Circuits and Signals Final Examination

Question 1 -

(a) The AC circuit in the diagram shown in Figure 1 is operating in the steady state. The supplied voltage is, V_s = 5cos(2t)[V] and the current supplied by the current source is i_c = 2 cos(2t + 45^o) [A]. The polarities are as indicated. Find the current supplied to the inductor, as a function of time.

(b) The circuit shown in Figure 2 is operating at a sinusoidal steady state with frequency 50 Hz. The voltage of the source indicated is rms.

Find:

(i) The current supplied by the source.

(ii) The power factor of the circuit.

(iii) The average power supplied by the source.

(iv) The reactive power supplied by the source.

(v) The value of a capacitor that could be connected in parallel across the 15? resistor to correct the power factor of the circuit to 1.

(c) Determine the maximum power that can be delivered to a load by the circuit shown in Figure 3 if

(i) The load can have any complex value.

(ii) The load must be a pure resistance.

Question 2 -

(a) (i) Show that the resonant frequency of the circuit in Figure 4 is given  by

ω_o = (1/√(LC)) √(1 - (L/R²C))

If  R→∞, show that the resonant frequency approaches that of an ideal series resonant circuit  ω_o = 1/√(LC).

(ii) For the series resonant circuit obtained in part (i) above (when R → ∞), when L = 250 mH, C = 25 pF, and R_s =500 Ω, calculate:

(1) The resonant frequency.

(2) The quality factor.

(3) The bandwidth and the half power frequencies.

(4) The power factor at the resonant frequency.

(iii) Show that the voltage across the inductor shown in Figure 4 is equal to 200V_s at the resonant frequency. (You may assume that the frequency of the source   is the same as the resonant frequency).

(b) For the circuit shown in Figure 5, V_i is given by

v_i(t) = 10 sin(1000t - 90^o)

Find the current through each element and construct a phasor diagram showing all currents and the source voltage.

Question 3 -

(a) (i) A reactance X_p when connected in parallel to a load Z = R + jX will raise the power factor of the composite load (Z//jX_p) to a new value pf_new. Show that the reactance X_p is given by

X_p = (V_s²/P)/(tan(cos^-1pf_new)-tan(cos^-1pf_old))

where P is the average AC power and pf_old and pf_new are, respectively, the values of the power factor before and after the connection of reactance in parallel and V_s is the source voltage (rms).

(ii) An industrial plant is powered from a 220-V (rms), 60-Hz source, and it consumes 50 kW with pf = 0.75, lagging. Using  the  equation  derived  in  part (i) above, find a suitable parallel capacitor  to  ensure  pf  =  0.95,  lagging. What current must this capacitor be able to withstand?

(b) (i) Obtain an expression for the input impedance Z_in as seen from the source (V) at angular frequency ω for Figure 6.

(ii) Hence, show that the reflected impedance Z_R to the primary is given by,

Z_R = (ω²M²)/(R₂+jωL₂+Z_L)

(iii) Using the equation obtained in part (i) above, show  that the current I₁ for   the circuit in Figure 7 is given by I₁ = 0.5∠113.1^o A.

Question 4 -

(a) Find the transfer function of the system described by the following differential equation

d²y/dt² + 5(dy(t)/dt) + 6y(t) = 2(dx(t)/dt) - x(t)

where y(t) is the output and x(t) is the input, y(0) = 0, y'(0) = 0 and x(0) = 0 .

(b) Find the capacitor voltage v_c(t) for all t for the circuit shown in Figure 8 (in s- domain), assuming the circuit is initially at rest and the source voltage is a unit step function u(t) and show that v_c(t) is a damped sinusoid.

(c) If the switch in Figure 9 has been closed for a long time before it is opened at t = 0, determine:

(i) The characteristic equation of the circuit

(ii) The voltage across the capacitor v(t) for t > 0.

Question 5 -

(a) Using the Fourier transform pair given below and the Fourier transform properties, determine the frequency domain  representation Y(ω) for the signal y(t) shown in Figure 10 and show that Y(ω) = 2j/ω[cos(ω) - sin(ω)/ω].

 (b) Is it possible to evaluate the Fourier transform of the Fourier transform of a real valued signal i.e., if F{·} denotes the Fourier transform operation and x(t) is a real valued signal, is it possible to evaluate F{F{x(t)}}? If not, explain why not and if so, what is F{F{x(t)}} in terms of x(t)?

(c) Assume that radio stations (AM stations) work by multiplying the audio signal (speech or music) to be transmitted, s(t), with a carrier signal of appropriate frequency, c(t), and then transmitting the output, x(t) as seen in Figure 11. You may also assume that the audio signal has frequency components between 0Hz and 20kHz.

A radio receiver could then work as given by the block diagram in Figure 12. A   good radio system will produce a recovered audio signal, r(t), that is very similar to the original audio signal, s(t).

Assume the system H₁ has an impulse response, h₁(t), given  by: h₁(t) = cos(2πf_ht)e^{-at^2}

where, f_h = 100kHz. Also, assume that the system H₂ is a low pass filter with a 3dB cut-off frequency of 20kHz.

If 2 radio stations are operating, one using a carrier frequency, f_c, of 702kHz, and the other using 1363kHz; and if you want to pick up the first radio station (f_c = 702kHz), answer the following:

(i) What value of f₁ will you use?

(ii) The system H₁ is used to isolate the desired radio signal and eliminate the undesired signals. What value of a will you choose in order to suppress the undesired signals as much as possible while not attenuating any component of the desired signal by more than 3dB?

(iii) What value of f₂ will you choose?

(iv) If you use an RC circuit, Figure 13, as H₂, what values of R and C will you choose such that the 3dB bandwidth of the low pass filter is 20kHz?

Attachment:- Assignment File.rar