Lecture 2

Derive the transmission line parameters (R, L, G, C) in terms of the electromagnetic fields

Rederive the telegrapher equations using these parameters

1.4 Field analysis of transmission lines

Example:

Voltage : V0ejz

Current: I0ejz

Work (W) and power (P)

H* Multiplies the two sides of the first Maxell’s equation:

E Multiplies the two sides of the conjugated second Maxell’s equation:

Add the above two equations and utilize

We obtain (J=Js+σE):

Integrate the above formula in volume V and utilize divergence theory,we have the following after reorganize the equation

Poynting law: )(2 emloS WWjPPP

Source power Ps:

Output power P0:

Loss power Pl:

Stored magnetic energy Wm:

Stored electric energy We:

dvMHJEPV ss

)(

2

1

)(2

1

2

1 HESsdSsdHEP

SSo

dvHEdvEPVVl )''''(

22

222

dvHWVm

2'

4

1

dvEWVe

2'

4

1

(Time averaged)

V

V

V

T

T

V

m

dvBH

dvBH

dvdttBtHT

dvdtBHT

W

Re4

1

)cos(cos4

1

cos)cos()cos(2

1

]Re[]Re[2

1

21

0

21

0

);( 21B

jH

j eeBBeeHH

)(cos BH ee

Calculate the time-average stored magnetic energy in an isotropic medium ( the results valid for any media )

Calculate magnetic energy

Surface resistance and surface current of metal

dsnHEPSSav

0

Re2

1Energy entering a conductor:

The contribution to the integral from the surface S can be made zero by proper selection of this surface. Therefore,

dszHEP tSav

0

Re2

1

From vector identity, we have .)()(

HHHEzHEz

dsJR

dsHR

PS s

s

S ts

av 00

22

22

The energy absorbed by a conductor:

ss jR

1

]2

)1Re[()Re((

)HnJ s

)/( EnH

Transmission line parameter: L

The time-average stored magnetic energy for 1 m long transmission line is

1.4 Field analysis of transmission lines

dsHHWS

m

4

And circuit line gives . Hence the self inductance could be identified as

4/2

ILW m

)/(2

0

mHdsHHI

LS

Appendix 1:

V

V

V

T

T

V

m

dvBH

dvBH

dvdttBtHT

dvdtBHT

W

Re4

1

)cos(cos4

1

cos)cos()cos(2

1

]Re[]Re[2

1

21

0

21

0

);( 21B

jH

j eeBBeeHH

)(cos BH ee

Calculate the time-average stored magnetic energy in an isotropic medium ( the results valid for any media )

Transmission line parameter: C

1.4 Field analysis of transmission lines

Similarly, the time-average stored electric energy per unit length can be found as

dsEEWS

e

4

Circuit theory gives , resulting in the following expression for the capacitance per unit length:

4/2

VCW e

)/(2

0

mFdsEEV

CS

Transmission line parameter: R

1.4 Field analysis of transmission lines

The power loss per unit length due to the finite conductivity of the metallic conductors is

dlHHR

PCC

sc

212

The circuit theory gives , so the series resistance R per unit length of line is

2/2

0IRPc

)/(21

2

0

mdlHHI

RR

CC

s

(Rs = 1/ is the surface resistance and H is the tangential field)

Transmission line parameter, G

1.4 Field analysis of transmission lines

The time-average power dissipated per unit length in a lossy dielectric is

.2

''dsEEP

S

d

Circuit theory gives , so the shunt conductance per unit length can be written as

2/2

0VGPd

)/(''

2

0

mSdsEEV

GS

Homework

1. The fields of a traveling TEM wave inside the coaxial line shown left can be expressed as

where is the propagation constant of the line. The conductors are assumed to have a surface resistivity Rs, and the material filling the space between the conductors is assumed to have a complex permittivity = ’ - j" and a permeability μ = μ0μr. Determine the transmission line parameters (L,C,R,G).

x

y

a ρ φb

μ,

zz eI

Heab

VE

2;

/ln00

2. For the parallel plate line shown left, derive the R, L, G, and C parameters. Assume w >> d.

d r

y

zx

w