11 ME1000 RF CIRCUIT DESIGN This courseware product contains scholarly and technical information and is protected by copyright laws and international treaties

11

ME1000 RF CIRCUIT DESIGN

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2

3B. RF Microwave Filters

3

1.0 Basic Filter Theory

4

Introduction

• An ideal filter is a linear 2-port network that provides perfect transmission of signal for frequencies in a certain passband region, infinite attenuation for frequencies in the stopband region, and a linear phase response in the passband (to reduce signal distortion).

• The goal of filter design is to approximate the ideal requirements within acceptable tolerance with circuits or systems consisting of real components.

5

Categorization of Filters

• Low pass filter (LPF), high pass filter (HPF), bandpass filter (BPF), bandstop filter (BSF), arbitrary type, etc.

• In each category, the filter can be further divided into active and passive types.

• In an active filter, there can be amplification of the signal power in the passband region; a passive filter do not provide power amplification in the passband.

• Filters used in electronics can be constructed from resistors, inductors, capacitors, transmission line sections, and resonating structures (e.g., piezoelectric crystal, Surface Acoustic Wave (SAW) devices, mechanical resonators, etc.).

• An active filter may contain a transistor, FET, and an op-amp.

Filter

LPF BPFHPF

Active Passive Active Passive

6

Filter Frequency Response

• Frequency response implies the behavior of the filter with respect to steady-state sinusoidal excitation (e.g., energizing the filter with a sine voltage or current source and observing its output).

• There are various approaches to displaying the frequency response:– Transfer function H() (the traditional approach)– Attenuation factor A()– S-parameters, e.g., s21()

– Others, such as ABCD parameters, etc.

7

Filter Frequency Response (cont’d)

• Low pass filter (passive)

Filter H()

V1() V2()ZL

c

A()/dB

0 c

3

10

20

30

40

50

1

21020A

V

VLognAttenuatio (1.1b)

1

2

V

VH (1.1a)

c

|H()|

1Transfer function

Arg(H())

Complex value

Real value

8


• Low pass filter (passive) continued...

• For the impedance matched system, using s21 to observe the filter response is more convenient, as this can be easily measured using a vector network analyzer (VNA).

Zc

01

221

01

111

22

aa a

bs

a

bs

ZcZc

Transmission lineis optional

c

20log|s21()|

0 dB

Arg(s21())

FilterZcZc

ZcVs

a1 b2

Complex value

9

• Low pass filter (passive) continued...


A()/dB

0 c

3

10

20

30

40

50

Filter H()

V1() V2() ZL

Passband

Stopband

Transition band

Cut-off frequency (3 dB)

10

• High pass filter (passive)


A()/dB

0 c

3

10

20

30

40

50

c

|H()|

1

Transfer function

Stopband

Passband

11


Bandpass filter (passive) Bandstop filter

A()/dB

40

1

3

30

20

10

0 2o

1

|H()|

1 Transfer function

2o

A()/dB

40

1

3

30

20

10

0 2o

1

|H()|

1

Transfer function

2o

12

Basic Filter Synthesis Approaches

• Image Parameter Method.

ZoZo Zo

ZoZo

Filter Zo

H1() H2() Hn() Zo

Zo

• Consider a filter to be a cascade of linear 2-port networks.• Synthesize or realize each 2-port network, so that the combine effect gives the required frequency response.• The ‘image’ impedance seen at the input and output of each network is maintained.

The combinedresponse

Response ofa singlenetwork

13

Basic Filter Synthesis Approaches (cont’d)

• Insertion Loss Method.

Filter Zo

Zo

Use the RCLM circuit synthesis theorem to come up with a resistive terminatedLC network that can produce theapproximate response. Zo

IdealApproximate with rational polynomialfunction

|H()|

obsbnsnbns

oasansnansKsH

11

1

11

1

We can also use Attenuation Factor or |s21| for this.

Approximate ideal filter responsewith polynomial function:

14

Our Scope

• Only concentrate on passive LC and stripline filters.

• Filter synthesis using the Insertion Loss Method (ILM). The Image Parameter Method (IPM) is more efficient and suitable for simple filter designs, but has the disadvantage that arbitrary frequency response cannot be incorporated into the design.

15

2.0 Passive LC Filter Synthesis Using the Insertion Loss Method

16

Insertion Loss Method (ILM)

• The insertion loss method (ILM) enables a systematic way to design and synthesize a filter with various frequency responses.

• The ILM method also enables a filter performance to be improved in a straightforward manner, at the expense of a ‘higher order’ filter.

• A rational polynomial function is used to approximate the ideal |H()|, A(), or |s21()|.

• Phase information is totally ignored.

• Ignoring phase simplifies the actual synthesis method. An LC network is then derived which will produce this approximated response.

• The attenuation A() can be cast into power attenuation ratio, called the Power Loss Ratio, PLR, which is related to A()2.

17

More on ILM

• There is a historical reason why phase information is ignored. Original filter synthesis methods are developed in the 1920s–60s, for voice communication. The human ear is insensitive to phase distortion, thus only the magnitude response (e.g., |H()|, A()) is considered.

• Modern filter synthesis can optimize a circuit to meet both magnitude and phase requirements. This is usually done using computer optimization procedures with ‘goal functions’.

Extra

18

Power Loss Ratio (PLR)

211

12

11

Load todeliveredPower network source from availablePower

AP

AP

LoadPincP

LRP

•PLR large, high attenuation•PLR close to 1, low attenuationFor example, a low passfilter response is shownbelow:

•PLR large, high attenuation•PLR close to 1, low attenuationFor example, a low passfilter response is shownbelow:

ZLVs

Lossless2-port network

1

Zs

PAPin

PL

PLR(f)

Low pass filter PLRf

1

0

Low attenuation

Highattenuation

fc

(2.1a)

19

PLR and s21

• In terms of incident and reflected waves, assuming ZL = Zs = ZC.

ZcVs

Lossless2-port network

Zc

PAPin

PL

a1

b1

b2

221

1

2

21

222

1

212

1

sLR

ba

b

a

LPAP

LR

P

P

(2.1b)

20

PLR for the Low Pass Filter (LPF)

• Since |1()|2 is an even function of , it can be written in terms of 2 as:

• PLR can be expressed as:

• Various types of polynomial function in can be used for P(). The requirement is that P() must either be an odd or even function. Among the classical polynomial functions are:– Maximally flat or Butterworth functions– Equal ripple or Chebyshev functions– Elliptic function– Many, many more

21 PPLR

22

22

NM

M

2

2

22

2

1

12

11

1 1

N

M

NM

MLRP

2

22

N

MP

(2.2)

(2.3a)

(2.3b)

This is also knownas Characteristic Polynomial

The characteristics we need from [P()]2 for LPF: • [P()]2 0 for < c

• [P()]2 >> 1 for >> c

The characteristics we need from [P()]2 for LPF: • [P()]2 0 for < c

• [P()]2 >> 1 for >> c

PP

21

Characteristic Polynomial Functions

• Maximally flat or Butterworth:

• Equal ripple or Chebyshev:

• Bessel or linear phase:

N

cP

2 , 2

1

21

1

0

nCCC

C

C

C

nnn

N

factor ripple , NCP

N = order of theCharacteristicPolynomial P()

N = order of theCharacteristicPolynomial P()

12 jBjBP

2 , 12

1

1

22

1

1

0

nsBssBssB

ssB

sB

sB

nnn

N

(2.4a)

(2.4b)

(2.4c)

22

Examples of PLR for the Low Pass Filter

• PLR of the low pass filter using 4th order polynomial functions (N = 4) – Butterworth, Chebyshev (ripple factor =1), and Bessel. Normalized to c = 1 rad/s, k = 1.

0 0.5 1 1.5 21

10

100

1 103

1 104

PLRbt ( )

PLRcb ( )

PLRbs ( )

Butterworth

Chebyshev

Bessel

222

)( 1481

cckP chebyshevLR

242

)( 1

ckP hButterwortLR

1051054510

11

234

1051

2)(

cs

cs

cs

cs

BesselLR

sB

jBjBkP

PLR

Ideal

If we convert into dB,this ripple is equal to3 dB

k=1

23

Examples of PLR for the Low Pass Filter (cont’d)

• PLR of the low pass filter using the Butterworth characteristic polynomial, normalized to c = 1 rad/s, k = 1. 2

2)( 1

N

chButterwortLR kP

0 0.5 1 1.5 21

10

100

1 103

1 104

1 105

PLR 2( )

PLR 3( )

PLR 4( )

PLR 5( )

PLR 6( )

PLR 7( )

N=2

N=6

N=4

N=5

N=3

N=7 Conclusion:The type of polynomialfunction and the orderdetermine the attenuation rate in the stopband.

Conclusion:The type of polynomialfunction and the orderdetermine the attenuation rate in the stopband.

24

Characteristics of Low Pass Filters Using Various Polynomial Functions

• Butterworth: Moderately linear phase response, slow cutoff, smooth attenuation in the passband.

• Chebyshev: Bad phase response, rapid cutoff for a similar order, contains ripple in the passband. May have impedance mismatch for N even.

• Bessel: Good phase response, linear. Very slow cutoff. Smooth amplitude response in the passband.

25

Low Pass Prototype Design

• A lossless linear, passive, reciprocal network that can produce the insertion loss profile for the low pass filter is the LC ladder network.

• Many researchers have tabulated the values for the L and C for the low pass filter with cut-off frequency c = 1 rad/s, that works with the source and load impedance Zs = ZL = 1 .

• This low pass filter is known as the Low Pass Prototype (LPP).

• As the order N of the polynomial P increases, the required element also increases. The no. of elements = N.

L1=g2 L2=g4

C1=g1 C2=g3 RL= gN+1

1

L1=g1L2=g3

C1=g2 C2=g4RL= gN+1g0= 1

Dual of eachother

26

Low Pass Prototype Design (cont’d)

• The LPP is the ‘building block’ from which real filters may be constructed.

• Various transformations may be used to convert it into a high pass, bandpass, or other filter of arbitrary center frequency and bandwidth.

• The following slides show some sample tables for designing LPP for Butterworth and Chebyshev amplitude response of PLR.

27

Table for the Butterworth LPP Design

N g1 g2 g3 g4 g5 g6 g7 g8 g91 2.0000 1.00002 1.4142 1.4142 1.00003 1.0000 2.0000 1.0000 1.00004 0.7654 1.8478 1.8478 0.7654 1.00005 0.6180 1.6180 2.0000 1.6180 0.6180 1.00006 0.5176 1.4142 1.9318 1.9318 1.4142 0.5176 1.0000

7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.0000

8 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 1.1111 0.3902 1.0000

See Example 2.1 in the following slides on how the constant values g1, g2, g3, … etc., are obtained.

28

Table for the Chebyshev LPP Design

• Ripple factor 20log10 = 0.5 dB

• Ripple factor 20log10 = 3.0 dB

N g1 g2 g3 g4 g5 g6 g7

1 0.6986 1.00002 1.4029 0.7071 1.98413 1.5963 1.0967 1.5963 1.00004 1.6703 1.1926 2.3661 0.8419 1.98415 1.7058 1.2296 2.5408 1.2296 1.7058 1.00006 1.7254 1.2479 2.6064 1.3137 2.4578 0.8696 1.9841

N g1 g2 g3 g4 g5 g6 g7

1 1.9953 1.00002 3.1013 0.5339 5.80953 3.3487 0.7117 3.3487 1.00004 3.4389 0.7483 4.3471 0.5920 5.80955 3.4817 0.7618 4.5381 0.7618 3.4817 1.00006 3.5045 0.7685 4.6061 0.7929 4.4641 0.6033 5.8095

29

Table for the Maximally-Flat Time Delay LPP Design

N g1 g2 g3 g4 g5 g6 g7 g8 g9 1 2.0000 1.0000 2 1.5774 0.4226 1.0000 3 1.2550 0.5528 0.1922 1.0000 4 1.0598 0.5116 0.3181 0.1104 1.0000 5 0.9303 0.4577 0.3312 0.2090 0.0718 1.0000 6 0.8377 0.4116 0.3158 0.2364 0.1480 0.0505 1.0000

7 0.7677 0.3744 0.2944 0.2378 0.1778 0.1104 0.0375 1.0000

8 0.7125 0.3446 0.2735 0.2297 0.1867 0.1387 0.0855 0.0289 1.0000

30

Example 2.1: Finding the Constants for the LPP Design

CRLjRLCR

RVRCjLjRR

RV

LjR

Vss

RCjR

sRCjR

V 221

1

211

222222

2

22

212

1

CRLRLC

RVRL

sVP

281

sRA VP

42

2222

41

222222

81

22

8

2

2

222222

2

2

1

222

LCR

R

CRLRLC

RVR

V

PP

LR

LCCRL

CRLLCRPs

s

L

A

andThus

Therefore we can compute the power loss ratio as:

[P()]2

R

RVs C

L R jL

RVs 1/jC V1

Consider a simple case of a 2nd order low pass filter:

Extra

31

42

21

42

2222

41 11 2 aaLCCRLP LCR

LR

42422)( 10111 hButterwortLRP

21122 LCa LC

Extra

CRLLC

LCCRLa

R

R2

21

22

41

1 00 2

PLR can be written in terms of polynomial of 2:

For Butterworth response with k = 1, c = 1:

(E1.1)

(E1.2)

Comparing equations (E1.1) and (E1.2):

Setting R = 1 for the Low Pass Prototype (LPP):

1R

CL

CL

LCCLCLLC

0

02

2

22221 4142.12

22 2

C

CLC

4142.1CL

(E1.3)(E1.4)

Thus from equation (E1.4):

Using (E1.3)

Compare this result withN=2 in the table for the LPP Butterworth response.This direct ‘brute force’approach can beextended to N=3, 4, 5…

Compare this result withN=2 in the table for the LPP Butterworth response.This direct ‘brute force’approach can beextended to N=3, 4, 5…

Example 2.1: Finding the Constants for the LPP Design (cont’d)

32

Example 2.1: Verification

Vin Vout

ACAC1

Step=0.01 HzStop=2.0 HzStart=0.01 Hz

AC

CC1C=1.4142 F

LL1

R=L=1.4142 H R

R1R=1 Ohm

RR2R=1 OhmV_AC

SRC1

Freq=freqVac=polar(1,0) V

Extra

33

Example 2.1: Verification (cont’d)

Eqn PLR=PA/PL

Eqn PA=1/8 EqnPL=0.5*mag(Vout)*mag(Vout)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.0 2.0

5.0E3

1.0E4

1.5E4

2.0E4

0.0

2.5E4

freq, Hz

PLR

Extra

m1freq=m1=-3.056

160.0mHz

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.0 2.0

-40

-30

-20

-10

0

-50

5

freq, Hz

dB(V

out/0

.5)

160.0m-3.056

m1

–3 dB at 160 mHz (milliHertz!!),which is equivalent to 1 rad/s

Power loss ratioversus frequency

34

Impedance Denormalization and Frequency Transformation of LPP

• Once the LPP filter is designed, the cut-off frequency c can be transformed to other frequencies.

• Furthermore the LPP can be mapped to other filter types such as high pass, bandpass, and bandstop.

• This frequency scaling and transformation entails changing the value and configuration of the elements of the LPP.

• Finally the impedance presented by the filter at the operating frequency can also be scaled, from unity to other values; this is called impedance denormalization.

• Let Zo be the new system impedance value. The following slide summarizes the various transformation from the LPP filter.

35

Impedance Denormalization and Frequency Transformation of LPP (cont’d)

212or 21 o

o 12

c

oLZ

coZ

C

ocLZ1

C

Z

c

o

o

oLZ

ooLZ

oo Z

C

C

Z

o

o

o

oZL

oo ZL1

C

Z

o

o

ooZ

C

LPP to Low Pass

LPP to High Pass

LPP toBandpass

LPP toBandstop

Note that the inductor always multiplies with Zo while the capacitor divides with Zo

Note that the inductor always multiplies with Zo while the capacitor divides with Zo

(2.5a) (2.5b)

L

C

Center frequency Fractional bandwidth

36

Summary of Passive LC Filter Design Flow Using the ILM Method

• Step 1: From the requirements, determine the order and type of approximation functions to use.– Insertion loss (dB) in the passband ?– Attenuation (dB) in the stopband ?– Cut-off rate (dB/decade) in the transition band ?– Tolerable ripple?– Linearity of phase?

• Step 2: Design a normalized low pass prototype (LPP) using the L and C elements.

L1=g2 L2=g4

C1=g1 C2=g3 RL= gN+1

1

|H()|

0

1

1

37

Summary of Passive Filter Design Flow Using the ILM Method (cont’d)

• Step 3: Perform frequency scaling, and denormalize the impedance.

• Step 4: Choose suitable lumped components, or transform the lumped circuit design into distributed realization.

|H()|

0

1

1 2

50

Vs15.916pF

0.1414pF79.58nH

0.7072nH 0.7072nH15.916pF50

RL

All uses the microstrip stripline circuitAll uses the microstrip stripline circuit

38

Filter vs. Impedance Transformation Network

• If we ponder carefully, the sharp observer will notice that the filter can be considered as a class of impedance transformation network.

• In the passband, the load is matched to the source network, much like a filter.

• In the stopband, the load impedance is highly mismatched from the source impedance.

• However, the procedure described here only applies to the case when both load and source impedance are equal and real.

Extra

39

Example 2.2A: LPF Design – Butterworth Response

• Design a 4th order Butterworth low pass filter, Rs = RL= 50 , fc = 1.5 GHz.

L1=0.7654H L2=1.8478H

C1=1.8478F C2=0.7654FRL= 1 g0= 1

L1=4.061 nH L2=9.803 nH

C1=3.921 pF C2=1.624 pFRL= 50 g0=1/50

noRZR

c

no

LZL

co

n

Z

CC

50Z

rad/s 104248.95.12

o

9

GHzc

Steps 1 & 2: LPP

Step 3: Frequency scalingand impedance denormalization

40

• Design a 4th order Chebyshev low pass filter, 0.5 dB ripple factor, Rs = 50 , fc = 1.5 GHz.

Example 2.2B: LPF Design – Chebyshev Response

L1=1.6703H L2=2.3661H

C1=1.1926F C2=0.8419FRL= 1.9841

g0= 1

L1=8.861 nH L2=12.55 nH

C1=2.531 pF C2=1.787 pFRL= 99.2

g0=1/50

noRZR

c

no

LZL

co

n

Z

CC

50Z

rad/s 104248.95.12

o

9

GHzc

Steps 1 & 2: LPP


41

Example 2.2 (cont’d)

0.5 1.0 1.5 2.0 2.50.0 3.0

-20

-10

0

-30

5

freq, GHz

dB(S

(2,1

))dB

(LP

F_b

utte

rwor

th..S

(2,1

))

Chebyshev

Butterworth

|s21|

Ripple is roughly 0.5 dB

0.5 1.0 1.5 2.0 2.50.0 3.0

-300

-250

-200

-150

-100

-50

-350

0

freq, GHz

Pha

se_c

heby

shev

Pha

se_b

utte

rwor

thArg(s21)

Chebyshev

Butterworth

Better phaselinearity for ButterworthLPF in the passband

Computer simulation resultusing AC analysis (ADS2003C)

Eqn Phase_chebyshev = if (phase(S(2,1))<0) then phase(S(2,1)) else (phase(S(2,1))-360)

Note: Equation used in Data Display of ADS2003Cto obtain a continuous phase display with the built-infunction phase( ).

42

Example 2.3: BPF Design

• Design a bandpass filter with Butterworth (maximally flat) response.

• N = 3

• Center frequency fo = 1.5 GHz

• 3 dB Bandwidth = 200 MHz or f1 = 1.4 GHz, f2 = 1.6 GHz

• Impedance = 50 Ω

43


• From table, design the low pass prototype (LPP) for 3rd order Butterworth response, c = 1.

Zo=1

g1 1.000F

g3 1.000F

g2 2.000H

g4

12<0o

Hz 1592.0

12

21

c

cc

f

f

Simulated resultusing PSpice

Voltage across g4

Steps 1 & 2: LPP

44


• LPP to bandpass transformation

• Impedance denormalization

133.0

497.1

6.12

4.12

12

21

2

1

o

GHzfff

GHz

GHz

o

50

Vs15.916 pF

0.1414 pF79.58 nH

0.7072 nH 0.7072 nH15.916 pF50

RL

o

oLZ

ooLZ

oo Z

C

C

Z

o

o


45


• Simulated result using PSpice:

Voltage across RL

46

All Pass Filter

• There is also another class of filter known as the All Pass Filter (APF).

• This type of filter does not produce any attenuation in the magnitude response, but provides phase response in the band of interest.

• APF is often used in conjunction with LPF, BPF, HPF, etc., to compensate for phase distortion.

Extra

Zo BPF APF

f0

|H(f)|

1

f

Arg(H(f))

Example of the APF response

f

|H(f)|

1

0 f

Arg(H(f))

f0

|H(f)|

1

f

Arg(H(f)) Linearphase inpassband

Nonlinearphase in passband

47

Example 2.4: Practical RF BPF Design Using SMD Discrete Components

VARVAR1

Ct_value2=2.9Ct_value=3.5Lt_value=4.8

EqnVar

b82496c3229j000L3param=SIMID 0603-C (2.2 nH +-5%)4_7pF_NPO_0603

C3

b82496c3229j000L2param=SIMID 0603-C (2.2 nH +-5%)

4_7pF_NPO_0603C2

CCt45C=Ct_value2 pF

CCt3C=Ct_value2 pF

S_ParamSP1

Step=1.0 MHzStop=3.0 GHzStart=0.1 GHz

S-PARAMETERS

CPWSUBCPWSub1

Rough=0.0 milTanD=0.02T=1.38 milCond=5.8E+7Mur=1Er=4.6H=62.0 mil

CPWSub

INDQL4

Rdc=0.1 OhmMode=proportional to freqF=800.0 MHzQ=90.0L=15.0 nH

CCt2C=Ct_value pF

CCt1C=Ct_value pF

LLt2

R=L=Lt_value nH

TermTerm2

Z=50 OhmNum=2

LLt1

R=L=Lt_value nH

TermTerm1

Z=50 OhmNum=1

CPWGCPW1

L=28.0 mmG=10.0 milW=50.0 milSubst="CPWSub1"

1_0pF_NPO_0603C1 CPWG

CPW2

L=28.0 mmG=10.0 milW=50.0 milSubst="CPWSub1"

48


BPF synthesisusing synthesistool E-synof ADS2003C

49


0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.80.0 3.0

-100

0

100

-200

200

freq, GHz

phas

e(S(

2,1)

)ph

ase(

RF_

BPF_

mea

sure

d..S

(2,1

))

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.80.0 3.0

-100

0

100

-200

200

freq, GHz

phas

e(S(

2,1)

)ph

ase(

RF_

BPF_

mea

sure

d..S

(2,1

))

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.80.0 3.0

-40

-20

-60

0

freq, GHz

dB(S

(2,1

))dB

(RF_

BPF_

mea

sure

d..S

(2,1

))

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.80.0 3.0

-40

-20

-60

0

freq, GHz

dB(S

(2,1

))dB

(RF_

BPF_

mea

sure

d..S

(2,1

))

|s21|/dB

Arg(s21)/degree

MeasuredSimulated

Measurement is performed with theAgilent 8753ES Vector NetworkAnalyzer, using Full OSL calibration

50

3.0 Microwave Filter Realization Using Stripline Structures

51

3.1 Basic Approach

52

Filter Realization Using Distributed Circuit Elements

• Lumped-element filter realization using surface mounted inductors and capacitors generally works well at lower frequencies (at UHF, say < 3 GHz).

• At higher frequencies, the practical inductors and capacitors loses their intrinsic characteristics.

• Also, a limited range of component values is available from the manufacturer.

• Therefore, for microwave frequencies (> 3 GHz), the passive filter is usually realized using distributed circuit elements such as transmission line sections.

• Here we will focus on stripline microwave circuits.

53

Filter Realization Using Distributed Circuit Elements (cont’d)

• Recall in the study of Terminated Transmission Line Circuit that a length of terminated Tline can be used to approximate an inductor and a capacitor.

• This concept forms the basis of transforming the LC passive filter into distributed circuit elements.

Zo

Zo

Zc ,

l

L

Zc ,

l

C

Zc

,

Zc ,

Zc ,

Zo

Zo

54

Filter Realization Using Distributed Circuit Elements (cont’d)

• This approach is only an approximation. There will be deviation between the actual LC filter response and those implemented with terminated Tline.

• Also, the frequency response of the distributed circuit filter is periodic. • Other issues are shown below.

Zc

,

Zc ,

Zc ,

Zo

Zo

How do we implement a series Tlineconnection? (only practical for certain Tline configuration)

Connection of physicallength cannot beignored at themicrowave region,comparable to

Thus some theorems are used to facilitate the transformation of the LCcircuit into stripline microwave circuits.Chief among these are the Kuroda’sIdentities (See Appendix 1)

Thus some theorems are used to facilitate the transformation of the LCcircuit into stripline microwave circuits.Chief among these are the Kuroda’sIdentities (See Appendix 1)

55

More on Approximating L and C with Terminated Tline: Richard’s Transformation

Zc ,

l

L

jLLjljZZ cin tan

Zin

LZ

l

c tan

(3.1.1a)

Zc ,

l

CZin

jCCjljYY cin tan

CY

l

cZc

1

tan (3.1.1b)

For LPP design, a further requirement isthat:

1tan cl (3.1.1c)82 1tan c

cll

Wavelength atcut-off frequency

Here, instead of fixing Zc and tuning l to approach an L or C,we allow Zc to be a variable too.

56

Example 3.1: LPF Design Using Stripline

• Design a 3rd order Butterworth low pass filter, Rs = RL= 50 , fc = 1.5 GHz.

Steps 1 & 2: LPP

Step 3: Convert to Tlines Zc =

0.500

Zc=

1.00

0

1

Zc=

1.00

0

1

Zo=1 g1 1.000H

g3 1.000H

g2 2.000F

g4

1

Length = c/8for all Tlinesat = 1 rad/s

500.0000.21

57


Length = c/8for all Tlinesat = 1 rad/s

Step 4: Add an extra Tline on the series connection and apply Kuroda’s 2nd Identity.

Zc =

0.500Z

c=1.

000

1

Zc=

1.00

01

Zc=1.0

Zc=1.0

Extra TlineExtra Tline

5.02

21 Zn

n2Z1=2

l

Z2=10.11 Z

Similar operation isperformed here

21

1

11

122

ZZ

n

Yc

58


Zc =

0.500

1

1

Zc=2.0 Zc=2.0

Zc =

2.000

Zc =

2.000

After applying Kuroda’s 2nd Identity

Length = c/8for all Tlinesat = 1 rad/s Since all Tlines have similar physical

length, this approach to stripline filterimplementation is also known as Commensurate Line Approach.

Since all Tlines have similar physicallength, this approach to stripline filterimplementation is also known as Commensurate Line Approach.

59


Zc =

25

50

50

Zc=100 Zc=100

Zc =

100

Zc =

100

Length = c/8for all Tlines atf = fc = 1.5 GHz

Zc/Ω /8 @ 1.5 GHz/mm W/mm 50 13.45 2.8525 12.77 8.00100 14.23 0.61

Microstrip line using double-sided FR4 PCB (r = 4.6, H=1.57 mm)

Step 5: Impedance and frequency denormalization Here we multiply allimpedance with Zo = 50

We can work out the correct width W given theimpedance, dielectric constant, and thickness.From W/H ratio, the effective dielectric constanteff can be determined. Use this together withfrequency at 1.5 GHz to find the wavelength.

We can work out the correct width W given theimpedance, dielectric constant, and thickness.From W/H ratio, the effective dielectric constanteff can be determined. Use this together withfrequency at 1.5 GHz to find the wavelength.

60


Step 6: The layout (top view)

61


m1freq=m1=-6.092

1.500GHz

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

-30

-20

-10

-40

0

freq, GHz

dB(S

(2,1

))

1.500G-6.092

m1

dB(B

utte

r_LP

F_L

C..S

(2,1

))

CC1C=4.244 pF

LL2

R=L=5.305 nH

LL1

R=L=5.305 nH

TermTerm2

Z=50 OhmNum=2

TermTerm1

Z=50 OhmNum=1

Simulated results

MSUBMSub1

Rough=0 milTanD=0.02T=0.036 mmHu=3.9e+034 milCond=1.0E+50Mur=1Er=4.6H=1.57 mm

MSubS_ParamSP1

Step=5 MHzStop=4.0 GHzStart=0.2 GHz

S-PARAMETERS

MTEETee3

W3=8.00 mmW2=0.61 mmW1=0.61 mmSubst="MSub1"

MLOCTL5

L=12.77 mmW=8.0 mmSubst="MSub1"

MTEETee2


MLOCTL7


MTEETee1


MLINTL2


TermTerm2

Z=50 OhmNum=2

MLOCTL6


TermTerm1

Z=50 OhmNum=1

MLINTL1


MLINTL3


MLINTL4


62

Conclusions for Section 3.1

• Further tuning is needed to optimize the frequency response.

• The method illustrated is good for the low pass and bandstop filter implementation.

• For high pass and bandpass, other approaches are needed.

63

3.2 Further Implementations

64

Realization of LPF Using the Step-Impedance Approach

• A relatively easy way to implement LPF using stripline components.

• Using alternating sections of high and low characteristic impedance Tlines to approximate the alternating L and C elements in an LPF.

• Performance of this approach is marginal as it is an approximation, where a sharp cutoff is not required.

• As usual, beware of parasitic passbands!!!

65

Equivalent Circuit of a Transmission Line Section

Z11 – Z12 Z11 – Z12

Z12

l

Zc

ljZZZ c cot2211

ljZZZ c cosec2112

oeoeo k

(3.2.1a)

(3.2.1b)

(3.2.1c)

Ideal lossless TlineT-network equivalent circuit

2

2cos2sin2

2sin2

2sin

2cos1

sincos

sin1

1211

tan

22

22

2

2

lc

cc

ll

lc

jZ

jZjZ

jZZZ

ll

l

l

l

Positive reactance

Positivesusceptance

66

Approximation for High and Low ZC

• When l < /2, the series element can be thought of as an inductor and the shunt element can be considered a capacitor.

• For l < /4 and Zc = ZH >> 1:

• For l < /4 and Zc = ZL 1:

2

tan

21211l

ZX

ZZ c l

ZB

Z c sin

11

12

lZX H 0B

0X lZ

BL

1

jX/2

jB

jX/2

X ZH l

B YLlWhen Zc 1l < /4

When Zc >> 1l < /4

Z11 - Z12 Z11 - Z12

Z12

67

Approximation for High and Low ZC (cont’d)

• Note that l < /2 implies a physically short Tline. Thus a short Tline with high Zc (e.g., ZH) approximates an inductor.

• A short Tline with low Zc (e.g., ZL) approximates a capacitor.

• The ratio of ZH/ZL should be as high as possible. Typical values: ZH = 100 to 150 , ZL = 10 to 15 .

H

cL Z

Ll

Lc

CCZ

l

(3.2.2a)

(3.2.2b)

68

Example 3.2: Mapping an LPF Circuit into a Step Impedance Tline Network

• For instance, consider the LPF Design Example 2.2A (Butterworth).

• Let us use the microstrip line. Since a microstrip Tline with low Zc is wide and a Tline with high Zc is narrow, the transformation from circuit to physical layout would be as follows:

L1=4.061 nH L2=9.803 nH

C1=3.921 pF C2=1.624 pFRL= 50 g0=1/50

69

Example 3.2: Physical Realization of LPF

• Using the microstrip line, with r = 4.2, d = 1.5 mm:

• L1 = 4.061 nH, L2 = 9.083 nH, C1 = 3.921 pF, C2 = 1.624 pF

W/d d/mm W/mm e

Zc = 15 10.0 1.5 15.0 3.68Zc = 50 2.0 1.5 3.0 3.21Zc = 110 0.36 1.5 0.6 2.83

19 307.60103356.32 sfk ceLoeLL

19 258.53103356.32 sfk ceHoeHH

70

Example 3.2: Physical Realization of LPF (cont’d)

l2l1

50 line 50 line

l4l3

0.6 mm15.0 mm

3.0 mm

To 50Load

mmZ

Ll

HH

c 5.611

mmZC

lL

Lc 2.912

mml 0.153

mml 8.34

Verification:

7854.0490.042 lL

7854.0202.044 lL

7854.0905.043 lH

7854.0392.041 lH

Nevertheless we stillproceed with the imple-mentation. It will be seenthat this will affect the accuracy of the –3 dB cut-offpoint of the filter.

71

Example 3.2: Step Impedance LPF Simulation with ADS Software

• Transferring the microstrip line design to ADS:

Microstrip line model

Microstrip step junctionmodel

Microstrip line substrate model

72

Example 3.2: Step Impedance LPF Simulation with ADS Software (cont’d)

m1freq=1.410GHzdB(S(2,1))=-3.051

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

-20

-15

-10

-5

-25

0

freq, GHz

dB(S

(2,1

))

m1

73

Example 3.2: Step Impedance LPF Simulation with ADS Software (cont’d)

• However if we extent the stop frequency for the S-parameter simulation to 9 GHz...

m1freq=1.410GHzdB(S(2,1))=-3.051

1 2 3 4 5 6 7 80 9

-15

-5

-25

0

freq, GHz

dB(S

(2,1

))

m1

Parasitic passbands,artifacts due to usingtransmission lines

74

Example 3.2: Verification with Measurement

The –3 dB point is around 1.417 GHz!

The actual LPF constructed in year 2000. The Agilent 8720D Vector Network Analyzer is used to perform the S-parameter measurement.

75

Example 3.3: Realization of BPF Using a Coupled Microstrip Line

• Based on the BPF design of Example 2.3:

50

Vs15.916 pF

0.1414 pF79.58 nH

0.7072 nH 0.7072 nH15.916 pF50 RL

J1

–90o

J2

–90o

J3

–90o

J4

–90o

4o

TlineAdmittanceinverter

To RL

To sourcenetwork

See appendix (using Richard’s transformationand Kuroda’s identities)

An array of coupledmicrostrip line

4o

o = wavelength at oo = wavelength at o

Section 1 Section 2 Section 3Section 4

An equivalent circuit model for coupled Tlineswith open circuit attwo ends.

Extra

76

Example 3.3: Realization of BPF Using the Coupled Microstrip Line (cont’d)

• Each section of the coupled stripline contains three parameters: S, W, d. These parameters can be determined from the values of the odd and even mode impedances (Zoo & Zoe) of each coupled line.

• Zoo and Zoe are in turn depend on the “gain” of the corresponding admittance inverter J.

• And each Jn is given by:

SW W

d

2

2

1

1

ooooo

ooooe

JZJZZZ

JZJZZZ

4,3,2for

1

1

1

21

1

21

21

1

NNo

nno

o

ggZN

ggZn

gZ

J

NnJ

J

Extra

133.0

497.1

6.12

4.12

12

21

2

1

o

GHzfff

GHz

GHz

o

From Example 2.3

77


009163.012

11

gZoJ

002969.021

21

2 ggZo

J

002969.032

21

3 ggZo

J

009163.0432

14

ggZoJ

588.371

403.831

2111

2111

ooooo

ooooe

ZJZJZZ

ZJZJZZSection 1:

Section 2:

Section 3:

Section 4:

680.431

523.581

2222

2222

ooooo

ooooe

ZJZJZZ

ZJZJZZ

680.43

523.58

4

4

oo

oe

Z

Z

588.37

403.83

3

3

oo

oe

Z

Z

Note:g1=1.0000g2=2.0000g3=1.0000g4=1.0000

Note:g1=1.0000g2=2.0000g3=1.0000g4=1.0000

Extra

78


• In this example, an edge-coupled microstrip line is used to implement the coupled transmission line structures needed in the BPF. Stripline does not suffer from dispersion and its propagation mode is pure TEM mode, however it is more difficult to implement physically due to the fact that the trace is buried within the dielectric.

• Design equations for coupled microstrip line implemented are widely tabulated.

• Here we will use FR4 (r = 4.6, r = 1.0) substrate with 1.0mm dielectric thickness, and 1 ounce copper (about 36m thick) copper laminate. The conductivity of copper is around 5.8107 Siemens/meter.

• Furthermore we will use the LineCal tool in Advanced Design System to work out the dimensions needed for the coupled microstrip line.

Extra

79


• Using the LineCal tool to work out the dimensions for sections 1 and 3.

Electrical length (l ), 90o for quarterwavelength.

Zoe

Zoo

Zo

Voltage couplingfactor in dB

Fix the frequency at1.5GHz, the centerof passband

Strategy:1) We ‘tune’ the W andS for the specifiedZoo and Zoe. 2) Then we ‘tune’ the length Lto meet the electrical length of/2 (quarter wavelength) at 1.5GHz.

Extra

80


• Using the LineCal tool to work out the dimensions for sections 2 and 4.

Extra

81


• Alternatively we can implement our own design tool, as shown below implemented on Microsoft Excel.

Based on design equations from Garg R., Bahl I. J.,”Characteristics of coupled microstriplines”, IEEETransaction on Microwave Theory and Techniques, MTT-27, No.7, pp. 700-705,July 1979.

Strategy:1) We ‘tune’ the W andS for the specified Zoo and Zoe. 2) Based on the width W of a single trace, we work out the effective permittivity, and use this to calculate the phase velocity. 3) From this we find the wavelength at 1.5GHz and work out the required quarter wavelength.

Extra

82

Example 3.3: Coupled-Line BPF Simulation with ADS Software

• Using ideal transmission line elements:

Ideal open circuit

Ideal coupled tline

Extra

83

Example 3.3: Coupled-Line BPF Simulation with ADS Software (cont’d)

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.51.0 10.0

0.2

0.4

0.6

0.8

0.0

1.0

freq, GHz

mag(S

(2,1

))

Parasitic passbands. Artifacts due to using distributed elements, these are not present if lumped components are used.

2fo

Extra

84

Example 3.3: Coupled-Line BPF Simulation With ADS Software (cont’d)

• Using a practical stripline model:

Coupled stripline model

Open circuitmodel

Stripline substrate model

Extra

85

Example 3.3: Coupled-Line BPF Simulation with ADS Software (cont’d)

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.51.0 10.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0

1.0

freq, GHz

mag(S

(2,1

))

Attenuation due to losses in the conductor and dielectric

Extra

86

Items for Self-Study

• Network analysis and realizability theory

• Synthesis of terminated RLCM 1-port circuits

• Ideal impedance and admittance inverters and practical implementation

• Periodic structures theory

• Filter design by the Image Parameter Method (IPM).

87

Other Types of Stripline Filter• LPF

• HPF:

SMD capacitor

BPF:

88

Other Types of Stripline Filter (cont’d)

• More BPFs:

• BSF:

89

Appendix 1 – Kuroda’s Identities

90

Kuroda’s Identities

• As extracted from Ref. [2]

122 1

ZZ

n

Z1

l

21

Z Z2/n2

l

nZ1

Z2

l

221

Znn2Z1

l

1Z

Z2

l

21

n

ZZ2/n2

l

1Z

1: n2

Z1

l

221

Znn2Z1

l

21

Z

n2: 1

Note: The inductor represents the shorted Tline while the capacitorrepresents the open-circuit Tline.

Note: Thelength of alltransmissionlines isl = /8

Note: Thelength of alltransmissionlines isl = /8

91

References

[1] R. E. Collin, “Foundations for Microwave Engineering”, 2nd Edition 1992, McGraw-Hill.

[2] D. M. Pozar, “Microwave Engineering”, 2nd Edition 1998, John-Wiley & Sons.* (3rd Edition 2005, John-Wiley & Sons is now available)

Other more advanced references:

[3] W. Chen (Editor), “The Circuits and Filters Handbook”, 1995, CRC Press.*

[4] I. Hunter, “Theory and Design of Microwave Filters”, 2001, The Institution of Electrical Engineers.*

[5] G. Matthaei, L. Young, E.M.T. Jones, “Microwave Filters, Impedance-Matching Networks, and Coupling Structures”, 1980, Artech House.*

[6] F. F. Kuo, “Network Analysis and Synthesis”, 2nd Edition 1966, John-Wiley & Sons.

* Recommended* Recommended

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