRATING VOLTAGES AND CURRENTS
In this Appendix, the voltages and current at the surface of grating due
to the zeroth order boundary layer fields are evaluated; two grating voltages will be
defined. First, consider the voltage difference between the strip grating and -oo
due to the vertical component o f the boundary layer electric fields. We will call this
the boundary-layer voltage.
From (2.104), the boundary-layer voltage at the strip grating with respect
to —oo, V h , which is function of y ' is
jri
i
znk +0
znrc - o
ifr)
2
[
2mt +0
2n7i - 0
With K n and L n given in (2.105), then (D .l) becomes
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pK„\ . „ n=co
-jin n y'
jin n y’
S — - +f — - » .+
2ntt+ 0
2n7t - 0
2 sa
zrO
J
= o n=® r- e ~j2 m y ' e j 2 m y ' ,
—------- / [—
+ - -------- ]cp
2
L 2n7t + e
2 n r t - 0 JVn
( ° - 2)
p E yi\x'
where the summation terms in (D.2) have been obtained in (B.7) and (B.8). Equation
(D.2) then becomes
v b, 0”)
J K
^
2 ea
(D.3,
4 smo7t
2
57t
2sm57t
At y ' = 0 , (D.3) becomes
=
2sa
4 sin57t
2
07t
2 sinS7t
( d .4)
The current If, flowing in z -direction on the strip centered at y ' = 0 due
to zeroth order boundary layer magnetic field can be obtained from integrating the
boundary layer current density over the strip, i.e.
h =
I
!
i
“ hy_]e~jQy/Pdy
on grating
168
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(D.5)
From equation (2.99), hy on the grating is
(D6)
h ^ = — 2 ][ —n +-j F n e~J2n7zy/p+ Gn a=l
2
2
where Fn and Ln are defined in (2.100). Substitute ( D .6 ) into (D .5 ), after some
manipulation we have (assume
_
P K h \ x ’=o
= |Iq )
V f . sinQi-SjA ^ sinpi+SjA^
2it
n -5
n+5
(D.7)
j p ( ^xi|jr'=0)
sin(n+5 )A sin(n-5 )A, . . .
---------------- 2 - l---------- ;----------------; — Jon(A)
Ho*
n=l
n+8
n -5
The summation terms in (D.7) can be evaluated by the following
identities. From (C.4), after a little arrangement in (C.4) we have
g - j l n (n + 5)y
£ i>n (cosA) ------ —
n +5
n=«
£ P n -i(cosA )
i£l
ej2n(n-5)y'
---n"5
Tte- ^ 6 n ,
., e ' W
= . g P - s (cos A )
----smSrt
6
Taking the imaginary part of (D .8 ) yields
169
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(D.8)
”f [/>.(«. ttfZ& ULpX. +
n=i
ri»2.c-<y
n +o
n -5
_ .
..
sin(27c8.y')
= n P _5 (cos A )
— —
(D.9)
By letting 8 -> -6 in (D.9) we have
V °rD (
AxSin27c(n-8)y
.. sin27t(n+5)y'
2 t^n(C0SA)
-V
+ ^n-l(C0S A)
]
nTj
n -5
n+5
(D.10)
= x/>5 ( c o s A ) - ^ ^
Substitute (D.9) and (D.10) by replacing y ' by b/2p into (D .7 ), the current on the
center strip due to zeroth order boundary layer magnetic field is obtained as
lh =
PKh\X’ = o
2sin(A5) 1 JP B xi^o
T
[« Rs (A) --------------- + ---------1— S5 (A)
lK
5
Ji0
(D. 11)
where
'
^ k = o = ( //? + + ^ + - ^ ? - V = °
(D .12)
4 y =0 = H o ( ^ ++ ^ + V=o =
Note that If, obtained in (D .ll) is equal to the integration o f the boundary-layer
surface current density shown in (2.97) with respect to y from -6 /2 to 6/2.
170
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Finally, consider the voltage difference due to the transverse component
o f the zeroth order boundary layer electric fields. Let Vi be the voltage on the strip
at y ' = 0 and Vie~J® is the voltage on the strip to the right o f y ' - 0 due to the
boundary layer electric fields, then we have the voltage difference between these two
strips
Ve(e~JQ - 1 ) =
where
J 1bj2ey+e~-iQy'd y
011 the strip
(D. 13)
is continuous in the gaps and zero on the strips and the fast variable y ' is
replaced by y [ p . Then substitute e® in equation (2.104) into (D.13), we have
Ve(e~JQ - 1)
=
J~
^
a e - j (2tat+Q)y/Pdy+
2
(D. 14)
r bl2ej(2m -Q)y/pdy]
*b/2
where the integrations in (D. 14) are carried out as
ra+b/2
...
.
-y'(n+5)(27t-A)_ -y'(n+S)A
A= j
e -y(2nji +Q)y/P(fy = j p t -------------------- e-----------Jb/2
r
2mt +0
°
r a+bn
/
^ (n -S )(2 7 i-A )_ ^ (n -S )A
B= I
eJ(2m ~Q)ylPdy = - j p —----------------------------I**
*
Jy
2 hk - 0
171
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(D.15)
Then (D. 14) becomes
U=«J T
r X e - f l - l ) * Z [ ± ? - ( A + B) + ^ - ( 3 - A ) l
n=l 2
2
(D.16)
where
■R«
* \ e -yn(27i-A) ^ n (2 * -A )
/nA
-ynA
A + B = jp { e -J 5 V n - V [ - -----------------] + e-yA5 |- _ £ ------------ e--------
2n?t +0
2n7C- 0
2n7t - 0
2nrc +0
•so* a\ eyn(27l-A) e- M 2n~ V
.AS J * *
e~jnA
B - A = -jp{e~J?>(27t ~A) [ +
] - e~J--- [— ------- + —------- ]}
2n7t - 0
2 nrc +0
2n7t - 0
2ntt +0
(D.17)
and L n and K n are shown in (2.105). Substitute (D.8), L n , and K n into (D.16)
with the aid of (B.7) and (B.8), after some manipulation we have
,-y'A5 _ ^/A5 g-ye
1
+
£%r J
]
k8
(D.18)
11=50 /X
2
n=l
-J ± ( B - A ) = o
2
Then (D. 16) becomes
,-y'A5 _ gj'AS e -ye
7t6
]
172
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(D.19)
or the voltage on the strip located at y ' - 0 due to the zeroth order boundary-layer
electric field e£ is
173
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