Direct Filter Synthesis Rhea Preview
Direct Filter Synthesis Rhea Preview
Direct Filter Synthesis Rhea Preview
xiv
References
[1] G. Matthaei, L. Young and E.M.T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Artech House, Dedham, MA, 1980.
[2] H. Orchard and G. Temes, Filter Design Using Transformed Variables, IEEE
Trans. on Circuit Theory, Vol. CT-15, December 1968, pp. 385-408.
[3] G. Szentirmai, FILSYN A General Purpose Filter Synthesis Program, Proc. of
the IEEE, Vol. 65, October 1977, pp. 1443-1458.
[4] A. Zverev, Handbook of Filter Synthesis, John Wiley, Hoboken, NJ, 1967
[5] G. Temes and S. Mitra, editors, Modern Filter Theory and Design, John Wiley,
New York, 1973.
[6] R. Rhea, HF Filter Design and Computer Simulation, SciTech Publishing,
Raleigh, NC, 1994.
[7] R. Rhea, Practical Issues in RF Design (Three CD Series), SciTech Publishing,
Raleigh, NC, 2003.
To Marilynn:
As just a boy, I made my choice,
I now look back, and proudly smile.
With wind in face, or at our backs,
hand in hand, we trod our paths,
some a thousand, some a mile.
Youthful dreams, some wise some poor,
but oh so sweet, the ones we store.
For Hera, only change endures,
not so we say, tis love evermore,
hand in hand, tis love evermore.
Contents
Preface
References
1 Transmission Zeros
1.1 Determining TZ by Inspection
1.2 Filter Degree
1.3 Canonical Realization
1.4 Influence of TZs on the Response
References
xiii
xiv
1
1
4
4
4
6
7
7
9
10
11
12
13
13
15
4 Conventional Bandpass
4.1 Bandpass Transform
4.2 Classification Symmetry or Antimetry
4.3 A 75- to 125-MHz Bandpass
4.4 A 96- to 104-MHz Bandpass Filter
4.5 Comparative Analysis of the Wide and Narrow Filters
Reference
17
17
17
18
19
19
21
5 Extraction Sequences
5.1 The Extraction Tab
Reference
23
23
27
29
29
33
34
34
7 Norton Transforms
7.1 Norton Series Transform
7.2 Removing a Transformer with the Series Norton
39
39
40
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43
44
45
46
47
47
49
51
53
55
56
57
58
58
59
61
61
63
67
9 TEM-Mode Resonators
9.1 Filter Insertion Loss
9.2 Filter Using 50-Ohm Coaxial Resonators
9.2.1 Lumped to Distributed Equivalents
9.2.2 The Convert Using Advanced Tline Routine
9.3 Generalized Bandpass Using Ceramic Resonators
9.3.1 Creating Parallel Resonators
9.3.2 Shifting the Internal Impedance Level
9.3.3 The Pi to Tee Transform: Increasing Coupling Caps
9.3.4 Converting the Parallel L-C to Coaxial Resonators
9.3.5 Optimizing the Values
9.4 Ceramic Bandpass with Two FTZs
References
69
69
70
70
72
74
75
76
77
77
77
78
81
10 Piezoelectric Devices
83
10.1 Quartz-Crystal Device Model
83
10.1.1 Physical Form of the Quartz Crystal
83
10.1.2 Insertion Response of a Quartz Crystal
84
10.1.3 Modeling the Quartz Crystal
84
10.1.4 Calculating Model Parameters from the Response 85
10.1.5 The Quartz-Crystal Model and Filter Design
86
10.2 Quartz-Crystal Filter Approximate Design
86
10.3 Nulling the Static Capacitance
90
10.4 Design of a Lower-Sideband Crystal Filter
91
10.5 Upper-Sideband Quartz-Crystal Filter
97
10.6 Filters with TZs Above and Below the Passband
103
Contents
10.7 Wide-Bandwidth Quartz-Crystal Filters
10.8 Very Wide-Bandwidth Quartz-Crystal Filters
10.9 Ceramic-Piezoelectric Resonators
Reference
ix
107
108
111
113
11 Symmetry
11.1 Physical Symmetry
11.1.1 A Lowpass Filter with FTZ Pairings
11.1.2 A Bandpass Filter with FTZ Pairings
11.2 Response Symmetry
11.2.1 All-Pole Symmetric Response Filters
11.2.2 Generalized Bandpass with Symmetric Response
11.2.3 Symmetry by FTZ Placement
11.3 Group-Delay Equalization
References
115
115
115
117
119
120
120
123
124
127
129
129
130
130
132
132
138
139
139
140
142
144
13 Distributed Filters
13.1 Comparing Distributed and Lumped Filters
13.2 The Genesys Microwave Filter Module
13.3 Distributed Synthesis Concepts
13.3.1 TLEs
13.3.2 Richards Transform
13.3.3 Kuroda Identities
13.3.4 Ikeno Transforms
13.3.5 Kuroda-Minnis Transform
13.3.6 Half-Angle Transform
13.3.7 Interdigital Transform
13.3.8 Combline Transform
13.4 Lumped to Distributed Equivalent Transforms
13.5 Inverters
13.6 The Convert Using Advanced TLine Routine
13.7 Box Modes
13.8 Introduction to Distributed Filter Examples
References
145
145
146
149
149
150
152
155
157
159
161
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162
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165
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166
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169
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204
205
211
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218
224
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228
233
238
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248
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249
252
258
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259
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261
263
266
268
Contents
17.2.1 Highpass with Three TZs at DC and a UE
17.2.2 Highpass with Three TZs at DC and Four UEs
17.3 The Highpass Synthesized as a Bandpass
17.3.1 Hybrid Highpass from an Eighth-Degree Bandpass
17.3.2 Hybrid Highpass from a 10th-Degree Bandpass
xi
268
270
272
272
275
18 Multiplexers
18.1 Contiguous Multiplexers
18.1.1 Contiguous Lowpass-Highpass Diplexer
18.1.2 Contiguous LP/BP/HP Multiplexer
18.2 Noncontiguous Multiplexers
18.2.1 Noncontiguous LP/HP Diplexer with FTZ
18.2.2 Noncontiguous Distributed Combline Diplexer
Reference
277
277
277
279
281
281
284
287
19 Electromagnetic Simulation
19.1 Overview
19.1.1 The EMPower Program
19.1.2 The Momentum Program
19.1.3 The EMPro Program
19.2 Box Modes
19.3 EM Simulation of Distributed Circuits
19.3.1 EM Simulation of Penetrating Stepped-Z Lowpass
19.3.2 EM Simulation of a Combline Bandpass
19.3.3 EM Simulation of a Direct-Coupled Bandpass
19.4 Classic Method of Bandpass Design
19.4.1 Classic Method Fundamentals
19.4.2 Example: Determining K Values
19.4.3 Example: Determining Q Values
19.4.4 Filter Example Using the Classic Method
References
289
289
290
291
292
292
295
295
298
300
302
302
304
307
307
310
313
313
315
316
317
319
323
Index
325
LOAD
Rsource
R=50
VS1
VDC=1V
Rload
R=50
129
130
The current is 1 volt over 100 ohms, or 10 mA. The voltage source
delivers 10 mW; 5 mW to the source resistance and 5 mW to the load
resistance. If the load is 25 ohms, the current increases to 13.33 mA, and
the 1-volt source delivers 13.33 mA. However, only 4.44 mW is delivered to
the load, while the majority is delivered to the source resistance. If the load
is 100 ohms, the current decreases to 6.67 mA, and the 1-volt source
delivers 6.67 mW. While the majority is delivered to the load, again it is
only 4.44 mW. Maximum transfer of power occurs when the load resistance
equals the source resistance.
From a wave perspective, when a signal propagating through a system
encounters an impedance shift, a portion of the signal is reflected, thus
reducing the transmitted signal. This is analogous to reflection in an optical
system. Mismatch occurs when the impedance shift is either resistive or
complex.
YL = GL + jBL =
and then setting
RL
RL2 + X L2
XL
RL2 + X L2
(12.1)
131
GL
GL2
R0
A=
(12.2)
B = RL ( R0 RL ) X L
(12.3)
C = RL ( R0 RL ) X L
(12.4)
L1 =
2f GL2 + A2
(12.6)
1
2f ( A + BL )
(12.7)
GL2 + A2
2fA
(12.8)
B
2f
(12.9)
C2 =
L3 =
C3 =
2f
L4 =
B + XL
+ (B + X L )2
RL2
(R
+ (C + X L )2
2f (C + X L )
2
L
C5a =
(12.10)
(12.11)
1
2fC
(12.12)
GL2 + A2
2fA
(12.13)
A + BL
2f
(12.14)
C4 =
C5b =
L6 a =
(12.5)
A BL
2f
C1 =
L2 =
2f Gl2 + A2
(12.15)
132
1
2f ( A BL )
L6b =
C7 a =
C7b =
2f
1
2fB
(12.17)
B + XL
RL2
+ (B + X L )2
L8a =
L8b =
(12.16)
C
2f
(R
+ ( X L + C )2
2f ( X L + C )
2
L
(12.18)
(12.19)
(12.20)
133
134
135
Figure 12.4 Matchable termination resistance ratios versus the ratio of the upper to lower
cutoff frequency for a passband return loss of 20 dB.
For matching, even quantities of TZs are more economic than odd
quantities. For example, 2/4 provides better bandwidth or higher
termination resistance ratios than 3/3. All even TZ choices of the same
quantity result in similar performance. For example, 2/6, 4/4, and 6/2
provide similar performance.
Fig. 12.4 is for the case with 20-dB passband return loss. A worse
return loss improves the bandwidth and increases the termination
resistance ratio.
With respect to matching, FTZs are not economic. For a given quantity
of elements, the specification of FTZs reduces the performance. FTZs are
employed in filters used for matching only if the stopband requirements
benefit.
The above discussion involves design to maximize the bandwidth or
termination resistance ratio. The general case of matching, however,
involves matching a source resistance to a specific load resistance. For
example, consider a required resistance ratio of 30 over an octave
bandwidth. This requirement lies between TZ placements of 4/4 and 6/6. A
number of techniques are employed:
1) If the design is purely a matching issue, the 6/6 synthesis is used and
the bandwidth is extended beyond the required bandwidth. This
reduces the sensitivity of the design to element tolerance.
136
Figure 12.5 Specification tab for matching example that exploits extraction sequences.
137
ZO=50
L2
L=25.78nH
L1
L=194.21nH
L3
L=10.02nH
C1
C=43.12pF
L4
L=26.96nH
C2
C=147.34pF
C3
C=48.34pF
ZO=50
L1
L=24.97nH
L2
L=12.03nH
C1
C=35.85pF
L3
L=22.44nH
C2
C=122.7pF
C3
C=59.11pF
Figure 12.7 Matching network with bandwidth expanded to avoid a Norton transform.
138
Figure 12.8 Responses of the 50- to 12.5-ohm matching network with expanded bandwidth.
ZO=100
L1
L=666.37nH
C3
C=4.87pF
L2
L=666.37nH
C2
C=18.22pF
ZO=100
L3
L=27.75nH
C4
C=89.29pF
T1
P=1
S=4.9
Figure 12.9 Schematic of the 88- to 108-MHz bandpass used as a matching network.
139
Notice that the transformer turns ratio is 4.9, thus indicating that
without a transformer, this network matches 100 ohms down to about 4.2
ohms. The application of a shunt Norton to the final shunt capacitor
supports matching any resistance from 4.2 ohms up to 100 ohms.
Therefore, the termination resistance in the Specification tab is set to the
desired 50 ohms and a Norton Shunt transform is applied to C4, selecting n
to remove the transformer. The filter matching 100 to 50 ohms is given in
Fig. 12.10.
L3
L=332.5nH
L2
L=666.37nH
L1
L=666.37nH
C5
C=10.4779pF
C3
C=5.6216pF
C1
C=4.671pF
ZO=100
ZO=50
C4
C=25.79pF
C2
C=18.22pF
Figure 12.10 Final schematic of the series-resonator bandpass used to match 100 to 50
ohms.
Qloaded =
f upper
f0
f lower
(12.21)
It is a finite value even if the circuit is built using elements with infinite
Q. Element Q, or unloaded Q, is a measure of element quality. It is defined
as 2f times the stored energy divided by the dissipated energy in the
element. It is as high as 200 for excellent inductors. Unloaded Q increases
with physical size, so modern miniature chip inductors have low unloaded
Q. Finally, Q of the load is a property of a complex termination. For series
impedance it is given simply by
140
Qof load =
XL
RL
(12.22)
Qof load =
BL
GL
(12.23)
min = e
Qloaded
Qof load
(12.24)
(12.25)
Qloaded =
Qof load =
Qloaded
ln min
(12.26)
(12.27)
141
Figure 12.11 Initial Specification tab for the 200- to 400-MHz power-amplifier matching
network.
The matching network must include a shunt capacitor at the input that
is sufficiently large to absorb the 40-pF transistor output capacitance. The
extraction must therefore begin with a shunt element first. Furthermore,
the extraction must result in a topology with a transformer turns ratio less
than unity so that a Norton transform is available to remove the
transformer. The quantity of nonfinite TZs is increased until a satisfactory
extraction sequence is achieved. The lowest degree is six. Since harmonic
suppression is desirable in a power amplifier, the extraction with one TZ at
DC and five at infinity is selected with the sequence DC .
Finally, to avoid adding an additional capacitor with a Norton
transform, to improve the design margin, and to retain the filtering
functions of the network, the requested passband ripple in the Specification
tab is reduced until the transformer turns ratio in the Extraction tab is
unity. The resulting passband ripple is 0.0282 dB, or a passband return
loss of 22 dB. The schematic for the power-amplifier matching network is
given in Fig. 12.12. Capacitor C1 is replaced with a 24.84-pF capacitor with
the remaining capacitance provided by the transistor output capacitance.
142
ZO=50
L2
L=7.68nH
L1
L=6nH
C1
C=64.84pF
C2
C=38.51pF
L3
L=17.5nH
C3
C=12.47pF
VSWR =
1 + S11
1 S11
(12.28)
143
Figure 12.13 VSWR responses of the antenna before (solid trace) and after matching (dashed
trace).
The goal is to achieve a VSWR under 3:1 across the frequency span of
3.5 to 4.0 MHz. Design begins with the specification of the 50-ohm source
and 34-ohm load, frequency cutoffs of 3.5 and 4.0 MHz, and a passband
ripple of 1.25 dB that corresponds to a VSWR of 3:1. The series L-C
component of the model is incorporated into the final series branch of the
matching network. The quantity of TZs is increased and an extraction
sequence is selected so that the 3:1 VSWR is achieved over the desired
band, and the matching network contains a final series L-C branch with the
inductor greater than the antenna 19.5 uH, and a capacitor smaller than the
antenna 87 pF. The extraction sequence selected is DC DC DC . Then
a Remove Transformer transform is applied to set the output resistance at
34 ohms. The final matching network could be realized by absorbing the
antenna L-C into the matching network values. However, to eliminate the
matching network capacitor and to afford margin in the design, the ripple is
reduced to 1.0776 dB to improve the VSWR maximum and the cutoffs are
broadened to 3.48 and 4.02 MHz to widen the frequency response. This
sets the capacitor C4 to exactly match the antenna capacitance. The network
is shown in Fig. 12.14. The capacitor C4 is absent in the final network and
the inductor is 20.857 uH minus 19.5 uH, or 1.357 uH. If the antenna is
physically lengthened, this inductor is eliminated as well. The final
response is given as a dashed trace in Fig. 12.13.
144
ZO=50
L1
L=30677.88nH
C4
C=87pF
L3
L=20857.56nH
L2
L=313.57nH
ZO=34
C3
C=7002.93pF
Figure 12.14 Matching and broadbanding network for the antenna. Refer to the text regarding
L3 and C4.
References
[1] R. Rhea, The Yin-Yang of Matching: Part 1 Basic Matching Techniques, High
Frequency Electronics, March 2006.
[2] R. Rhea, The Yin-Yang of Matching: Part 2 Practical Matching Techniques,
High Frequency Electronics, April 2006.
[3] P. Smith, Electronic Applications of the Smith Chart, SciTech Publishing,
Raleigh, NC, 1995.
[4] R. Fano, Theoretical Limitations on the Broadband Matching of Arbitrary
Impedances, Jour. Franklin Institute, January, 1950.