(a) For the circuit shown below, calculate the output voltage :
R₁ = 1 kΩ R₂ = 2 kΩ R₃ = 3 kΩ
R₄ = 10 kΩ R₅ = 10 kΩ R₆ = 100 kΩ

V₁ = -1 V V₂ = -2 V V₃ = 8 V

(b) A signal xₐ(t) is band-limited to the range 900 Hz ≤ f ≤ 1100 Hz (assum

Question

(a) For the circuit shown below, calculate the output voltage :
R₁ = 1 kΩ    R₂ = 2 kΩ    R₃ = 3 kΩ
R₄ = 10 kΩ   R₅ = 10 kΩ   R₆ = 100 kΩ

V₁ = -1 V    V₂ = -2 V    V₃ = 8 V

(b) A signal xₐ(t) is band-limited to the range 900 Hz ≤ f ≤ 1100 Hz (assume the shape of an isosceles triangle for continuous Fourier transform and |Xₐ(f)| = 1 and f = 1000 Hz). It is used as an input to the system shown below :

In this system, H(ω) is a low-pass filter with a discrete cut-off frequency equivalent to f꜀ = 125 Hz (normalized w.r.t. the sample rate at the point in the block diagram). Determine and sketch the spectra of X(ωₓ), W(ωᵥ), V(ωᵥ) and Y(ωᵧ) w.r.t. ωₓ, ωᵥ, ωᵥ and ωᵧ respectively for -π < ω < π.

(c) For the system shown in the figure below, the step response of G(s) is given by (1·5 - 2e⁻ᵗ + 0·5e⁻²ᵗ)u(t) and K(s) is the integral controller with K(s) = K/s. Sketch the approximate root locus of the closed-loop system poles as K varies from 0 to ∞. Also calculate the real part of poles when K becomes ∞ :

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating approximately 3 marks worth of effort to each sub-part. For (a), apply superposition or nodal analysis to find the op-amp output voltage. For (b), trace the spectral transformations through sampling, filtering, and decimation stages with clear frequency axis annotations. For (c), derive G(s) from the given step response, then construct the root locus for the integral-controlled system showing asymptotes and breakaway points. Present derivations first, followed by numerical results and clearly labeled sketches.
- Part (a): Correct application of superposition theorem or virtual ground concept for the multi-input op-amp circuit; accurate calculation of contribution from each voltage source V₁, V₂, V₃
- Part (a): Proper handling of resistor ratios (R₄/R₁, R₄/R₂, R₄/R₃) with correct polarity for inverting summer configuration; final output voltage calculation with sign
- Part (b): Correct determination of sampling rate and resulting spectral replicas; sketch of X(ωₓ) showing triangular spectrum centered at ω = 0.2π (normalized) with replicas at 2π intervals
- Part (b): Accurate depiction of low-pass filtering effect on W(ωᵥ), decimation-induced spectral stretching in V(ωᵥ), and final output Y(ωᵧ) with proper frequency axis scaling
- Part (c): Derivation of G(s) = 3(s+1)/[s(s+2)] from the given step response via Laplace transform; identification of poles at s = 0, -2 and zero at s = -1
- Part (c): Construction of root locus showing: branches starting at s = 0 and s = -2, meeting at breakaway point, then becoming complex; asymptotes at ±90°; real part of poles at -1.5 when K → ∞

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	2	Demonstrates correct theoretical foundation: superposition/virtual ground for (a), sampling theorem and spectral effects for (b), inverse Laplace and root locus rules for (c); no conceptual errors in any sub-part	Mostly correct concepts with minor errors: e.g., correct op-amp analysis but wrong polarity; understands sampling but confuses normalized frequency; derives G(s) correctly but applies wrong root locus rule	Fundamental conceptual errors: treats op-amp as simple summer without inversion, misunderstands aliasing vs. imaging, or cannot relate step response to transfer function
Numerical accuracy	20%	2	All calculations precise: correct output voltage in (a), accurate frequency values and amplitudes in (b) spectra, correct breakaway point (-0.586) and asymptote intersection (-1.5) in (c)	Correct approach with arithmetic slips: e.g., resistor ratio errors, off-by-factor-of-2 in sampling rate, or minor calculation errors in root locus features	Severe numerical errors: wrong final answer despite correct method, completely wrong frequency scaling, or inability to compute breakaway point/asymptote locations
Diagram quality	20%	2	Clear, labeled sketches for (b) and (c): frequency axes properly normalized (-π to π), triangular spectra with correct peak locations, root locus with poles, zeros, arrows, and asymptotes clearly marked; circuit diagram implied or drawn for (a)	Adequate diagrams with minor issues: missing labels, approximate rather than precise frequency locations, or root locus missing some features like direction arrows	Poor or missing diagrams: unrecognizable spectra, no indication of frequency scaling, or root locus showing incorrect pole-zero configuration
Step-by-step derivation	20%	2	Complete, logical derivation flow: explicit superposition steps in (a), clear sampling-filtering-decimation chain in (b), Laplace inversion and angle/magnitude condition application in (c); examiner can follow without gaps	Most steps shown but some skipped: jumps from input to output without intermediate expressions, or assumes root locus rules without stating them	Missing derivations: only final answers given, or incoherent logical flow that prevents verification of method
Practical interpretation	20%	2	Insightful practical connections: discusses op-amp saturation limits for (a), explains anti-aliasing necessity and filter selectivity trade-offs for (b), comments on stability margins and integral windup for (c); references relevant applications (e.g., signal processing in Indian communication systems, control in power plants)	Brief practical mention: notes that integral control eliminates steady-state error, or mentions Nyquist criterion without elaboration	No practical context: purely mathematical treatment with no engineering significance discussed

Q3

Directive word: Solve

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Electrical Engineering 2022 Paper II