Assignment - Transmission Line Modelling

For all questions "explain" your answer. "Derive" and "Describe" requires that you show the essential steps in a logical sequence including the invocation of a fundamental result/theorems as required. "Estimate" requires a statement of the method used to estimate the required quantity and a statement of why the method has useful characteristics.

Question 1 - Assume that the circuit shown in Figure 1 can be used to model small section of length Δx of a uniform lossless transmission line. The inductance L = L₀Δx, where L₀ is the per unit length inductance of the transmission line and the capacitance C = C₀Δx, where C₀ is the per unit length capacitance of the transmission line. For all that follows assume that

R₀ = √(L₀/C₀) = 100 Ω, C₀ = 10 pF/m

(A) Derive expressions for the magnitude, phase of the voltage gain from the ab port to the output port (potential difference across R₀) as a function of frequency.

(B) From the phase of the voltage gain derive (**) an expression for the group delay evaluated at frequency ω = 0 radians/sec in terms R₀, L and C. What is the significance of group delay in the time domain, i.e. what does it tells us about the time domain input to output characteristics of the network?

(C) Consider the circuit shown Figure 2 with Δx =1.

Use Orcad PSPICE AC Sweep analysis to the magnitude, phase of the voltage gain from the ab port to the output port over the interval [10Hz, 500MHz].

Is the result consistent with you result in Q1 (A)? Explain.

(D) Consider the circuit shown Figure 2 with Δx =1.

Use Orcad PSPICE AC Sweep analysis to plot the real and imaginary part of the impedance at the terminal pair a-b over the interval [10Hz, 500MHz]

How is the value of this impedance relevant when sections are cascaded?

(E) Consider the eight stage model shown in Figure 3.

Use Orcad PSPICE Transient analysis to obtain the waveform for the potential difference between x and y for the time interval [0, 40nS]. The waveform of the ideal voltage source is shown in Figure 4. Use the parameter sweep option in PSPICE to obtain the required waveform for rise time values r = 0, 2nS, 4nS, 6nS

Estimate the total delay and the delay per section from source to x-y.

How are these delay values related to the parameters of components in the eight stage model? Use your results from Q1A and transmission line modelling results you have seen in the course.

What is the effect of rise time on the observed waveforms?

(F) Consider the two instances, i.e. with C₀ and C₀/2 of the eight stage model shown in Figure 5.

Use Orcad PSPICE Transient analysis to obtain the waveform for the potential difference v_ab and v_xy, for the time interval [0, 40nS]. The waveform of the ideal voltage source is shown in Figure 4. Use the parameter sweep option in PSPICE to obtain the required waveform for rise time values r = 0, 2nS, 4nS, 6nS.

How do the waveforms obtained compare with what you would predict if the eight stages are considered to be a uniform transmission line with characteristic impedance R₀?