Electromagnetics II

Unit 5：Single-stub and Double-stub Impedance TransformersLecturer: Professor Jean-Fu Kiang

Text book: N. N. Rao, “Elements of Engineering Electromagnetics,” sixth ed., Pearson, 2004.

1Unless noted, the course materials are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan (CC BY-NC-SA 3.0)

Single-stub Transformer (1/2)

2

1

1'1

' '' 1 11 ' '

1 1

2 2'1

01

1

1 11 1

2s sj d j d j

L L

y

y jb

Z yZ y

jbe e ejb

2

1

4

tan 24 2 2

L

s

b

bbd n

Single-stub Transformer (2/2)

3

2in

inin

in

2

in2

0

0

11

11-

111 cos sin1 cos sin11

tantan

LL

Lj l

L

j lL

L L

j lL L

L

L

L

ZZ

e

Z

Z eZ Z l j lZZ l jZ leZ

Z jZ lZ jZ l

Z (z) 1 (z)1(z)

(z) Le j 2z Z (z) 1Z (z) 1

Double-stub Transformer

4

2 0y2 1y2

' y2 jb2

2'

1 y2'

1 y2'

1 2' e j 2d12

jb2

2 jb2

e j 2d12

y1 11

1 1

2' ' 2 12 2 121 1 1 12 2

2 12 2 12

2in 2

2

sin 2 cos 22 2 sin 2 2 sin

1 tan tantan

L

L

b d b dy y jb g j bb d b d

jZ ljb Y j lZ j l

g' 11 b2 sin2l12 b2

2 sin2l12

b2 cosd12 1/g'sin2 d12

sind12

b1

Smith Chart

5

2 2

2 2 2 2

2 2

2 2 2 2

2 22

2

( ) 0,( ) 1 -1 1 2( )( ) 1 +1 ( 1) ( 1)

1 1 211 1 (1 ) (1 )

11 1

( 1)

r i

r i r i i

r i r i r i

r i

r i

Z z r jx r xZ z r jx r x xz jZ z r jx r x r xj

jr jx jj

rr r

2 21 1

x x

Constant r circles

Constant x circles

Example: Single-stub Transformer (1/2)

6

L 1'e j 2d s

1'

1 y1'

1 y1'

y1' 1 jb

y1 1in 0

Example: Single-stub Transformer (2/2)

7

0

'1

30 40500.6 0.8 (A)

( ) ( / 4)0.6 0.8 (B)

1 1.16 (C)1.16

0.113

L

L

L

s

Z jZZ jY z Z zy j

y jbl

Example: Double-stub Transformer (1/2)

8

y1' y1 jb1 yL

y1 11

1 1

1 2' e j 2d12

2'

1 y2'

1 y2'

y2' 1 jb2

y2 1in 0

d1 0

Example: Double-stub Transformer (2/2)

9

0

'1

1'2

'1 1 1

2

30 40500.6 0.8 (A)

( ) ( / 4)0.6 0.8 (B)

0.6 0.09 (C)

1 0.53 (D)

0.890.53

L

L

L

Z jZZ jY z Z zy y jy j

y j

jb y y jjb j

Copyright Declaration

10

Page Work Licensing Author/Source

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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

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Copyright DeclarationPage Work Licensing Author/Source

5

Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

6

Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

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Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

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Copyright DeclarationPage Work Licensing Author/Source

8

Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan

9

Jean-Fu Kiang / National Taiwan UniversityThis work is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan