Chapter 11 Flashcards
From Kirchoffs Voltage Law (KVL) and Kirchoff’s Current Law (KCL) what can one derive?
The transmission line (T-Line) equation for time harmonic voltage and current waves in the form of the 1D Helmotz equation.
What is the 1D Helmholtz equation?
(d^2V/(dz^2)) - y^2 * V = 0
d^2I / (dz^2) - y^2 * I = 0
What is the complex propagation constant in helmotz equation?
y^2 = (R + jwL)(G+jwC)
How are R, L, G, and C measured in the complex propagation constant?
R and L are measured along the length of the T-line (series).
G and C are measured between the two conductors of the T-line (shunt).
What is the voltage solution to the 1D Helmholtz equation?
V(z) = (V0+)e^-yz + (V0-)e^yz,
What are (V0+) and (V0-)?
amplitudes of the wave traveling in the +z direction and the wave traveling in the -z direction.
Current solution to the helmhotz equation?
I(z) = (I0+)e^-yz + (I0-)e^yz
Or the voltage solution divided by the impedance Z0
What does Z0 equal (characteristic impedance)
Z0 = sqrt((R+jwL)/(G+jwC))
Complex propagation constant
y = alpha + j*beta
Lossless T-line vs low-loss T-line
Lossless:
- R=G=0
- alpha = 0
- beta = w*sqrt(LC)
- Z0 = sqrt(L/C)
Low-loss:
- R«jwL and G«jwC
- alpha = 1/2((R/Z0) + GZ0)
- beta = w*sqrt(LC)
- Z0 = sqrt(L/C)
For time harmonic waves, how is time dependence incorporated in the fields for a wave traveling in the +z direction?
V(z) = (V0+)(e^-az)(e^j(wt-bz)) or
V(z) = (V0+)(e^-az)cos(wt-bz)
For time harmonic waves, how is time dependence incorporated in the fields for a wave traveling in the -z direction?
V(z) = (V0-)(e^az)(e^j(wt+bz))
or
V(z) = (V0-)(e^az)cos(wt+bz)
What direction does the wave decay exponentially in?
Z
In a lossless line, does the amplitude stay constant?
yes
What is the reflected wave ampitude described in terms of?
Reflection coefficient (Γ)
Obtained by applying ohms law across the terminals of the load impedance.
The reflection coefficient (Γ) is given by:
Γ = (V0-)/(V0+) = (ZL - Z0)/(ZL+Z0)
Incident voltage wave vs Reflected voltage wave
Incident: (V0+)
Reflected: (V0-)
If gamma is complex, what does that mean?
There will be a phase difference between the incident voltage wave (V0+) and reflected voltage wave (V0-).
Does the reflected wave always have to be less than the incident wave for a passive load?
What does Γ equal?
yes
|Γ| <= 1
The reflection coefficient may be defined anywhere along the length of the line as:
Γ(-l) = ((V0-)(-l)) / ((V0+)(-l)) = (ZL-Z0) / (ZL+Z0)
Depending on Z means what?
Γ might be complex, which results in a phase difference between incident and reflected voltage wave.
Input impedance Zin can be written as:
Zin(-l) = Z0* (((ZL+jZ0tan(betal)) / (Z0+jZLtan(betal)))
Also known as impedance transformation equation.
Special cases of input impedance:
At the load (l = 0): Zin = ZL
Shorted line (ZL=0): Zin=jZ0tan(betal)
Open line (ZL=∞): Zin=-jZ0cotan(betal)
Half-wave line (l = ʎ/2): Zin=ZL
Quarter-wave line (I = ʎ/4): Zin=Z0^2 / ZL
Impedance matching
Provides the maximum available power from the source to the load.
Is achieved by setting the input impedance to be equal to the complex
conjugate of the source impedance Zin=Z*s
Can Zin be used by itself to match a complex load< (RL+jXL)?
No
How does a standing wave occur
Incident and reflected wave combine along the section of the T-line
Standing wave ratio (SWR)
s = Vmax / Vmin = (1+|Γ|) / (1 - |Γ|)
SWR is…
A measure of how well the load is matched the T-line and is in the range, s >= 1.
Average incident, reflected, and load powers are given by:
Pi = |V0+|^2 / 2Z0
Pr = |V0-|^2 / 2Z0 = |ΓV0+|^2 / 2Z0
PL = (1-|Γ|^2)Pi
What does smith chart show?
The Smith chart shown below is a graphical tool that may be used to find input impedances,
reflection coefficients, lengths of lines, matching circuits etc. A separate document that
illustrates the use of the Smith chart is also available.
