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Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Ch. 5. Overview of RF filter design
Microstrip line low-pass filter implementation
Filters are important circuit elements used to enhance or attenuate certain ranges of frequencies.
This chapter presents basic concepts and definitions related to filters and resonators.
Apply one- and two-port networks and transmission lines to develop RF filters.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Basic filter typesIdealized low-pass, high-pass, band-pass and band-stopfilters.
Use normalized frequencies Ω = ω/ωc.
ωc is the cut-off frequency for LP and HP filters and center frequency for BP and BS filters.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Standard filter types (low-pass)
Butterworth (binomial) filter+ Monotonic+ Easy to implement– Steep transition
requires a large number of elements
Chebyshev filter+ Steep transition – Ripples in the
passband + Ripple control,
equal ripples foroptimization
Elliptic (Cauer) filter+ Steepest transition – Finite attenuation in
stopband– Ripples in passband
and stopband – Complex
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Bandwidth:
Shape factor:
Rejection:Attenuation required in stopband, typically 60 dB
Quality factor Q:Describes selectivity of filter( )2
21
10 log
10log 1 20log
in
L
in
PILP
S
α⎛ ⎞
= = ⎜ ⎟⎝ ⎠
= − − Γ = −
dBl
dBu
dB ffBW 333 −=
dBl
dBu
dBl
dBu
dB
dB
ffff
BWBWSF 33
6060
3
60
−−
==
Insertion loss:
Filter parameters
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Quality factor Q
cc
c
loss
stored
PW
Q
ωωωω
ωω
ωω
π
==
=
==
=
losspower energystoredaverage
cycleperlossenergyenergystoredaverage2
Distinguish between loaded and unloaded QLoaded Q = QLD : including load ZL
c
dB
EFLD fBW
QQQ
3111=+=
QF : filter Q, QE : external Q
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
First-order filter realizations
Low-pass
High-pass
Band-pass
Band-stop
Consider each as cascade of ABCD-networks
[ ] ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡1101
101
L
G
ZZ
DCBA
Z
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Low-pass filter
Consider as cascade of 4 ABCD-networks
( ) ( ) ( )( )210 0
2 221 1
S HA R Z j C Z
ω ωω
= = =+ + +
For ω → 0:S21(ω) → 2Z0/(R +2Z0)
For ω →∞S21(ω) → 0
( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+++
=⎥⎦
⎤⎢⎣
⎡
11
11
L
LGL
G
RCj
RRR
CjRR
DCBA
ω
ω
Transfer function H(ω ) =V2/VG = 1/A) (matching: ZG = ZL = Z0 ):
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
LP filter response
( ) ( )210
21
Sj C R Z
ωω
→+ +
High-frequency limit: Attenuation factor:
Phase angle:
Group delay:
( ) ( )2120log Sα ω ω= −
( ) ( ) ( )
211
21
Imtan
ReSS
ωϕ ω
ω−⎡ ⎤
= ⎢ ⎥⎢ ⎥⎣ ⎦( ) ωωφ ddtg =
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
High-pass filter
Consider as cascade of 4 ABCD-networks
( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+++
=⎥⎦
⎤⎢⎣
⎡
111
111
L
LGL
G
RLj
RRRLj
RR
DCBA
ω
ω
( )( )
21
00
2 21 11
SA
R Zj L Z
ω
ω
= =⎛ ⎞
+ + +⎜ ⎟⎝ ⎠
For ω → 0S21(ω) → 0
For ω →∞S21(ω) → 2Z0/(R + 2Z0)
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
HP filter response
Low-frequency limit: ( ) ( )210
21
Sj R Z L
ωω
→− +
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Band-pass filter
Band-pass filter in series config-uration. Consider as cascade of 3 ABCD-networks where:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ ++
+=⎥
⎦
⎤⎢⎣
⎡
11
1
L
GL
G
R
ZRR
ZR
DCBA
( ) ( )0
210
22 1
ZSZ R j L C
ωω ω
=+ + −
( )CLjRZ ωω 1−+=
Z
ZL = ZG = 50 ΩR = 20 ΩL = 5 nHC = 2 pF
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Band-stop filter
Band-stop filter in parallel configuration. Consider as cascade of 3 ABCD networks. G = 1/R
( )0
21
0
12
11 2
Z G j CLS
Z G j CL
ωω
ωω
ω
⎡ ⎤⎛ ⎞+ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=⎡ ⎤⎛ ⎞+ + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Quality factor
GGC
GC
GC
RRL
RL
RL
QQQ
EE
EE
LDFE
+
+000
000
Parallel
Series
LoadedFilter External
ωωω
ωωωResonance frequency:
LC1
0 ≈ω
Generally:
00
ImRe2
orImRe2
00ffffLD df
YdY
fdf
ZdZ
fQ ===
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Series and parallel resonators
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Insertion lossQ-factors easier to measure than impedances and admittances.
( ) ⎥⎦
⎤⎢⎣
⎡++=⎟
⎠⎞
⎜⎝⎛ −+= ε
ωω LD
F
LDE jQ
QQRR
CLjRZ 1Series resonance:
( ) ⎥⎦
⎤⎢⎣
⎡++=⎟
⎠⎞
⎜⎝⎛ −+= ε
ωω LD
F
LDE jQ
QQGG
LCjGY 1Parallel resonance: ω
ωωωε 0
0−=
02 8ZVPP GinL ==
( )( )2220
2
0 11
221
LDELDin
GL QQQ
PZZZ
VPε+
=+
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ += 22
221log10ELD
LD
QQQIL εInsertion loss:
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Butterworth filters
For LP filter:
IL for low-pass Butterworth filtera = 1
( ) ( ) ( )Nin aLFIL 222 1log10log101log10 Ω+==Γ−−=
1at 3dB gives1filteroforder
1 :factor Loss 22
=Ω===
Ω+=
=Ω
ILaN
aLF Ncωω
Maximally flat filters (no ripples)
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Filter elementsEquivalent realizations of generic multisection LP filters with normalized elements
⎪⎪⎩
⎪⎪⎨
⎧
=
⎪⎩
⎪⎨
⎧=
+
inductor seriesafter econductanc load-
capacitorshunt after resistance load-
(b) econductancor (a) resistance
generator internal
1
0
Ng
g
⎩⎨⎧
== capacitorshunt for ecapacitancinductor seriesfor inductance
,..,1 Nmmg
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Butterworth LP filter coefficients3 dB design (a = 1)
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Attenuation versus order of LP filter
circuits)mixer andmodulationfor (useful
order low requires phaselinear :Note
dB/decade02
as increases ,1For
2
N
N
LFN →Ω
>>Ω
3 dB design (a = 1)
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Chebyshev filters
For LP filter: ( ) Ω+== 221log10log10 NTaLFIL
Equi-ripple filters
( )( )[ ]( )[ ]⎪⎩
⎪⎨⎧
≥ΩΩ
≤ΩΩ=Ω
−
−
1for ,coshcosh
1for ,coscos1
1
N
NTN
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
Ω+Ω−=
Ω+Ω−=
Ω+−=
Ω==
424
33
22
1
0
881
43
21
1
T
T
T
TT
In the range:-1 < Ω < 1
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
LP Chebyshev filter properties
IL and attenuation response for 3dB design (a = 1)( )122 ,1For −≈>>Ω N
BC LFLF
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Chebyshev LP filter coefficients3 dB design
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Comparison of filtersThird order Butterworth and Chebyshev filters
VG
RG L1 L2
C1 RLV2V1 V2
RG = RL = 1 Ω
Standard 3 dB Butterworth: L1 = L2 = 1 H, C1 = 2 F
Linear phase Butterworth:L1 = 1.255 H, C1 = 0.5528 F, L2 = 0.1922 H
3 dB Chebyshev:L1 = L2 = 3.3487 H, C1 = 0.7117 F
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Filter transformation
• Start with standard, normalized Chebyshev LP filter.• Apply appropriate frequency and impedance scaling.• Generate real filters of all four types (LP, HP, BP, BS). • Use simple cook-book approach.
3dB normalized Chebyshev LP filter shown for positive and negative frequencies Ω = ω/ωc.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Low-pass filter transformation
GHz1h filter wit-LP =cω
Scaled frequency: cωω Ω=
Scaled reactances:
( )
( )c
cC
c
cL
CC
CjCjCjjX
LL
LjLjLjjX
ω
ωωω
ω
ωωω
=⇒
==Ω
=
=⇒
==Ω=
~
~111
~
~
⇒
⇒
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
High-pass filter transformationScaled frequency: Ω−= cωω
Scaled reactances:
( )
CL
LC
LjCjCj
jX
CjLjLjjX
cc
cC
cL
ωω
ωωω
ωωω
1~,1~
~1
~1
==⇒
=−=Ω
=
=−=Ω=
GHz1h filter wit-HP =cω
⇒
⇒
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Band-pass filter transformationScaled and shifted frequency:
LU
cc
cLU
c
ωωεω
ωω
ωω
ωωω
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=Ω
Scaled reactances:
LU
LU
LUC
LU
LU
LUL
CCC
L
LjCjCjCjjB
LCLL
CjLjLjLjjX
ωωω
ωωω
ωωωεω
Ω
ω
ωωωω
ωω
ωωεω
Ω
−=
−=
+=−
==
−=
−=
+=−
==
~,~
~1~
~,~
~1~
20
0
20
0
GHz 1
:frequencycenter filter -BP
0 === LUc ωωωω
⇒
⇒
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Band-stop filter transformation
( )
( )( )
20
20
~,1~:capacitorShunt
1~,~:inductor Series
ωωω
ωω
ωωωωω
CCC
L
LCLL
LU
LU
LU
LU
−=
−=⇒
−=
−=⇒
GHz 1:frequencycenter filter -BS
0 == cωω
⇒
⇒
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Summary of transformationsLUBW ωω −=
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Impedance transformationHave so far assumed that the generator resistance g0 =1. If not, all impedances have to be scaled according to:
GLLG
GGG
RRRRCC
LRLRR
==
==~,~
~, 1~
RG = 50 Ω L1 L3
C2RL =50 Ω
L1 = L3 = 167.4 H C2 = 14.23 mF
Example: N = 3 Chebyshev 3 dB filterg0 = g4 = 1, g1= g3 = 3.3487, g2 = 0.7117
BP-filter withf0 =2.4 GHzBW = 0.2 f0
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
RF filter implementationRF filters difficult to realize with discrete devices because of physical dimensions. Have to use distributed transmission elements lines based on:
• Richard’s transformation• Unit elements• Kuroda’s identities
Apply the property that short- or open-circuit transmission lines behave as reactive elements:
( ) ( )λπβ ljZljZZ shortin 2tantan 00 ==
( ) ( )λπβ ljYljYY openin 2tantan 00 ==
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Richard’s transformationChoose arbitrarily a line segment of length l = λ0/8 at a reference frequency f0 = vp/λ0. Use:
00 4tan SZjZLjjXZ Lin =⎟
⎠⎞
⎜⎝⎛ Ω===πω
Ω===444
20
0 ππλλπ
λπ
ffl
Short-circuit:
00 4tan SYjYCjjBY Cin =⎟
⎠⎞
⎜⎝⎛ Ω===πωOpen-circuit:
Note: Richard’s transformation maps the lumped element frequency response for 0 ≤ f ≤ ∞ into the range 0 ≤ f ≤ 4 f0. Short-circuit inductive and open-circuit capacitive for 0 ≤ f ≤ 2 f0.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Unit elementsHave to separate transmission line elements spatially to achieve practical circuit configurations. Accomplished by inserting unit elements (UEs) of electrical length Ωπ/4 and characteristic impedance ZUE. Represent as chain-parameter two-port:
[ ]( ) ( )( ) ( ) ⎥
⎥
⎦
⎤
⎢⎢
⎣
⎡
−=
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
ΩΩ
ΩΩ=⎥
⎦
⎤⎢⎣
⎡= 1
1
1
14cos4sin
4sin4cos
2UE
UE
UE
UE
UEUE
UEUE
ZS
SZ
SZj
jZ
DCBA
UE ππππ
( ) ( )4cos1
1,4tan2
ππ Ω=−
Ω=S
jS
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Kuroda’s identities
These identities are used to facilitate practical implementations.
For example:Open shunt stub lines easier to realize than shorted series lines.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Microstrip filter design
Procedure
• Select the normalized filter parameters for the design.
• Use Richard’s transformation to replace Ls and Cs byequivalent λ0/8 transmission lines.
• Convert series stub lines to shunt stubs using Kuroda’s identities.
• De-normalize and and select equivalent microstrip lines
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Example: Low-pass microstrip filter
Specifications:• Input and output matched to 50 Ω.
• Cut-off frequency: 3 GHz
• Equi-ripple of 0.5dB• Rejection of 25 dBat 4.5 GHz (N = 5)
• Dielectric with vp = 0.6 c
Step 1:Fig. 5-22 gives N = 5. From Table 5-4 (b): g1 = g5 = 1.706, g2 = g4 = 1.230, g3 = 2.541, g6 = 1
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Microstrip LP-filter, step 2
Use Richard’s transformation to replace Ls and Cs by open and short series and shunt TL stubs. Characteristic line impedances:
Y1 = Y5 = g1, Y3 = g3, Z2 = Z4 = g2
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Microstrip LP-filter, step 3
• Introduce matched UEs.
• Apply Kuroda’s 1, and 2. identities to convert shunts 1 and 3 to seriesinductors.
• Resulting circuit not realizable in microstrip.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Microstrip LP-filter, step 3 cont.
• Again introduce matched UEs.
• Apply Kuroda’s 1, and 2. identities to convert all seriesinductors to shunts.
• Resulting circuit isnow realizable inmicrostrip.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Microstrip LP-filter, step 4Denormalization:
• Use RG = RL = 50 Ω.
• Scale all other elements according totransformation rules.
• Determine line lengths: l = λ0/8 = vp/8f0 = 7.5 mm.
• Scale line widths according to expressions presented in Chapter 2.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Coupled-line filtersUtilize coupling of microrstip lines through a common ground plane.
Describe the overall system impedance in terms of a two-port chain matrix formalism.
C12 = C21 and L12 = L21 give rise to coupling between the lossless lines.
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Single bandpass filter section
( ) ( ) ( )( )
12221211
2211
00
2200
200
22
:modes (o) odd and (e)even with lines identicalFor
1,1sin2
cos
CCCCCCCC
CvZ
CvZ
llZZZZ
Z
o
e
opoo
epee
oeoein
+=+===
==
+−−=
ββ
Institutt for InformatikkINF5481: RF kretser, teori og design Svein-Erik Hamran
Cascading filter elements
Gives steeper passband and stopband transitions.
N = 5 coupled-line 3 dB Chebyshev filter.
f0 = 5 GHz