notes 3 - smith chart examples

7/23/2019 Notes 3 - Smith Chart Examples

1/17

Prof. David R. Jackson

Dept. of ECE

Notes 3

ECE 5317-6351

Microwave Engineering

Fall 2011

Smith ChartExamples

1


2/17

0 50

100 50L

Z

Z j

=

= +

I(-d)

V(-d)

+

d

ZL-

0 ,Z

FindZ(-d)

/ 1/ 4, 3/ 8, 1/ 2g

d =at

,

0

2 1LL nZ

Z jZ

= = +

( )

/ 4

0.4 0.2

/ 4 20 10

g

n

g

d

Z j

Z j

=

nLZ

3 / 8g

d =

/ 4g

d =

a

b

0

1/2 gd =

or

Example 1

a

2

Impedance chart


3/17

( )

3 / 8

0.5 0.53 / 8 25 25

g

n

g

d

Z jZ j

=

+

+

b

( )

/ 2

2 1

/ 2 100 50

g

n

g

d

Z j

Z j

=

+

= +

c

nLZ

3 / 8g

d =

/ 4g

d =

a

b

0

/2gd =

=0.087g

c0.5g

0.462g

0.212g

Example 1 (cont.)

3

I(-d)

V(-d)

+

d

ZL-

0 ,Z


4/17

( )0 050 20 mS

8mS 4 mSL

Z Y

Y j

= =

=

Find Y(-d)

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

,

0

0.4 0.2LL n YY jY

= =

( )

/ 4

2 1

/ 4 40 mS 20 mS

g

n

g

d

Y j

Y j

=

+

+

nLY

3 / 8 gd =

/ 4gd =

a

b

0

1/2

gd =

or

c

Example 2

a

4

I(-d)

V(-d)

+

d

ZL-

0 ,Z

Admittance chart


5/17

( )

3 / 8

1 1

3 / 8 20 mS 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

5

I(-d)

V(-d)

+

d

ZL-

0 ,Z

nLY

3 / 8g

d =

/ 4gd =

a

b

0

1/2 g

d =

=

c

Admittance chart


6/17

Simple answer:

* When adding elements in series useZ-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?

6


7/17

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

d

d = 0

nLX

( )nX d

Using Reactive Loads

7

Impedance chart


8/17

d

YL0 ,Z

( )Y d

L LY j=

SC OC

d

d = 0

nL

Y

( )nY d

( ) ( )

( )

0

0/

n

n

Y d Y d Y

Y d Z

=

=

Using Reactive Loads (cont.)

8

Admittance chart


9/17

Use a short-circuited section of air-filled TEM, 50 transmission line

(= k0, g=0) to create an impedance ofZin= -j25

atf= 10 GHz.

SC

-1/2

0.42

6 g

50

0 g L

25in

Z j=

SC50 ,k0

( ),25

1/ 250

in nZ j j= =

0.426 0 0.426g g gL = =

Example 3

L= 1.28 cm

90 0 0 0

2 2c

f k

= = =

0 3.0 cm =

Impedance chart


10/17

Use an open circuited section of 75 (Y0= 1/75 S) air-filled

transmission line atf =10 GHz to create an admittance of

j1

1/75 S

OC

L

L

( )1/75inY j= +

OC75 , k0

( ), 1in nY j=

0

1

75

Y =

1S 13.3mS

75inY j j= + =

0.375cmL =

Example 4

10

00.125L = 0 3.0 cm =

Admittance chart


11/17

Y0

d

1

njb+

,A nY

,L nY

1n

G =

,in in

Z Y d

ZLYS

AY

0Z

( )0inZ Z=

,,

0 0 ,

1LL n

L

L n

L n

YY

Y Z

ZZ

Z == =

, , ,in n A n S nY Y Y= +

Note: At d we have

Want to pick dand Yssuch that Yin= Y0.

0

0

1YZ

=

Matching Circuit

11

Admittance chart


12/17

,S n n

Y jb=

, ,1 1 n S nin n YY jb= + +=

We want

Choose

( )

0

,

0

1

1 1in n n n

in

in

in

Y jb b

Y

j

Y Y

Z Z

= + =

= =

=

,in in

Z Y d

ZLYS

AY

0Z

Y0

d

1

njb+

,A nY

,L nY

1n

G =

Matching Circuit (cont.)

12Admittance chart


13/17

Example 5

ZLZ0

ls

Z0s

d

0 50[ ]Z =

100 100 [ ]LZ j= +

, 2 2L nZ j= +

( ), 1 0.25 .252 2L nY j

j= = +

/6 o0.62 0.62 30j

L e = = 0

0

1

1

L LnL

L Ln

Z Z Z

Z Z Z

= =

+ +13

In this example we will

use the usual Smith

chart, but as an

admittance calculator.


14/17

Example 5 (cont.)

0.178 g

,Y 0.25 0. 52L n j=

1 1.57j+

1 1.57j

0.322 g

0.363 g

0.219 g

0.041 g

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 load

wavelengths toward generator

Smith chart scale:

Admittance calculator

14

plane

SCOC

(Well use the first choice.)


15/17

Example 5 (cont.)

S / C

0 1.57j

0.09

O / C

Admittance calculator

( ), cot

1.57 cot

1cot 1.57; tan

1.57

2 0.567 [radians]

s nB l

l

l l

l l

=

=

= =

= =

0.09s gl =

From the Smith chart:

0.0903s gl =

Analytically:

15


16/17

00

0

tantan

L Tin T

T L

Z jZZ ZZ jZ

+= +

( ) 2

/ 4

4 2

g

g

g

= = =

[ ]

2

0

120

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, in ZL is real

/ 4g

Quarter-Wave Transformer

At f0 :

For in= 0

16


17/17

Match 100 load to 50 transmission Line at f0.

0 100 50

70.7

TZ = =

0

0

0

0

2 2 2g f

r rf k k

= = = =

Example 6

0 70.7TZ =

17

( )1r =, 50 Z0T

/ 4g

100

notes 3 - smith chart examples

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