Electrical Engineering 2024 Paper I 50 marks Solve

Q3

(a) Determine the time signal that corresponds to the following bilateral Laplace transform and the ROCs given below by using the method of partial fractions: $$X(s) = \frac{4s^2 + 8s + 10}{(s+2)(s^2 + 2s + 5)}$$ (i) With ROC Re(s) < -2 (ii) With ROC Re(s) > -1 (iii) With ROC -2 < Re(s) < -1 (b) Explain the working of the given OPAMP circuit. Draw the output waveforms at points A and B showing the time and voltage. Given that, $V_{Z_1} = V_{Z_2} = 3.3$ V, $C_1 = 1 \mu$F, the power supply voltage to OPAMPs is $\pm 12$ V and $R_1 = R_2 = R_3 = R_4 = R_5 = R_6 = 2$ k$\Omega$ : Suggest to replace suitable resistances so that the output voltages at A and B are having swing of $\pm 6$ V and the output frequency is fixed to 500 Hz. (c) Find the logic equations for the outputs in the concise form and write the corresponding truth table for the circuit given below :

हिंदी में प्रश्न पढ़ें

(a) नीचे दिए गए द्विपक्षीय लाप्लास रूपांतर और ROCs के अनुरूप समय संकेत, आंशिक भिन्न विधि प्रयोग करते हुए ज्ञात कीजिए : $$X(s) = \frac{4s^2 + 8s + 10}{(s+2)(s^2 + 2s + 5)}$$ (i) ROC Re(s) < -2 के साथ (ii) ROC Re(s) > -1 के साथ (iii) ROC -2 < Re(s) < -1 के साथ (b) दिए गए OPAMP परिपथ की कार्यप्रणाली की व्याख्या कीजिए। समय व बोल्टता प्रदर्शित करते हुए बिंदु A तथा B पर निर्गत तरंग रूप आरेखित कीजिए। दिया गया है, $V_{Z_1} = V_{Z_2} = 3.3$ V, $C_1 = 1 \mu$F, OPAMP को प्रदत्त शक्ति प्रदाय बोल्टता $\pm 12$ V है तथा $R_1 = R_2 = R_3 = R_4 = R_5 = R_6 = 2$ k$\Omega$ है : उपयुक्त प्रतिरोधों में बदलाव प्रस्तावित कीजिए ताकि A और B पर निर्गत बोल्टता परास $\pm 6$ V हो जाए तथा निर्गत आवृत्ति 500 Hz हो जाए। (c) नीचे दिए गए परिपथ के आउटपुट के लिए संक्षिप्त तार्किक समीकरण निकालिए और तदनुरूप सत्य-सारणी लिखिए :

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Approach

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) for partial fraction decomposition and three ROC analyses, 35% to part (b) for OPAMP circuit analysis, waveform sketching and redesign calculations, and 25% to part (c) for logic equation derivation and truth table construction. Begin with clear partial fraction expansion in (a), apply bilateral Laplace transform pairs correctly for each ROC, then explain the astable multivibrator operation in (b) with proper timing calculations, and finally use Boolean algebra or K-map simplification in (c) before presenting the truth table.

Key points expected

  • Part (a): Correct partial fraction decomposition of X(s) into A/(s+2) + (Bs+C)/(s²+2s+5) with A=2, B=2, C=0, yielding poles at s=-2 and s=-1±2j
  • Part (a)(i)-(iii): Correct time-domain signals for each ROC — left-sided for Re(s)<-2, right-sided for Re(s)>-1, and two-sided for -2<Re(s)<-1 with proper handling of causal/anti-causal components
  • Part (b): Identification of the circuit as an astable multivibrator (square wave generator) with Schmitt trigger and RC timing network, correct threshold voltages ±βVz where β=R2/(R1+R2)
  • Part (b): Correct waveform sketches showing square waves at A (±Vz=±3.3V) and B (±Vsat≈±12V) with proper time periods, and redesigned values for ±6V swing at 500Hz using R5=R6=1kΩ or modified timing resistors
  • Part (c): Derivation of simplified logic equations using Boolean algebra or K-maps, identification of circuit as a 3-bit binary to Gray code converter or similar standard combinational circuit
  • Part (c): Complete truth table with all 8 input combinations showing correct output values for the derived logic equations

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Demonstrates mastery of bilateral Laplace transform theory (ROC determines causality), astable multivibrator operation with hysteresis, and combinational logic analysis; correctly identifies that complex conjugate poles in (a) yield damped sinusoids, that β=R2/(R1+R2) sets thresholds in (b), and applies De Morgan's laws or K-maps effectively in (c)Shows basic understanding of unilateral Laplace transforms but confuses ROC implications in (a); identifies OPAMP as oscillator but miscalculates thresholds or frequency; derives some logic equations correctly but misses simplification opportunities in (c)Treats all ROCs identically in (a), confuses Schmitt trigger with integrator in (b), or fails to recognize basic logic gates in (c); fundamental conceptual errors in multiple parts
Numerical accuracy20%10All calculations precise: partial fraction coefficients A=2, B=2, C=0 verified; time constants T=2RC·ln[(1+β)/(1-β)] computed accurately; redesigned resistances calculated for exact 500Hz with ±6V swing; logic minimization verified with truth tableMinor arithmetic errors in partial fractions or time constant calculations; correct methodology but wrong final values for redesigned components; truth table mostly correct with 1-2 entry errorsMajor calculation errors in residue computation, frequency formula application, or logic function evaluation; inconsistent numerical results across the answer
Diagram quality20%10Clear, labeled circuit diagrams with all component values; waveforms in (b) show proper voltage levels (±3.3V at A, ±12V at B), time axis with T/2 markings, and charging/discharging exponential sketches; s-plane ROCs sketched for all three cases in (a)Diagrams present but missing some labels or incorrect voltage levels shown; waveforms recognizable but missing time/voltage annotations; minimal or no s-plane sketchesMissing circuit diagram or unrecognizable waveforms; no indication of time scales or voltage levels; diagrams fail to convey circuit operation
Step-by-step derivation20%10Systematic presentation: partial fraction setup with coefficient comparison, clear ROC-to-causality mapping for each case, complete charging equation derivation for multivibrator timing, explicit threshold calculation, and step-by-step Boolean simplification with K-map or algebraic manipulation shownSome steps shown but gaps in derivation; jumps from setup to answer without intermediate working; correct final answers but unclear how obtainedNo derivations shown—only final answers stated; or incorrect/incomplete steps that don't lead to stated results
Practical interpretation20%10Relates bilateral Laplace to signal processing applications (stable/unstable system identification); discusses practical limitations of zener-based multivibrators (frequency stability, temperature drift); identifies the logic circuit's practical use (e.g., in ADCs, error detection); suggests component tolerances for 500Hz stabilityBrief mention of practical relevance without elaboration; generic statements about OPAMP applications or logic circuits without specific connection to the given circuitsNo practical context provided; purely mathematical treatment without engineering insight; fails to recognize real-world significance of ROC choices or circuit configurations

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