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CHAPTER 5
TUNABLE FILTERS
5.1
TUNABLE FILTERS IN COMMUNICATION SYSTEMS
There are two kinds of filters in electrical circuit design: fixed and tunable. A fixed
filter pass band is fixed and well-defined, while a tunable filter pass band can be
tuned over a certain frequency range.
The theory for a fixed filter design was comprehensively developed many decades
ago. The engineering developments of passive filters, including the LPF (Low-Pass
Filter), HPF (High Pass Filter), BPF (Band Pass Filter), and BRF (Band Reject
Filter), have been formulized and tabulated for the design with discrete parts.
This simplifies the design procedures. The only problem remaining is how to
convert the values of parts into reasonable ones through the so-called “Δ-*” or
“π-T ” transformation.
In past years, the frequency bands assigned for a communication system rose
higher and higher. It became more and more difficult to implement filters by discrete
parts because the required values of either capacitors or inductors became too small
for the production line. Many types of filters with distributed parameters were
developed, such as the crystal filter, ceramic block filter, micro strip line filter, and
SAW filter.
The theory for a tunable filter design was developed many decades ago also. Its
implementation, however, faces some hurdles. This is one of the reasons why it, and
not another type of filter, was selected as the topic of this book. Today, the SAW
filter is very often applied to a communication system, whether a dual conversion
or a direct conversion system. As a matter of fact, a lot of effort is being put into
the development of a SAW tunable filter at present.
RF Circuit Design, by Richard Chi-Hsi Li
Copyright © 2009 by John Wiley & Sons, Inc.
219
220
TUNABLE FILTERS
In this chapter we are going to show how to remove the main obstacles in the
tunable filter design. It might be helpful to provide some clues to assist in the design
of a SAW tunable filter or other type of tunable filter, although in our example the
tunable filter is implemented by discrete parts. This is the second reason to include
this chapter. (The content of this chapter is abstracted from an US patent which was
obtained by the author in 1992.)
5.1.1
Expected Constant Bandwidth of a Tunable Filter
In a communication system, a customer may occupy only one channel, although
the number of available channels is usually huge. In other words, the bandwidth
that a customer needs is much less than the entire bandwidth of the system. For
example, in a UHF portable radio, the entire bandwidth is assigned to be from 403
to 520 MHz, while a customer occupying bandwidth for voice conversation needs
no more than 25 kHz. To approach good performance, the ideal condition is to
assign 25 kHz only to each customer by a tunable filter while the rest of the bandwidth is rejected. All of the hardware is the same for each customer: the difference
lies only in the control voltage being provided to the tunable filter for each individual customer. Consequently, the goals of good performance and mass production
can be realized simultaneously.
In practical RF tunable filter design, it is possible to request that the bandwidth
be narrowed down to 10 MHz or so, but it is almost impossible to narrow the bandwidth down to the order of around 25 kHz. Nevertheless, narrowing the bandwidth
is still very helpful in improving the selectivity of the receiver, reducing noise, and
preventing a variety of spurious interference. As long as the bandwidth can be narrowed down somewhat from the entire bandwidth of the system, the performance
of the portable radio can be appreciably improved. Without a tunable filter functioning in the front end of the receiver, spurious interference and noise may be
significant.
Figure 5.1 shows how the tunable filter operates in a portable radio. When the
control voltage, CV, is moved from its minimum, CVL, to its maximum, CVH, the
capacitance of the varactor, C, is changed from its maximum, CH, to its minimum,
CL, and the central frequency of the tunable filter, fo, moves from its minimum, foL,
to its maximum, foH, correspondingly.
To enable a tunable filter tuned over a wide frequency tuning range to be realistic,
the bandwidth should be kept constant as shown in Figure 5.1 as much as possible
when the central frequency, fo, is tuned.
5.1.2
Variation of Bandwidth
Unfortunately, many tunable filter designs are far from ideal; the bandwidth
varies significantly as shown in Figure 5.2 when the tuning frequency is tuned. In
Figure 5.2(b), at the low end of the frequency range, the bandwidth of the tunable
filter is narrow, as expected. As the control voltage is increased, the capacitance of
the varactor is decreased and hence the central frequency of the tunable filter
increases. On the other hand, its bandwidth also increases. Up to the high end of
the frequency range, the bandwidth could be widened to an unacceptable amount,
and the “tunable” in “tunable filter” becomes meaningless. In Figure 5.2(c), the
221
COUPLING BETWEEN TWO TANK CIRCUITS
CH
Capacitance of varactor, C,
Central frequency, fo
foH
C
fo
foL
CL
CV
CVL
CVH
(a) Variation of the varactor capacitance and the central frequency as the control voltage is varied
P
fo,L
fo,H
fo,i
fo
(b) Central frequency is moved to higher frequency with constant bandwidth as the control voltage is
increased
Figure 5.1 Expected constant bandwidth as the control voltage is tuned from low to high
voltage.
variation of bandwidth is just opposite from that in Figure 5.2(b). At the high end
of the frequency range, the bandwidth of the tunable filter is narrow, as expected.
As the control voltage is decreased, the capacitance of the varactor is increased and
the central frequency of the tunable filter is lowered, but, on the other hand, its
bandwidth is increased. At the low end of the frequency range, the bandwidth could
be widened to an unacceptable amount, by which the “tunable” in “tunable filter”
likewise becomes meaningless.
5.2
COUPLING BETWEEN TWO TANK CIRCUITS
A RF tunable filter is a BPF (Band Pass Filter) with a variable central frequency.
Typically, an RF tunable filter in a communication system consists of two tank
circuits coupled by one or more coupling parts. Figure 5.3 shows its blocks.
The input and output tank circuits are implemented by an inductor and a capacitor in parallel. In order to tune the central frequency, the capacitor is a special one
called a varactor, in which its capacitance is varied and tuned by a common control
voltage, CV. Both the input and output tank circuits must be resonant at the same
frequency and hence must be identical. Should the tunable filter be built by more
than one tank circuit, its frequency response would be rolled up and down more
sharply, and within the bandwidth of the pass band its frequency response would be
222
TUNABLE FILTERS
CH
Capacitance of varactor, C,
Central frequency, fo
foH
C
fo
foL
CL
CVH
CV
CVL
(a) Variation of the varactor capacitance and the central frequency as the control voltage is varied
P
fo,i
fo,L
fo,H
fo
(b) Central frequency is moved to higher frequency with wider bandwidth as the control voltage is
increased
P
fo,i
fo,L
fo,H
fo
(c) Central frequency is moved to higher frequency with narrower bandwidth as the control voltage is
increased
Figure 5.2 Variation of the bandwidth as the control voltage is changed.
CV
Input
Input
Tank
Circuit
Coupling
parts
Output
Tank
Circuit
Figure 5.3 Blocks of a RF tunable filter.
Output
COUPLING BETWEEN TWO TANK CIRCUITS
223
CV
Co
R1
R1
Out
In
L1
VR1
(C1)
VR1 L2
(C1)
L2
C3
VR1
(C1)
VR1
(C1)
L1
(a) Tunable filter implemented by two tank circuits and coupled by capacitor only
CV
Co
R1
R1
L3
In
L1
VR1
(C1)
C3
VR1 L2
(C1)
Out
L2
VR1
(C1)
VR1
(C1)
L1
(b) Tunable filter implemented by two tank circuits and coupled by LC in series
CV
Co
R1
R1
L3
In
L1
VR1
(C1)
VR1 L2
(C1)
Out
L2
C3
VR1
(C1)
VR1
(C1)
L1
(c) Tunable filter implemented by two tank circuits and coupled by LC in parallel
Figure 5.4 Three coupling types of a tunable filter.
more flattened. However, the uniform requirement of the tank circuits restricts their
number. The more the number of tank circuits, the more difficult it is to maintain
their uniformity. This is why most RF tunable filters are implemented by only two
tank circuits.
Figure 5.4 shows the schematics of an RF tunable filter with three different coupling styles. Instead of one inductor and one capacitor applied in one tank circuit,
the tank circuit consists of two inductors and two varactors. This makes it convenient
to apply the control voltage to the varactors and has the additional advantage of
simplifying impedance matching to the input and output terminals. Figure 5.4(a)
shows a tunable filter implemented by two tank circuits and coupled only by
224
TUNABLE FILTERS
capacitor. Figure 5.4(b) and (c) shows tunable filters implemented by two tank circuits and coupled by an LC in series and in parallel respectively.
It should be noted that the tank circuits shown in Figure 5.4 have the same topology. The capacitors or varactors of the tank circuit are arranged as an arm in series
with input or output terminals, while most of the inductors are connected as branches
in parallel with the input or output terminals. Such a topology leads to a frequency
response with a higher slope to the lower side of the pass band and a response with
a lower slope to the higher side of pass band. Such a tunable filter is therefore more
appropriate to be cooperated with a low-side injection mixer. On the contrary, if
the capacitors or varactors of the tank circuit are arranged as a branch in parallel
with the input or output while most of inductors are connected as arm in series with
the input or output, it would result in a frequency response with a higher slope in
the higher side of the pass band and a lower slope in the lower side of the pass band.
Such a tunable filter is more appropriate to be cooperated with a high side injection
mixer. In the following discussion, we will focus on the topology appropriate to the
low-side injection mixer only.
5.2.1
Inappropriate Coupling
The tunable filter with three coupling types as shown in Figure 5.4 was designed in
years past. One of their common problems is, as mentioned above, that the bandwidth changes significantly from the low end to high end of the frequency range.
Through a simple circuit analysis one can understand why this is so. As a matter
of fact, the tank circuits are always tuned up to resonance at the central frequency.
Therefore, the two tank circuits look like two small resistors, ro, connected with the
main coupling parts in series since the inductor and the capacitor are in resonant
status so that their reactances are “neutralized” with each other. The tunable filters
with three kinds of main coupling types as shown in Figure 5.4 can be replaced by
their equivalents shown in Figure 5.5 when the tank circuits are resonant at the
central frequency.
The bandwidth of the tunable filter is mainly determined by the Q value of the
tank circuit. In cases where the main coupling part is the capacitor C only, as in
Figures 5.4 (a) and 5.5(a),
Q=
fo
1
1
=
=
,
BW 2roCω o 4π roCfo
(5.1)
where
Q = unloaded Q of the tunable filter;
fo = central frequency of the tunable filter;
ωo = central angular frequency of the tunable filter;
BW = frequency bandwidth of the tunable filter;
ro = equivalent resistor when tank circuit is resonant at the central frequency.
From (5.1),
BW = 4π roCfo2,
(5.2)
225
COUPLING BETWEEN TWO TANK CIRCUITS
Out
In
ro
ro
C3
(a) At fo, two tank circuits are in resonant state and coupled by C only
L3
In
C3
ro
Out
ro
(b) At fo, two tank circuits are in resonant state and coupled by LC in series
L3
In
Out
ro
ro
C3
(c) At fo, two tank circuits are in resonant state and coupled by LC in parallel
Figure 5.5 Equivalent circuit of a tunable filter with three coupling types in resonant state
when f = fo.
Then,
∂ ( BW )
= 8π roCfo.
∂fo
(5.3)
In cases where the main coupling part is an LC in series as in Figures 5.4(b) and
5.5(b),
1
Lω o −
fo
Cω o 4 LCπ 2 fo2 − 1
Q=
=
=
,
(5.4)
BW
2ro
4π roCfo
From (5.4),
BW =
4π roCfo2
,
4π 2 LCfo2 − 1
(5.5)
226
TUNABLE FILTERS
Then,
∂ ( BW )
8π roCfo
=−
.
2
2
∂fo
( 4π LCfo2 − 1)
(5.6)
In cases where the main coupling part is an LC in parallel, as in Figures 5.4(c) and
5.5(c),
Lω o
π Lfo
fo
1 − LCω o2
Q=
=
=
.
(5.7)
2ro
BW
ro (1 − 4π 2 LCfo2 )
From (5.4),
BW =
then,
(1 − 4π 2 LCfo2 ) ro ,
πL
∂ ( BW )
= −8π roCfo.
∂fo
(5.8)
(5.9)
As shown in equations (5.3), (5.6), and (5.9), in all three coupling types of tunable
filters shown in Figure 5.4, the variation of the bandwidth is dependent on the
central frequency, fo.
Where the main coupling part is the capacitor C only, as shown in Figures 5.4(a)
and 5.5(a), the variation of bandwidth is proportional to the central frequency fo as
shown in equation (5.3). The bandwidth becomes wider and wider as the central
frequency increases. It may become unacceptably wide at the high-frequency end
as shown in Figure 5.2(b). In the case where the main coupling part is an LC in
series, as shown in Figures 5.4(b) and 5.5(b), the variation of bandwidth is dependent
on the central frequency by a complicated function. In the case where the main
coupling part is an LC in parallel as shown in Figure 5.5(c), the variation of bandwidth is negatively proportional to the central frequency. In both the cases shown
in Figures 5.4(b) and 5.5 (b), and 5.4(c) and 5.5(c), the bandwidth becomes wider
and wider as the central frequency is decreased. It may become unacceptably wide
at the low-frequency end as shown in Figure 5.2(c).
It is therefore concluded that, generally speaking, all three coupling styles shown
in Figure 5.4 are inappropriate in the tunable circuit design.
5.2.2
Reasonable Coupling
A reasonable coupling approach is shown in Figure 5.6, in which the coupling
part between the two tank circuits consists of only an inductor L. When this filter
is operating at its central frequency, its equivalent circuit operates as shown in
Figure 5.7.
Similar to the derivation as in Section 5.2.1, we have
Q=
fo
Lω o Lπ fo
.
=
=
2ro
BW
ro
(5.10)
From (5.10),
BW =
ro
,
Lπ
(5.11)
227
CIRCUIT DESCRIPTION
CV
Co
R1
R1
L3
In
L1
VR1
(C1)
Out
VR1 L2
(C1)
L2
VR1
(C1)
VR1
(C1)
L1
Figure 5.6 Tunable filter implemented by two tank circuits and coupled by L only.
L3
In
Out
ro
ro
Figure 5.7 At fo, two tank circuits are in resonant state and coupled by L only.
then,
∂ ( BW )
= 0.
∂fo
(5.12)
The expression (5.12) leads to an important conclusion: The bandwidth of a tunable
filter can be kept unchanged over the entire frequency range if the main coupling
parts between the two tank circuits consist of only a pure inductor L.
5.3
CIRCUIT DESCRIPTION
Figure 5.8 shows the schematic of a tunable filter which will be cooperated with a
low-side injection mixer in a UHF portable radio. The main feature of the circuit is
the pair of identical tank circuits coupled by an inductor, L3. The simplest tank
circuit consists of one inductor and one capacitor in parallel. As shown in Figure
5.8, the tank circuit consists of three inductors, two L1 inductors, and one L2 inductor, as well as two VR1 varactors. The purpose of applying three inductors, instead
of only one, is twofold: 1) To match the impedance of the input or output, 50 Ω,
without additional impedance matching parts; 2) To avoid too tight a coupling
between the two tank circuits. In each tank circuit, two identical varactors are piggybacked together. The tuning function is performed by these varactors. The capacitances of these varactors are controlled by the control voltage, CV, via the resistor
R1 = 20 kΩ, through which there is no current flowing.
The capacitor Co is a “zero” capacitor in the UHF frequency range.
Another remarkable achievement in this tunable circuit design is that there is a
second coupling capacitor, C2, which creates two “zeros” in the skirt portion of the
228
TUNABLE FILTERS
CV
Co
R1
R1
C2
L1
L1
In
L1
L1
L2
L3
C1
:
:
:
:
VR1
(C1)
A VR1
(C1)
L2
Inductor 4.5 nH // 27 kW // 0.2 pF
Inductor 8.8 nH // 27 kW // 0.2 pF
Inductor 40.5 nH // 27 kW // 0.2 pF
Capacitance of varactor 13.75 pF
when CV=5V
L3
L2
VR1
(C1)
B
Out
VR1
(C1)
C2 : Capacitor 2.4 pF
Co : Capacitor 150 pF
L1
R1 : Resistor 20 kW
CV : 5 V
Figure 5.8 Schematic of a UHF tunable filter.
frequency response plot. One is located below the pass-band and another above.
These two “zeros” can be adjusted by C2, L2, L3, and VR1, and can be applied to
trace the imaginary spurious products in either the high side or low-side injections
if the receiver is not operating in direct conversion mode. In the pass-band, the effect
of the second coupling is negligible. Now, let’s analyze how the second coupling
works.
5.4
EFFECT OF SECOND COUPLING
Let’s introduce the “π-T” or “Δ-*” transformation of the impedance.
Figure 5.9 shows the process of impedance transformation from the original
network (a) to the final network (c).
The corresponding expressions between the impedances shown in Figure 5.9
are:
Z2′ =
Z2 Z3
,
Z2 + Z3 + Z4
(5.13)
Z3′ =
Z3 Z4
,
Z2 + Z3 + Z4
(5.14)
Z4′ =
Z4 Z2
,
Z2 + Z3 + Z4
(5.15)
and
Za = Z1 + Z2′ + Z4′ + Z4′
Z1 + Z2′
,
Z3′ + Z5
(5.16)
EFFECT OF SECOND COUPLING
Z3
Z1
229
Z5
Z2
Z4
(a) Original network
Z1
Z2′
Z3′
Z5
Z 4′
(b) Intermediate network
Zm
Za
Zb
(c) Final network
Figure 5.9 Z2, Z3, and Z4 should have primes, not apostrophes.
Z3′ + Z5
,
Z1 + Z2′
(5.17)
( Z1 + Z2′ ) ( Z3′ + Z5 )
.
Z4′
(5.18)
Zb = Z5 + Z3′ + Z4′ + Z4′
Zm = Z1 + Z5 + Z2′ + Z3′ +
If
Z1 = Z5,
(5.19)
Z2 = Z4,
(5.20)
then
Z2′ = Z3′ =
Z4′ =
Z2 ⋅ Z3
,
2Z2 + Z3
Z22
,
2Z2 + Z3
Za = Zb = Z1 + Z2,
(5.21)
(5.22)
(5.23)
230
TUNABLE FILTERS
2
Z
Z ⎤
⎡ Z
Zm = 2Z1⎛ 1 + 3 ⎞ + Z3 ⎢1 + 12 ⎛⎜ 1 + 2 2 ⎞⎟ ⎥.
⎝
Z2 ⎠
Z3 ⎠ ⎦
⎣ Z2 ⎝
(5.24)
Now let’s apply the “π-T” or “Δ-*” transformation of impedance to the network
between the nodes of A and B in Figure 5.8. The corresponding impedances are
1
,
jω C1
(5.25)
Z2 = Z4 ⇒ jω L2,
(5.26)
Z3 ⇒ jω L3,
(5.27)
Z1 = Z5 ⇒
Substituting the relationships (5.25), (5.26), and (5.27) into (5.23) and (5.24), we
have
Za = Zb =
Zm =
1
+ jω L3,
jω C1
1
+ jω Lm,
jω Cm
where
Cm =
C1
L
2⎛1+ 3 ⎞
⎝
L2 ⎠
,
L
⎛
1+ 2 2 ⎞
⎜
L ⎟
Lm = ⎜ 1 + 2 2 34 ⎟ L3.
L2C1 ω ⎟
⎜
⎝
⎠
(5.28)
(5.29)
(5.30)
(5.31)
The equivalent schematic of the tunable filter for UHF portable radio as shown in
Figure 5.8 can be depicted as in Figure 5.10, which implies two possible “zeros” in
the tunable filter. One of them is contributed by the series network, Lm, Cm, and C2
and another exists in the parallel branch, VR1(C1) and L2. As a matter of fact, the
original idea to have the second coupling is due to the recognition of these two
“zeros”.
The equivalent coupling network consists of Lm, Cm, and C2. Its maximum impedance corresponds to a “zero” of the tunable filter at the frequency,
2
ω Lo
=
C2 + Cm
.
LmC2Cm
(5.32)
It is always lower than the central frequency of the tunable filter, ωo, that is,
ω Lo < ω o,
(5.33)
ωLo is therefore called the low side “zero” frequency where a “zero” appears.
In a practical design, the difference between ωo and ωLo is adjusted to about the
value of 2ωIF, so that ωLo could be treated as the imaginary frequency of the RF
EFFECT OF SECOND COUPLING
231
CV
Co
R1
R1
C2
L1
L1
A
In
L1
VR1
(C1)
B
VR1
(C1)
Cm
Out
VR1
(C1)
Lm
VR1
(C1)
L1
L2
L2
Figure 5.10 Equivalent schematic of a UHF tunable filter.
input, ωLir ,when a mixer is operated in a low-side injection mode. If the values of
the parts in the tunable filter are selected and adjusted carefully, the difference
between ωo and ωLo is kept almost unchanged when the central frequency is tuned
from the low-frequency end to the high-frequency end. This is a useful behavior of
the imaginary rejection in co-operation with a mixer design.
Another “zero” of the tunable filter is created by the equivalent branch in parallel
at either node A or node B as shown in Figure 5.10, which is formed by the parts
C1 and L2 that is,
2
ω Ho
=
1
.
L2C1
(5.34)
It should noted that the frequency ωHo is always higher than ωo, that is,
ω Ho > ω o,
(5.35)
ωHo is therefore called the high side “zero” frequency where a “zero” appears.
When the tunable filter is cooperated with a mixer operating in the low-side
injection mode, the “zero” at the frequency ωHo is not helpful to the imaginary rejection. However, it does help narrow the skirt of the entire frequency response
curve.
Figure 5.11 compares the frequency responses between two cases with and
without second coupling or, in other words, with or without the existence of the
capacitor C2. Both frequency response curves are tested when fo = 435.43 MHz.
Without second coupling, the skirt of the frequency response curve is wide open
and the corresponding low-side imaginary rejection is poor, that is,
Imag_Rejat low side = −29.2 dB,
down from the input signal level.
(5.36)
232
TUNABLE FILTERS
S21, dB
10
fo = 435.21 MHz
0
Lowside Imag_Rej.
= - 29.2 dB
-10
-20
-30
-40
-50
-60
-70
-80
Lowside Imag_Rej.
= - 74.2 dB
-90
fLo fLir
200
fo
300
400
fHir
500
600
fH o
700
f , MHz
Figure 5.11 Comparison of frequency response of tunable filter.
with second coupling
without second coupling
With second coupling, the skirt of the frequency response curve is much narrowed since two “zeros” are created. The corresponding low-side imaginary rejection is much better, that is,
Imag_Rejat low side = −74.2 dB,
(5.36)
down from the input signal level.
This is a 45 dB improvement! It is quite encouraging that such a small capacitor
C2 brings about such a huge advantage!
5.5 PERFORMANCE
The tested results of the tunable filter for a UHF portable radio are presented with
four cases when the central frequency is tuned at
Low end of frequencies:
Intermediate frequency 1:
fo1 = 403 MHz,
fo 2 = fo1 fo 3 = 435.21 MHz,
PERFORMANCE
S21, S11, dB
233
IL=- 1.88 dB
10
S21
0
-10
-20
-30
S11
-40
-50
Highside Imag_Rej.
= - 29.5 dB
Lowside Imag_Rej.
= - 74..2 dB
-60
-70
High side “zero”
= - 54.1dB
Low side “zero”
= - 85.1 dB
-80
-90
fLir fLo
200
fo
300
fHir
400
500
fH o
600
700
f , MHz
Figure 5.12 Frequency response of tunable filter when fo = 403.00 MHz.
Intermediate frequency 2:
High end of frequencies:
fo3 = 470 MHz,
fo4 = 512 MHz.
The frequency response of this tunable filter when fo = 403.00 MHz is depicted in
Figure 5.12. It shows that
•
•
•
•
Central frequency,
fo = 403.00 MHz,
Bandwidth,
BW = 33.5 MHz,
Insertion loss,
IL = −1.88 dB,
Image rejection applied for low-side injection, when fIF = 73.35 MHz,
Low side
at fLir = 256.3 MHz,
Imag_Rej = −74.2 dB,
“zero” = −85.1 dB,
at fLo = 267.5 MHz,
High side
Imag_Rej = −29.5 dB,
at fHir = 549.7 MHz.
at fHo = 653.1 MHz.
“zero” = −54.1 dB,
The frequency response of this tunable filter when fo = 435.43 MHz is depicted in
Figure 5.13. It shows that
234
TUNABLE FILTERS
S21, S11, dB
IL=- 1.76 dB
10
S21
0
-10
-20
S11
-30
Highside Imag_Rej.
= - 27.3 dB
-40
-50
Lowside Imag_Rej.
= - 74.2 dB
High side “zero”
= - 48.8 dB
-60
-70
-80
Low side “zero”
= - 76.8 dB
-90
fLo fLir
200
300
fo
400
fHir
500
600
fH o
700
f , MHz
Figure 5.13 Frequency response of tunable filter when fo = 435.21 MHz.
•
•
•
•
Central frequency,
fo = 435.21 MHz,
Bandwidth,
BW = 34.3 MHz,
Insertion loss,
IL = −1.76 dB,
Image rejection applied for low-side injection, when fIF = 73.35 MHz,
Low side
Imag_Rej = −74.2 dB,
at fLir = 288.73 MHz,
“zero” = −76.8 dB,
at fLo = 267.5 MHz,
High side
Imag_Rej = −27.3 dB,
at fHir = 582.13 MHz.
“zero” = −48.8 dB,
at fHo = 653.1 MHz.
The frequency response of this tunable filter when fo = 470.00 MHz is depicted in
Figure 5.14. It shows that
•
•
•
•
Central frequency,
fo = 470.00 MHz,
Bandwidth,
BW = 35.1 MHz,
Insertion loss,
IL = −1.56 dB,
Image rejection applied for low-side injection, when fIF = 73.35 MHz,
Low side
Imag_Rej = −63.1 dB,
at fLir = 323.3 MHz,
“zero” = −72.5 dB,
at fLo = 267.5 MHz,
PERFORMANCE
S21, S11, dB
235
IL=- 1.56 dB
10
S21
0
-10
-20
S11
-30
Highside Imag_Rej.
= - 27.3 dB
-40
-50
High side “zero”
Lowside Imag_Rej.
= - 63.1 dB
-60
-70
Low side “zero”
= - 72.5 dB
-80
-90
fLir fLo
200
300
fo
400
fHir
500
600
700
f , MHz
Figure 5.14 Frequency response of tunable filter when fo = 470.00 MHz.
High side
Imag_Rej = −24.8 dB,
“zero” = −? dB,
at fHir = 65.7 MHz.
at fHo = ? MHz.
The frequency response of this tunable filter when fo = 512.00 MHz is depicted
in Figure 5.15. It shows that
•
•
•
•
Central frequency,
fo = 512.00 MHz,
Bandwidth,
BW = 35.8 MHz,
Insertion loss,
IL = −1.42 dB,
Image rejection applied for low-side injection, when fIF = 73.35 MHz,
Low side
Imag_Rej = −65.1 dB,
at fLir = 365.3 MHz,
“zero” = −70.5 dB,
at fLo = 377.5 MHz,
High side
Imag_Rej = −24.2 dB,
at fHir = 658.7 MHz.
“zero” = −? dB,
at fHo = ? MHz.
It can be concluded from Figures 5.12 to 5.15 that when 403 < fo < 512 MHz,
236
TUNABLE FILTERS
S21, S11, dB
IL=- 1.42 dB
10
S21
0
-10
S11
-20
-30
Highside Imag_Rej.
= - 24.2 dB
-40
-50
Lowside Imag_Rej.
= - 65.1 dB
-60
High side “zero”
-70
Low side “zero”
= - 70.5 dB
-80
-90
fLir fLo
200
300
fo
400
500
fHir
600
700
f , MHz
Figure 5.15 Frequency response of tunable filter when fo = 512.00 MHz.
1) The insertion loss is low;
IL < l.88 dB,
(5.37)
2) The low side imaginary rejection is excellent if fIF = 73.35 MHz.
Imag_Rej, Lowside < −60 dB.
(5.38)
3) The insertion loss and the imaginary rejection are tunable or traceable!
REFERENCES
[1] William H. Beyer, Standard Mathematical Tables, CRC Press, 24th ed., 1976.
[2] Donald R. J. White, Electrical Filters, Don White Consultants, Inc., 1980.
[3] Richard Chi-Hsi Li, “Tunable Filter Having Capacitive Coupled Tuning Elements,” U.S.
Patent 5,392,011, Motorola Inc., 1992.
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