Transmission

Question 1

How do you draw the circuit model for a transmission line?

Answer 1

A transmission line has a forward and return line, so two parallel lines of length dx will be needed. A capacitance C is formed between any two conductors, so for each length dx there should be a capacitance Cdx. An inductance is also a part of any transmission wire, and so an inductance Ldx should be added to one of the parallel edges.

Question 2

How does voltage change with respect to x along a transmission line? (Telegrapher’s equation.)

Answer 2

[dV/dx]_t = -L[dI/dt]_t

Question 3

How does current change with respect to x along a transmission line?

Answer 3

[dI/dx]_t = -C[dV/dt]_x

Question 4

What are the telegrapher’s equations?

Answer 4

[dV/dx]_t = -L[dI/dt]_x
[dI/dx]_t = -C[dV/dt]_x

Question 5

By the way, all these [dV/dx]_t things should really be partial differentials.

Answer 5

I can’t do deltas very easily here.

Question 6

How do you solve the telegraphers equations?

Answer 6

Differentiate both sides of the dV/dx equation with respect to x to give d²V/dx² = -Ld/dx dI/dt = -Ld/dt dI/dx
Then sub in dI/dx = CdV/dt
This gives [d²V/dx²] _t = LC [d²V/dt²]_x

Similarly, the opposite can be done to form an identical equation for current (directly swap V for I in the result).

Question 7

What is the general form of the wave equation?

Answer 7

Here, d represent a partial differential

d²f/dx² = 1/c² d²f/dt²

Question 8

What is the solution to the wave equation?

Answer 8

f = f₀ cos(wt - Bx)

Question 9

Why is x negative in the wave equation solution?

Answer 9

The wave progresses with time, and in the positive x direction. However, this means that a step forward in x is effectively a step backwards in time for a continuous wave, so x and t must be opposite signs.

Question 10

What determines c in a wire?

Answer 10

C = 1/sqrt(LC), since LC takes the place of 1/c² in the waveform equation. L and C both contain on and Pi terms that cancel, leaving only LC = permittivity x permeability (relative x free space for the dielectric between the wires)
Therefore c = 1/sqrt(permittivity x permeability)
Therefore speed of conduction only dependent on the property of the dielectric medium between the conductors.

Question 11

How do you find wavelength in a wave in a wire?

Answer 11

Use c/f to find the wavelength. If the wavelength is long compared to the length of the wire, we do not need to think in transmission line terms, and can consider the flow “incompressible”.

Question 12

How should the circuit model be modified for a lossy transmission line?

Answer 12

A resistance Rdx should be added in series with the inductance on one side of the wire, and a resistance 1/(Gdx) (1/conductance) should be added in parallel with the capacitor between the wires.

Question 13

What is the form of the general expression for voltage in a wire?

Answer 13

The general expression, found in the Databook, is (V_Fe^-jBx + V_B e^jBx) e^jwt

Question 14

How does resistance/loss effect the transmission equations?

Answer 14

Loss introduced an attenuation constant a, as an exponential with respect to distance, e^-ax. The B terms in the exponential of the transmission expression may be simply replaced with Y terms, where Y = a + Bj

Question 15

How can transmission lines be analysed to find a relationship between B, w and C?

Answer 15

The forward and backward waves of voltage and current can be separated, and it can be shown that V_F/I_F = B/wC = V_B/I_B

Question 16

What is the characteristic impedance of a transmission line?

Answer 16

The characteristic impedance of a transmission line is the ratio between voltage and current of a unidirectional wave (either the forward or the backward wave) at any point on the line.
Z₀ = V_F/I_F = V_B/I_B = B/wC = rt(L/C)

Z₀ for real transmission line is dependent on a “geometry factor”.

Question 17

What does Z₀, the characteristic impedance, represent?

Answer 17

Z₀ is always a real number for an ideal lossless line. It does not dissipate power. It is the apparent impedance “seen” if looking into an infinitely long line at x = 0.

Question 18

What does Z₀, the characteristic impedance, represent?

Answer 18

Z₀ is always a real number for an ideal lossless line. It does not dissipate power. It is the apparent impedance “seen” if looking into an infinitely long line at x = 0.

Question 19

What is the relevance of Z₀ for circuit design?

Answer 19

Z₀ should be matched to source impedance, Z_S, for maximum power transfer into the cable (see power division across a potential divider). (Usually this is designed to be 50 or 75 ohms.)

Question 20

What is the voltage reflection coefficient?

Answer 20

P_L, the voltage reflection coefficient, is the ratio of the voltage of the reflected wave to the forward wave, V_B/V_F. It is equal to [Z_L - Z₀]/[Z_L + Z₀]

Question 21

What is the voltage standing wave ratio?

Answer 21

The VSWR is the ratio of maximum standing wave amplitude to minimum standing wave amplitude.
VSWR = [V_F + V_B]/[V_F - V_B] = [1 + P_L]/[1 - P_L]

Question 22

What is the effect on the reflection coefficient for open and closed circuits?

Answer 22

For open circuit, P_L = 1 since Z_L is infinite. Wave is fully reflected with no phase change and a perfect standing wave is formed with an anti-node at the boundary.
For closed circuit, Z_L = 0 so P_L = -1. Wave is reflected in anti-phase creating a node at the boundary.
For Z_L = Z₀, P_L = 0. No wave is reflected and all power dissipated in load.

Question 23

How is quarter wave matching used?

Answer 23

Quarter wave matching is used to connect two lines of different impedances. A line of length [wavelength]/4 and characteristic impedance Z₀ = sqrt(Z_in Z_L) (the geometric mean) is used to connect the input and load lines. It will appear to have an impedance matching each end, and a smoothly varying impedance throughout, removing reflections.