FIELD ANALYSIS OF LINES
The idea is to find relations between and with R, L, G, C.
where and We, Wm, Pd, PC 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)
Converting to time domain by using
where is the phase.
Lossless Transmission Lines
When the lossless line is terminated by a load ZL.
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 Pav is constant at anywhere on the line. When the line is matched
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, Vmax/Vmin 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:
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 .
These are two families of circles as rL and xL. Superposition of Smith Chart and its 180º rotated version is known as Combined Impedance-Admittance Smith Chart.
This device is used to find ZL as first Vmin.
Generator & Load Mismatches
Then, using this
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