101S109 AA02L01.ppt [相容模式]

Electromagnetics II

Unit 2：Line Terminated by Resistive Load, Transmission Line Discontinuity

Lecturer: Professor Jean-Fu KiangText book: N. N. Rao, “Elements of Engineering

Electromagnetics,” sixth ed., Pearson, 2004.1

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

Voltage and Current in Transmission Line

2

V1(z) V1e jk1z V1e

jk1z

I1(z) V1

Z01

e jk1z V1

Z01

e jk1z

V2 (z) V2e jk2z

I2 (z) V2

Z02

e jk2z

Reflection and Transmission Coefficients between Two Transmission Lines (1/2)

3

1

1

1 2/

1 1 2

1 2/

1 1 2

01 02 01 02

At 0, KVL: (0) (0)

11

KCL: (0 ) (0 )1

2

V

v v

Vv v

zV V

V V V

I IV V V

Z Z Z Z

Reflection and Transmission Coefficients between Two Transmission Lines (2/2)

4

v Z02 Z01

Z02 Z01

v 2Z02

Z02 Z01

solve 1 , 2

Example 1 - Three Transmission Lines Matched at source and load

5

Time-space Diagram of Example 1

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Output Voltage with Delta Input in Example 1

7

0 0 2 0 22 3

0 20

4 4 4( ) ( ) ( 2 ) ( 4 ) ...9 9 9

4 1 ( 2 ) 9 9

( ): impulse response function

o

n

n

v t t T t T T t T T

t T nT

R t

vi (t) (t)

0 In general : ( ) ( ) ( )o iv t R t v d

Output Voltage with Harmonic Inputin Example 1

8

V o( ) (4 / 9)e jT0

1 19

e j 2T2

Frequency response function0 2

0 20

( 2 )

0

If ( ) cosConsider a dispersion-free system

4 1( ) cos ( 2 )9 9

4 1Re9 9

i

n

on

nj T nT j t

n

v t t

v t t T nT

e e

Transmission-line Discontinuity with Inserted Resistor

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V1(z) V1e jk1z V1e

jk1z

I1(z) V1

Z01

e jk1z V1

Z01

e jk1z

V2 (z) V2e jk2z

I2 (z) V2

Z02

e jk2z

Evaluation of Reflection and Transmission Coefficients

10

B.C. at z 0, KVL: V1(0) V2 (0)

V1 V1 V2 1 v v (1)

KCL: I1(0) I2 (0) IL

IL

V2ZL

V1 V1

Z01

1

Z02

1ZL

1/ Z02||ZL

V2 /V1 1 v

Z01

v

Z02 || ZL

(2)

Equivalent Lumped-circuit of an Infinitesimal Parallel-plate Waveguide

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v (ZL || Z02 ) Z01

(ZL || Z02 ) Z01

v 1 v

Time-domain Reflectometry

12

ZL 120, Z02 60ZL || Z02 40

v 40 6040 60

15

v 1 v 45

Example 3: Inductive Load

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Example 3: Solution

14

0 0

0 0

0

0

0

, at

2 2

2( ) 0

( )2

L

t T z l

V V VdV Ldt Z Z

VdVL VZ dtI t T

VV t T

V V0

2 Ae(tT )/(L/Z0 ),

let L LZ0

I.C. : V0

2

V0

2 A

A V0

V V0

2V0e

(tT )/ L , t T ; I V0

2Z0

V0

Z0

e(tT )/ L , t T

} initial condition

Exercise: Capacitive Load

15

0

0 0

0 0

00

0

0

( )/( )0

0

( )/ ( )/0 0 00

0 0

( )

2 2

2

( ) 0 02

( ) (I.C.)2

2(I.C.)

, 2 2

c

c c

CT

t Z C

t t

t T z lV V VdC V

dt Z ZVdVCZ V

dtVV t T V

VV t T

VV Ae

A VV V VV V e I e

Z Z

Voltage and Current at z=l

16

tT

V0 / Z0

V0

I z l

V z l

Propagation of V and I Waves

17

V (z,0) 12

V (z,0) Z0I (z,0)

V (z,0) 12

V (z,0) Z0I (z,0)

{

V (z,0) V (z,0)V (z,0)Z0I (z,0) V (z,0)V (z,0)

t 0

V z,t

V z,t t

t

Propagation of V and I Waves

18

I z,t

I z,t

t

t

Example 4

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Z0 50, RL 150

T 1s, V0 100V

At t 0 :

KVL:RLI V0 V

RLV

Z0

V0 V

00

0

00

25V

1 1 A2

L

L

ZV VR Z

I VR Z

Conservation of Energy

20

We 12

CV 20l

12

CV 20vpT

12

CV 20

1LC

T 12

V 20

Z0

T 104 J

Wm 12

LI 20l 0, at t 0

Wd dt0

V 2 (0,t)

RL

104 J

Copyright Declaration

21

Page Work Licensing Author/Source

2, 3

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

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101S109 AA02L01.ppt [相容模式]