1

Prof. David R. JacksonDept. of ECE

Notes 3

ECE 5317-6351 Microwave Engineering

Fall 2011

Smith ChartExamples

2

0 50

100 50L

Z

Z j

I(-d)

V(-d)+

d

ZL-

0,Z

Find Z(-d)

/ 1/ 4, 3 / 8, 1/ 2gd at

,0

2 1LL n

ZZ j

Z

/ 4

0.4 0.2

/ 4 20 10

g

n

g

d

Z j

Z j

nLZ

3 / 8 gd

/ 4gd

a

b

01/ 2 gd

or

Example 1

a

Impedance chart

3

3 / 8

0.5 0.5

3 / 8 25 25

g

n

g

d

Z j

Z j

b

/ 2

2 1

/ 2 100 50

g

n

g

d

Z j

Z j

c

nLZ

3 / 8 gd

/ 4gd

a

b

0/ 2

gd

0.087g

c0.5g

0.462g

0.212g

Example 1 (cont.) I(-d)

V(-d)+

d

ZL-

0,Z

4

0 050 20mS

8mS 4mSL

Z Y

Y j

Find Y(-d)

/ 1/ 4, 3 / 8, 1/ 2gd at

,0

0.4 0.2LL n

YY j

Y

/ 4

2 1

/ 4 40mS 20mS

g

n

g

d

Y j

Y j

nLY

3 / 8 gd

/ 4gd

a

b

01/ 2 gd

or

c

Example 2

a

I(-d)

V(-d)+

d

ZL-

0,Z

Admittance chart

5

3 / 8

1 1

3 / 8 20mS 20mS

g

n

g

d

Y j

Y j

/ 2

0.4 0.2

/ 2 8mS 4mS

g

n

g

d

Y j

Y j

Example 2 (cont.) b

c

I(-d)

V(-d)+

d

ZL-

0,Z

nLY

3 / 8 gd

/ 4gd

a

b

0

1/ 2 gd

c

Admittance chart

6

Simple answer:

* When adding elements in series use Z-chart

* When adding elements in parallel use Y-chart

A B CZ Z Z

A B CY Y Y

ZA ZB ZC

YA YB YC

Which Chart to Use?

7

Use reactively loaded section of transmission line.

most common to use open or short load

d

ZL0 ,Z

Z d

L LZ jX

LX 0LX

SC OC

dd = 0

nLX

nX d

Non-absorbing load

(RL = 0)

Using Reactive Loads

Impedance chart

Reactance

8

d

YL0 ,Z

Y d

L LY jSC OC

dd = 0

nLY

nY d

Susceptance

0

0/

n

n

Y d Y d Y

Y d Z

Using Reactive Loads (cont.)

Admittance chart

9

Use a short-circuited section of air-filled TEM, 50 transmission line ( = k0, g =0) to create an impedance of Zin = -j25 at f = 10 GHz.

SC

-1/2

0 .426 g

50

0 g L

25inZ j

SC50 Ω , k0

,

251/ 2

50in nZ j j

0.426 0 0.426g g gL

Example 3

L = 1.28 cm

00 0 0

2 2c

f k

0 3.0 cm

Impedance chart

10

Use an open circuited section of 75 (Y0 = 1/75 S) air-filled transmission line at f = 10 GHz to create an admittance of

j1

1/75 S

OC

L

L

1/ 75inY j

OC75 Ω , k0

, 1in nY j

0

1

75Y

1S 13.3mS

75inY j j

0.375cmL

Example 4

00.125L 0 3.0 cm

Admittance chart

11

Y0

d1

njb ,A nY

,L nY

1nG

,in inZ Y d

ZLYS

AY

0Z

0inZ Z

,,0 0 ,

1LL n

L

L nL n

YY

Y Z

ZZ

Z

, , ,in n A n S nY Y Y

Note: At d we have

Want to pick d and Ys such that Yin = Y0.

00

1Y

Z

Matching Circuit

Admittance chart

12

,S n nY jb

, ,1 1 n S nin n YY jb

We want

Choose

0

,

0

1

1 1in n n n

in

inin

Y jb b

Y

j

Y Y

Z Z

,in inZ Y d

ZLYS

AY

0Z

Y0

d1

njb ,A nY

,L nY

1nG

Matching Circuit (cont.)

Admittance chart

13

Example 5

ZL Z0

ls

Z0s

d

0 50[ ]Z

100 100 [ ]LZ j

, 2 2L nZ j

,

10.25 .25

2 2L nY jj

/6 o0.62 0.62 30jL e 0

0

1

1L Ln

LL Ln

Z Z Z

Z Z Z

In this example we will use the “usual” Smith chart, but as an admittance calculator.

14

Example 5 (cont.)

X

X

0.178 g

,Y 0.25 0. 5 2L n j

1 1.57j

1 1.57j

0.322 g

0.363 g

0.219 g

0.041 g

X

0.219 -

0.36

1.57

1.57 3

g

g

d

d

j

j

Solution :

Add at

or at

0.170 0.2.0 981 14 g 0.320 0.3.0 321 64 g

wavelengths toward loadwavelengths toward generator

Smith chart scale:

Admittance calculator

plane

SCOC

(We’ll use the first choice.)

15

Example 5 (cont.)

S / C

X0 1.57j

0.09

O / C

Admittance calculator

, cot

1.57 cot

1cot 1.57; tan

1.572

0.567 [radians]

s nB l

l

l l

l l

0.09s gl From the Smith chart:

0.0903s gl

Analytically:

16

00

0

tan

tanL T

in TT L

Z jZZ Z

Z jZ

2/ 4

4 2g

gg

20

12

0

Tin

L

T in L

ZZ

Z

Z Z Z

0

12

0 0

in

T L

Z Z

Z Z Z

ZLZ0 Z0T

Zin, inZL is real

/ 4g

Quarter-Wave Transformer

At f0 :

For in = 0

17

Match 100 load to 50 transmission Line at f0.

0 100 50

70.7TZ

0

0

0

0

2 2 2g f

r rfk k

Example 6

0 70.7TZ

1r , 50 Ω Z0T

/ 4g

100 Ω