ch. 2. transmission line analysis - universitetet i oslo€¦ · design dag t. wisland ch. 2....

Institutt for InformatikkINF5481: RF kretser, teori og design Dag T. Wisland

Ch. 2. Transmission Line Analysis

Phase velocity

rp cv eµebw === 1

Traveling voltage wave

( ) ( )0, sinxEV z t t zw bb

= -

• Voltage has a time and space variation• Space is neglected for low frequency applications• For RF there can be a large spatial variation


Two-wire transmission line• Alternating electric

field betweenconductors

• Alternating magneticfield surroundingconductors

• Dielectric medium tends to confine fieldinside material


Coaxial line

• Electric field contained between conductors

• Perfect shielding of magnetic field

• TEM mode up to a certain cutoff frequency

Always used for externally connected RF systems or measuring equipment. Also LAN.


Microstrip lines

Low dielectric medium High dielectric medium

Printed circuit board (PCB) section with ground plane to prevent excessive fieldleakage, interference, and radiation loss


Other TEM configurations

Triple-layer lineReduced radiation losses

Parallel plate lineLow impedance, high power


Transmission line representation• Detailed analysis is based on differential section

• Analysis applies to many types of transmission lines such as coax cables, two-wire, microstrip, etc.


Teminated lines - Voltage reflection coefficient

Open line: ZL ®¥, Γ0 = 1 Wave fully reflected with same polarity as incident wave

Short circuit: ZL = 0, Γ0 = -1 Wave fully reflected with opposite polarity of incident wave

Load match: ZL = Z0, Γ0 = 0 No reflection when load matches line impedance

0

00 ZZ

ZZVV

L

L

+-

=ºG +

-

Load impedance:

( )( ) ( ) 0

z z

z z

V z V e V e

I z V e V e Z

g g

g g

+ - - +

+ - - +

= +

= -

Reflection coefficient:

( ) ( )( ) 0

000 11

000

G-G+

=-+

=== -+

-+

ZVVVVZ

IVZZ L


Standing wavesShort circuit: ZL = 0, Γ0 = -1 Wave fully reflected with opposite polarity of incident wave

( ) ( ) ( ){ } ( ) ( )2cossin2Re),(

sin2

pwb

bw

bb

+==

=-=+

+-++

tdVVetdv

djVeeVdVtj

djdj

Þ Standing wave pattern:

d = λ

Note: d = -z


Standing wave ratio (SWR)Generally: ( ) ( ) ( ) ( )[ ]

( ) ( ) ( )[ ]dZdAdI

ddAeeVdV djdj

G-=

G+=G+= -++

1

11

0

20

bb

( ) djed b20

-G=GReflection coefficient:

SWR is a measure of mismatch of the load to the line

SWR=1 (matched)

SWR ®¥ (total mismatch)

0

0

min

max

min

max11

GG

-

+===

II

VV

SWRmatch

Note: SRW applies to lossless lines, but also works well in low-loss cases


Transformation of load impedance• Terminated lossless transmission line ( )- Input impedance:

- Used:

CLZ =0

( ) ( )( )

( ) ( )( ) ( ) lj

Llj

L

ljL

ljL

ljlj

ljlj

in eZZeZZeZZeZZZ

eeeeZ

lIlVldZ bb

bb

bb

bb

-

-

-

-

--+-++

=G-G+

===00

000

0

00

0

00 ZZ

ZZVV

L

L

+-

=ºG +

-

Þ ( ) ( )( )djZZdjZZZdZ

L

Lin b

btantan

0

00 +

+=


Quarter-wave transmission line

0 /4 L inZ Z Zl =

For d = λ/4 we have:

20 /40 /4

0 /4

0 /4

2tan4

24 tan4

L

inL

L

Z jZZZ d ZZZ jZ

ll

l

l

p ll l

p ll

æ ö+ ç ÷æ ö è ø= = =ç ÷ æ öè ø + ç ÷è ø

Lamda-quarter transformermatches given input and output impedances by choosing a line with characteristic impedance (narrowband matching, Zin =Z0):

0 /4 0line LZ Z Z Zl º =

500MHz 1.5GHz


Return and insertion losses

inini

r

PPRL G-=G-=÷÷ø

öççè

æ-= log20log10log10 2

( )21log10log10log10 ini

ri

i

t

PPP

PPIL G--=÷÷

ø

öççè

æ --=÷÷

ø

öççè

æ-=

IL¥ dB0 dB


Ch. 3. The Smith Chart

Mapping of the reflection coefficient in the complex domain

ljir

L

L ejZZZZ q

0000

00 G=G+G=

+-

=G

d

d = l 0


Combined diagram: Smith Chart

11

++-+

=Gjxrjxr

jxrz +=


How to use the Smith Chart

• Normalize load impedance: ZL ® zL

• find reflection coefficient: zL ® Γ0

• rotate reflection coefficient: Γ0 ® Γ(d)

• find normalized input impedance: zin(d)

• de-normalize input impedance: zin(d) ® Zin(d)

Example: determine input impedance Zin(d)

d

d = l 0

Γ(d)


Impedance transformation using Smith chart

1) Normalize load impedance:

ZL ® zL = ZL/Z0

2) Find reflection coefficient:

zL ® Γ0 = (zL-1)/(zL+1)

3) Rotate reflection coefficient:

Γ0 ® Γ(d) = Γ0 exp(-j2βd)

4) Find normalized input impedance:

Γ(d) ® zin(d) = (1+ Γ(d))/(1- Γ(d))

5) De-normalize input impedance:

zin(d) ® Zin(d) = Z0 zin(d)

zL=0.6+j1.2 Γ0 =0.2+j0.6

zin=0.3-j0.53

Γ=-0.32-j0.55

ZL=30+j60 ΩZ0=50 Ωd=l=2cmf=2GHz vp=c/2


Generalized standing wave ratio

11)(

)(1)(1

)(

)( 20

+-

=GÞ

G-G+

=

G=G -

SWRSWRd

dd

dSWR

ed dj b

Can determine SWR for a given Γ(d) by drawing circle with center at Γ = 0 through Γ(d) in the Smith chart.


Admittance transformation

)(1)(1

)(1)(11

)(1)(111

0

dede

dd

zy

dd

ZZY

j

j

inin

inin

G-G+

=G+G-

==

G+G-

==

-

-

p

p

e-jπ Γ(d) corresponds to 180º rotation of Γ(d) in Smith chart. This converts impedance to admittance

Alternatively: Rotate Smith chart by 180º : Admittance Smith chart

zin = 1+j

yin = 0.5-j0.5


Admittance Smith chart


ZY Smith chart

2222 ,

11

xrxb

xrrg

jxrzjbgy

inin

+-

=+

=Þ

+==+=

Use original Smith chartto display impedancesand rotated chart to display admittances.


A

B

C

D

EA: gA = Z0/RL = 1.6B: yB = gA + jZ0ωCL = 1.6 + j1.2Þ zB = 0.4 - j0.3C: zC = zB + jωL1/Z0 = 0.4 + j0.8Þ yC = 0.5 – j1.0D: yD = yC + jZ0ωC = 0.5 + j0.5Þ zD = 1 – j1E: zE = zD + jωL2/Z0 = 1

Z0 = 50 Ω f = 2 GHz

Þ Zin = Z0 = 50 Ω : Match at 2 GHz

Example: T-type network


Ch. 4. Single and multiport networks

• ”Black box” appoach forrestructuring andsimplifying complicatedcircuits

• Establish basic input andoutput relations for- Z parameters, - Y parameters, - h parameters, - ABCD parameters,- S parameters


Basic definitions

ïïþ

ïïý

ü

ïïî

ïïí

ì

úúúú

û

ù

êêêê

ë

é

=

ïïþ

ïïý

ü

ïïî

ïïí

ì

NNNNN

N

N

N i

ii

ZZZ

ZZZZZZ

v

vv

!"

!#!!""

!2

1

21

22221

11211

2

1

ïïþ

ïïý

ü

ïïî

ïïí

ì

úúúú

û

ù

êêêê

ë

é

=

ïïþ

ïïý

ü

ïïî

ïïí

ì

NNNNN

N

N

N v

vv

YYY

YYYYYY

i

ii

!"

!#!!""

!2

1

21

22221

11211

2

1

{ } [ ]{ }{ } [ ]{ }[ ] [ ] 1-=

==

YZ

VYIIZV

Z-matrix form Y-matrix form


Additional networks

þýü

îíì-ú

û

ùêë

é=

þýü

îíì

2

2

1

1

iv

DCBA

iv Chain or ABCD network

(often used for cascading)

þýü

îíìúû

ùêë

é=

þýü

îíì

2

1

2221

1211

2

1

vi

hhhh

iv Hybrid or h-network

(often used for active devices)

Example: BJT small-signal, low-frequencyh-network.Common-emitter


Series connection of networks

[ ] [ ] [ ] úû

ùêë

é¢¢+¢¢¢+¢¢¢+¢¢¢+¢

=¢¢+¢=22222121

12121111

ZZZZZZZZ

ZZZ

[ ]þýü

îíì

=þýü

îíì

¢¢+¢¢¢+¢

=þýü

îíì

2

1

22

11

2

1

ii

vvvv

vv

Z


Hybrid representation

Example:Darlingtontransistor pair

þýü

îíìúû

ùêë

é¢¢+¢¢¢+¢¢¢+¢¢¢+¢

=

þýü

îíì

¢¢+¢¢¢+¢

=þýü

îíì

2

1

22222121

12121111

22

11

2

1

vi

hhhhhhhh

iivv

iv


Parallel connection of networks

[ ] [ ] [ ] úû

ùêë

é¢¢+¢¢¢+¢¢¢+¢¢¢+¢

=¢¢+¢=22222121

12121111

YYYYYYYY

YYY

[ ]þýü

îíì

=þýü

îíì

¢¢+¢¢¢+¢

=þýü

îíì

2

1

22

11

2

1

vv

iiii

ii

Y


Cascading of networks using ABCD matrices

þýü

îíì

¢¢-

¢¢úû

ùêë

é¢¢¢¢¢¢¢¢

úû

ùêë

é¢¢¢¢

=þýü

îíì¢¢¢¢

úû

ùêë

é¢¢¢¢

=þýü

îíì

¢-

¢úû

ùêë

é¢¢¢¢

=þýü

îíì¢¢

=þýü

îíì

2

2

1

1

2

2

1

1

1

1

iv

DCBA

DCBA

iv

DCBA

iv

DCBA

iv

iv


Scattering parameters- At RF and microwave frequencies, open and short circuit conditions needed for measuring Z-, Y-, h-, and ABCD-parameters are no longer guaranteed!

- Also phenomena associated with wave propagation (reflections, oscillations, etc) may create experimental difficulties.

- Introduce S-parameters to characterize RF circuits and devices. Based on properly terminated transmission lines.

- Use power wave description to define input - output relations in terms of incident and reflected waves


S-parameter definitions

2portatwavepowerincident2portatwavepowerreflected

2portatwavepowerincident1portatwavepowerdtransmitte

1portatwavepowerincident2portatwavepowerdtransmitte

1portatwavepowerincident1portatwavepowerreflected

02

222

02

112

01

221

01

111

1

1

2

2

==

==

==

==

=

=

=

=

a

a

a

a

abS

abS

abS

abS

Incident and reflected normalized power waves:

--

++

-==-

=

==+

=

nnnn

n

nnnn

n

IZZV

ZIZVb

IZZV

ZIZVa

000

0

000

0

2

2

Power:

{ } ( )22*

21Re

21

nnnnn baIVP -==

þýü

îíìúû

ùêë

é=

þýü

îíì

Þ2

1

2221

1211

2

1

aa

SSSS

bb


Generalized S-parametersConsider two ports connected to different line impedances

( ) ( )jnVjj

iijna

j

iij nn ZV

ZVabS ¹=+

-

¹= +== 00

00

Note scaling by appropriate line impedances!


Analysis of two-port network

( ) ( ) ( )1111111 0 lj

inin eVlzVVzV --++++ =-=Þ== bIncident voltage:

( ) ( ) ( )1 11 1 1 1 10 j l

in inV z V V z l V e b- -- - - -= = Þ = - =Reflected voltage:

( )( ) þ

ýü

îíìúû

ùêë

é=

þýü

îíì

þýü

îíìúû

ùêë

é=

þýü

îíì

--

+

+

-

-

±

±

±

±

±

±

2

1

2221

1211

2

1

2

1

2

1 and0

022

11

VV

SSSS

VV

VV

ee

lVlV

lj

lj

out

inb

b

( )( )

( )

( )( )( )ïþ

ïýü

ïî

ïíì

--

úúû

ù

êêë

é=

ïþ

ïýü

ïî

ïíì

--

+

+

b-b+b-

b+b-b-

-

-

2

12

2221

122

11

2

1222211

221111

lVlV

eSeSeSeS

lVlV

out

inljllj

lljlj

out

in


Set-up for measuring S-parameters

System for mesuring S-parameters using a network analyzer

Requires measurements of traveling wave reflections and transmissions at both ports.

• RF output port• R reference input port• A measure reflected wave

S11 = A/R• B measure transm. wave

S21 = B/R• Reverse DUT for S22, S12• Correct for system effects


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

10log

10log 1 20log

in

L

in

PILP

S

aæ ö

= = ç ÷è ø

= - - G = -

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

wwww

ww

ww

p

==

=

==

=

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

[ ] úû

ùêë

éúû

ùêë

é=ú

û

ùêë

é1101

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

w ww

= = =+ + +

For ω ® 0:S21(ω) ® 2Z0/(R +2Z0)

For ω ®¥S21(ω) ® 0

( )

úúúú

û

ù

êêêê

ë

é

+

+÷÷ø

öççè

æ+++

=úû

ùêë

é

11

11

L

LGL

G

RCj

RRR

CjRR

DCBA

w

w

Transfer function H(ω ) =V2/VG = 1/A) (matching: ZG = ZL = Z0 ):


High-pass filter

Consider as cascade of 4 ABCD-networks

( )

úúúú

û

ù

êêêê

ë

é

+

+÷÷ø

öççè

æ+++

=úû

ùêë

é

111

111

L

LGL

G

RLj

RRRLj

RR

DCBA

w

w

( )( )

21

00

2 21 11

SA

R Zj L Z

w

w

= =æ ö

+ + +ç ÷è ø

For ω ® 0S21(ω) ® 0

For ω ®¥S21(ω) ® 2Z0/(R + 2Z0)


Band-pass filter

Band-pass filter in series config-uration. Consider as cascade of 3 ABCD-networks where:

úúúú

û

ù

êêêê

ë

é ++

+=ú

û

ùêë

é

11

1

L

GL

G

R

ZRRZR

DCBA

( ) ( )0

210

22 1

ZSZ R j L C

ww w

=+ + -

( )CLjRZ ww 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

ww

ww

w

é ùæ ö+ -ç ÷ê úè øë û=é ùæ ö+ + -ç ÷ê úè øë û


Insertion lossQ-factors easier to measure than impedances and admittances.

( ) úû

ùêë

é++=÷

øö

çèæ -+= e

ww LD

F

LDE jQ

QQRR

CLjRZ 1Series resonance:

( ) úû

ùêë

é++=÷

øö

çèæ -+= e

ww LD

F

LDE jQ

QQGG

LCjGY 1Parallel resonance: w

wwwe 0

0-=

02 8ZVPP GinL ==

( )( )2220

2

0 11

221

LDELDin

GL QQQ

PZZZ

VPe+

=+

=

÷÷ø

öççè

æ += 22

221log10ELD

LD

QQQIL eInsertion loss:


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 W = w/wc.


Low-pass filter transformation

GHz1h filter wit-LP =cw

Scaled frequency: cww W=

Scaled reactances:

( )

( )c

cC

c

cL

CC

CjCjCjjX

LL

LjLjLjjX

w

www

w

www

=Þ

==W

=

=Þ

==W=

~

~111

~

~

Þ

Þ


High-pass filter transformationScaled frequency: W-= cwwScaled reactances:

( )

CL

LC

LjCjCj

jX

CjLjLjjX

cc

cC

cL

ww

www

www

1~,1~

~1

~1

==Þ

=-=W

=

=-=W=

GHz1h filter wit-HP =cw

Þ

Þ


Band-pass filter transformationScaled and shifted frequency:

LU

cc

cLU

c

wwew

ww

ww

www

-=÷÷ø

öççè

æ-

-=W

Scaled reactances:

LU

LU

LUC

LU

LU

LUL

CCC

L

LjCjCjCjjB

LCLL

CjLjLjLjjX

www

www

wwwew

W

w

wwww

ww

wwew

W

-=

-=

+=-

==

-=

-=

+=-

==

~,~

~1~

~,~

~1~

20

0

20

0

GHz 1

:frequencycenter filter -BP

0 === LUc wwww

Þ

Þ


Band-stop filter transformation

( )

( )( )

20

20

~,1~:capacitorShunt

1~,~:inductor Series

www

ww

wwwww

CCC

L

LCLL

LU

LU

LU

LU

-=

-=Þ

-=

-=Þ

GHz 1:frequencycenter filter -BS

0 == cww

Þ

Þ


Summary of transformationsLUBW ww -=


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:

( ) ( )lpb ljZljZZ shortin 2tantan 00 ==( ) ( )lpb ljYljYY openin 2tantan 00 ==


Richard’s transformationChoose arbitrarily a line segment of length l = l0/8 at a reference frequency f0 = vp/l0. Use:

00 4tan SZjZLjjXZ Lin =÷

øö

çèæ W===pw

W===444

20

0 ppllp

lp

ffl

Short-circuit:

00 4tan SYjYCjjBY Cin =÷

øö

çèæ W===pwOpen-circuit:

Note: Richard’s transformation maps the lumped element frequency response for 0 £ f £ ¥ into the range 0 £ f £ 2 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 Wp/4 and characteristic impedance ZUE. Represent as chain-parameter two-port:

[ ]( ) ( )( ) ( ) ú

ú

û

ù

êê

ë

é

-=

úú

û

ù

êê

ë

é

WW

WW=ú

û

ùêë

é= 1

1

1

14cos4sin4sin4cos

2UE

UE

UE

UE

UEUE

UEUE

ZS

SZ

SZj

jZ

DCBA

UE pppp

( ) ( )4cos1

1,4tan2

pp W=-

W=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 l0/8 transmission lines.

• Convert series stub lines to shunt stubs using Kuroda’s identities.

• De-normalize and and select equivalent microstrip lines

Institutt for Informatikk

Ch. 8. Matching Networks

• Matching networks are critical for at least tworeasons:– Maximize power transfer– Minimize SWR

• Primary goal of a matching network is to get noreflection

INF5481: RF kretser, teori og design Svein-Erik Hamran


Two Component Matching Network

Also called L-sections.



Impedance effect of series and shunt

• The addition of a reactanceconnected in series with a complex impedance results in motion along a constant-resistance circle in the combinedSmith Chart.

• A shunt connection producesmotion along a constantconductance circle.



Matching network using Smith Chart


1. Find normalized source and loadimpedances

2. Plot circles of constant resistance and conductance that pass through

3. Plot circles of constant resistance and conductance that pass through

4. Identify intersection points.5. Find values of normalized reactances

and suceptances of inductors and capacitors by tracing a path along thecircles.

6. Determine the actual values ofinductors and capacitors fror a given frequency.


Two Designs of an L-type Matching



T-type Matching Network

• The loaded nodal quality factor of the matching networkcan be estimated from the maximum nodal

• Addition of a third element into the matching networkintroduces an additional freedom that allows us to controlthe value of by chosing an approriate intermediateimpedance



T-type Matching Network for Qn = 3



Mixed Design TL and Discrete• At increasing frequency and reduced wavelength the parasitics in the

discrete elements become noticable.• Need to take parasitics into account.• Discrete components only available in certain values.• A solution is a mix with transmission lines and capacitors.• Possible to tune the design below by changing capacitor value and

positions.



Matching network with lumped and distributed components


• Draw a SWR circle through• Draw a SWR circle through• Transition between the two circles can

be made arbitrarly. Chose A and B.• Read off the two transmission line

lengths needed.


Single-stub macthing network

• Select the length of the stub suchthat it produces a susceptancesufficient to move the loadadmittance to the SWR circle thatpasses through the normalized input impedance.


ch. 2. transmission line analysis - universitetet i oslo€¦ · design dag t. wisland ch. 2....

Documents