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5.2 Single-Stub Matching

Matching using TL L, C not required.

Hi h Q f t (l l )

Open or shorted stub (TL) Zin = pure reactance

Y t High Q-factor (low loss) Easy fabrication Big size

Yin = pure suceptance

itiit binductive is stub shorted ,4/ if

d=0.425-0.315=0.110 d=0.5+0.075-0.315=0.260y=1+j1 47 -j1 47 needed y=1 j1 47 j1 47 needed

1st solution 2nd solution

Microwave Engineering, JJEONG3

y=1+j1.47 j1.47 needed y=1-j1.47 j1.47 neededShort-circuited stub l=0.095 Short-circuited stub l= 0.405

Solutions

Line length metal loss bandwidth

Figure 5.5b (p. 231)(b) The two shunt-

stub tuning solutions.

fvp 2==

2 (c) Reflection

coefficient magnitudes versus frequency for the -> Use short line if possible

the longer , the larger phase deviationpf v

l

=


tuning circuits of (b).p

How to Make Open/Short Stubs

Microstrip line Open stub : preferred.

Sh t t b i h l d d Short stub : via-hole needed. Coaxial cable and waveguide

Open stub : large size antenna Short stub preferred.


Analytic Solutions


HW 5.1 Use ADS to solve the following problemsSmall-signal equivalent circuit of MOSFET are shown below with parameter values. You design an amplifier at 10 GHz using this MOSFET A Z is 50 ohmdesign an amplifier at 10 GHz using this MOSFET. A Z0 is 50 ohm.1) Compute the input and output impedance (Zin and Zout) at 10 GHz. (Zin =10-j15.9, Zout

=7.6-j38.2)2) Design input matching network using shunt C-series L.2) Design input matching network using shunt C series L.3) Design output matching circuit using series L-shunt C.4) For the designed input/output matching circuits, find ZS and ZL. What relationship do ZS

and Zin have? What relationship do ZL and Zout have?

VC

G D

dsR

Small-signal equivalent circuitin p L out

output matchingnetworkZ

cmVgcVgsC

iR

ds

dsC

Input matchingnetwork

network

SZLZ

= 50Z

= 500Z iLoL

oC= 10iR = 200dsR

pF1=gsC pF4.0=dsCinZ0Z outZ

= 500Z

sViC


mS50=mg

Remarks


Quarter-wave Transformer

ljZZljZZZZ

L

Lin

tantan

1

11 +

+=in

Z

0 : matching ZZin =L

in ZZlZ

21)4/( ==

Figure 5.10 (p. 241) A single-section quarter-wave

242 ==

40/=matching transformer. at the design frequency f0. real. be should 01 LL ZZZZ =

series stub

TL

shunt stub

ZZ

180 rotation real axisLZLZ0Z

4

Real impedance to TL Complex ZL : use TL/series/shunt stub first (bandwidth reduced)


Multi-section : broader bandwidth

Multiple Reflections (1) Find for the right figure (quarter-wave impedance transformer)

1) Determine Zin) in

2

22 ,tan

tanjZZjZZZZ

L

Lin =+

+=

Figure 5.13 (p. 244)Partial reflections and transmissions on a

single-section matching transformer0

0

2 tan

ZZZZ

jZZ

in

in

L

+

=

+

then,

single-section matching transformer.

2) Multiple reflections

2 1 1 2 21 2 1 3

2 1 1 2 2

, ,

2

LL

Z Z Z Z Z ZZ Z Z Z Z Z

Z

= = = =

+ + +

221 1

1 2

112 2

21

21

ZTZ Z

ZTZ Z

= + =+

= + =


12 21 2Z Z+

Multiple Reflections (2)assume Vinc=1

12222112221123211 +++= ATAATATATTeeT jj

2

22

2212211

12222112221123211

1(

++++= eTT

AAATTj

A

( )1121211222

31

02

32

321121

232

2312211

1,1,1

1

=+==+

=

+=+=

=

TTeeeTTeeTT

j

j

nj

jnjnnj


( )2311 + e

j

Multiple Reflections (3)

231

231

1 jj

ee

++

=

/4 transformer

312

2

112 ===

LL Z

ZZZZZZ

12

2

== jjL

ee ( )matchingperfact01 2111 =

=

fl i ll F

or ,reflection small For

22

31

31 .1,1

jj

ee

++

=

Multiple Reflections (4)

( )3232132132121 eeeTeeTeTeTV

j

jjjjjjjL ++++=

Voltage across ZL

( )( ) ( )

( )

232

32132321

4

1111

eeTeeT j

jjj

==

+

=+++=

rtransforme for

( )

( ) ( )21

2

1

1

1

12

1

21

31

11

11

11

,42

ZZ

ZZVjjV LLL ==

+=

+

=+

=

== rtransforme for

21111

power to load = power from TL1 (Z1)22

2

21

21

ZZ

ZV L

L

L =1

121

Z

Thus, all power incident to Z1 Transmission line delivered to load ZL !

( )( )

=> load thein phasein added 10,0 223321

jeZZZ

/4 transformerreflected waves added out-of phase at the interface between Z1 and Z2 lines


p 1 2incident waves added in-phase in the load ZL

Anti-reflection Coatings

Comparison between a glasses lens without anti-reflective coating (top) and a lens with anti reflectivea lens with anti-reflective coating (bottom). Note the reflection of the photographer in the top lens and the tinted


lens and the tinted reflection in the bottom.

Anti-reflection Coatings - Transformer

Single layer coating : eliminates the reflection at one wavelength

Quarter-wave impedance transformation

>> 210 ZZZ

eliminates the reflection at one wavelengthMulti-layer coatings :

eliminates the reflection over the visible spectrum.

Bandwidth of Quarter-wave Transformer (1) At the design frequency of f0, the electrical length of the matching section is

0/4, but at other frequencies the length is different, so a perfect match is no longer obtained.g

Bandwidth of the transformer ?

( )1

1 ,tan

jZZjZZZfZ Lin

+=( )

02

10

0

11

,

tan

ZZZZZZZ

jZZf

Lin

in

Lin

=+

=

+

ffrequencydesignaround:ionapproximat2200 sec])/(4[1

1ZZZZ LL +

=

1sec2

f frequency,design around :ion approximat

2

0

ZZ

>> ==

00

0

2

,4 f

c

f

cos2 0

0

L

L

ZZZZ

Figure 5.11 (p. 242)

Approximate behavior of the reflection coefficient magnitude for a single-section

==

==

022

2

fff

c

0ff


g gquarter-wave transformer operating near its

design frequency.frequency0f

Bandwidth of Quarter-wave Transformer (2)

|| vs. f/f0 as a function of ZL/Z0 The ZL/Z0 closer to 1 the wider bandwidthL

Z0Z 4/ The ZL/Z0 closer to 1, the wider bandwidth(increased bandwidth for smaller load mismatches)

Use multi-section transformer for wider bandwidth

For allowable m, the fractional BW f/f0 isi bgiven by

=

0

0

2

1

0

2

1cos42

ZZZZ

ff Lm

Ex 5.5) /4-transformer from 10 to 50 at 3 GHz. Bandwidth for VSWR 1.5

00 1 ZZf Lm

Figure 5.12 (p. 243)Reflection coefficient magnitude versus frequency

for a single section quarter wave matching

3 GHz. Bandwidth for VSWR 1.5

( )%2929020136.221050. 01

===

fZZZ L

VSWRans)


for a single-section quarter-wave matching transformer with various load mismatches.

( )%2929.0,2.01 0

==+

=fm VSWR

Multi-section Transformer

?nZ

Fi 5 14 ( 245)

Single sectionLower ZL/Z0, wider bandwidth

Figure 5.14 (p. 245)Partial reflection coefficients for a multisection matching transformer.

LZ1Z0Z

section-single

100

=ZZL

Lets use 10-sections with a same impedance transformation ratio for each section. Then,

23.1

10, 11011

101

2

0

1

=

======

X

XZXZXZZ

ZZ

ZZ

LL

2


Multi-section low |Zn+1-Zn| broad bandwidth23.1 X

Binomial Multi-Section Transformer

Maximally flat performance2ln0

01 CZZZZ

ZZ N

nL

LN

n

n

+

!)!(!,

nnNNC Nn

=

Figure 5.15 (p. 250)Reflection coefficient magnitude versus frequency for

lti ti bi i l t hi t f f E l


multisection binomial matching transformers of Example 5.6 ZL = 50 and Z0 = 100.

Chebyshev Multi-Section TransformerEqui-ripple performance : optimize the bandwidth at the expense of passband ripple.

sections, of numbe for table Chebyshev to Refer

:(NZn

) allowable minimumm :

Figure 5.17 (p. 255)R fl ti ffi i t it d f f th


Reflection coefficient magnitude versus frequency for the multisection matching transformers of Example 5.7.

Tapered Lines

Multi-section matching transformersection # BWnn ZZ +1infinite # of sections

C ti l t d li

01 + nn ZZ

Continuously tapered line

ZZ

ZZZZZZ

2

+++

=

dzdz

ZZd

ZdZdz

)ln(

21

2,0 0== then

Figure 5.18 (p. 256)A tapered transmission line matching

section and the model for an incremental

Theory of small reflection

( ) = ZdLz j1 2 length of tapered line. (a) The tapered

transmission line matching section. (b) Model for an incremental step change

in impedance of the tapered line.

( )

( ) ( )

= =

zZ

dzZZ

dzde

Lz

z

zj

:

)ln(21

00

2


Tapered Lines : Exponential Taper Exponential taper

( )

Tapered Lines : Klopfenstain Taper Klopfenstain taper

which design is best? Klopfenstain taper minimum || over the passband for a given length. minimum || over the passband for a given length.

( ) ( ),1/2cosh

ln21ln 200 ALzAA

ZZzZ L

+=

( ) ( ) ( ) .11

1,,

cosh2

0 2

21 xdy

yAyAI

AxAx

Ax

for

==

: 0.02 for 1.13

mK

=

( )

( ) ( )cos:

22

1

ALAL

e

xIy

Lj for

function. Bessel modified

>

=

( )

ln21

.cosh

00

00

0

ZZ

ZZZZ

ALA

e

L

L

L DC at

for

+

==

>=

( ) 0cosh

0max

00

LzZA

L

and z at steps has

=


Figure 5.21 (p. 260) Solution to Example 5.8. (a) Impedance variations for the triangular, exponential, and Klopfenstein tapers. (b) Resulting reflection coefficient

magnitude versus frequency for the tapers of (a).

( ) .0 LzZ and z at steps has =

Bode-Fano Criterion (1)0RC=Q Theoretical limits of the performance of

the impedance matching network:

1RC

=Q

Can we achieve a perfect match over a specified bandwidth?

If h ll d ? Wh i h 0RC If not, how well can we do? What is the trade off between m, the maximum allowable reflection in the passband, and the bandwidth?

0

RL

=Q

the bandwidth?

How complex the matching network be for a given specification?

0L=Q

Bode Fano limit

R=Q


Bode-Fano Criterion (2)

dd == 1ln1ln1ln

( ) figure. rightin and a type Consider ex) )(

m 0 0 or perfect match

RCdd

mm

lnlnln0

m = 0 = 0 or perfect match at a finite # of freq. As RC increases, the quality of match(, 1/m ) must decrease.( , m )

Qm

1ln0

Figure 5.23 (p. 263)Illustrating the Bode-Fano criterion. (a) A possible

reflection coefficient response. (b) Nonrealizable and realizable reflection coefficient responses.

HigherQ circuits are intrinsically harder to match than are lowerQ circuits.Higher-Q load : narrower band impedance p

Optimum case ||= over the passband and ||=1 elsewhere like Fig (a) (sharp transition)

matching.


||=m over the passband and ||=1 elsewhere like Fig. (a) (sharp transition) but, impractical Chebyshev type

Q-factor Lines on Smith Chart

The lower Q, the wider bandwidth Keep the Q low while impedance

transformation!!!

Q-factor Series elements : Q = Im[Z]/Re[Z]

Q of inductor : Ls/Rs transformation!!!s sRs Ls

Q of capacitor : 1/(RsCs)

R C

Q /

Rs Cs

Parallel elements : Q = Im[Y]/Re[Y] Q of inductor : Rp/Lp Q of capacitor : RpCp

RRp Rp


Lp Cp

Broadband Match Using Multi-Section

1. Shunt C-series L2. Shunt L-series C3 Series TL-series L3. Series TL series L4. Series TL-shunt L

Which is the most broadband?

Think of the BW of Series L-shunt C-series L-shunt C-series L


Series L shunt C series L shunt C series Lmatching circuits.

HW 5.2

bandwidth : HW 5.1(S S ) ( 10 GH )(S11, S21). ( = 10 GHz)

1. 1 lumped elements2. TL (series-L ) + lumped L or C3 TL (series-L) + stub3. TL (series-L ) + stub4. 2 lumped elements5. 3 lumped elements6. .6. .


Remarks


Impedance Matching Using Transmission Lines

Lumped elements (L, C) Low quality factor : high loss Small size

Transmission lines Transmission lines High quality factor : low loss Easy fabrication : well-controlled Big size

Can we replace the lumped elements with transmission lines? Can we replace the lumped elements with transmission lines?


Impedance Synthesis using Short T/L (1)

ljZZZZ L tan0+=

, l Z0Z Z 12/6/ inZ ljZZljZZ

ljZZZZin

1001

101 ++

+

1Z0Z0Z

ljZZ 01 +

peq v

lZL 1= High impedance (Z1) line

inductive (series) Z L

l

shunt C

p Z1 Leq

ljYYljYYljYYYYin

10

01

101 ++

+

shunt C

01 YY >>inY

1Z0Z0Z

lp

eq vl

ZC

1

1=

Low impedance (Z1) line capacitive (shunt) Z1 Ceq

l

series C : gap


What happens on Smith Chart?

5.05.0 jzL +=

Series High/low impedance line Cf)quarter-wave impedance transformer

5.05.0 jzL +

inductor series~01 ZZ >

capacitorshunt ~01 ZZ distributed Multi-stage > single-stage Short TL > long TL Some mismatch at center frequency > perfect match at center frequencySome mismatch at center frequency > perfect match at center frequency


Remarks


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