Field Analysis of Lines
The field analysis of lines is a fundamental concept in microwave engineering that deals with the study of electromagnetic waves propagating through transmission 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
Wave propagation along transmission lines is a fundamental concept in electrical engineering and plays a crucial role in the design and optimization of microwave circuits and systems. It refers to the behaviour of electromagnetic waves as they travel through transmission lines.
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
Lossless transmission lines are an important concept in the field of electrical engineering. They are idealized transmission lines that do not dissipate any energy due to resistance, making them valuable for high-frequency applications.
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 
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, Vmax/Vmin increases as a measure of mismatch.
- Standing Wave Ratio (
) is
) 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
Lossy transmission lines are an important topic in the field of electrical engineering. These lines are characterized by a significant amount of energy loss due to resistance, dielectric losses, or radiation.In practice, finite conductivity (or lossy dielectrics) lines can be evaluated as a Lossy Lines.
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
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
The Smith Chart is a graphical tool used in microwave engineering to simplify calculations and aid in the design of radio frequency (RF) and microwave circuits. It provides a visual representation of complex impedance and reflection coefficients.
where R is Resistance, X is Reactance, G is Conductance and B is Susceptance. Whenever z = Z/Z0 is the normalized impedance
The apsis and ordinate of Smith chart are 
and 
.
Rearranging them
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.
Slotted Line
This device is used to find ZL as first Vmin.
Generator Load Mismatches
Then, using this
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