Microwave-Engineering : Field Analysis of Lines

FIELD ANALYSIS OF LINES

The idea is to find relations between and with R, L, G, C.

where and W_e, W_m, P_d, P_C are time-averaged values calculated for 1m length line and S is the line cross-sectional area.

Wave Propagation along the Line

Using the frequency domain Telegrapher equation

where is propagation constant.

The solution of Telegrapher equation is:

Then, taking the z derivation of v(z), the calculation of i(z)

Then

Converting to time domain by using

where is the phase.

Lossless Transmission Lines

When the lossless line is terminated by a load Z_L.

Reflected waves occur.

Reflection coefficient at the load, z = 0

where v(z) and i(z) consists of a superposition of an incident and reflected waves called Standing Waves

Time Average Power Flow:

This shows P_av is constant at anywhere on the line. When the line is matched

is constant.

When the line is mismatched (Return Loss

Matched load (No reflected power, maximum power is delivered).

Total reflection, (All power reflected).

The maximum value occurs when The minimum value occurs when When increases, V_max/V_min increases as a measure of mismatch.

Standing Wave Ratio () is

When SWR = 1 means matched line. In that case at z = -l, the reflection coefficient and input impedance

Using the definition of , more useful form known as Transmission Line Impedance Equation as

Transmission Coefficient: Some part of EM wave is also transmitted to the second region as

Insertion Loss:

Short Circuit:

Open Circuit:

The proper length of open or short circuited transmission line can provide any desired reactance or susceptance.

The same impedance is observed at the input.

(Quarter Wave Transform)

Lossy Transmission Lines

In practice, finite conductivity (or lossy dielectrics) lines can be evaluated as a Lossy Line.

In the lossy line; can be approximated to the lossless line.

Distortionless Line: For the lossy line, in fact the exact is not a linear function of frequency means dispersive. But specifically, if the following condition holds

then mean that the lossy line behaves as a lossless (distortionless) line.

Terminated Lossy Line: Loss is assumed small that

Power lost in the line:

Perturbation Method for Calculating Attenuation

Attenuation constant:

Taylor series Inductance Rule:

Smith Chart

where R is Resistance, X is Reactance, G is Conductance and B is Susceptance. Whenever z = Z/Z₀ is normalized impedance

The apsis and ordinate of Smith chart are and .

Rearranging them

These are two families of circles as r_L and x_L. Superposition of Smith Chart and its 180º rotated version is known as Combined Impedance-Admittance Smith Chart.