(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), calcu

Question

(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)

UPSC Answer Check · Accepted Answer

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

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies and applies: two-port network theory for (a), signal energy definition and time transformation properties for (b), DC motor equivalent circuit and power flow for (c), BJT current equations in active region for (d), and Boolean algebra theorems for (e); no conceptual errors in any sub-part	Mostly correct concepts with minor errors in one sub-part, such as confusing h-parameters with y-parameters, or misapplying time-scaling vs time-shifting, or incorrect power flow direction in motor	Fundamental conceptual errors in multiple sub-parts, such as using Z-parameters instead of Y-parameters, calculating power as V×I without considering losses, or incorrect Boolean simplification approach
Numerical accuracy	20%	10	All calculations accurate to appropriate significant figures: load voltage in (a), energy value in (b), all four motor parameters in (c), I_S in (d); correct arithmetic with proper unit conversions (mV to V, rpm to rad/s)	Correct methodology but arithmetic errors in 1-2 sub-parts, such as calculation mistakes in back EMF or torque, or incorrect exponent handling in I_S calculation	Multiple calculation errors, wrong order of magnitude results, or missing units; significant errors in motor efficiency or completely wrong I_S value
Diagram quality	15%	7.5	Clear sketch for (b) showing original X(t), intermediate steps (X(10t) compression, then X(10t-5) shift), with labeled axes, amplitude scaling, and time markers; circuit diagram for (e) with proper gate symbols and connections	Sketch present but missing intermediate steps or unclear time axis labels; circuit diagram for (e) functional but with minor drawing issues	Missing sketch for (b), or completely wrong transformation shown; no circuit diagram for (e), or diagram that cannot implement the function
Step-by-step derivation	25%	12.5	Complete derivation chain visible: y-parameter matrix setup → load condition equations → solution for (a); integral setup with limits → evaluation for (b); equivalent circuit → current distribution → sequential power calculations for (c); KVL → diode equation → algebraic solution for (d); K-map/algebraic steps → minimized expression → gate realization for (e)	Most steps shown but with gaps in derivation, such as jumping from y-parameters to answer without showing load equation, or stating final Boolean form without simplification steps	Minimal working shown, mostly final answers without derivation, or incorrect steps that don't logically follow; 'answer only' approach with no method visibility
Practical interpretation	20%	10	Interprets results physically: comments on unilateral nature of network in (a), discusses energy concentration in (b), assesses motor efficiency typicality (~80% for small DC motor) and torque-speed implications in (c), notes reasonableness of I_S magnitude (pA to nA range) in (d), and explains gate minimization importance for IC design in (e)	Brief comment on 1-2 sub-parts without depth, such as noting efficiency value is reasonable, or mentioning that fewer gates reduce cost	No physical interpretation provided; purely mathematical treatment without connecting to engineering significance or real-world device behavior

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