ECSE 354 - Laboratory 2 - Reflection on a Transmission Line - Solved

 

In this laboratory, we will develop several simple functions to understand the influence of the reflection coefficient Γ arising from a load impedance ZL at z = 0 on the voltage and current along a transmission line.  

 

1.  Phasor voltage and current in the presence of reflection
 

Let us begin with a phasor description of the problem. Write a function that calculates the voltage phasor Vs(z) and the current phasor Is(z) on a transmission line in the presence of an incident wave and a reflected wave using the same convention as in the lecture notes (load impedance ZL located at z = 0),

 

[ Vs Is ]= linephasor(V0,Gamma,gamma,Z0,z)

 

as defined by the following equations,

 

                                                                                                                                             𝑉                           𝑉

𝑉 (𝑧) = 𝑉 exp(−𝛾𝑧) + Γ 𝑉 exp(+𝛾𝑧)                   𝐼 (𝑧) =   exp(−𝛾𝑧) − Γ   exp(+𝛾𝑧)

                                                                                                                                             𝑍                            𝑍

 

where V0 is the complex voltage phasor amplitude of the incident wave, γ is the phase constant of the line (where γ = α + j β in general), and Z0 is the characteristic impedance of the transmission line.

 

 

2.  Phasor magnitude on a lossless transmission line with resistive loads
 

Consider first a lossless transmission line with Z0 = 100 Ω and γ = j 2π rad/m, excited with an incident wave amplitude V0 = 1 V. Restrict your attention to the length –2 m ≤ z ≤ 0 m. Plot the absolute value of the voltage and current phasors,  |Vs(z)| and I Is(z) |, versus position z on the same figure in different colour so that you can compare both. You will find it useful to plot 100 x I Is(z) | such that the maxima in voltage and current are of comparable size in your figure. When reading your figures, take care that the units for |Vs(z)| and I Is(z) | are different. Consider the following scenarios for the reflection coefficient, plotting a separate figure for each:

 

a)       Γ = 0, corresponding to a matched load ZL = Z0 = 100 Ω.

b)      Γ = +1/3, corresponding to the resistive load ZL = 2Z0 = 200 Ω.

c)       Γ = +2/3, corresponding to the resistive load ZL = 5Z0 = 500 Ω.

d)      Γ = +1, corresponding to the open circuit load ZL → ∞.

 

Do you observe the expected trend for the voltage standing wave ratio (VSWR) for a) to d) ? Recall that VSWR can be expressed as s = max|Vs(z)|/min|Vs(z)|.

 

Is there a need to define an independent quantity called the current standing wave ratio? Or would this quantity be identical to the VSWR?

 

Calculate the ratios Vs(z=0) / Is(z=0) for a) to d). How do the ratios compare with the load impedances ZL ?

 

№ 1: Show your results to the teaching assistant.

 

 

 

3.  Phasor magnitude on a lossless transmission line with inductive loads
 

Use the same transmission line parameters and incident wave amplitude as above. Plot the absolute value of the voltage phasors |Vs(z)| and I Is(z) | for the following four additional scenarios for the reflection coefficient:

 

e)       Γ = exp( j π/4 ), corresponding to the inductive load ZL ≈ 2.4j Z0 = 240 j Ω.

f)       Γ = exp( j π/2 ) = +j, corresponding to the inductive load ZL = j Z0 = 100 j Ω.

g)      Γ = exp( j 3π/4 ), corresponding to the inductive load ZL ≈ 0.41j Z0 = 41 j Ω.

h)      Γ = exp( j 4π/4 ) = -1, corresponding to the short circuit load ZL = 0 Ω.

 

Comparing the sequence d) to h), what is changing in |Vs(z)| and I Is(z) | ?

 

Calculate the ratios Vs(z=0) / Is(z=0) for e) to h). How do the ratios compare with the load impedances ZL ?

 

№ 2: Show your results to the teaching assistant.

 

You may wish to continue the exercise, and investigate what happens for capacitive loads, and more general load impedances of the form ZL = R + j X.

 

 

 

4.  Phasor magnitude on a lossy transmission line with matched load
 

Consider now a transmission line with loss with a characteristic impedance Z0 = 100 Ω *, excited with an incident wave amplitude V0 = 1 V, and terminated with a matched load ZL = Z0 such that Γ = 0. Restrict your attention to the length –2 m ≤ z ≤ 0 m. Plot the absolute value of the voltage and current phasors |Vs(z)| and I Is(z) | for the following scenarios of transmission line propagation constant:

 

a)       α  =  0.050 Np/m and β = 2π rad/m.

b)      α  =  0.347 Np/m and β = 2π rad/m.

c)       α  =  0.693 Np/m and β = 2π rad/m.

d)      α  =  2 Np/m and β = 2π rad/m.

 

How is the voltage V(z=0) at the load affected by the increased attenuation in the sequence a) to d)?

 

Are the ratios Vs(z=0) / Is(z=0) equal to the load impedance ZL = Z0 ?

 

 

№ 3: Show your results to the teaching assistant.

 

 

* While Z0 is in general a complex quantity for a transmission line with loss, it is possible to tune the parameters R, L, G, C of a transmission line with loss such that Z0 is real. This is called a dispersionless transmission line. Can you derive the conditions for the dispersionless transmission line?

 

 

 

5.  Phasor magnitude on a lossy transmission line with an open circuit load
 

Repeat exercise 4 above, now with an open circuit load such that Γ = +1. Plot the absolute value of the voltage and current phasors |Vs(z)| and I Is(z) | for the same scenarios of transmission line propagation constant:

 

a)       α  =  0.050 Np/m and β = 2π rad/m.

b)      α  =  0.347 Np/m and β = 2π rad/m.

c)       α  =  0.693 Np/m and β = 2π rad/m.

d)      α  =  2 Np/m and β = 2π rad/m.

 

How is the standing wave component of the voltage on the line affected by the increased attenuation in the sequence a) to d)?

 

№ 4: Show your results to the teaching assistant.

 

 

6.  Instantaneous voltage and current on a lossless transmission line with reflections
 

Let us now consider the behaviour of the instantaneous voltage vs(z,t) and current is(z,t) on a lossless transmission line. This can be done by first calculating the phasor representation of the harmonic waves as you have done in the preceding exercises, and then converting the phasors to instantaneous time representations using the ph2inst function that you developed in laboratory 1, exercise 4 for precisely this purpose.

 

Assume an incident wave amplitude V0 = 1 V, angular frequency ω = 2π x 200 x 106 rad/s, characteristic impedance Z0 = 100 Ω, and phase constant β = 2π rad/m. Restrict your attention to the length –2 m ≤ z ≤ 0 m and the time interval 0 s ≤ t ≤ 20 ns.

 

Make movies of the instantaneous voltage vs(z,t) and current is(z,t) for the following three scenarios:

 

a)       Γ = +2/3, corresponding to the resistive load ZL = 5Z0 = 500 Ω.

b)      Γ = +1, corresponding to the open circuit load ZL → ∞.

c)       Γ = exp( j π/2 ) = +j, corresponding to the inductive load ZL = j Z0 = 100 j Ω.

 

Carefully observe the simultaneous travelling wave components and standing wave components in the voltage and current of scenario a).

 

Carefully observe the standing waves in the instantaneous voltage and current of scenarios b) and c). What difference do you observe at the load (z = 0) between open circuit and inductive load ?