325 Smith Chart

7/29/2019 325 Smith Chart

1/13

Dean P. Neikirk 2004, last update May 5, 2004 1 EE 325, Dept. of ECE, Univ. of Texas at Austin

Smith charts

the load reflection coefficient behavior is given

as r and x vary, what exactly does L do? lets plot in the complex plane

for any non-negative value of Re(Z) (i.e., r 0) and any value ofIm(Z), falls on or within the unit circle in the complex plane

Notes on the reflection coefficient on lossless lines( )

( )

1

1

L oL

o L

r jxZ Z

Z Z r jx

+ = =

+ + +

0 1L

x = 0

r = 10

-j

j

-1 Re()

Im()

x=-20

r = 2

r = 0r = .5

r = 1

x=20

x = +2

r = 0 r = 100

r = 0r = 0

1

-j

j r = 0

-1 Re()

Im()

r = 0r = 0

r = 0

x = -2

x = 0

x = .5

x = -.5

x = -1

x = 1

pick a value of x, then vary r pick a value of r, then vary x


2/13


Smith charts

Relationships between and Z note relation between and the normalized impedance

u j v = +

( )( ) ( ) 222222

12

11

vuvj

vuvuxjr

+

++

+=+

( )

( )

( )( )

11

11

1 111

r j x

r j x

r j xr j x

+ ++ + + =

+ + +

consider in complex plane

so the relationship between the normalized impedance and the

real and imaginary parts of is

1

1r j x

+ + =

( )[ ]

( )[ ]1

1

++

++

xjr

xjr ( )[ ] ( )[ ]

( )[ ] ( )[ ]11

11

+++

++++=

xjrxjr

xjrxjr

2

22 xjr +=

1

1

L

o

Zr j x

Z

+ = + =

( )( )vju

vju

+

++=

1

1 ( )( )

( )( ) vju

vju

vju

vju

+

+

++=

1

1

1

1

( ) 2222

1

21

vu

vjvu

+

+=


3/13


Smith charts

Relationships between and Z

from the real part:

( )22

2

1

1

1 rvr

r

u +=+

+

( )( ) ( ) 222222

12

11

vuvj

vuvuxjr

++

++=+

( ) ( )[ ] ( )( ) ( )

( ) ( )( )

( )222

2

2

2

22

2

2

2

2

2

22

22

2222

2222

1

1

1

1

11

1

11

1

112

1

1

12

1121

121

11

+

+

=+

+

+

++=+

+

++

+

=+

++

+

+=+

+

=+++

+=++

+=+

r

rr

vr

r

u

r

rrrv

r

ru

r

r

r

rv

r

r

r

ruu

r

rvr

ruu

rrvurru

vuvuur

vuvur

u j v = +

( )

( )

2 2

2 2

1

1

u vr

u v

+=

+


4/13


Smith charts

Plot of reflection coefficient in complex

plane

-1

1

u, Re(reflection coef.)

v

,Im(reflect

ioncoef.)

1j

-1jcurves of constant r = Re(Z)

( )22

2

1

1

1 rv

r

ru+

=+

+

plot as a function of r these are circles!

2

2

2

2

1

2

1

=+

vu

r = 1

( )2

22

1

10

=+ vu

0 < r < 1

r > 1

r = 0

radius

+0,

1 r

r

r+1

1

center

u j v = +


5/13


Smith charts

Plot of reflection coefficient

-1

1


v

,Im(reflect

ioncoef.)

j

-j

plot as a function of x x =

x < 0

x > 0

x = 0

from the Im part:

( )2

2

2

0

1

0

11 =

+ vu

( )

=

+

111

2

2vu

curves of constant x = Im(Z)

( )

( ) ( )2222

22

1

2

1

1

vu

vj

vu

vuxjr

+

+

+

+=+

u = 1

these are alsocircles!

( )2

2

2 111

xxvu =

+

radius

x

1,1

x

1 center

u j v = +


6/13


Smith charts

1 .

0

1

.21.4

1.6

1.8

2.0

0.5

0.6

0.7

0.8

0.9

0.4

0.3

0.2

0.1

3.0

4.0

10

20

5.0

1.0

0.5

0.6

0.7

0.8

0.9

0.4

0.3

0.2

0.1

1.2

1.4

1.6

1.8

2.0

4.0

10

20

5.0

3.0

0.2

0.2

0.6

0.8

1

.0

1.0

0.4

0.4

0.6

0.8

0.2

0.2

0.6

0.8

1.0

1.0

0.4

0.4

0.6

0.8

0.1

0.2

0. 3

0.4

0.5

0.6

0.7

0. 8

0 .

9

1 .

0

1 .

2

1 .

4

1 .

6

1 .

8

2 .

0 4

. 0

3 .

0

5 .

0

1 0

2 0

0

neg

ativ

ere

activ

ecom

ponent

positiv

ere

activ

ecompo

nent


plane


v

,Im(reflectioncoef.)

curves of constant r = Re(Z)

curves of constant x = Im(Z)

the Smith Chart isa plot of the

reflection

coefficient in the

complex plane,with contours of

constant load

resistance and

load reactancesuperimposed

( )

( )

1

1

L oL

o L

r jxZ Z

Z Z r jx

+ = =

+ + +( ) 2

2

2 111xx

vu = +

( )22

2

1

1

1 rv

r

ru

+=+

+


7/13Dean P. Neikirk 2004, last update May 5, 2004 7 EE 325, Dept. of ECE, Univ. of Texas at Austin

Smith charts

lets plot in the

complex plane

example:

ZLn = 0.5 + j0.5

you go all the way

around the circlewhen l is such that2l= 2 recall = 2/, so

you go all the wayaround when

l= /2 !


plane

0.25

0.3

0

0.350.4

0

0.4

5

0.

49

0.2

0

0.150.10

0.0

5

0.0

1

0

0.2

9

0.28

0.27

0.26

0.3

1

0.3

2

0.33

0.34

0.360.370.38

0.39

0.41

0.42

0.43

0.4

4

0.4

6

0.4

7

0.4

8

0.0

2

0.03

0.04

0.06

0.07

0.08

0.09

0.110.12 0.13 0.14

0.16

0.17

0.18

0.19

0.21

0.2

2

0.2

3

0.2

4

negativ

ereactiv

ecom

pone

nt

positiv

ere

active

compon

ent

wa

ve

leng

ths

toward

generat

or

wavelength

s

towar

d

load

Re()

Im()

( )

( )

1

1

r j x

r j x

+ =

+ +

( )j 0.650.2 0.4 0.2L j e = + =

j

-j

01-1

x = 0.5

r = 0.5

( )2 21

1

j l j lnL

L

nL

Z

l e eZ

= = +



Smith charts

recall the input

impedance is

tangent is periodic

with period , soyou repeat every

time l= m recall = 2/, so

you get back to

the same

impedance when

l= m/2 ! you can find Z(l)

directly from the

Smith chart!

applet


plane

0.25

0.3

0

0.350.4

0

0.4

5

0.

49

0.2

0

0.150.10

0.0

5

0.01

0

0.2

9

0.28

0.27

0.26

0.3

1

0.3

2

0.33

0.34

0.360.370.38

0.39

0.41

0.42

0.43

0.4

4

0.4

6

0.4

7

0.4

8

0.0

2

0.03

0.04

0.06

0.07

0.08

0.09

0.110.12 0.13 0.14

0.16

0.17

0.18

0.19

0.21

0.2

2

0.2

3

0.2

4

negativ

ereactiv

ecom

pone

nt

positiv

ere

active

compon

ent

wa

ve

leng

ths

toward

generat

or

wavelength

s

towar

d

load

Re()

Im()

j

-j

01-1

( )( )

( )

tan

tan

L o

o

o L

Z jZ lZ l Z

Z jZ l

+ =

+

( ) 2 21

1

j l j lnLL

nL

Zl e e

Z

= =+



Smith charts

The Smith

Chart

the Smith chartalso has a phase

angle scale

measured using

distance inunits of

wavelengths

link to high res

smith chart

why? because

the angle of is

( ) 2 21

1

j l j lnLL

nL

Zl e e

Z

= =+

( )N

2

2 4 ll l

= =

direction of increasing l, orequivalently

away from load

towards generator



Smith charts

recall the inputimpedance is

example:

ZLn = 0.5 + j0.5 mark on Smith

chart

angle ofL is0.088

what is the inputimpedance ifl= /4? angle of at this

distance is(0.088 + 0.25) =0.338

read the Smith chart

Z( l= /4) =1 j 0.9

applet

Example 1

0.25

0.3

0

0.350.4

0

0.4

5

0.

49

0.2

0

0.150.10

0.0

5

0.01

0

0.2

9

0.28

0.27

0.26

0.3

1

0.3

2

0.33

0.34

0.360.370.38

0.39

0.41

0.42

0.43

0.4

4

0.4

6

0.4

7

0.4

8

0.0

2

0.03

0.04

0.06

0.07

0.08

0.09

0.110.12 0.13 0.14

0.16

0.17

0.18

0.19

0.21

0.2

2

0.2

3

0.2

4

negativ

ereactiv

ecom

pone

nt

positiv

ere

active

compon

ent

wa

ve

leng

ths

toward

generat

or

wavelength

s

towar

d

load

Re()

Im()

j

-j

01-1

( )( )

( )

tan

tan

L o

o

o L

Z jZ lZ l Z

Z jZ l

+ =

+

0.088

0.338

( ) 2 21

1

j l j lnLL

nL

Zl e e

Z

= =+

x = 0.5

r = 0.5

l= 0.25



Smith charts

recall the input

impedance is

example:

ZLn

= 2 + j(-1.2)

mark on Smith chart

angle ofL is0.288

what is the input

impedance ifl=0.232? angle of at this

distance is

(0.288 + 0.232) =(0.52) =(0.5+0.02)

read the Smith chart

Z( l= 0.232) =0.4 + j 0.1

applet

Example 2

0.25

0.3

0

0.350.4

0

0.4

5

0.

49

0.2

0

0.150.10

0.0

5

0.01

0

0.2

9

0.28

0.27

0.26

0.3

1

0.3

2

0.33

0.34

0.360.370.38

0.39

0.41

0.42

0.43

0.4

4

0.4

6

0.4

7

0.4

8

0.0

2

0.03

0.04

0.06

0.07

0.08

0.09

0.110.12 0.13 0.14

0.16

0.17

0.18

0.19

0.21

0.2

2

0.2

3

0.2

4

negativ

ereactiv

ecom

pone

nt

positiv

ere

active

compon

ent

wa

ve

leng

ths

toward

generat

or

wavelength

s

towar

d

load

-j

( )( )

( )

tan

tan

L o

o

o L

Z jZ lZ l Z

Z jZ l

+ =

+

0.288

0.5

( ) 2 21

1

j l j lnLL

nL

Zl e e

Z

= =+

x = -1.2

r = 2+0.02

S i h h



Smith charts

Transmission line applets

various http://www.eecg.toronto.edu/~bradel/projects/transmissionline/

http://fermi.la.asu.edu/w9cf/tran/

matching (single stub)

http://home3.netcarrier.com/~chan/EM/PROGRAMS/STUBMATCH/

Smith chart

pointer on chart gives gamma and impedance:

http://www.ntu.edu.sg/eee/eee5/el/smithchart/ http://www.ee.ust.hk/eee/course/elec241/java/SmithChart.html

local link

how tos http://www.web-ee.com/primers/files/SmithCharts/smith_charts.htm

http://cnx.rice.edu/content/m1060/latest/

S ith h t


13/13Dean P Neikirk 2004 last update May 5 2004 13 EE 325 Dept of ECE Univ of Texas at Austin

Smith charts

L-C transmission line summary

lossless L-C transmission line

dz

Z()

Y()Z Y j L j C j LC j j = = = = =

oo

o

V Z j L LZ Y j C C

+

+= = = =o

o

o

VZI

=

( ) j zoV z V e+ + =

( ) j zoV z V e +=

( ) j zoI z I e+ + =

( ) j zoI z I e +=

z = 0

V(z = -l, )

I(z = -l, )

z = -l

l

ZL

( ) 2 2 2 21

1

l j l j l j l L o nLL L

o L nL

Z Z Zl e e e e

Z Z Z

= = = =+ +

( )

( )

1 1

1 1

L oo L o nLL

o o L L o nL

Z ZV Z Z Z

V Z Z Z Z Z

+

= = =

+ + +

V+V-

( ) ( )( )

( )( )

( )( )

tanh tanh tan

tanh tanh tan

L o L o L o

o o o

o L o L o L

Z Z l Z Z j l Z jZ lZ l Z Z Z

Z Z l Z Z j l Z jZ l

+ + + = = =+ + +

1

1

L

o

Zr j x

Z

+ = + =

325 Smith Chart

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