Circuits and Signals - Final Examination

Question 1-

(a) In a linear circuit voltage across and current through the total impedance are given as

v(t) = 50 cos(10t + 10^o) V and i(t) = 25 cos(10t + 41^o) A.

Is the total impedance Z of the circuit inductive or capacitive? Explain your answer.

(b) At ω = 314 rad/s, the equivalent admittance Y for the circuit in Figure 1 is (0.32 + j0.64) mS. Calculate values of R and C.

(c) Find the source voltage v_s in the circuit of Figure 2 if the current I, through the 1-? resistor is (0.5 sin 200t) A.

(d) Determine current i(t) in the circuit of Figure 3 which have two voltage sources of two different frequencies.

Question 2 -

(a) Calculate the value of impedance Z_L in the circuit of Figure 4 in order for Z_L to receive maximum average power. What is the maximum average power received by Z_L when Thevenin's equivalent voltage across the terminals of Z_L is given as V_Th = (10+j70) V?

You may use P_max = |VTh|²/8R_L to calculate maximum average power.

(b) In the circuit of Figure 5 device A receives 2 kW at 0.8 pf lagging, device B receives 3 kVA at 0.4 pf leading, while device C is inductive and consumes 1 kW and re 500VAR.

(i) Determine the power factor of the entire system.

(ii) Find current I (rms) given that V_s = 120∠45^o V (rms).

(c) (i) Discuss mutual inductance M and its effects in two magnetically coupled coils.

(ii) For the coupled coils in Figure 6 show that equivalent inductance is

L_eq = (L₁L₂ - M²)/(L₁ + L₂ - 2M)

Question 3 -

(a) If the switch in Figure 7 has been closed for a long t me before it is opened at t = 0, determine-

(i) the characteristic equation of the circuit,

(ii) i_x for t > 0

You may take help of following formula:

Over-damped case: i(t) or v(t) = A₁e^s_1t + A₂e^s_2t , where s_1,2 = -α ± √(α² - ω₀²)

Critically damped case: i(t) or v(t) = (A₂ + A₁t)e^-αt, where s_1,2 = - α

Under damped case: i(t) or v(t) = e^-αt(B₁cosω_dt + B₂sinω_dt), where ω_d = √(ω₀² - α²)

(b) For the network given in Figure 8,

(i) Draw the equivalent s-domain circuit. Assume at t < 0 voltage and current in the circuit are zero.

(ii) Write the mesh analysis equations of the circuit in s-domain.

(iii) Comment on difference between s domain (Laplace transformation) and frequency domain (phasors) analysis.

(c) A circuit is known to have its transfer function as,

H(s) = (s + 3)/(s² + 4s + 5)

Find its output in time domain when the input is a unit step function u(t).

You may use Laplace transformation tables provided at the back of the question paper.

Question 4 -

(a) (i) What are the half power frequencies in a resonant circuit? Explain with help of a sketch, labelling all the important points.

(ii) Starting from  the condition of half power |Z| = √2R,  express  the two  half power frequencies ω₁ and ω₂ in terms of circuit parameters R, Land C of a series resonant circuit.

(iii) Show that bandwidth B in a series RLC resonant circuit is R/L.

(iv) If a series RLC circuit has B = 8 rad/s and resonance frequency ω_o = 1,500 rad/s and R = 10 Ω, find values of L and C.

(v) Also find the quality factor Q of the circuit defined in part (iv).

(b) Show that the trigonometric Fourier series expansion of the backward sawtooth waveform of Figure 9 is  f(t) = 5 + 10/π _n=1∑^∞ 1/n sin(nπt)

(c) Find the response v₀(t) of the circuit of Figure 10 when the voltage source v_s(t) is given by the backward sawtooth waveform f(t) of Figure 9.

Attachment:- Assignment File.rar