Transmission Line Eq Fixed

8/2/2019 Transmission Line Eq Fixed

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TRANSMISSION LINE EQUATION


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Quotes

No mathematical purist could ever dothe work involved in Maxwells

treatise. He might have all the

mathematic, and much more, but it

would be to no purpose, as he could

not put it together without the

physical guidance. This is in no way

to his discredit, but only illustrates

different ways of thought.

Oliver Heaviside


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Introduction

A transmission line basically consists of two or moreparallel conductors used to connect a source to a load.

The source may be a hydroelectric generator, a

transmitter, or an oscillator; the load may be a factory, an

antenna, or an oscilloscope, respectively. Typical transmission lines include coaxial cable, a two-wire

line, a parallel-plate or planar line, a wire above the

conducting plane, and a microstrip line.

Transmission line problems are usually solved using EM fieldtheory and electric circuit theory, the two major theories

on which electrical engineering is based.

Basically, all engineering problem can be solved employing

field theory equations


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Key point about transmission line

operation

The major deviation from circuit theory with transmission line,

distributed networks is this positional dependence of voltage and

current! Must think in terms of position and time to understand

transmission line behavior

This positional dependence is added when the assumption of

the size of the circuit being small compared to the signaling

wavelength

Voltage and current on a transmission line isa function of both time and position.

V1 V2

dz

I2I1

tz,fI

tz,fV


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Presence of Electric and Magnetic

Fields

V

I

I

E

+

-

+

-

+

-

+

-

V + DV

I + DI

I + DIV

IH

IH

V + DV

I + DI

I + DI

Both Electric and Magnetic fields are present in the transmissionlines

These fields are perpendicular to each other and to the direction of wavepropagation for TEM mode waves, which is the simplest mode.

Electric field is established by a potential difference between twoconductors.

Implies equivalent circuit model must contain capacitor.

Magnetic field induced by current flowing on the line

Implies equivalent circuit model must contain inductor.

dlEV . dlHI .


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Kirchhoffs Law

Current Law I=0

Dividing by x, we obtain:

.(2)


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Transmission Line Equation

(f-Domain)

dxdxandjtxItxixVtxv ///);(),();(),( Substituting

.(4)

.(3)

IGRIGLCRjIjCLdx

Id

VGRVGLCRjVjCLdx

Vd

'')''''()(''

'')''''()(''

2

2

2

2

2

2

Become

.(6)

.(5)


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The Solution of Transmission Line

Equation

Idx

VdICjRLjR

dx

Id

Vdx

VdVCjRLjR

dx

Vd

2

2

2

2

2

2

2

2

2

2

or)'')(''(

or)'')(''(

xx

eAeAxV

21)(

VeAeAxdx

Vd xx 22

2

1

2

2

2

)(

)'')(''( CjRLjR

General exponential solution ofdAlembert


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The Solution of Transmission Line

Equation

ILIRdx

Vd

j

''

0''

''''Z

CjG

LjRLjR

General current Solution

The Ratio

Characteristic impedance of

the transmission line

is the attenuation (loss) factor

is the phase (velocity) factor

)(''

)( 21xx

eAeALjR

xI


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The distinction between electrically

long and electrically short

Definition in frequency domain (time harmonic quantities)

c

x Definition in frequency domain (time harmonic quantities)

(x)VV

r

c

vC

c/fcT


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The distinction between electrically

long and electrically short

Definition in time domin),()( txvtv

xconsttvtv )()(

v

ac

lT

Electrically short if the voltage at a given instant

possesses the same value everywhere along a line

i.e. If the rise time of the travelling wave is very

large compared with the propagation time or delaytime


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fin

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Transmission Line Eq Fixed