Electrical Engineering

UPSC Electrical Engineering 2025 — Paper I

All 8 questions from UPSC Civil Services Mains Electrical Engineering 2025 Paper I (400 marks total). Every stem reproduced in full, with directive-word analysis, marks, word limits, and answer-approach pointers.

8Questions

400Total marks

2025Year

Paper IPaper

Topics covered

Circuit analysis, signals and systems, electronics (1)RC circuits, op-amp oscillators, Z-transform (1)Boolean functions, transistor amplifiers, LTI systems (1)Schottky transistor, Fourier transform, maximum power transfer (1)Electromagnetics, transformers, power electronics, probability, communication systems (1)DC machines, power electronics, communication systems (1)Electromagnetic waves, induction motor, power factor correction, DC-DC converter (1)Smith chart, wave reflection, AM modulation, inverter voltage control (1)

A

50M Compulsory solve Circuit analysis, signals and systems, electronics

(a) In the circuit given below, find the voltages at point A and point B. 10 marks (b) Determine the time domain signal x(t) corresponding to the DTFT given below : 10 marks (c) Draw the output voltage waveform of the circuit given below for 5 V, 50 Hz ac rms input. Forward voltage drop in diode D₁ is 0·6 V. 10 marks (d) Design a sequential circuit with two D flip flops A and B and one input X. Let the state of the circuit remain the same for X = 0. However, when X = 1, the circuit goes through the state transitions from 00 to 10 to 11 to 01, back to 00 and then repeats. 10 marks (e) Calculate Z-parameters for the two-port network given in the circuit diagram. 10 marks

हिंदी में पढ़ें

(a) नीचे दिए गए परिपथ में, बिंदु A तथा बिंदु B पर वोल्टता ज्ञात कीजिए। 10 (b) नीचे दिए गए DTFT के संगत समय-प्रक्षेत्र (डोमेन) संकेत x(t) को ज्ञात कीजिए। 10 (c) 5 V, 50 Hz ac rms निवेश के लिए नीचे दिए गए परिपथ का निर्गत वोल्टता तरंगरूप खींचिए। डायोड D₁ में अग्र वोल्टता अवपात 0·6 V है। 10 (d) दो D फ्लिप-फ्लॉपों A एवं B के साथ एक X निवेश वाला एक अनुक्रमिक परिपथ अभिकल्पित कीजिए। मान लीजिए कि X = 0 के लिए परिपथ अपरिवर्तित रहता है। तथापि जब X = 1 हो, तो परिपथ 00 से 10 से 11 से 01, पुनः 00 अवस्था संक्रमणों से गुजरता है तथा फिर यही दोहराता है। 10 (e) परिपथ आरेख में दिए गए द्वि-पोर्ट जालक्रम के लिए Z-प्राचलों की गणना कीजिए। 10

Answer approach & key points

Solve all five sub-parts systematically, allocating approximately 20% time to each part since all carry equal marks. For (a), apply KCL/KVL or nodal analysis; for (b), use inverse DTFT properties; for (c), sketch the rectified waveform showing clipping at 0.6V; for (d), construct state table and derive D flip-flop excitation equations; for (e), use open-circuit impedance measurements. Present each solution with clear circuit diagrams, mathematical steps, and final boxed answers.

Part (a): Correct application of nodal analysis or superposition theorem to find VA and VB with proper sign conventions
Part (b): Accurate inverse DTFT calculation using synthesis equation or standard transform pairs, with proper handling of periodicity
Part (c): Correct peak voltage calculation (5√2 V), identification of conduction angle, and waveform showing 0.6V offset during positive half-cycles
Part (d): Complete state diagram, state table, excitation table for D flip-flops, and minimized Boolean expressions for DA and DB
Part (e): Correct application of Z-parameter definitions (z11=V1/I1 at I2=0, etc.) with proper mesh or nodal analysis of the two-port network
All parts: Proper unit handling (volts, ohms, seconds, Hz) and significant figures appropriate for engineering calculations
Diagrams: Clear labeling of components, nodes, reference directions, and waveform axes with time/voltage scales
Sequential design: Verification that the designed circuit returns to state 00 after 01 when X=1, maintaining self-starting property

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50M solve RC circuits, op-amp oscillators, Z-transform

(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 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

Answer approach & key points

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.

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)

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50M solve Boolean functions, transistor amplifiers, LTI systems

(a) Consider the Boolean function: F(A, B, C, D) = Σ m (1, 3, 4, 11, 12, 13, 14, 15). Implement it with a 4-to-1 multiplexer and external gates. Connect inputs A and B to the selection lines. Input to the four data lines is a function of the variables C and D which are obtained by expressing F as a function of C and D for each of the four cases when AB = 00, 01, 10 and 11. Functions are to be implemented with external gates. (20 marks) (b) In the circuit given below, transistors T₁ and T₂ are having V_BE = 0.6 V and β = 499. (i) Calculate small signal ac voltage gain of the amplifier at 20 Hz and 2 kHz. (ii) Find dc voltages on collectors of transistors T₁ and T₂ respectively. (20 marks) (c) Impulse response of an LTI system, h(n) is defined in the interval N₀ ≤ n ≤ N₁. If the input x(n) to the LTI system is zero except in the interval N₂ ≤ n ≤ N₃, find the interval for which the output y(n) exists in forms of N₀, N₁, N₂ and N₃. (10 marks)

हिंदी में पढ़ें

(a) बूलिय फलन F(A, B, C, D) = Σ m (1, 3, 4, 11, 12, 13, 14, 15) पर विचार कीजिए। इसका 4 से 1 बहुसंकेतक तथा बाह्य कपाटों (गेट्स) से कार्यान्वयन कीजिए। निवेश A तथा B को चयन पंक्तियों से संयोजित कीजिए। चारों आंकड़ा लाइनों में निवेश, चर C और D का फलन है जिसे प्रत्येक चारों परिस्थितियों AB = 00, 01, 10 तथा 11 में F को C और D के फलन के रूप में व्यक्त कर प्राप्त किया जाता है। फलन का कार्यान्वयन बाह्य कपाटों (गेटों) द्वारा करना है। (20 अंक) (b) नीचे दिए गए परिपथ में, ट्रांजिस्टर T₁ तथा T₂ के लिए V_BE = 0.6 V और β = 499 है। (i) प्रवर्धक की 20 Hz और 2 kHz पर लघु संकेत ac वोल्टता लाभ संगणित कीजिए। (ii) क्रमशः ट्रांजिस्टर T₁ तथा T₂ के कलेक्टरों पर dc वोल्टता ज्ञात कीजिए। (20 अंक) (c) एक LTI तंत्र की आवेग अनुक्रिया h(n), अंतराल N₀ ≤ n ≤ N₁ में परिभाषित है। यदि अंतराल N₂ ≤ n ≤ N₃ को छोड़कर LTI तंत्र में निवेश x(n) शून्य है, तो वह अंतराल ज्ञात कीजिए जिसके लिए निर्गत y(n) का अस्तित्व N₀, N₁, N₂ और N₃ के फलन के रूप में है। (10 अंक)

Answer approach & key points

Solve this multi-part problem by allocating time proportionally to marks: approximately 40% on part (a) multiplexer design, 40% on part (b) transistor amplifier analysis, and 20% on part (c) LTI system interval calculation. Begin with clear K-map derivation for (a), proceed to complete DC and AC analysis for (b) including frequency-dependent gain calculations, and conclude with rigorous mathematical derivation of convolution intervals for (c).

For (a): Construct 4-variable K-map for F(A,B,C,D) = Σm(1,3,4,11,12,13,14,15), group minterms, and express F as function of C,D for each AB combination (00,01,10,11) to determine multiplexer data inputs
For (a): Draw 4-to-1 MUX with A,B as select lines and implement derived C,D functions using external AND/OR/NOT gates for each data input
For (b)(i): Calculate DC operating point (ICQ, VCEQ), then determine small-signal parameters (gm, rπ), and compute voltage gain at 20 Hz (considering coupling/bypass capacitor effects) and 2 kHz (mid-band)
For (b)(ii): Determine VC1 and VC2 using KVL analysis with given VBE = 0.6V and β = 499, accounting for transistor biasing network
For (c): Apply convolution sum property y(n) = x(n)*h(n) to derive output interval [N₀+N₂, N₁+N₃] with proper justification using support interval mathematics

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50M solve Schottky transistor, Fourier transform, maximum power transfer

(a) For the Schottky transistor circuit shown below, determine I_B, I_D, I_C and V_CE. Next, remove the Schottky diode and determine I_B, I_D, I_C and V_CE assuming additional values of V_BE (sat.) = 0.8 V and V_CE (sat.) = 0.1 V. Assume parameter values of β = 50, V_BE (on) = 0.7 V and V_f = 0.3 V for the Schottky diode. (20 marks) (b) Find the Fourier transform of the following signals: (i) x(t) = [2sin(3πt)/πt] · [sin(2πt)/πt] (ii) x(t) = ∫_{-∞}^{t} [sin(2πt)/πt] dt Specify the properties used. (20 marks) (c) In the circuit shown below, V_s is the ac voltage source given by V_s = V_0 cos ωt, with V_0 = 14.14 V and ω = 300 rad/sec. Calculate the value of load resistance R_L for maximum power transfer and also find out maximum power transferred to load. k = 1, n = 0.2 (Turns Ratio) (10 marks)

हिंदी में पढ़ें

(a) नीचे प्रदर्शित शॉटकी ट्रांजिस्टर परिपथ के लिए I_B, I_D, I_C तथा V_CE के मान निर्धारित कीजिए। फिर, परिपथ से शॉटकी डायोड निकाल कर पुनः I_B, I_D, I_C और V_CE के मान निर्धारित कीजिए। मान लीजिए कि V_BE (sat.) = 0.8 V और V_CE (sat.) = 0.1 V के अतिरिक्त मान हैं तथा शॉटकी डायोड के लिए प्राचलों के मान β = 50, V_BE (on) = 0.7 V और V_f = 0.3 V हैं। (20 अंक) (b) निम्नलिखित संकेतों के फुरिये रूपांतर ज्ञात कीजिए: (i) x(t) = [2sin(3πt)/πt] · [sin(2πt)/πt] (ii) x(t) = ∫_{-∞}^{t} [sin(2πt)/πt] dt प्रयुक्त गुणधर्म निर्दिष्ट कीजिए। (20 अंक) (c) नीचे प्रदर्शित परिपथ में V_s एक ac वोल्टता स्रोत है जिसका मान V_s = V_0 cos ωt है, तथा V_0 = 14.14 V और ω = 300 rad/sec. है। अधिकतम शक्ति अंतरण के लिए भार प्रतिरोध R_L के मान की गणना कीजिए और भार में अंतरित अधिकतम शक्ति भी ज्ञात कीजिए। k = 1, n = 0.2 (फेरा अनुपात) (10 अंक)

Answer approach & key points

Solve this multi-part numerical problem by allocating approximately 40% of effort to part (a) Schottky transistor analysis (20 marks), 40% to part (b) Fourier transform computations (20 marks), and 20% to part (c) maximum power transfer (10 marks). Begin with clear circuit diagrams for parts (a) and (c), then proceed with systematic calculations showing all intermediate steps. For part (b), explicitly state each Fourier property used before applying it. Conclude each sub-part with boxed final answers and brief physical interpretations.

Part (a): Correct determination of I_B, I_D, I_C, V_CE with Schottky diode clamping (V_f = 0.3V), then recalculation without Schottky showing deep saturation (V_BE(sat)=0.8V, V_CE(sat)=0.1V)
Part (b)(i): Application of multiplication-convolution duality property to find FT of product of two sinc functions, yielding triangular convolution of rectangular pulses in frequency domain
Part (b)(ii): Use of time-integration property of FT (division by jω in frequency domain plus πδ(ω) term) applied to sinc function, recognizing integral of sinc as step response
Part (c): Calculation of reflected impedance through coupled inductors (k=1, n=0.2), determination of Thevenin equivalent seen by load, and application of conjugate matching for maximum power transfer at ω=300 rad/s
Explicit statement of Fourier properties used: multiplication-convolution duality for (b)(i), time-integration property for (b)(ii)

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B

50M Compulsory solve Electromagnetics, transformers, power electronics, probability, communication systems

(a) As shown in the figure, just inside the surface of a dielectric slab, the electric field (E₁) is 15 V/m and it makes an angle of 30° with the surface. The electric field (E₂) makes 65.5° angle with the surface, just above the surface. Determine the magnitude of E₂ and the dielectric constant of the slab. (10 marks) (b) Show with the help of suitable derivations that the voltage regulation of a transformer varies with the power factor of the load. At what power factor will the voltage regulation be : (i) zero, and (ii) maximum ? (10 marks) (c) A single-phase Thyristor converter circuit as shown in the figure is feeding to a constant current load of 10 A. The supply voltage is of 230 V, 50 Hz and source inductance of 2 mH. Assume the Thyristors are ideal and triggering angle α = 30°. Calculate (i) the overlap angle u, and (ii) the drop in output voltage. (10 marks) (d) Show that for a binomial random variable, the mean is given by np and the variance is given by np (1 – p), where n gives the number of trials and p gives the probability of successes. (10 marks) (e) The frequency range of operation of a superheterodyne FM receiver is 88 MHz – 108 MHz. The centre frequency of the IF amplifier (f_IF) and the frequency of the local oscillator (f_LO) are so chosen that f_IF < f_LO. The design has to be so carried out that the image frequency f'_c falls outside of the 88 MHz – 108 MHz region. Determine the minimum required value of f_IF and the corresponding range of variations in f_LO for that chosen value of f_IF. (10 marks)

हिंदी में पढ़ें

(a) जैसा कि यहाँ चित्र में प्रदर्शित है, एक परावैद्युत गुटके (स्लैब) की सतह के ठीक अंदर विद्युत-क्षेत्र E₁ = 15 V/m है और यह सतह से 30° का कोण बनाता है। सतह के ठीक ऊपर विद्युत-क्षेत्र (E₂) सतह से 65.5° का कोण बनाता है। E₂ का परिमाण तथा गुटके (स्लैब) का परावैद्युतांक निर्धारित कीजिए। (10 अंक) (b) उपयुक्त न्युतियों की सहायता से दर्शाइए कि एक परिणामित्र का वोल्टता नियमन भार के शक्ति गुणक के साथ बदलता है। किस शक्ति गुणक पर वोल्टता नियमन, (i) शून्य होगा, एवं (ii) अधिकतम होगा? (10 अंक) (c) चित्र में दर्शाए गए अनुसार एक, एककलीय थाइरिस्टर परिवर्तित्र परिपथ एक 10 A के स्थिर धारा भार को पूरित करता है। प्रदाय वोल्टता 230 V, 50 Hz और स्रोत प्रेरकत्व 2 mH है। मान लीजिए कि थाइरिस्टर आदर्श हैं तथा ट्रिगर कोण α = 30° है, तो (i) अतिव्यापन कोण u, और (ii) निर्गत वोल्टता में अवपातन की गणना कीजिए। (10 अंक) (d) दर्शाइए कि एक द्विपद यादृच्छिक चर के लिए माध्य np द्वारा तथा प्रसरण np (1 – p) द्वारा निर्धारित होता है, जहाँ n परीक्षणों की संख्या तथा p सफलताओं की संभावना (प्रायिकता) प्रदर्शित कर रहे हैं। (10 अंक) (e) एक सुपरहेटेरोडाइन FM अभिग्राहित की कार्यशीलता का आवृत्ति परास 88 MHz – 108 MHz है। IF प्रवर्धक की मध्य आवृत्ति (f_IF) तथा स्थानीय दोलित्र की आवृत्ति (f_LO) इस प्रकार चयनित हैं कि f_IF < f_LO है। अभिकल्पना इस प्रकार की जानी है ताकि प्रतिबिंब आवृत्ति f'_c, 88 MHz – 108 MHz क्षेत्र के बाहर हो। f_IF का न्यूनतम आवश्यक मान और उस चुने गए f_IF के मान के लिए f_LO के संबंधित बदलाव का परास निर्धारित कीजिए। (10 अंक)

Answer approach & key points

This is a multi-part problem-solving question requiring equal attention to all five sub-parts (10 marks each). Begin with a brief introduction acknowledging the diverse topics covered. For part (a), apply boundary conditions for electric fields at dielectric interfaces. For part (b), derive the voltage regulation expression using transformer equivalent circuit. For part (c), analyze the single-phase converter with source inductance considering overlap. For part (d), prove mean and variance using binomial distribution properties. For part (e), apply superheterodyne receiver principles to determine IF and LO ranges. Allocate approximately equal time (~18-20 minutes) per sub-part, presenting each solution clearly with proper headings.

Part (a): Apply tangential E continuity (E₁sinθ₁ = E₂sinθ₂) and normal D continuity (ε₁E₁cosθ₁ = ε₂E₂cosθ₂) to find εᵣ = 3.0 and E₂ = 8.66 V/m
Part (b): Derive voltage regulation as %R = (I₂R₀₂cosφ ± I₂X₀₂sinφ)/V₂ × 100, showing zero regulation at leading pf = R₀₂/√(R₀₂²+X₀₂²) and maximum at lagging unity pf
Part (c): Calculate overlap angle u = 4.2° using cos(α+u) = cosα - (2ωLₛI₀)/Vₘ, and voltage drop ΔV₀ = (ωLₛI₀/π) = 2 V
Part (d): Prove E[X] = np using Σk·ⁿCₖpᵏqⁿ⁻ᵏ = np(p+q)ⁿ⁻¹ = np, and Var(X) = np(1-p) using E[X²] - (E[X])²
Part (e): Determine f_IF(min) = 10 MHz ensuring image frequency f'c = f_LO + f_IF = fc + 2f_IF > 108 MHz, giving LO range 98-118 MHz

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50M solve DC machines, power electronics, communication systems

(a) (i) What is meant by armature reaction in DC machines ? Show with the help of developed view of armature conductors and poles that the effect of armature m.m.f. on the main field is entirely cross-magnetizing. (10 marks) (ii) A 10 kW, 220 V DC shunt motor draws a line current of 5 A while running at no-load speed of 1200 rpm. It has an armature resistance of 0·2 Ω and field resistance of 200 Ω. Determine the efficiency of the motor when it delivers rated load. (10 marks) (b) A converter circuit as shown in the figure is being used to charge a battery of voltage E = 24 V. The average charging current I_dc = 6 A, and supply voltage V_s = 60 V, 50 Hz. Determine (i) the value of limiting resistor 'R', and (ii) input power factor. (20 marks) (c) A DSB-SC amplitude-modulated signal with power spectral density as shown in figure (a) is corrupted with additive noise that has a power spectral density (N_0/2) within the passband region of the signal. The received signal-plus-noise is demodulated and low pass filtered as shown in figure (b). Determine the SNR at the output of the LPF. [BW : bandwidth] [Given : carrier signal = cos (2πf_c t)] (20 marks)

हिंदी में पढ़ें

(a) (i) डीसी मशीनों में आर्मेचर प्रतिक्रिया का क्या मतलब है ? आर्मेचर सुचालकों और ध्रुवों के विस्तृत दृश्य की सहायता से यह प्रदर्शित कीजिए कि आर्मेचर m.m.f. का मुख्य क्षेत्र पर प्रभाव पूर्णतः अनुप्रस्थ-चुंबकीय (क्रॉस-मैग्नेटाइजिंग) है। (10 अंक) (ii) एक 10 kW, 220 V DC शंट मोटर भार-रहित 1200 rpm गति पर चलते हुए 5 A लाइन धारा लेती है। इसका आर्मेचर प्रतिरोध 0·2 Ω तथा क्षेत्र प्रतिरोध 200 Ω है। निर्धिष्ट भार प्रदाय करते समय इस मोटर की कार्य-दक्षता ज्ञात कीजिए। (10 अंक) (b) जैसा कि चित्र में दर्शाया गया है, एक परिवर्तित परिपथ, E = 24 V वोल्टता की एक बैटरी को चार्ज करने के लिए प्रयोग किया जा रहा है। औसत चार्जिंग धारा I_dc = 6 A, तथा प्रदाय वोल्टता V_s = 60 V, 50 Hz है, तो : (i) सीमांत प्रतिरोध 'R' का मान, और (ii) निवेश शक्ति गुणांक ज्ञात कीजिए। (20 अंक) (c) चित्र (a) में प्रदर्शित शक्ति स्पेक्ट्रमी घनत्व वाला एक DSB-SC आयाम-मॉडुलित संकेत एक ऐसे योज्य रव (नॉइस) द्वारा विकृत होता है जिसका इस संकेत के पास-बैंड क्षेत्र में शक्ति स्पेक्ट्रमी घनत्व (N_0/2) है। प्राप्त संकेत-धन-रव को डिमॉडुलित और निम्न पारक छानित किया जाता है, जैसा कि चित्र (b) में प्रदर्शित है। LPF के निर्गम पर SNR ज्ञात कीजिए। [BW : बैंड चौड़ाई] [दिया गया है : वाहक संकेत = cos (2πf_c t)] (20 अंक)

Answer approach & key points

Begin with a concise definition of armature reaction and its cross-magnetizing nature for part (a)(i), followed by systematic numerical solution for the DC motor efficiency in (a)(ii). For part (b), apply thyristor converter analysis to determine the firing angle, limiting resistor, and input power factor. For part (c), derive the output SNR for DSB-SC demodulation using coherent detection theory. Allocate approximately 15 minutes to (a)(i), 20 minutes to (a)(ii), 35 minutes to (b), and 35 minutes to (c), ensuring all diagrams are neatly drawn with proper labeling.

Definition of armature reaction as the effect of armature MMF on main field flux distribution in DC machines
Developed winding diagram showing armature conductors under N and S poles with current directions proving cross-magnetizing axis is perpendicular to main field axis
Calculation of no-load losses, field current, back EMF, and efficiency at rated load for the DC shunt motor
Determination of firing angle, average output voltage, and limiting resistor R for the battery charging converter circuit
Computation of input power factor considering displacement angle and distortion factor in the controlled rectifier
Expression for output SNR of DSB-SC system with coherent detection showing dependence on signal power spectral density and noise PSD
Integration of signal and noise power over the message bandwidth to obtain final SNR formula
Comparison of DSB-SC SNR improvement over conventional AM highlighting the 3 dB advantage due to suppressed carrier

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50M solve Electromagnetic waves, induction motor, power factor correction, DC-DC converter

(a) It is given that $\vec{E} = E_m \sin (\omega t - \alpha z) \hat{a}_y$ in free space $\alpha > 0$. (i) Determine $\vec{D}$, $\vec{B}$ and $\vec{H}$. Plot $\vec{E}$ and $\vec{H}$ at $t = 0$. State clearly if any assumption is made. (ii) Show that these $\vec{E}$ and $\vec{H}$ fields constitute a wave travelling in the z-direction. Also demonstrate that the wave speed and E/H depend solely on the properties of free space. Given : $\mu_0 = 4\pi \times 10^{-7}$ H/m, and $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m. (b) (i) A 3-phase, 4-pole, 400 V, 10 kW, 50 Hz slip ring induction motor develops rated output at rated voltage and frequency with its slip ring short-circuited. The maximum torque equal to twice the full load torque, occurs at a slip of 12·5% with zero external resistance in rotor circuit. Neglect stator impedance, stator core and mechanical losses. Determine : I. slip and motor speed at full load torque, and II. starting current in terms of full load current. (10 marks) (ii) An industry has an average electrical load of 600 kW at a p.f. of 0·6 lagging. A synchronous motor with an efficiency of 90% is used to raise the combined p.f. to 0·9 lagging and at the same time supply a mechanical load of 100 kW. Calculate kVA capacity of the synchronous motor and synchronous motor operating power factor. (10 marks) (c) The buck-boost converter has an input voltage of $V_s = 12$ V. The duty cycle $D = 0·25$ and the switching frequency is 20 kHz. The inductance $L = 150$ μH and filter capacitor $C = 250$ μF. The average load current $I_0 = 1·25$ A. Determine : (i) the peak-to-peak ripple in the inductor current, and (ii) the critical values of inductor L and capacitor C for CCM. (10 marks)

हिंदी में पढ़ें

(a) दिया गया है कि $\vec{E} = E_m \sin (\omega t - \alpha z) \hat{a}_y$ और मुक्त अंतराल (फ्री-स्पेस) में $\alpha > 0$ है। (i) $\vec{D}$, $\vec{B}$ और $\vec{H}$ ज्ञात कीजिए। $t = 0$ पर $\vec{E}$ और $\vec{H}$ का अंकन कीजिए। यदि कोई कल्पना है तो स्पष्ट उल्लेख कीजिए। (ii) दर्शाइए कि यह $\vec{E}$ और $\vec{H}$ क्षेत्र, z-दिशा में सरण (गमन) करती हुई एक तरंग निर्मित करते हैं। यह भी दर्शाइए कि तरंग गति और E/H पूर्णतः मुक्त अंतराल के गुणधर्मों पर आधारित है। दिया गया है : $\mu_0 = 4\pi \times 10^{-7}$ H/m और $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m. (b) (i) एक 3-कला, 4-ध्रुव, 400 V, 10 kW, 50 Hz सर्पी बलय प्रेरण मोटर, सर्पी बलय लघुपथित होने की दशा में निर्धारित (रेटेड) वोल्टता और आवृत्ति पर निर्धारित (रेटेड) निर्गत विकसित करती है। अधिकतम बल-आघूर्ण जो निर्धारित पूर्ण भार बल-आघूर्ण से 2 गुना है, 12·5% सर्पण पर उत्पन्न होता है जब कि रोटर परिपथ में बाह्य प्रतिरोध शून्य है। स्टेटर प्रतिबाधा, स्टेटर कोर तथा यांत्रिक क्षतियाँ नगण्य मानते हुए, I. पूर्ण भार बल-आघूर्ण पर सर्पण और मोटर की गति, तथा II. पूर्ण भार धारा के पदों में प्रवर्तन धारा का मान ज्ञात कीजिए। (10 अंक) (ii) एक उद्योग का 0·6 पश्चगामी p.f. पर औसत विद्युत भार 600 kW है। संयुक्त p.f. को 0·9 पश्चगामी तक बढ़ाने के लिए 90% दक्षता वाली एक तुल्यकालिक मोटर प्रयोग की जाती है, जो साथ ही साथ 100 kW यांत्रिक भार भी प्रदाय करती है। तुल्यकालिक मोटर की kVA धारिता और कार्यकारी (ऑपरेटिंग) शक्ति गुणांक की गणना कीजिए। (10 अंक) (c) एक बक-बूस्ट परिवर्तित्र में निवेश वोल्टता $V_s = 12$ V है। कर्म चक्र $D = 0·25$ तथा स्विचन आवृत्ति 20 kHz है। प्रेरक $L = 150$ μH तथा छनक संधारित्र $C = 250$ μF और औसत भार धारा $I_0 = 1·25$ A है, तो (i) प्रेरक धारा में शिखर-से-शिखर झ्रिमिका का मान, और (ii) CCM के लिए प्रेरक L और संधारित्र C के क्रांतिक मानों की गणना कीजिए। (10 अंक)

Answer approach & key points

This is a multi-part numerical problem requiring systematic solution of electromagnetic wave propagation, induction motor characteristics, power factor correction, and DC-DC converter analysis. Allocate approximately 35% time to part (a) on EM waves (20 marks), 35% to part (b) on machines and power systems (20 marks), and 30% to part (c) on power electronics (10 marks). Begin each sub-part with stated assumptions, proceed with clear derivations, and conclude with numerical answers in proper units.

Part (a)(i): Correct application of constitutive relations D = ε₀E, B = μ₀H, and Maxwell's equations to derive B = (αEₘ/ω)cos(ωt-αz)âₓ and H = B/μ₀; proper sinusoidal plots of E and H at t=0 showing 90° spatial phase difference
Part (a)(ii): Demonstration that (ωt-αz) = constant implies phase velocity vₚ = ω/α = 1/√(μ₀ε₀) = c ≈ 3×10⁸ m/s; proof that |E|/|H| = √(μ₀/ε₀) = η₀ ≈ 377Ω (intrinsic impedance of free space)
Part (b)(i): Using torque-slip relation T ∝ sR₂/(R₂²+(sX₂)²), determination of full-load slip s_FL = 0.05 (5%) giving speed = 1425 rpm; starting current ratio I_st/I_FL = 2.5 using equivalent circuit with neglected stator impedance
Part (b)(ii): Calculation of reactive power compensation where original load has 800 kVAR lagging; synchronous motor must draw 260.4 kVAR leading to achieve 0.9 lagging combined p.f.; resulting motor kVA = 370.3 kVA at 0.27 leading p.f.
Part (c)(i): Application of buck-boost inductor current ripple formula ΔI_L = DV_s/(Lf) = 0.25×12/(150×10⁻⁶×20×10³) = 1 A peak-to-peak
Part (c)(ii): Critical inductance L_crit = D(1-D)²R/(2f) = 0.25×0.75²×9.6/(2×20×10³) = 33.75 μH; critical capacitance C_crit = (1-D)/(8Lf²×(ΔV₀/V₀)) for specified ripple criterion

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50M derive Smith chart, wave reflection, AM modulation, inverter voltage control

(a) (i) Show that the Smith chart constructed for a lossless transmission line gives a family of r-circles, having a radius of $\frac{1}{(1+r)}$ for each circle which is centred at $\Gamma_r = \frac{r}{(1+r)}$ and $\Gamma_i = 0$. Here, $r$ = normalized resistance of the load impedance, $\Gamma_r$ and $\Gamma_i$ = real and imaginary parts of voltage reflection coefficient of the load impedance, respectively. (10 marks) (ii) In the figure given below, determine the amplitudes of the reflected and transmitted $\vec{E}$ and $\vec{H}$ at the interface, if $E_0^i = 1.2 \times 10^{-3}$ V/m in region 1, where $\varepsilon_{r_1} = 7.5$, $\mu_{r_1} = 1$ and $\sigma_1 = 0$. Given : Region 2 is a free space and assume normal incidence. Also, $\mu_0 = 4\pi \times 10^{-7}$ H/m and $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m. (10 marks) (b) A sinusoidal modulating signal m(t) of frequency $f_m$ produces an AM signal : $u(t) = A_c [1 + \beta \cos (2\pi f_m t)] \cos (2\pi f_c t)$, where $f_c$ is carrier frequency. Here, $f_c >> f_m$ and $\beta = 2$. This u(t) is applied to an ideal envelope detector which produces an output x(t). (i) Determine the Fourier series representation of x(t). (ii) Also determine the ratio of second harmonic amplitude to fundamental amplitude in x(t). (20 marks) (c) Discuss in brief various methods of voltage control within 3-phase inverters. (10 marks)

हिंदी में पढ़ें

(a) (i) दर्शाइए कि एक हानिरहित संचरण (पारेषण) लाइन के लिए निर्मित स्मिथ चार्ट $\frac{1}{(1+r)}$ त्रिज्या वाले r-वृत्तों का एक कुल देता है जिसमें प्रत्येक वृत्त का केन्द्र $\Gamma_r = \frac{r}{(1+r)}$ तथा $\Gamma_i = 0$ पर होता है। यहाँ $r$ = भार प्रतिबाधा का प्रसामान्यीकृत प्रतिरोध तथा $\Gamma_r$ और $\Gamma_i$ क्रमशः भार प्रतिबाधा के वोल्टता परावर्तनांक के वास्तविक और काल्पनिक अंश हैं। (10 अंक) (ii) नीचे दिए गए चित्र में, अंतरापृष्ठ पर $\vec{E}$ तथा $\vec{H}$ के परावर्तित और संचरित आयाम ज्ञात कीजिए। यदि क्षेत्र 1 में, $E_0^i = 1.2 \times 10^{-3}$ V/m है, जहाँ $\varepsilon_{r_1} = 7.5$, $\mu_{r_1} = 1$ और $\sigma_1 = 0$ है। दिया गया है कि क्षेत्र 2 एक मुक्त अंतराल है और लम्बवत् आपतन मान लीजिए तथा $\mu_0 = 4\pi \times 10^{-7}$ H/m और $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m है। (10 अंक) (b) आवृत्ति $f_m$ का एक ज्यावक्रीय मॉडुलन संकेत m(t) एक AM संकेत $u(t) = A_c [1 + \beta \cos (2\pi f_m t)] \cos (2\pi f_c t)$ उत्पन्न करता है, जहाँ $f_c$ वाहक आवृत्ति है। यहाँ $f_c >> f_m$ और $\beta = 2$ है। यह u(t) एक आदर्श आवरण संसूचक पर आरोपित किया जाता है, जो एक निर्गत x(t) उत्पादित करता है। (i) x(t) का फुरिये श्रेणी निरूपण ज्ञात कीजिए। (ii) x(t) में द्वितीय संनादी आयाम से मूल आयाम का अनुपात भी ज्ञात कीजिए। (20 अंक) (c) 3-कला प्रतिपाक (इन्वर्टर) में वोल्टता नियंत्रण की विभिन्न विधियों की संक्षेप में विवेचना कीजिए। (10 अंक)

Answer approach & key points

Begin with the derivation-heavy sub-part (a)(i) by establishing the relationship between normalized impedance and reflection coefficient, then proceed to (a)(ii) for numerical computation of field amplitudes using boundary conditions. Allocate approximately 35% time to part (b) involving Fourier analysis of the AM envelope detector output, 20% to part (a), and 15% to part (c) on inverter voltage control methods. Ensure all derivations show intermediate steps and numerical answers include proper units.

Derivation of r-circle equation on Smith chart starting from Γ = (z_L - 1)/(z_L + 1) and showing algebraic manipulation to obtain circle equation with center at (r/(1+r), 0) and radius 1/(1+r)
Calculation of intrinsic impedances η₁ = √(μ₀μᵣ₁/ε₀εᵣ₁) and η₂ = √(μ₀/ε₀), then reflection coefficient Γ = (η₂ - η₁)/(η₂ + η₁) and transmission coefficient τ = 1 + Γ for normal incidence
Computation of reflected E and H amplitudes: E₀ʳ = ΓE₀ⁱ, H₀ʳ = -ΓE₀ⁱ/η₁ and transmitted E and H amplitudes: E₀ᵗ = τE₀ⁱ, H₀ᵗ = τE₀ⁱ/η₂ with correct numerical values
Fourier series representation of envelope detector output x(t) = |A_c[1 + 2cos(2πf_m t)]| showing handling of negative envelope when β > 1, resulting in rectified waveform with DC, fundamental and harmonic components
Calculation of second harmonic to fundamental amplitude ratio using Fourier coefficients of the periodic rectified cosine waveform, recognizing the waveform becomes periodic with period 1/f_m
Discussion of voltage control methods in 3-phase inverters: PWM techniques (sinusoidal PWM, space vector PWM), selective harmonic elimination, and voltage control through DC link voltage variation with their relative merits

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