ELEN30009

Electrical Network Analysis and Design

Final Examination, Semester 1 2024

Question 1 (20 marks)

Consider the second-order circuit shown in Fig. 1. The switch has been closed for a long time, and then it opens at t = 0.

Fig. 1.

(a) [8 marks] Derive the second-order differential equation that i(t) must satisfy for t  ≥ 0.

(b) [12 marks] Consider Vs  = 24 V and R2  = 20 Ω . We wish to design this circuit to obtain the following inductor current:

The only capacitor available is 5 mF. Determine the circuit parameters R1, R3 , and L.

Question 2 (25 marks)

Consider the circuit shown in Fig. 2 with input vi (t) and output vo (t).

Fig. 2.

(a) [8 marks] Show that the impulse response of this circuit is given by

(b) [14 marks] Use the convolution method to compute the output response when the input is given by

vi (t) = 10u(t) — 5u(t — 4) — 5u(t — 8) V

(c) [3 marks] Now, consider the area under the impulse response from (a) stays the same but the time constant of the circuit increases to τ  = 100 ms. Which output waveform is closer to replicating the input waveform. using h(t) from (a) or the new h(t) with τ  = 100 ms? Explain.

Question 3 (30 marks)

The circuit in Fig. 3a with input vi  and output vo  shows the interconnection of three op amp stages.

Fig. 3.

(a) [3 marks] Does this interconnection involve loading? Justify your answer.

(b) [11 marks] Show that the overall transfer function of the circuit in Fig. 3 is given by:

(c) [8 marks] Draw the Bode magnitude plot of this circuit.

(d) [5 marks] Determine the steady-state output voltage of this circuit when

vi (t) = 2 cos(500t + 10o) V

(e) [3 marks] Can you think of a use for this circuit? Justify your answer.

Question 4 (30 marks)

A three-way crossover network is used to couple an audio amplifier to speakers. By replacing the amplifier with a voltage source and each speaker with an 8Ω resistor, the approximate equivalent circuit of the three- way crossover network is composed of three circuits, as shown in Fig 4. Crossover networks are used in audio systems to ensure that the proper frequencies are channeled to the appropriate speaker. A woofer is a low-frequency speaker, and a tweeter is a high-frequency speaker. The circuit also includes a mid-range speaker. The combination of all three speakers allows the entire audio range of interest to be reproduced.

Fig. 4a.

(a) [14 marks] Find the transfer function H(s) = Vo (s)/Vi (s) for each circuit shown in Fig. 4, where the output voltage is taken across each of the three speakers. Discuss the type of filter each of these circuits implements (justify your answer). Using this information, determine which circuit is used with the woofer, tweeter, and midrange speakers, respectively.

(b) [3 marks] For the low-pass and high-pass filters, determine the cut-off frequencies of the circuits in Hz.

(c) [5 marks] For the band-pass filter, determine the quality factor and bandwidth (in Hz) of the circuit. Based on the calculated quality factor, is the circuit overdamped, underdamped, or critically damped?

(d) [8 marks] A fellow student engineer at Unimelb has developed the circuit shown in Fig. 4d. The student argues that the circuit in Fig. 4d can also be used to implement a crossover network with a similar behaviour to the circuit in Fig. 4a. To confirm this statement, determine the behaviour (i.e., filter type) of each of the circuits in Fig. 4d. You must justify your answers for full marks.

Fig. 4d.

Question 5 (30 marks)

(a) [5 marks] Calculate the h parameters for the circuit shown in Fig. 5a.

Fig. 5a.

(b) [7 marks] Calculate the z parameters for the circuit shown in Fig. 5b.

Fig. 5b.

(c)  [4 marks] The following experimental data has been obtained from dc measurements made on a resistive two-port network. Choose one of the six possible sets of equations for two-port networks and determine the corresponding parameters.

(d) [14 marks] The three two-port circuits from items (a), (b) and (c) are interconnected in cascade. The input port of the circuit from (a) is driven by a 5 V dc voltage source in series with an internal resistance of 10 Ω . At the output port of the circuit from (c) is a load resistor ZL  that is adjusted until maximum power is delivered to the load. Calculate the maximum power delivered to this load.

Question 6 (45 marks)

(a) [15 marks] Design a Butterworth band-reject op amp filter with a lower cutoff frequency of 100 Hz and an upper cutoff frequency of 400 Hz. The passband gain of the filter is 0 dB. The filter must be designed with a gain of no more than -20 dB at  180 Hz. Use 39.8 nF capacitors in the high-pass circuit and  10 kΩ resistors in the low-pass circuit.

(b) [7 marks] Draw a circuit diagram of the filter designed in (a) and label all the component values.

(c) [10 marks] A Melbourne-based company designed the passive RLC filter with an op amp buffer circuit shown in Fig. 6c. The supplier of the inductor is no longer in business and a suitable replacement is not available. You have been asked to design a suitable replacement without inductors. To minimise production changes, your design must use the existing op amp and as many  1 kΩ resistors as possible. Draw the replacement circuit and label all the component values.

Hint: determine the behaviour of the circuit and then design a suitable active Butterworth filter replacement.

Fig. 6c.

(d) [13 marks] The op amp in Fig. 6d has the characteristics described in Table 1, as obtained from the op amp datasheet.

Fig. 6d.

Table 1

(i) [3 marks] Consider vin (t) = 2 cos(100t) V, Vcc  = 12 V and —VEE  = —12 V. What is the maximum op-amp gain before saturation occurs? Show your calculations.

(ii) [3 marks] Consider Rf   = 10 kΩ, determine the effect of the input offset current on the output voltage. (iii) [3 marks] Consider Rf /Rin  = 99, determine the effect of the input offset voltage on the output voltage. (iv) [4 marks] Consider vin (t) = sin(2π × 105 t) V and Rf /Rin  = 10. Is the output voltage affected by the slew rate specification? If so, what is the impact on the output waveform? Justify your answer.