Plot the amplitude y versus distance x along the line from

Question

Question 1:

A transmission line of length L meters has a source at one end, and is terminated with an open-circuit at the other end.

The waveform along a transmission line of length L at time t and offset x is given by

y_src(x, t) = Asin(βx - ωt)

where ω is the frequency of the source waveform in radians/second, the velocity is v = fλ m/s, and the phase constant β = 2Π/λ radians/meter. As always, the conversion from Hertz frequency to radian frequency is given by ω = 2Πf radians.

Use the following simplified parameters (you might like to think about what would be more realistic parameter values).

Variable Name Value

f Frequency of Waveform 2 Hz

L Length of Line 1 m

v Speed of travel 1 m/s

A Peak amplitude 1 V

Plot the amplitude y versus distance x along the line, from 0 at the left to L at the right, at time t = 0.

Plot waveform & comment 20

MATLAB or Excel code/formula 20

Total 40

Question 2

The reflected waveform travelling from the termination (an open-circuit) back towards the source is found by replacing x with L x in the above equation. Thus the reflected waveform equation is

y_ref(x, t) = A sin (β(L - x) - ωt)

Plot the amplitude y versus distance x along the line, from L at the right to 0 at the left, at time t = 0.

The resulting waveform on the transmission line is the sum of the forward and reflected waves, that is

y(x, t) = y_src(x, t) + y_ref(x, t) (3)

Plot the reflected wave, and the net wave resulting (source plus reflected), at time t = 0.

The type of plot you should obtain for this question is similar to that shown in Figure 1, except that the frequency of the waveform is different, and the time is t = 0:2.

Figure 1: Transmission line voltages for f=1 Hz at t = 0:2 seconds.

Plot reflected waveform & comment 20

Plot overall waveform & comment 10

MATLAB or Excel code/formula 10
Total 40

Question 3

Repeat the above source and reflected plots for t = 0:2, t = 0:4, t = 0:6 and t = 0:8 seconds.

Plot all five overall (sum) waveforms on the same axes, and clearly indicate which is which using a legend or arrows. This shows the standing-wave pattern. Where do the minima and maxima of the upper envelope1occur? Define these in terms of wavelength of the source.

The type of plot you should obtain for this question is similar to that shown in Figure 2, except that the frequency of the waveform is different.

Plot all resultant waveforms 20

Find maxima and minima & discuss 20

Total 40

Question 4

The far-field pattern of a simple dipole antenna of length L may be approximated by the equation

where β = 2Π/λ radians/m.

This gives the electric field E at angle θ from a line parallel to the antenna. The key question is: what is the best length of the dipole antenna L so as to direct the energy in the most efficient way?

We wish to determine the radiation pattern, and particularly the power P(θ) = E²(θ).

Plot the power P versus angle θ for L = λ/2 . Note that there may be difficulties when sin θ = 0, and you will need to evaluate at a small, but non-zero, angle.

The type of plot you should obtain for this part is similar to that shown in Figure 3, except that the length of the dipole antenna is different.

Correct plot & explanation 30

MATLAB or Excel code/formula 10

Question 5

Repeat the above plot, but for a length of L = 3λ/2.

Briefly comment on your results { what is this and the previous plot telling you?

Correct plot 20
Comment on results 20
Total 40