Q3
(a) (i) Explain what happens when a circuit shown in Figure 3(a)(i) below is constructed using logarithmic amplifier. 10 Figure 3(a)(i) (ii) Explain what happens if the topology is modified as shown in Figure 3(a)(ii) below. 10 Figure 3(a)(ii) (b) For the circuit shown in Figure 3(b), calculate the voltage V₀(t) as function of time, Figure 3(b) where V(t) = 10 sin (6t + 60°) V and I(t) = 5 cos (4t + 30°) A. 20 (c) A mixer (analog multiplier) is used as a process in some analog communication systems. Two signals X₁(t) and X₂(t) are mixed to produce the output y(t) = X₁(t) X₂(t). If X₁(t) = 10 sin c (10t) and X₂(t) = 2 cos (1000 πt), then calculate and plot the magnitude of the Fourier transform of output signal. Further, specify and prove the property of Fourier transform used in calculations. 10
हिंदी में प्रश्न पढ़ें
(a) (i) समझाइए कि जब चित्र 3(a)(i) में प्रदर्शित परिपथ को लघुगणकीय प्रवर्धक का प्रयोग करते हुए निर्मित किया जाएगा, तो क्या होगा । चित्र 3(a)(i) (ii) समझाइए कि यदि संस्थितिकी को चित्र 3(a)(ii) में दर्शाए अनुसार परिवर्तित कर दिया जाए, तो क्या होगा । चित्र 3(a)(ii) (b) चित्र 3(b) में प्रदर्शित परिपथ के लिए वोल्टता V₀(t) की गणना समय के फलन के रूप में कीजिए, चित्र 3(b) जहाँ V(t) = 10 sin (6t + 60°) V और I(t) = 5 cos (4t + 30°) A हैं। (c) एक मिश्रक (अनुकृप गुणक) को किसी अनुकृप संचार प्रणाली के एक प्रक्रम के रूप में प्रयुक्त किया गया है । निर्गत y(t) = X₁(t) X₂(t) निर्मित करने के लिए दो संकेतों X₁(t) तथा X₂(t) को मिश्रित किया गया है । यदि X₁(t) = 10 sin c (10t) तथा X₂(t) = 2 cos (1000 πt) हो, तो निर्गत संकेत के फुरिये रूपांतर के परिमाण की गणना कीजिए तथा उसका आरेखण कीजिए । गणना में प्रयुक्त फुरिये रूपांतर के गुणों का उल्लेख कीजिए तथा उन्हें सिद्ध कीजिए ।
Directive word: Calculate
This question asks you to calculate. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.
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How this answer will be evaluated
Approach
The directive 'calculate' dominates part (b) carrying 20 marks, so allocate ~50% effort there with rigorous time-domain analysis; spend ~25% each on (a)(i) and (a)(ii) explaining logarithmic/antilogarithmic amplifier behavior with circuit diagrams; reserve ~10% for (c) applying the multiplication/convolution property of Fourier transforms. Structure: begin with conceptual explanations of log-amp configurations, proceed to detailed circuit analysis for V₀(t), and conclude with spectral analysis of the mixer output.
Key points expected
- For (a)(i): Explain that Figure 3(a)(i) realizes a log-amp producing V₀ = -V_T ln(V_i/R_1I_S), converting multiplication to addition for analog computation
- For (a)(ii): Explain the modified topology as an antilogarithmic (exponential) amplifier giving V₀ = -R_f I_S exp(-V_i/V_T), enabling division and power operations
- For (b): Apply KCL/KVL to derive the differential equation and solve for V₀(t) = 10 sin(6t + 60°) - L(di/dt) or equivalent, handling the frequency mismatch (6 rad/s vs 4 rad/s) correctly
- For (b): Show proper handling of phase relationships and superposition when input voltage and current have different frequencies
- For (c): Apply the multiplication property: F{x₁(t)x₂(t)} = (1/2π)[X₁(ω) * X₂(ω)], proving convolution in frequency domain shifts the sinc spectrum to ±1000π
- For (c): Sketch the resulting spectrum showing two sinc-shaped bands centered at ω = ±1000π with appropriate scaling and bandwidth 20π rad/s
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 8 | Correctly identifies (a)(i) as log-amp and (a)(ii) as antilog-amp with proper transfer equations; for (b) recognizes the non-sinusoidal steady-state due to different frequencies; for (c) correctly states and proves the multiplication-convolution property of Fourier transforms | Identifies basic log-amp function but confuses (i) and (ii) or omits diode current equation; attempts (b) with phasor methods incorrectly; states convolution property without rigorous proof | Misidentifies both circuits as simple inverting amplifiers; applies phasor analysis blindly to mismatched frequencies; confuses multiplication with modulation property |
| Numerical accuracy | 20% | 8 | For (b) derives exact V₀(t) with correct amplitude, phase and frequency components; for (c) calculates precise Fourier transform magnitude including (1/2π) scaling factor and correct sinc amplitude 10/10 = 1 | Correct approach but arithmetic errors in phase calculation or missing time-derivative term; approximate sinc spectrum with correct shape but wrong amplitude scaling | Major calculation errors in differentiation; incorrect frequency translation in (c); omits scaling constants entirely |
| Diagram quality | 20% | 8 | Clear labeled diagrams for (a)(i) and (a)(ii) showing op-amp, diode/transistor, input resistor with proper polarity; for (c) accurate magnitude spectrum plot with labeled axes, center frequencies ±1000π, nulls at ±(1000π±10), and amplitude scale | Basic circuit sketches missing component labels or current directions; spectrum plot shows correct shape but missing frequency markers or amplitude scale | Unrecognizable circuits; no diagram for (c) or completely misaligned frequency axis; missing essential features like diode orientation |
| Step-by-step derivation | 20% | 8 | For (a): Full derivation using diode equation I = I_S(e^(V_D/V_T) - 1) ≈ I_S e^(V_D/V_T); for (b): Complete KCL setup, differential equation formulation, and integration; for (c): Rigorous proof of multiplication property using inverse Fourier transform and convolution integral | States key equations without full derivation; skips integration steps in (b); outlines proof structure but omits key integral manipulations | No derivations shown; only final answers stated; confuses derivation with description |
| Practical interpretation | 20% | 8 | For (a): Explains applications in analog computation (multipliers, dividers, RMS converters) and temperature compensation needs; for (b): Discusses physical realizability with given mismatched sources; for (c): Relates mixer output to DSB-SC modulation with practical bandwidth considerations for Indian communication systems | Mentions general applications without specific context; limited discussion of practical constraints like diode matching or frequency separation | No practical context provided; fails to recognize (c) as modulation process; ignores temperature effects in log-amps |
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