Ch. 5. Overview of RF filter design · INF5481: RF kretser, teori og design Institutt for...

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.


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.


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


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


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


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


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 ):


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 =


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)


HP filter response

Low-frequency limit: ( ) ( )210

21

Sj R Z L

ωω

→− +


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


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

⎡ ⎤⎛ ⎞+ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=⎡ ⎤⎛ ⎞+ + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦


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 ===


Series and parallel resonators


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:


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)


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


Butterworth LP filter coefficients3 dB design (a = 1)


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)


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


LP Chebyshev filter properties

IL and attenuation response for 3dB design (a = 1)( )122 ,1For −≈>>Ω N

BC LFLF


Chebyshev LP filter coefficients3 dB design


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


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.


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

~

~

⇒

⇒


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

⇒

⇒


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

⇒

⇒


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

⇒

⇒


Summary of transformationsLUBW ωω −=


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


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 ==


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.


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


Kuroda’s identities

These identities are used to facilitate practical implementations.

For example:Open shunt stub lines easier to realize than shorted series lines.


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


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


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


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.


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.


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.


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.


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


Cascading filter elements

Gives steeper passband and stopband transitions.

N = 5 coupled-line 3 dB Chebyshev filter.

f0 = 5 GHz

Ch. 5. Overview of RF filter design · INF5481: RF kretser, teori og design Institutt for...

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