Q1 50M Compulsory solve Network analysis, signals, DC machines, transistors, digital logic
(a) In Figure 1(a) shown below, the two-port network is characterized in terms of y-parameters with y₁₁ = 3·3 × 10⁻³ S, y₂₂ = 5 × 10⁻³ S and y₁₂ = y₂₁ = 0. Find the voltage across 200 Ω load. (10 marks)
(b) For the signal shown in Figure 1(b), calculate the total energy of the signal X(t). Also sketch y(t) = X(10t – 5). (10 marks)
(c) A 220 V dc shunt motor has armature resistance Rₐ = 0·13 Ω, field resistance Rf = 250 Ω and rotational loss 230 W. On full-load, the line current is 9·5 A with the motor running at 1440 rpm. Determine the following:
(i) The mechanical power developed
(ii) The power output
(iii) The load torque
(iv) The full-load efficiency (10 marks)
(d) For the transistor circuit shown in Figure 1(d), determine the value of reverse saturation current, I_S, that would give a collector current of 1 mA, if β = 80, V_A = ∞ and V_T = 26 mV at T = 300 K. (10 marks)
(e) Consider the four variables logic function defined as follows:
F (A, B, C, D) = ĀC + ĀD + B̄C + B̄D + ABC̄ D̄
Assuming input variables as A, B, C and D, propose a logic circuit using only three logic gates to implement the function. (10 marks)
Answer approach & key points
Solve each sub-part systematically with clear step-by-step calculations. For part (a), apply y-parameter equations to find load voltage; for (b), compute energy using ∫|x(t)|²dt and apply time-scaling/shifting for the sketch; for (c), calculate motor performance parameters using DC machine equations; for (d), use Ebers-Moll model with given β; for (e), simplify the Boolean expression using K-map or algebraic manipulation to implement with only three gates. Allocate approximately 15% time to (a), 15% to (b), 25% to (c), 15% to (d), and 30% to (e) due to its simplification complexity.
- Part (a): Correct application of y-parameter equations I₁ = y₁₁V₁ + y₁₂V₂ and I₂ = y₂₁V₁ + y₂₂V₂ with y₁₂ = y₂₁ = 0, leading to V₂ = -I₂R_L and solving for load voltage
- Part (b): Energy calculation using E = ∫_{-∞}^{∞} |X(t)|² dt for the given waveform, correct application of time scaling (compression by 10) and time shifting (advance by 0.5s) for y(t) = X(10t-5)
- Part (c)(i)-(iv): Correct calculation of field current I_f = V/R_f, armature current I_a = I_L - I_f, back EMF E_b = V - I_aR_a, mechanical power P_mech = E_b × I_a, output power P_out = P_mech - rotational losses, torque T = P_out/ω, and efficiency η = P_out/(V×I_L)
- Part (d): Application of I_C = βI_B with I_B = I_S(e^{V_BE/V_T} - 1), using active mode assumption and given β = 80, V_T = 26mV to solve for I_S
- Part (e): Boolean simplification of F = ĀC + ĀD + B̄C + B̄D + ABC̄D̄ to minimal form using consensus theorem or K-map, resulting in implementation using only three logic gates (e.g., two AND-OR or NAND-NAND structure)
- Correct unit handling throughout (Siemens, Volts, Amperes, Watts, rad/s, Nm) and proper significant figures in final answers
Q2 50M solve Thevenin's theorem, convolution, sequential circuits
(a) Find the Thevenin's equivalent of the circuit shown in Figure 2(a) below as seen from the load impedance Z_L. Also find the value of Z_L for maximum power transfer. (20 marks)
(b) (i) Compute the convolution X[n] * h[n], where
X[n] = (1/2)^(-n) u[-n-2]
h[n] = u[n-2].
(ii) Consider the signal X(t) shown in Figure 2(b)(ii) below. Represent the signal X(t) in terms of rectangular pulse signal V(t) shown in the same figure. (20 marks)
(c) Consider the circuit shown in Figure 2(c) below. Let inputs A, B and C be all initially LOW. Output Y is supposed to go HIGH only when A, B and C go HIGH in a certain sequence. Determine the sequence that will make Y go HIGH. Modify this circuit to use D-Flip-flops. (10 marks)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) Thevenin's equivalent and maximum power transfer, 40% to part (b) convolution and signal representation, and 20% to part (c) sequential circuit analysis and D-flip-flop modification. Begin each sub-part with the relevant circuit diagram or signal sketch, show all mathematical steps clearly, and conclude with the final numerical answer or circuit modification. For part (b)(i), carefully handle the time-reversed nature of x[n] and the shifted unit step functions.
- Part (a): Correct calculation of Thevenin's equivalent voltage (V_th) by open-circuit analysis and Thevenin's equivalent impedance (Z_th) by deactivating independent sources
- Part (a): Application of maximum power transfer theorem stating Z_L = Z_th* (complex conjugate) for AC circuits, or Z_L = Z_th for purely resistive circuits
- Part (b)(i): Proper handling of convolution with anti-causal signal x[n]=(1/2)^(-n)u[-n-2] rewritten as 2^n u[-n-2], correct determination of overlap regions for n<0 and n≥0
- Part (b)(ii): Expression of x(t) as weighted sum of shifted rectangular pulses V(t), identification of amplitude and time-shift parameters from figure
- Part (c): Analysis of sequential circuit to determine required input sequence (likely A→B→C or specific order) that activates Y through state transitions
- Part (c): Correct modification to D-flip-flops: converting existing flip-flops or designing equivalent state machine with proper excitation table and next-state logic
Q3 50M calculate Analog circuits and signal processing
(a) (i) Explain what happens when a circuit shown in Figure 3(a)(i) below is constructed using logarithmic amplifier. 10
Figure 3(a)(i)
(ii) Explain what happens if the topology is modified as shown in Figure 3(a)(ii) below. 10
Figure 3(a)(ii)
(b) For the circuit shown in Figure 3(b), calculate the voltage V₀(t) as function of time,
Figure 3(b)
where V(t) = 10 sin (6t + 60°) V and I(t) = 5 cos (4t + 30°) A. 20
(c) A mixer (analog multiplier) is used as a process in some analog communication systems. Two signals X₁(t) and X₂(t) are mixed to produce the output y(t) = X₁(t) X₂(t).
If X₁(t) = 10 sin c (10t) and X₂(t) = 2 cos (1000 πt), then calculate and plot the magnitude of the Fourier transform of output signal. Further, specify and prove the property of Fourier transform used in calculations. 10
Answer approach & key points
The directive 'calculate' dominates part (b) carrying 20 marks, so allocate ~50% effort there with rigorous time-domain analysis; spend ~25% each on (a)(i) and (a)(ii) explaining logarithmic/antilogarithmic amplifier behavior with circuit diagrams; reserve ~10% for (c) applying the multiplication/convolution property of Fourier transforms. Structure: begin with conceptual explanations of log-amp configurations, proceed to detailed circuit analysis for V₀(t), and conclude with spectral analysis of the mixer output.
- For (a)(i): Explain that Figure 3(a)(i) realizes a log-amp producing V₀ = -V_T ln(V_i/R_1I_S), converting multiplication to addition for analog computation
- For (a)(ii): Explain the modified topology as an antilogarithmic (exponential) amplifier giving V₀ = -R_f I_S exp(-V_i/V_T), enabling division and power operations
- For (b): Apply KCL/KVL to derive the differential equation and solve for V₀(t) = 10 sin(6t + 60°) - L(di/dt) or equivalent, handling the frequency mismatch (6 rad/s vs 4 rad/s) correctly
- For (b): Show proper handling of phase relationships and superposition when input voltage and current have different frequencies
- For (c): Apply the multiplication property: F{x₁(t)x₂(t)} = (1/2π)[X₁(ω) * X₂(ω)], proving convolution in frequency domain shifts the sinc spectrum to ±1000π
- For (c): Sketch the resulting spectrum showing two sinc-shaped bands centered at ω = ±1000π with appropriate scaling and bandwidth 20π rad/s
Q4 50M calculate Digital signal processing and digital electronics
(a) Consider a discrete time system with transfer function given by
H(z) = Y(z)/R(z) = (z⁻¹ - ½z⁻²)/(1 - z⁻¹ + 2/9 z⁻²).
Calculate the following :
(i) The impulse response of the system
(ii) The step response of the system with zero initial conditions
(iii) The step response of the system with initial conditions y[-1] = 1 and y[-2] = 2 20
(b) (i) Verify by determining the logic equation for the output and by constructing the truth table for the logic circuit shown in Figure 4(b).
(ii) Use an 8 to 1 multiplexer and logic gates to implement the following function :
F(A, B, C, D, E) = Σ m (0, 1, 2, 4, 5, 6, 7, 13, 14, 20, 21, ..., 28, 29, 30, 31)
20
Figure 4(b)
(c) Determine the closed loop gain of the inverting amplifier shown in Figure 4(c) below. Explain the result if R₁ → 0 or R₃ → 0.
10
Answer approach & key points
Calculate the required responses and circuit parameters systematically. For part (a) [20 marks], perform partial fraction expansion on H(z) and apply Z-transform properties for impulse and step responses, handling initial conditions via unilateral Z-transform. For part (b) [20 marks], derive the logic equation from Figure 4(b), construct truth table, then implement F(A,B,C,D,E) using 8:1 MUX with A,B,C as select lines. For part (c) [10 marks], apply ideal op-amp assumptions to find closed-loop gain and analyze limiting cases. Allocate approximately 40% time to (a), 35% to (b), and 25% to (c).
- Part (a)(i): Factor denominator, perform partial fraction expansion, identify poles at z=1/3 and z=2/3, obtain h[n] = [3(1/3)^n - 3(2/3)^n]u[n-1] or equivalent causal form
- Part (a)(ii): Apply step input R(z)=z/(z-1), use final value theorem or convolution sum, obtain y_step[n] with zero initial conditions
- Part (a)(iii): Apply unilateral Z-transform accounting for y[-1]=1, y[-2]=2, separate zero-state and zero-input responses, combine for total response
- Part (b)(i): Derive Boolean expression from Figure 4(b) circuit topology, verify with complete truth table showing all input combinations and output
- Part (b)(ii): Implement 5-variable function using 8:1 MUX with A,B,C as select inputs, determine D,E combinations for each minterm group (0-7, 13-14, 20-21, 28-31), connect appropriate logic to data inputs
- Part (c): Apply virtual ground concept, derive V_o/V_i = -R_f/R_1 where R_f involves R_2,R_3 network, analyze R_1→0 (infinite gain/saturation) and R_3→0 (gain becomes -R_2/R_1) cases with practical implications
Q5 50M Compulsory calculate DC chopper, electromagnetic waves, Scott transformer, AM modulation, circuit analysis
(a) A step down dc chopper is feeding a load of R = 10 Ω and L = 20 mH. The dc supply voltage is 100 V. The chopper is switching at a frequency of 2 kHz with a duty cycle of 50%. Determine the load current and the peak-to-peak ripple current as an absolute value and as percentage of dc value. (10 marks)
(b) In a certain material with σ = 0, ε = ε₀ εᵣ and μ = μ₀ μᵣ, the magnetic field intensity component is given by H = 10 sin (10⁸ t – 2x) aᵤ A/m. Find the following: (i) Displacement current density (ii) Electric field intensity (10 marks)
(c) A Scott connected transformer shown in Figure 5(c) is supplied from 11 kV, 3-phase, 50 Hz mains. Secondaries are series connected and supply 1100 A at a voltage of 100√2 V to a resistive load. The phase sequence of the 3-phase supply is ABC. (i) Calculate the turns ratio of the teaser transformer. (ii) Calculate the line current I_B and its phase angle with respect to the voltage of phase A to neutral on the 3-phase side. (10 marks)
(d) A transmitter with a 10 kW carrier transmits 11·2 kW when modulated with a single sine wave. Calculate the modulation index. If the carrier is simultaneously modulated with two other sine waves also at 50% modulation, calculate the total power transmitted. (10 marks)
(e) For the circuit shown in Figure 5(e), v_C(0+) = 2 V and i(0+) = 2/3 A. Calculate the value of v_C(t) for t > 0. (10 marks)
Answer approach & key points
This is a multi-part numerical problem requiring systematic calculation across five distinct areas of electrical engineering. Allocate approximately 20% time to each sub-part: (a) DC chopper analysis using duty cycle and ripple current formulas, (b) electromagnetic wave propagation applying Maxwell's equations, (c) Scott transformer phasor analysis with 90° phase relationships, (d) AM power calculations using modulation index formulas, and (e) transient circuit analysis using Laplace transforms or classical methods. Begin each part with the relevant governing equation, show substitution of values with units, and conclude with clear numerical answers.
- Part (a): Correct application of step-down chopper duty cycle formula V₀ = δVₛ, average load current I₀ = V₀/R, and ripple current ΔI = Vₛδ(1-δ)/fL with proper unit handling
- Part (b): Application of Maxwell's equations to find displacement current density Jd = ∂D/∂t and E-field using intrinsic impedance η = √(μ/ε) for lossless medium
- Part (c): Scott transformer teaser transformer turns ratio calculation (0.866 factor) and phasor diagram construction for 90° phase shift between teaser and main transformer
- Part (d): AM power calculation using Pₜ = Pc(1 + m²/2) for single tone and extension to multiple tones with Pₜ = Pc(1 + m₁²/2 + m₂²/2 + m₃²/2)
- Part (e): Second-order circuit transient analysis using characteristic equation, damping classification, and complete solution form with initial condition application
- Proper handling of per-unit and absolute values for ripple current percentage calculation in part (a)
- Recognition that σ = 0 implies purely displacement current with no conduction current in part (b)
Q6 50M solve Plane wave propagation, three-phase bridge inverter, probability density function
(a) The magnetic field intensity of a linearly polarized uniform plane wave propagating in the +Y-direction in sea water (ε_r = 80, μ_r = 1, σ = 4 S/m) is H = 0·1 sin (10^10 πt - π/3) a_x A/m. At Y = 0, determine the following: (i) The attenuation constant, intrinsic impedance, the wavelength and skin depth. (ii) The location at which the amplitude of H is 0·01 A/m. (iii) The expression for E(y, t) and H(y, t) at Y = 0·5 (m) as functions of t. (20 marks)
(b) A three-phase bridge inverter shown in Figure 6(b) is used to feed a Y-connected resistive load with R = 10 Ω per phase. The dc input to the inverter V_S = 400 V and the output frequency is 50 Hz. If the inverter is operating with 180° conduction mode, (i) compute the rms value of the load current, (ii) compute the rms value of the current in each switching device, (iii) calculate the output power, and (iv) draw the waveforms of phase and line voltages. (20 marks)
(c) Let the measurement error of a physical quantity be defined by a random variable X and its density function as follows: f(x) = {K(3-x²) for -1≤x≤1, {0 elsewhere. Determine the value of 'K' and find the probability that a random error in measurement is less than 1/2. (10 marks)
Answer approach & key points
This is a computational problem requiring systematic solution of three distinct parts. Allocate approximately 40% of time to part (a) given its 20 marks and complexity involving good conductor analysis; 40% to part (b) for inverter calculations and waveform sketching; and 20% to part (c) for probability determination. Begin each part with stated assumptions and relevant formulas, proceed through step-by-step calculations with proper units, and conclude with boxed final answers.
- Part (a): Calculate attenuation constant α, intrinsic impedance η, wavelength λ, and skin depth δ for sea water at given frequency, identifying it as a good conductor (σ/ωε >> 1)
- Part (a)(ii): Determine propagation distance for amplitude decay from 0.1 A/m to 0.01 A/m using exponential attenuation formula
- Part (a)(iii): Derive time-domain expressions for E(y,t) and H(y,t) at y=0.5m, accounting for phase shift and attenuation
- Part (b): Compute RMS load current, device current, output power for 180° conduction mode three-phase bridge inverter with Y-connected resistive load
- Part (b)(iv): Sketch phase voltages (V_AN, V_BN, V_CN) and line voltages (V_AB, V_BC, V_CA) showing 120° phase displacement and six-step waveform
- Part (c): Determine normalization constant K by integrating PDF over [-1,1], then calculate P(X < 0.5) through proper integration
Q7 50M solve Synchronous machines, converters, transmission lines
The following test data are obtained for a three-phase, 195 MVA, 15 kV, 50 Hz star connected synchronous machine.
Open circuit test :
| I_f (A) | 150 | 300 | 450 | 600 | 750 | 900 | 1200 |
|---------|-----|-----|-----|-----|-----|-----|------|
| V_LL (kV) | 3·75 | 7·5 | 11·2 | 13·6 | 15 | 15·8 | 16·5 |
Short circuit test :
I_f = 750 A, I_a = 7000 A
The armature resistance is small.
(i) Draw the open circuit characteristic, the short circuit characteristic, the airgap line and the modified airgap line.
(ii) Determine the unsaturated and saturated values of the synchronous reactance in pu.
(iii) Find the field current required, if the synchronous machine is to deliver 100 MVA at rated voltage, at 0·8 leading power factor. (20 marks)
(b) A three-phase, full-wave thyristor bridge converter is operated from a three-phase, Y-connected 220 V, 50 Hz supply and the load resistance is 20 Ω.
It is required to obtain an average output voltage of 50% of the maximum possible output voltage. Determine the following :
(i) The delay angle α
(ii) The rms and average output currents
(iii) The rms and average thyristor currents
(iv) The rectification efficiency
(v) The input PF (20 marks)
(c) A lossless transmission line has characteristic impedance Z₀ = 50 Ω. Its length is 30 m and operates at 5 MHz. The line is terminated with a load Zₗ = 60 + j50 Ω. If the phase velocity u = 0.6c on the line, find the following :
(i) The reflection coefficient 'Γ'
(ii) The standing wave ratio 'S'
(iii) The input impedance 'Zᵢₙ' (10 marks)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) synchronous machine analysis (20 marks), 40% to part (b) thyristor converter calculations (20 marks), and 20% to part (c) transmission line parameters (10 marks). Begin with clear identification of given data, proceed with systematic derivations for each sub-part, and conclude with verification of results against physical constraints.
- Part (a)(i): Correct plotting of OCC, SCC, airgap line and modified airgap line with proper scaling and identification of knee point
- Part (a)(ii): Calculation of unsaturated and saturated synchronous reactance in per-unit using V_OC/I_SC at appropriate field currents
- Part (a)(iii): Determination of field current for 100 MVA, 0.8 leading PF using Potier triangle or ASA method with correct phasor diagram
- Part (b)(i)-(v): Complete converter analysis including delay angle α=60°, rms/average output currents, thyristor currents, rectification efficiency, and input power factor
- Part (c)(i)-(iii): Transmission line calculations with correct electrical length (βl), reflection coefficient Γ=0.326∠56.3°, SWR S=1.97, and input impedance Z_in using Smith chart or analytical formulas
Q8 50M calculate FM communication, induction motor, DC motor control
For an FM communication system with β = 2 and white channel noise with PSD Sₙ(ω) = 10⁻¹⁰, the output SNR is found to be 28 dB. The base band signal m(t) is Gaussian, band-limited to 15 kHz, and 3σ loading is used.
Determine the following :
(i) The received signal power 'Sᵢ'
(ii) The output signal power 'S₀'
(iii) The output noise power 'N₀' (20 marks)
(b) A three-phase, 4-pole, 50 Hz induction motor has a rotor resistance of 4·5 Ω/phase and a standstill reactance of 8·5 Ω/phase with no external resistance in the rotor circuit. The starting torque of the motor is 85 Nm. Neglecting stator voltage drop, determine the following :
(i) The rotor voltage at standstill
(ii) The starting torque, if a 3 Ω resistance were added in each rotor phase
(iii) The rotor induced voltage and the torque at a slip of 0·03 (20 marks)
(c) A 220 V, 1500 rpm, 10 A separately excited dc motor has an armature resistance of 1 ohm. It is fed from a single phase fully-controlled bridge rectifier with an ac source voltage of 230 V, 50 Hz. Assuming continuous load current, determine the following :
(i) Motor speed at the firing angle of 30° and torque of 5 Nm
(ii) Developed torque at the firing angle of 45° and speed of 1000 rpm (10 marks)
Answer approach & key points
Calculate the required parameters for all three parts systematically. Spend approximately 40% of time on part (a) FM communication (20 marks), 40% on part (b) induction motor (20 marks), and 20% on part (c) DC motor control (10 marks). Begin with stating relevant formulas, substitute given values with proper units, perform step-by-step calculations, and conclude with final numerical answers in appropriate units (W, dB, V, Nm, rpm).
- Part (a): Apply FM SNR formula (S₀/N₀) = 3β²(Δf/fm)²(Si/Ni) with β=2, 3σ loading factor, and convert 28 dB SNR to linear scale; calculate Si using noise power Ni = η×B
- Part (a): Compute output signal power S₀ using S₀ = (kf²/2π)×Pm and output noise power N₀ from total SNR relationship
- Part (b): Calculate rotor standstill voltage E₂ using starting torque formula Tst = (3/ωs)×(E₂²R₂)/(R₂²+X₂²) with given 85 Nm torque
- Part (b): Determine modified starting torque with added 3Ω resistance using Tst ∝ R₂/(R₂²+X₂²) and find slip=0.03 values using torque-slip characteristics
- Part (c): Compute motor back EMF constant Kφ from rated conditions, then find speed at α=30° using Vt = (2√2×230/π)cosα = Ea + IaRa with T=5 Nm
- Part (c): Calculate developed torque at α=45° and 1000 rpm using rectifier output voltage equation and torque-speed relationship