Chapter 11

Question 1

From Kirchoffs Voltage Law (KVL) and Kirchoff’s Current Law (KCL) what can one derive?

Answer 1

The transmission line (T-Line) equation for time harmonic voltage and current waves in the form of the 1D Helmotz equation.

Question 2

What is the 1D Helmholtz equation?

Answer 2

(d^2V/(dz^2)) - y^2 * V = 0
d^2I / (dz^2) - y^2 * I = 0

Question 3

What is the complex propagation constant in helmotz equation?

Answer 3

y^2 = (R + jwL)(G+jwC)

Question 4

How are R, L, G, and C measured in the complex propagation constant?

Answer 4

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).

Question 5

What is the voltage solution to the 1D Helmholtz equation?

Answer 5

V(z) = (V0+)e^-yz + (V0-)e^yz,

Question 6

What are (V0+) and (V0-)?

Answer 6

amplitudes of the wave traveling in the +z direction and the wave traveling in the -z direction.

Question 7

Current solution to the helmhotz equation?

Answer 7

I(z) = (I0+)e^-yz + (I0-)e^yz
Or the voltage solution divided by the impedance Z0

Question 8

What does Z0 equal (characteristic impedance)

Answer 8

Z0 = sqrt((R+jwL)/(G+jwC))

Question 9

Complex propagation constant

Answer 9

y = alpha + j*beta

Question 10

Lossless T-line vs low-loss T-line

Answer 10

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)

Question 11

For time harmonic waves, how is time dependence incorporated in the fields for a wave traveling in the +z direction?

Answer 11

V(z) = (V0+)(e^-az)(e^j(wt-bz)) or
V(z) = (V0+)(e^-az)cos(wt-bz)

Question 12

For time harmonic waves, how is time dependence incorporated in the fields for a wave traveling in the -z direction?

Answer 12

V(z) = (V0-)(e^az)(e^j(wt+bz))
or
V(z) = (V0-)(e^az)cos(wt+bz)

Question 13

What direction does the wave decay exponentially in?

Answer 13

Z

Question 14

In a lossless line, does the amplitude stay constant?

Answer 14

yes

Question 15

What is the reflected wave ampitude described in terms of?

Answer 15

Reflection coefficient (Γ)

Obtained by applying ohms law across the terminals of the load impedance.

Question 16

The reflection coefficient (Γ) is given by:

Answer 16

Γ = (V0-)/(V0+) = (ZL - Z0)/(ZL+Z0)

Question 16

Incident voltage wave vs Reflected voltage wave

Answer 16

Incident: (V0+)
Reflected: (V0-)

Question 16

If gamma is complex, what does that mean?

Answer 16

There will be a phase difference between the incident voltage wave (V0+) and reflected voltage wave (V0-).

Question 16

Does the reflected wave always have to be less than the incident wave for a passive load?
What does Γ equal?

Answer 16

yes
|Γ| <= 1

Question 17

The reflection coefficient may be defined anywhere along the length of the line as:

Answer 17

Γ(-l) = ((V0-)(-l)) / ((V0+)(-l)) = (ZL-Z0) / (ZL+Z0)

Question 18

Depending on Z means what?

Answer 18

Γ might be complex, which results in a phase difference between incident and reflected voltage wave.

Question 19

Input impedance Zin can be written as:

Answer 19

Zin(-l) = Z0* (((ZL+jZ0tan(betal)) / (Z0+jZLtan(betal)))

Also known as impedance transformation equation.

Question 20

Special cases of input impedance:

Answer 20

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

Question 21

Impedance matching

Answer 21

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

Question 22

Can Zin be used by itself to match a complex load< (RL+jXL)?

Answer 22

No

Question 23

How does a standing wave occur

Answer 23

Incident and reflected wave combine along the section of the T-line

Question 24

Standing wave ratio (SWR)

Answer 24

s = Vmax / Vmin = (1+|Γ|) / (1 - |Γ|)

Question 25

SWR is…

Answer 25

A measure of how well the load is matched the T-line and is in the range, s >= 1.

Question 26

Average incident, reflected, and load powers are given by:

Answer 26

Pi = |V0+|^2 / 2Z0
Pr = |V0-|^2 / 2Z0 = |ΓV0+|^2 / 2Z0
PL = (1-|Γ|^2)Pi

Question 27

What does smith chart show?

Answer 27

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.