Electrical Engineering Paper I

(a) For the circuit shown below, if the power loss in 1 Ω resistor is 16 W, find the value of k in the dependent source using nodal method : (10 marks) (b) Consider the continuous-time system shown in the figure above, which is characterized by the following input-output relationships : System 1 : w(t) = ∫(t−3 to t) x(T) dT System 2 : y(t) = ∫(t−6 to t) w(T) dT (i) Determine whether the overall system is linear and time-invariant. (ii) Determine and plot impulse response of the overall system. (10 marks) (c) Two line currents of infinite extent in the z-direction are at distance d apart along the y-axis. The current I1 is located at y = d/2 and the current I2 is located at y = −d/2 as shown in the figure. Find the magnetic field at any point in the y = 0 plane, when the currents are equal, i.e., I1 = I2 = I : (10 marks) (d) A combinational logic circuit consisting of two numbers of 2 to 4 decoders is shown in the figure below : In this circuit, (A1, B1) are the inputs and (D0 to D3) are the outputs of the first decoder, while (A2, B2) are the inputs and (Y0 to Y3) are the outputs of the second decoder. Determine the logic expressions for Y1 and Y2 in terms of X and Y. (10 marks) (e) A 1250 r.p.m., 220 V, 4·5 kW, DC shunt machine is running as a generator delivering rated load at rated speed and voltage. The machine has rated core loss of 250 W, rated armature copper loss of 200 W and field copper loss of 50 W. Now, the prime mover speed is increased by 20% and simultaneously an external resistance is added in the field circuit so that the developed voltage in armature remains the same and load remains unchanged. Find the— (i) value of the added resistance; (ii) loss in external resistance and field; (iii) input power to the generator; (iv) armature developed voltage. Neglect friction and windage losses. Assume constant core loss and linear magnetic circuit. (10 marks) [Figure in source booklet]

(a) (i) Find the values of R and C in the figure shown below, so that Vb = 3Va, also V̄b and V̄a are in quadrature. Find also the phase relation between V and Vb, and Va and I : (10 marks) (ii) For the circuit shown below, find the value of R, if the power supplied by both the sources is equal : (10 marks) (b) (i) A continuous-time signal x(t), shown in the figure below, has a Fourier transform X(ω). Without explicitly determining X(ω), compute the following quantities : 1. The value of X(0) 2. The value of the integral ∫(−∞ to ∞) X(ω) dω 3. The value of the integral ∫(−∞ to ∞) |X(ω)|² dω 4. The phase angle of X(ω) (10 marks) (ii) Determine the Z-transform and the corresponding region of convergence (ROC) for the discrete-time sequence x(n) = |n|(½)^|n|. (10 marks) (c) A uniform plane wave in air is normally incident onto a lossless dielectric plate of thickness λ/8 and intrinsic impedance η = 260 Ω. Determine the standing wave ratio in front of the plate. (10 marks) [Figure in source booklet]

(a) (i) For the circuit shown below, steady state is achieved with switch K open. At t = 0, switch K is closed. Find the time expression of current in RL. Also find its initial and final values : (10 marks) (ii) Find the y-parameters of the two-port network shown below : (10 marks) (b) For the following circuit given in Fig. (1) below, the diode has the I-V characteristic in forward direction as shown in Fig. (2) : Find the (i) diode current, (ii) diode power loss and (iii) power supplied by the source. (20 marks) (c) The outer diameter of the inner conductor and the inner diameter of the outer conductor of a coaxial copper transmission line are 2 mm and 7 mm, respectively. Both the conductors have thickness much greater than the skin depth. The dielectric is lossless and the operating frequency is 400 MHz. Calculate the resistance per meter length of the transmission line. The resistivity and relative permeability of the copper are approximately ρ = 1·724×10⁻⁸ Ω-m and 1, respectively. (10 marks) [Figure in source booklet]

(a) (i) The continuous-time signal x(t) = cos(20πt + π/4) is ideally sampled at a rate of 30 samples per second. The resulting sampled signal is then passed through a filter whose impulse response is h(t) = sin(πt)/(πt)·cos(80πt − π/2). Find the expression for the output signal of the filter. (15 marks) (iii) A discrete-time sequence x(n) = {3, 5, 1, 4, −3, 5, −2, −2, 4} of length 9 has a 9-point DFT denoted by X(k), where k = 0, 1, 2, ..., 8. Another discrete-time sequence y(n) has a 9-point DFT given by Y(k) = e^(j4πk/3) X(k), k = 0, 1, 2, ..., 8 Without explicitly computing DFT or inverse DFT, determine the sequence y(n). (5 marks) (b) (i) For the circuit shown in the figure below, the two transistors Q1 and Q2 have β1 = 35 and β2 = 30. The resistance R2 is variable. The output voltage Vo across resistance Ro is +5·0 V. If the voltage VCC is kept at +12·0 V, find the value of R2. Derive any formula used. Assume VBE = 0·7 V for both the transistors : (10 marks) (ii) A single-phase, full-wave bridge diode rectifier supplied from 240 V, 50 Hz AC source is feeding a resistive load RL = 100 Ω as shown in the figure below. Due to fault in the diode D2, it behaves like a resistance of 1 kΩ. Sketch the load voltage, load current and source current waveforms under faulty condition. Also calculate the load average voltage and r.m.s. source current under this situation : (10 marks) (c) A lossless transmission line is 50 cm in length and operating at a frequency of 100 MHz. The line parameters are L = 0·2 µH/m and C = 80 pF/m. The line is terminated in a short circuit at z = 0 and there is a load ZL = 50 + j20 Ω across the line at z = −20 cm. What average power is delivered to the load ZL, if the input voltage is 100 V? (10 marks) [Figure in source booklet]

(a) For the circuit given in the figure below, the input voltage signal is given by Vin = 1·5 sin 62·8t + 0·5 sin 188·4t volts. Determine the output voltage expression for each input component indicating their magnitude and phases. Derive the frequency domain expression for the output voltage : (10 marks) (b) A 110/220 V, 1·5 kVA, 50 Hz, single-phase transformer absorbs 150 W power when supplied from a 10 V DC source to its LV side with HV side shorted. It absorbs 10 W at 0·4 p.f. lag when the source voltage is changed to 10 V, 50 Hz AC at the LV side and HV side shorted. It takes 50 W power at 0·2 p.f. lag with rated voltage at LV side and HV side open. Now, if the transformer is to be used at 60 Hz frequency at rated voltage and kVA, then find (i) the maximum efficiency at unity power factor load and (ii) the no-load current, input power and power factor at the LV side with rated voltage. Neglect skin effect and stray losses. Assume linear magnetic circuit, core loss without hysteresis component and same R/X ratio at both LV and HV side. (10 marks) (c) The thyristor T in the circuit shown below is turned on at t = 0, when key K is switched off. Determine the conduction time of T and the capacitor voltage after T is turned off. Given that the inductor carries an initial current of 250 A and initial capacitor voltage is zero : (10 marks) (d) A discrete random variable has two possible values X = 1 or X = 0 with probabilities p and 1 − p, respectively. The probability density function of X is shown in the figure : If Y = Σ(i = 1 to n) Xi, where Xi, i = 1, 2, ..., n are statistically independent and identically distributed random variables, what is the probability distribution function of Y? (10 marks) (e) A circuit consists of a resistor, a variable inductor and a capacitor, all in series. When the circuit is connected across a 300 V, 50 Hz supply, the maximum current is recorded as 0·3 A by varying the inductor. The voltage across the capacitor is then recorded as 900 V. Determine the resistance, inductance and capacitance of the circuit. (10 marks) [Figure in source booklet]

(a) Design a MOD-6 synchronous DOWN counter using J-K flip-flops. (20 marks) (b) A three-phase, star-connected, 415 V, 50 Hz, cage-type induction motor takes a starting current equal to 10 times of its no-load current magnitude. The starting torque is one-third (⅓ times) of its pull-out torque (POT), all under rated conditions. The motor takes 300 W input power at 0·2 p.f. lag with rated supply voltage and frequency. Its core loss has no hysteresis component. Find the starting current and input p.f. of the motor at 5 Hz supply voltage maintaining rated V/f ratio. Neglect stator leakage impedance and rotor current at no load. Assume linear magnetic circuit. (20 marks) (c) A three-stage cascade amplifier given below receives a noise power of 1×10⁻¹⁶ W. What will be the output noise power? where Fi and Gi are the noise figure and gain of the i th amplifier. (10 marks) [Figure in source booklet]

(a) Design a Programmable Logic Array (PLA) for a combinational logic circuit that computes the square of 3-bit binary numbers. Determine the minimum number of product terms required and construct the corresponding PLA programming table for the implementation. (20 marks) (b) (i) A single-phase, one-pulse, SCR-controlled converter with R-L-E load has the following data : Supply voltage Vs = 230 V at 50 Hz, R = 2 Ω, L = 1 mH, E = 120 V, firing angle α = 25° and extinction angle β = 220° 1. Calculate the voltage across the SCR at the instant SCR is triggered. 2. Find the voltage that appears across the SCR when current decays to zero. 3. Find the peak inverse voltage for the SCR. (10 marks) (ii) For a type-A chopper feeding an R-L-E load, show that the maximum value of average current rating for the freewheeling diode, in case the load current remains continuous, is given by Ifd max = [ (Vs/4R) (1 − E/Vs)² ] where Vs is the input voltage to the chopper. (10 marks) (c) The signal m(t) shown in the following figure is used once to frequency modulate a carrier and once to phase modulate the same carrier : Find the relation between kp and kf such that the maximum phase of the modulated signal in both the cases is equal, where kp and kf are phase and frequency sensitivities, respectively of the modulator. (10 marks) [Figure in source booklet]

(a) A 6·6 kV, three-phase, star-connected, cylindrical rotor alternator develops rated voltage with field current of 5 A on open circuit. It takes 100 A armature current at 0·2 p.f. lag for 1·0 A field current with its armature symmetrically shorted, all at rated speed. The alternator is delivering maximum power at rated speed to 6·6 kV bus with a field current of 4·5 A. Find the (i) power generated and delivered by the alternator, (ii) load angle and (iii) power factor and reactive power of the alternator. Assume linear magnetic circuit. (20 marks) (b) (i) A 240 V, 1200 r.p.m., 15 A, separately excited DC motor has armature resistance of 1·0 Ω. The motor is operated under dynamic braking with chopper control. The braking resistance has a value of 20 Ω. 1. Calculate the duty ratio of the chopper for the motor speed of 1000 r.p.m. and braking torque equal to 1·5 times the rated motor torque. 2. What will be the motor speed for the duty ratio of 0·5 and motor torque equal to its rated torque? (10 marks) (ii) A 200 V, 875 r.p.m., 150 A, separately excited DC motor has an armature resistance of 0·06 Ω. It is fed from a single-phase fully controlled rectifier with an AC source voltage of 230 V, 50 Hz. Assuming the continuous conduction, calculate the— 1. firing angle (α) for rated motor torque and 750 r.p.m.; 2. firing angle for rated motor torque and (−500) r.p.m.; 3. motor speed for α = 160° and rated torque. (10 marks) (c) An FM signal with a frequency deviation of 15 kHz at a modulation frequency of 5 kHz is applied to two frequency multipliers connected in cascade. The first multiplier multiplies the frequency by 1·5 times and the second multiplier multiplies it by 4 times. Determine the frequency deviation and the modulation index of the FM signal obtained at second multiplier output. Also compute the approximate bandwidth of the FM signal. (10 marks)