LINE EQUATIONS OF ACCOUNTING

In ordinay circuit theory it is assumed that all impedence elements are lumped constants.This is not true for a long transmission line over a wide range of frequencies Frequencies of operation are so high that inductances of short lengths of conductors and capacitors between short conductors and their surroundings can not be neglected.These inductances and capacitances are distributed along the length of a conductor ,and their effects combine at each point of the conductor.Since the wavelength is short in comparison to the physical length of the line ,distributed parameters can not be represented accurately by means of a lumped parameter equivalent circuit.Thus microwave transmission lines can be analyzed in terms of voltage,current and impedence only by distributed circuit theory.

 

 

1.1.Transmission line equations—

 

A transmission line can be analyzed either by the solution of Maxwell’s field equations or by the methods of distributed circuit theory.The solution of Maxwell’s equations involves three space variables in addition to the time variable in addition to the time variable.

Based uniformly distributed circuit theory ,the schematic circuit of a conventional two conductor transmission line with constant parameters R,L,G,and C is shown in figure-

 

 

 

Where V=by generator,

R’=R*delta(z)

L’=L*delta(z)

C’=C*delta(z)

G’=G*delta(z)

The parameters are expressed in their respective names per unit length ,and the wave propogation is assumed in the positive z direction.

 

By kirchoff’s voltage law ,the summation of the voltage drops around the central loop is given by ,

 

 

 

Rearringing, this equation ,dividing it by delta(z),and then omitting the argument (z,t),we obtain,

 

…………………………………..(1)

 

Using Kirchoff’s current law ,the summation of the currents at point B in figure above can be expressed as ,

 

Equation ,

Or

By rearranging the preceding equation ,dividing it by delta(z) ,omitting (z,t) and assuming delta(z) equal to zero,we have

………………………………………(2)

 

Then by differentiating  equation (1) with respect to z and and equation (2)with respect to t and combining the results ,the final transmission line equation in voltage form is found to be

 

 

 

Also ,by differentiating equation (1) with respect to t and equation (2) with respect to z and combining the results,the final transmission line equation in current form is,

 

 

 

All these transmission line equations are applicable to the genral transient solution.The voltage and current on the line are the functions of both position z and time t.

The instantaneous line voltage and current can be expressed as ,

 

 

The factors V(z) and I(z) are complex quantities of the sinusoidal functions of position z on the line and are known as phasors.The phasors give the magnitude and phases of the sinsusoidal  function at each position of z, and they can be expressed as ,

 

 

And    γ =α  +jβ    (propagation constant)

 

Where α is attenuation constant in nepers per unit length ,and β is the phase constant in radians per unit length .

 

If we substitute jω for  in above equations ,and divide each equation by

The transmission line equations in phasor form of the frequency domain become ,

 

 

 

……………………..(3)

 

 

1.2.Solutions Of Transmission Lines Equations—

 

The one possible solutions is

 

 

Similarly

 

 

 

Solving equation(3)

We have ,

 

 

Phase velocity is vp=ω/β

 

gamma=3;

Vf=5;

z=0:0.1:1;

Vz=Vf.*(exp(-(gamma).*z));

plot(z,Vz);

xlabel(‘z’);

ylabel(‘Vz’);

 

 

 

 

gamma=3;

Vf=5;

z=0:0.1:1;

Vz=Vf.*(exp((gamma).*z));

plot(z,Vz);

xlabel(‘z’);

ylabel(‘Vz’);

 

 

 

2.Reflection Coefficient —

 

In the analysis of the solution of transmission line equation ,the travelling wave along the line contains two components ,one travelling in the positive z direction and the other travelling in the negative z direction .If the load impedence is equal to the line characteristic impedence ,however,the reflecting wave does not exist.

 

If a transmission line terminated in an impedence Zl.It is usually more convenient to start solving the transmission line problem from the receiving rather than the sending end,since the voltage to current relationship at the load point is fixed by the load impedance.The incident voltage and current waves travelling along the transmission line are given by,

 

 

 

 

 

If the line has a length of l,the voltage and current at the receiving end become

 

 

 

 

 

Thus load impedence is ,

 

Zl=Vl/Il

 

The reflection coefficient ,which is designated by Γ is defined as,

 

Reflection coefficient = reflected voltage or current/incident voltage or current

 

Γ=Vref/Vinc = -Iref/Iinc

 

The reflection coefficient at the receiving end is,

 

Γl=(Zl-Zo)/(Zl-Zo)

 

For reflection coefficient at any point is calculated by z=l-d

 

 

 

 

3.Standing Wave and Standing Wave Ratio—

 

3.1.Standing Wave –

 

The genral solutions of the transmission line equation consist of two waves travelling in opposite directions with unequal amplitude ,

 

We have

 

 

 

 

By ,    γ =α  +jβ

 

And separating real and imaginary parts ,

 

We can write

,

which is called equation of standing wave ,

Where

 

 

 

Which is called standing wave pattern of voltage wave or the amplitude of the standing wave , and

 

 

 

Which is called phase pattern of the standing wave.

 

By maxima and minima,

 

1. The maximum amplitude is ,

Vmax at βz=nπ where n=0,+-1,+-2,……….

 

 

 

2.Minimum amplitude is at    βz=(2n-1)π/2 ,where n=0,+-1,+-2,……

 

 

 

3.Similar equations exists for currents.

 

4.The distance between two successive maxima and minima is called one half wave length.

For, βz=nπ,   z=nπ/β = nπ/(2π/λ)=n λ/2

 

For n=0,+-1,+-2,………..

 

5.When magnitude of positive wave and negative wave have equal amplitudes ,the magnitude of reflection coefficient is unity,the standing wave pattern with zero phase is called “pure standing wave”.

 

 

 

 

 

 

3.2.Standing Wave Ratio—

 

Standing waves result from the simultaneous presence of waves travelling in opposite directions on a transmission line .The ratio of the maximum of the standing wave pattern to the minimum is defined as the standing wave ratio,

 

Standing wave ratio= (maximum voltage or current)/(minimum voltage or current)

 

We have ,  ρ=|Vmax|/|V(min)|  =  |I(max)|/|I (min)|  ,

 

The standing wave ratio results from the fact that the two travelling wave components add in phase at some points and subtract at other points.The distance between two successive maxima or minima is λ/2 .The standing wave ratio of a pure travelling wave is unityand that of a pure standing wave is infinite .It should be noted that since the standing wave ratios of voltage and current are identical ,no distinctions are made between VSWR and ISWR.

 

When the standing wave ratio is unity ,there is no reflected wave and the line is called “flat line”.The standing wave ratio can not be defined on a loosy line because the standing wave pattern changes markedly from one position to another.On a low loss line the ratio remains fairly constant ,and it may be defined for some region.For a lossless line the ratio stays the same throughout the line.

 

In terms of reflection coefficient ,

 

We have    ρ=(1+|Γ|)/(1+| Γ|)

 

Since Γ<=1,  ρ>=1.

 

 

 

 

 

 

 

 

gamma=0:0.01:1;

rho=(1+abs(gamma))./(1- abs(gamma));

plot(gamma,rho);

xlabel(‘reflection coificient’);

ylabel(‘VSWR’);