서강대학교 전자공학과 윤상원 교수 2. smith chart. microwave & millimeter-wave...

서강대학교 전자공학과윤상원 교수

2. Smith Chart

Microwave & Millimeter-wave Lab. 2

차 례

1. Smith chart ; introduction --------------------------------2. Reflection coeff. and impedance ----------------------3. Normalized impedance equation ----------------------4. Impedance Transformation ------------------------------5. Some transmission line example -------------------6. Admittance transformation ------------------------------7. Parallel and series connections ------------------------8. Single stub matching ----------------------------------------

3 4 61020222632


1. Smith chart ; introduction

Distributed nature of the transmission line Impedance is repeated periodically

One-to-one correspondence between the impedance and the reflection coeff. Impedances of the passive circuits are mapped into unit circle of the reflection coefficient plane. Impedance is treated as a reflection coefficient.

Smith chart ; a unit circle which contains all the impedance of the passive circuits


2. Reflection coeff. and impedance

Reflection coefficient in phasor form

example

riLj

irL

L LejZZ

ZZ00

1000

0

00 /tan where

1 0 0 LZ

0 50 0 LZ1 0 LZ

o0 34 83.0 )15050( jZL


Reflection coefficient and impedance(2)

0

30

60120

150

180

210

240 300

330

0=+10=- 1

0=0

0=0.83<34o0.8

0.4


3. Normalized impedance equation

Input impedance ; reflection coefficient

z=- d z=0

Z0 Z L

Z in 0d

irdjj

d ee L 20

ir

irin j

jZdZ

1

10

Normalize the impedance as

ir

irinin j

j

d

djxrzZdZ

1

1

1

1/)( 0

A mapping from zin-plane to -plane

Microwave & Millimeter-wave Lab.

Reflection Coeff. Plane

7


Normalized impedance equation(2)

Normalized impedance vs. reflection coeff.

22

22

)1(

21

ir

iirin

jjxrz

Real and imaginary terms ;

, )1(

122

22

ir

irr

22)1(

2

ir

ix

Normalized impedance is represented in terms of the reflection coefficient. impedance can be plotted in the reflection coefficient plane.


Smith Chart

9

, 1

1

1

22

2

rr

rir

22

2 111

xxir

Normalized impedance is transformed to ;

Real Part(resistance)

Imaginary Part(reactance)


Real Part(resistance)

10


Imaginary Part(reactance)

11


Smith Chart(1)

12


Smith Chart(2)

Normalized impedance is transformed to ;

, 1

1

1

22

2

rr

rir

222 11

1

xxir


The Smith Chart(3)


4. Impedance Transformation

For general load ; reflection coefficient

z=- d z=0

Z0 Z L

Z in 0d djjd ee L 2

0

Only phase changes as d increases or decreases for lossless transmission line.But, impedance changes its magnitude and phase

Calculation of input impedance using reflection coefficient on the Smith Chart ; “ 6 steps in the textbook “


Impedance Transformation(2)

1. Normalize the load impedance ZL with Z0 ;

2. Locate zL in the Smith Chart ;

3. Identify the load reflection coefficient in the Smith Chart in terms of its magnitude and phase.

4. Rotate 0 by its electrical length 2d to obtain in(d).

5. Record the normalized input impedance at this spatial location d ;

6. Convert into the actual impedance ;

inz

inz inZ



Example 3-3 ;

z=- d z=0

Z0 Z L

Z in 0d 6030 jZL

GHzcmdZ 2at 2 , 500

odm 99.1912 , 77.832 1

1. Normalize the load impedance ZL with Z0 ;

2. Locate zL in the Smith Chart ;

2.16.050/6030 jjzL



3. Identify the load reflection coefficient in the Smith Chart in terms of its magnitude and phase.

4. Rotate 0 by its electrical length 2d to obtain in(d) ;

5. Record the normalized input impedance at this spatial location d ;

6. Convert into the actual impedance ;

* exact calculation leads to

od 99.1912

53.03.0 jzin

inz

5.2615 jZin 7.267.14 jZin


Example 10.10

The use of the transmission line chart is best shown by example . Let us again consider a load impedance, ZL=25+j50 , terminating a 50- line. The line length is 60cm and the operating frequency is such that the wavelength on theline is 2m. We desire the input impedance.

Solution. We have , and we readBy drawing a straight line from the origin through A to the circumference, we note a reading of 0.135On the wtg scale. We have And it is, therefore, 0.3 from the load to the inputWe therefore find on the circle oppositeWtg reading of 0.135+0.300=0.435. And the pointlocating the input impedance is marked B. TheNormalized input impedance is read as 0.28-j0.40,and thus . A more accurate analyticalCalculation gives .

20


D10.6

A load is located at z=0 on a lossless 50- line. The operating frequency is 200MHz and the wavelength on the line is2m.(a)If the line is 0.8m in length, use the Smith chart to find the input Impedance.

Therefore 0.4 from the load to theinput. Using smith chart tool, the pointlocating the input impedance ismarked B.The input impedance is read 79+j99

21


D10.6

(b)What is s?


2-6. Admittance transformation

Parametric admittance equation Normalized admittance equation

The conventional Smith Chart(Z-Smith Chart) gives

on the Smith Chart admittance located at 180o

away from the impedance point.

)(1

)(11

0 d

d

zY

Yy

in

inin

)(1

)(1

)(1

)(1

de

de

d

dj

j


Admittance transformation(2)

Example 3-6 ; convert the normalized input impedance given as

to normalized admittance.<solution> by direct inversion

4211

j

in ejz

2

1

2

1

2

1 4 jeyj

in



Admittance Smith Chart(Y-Smith Chart)



ZY-Smith Chart


2-7. Parallel and series connections

Parallel connection of R and L elements


Parallel and series connections(2)

Parallel connection of R and C elements



Series connection of R and L elements



Series connection of R and C elements



Example of a T-network


2-8. Single stub matching

Impedance matching using distributed element

d1

yL

y in y'in

y0 =1

jby in 1'

Input admittance at d1 ;


Single stub matching(2)

Stub length

d

yL

y in y'in

y stub

y0 =1

jbystub

Stub length l1 ;

서강대학교 전자공학과 윤상원 교수 2. smith chart. microwave & millimeter-wave...

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