1

Prof. Ji Chen

Notes 12 Transmission Lines

(Smith Chart)

ECE 3317

Spring 2014

2

Smith Chart

The Smith chart is a very convenient graphical tool for analyzing transmission lines and studying their behavior.

A network analyzer (Agilent N5245A PNA-X) showing a Smith chart.

3

Smith Chart (cont.)

Phillip Hagar Smith (April 29, 1905–August 29, 1987) was an electrical engineer, who became famous for his invention of the Smith chart.Smith graduated from Tufts College in 1928. While working for RCA, he invented his eponymous Smith chart. He retired from Bell Labs in 1970.

Phillip Smith invented the Smith Chart in 1939 while he was working for The Bell Telephone Laboratories. When asked why he invented this chart, Smith explained: “From the time I could operate a slide rule, I've been interested in graphical representations of mathematical relationships.”

In 1969 he published the book Electronic Applications of the Smith Chart in Waveguide, Circuit, and Component Analysis, a comprehensive work on the subject.

From Wikipedia:

4

Smith Chart (cont.)

1

1Nin

zZ z

z

The Smith chart is really a complex plane:

Re z

Im z

1

2

2

j zL

j lL

z e

e

L z2 l

0

0

ZΓ

ZL

LL

Z

Z

z l

5

Smith Chart (cont.)

1

1N Nin in

zR jX

z

Denote z x jy (complex variable)

1

1N Nin in

x jyR jX

x jy

1 1N Nin inx jy R jX x jy

1 1

1

N Nin in

N Nin in

x R yX x

x X yR y

Real part:

Imaginary part:

so

6

1 1

1

N Nin in

N Nin in

x R yX x

x X yR y

Smith Chart (cont.)

From the second one we have

11

N Nin in

yX R

x

Substituting into the first one, and multiplying by (1-x), we have

1 1 11

N Nin in

yx R y R x

x

2 21 1 1 1N Nin inx R y R x x

7

Smith Chart (cont.)

2 21 1 1 1N Nin inx R y R x x

2 2 21 1 1N Nin inx R y R x

2 21 2 1 1 0N N N Nin in in inx R xR R y R

2 21 2 1 1N N N Nin in in inx R xR y R R

2 2 12

1 1

N Nin in

N Nin in

R Rx x y

R R

Algebraic simplification:

8

Smith Chart (cont.)

2 2 12

1 1

N Nin in

N Nin in

R Rx x y

R R

2 2

2 1

1 1 1

N N Nin in in

N N Nin in in

R R Rx y

R R R

22

22

1 1

1 1

N N NNin in inin

N Nin in

R R RRx y

R R

2

22

1

1 1

Nin

N Nin in

Rx y

R R

9

Smith Chart (cont.)

2

22

1

1 1

Nin

N Nin in

Rx y

R R

This defines the equation of a circle:

, ,01

Nin

c c Nin

Rx y

R

center: radius:1

1 Nin

RR

Re z

Im z

0NinR

1NinR

3NinR

0.2NinR

NinR

1cx R Note:

10

Smith Chart (cont.)

1 1

1

N Nin in

N Nin in

x R yX x

x X yR y

Now we eliminate the resistance from the two equations.

From the second one we have:

1 NinN

in

x X yR

y

Substituting into the first one, we have

11 1

Nin N

in

x X yx yX x

y

11

Smith Chart (cont.)

Algebraic simplification:

11 1

Nin N

in

x X yx yX x

y

2 21 1 1 0N Nin inx X y x y X y x

2 21 2 0N Nin inx X y y X

2 221 0

Nin

x y yX

12

Smith Chart (cont.)

2 221 0

Nin

x y yX

2 2

2 1 11

N Nin in

x yX X

This defines the equation of a circle:

1, 1,c c N

in

x yX

center: radius:1

Nin

RX

cy RNote:

13

Smith Chart (cont.)

This defines the equation of a circle: 1, 1,c c N

in

x yX

1Nin

RX

Re z

Im z1N

inX

1NinX

0NinX

NinX

3NinX

3NinX

0.5NinX

0.5NinX

14

Smith Chart (cont.)

Re z

Im z

z

Actual Smith chart:

15

Smith Chart (cont.)

Important points:

S/C O/CPerfect Match

1circle

jX

= 0

( 1)

NinR

z

( 0)z

1

1

NinNin

Z zz

Z z

16

Smith Chart (cont.)

( )Lz Movement in negative direction Clockwise motion on

t

circle of co

oward genera

nstan

tor

t

2 2

2 2

1 1 1( )

1 1 1

j z j lN L Lin j z j l

L L

z e eZ z

z e e

ΓL

Im z

Re z

L

z

angle change = 2l

Transmission LineGeneratorLoad

z = -l

To generator

Zg

z

ZL

z = 0

Z0

S

17

Smith Chart (cont.)

2

2

1

1

j zN Lin j z

L

eZ z

e

We go completely around the Smith chart when

/ 2l

22 2 2 2

2z l

ΓL

Re z

Im z

18

Smith Chart (cont.)

2

22

4

4

z

z

z

l

In general, the angle change on the Smith chart as we go towards the generator is:

2 2j z j j z jL L Lz e e e e

4l

ΓL

Re z

Im z

The angle change is twice the electrical length change on the line: = -2( l).

19

Smith Chart (cont.)

Note:

The Smith chart already has wavelength scales on the perimeter for your convenience (so you don’t need to measure angles).

The “wavelengths towards generator” scale is measured clockwise, starting (arbitrarily) here.

The “wavelengths towards load” scale is measured counterclockwise, starting (arbitrarily) here.

Reciprocal Property

2

2

1

1

j zN Lin j z

L

eZ z

e

Go half-way around the Smith chart:

/ 4l

22 2

4z

ΓL

Re z

Im z

1( )

( )Nin N

in

Z BZ A

10

1N Lin

L

Z

1

1N Lin

L

Z l

B

A

0

0

0

1 1

/

1/

1/

Nin in

in

in

Nin

Z z Z z Z

Z z

Z

Y z

Y

Y z

20

Normalized impedances become normalized admittances.

21

Normalized Voltage

max max

min min

V 1

V 1-

L

L

z

V z

V

+ 2V( ) = V 1+Γ 1+Γj zLz z e z

Normalized voltage

We can use the Smith chart as a crank diagram.

maxVminV

V( )z

ΓL

Γ z

+V 1z Assume

22

SWR

ΓL

2Γ Γ j zLz e Γ z

The SWR is read off from the normalized resistance value on the positive real axis.

As we move along the transmission line, we stay on a circle of constant radius.

0

Ninin

RSWR R

Z

0N Nin inZ R Z real

On the positive real axis:

(from the previous property proved about a real load)

Positive real axis

23

Example

=0.707 45L

maxVminVΓL

V z 45ZN

L

X

Given:

1Z 1 2

1N LL

L

j

45 / 4 rad

Use the Smith chart to plot the magnitude of the normalized voltage, find the SWR, and find the normalized load admittance.

1.707

0.293

load

5

16

V(z)

16

z

=0.707 L

V (z) 1

22 2 =

4l l

16l

1 =1.707 L

Set

24

Example (cont.)

SWR = 5.8

/ 16 0.1875

0.2 0.4 NLY j

=0.707 45

1 2

L

NLZ j

45

1NinR

2NinX

25

Smith Chart as an Admittance Chart

The Smith chart can also be used as an admittance calculator instead of an impedance calculator.

1

1Nin

zZ z

z

0 0 0

1/ 1 1

1/ /in inN

in Nin in

Y z Z zY z

Y Z Z z Z Z z

1

1N

in

zY z

z

1

1N

in

zY z

z

where z z

Hence

or

26

Admittance Chart (cont.)

Re z

Im z

z

0/ 0.5inB Y

0/ 0.5inG Y

27

Comparison of Charts

Impedance chart

Inductive region

0NinX

Capacitive region

0NinX

Im z

28

Comparison of Charts (cont.)

Admittance chart

Capacitive region

0NinB

Inductive region

0NinB

Im z

29

Using the Smith chart for Impedance and Admittance Calculations

We can convert from normalized impedance to normalized admittance, using the reciprocal property (go half-way around the smith chart).

We can then continue to use the Smith chart on an admittance basis.

We can use the same Smith chart for both impedance and admittance calculations.

The Smith chart is then either the plane or the - plane, depending on which type of calculation we are doing.

For example:

30

Example

1 2 NLZ j

Find the normalized admittance l/8 away from the load.

Admittance Chart (cont.)

31

0.2 . 0 4NLY j

1 2 NLZ j

0.23 . 0 48NinY j

o90

2 4

14

8

2

ll

0.23 0.48 NinY j

Answer:

Re Rez z or

Im Imz z or

o90