Electrical Engineering 2025 Paper II 50 marks Solve

Q3

(a) (i) Sketch the approximate root locus plot for a time-delay system approximated by the transfer function $$G(s) = \frac{K\left(1-\frac{s}{2}\right)}{s(s+1)\left(1+\frac{s}{2}\right)}$$ Also compute the largest value of K for which the system is stable under unity feedback. Verify this value from the root locus plot. 10 (ii) The signal flow graph of a system is as shown below : [Diagram] Determine the overall transmission $\frac{R(s)}{Y(s)}$, and evaluate the sensitivity of the output to variations in $K_1$ at $s=10$. What would be the value of sensitivity obtained under DC condition, i.e., $s=0$? 10 (b) A bridge consists of the following configurations : Arm AB : A choke coil of unknown resistance R₁ and unknown inductance L₁ Arm BC : A non-inductive resistance R₃ Arm DA : A non-inductive resistance R₂ Arm CD : A mica condenser with capacitance C₄ in series with a non-inductive resistance R₄ Bridge balance is obtained at 400 Hz with the following component values : R₂ = 2000 Ω, R₃ = 500 Ω, C₄ = 0·2 μF, R₄ = 70·9 Ω Assume that capacitor has a series resistance of 0·1 Ω. Calculate the resistance and inductance of the choke coil. Also sketch the phasor diagram for the bridge under balanced conditions, and evaluate Q factor of the choke coil. 20 (c) Explain maskable interrupt. Draw the timing diagram for the maskable interrupt acknowledgement cycle. List the activities in each clock cycle. 10

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(a) (i) एक काल-विलम्ब (टाइम-डिले) तंत्र, जो कि निम्नांकित अंतरण फलन द्वारा अनुमानित है, के लिये अनुमानित मूल बिन्दुपथ (रूट लोकस) आलेख खींचिये : $$G(s) = \frac{K\left(1-\frac{s}{2}\right)}{s(s+1)\left(1+\frac{s}{2}\right)}$$ K के उस अधिकतम मान की भी गणना कीजिये, जिसके लिये तंत्र, इकाई पुनर्निवेश के तहत स्थिर हो। इस मान की पुष्टि मूल बिन्दुपथ आलेख से भी कीजिये। 10 (ii) एक तंत्र का संकेत प्रवाह आलेख (सिग्नल फ्लो ग्राफ) नीचे दर्शाया गया है : [Diagram] समग्र अंतरण $\frac{R(s)}{Y(s)}$ का निर्धारण कीजिये, और $s=10$ पर $K_1$ में उतार-चढ़ाव के लिये उत्पादन (आउटपुट) की संवेदनशीलता का मान निकालिये। दिष्ट धारा (डी. सी.) स्थिति (यानि $s=0$) के तहत संवेदनशीलता का क्या मान प्राप्त होगा? 10 (b) एक सेतु के निम्नलिखित विन्यास हैं : भुजा AB : एक चोक कुण्डली, जिसके प्रतिरोध R₁ एवं प्रेरकत्व L₁ अज्ञात हैं भुजा BC : एक गैर-प्रेरणिक प्रतिरोध R₃ भुजा DA : एक गैर-प्रेरणिक प्रतिरोध R₂ भुजा CD : एक माइका संधारित्र, जिसकी धारिता C₄ है, के श्रेणीक्रम में एक गैर-प्रेरणिक प्रतिरोध R₄ सेतु का संतुलन 400 Hz पर, घटकों के निम्नांकित मानों के लिये प्राप्त होता है : R₂ = 2000 Ω, R₃ = 500 Ω, C₄ = 0·2 μF, R₄ = 70·9 Ω मान लीजिये कि संधारित्र का श्रेणीक्रम प्रतिरोध 0·1 Ω है। चोक कुण्डली के प्रतिरोध एवं प्रेरकत्व की गणना कीजिये। संतुलन की अवस्था में सेतु का कला (फेजर) आरेख भी रेखांकित कीजिये तथा चोक कुण्डली के Q गुणांक का मान निकालिये। 20 (c) नकाबपोश क्रमभंजक (मास्केबल इंटरप्ट) की व्याख्या कीजिये। नकाबपोश क्रमभंजक के पावती चक्र के लिये काल-आरेख आरेखित कीजिये। प्रत्येक घड़ीका चक्र में गतिविधियों को क्रमबद्ध कीजिये। 10

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Approach

Solve this multi-part technical question by allocating approximately 40% effort to part (a) root locus and signal flow graph (20 marks), 40% to part (b) AC bridge calculations (20 marks), and 20% to part (c) 8085/8086 maskable interrupt explanation (10 marks). Begin with clear problem statements for each sub-part, present systematic derivations with intermediate steps, verify numerical results through cross-checks, and conclude with physical interpretations of stability margins, bridge balance conditions, and interrupt handling in Indian microprocessor-based systems.

Key points expected

  • Part (a)(i): Correct identification of poles at s=0, s=-1, s=-2 and zero at s=2; proper sketch of root locus showing RHP zero effect and asymptotic behavior; Routh-Hurwitz or direct substitution to find K_max=2 for stability
  • Part (a)(ii): Application of Mason's gain formula to signal flow graph; correct forward paths and loop identification; sensitivity calculation ∂T/∂K₁ · K₁/T at s=10 and s=0 showing unity sensitivity at DC
  • Part (b): Correct impedance expressions for each arm; balance condition Z₁Z₄ = Z₂Z₃ with complex algebra; calculation of R₁=500Ω, L₁=0.5H accounting for capacitor series resistance; proper phasor diagram with current and voltage relationships; Q-factor ≈ 6.28
  • Part (c): Clear distinction between maskable (INTR) and non-maskable interrupts; 8085-specific timing diagram showing T1-T4 states with INTA generation; clock-wise listing of PC save, ISR address fetch, and execution transfer
  • Cross-verification: Stability limit K=2 confirmed by root locus crossing imaginary axis; bridge balance verified by phase angle cancellation; interrupt cycle matched to standard 8085 microprocessor textbook specifications

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies RHP zero in (a)(i) causing non-minimum phase behavior; applies Mason's rule accurately in (a)(ii); recognizes Maxwell-Wein bridge configuration in (b); distinguishes 8085 maskable interrupt with proper EI/DI context in (c)Minor errors in pole-zero identification or bridge arm configuration; Mason's rule applied with some loop omissions; basic interrupt concept correct but timing details vagueFundamental errors like treating RHP zero as LHP, incorrect bridge balance equation, or confusing maskable with non-maskable interrupt types
Numerical accuracy20%10Precise calculation: K_max=2.0, R₁=500Ω, L₁=0.5H, Q=6.28; sensitivity values computed correctly at both s=10 and s=0; accounts for 0.1Ω capacitor resistance in bridge balanceCorrect order of magnitude but arithmetic slips; sensitivity formula correct with substitution errors; bridge calculations with minor complex number handling issuesOrder-of-magnitude errors in K, R, L values; incorrect sensitivity formula application; failure to account for capacitor series resistance
Diagram quality20%10Clear root locus with asymptotes, breakaway points, and jω crossing marked; labeled signal flow graph with paths identified; neat bridge phasor diagram showing I₁, I₂, V_AB, V_AD; accurate 8085 interrupt timing with T-states, ALE, INTA, and address bus shownDiagrams present but poorly labeled or missing key features like angle of departure in root locus; phasor diagram without proper phase relationships; timing diagram missing some control signalsMissing diagrams or sketches too rough to interpret; incorrect topology in bridge or signal flow representation; timing diagram confused with normal machine cycle
Step-by-step derivation20%10Systematic Routh array construction for stability; complete Mason's formula expansion with Δ and Δk calculations; explicit complex algebra for bridge balance Z₁ = Z₂Z₃/Z₄; detailed T-state analysis for interrupt acknowledgmentSome steps skipped but key intermediate results shown; Mason's rule applied without showing cofactor calculations; bridge derivation jumps to final answerNo derivation shown—only final answers; or incorrect/irrelevant steps that don't lead to solution; missing critical steps like characteristic equation formation
Practical interpretation20%10Explains physical significance of K_max as stability margin; interprets sensitivity variation with frequency for control system robustness; discusses Q-factor implications for choke coil applications; relates maskable interrupt to priority-based real-time systems like Indian railway signaling or process controlBrief mention of stability or measurement accuracy without elaboration; generic statement about interrupts without application contextNo physical interpretation provided; purely mathematical treatment without connecting to engineering practice or real-world implications

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