Electrical Engineering 2025 Paper I 50 marks Solve

Q2

(a) In the circuit shown in the diagram, initially key K₁ is closed and capacitor has no charge (at time t = 0). Now at time t = 10 seconds, key K₁ is opened and at t = 18·68 seconds it is again closed. Plot output voltage across the capacitor with respect to time and find output voltage values at time 10 seconds, 18·68 seconds and 28·68 seconds. 20 marks (b) Consider the circuit of an operational amplifier given here in which Zener diodes Z₁ and Z₂ are having reverse breakdown voltage = 7·4 V and forward voltage drop = 0·6 V. (i) Draw the output voltage waveform showing voltage value with time and calculate frequency of output waveform. (ii) Modify the circuit for duty cycle factor D = 0·25 by replacing R₁ from combination of suitable resistances and diodes, so that output frequency is not changed. 20 marks (c) Determine the causal signal x[n] if its z-transform X(z) is specified by a pole-zero pattern shown in the figure below. Take the constant G = 1/4. 10 marks

हिंदी में प्रश्न पढ़ें

(a) आरेख में प्रदर्शित परिपथ में, आरंभ में कुंजी K₁ संयोजित है तथा संधारित्र में कोई आवेश नहीं है (समय t = 0 पर)। अब समय t = 10 सेकंड पर कुंजी K₁ को विचोजित कर दिया जाता है और t = 18·68 सेकंड पर पुनः संयोजित कर दिया जाता है। समय के सापेक्ष संधारित्र के आर-पार निर्गत वोल्टता आरेखित कीजिए तथा समय 10 सेकंड, 18·68 सेकंड और 28·68 सेकंड पर निर्गत वोल्टता मान ज्ञात कीजिए। 20 (b) यहाँ दिए गए एक संक्रियात्मक प्रवर्धक के परिपथ पर विचार कीजिए, जिसमें जेनर डायोड Z₁ और Z₂ की प्रतिप भंजन (ब्रेकडाउन) वोल्टता = 7·4 V तथा अग्र वोल्टता अवपातन = 0·6 V है। (i) समय के साथ वोल्टता का मान प्रदर्शित करते हुए, निर्गत वोल्टता तरंगरूप को आरेखित कीजिए तथा निर्गत तरंगरूप की आवृत्ति की गणना कीजिए। (ii) R₁ को उपयुक्त प्रतिरोधों और डायोडों के संयोजन से बदल कर परिपथ को कर्म चक्र गुणांक D = 0·25 के लिए इस प्रकार रूपांतरित कीजिए ताकि निर्गत आवृत्ति अपरिवर्तित रहे। 20 (c) यदि हेतुक संकेत x(n) का z-रूपान्तर X(z) नीचे दिए गए चित्र में प्रदर्शित ध्रुव-शून्यक प्रतिरूप के द्वारा निर्दिष्ट होता है, तो हेतुक संकेत x(n) ज्ञात कीजिए। स्थिरांक G = 1/4 लीजिए। 10

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Approach

Solve this multi-part numerical problem by first analyzing the RC transient circuit in (a) with proper time-constant calculations for charging/discharging phases, then analyze the op-amp astable multivibrator in (b)(i)-(ii) for frequency and duty cycle modification, and finally perform inverse Z-transform for (c). Allocate approximately 40% time to part (a) given its 20 marks and complex multi-interval analysis, 40% to part (b) covering both frequency calculation and circuit redesign, and 20% to part (c) for the pole-zero inversion. Present each part with clear sectional headings, state assumptions explicitly, show all formulae before substitution, and conclude with verified numerical answers.

Key points expected

  • Part (a): Correct application of RC charging equation Vc = V(1-e^(-t/RC)) for 0-10s, discharging equation Vc = V₀e^(-t/RC) for 10-18.68s, and recharging from 18.68s with appropriate initial conditions; identification that 18.68s equals one time constant for discharge
  • Part (b)(i): Recognition of op-amp as astable multivibrator with Zener clamping; correct calculation of threshold voltages using Zener breakdown (7.4V) and forward drop (0.6V); derivation of frequency formula f = 1/(2R₁C₁ln((1+β)/(1-β))) or simplified form with β = R₂/(R₁+R₂)
  • Part (b)(ii): Design of asymmetric timing circuit using parallel branches with diodes to create different charging/discharging resistances while maintaining same total period; selection of resistor values to achieve D = 0.25 (25% duty cycle) with ton = T/4 and toff = 3T/4
  • Part (c): Identification of poles and zeros from given pattern; construction of X(z) = G·(z-z₁)(z-z₂).../((z-p₁)(z-p₂)...); partial fraction expansion and inverse Z-transform using standard pairs; causal sequence verification through ROC analysis (outside outermost pole)
  • Cross-cutting: Proper unit handling, significant figures consistent with given data (2 decimal places for time values), and physical verification of results (e.g., capacitor voltage cannot exceed supply, frequency in practical audio/radio range)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Demonstrates flawless understanding: for (a) recognizes piecewise RC transient behavior with correct initial/final value theorem; for (b) identifies astable multivibrator operation with proper Zener voltage limiting and hysteresis window calculation; for (c) correctly interprets pole-zero plot for causal system with ROC |z| > r_maxShows basic understanding of RC transients and op-amp oscillators but confuses charging/discharging time constants in (a), misapplies threshold voltage formula in (b), or makes sign errors in partial fraction expansion for (c)Fundamental conceptual errors: treats RC circuit as linear time-invariant without interval analysis, confuses astable with monostable operation, or applies bilateral Z-transform instead of unilateral for causal signal
Numerical accuracy20%10All calculations precise to appropriate significant figures: for (a) V(10s), V(18.68s), V(28.68s) exactly match expected values (typically 63.2% of final for one τ); for (b) frequency calculation within 1% tolerance; for (c) exact sequence coefficientsMinor calculation errors in one sub-part (e.g., arithmetic slip in exponent evaluation or resistor ratio), or correct method with final answer off by factor of 2 due to threshold voltage miscalculationMultiple gross errors: wrong time constant extraction, incorrect logarithmic evaluation, or algebraic mistakes leading to physically impossible results (negative voltages, imaginary frequencies)
Diagram quality20%10Professional-quality diagrams: for (a) properly labeled Vc vs t plot showing exponential curves with marked time constants, switching instants, and asymptotes; for (b)(i) square wave with accurate voltage levels (±8V or ±7.4V depending on configuration), marked ton/toff; for (b)(ii) clear modified circuit with diode-resistor branchesRecognizable sketches with basic labels but missing key details: unmarked axes, missing voltage levels on waveforms, or unclear component connections in modified circuitPoor or absent diagrams: rough freehand curves without scale, wrong waveform shapes (sine instead of square for oscillator), or completely missing required circuit modifications
Step-by-step derivation20%10Complete mathematical rigor: states general differential equation for RC circuit, applies appropriate boundary conditions for each time interval in (a); derives frequency formula from capacitor charging equation through threshold crossings in (b); shows partial fraction expansion and residue calculation for (c)Shows key steps but skips some intermediate algebra or assumes standard results without brief derivation; correct final expressions with gaps in logical flowJumps to answers without derivation, writes formulae without defining variables, or presents disconnected equations without logical progression
Practical interpretation20%10Insightful physical interpretation: for (a) explains why 18.68s represents one time constant and verifies capacitor voltage continuity; for (b) discusses practical component selection (standard E12 resistor values), Zener power dissipation, and frequency stability; for (c) verifies causality and identifies signal type (exponential, sinusoidal, etc.)Brief mention of physical significance without elaboration; standard component values mentioned without justification, or missing verification of results against physical constraintsPurely mathematical treatment with no physical insight; no discussion of practical implementation issues, component tolerances, or result validation

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