Slide Presentations for ECE 329,Slide Presentations for ECE 329,Introduction to Electromagnetic Fields,Introduction to Electromagnetic Fields,

to supplement “Elements of Engineering to supplement “Elements of Engineering Electromagnetics, Sixth Edition”Electromagnetics, Sixth Edition”

byby

Nannapaneni Narayana RaoNannapaneni Narayana RaoEdward C. Jordan Professor of Electrical and Computer EngineeringEdward C. Jordan Professor of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, Urbana, Illinois, USAUniversity of Illinois at Urbana-Champaign, Urbana, Illinois, USADistinguished Amrita Professor of EngineeringDistinguished Amrita Professor of Engineering

Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, IndiaAmrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India

6.1

Transmission Line

6.1-3

Parallel-Plate Line

Neglect fringing (d << w).

x

y

z

+-

w

d

gV t

6.1-4

, ; ,x x y yE z t H z t E a H a

dx

z

H , cE J, ,

+

- - - - -

+ + + +

6.1-5

0, ,

,

d

xx

x

V z t E z t dx

d E z t

,,x

V z tE z t

d

, ,

,s

y

I z t w J z t

w H z t

,,y

I z tH z t

w

6.1-6

yxHE

z t

-

y xx

H EEz t

- -

V Iz d t w

-

I V Vz w d t d

- -

6.1-7

Consider the circuit

V d Iz w t

-

I w w VVz d d t

- -

z

,V z t

,I z t

z z z

,V z z t

+ +

- -

,I z z tL z

G z C z

6.1-8

Then

TransmissionLine Equations

L = , inductance per unit lengthdw

G = , conductance per unit lengthwd

C = , capacitance per unit lengthwd

LV Iz t

-

G CI VVz t

- -

6.1-9

Transverse Electromagnetic (TEM) Waves

0z zE H

, ,b

aV z t z t dE l

, ,C

I z t z t dH l

a

b

Cx

yz

6.1-10

Distributed Equivalent Circuit

Assumes perfect conductors but lossy dielectric.

z

C z

L z

G z

L z

C zG z

6.1-11

Transmission-Line Equations

L = Inductance per unit length (H/m)

C = Capacitance per unit length (F/m)

G = Conductance per unit length (S/m)

LV Iz t

-

G CI VVz t

- -

6.1-12

In general, conductors are also lossy. Then, the wavesare not exactly TEM waves.

R = Resistance per unit length (/m)

LV IRIz t

- -

G CI VVz t

- -

6.1-13

Lossless Line (Perfect Conductors, Perfect Dielectric)

WaveEquation

Combining, we obtain

LV Iz t

-

CI Vz t

-

2 2

2 2LCV Vz t

6.1-14

Solution:

,

wave wave

p pV z t Af t z v Bg t z v -

-

velocity of propagation

Note, LC =

1 1LCpv

6.1-15

0

1, p pI z t Af t z v Bg t z vZ

- -

From L ,V Iz t

-

1L

I Vt z

-

1L p p

p

Af t z v Bg t z vv

- -

0Lwhere characteristic impedance of the lineC

Z

6.1-16

Thus

p pV V t z v V t z v - -

0 0

wave wave

p pV t z v V t z vI

Z Z

-- -

-

1LCpv

0LC

Z

6.1-17

Parallel-Plate Line

0dZw

w

d

d << w

L dw

C wd

6.1-18

Coaxial Cable

0 1n2bZ a

ab

L 1n2ba

2C1n b

a

6.1-19

Parallel-Wire Line

10 cosh dZ a

-

aa

2d

1L cosh da

-

1C

cosh da

-

6.1-20

Parallel-Strip Line (d/w 1)

imbedded in a homogeneous medium

w

2d

081n

4d wZw d

6.1-21

Microstrip Line

Ground Plane

Substrate, r

w

d

1 21 1 1012 2eff

r rr

dw

- -

060 81n 4

effr

d wZ w d

6.1-22

Shielded Strip Line