Electrical Engineering 2025 Paper I 50 marks Derive

Q8

(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 प्रश्न 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)

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Directive word: Derive

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Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Correctly identifies that Smith chart r-circles represent constant resistance loci in Γ-plane; properly applies boundary conditions for electromagnetic wave reflection/transmission; recognizes envelope detector distortion when modulation index β > 1; accurately describes PWM and other voltage control strategies for 3-phase invertersMinor errors in Smith chart geometry interpretation or boundary condition application; incomplete understanding of envelope detector behavior with overmodulation; partial coverage of inverter control methods with some confusion between techniquesFundamental misunderstanding of reflection coefficient mapping onto Smith chart; incorrect application of Snell's law or boundary conditions; failure to recognize envelope detector limitation with β > 1; confused or incorrect description of inverter control methods
Numerical accuracy20%10Precise calculation of η₁ = 377/√7.5 ≈ 137.7 Ω and η₂ = 377 Ω; correct Γ ≈ -0.465, τ ≈ 0.535; accurate field amplitudes E₀ʳ ≈ -0.558 mV/m, E₀ᵗ ≈ 0.642 mV/m with proper H field calculations; correct Fourier coefficients for the envelope waveformMinor arithmetic errors in impedance or field calculations; correct approach but incorrect final values; approximate Fourier coefficients with some calculation errorsMajor calculation errors in intrinsic impedance or reflection coefficient; incorrect field amplitude relationships; wrong numerical values for Fourier components; missing units or incorrect orders of magnitude
Diagram quality15%7.5Clear sketch of Smith chart showing r-circle family with labeled centers and radii for r = 0, 1, ∞; proper diagram of normal incidence wave reflection showing incident, reflected, transmitted waves with field directions; waveform sketch of AM signal and distorted envelope detector output when β > 1; schematic of 3-phase inverter with voltage control indicationAdequate diagrams with some missing labels or incomplete Smith chart circles; basic wave diagram without field vector directions; rough sketch of envelope waveformMissing essential diagrams; poorly drawn or unlabeled figures; incorrect Smith chart geometry; no waveform illustrations where clearly needed
Step-by-step derivation25%12.5Complete algebraic derivation from Γ = (z-1)/(z+1) to circle equation form with clear identification of real and imaginary parts; systematic application of tangential E and H continuity at interface; explicit Fourier series derivation showing periodic extension of rectified waveform and integration for coefficients; structured presentation of each control methodDerivations with some skipped steps but correct overall flow; missing intermediate algebraic steps in Smith chart proof; partial Fourier series derivation without full coefficient calculationMissing derivations entirely or only stating final results; disorganized mathematical presentation; incorrect or circular derivations; no attempt to show how results are obtained
Practical interpretation15%7.5Explains significance of Smith chart for impedance matching in RF systems like antenna design; discusses practical implications of wave reflection at dielectric interfaces for optical fiber or radome design; interprets envelope distortion in AM broadcasting with overmodulation; relates inverter voltage control to variable frequency drives in Indian textile/cement industriesBrief mention of practical applications without elaboration; generic statements about importance of topics; limited connection to real-world systemsNo practical context provided; purely mathematical treatment without physical interpretation; failure to recognize engineering significance of results

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