Transient Analysis of Transmission Lines Flashcards
Transmission Line
A collection of one or more conductors and dielectrics that impose boundary conditions on the EM wave equations, effectively guiding waves along the length of the line
Electrically Small
dmax «_space;λ/2pi (low-loss medium)
TEM
Transverse electromagnetic, E and B are perpendicular to the direction of propagation. TEM lines have electrically small cross-sections
Quasi-static approximations conditions
TEM lines have electrically small cross sections that permits quasi-static approximations
MQS
magneto-quasi-static approximation - displacement currents have a negligible impact on the magnetic field
EQS
electro-quasi-static approximation - time-varying magnetic fields have a negligible impact on the electric field. The E-field is assumes to be irrotational/conservative)
Losses in the line are due to
current flow in the line, both conduction and displacement currents
R’
(ohms/m) - loss due to conduction current flow (conductor losses, skin effect)
G’
(S/m) - loss due to displacement currents
Lossless line
R’=0 and G’=0
Zo
Characteristic Impedance = √(L’/C’)
V-
Is the reverse travelling wave along the transmission line, it is equal to the forward travelling component V= multiplied with the reflection coefficient
Short circuit termination
RL = 0 and ΓL = -1
Matched termination
RL = Z0 and ΓL = 0
Open circuit termination
RL = ∞ and ΓL = +1
How to calculate the reflection coefficient
Where you’re going minus where you’re coming from
What occurs when there are non zero reflection coefficients at both the load and the generator
Reflections go on indefinitely with the line voltage asymptotically approaches a steady state value, this value should approach what you would get from DC analysis
When is it
reasonable to ignore transmission line effects when analyzing the circuit:
l≪ λ/10
Wavelength equation
λ = c/f
quasi-static approximation
The quasi-static approximation implies that the equation of continuity can be written as ∇ ⋅ J = 0 and that the time derivative of the electric displacement ∂D/∂t can be disregarded in Maxwell-Ampère’s law
L’
Inductance per unit length of the line (H/m), defined from MQS
C’
Capacitance per unit length of the line (F/m), defined from EQS
When the length is not electrically small
we can break it into electrically small segments Δz in z (length)
How were the telegrapher’s equations derived?
Look at the voltages on the input and output of one electrically small segment of transmission line. The two voltages are related by the voltage drop across the series elements. Rearrange and take the limit as Δz goes to 0
Where is the tx line does the EM travel
through the dielectric surrounding the conductors
TDR
Time Domain Reflectometry - Analyzing the transient response of a Tx line is a common way to assess/diagnose damage
How does TDR work?
Apply a high speed voltage step (sub-nanosecond) and monitor the response of the line at the input. Record with a high speed scope over a time interval and convert to distance.
TDR Response for a matched line
V(t) = Vg/2 for all of z >0
TDR Response for a break halfway down the line
V(t) = Vg/2 for all of z >l/2 then V(t) = 0 for short or V(t) = Vg for open
TDR response capacitive load
Short circuit at load and charging curve up to Vg. Approaches steady state with a time constant of ZoC
TDR response inductive load
Open circuit at load and curve down to 0. Approaches steady state with a time constant of L/Zo
VNA
Vector Network Analyzer, measures the complex Scattering- or S-parameters of RF components in the frequency
domain
Steady state for Bounce Diagram
V = (VgRL)/Rg+RL
TDR response mismatched loading
For RL>Zo the voltage will be in between Vg/2 and Vg
For RL<Zo the voltage will be in between 0 and Vg/2
The characteristic impedance on a transmission line can be calculated by ratio of
forward travelling voltage to forward travelling current on the line
