UPSC Electrical Engineering 2022

Solve this multi-part numerical problem by allocating approximately 20% time to each sub-part (a)-(e), with sub-part (d) requiring additional attention for its three sub-sections. Begin with clear circuit diagrams for parts (a) and (d), apply fundamental laws (KCL/KVL, diode equation, generator/motor relations, op-amp virtual ground), show all derivation steps with proper units, and conclude with practical significance of results. For part (e), include truth tables and XOR gate diagrams for parity circuits.

• Part (a): Correct application of KCL/KVL to find V_in, V_S and power from dependent source; identification of controlling variable for dependent source
• Part (b): Accurate calculation of thermal voltage V_T at 22°C (295K), correct substitution in diode equation I = I_s[exp(V/nV_T) - 1] with ideality factor n=1
• Part (c): Proper calculation of armature current, series field drop, and efficiency for long shunt compound generator; correct treatment of no-load motor data for rotational losses
• Part (d)(i)-(iii): Application of ideal op-amp assumptions (virtual short, zero input current); correct nodal analysis for I_1, I_2, I_3, V_X; determination of R_L(max) from saturation limit; analysis of load regulation
• Part (e): Clear explanation of parity generation and checking for error detection; complete realization using XOR gates with proper logic diagrams and truth tables for even parity
• Consistent use of proper units (V, A, Ω, W) and significant figures throughout all numerical calculations
• Clear circuit diagrams for parts (a), (d), and (e) with labeled components and current/voltage polarities
• Physical interpretation of results: power flow direction, diode conduction regime, generator efficiency implications, op-amp operating limits, and parity scheme applications in data communication

Solve this multi-part numerical problem by allocating time proportionally to marks: approximately 40% on part (a) [20 marks], 40% on part (b) [20 marks], and 20% on part (c) [10 marks]. Begin with clear circuit diagrams for (a)(i), (a)(ii) and (c), then proceed with systematic mathematical derivations. For transient analysis in (a)(i), establish the differential equation and apply initial conditions. For Thevenin's equivalent in (a)(ii), show open-circuit voltage and short-circuit current calculations. In part (b), demonstrate partial fraction expansion for (i), apply Initial and Final Value Theorems correctly for (ii), and use Laplace transforms to solve the differential equation in (iii). Conclude with practical interpretations of time constants, steady-state values, and fault implications in power electronics.

• For (a)(i): Derive i(t) = (V/R)(1 - e^(-t/τ)) for RL circuit, calculate time constant τ = L/R, determine 95% settling time as t = 3τ, and state final value I_final = V/R
• For (a)(ii): Calculate Thevenin voltage V_TH using mesh/nodal analysis and Thevenin resistance R_TH by deactivating independent sources, presenting equivalent circuit with values
• For (b)(i): Perform polynomial long division to make F(s) proper, factor denominator (s+1)(s+2)(s+3), apply partial fraction expansion, and obtain inverse Laplace transform
• For (b)(ii): Verify applicability of Initial Value Theorem (proper rational function) and Final Value Theorem (poles in LHP), then apply limits as s→∞ and s→0
• For (b)(iii): Take Laplace transform of differential equation, apply initial conditions i(0-) = 4 and i'(0-) = -2, solve for I(s), decompose by partial fractions, and invert to get i(t)
• For (c): Analyze single-phase bridge rectifier with T₃ open (fault condition), determine conduction pattern with only two thyristors firing, derive average output voltage V_o = (V_m/π)(1+cosα) for half-wave equivalent, and solve for R = V_o/I_o

Derive expressions systematically across all sub-parts, allocating approximately 35% time to part (a) covering amplifier analysis, 25% to Routh-Hurwitz and impulse response in part (b), and 15% to induction motor calculations in part (c). Begin with small-signal equivalent circuit construction for the common-emitter amplifier, proceed through frequency response analysis with proper pole-zero identification, then apply R-H criterion with complete array construction, solve for network impulse response using Laplace transforms, and conclude with torque-slip characteristics for the induction motor. Ensure all sketches include labeled axes and critical frequency points.

• Small-signal equivalent circuit of common-emitter amplifier with r_π, g_m, and neglecting r_x and r_o as specified; correct identification of input and output resistances
• Midband gain expression: A_M = -g_m(R_C||R_L) · [r_π/(R_sig+r_π)] with proper sign convention for inverting amplifier
• Break frequency expressions: ω_L1 = 1/[C_E(R_E||(r_π+R_sig)/(1+β))] for emitter bypass and ω_L2 = 1/[C_C(R_C+R_L)] for coupling capacitor
• Complete transfer function A(s) = A_M · s²/[(s+ω_L1)(s+ω_L2)] showing second-order high-pass characteristic
• Numerical calculation: g_m = I_C/V_T = 40 mA/V, r_π = β/g_m = 2.5 kΩ, yielding A_M ≈ -160 V/V or 44 dB
• Capacitor selection: C_E and C_C values satisfying ω_L2 = 10ω_L1 with f_L = 100 Hz, minimizing C_total = C_E + C_C
• Routh-Hurwitz array construction showing no sign changes in first column, two pairs of complex conjugate roots with negative real parts, indicating marginal stability with oscillatory response
• Induction motor: s_maxT = R₂/X₂ = 0.04, n_maxT = n_s(1-s_maxT) = 1440 rpm for 4-pole 50 Hz machine

Solve this multi-part problem by allocating approximately 35% time to part (a) covering all three convolution approaches, 30% to part (b) including both the numerical calculation and chopper description, and 35% to part (c) for amplifier analysis. Begin with a brief statement of the network parameters and circuit configurations, then present systematic derivations for each sub-part with clear sectional headings, and conclude with verification of results against physical constraints.

• For 4(a): Derive impulse response h(t) from network in Figure 4(a)(i), then apply convolution integral v(t) = ∫e(τ)h(t-τ)dτ using s-domain (Laplace), direct time-domain integration, and graphical superposition methods
• For 4(b)(i): Calculate average output voltage V_d = (2V_m/π)cosα, determine back EMF E_b = V_d - I_aR_a, and solve for motor speed N = E_b/Kφ using given 240V supply, α=50°, and motor parameters
• For 4(b)(ii): Specify chopper requirements including fast switching capability, proper gate drive isolation, snubber circuits for dv/dt protection, current limiting, and EMI filtering for PWM/variable frequency operation
• For 4(c)(i): Analyze CS amplifier midband gain A_M = -g_m(R_D||R_L) using given g_m = 1mA/V and appropriate load resistance from Figure 4(c)
• For 4(c)(ii): Calculate source bypass capacitor C_S using f_L = 1/(2πR_S'C_S) where R_S' is equivalent resistance seen by C_S, targeting f_L = 10Hz

Solve all five sub-parts systematically, allocating approximately 20% time to each part given equal 10-mark weighting. For (a), clearly show the master-slave timing relationship with proper edge triggering; for (b)-(e), present formulas first, then substitution, then final numerical answers with units. Use separate sections for each sub-part with clear labels.

• (a) Master-Slave S-R flip-flop: Correct identification of level-triggered master and edge-triggered slave operation; output changes only on falling edge of clock; proper handling of S=R=1 forbidden state
• (b) Quarter-wave transformer: Application of Z_in = Z₀²/Z_L formula; recognition that matched load (Z_L = Z₀) yields Z_in = Z₀ = 60 Ω regardless of line length
• (c) Induction motor power flow: Air-gap power P_g = P_in - P_stator = 38.5 kW; rotor Cu loss = sP_g = 1.54 kW; P_mech = (1-s)P_g = 36.96 kW; shaft power = P_mech - P_fw = 36.16 kW; at new slip s' = 0.6, maintaining same torque implies same P_g, hence new rotor Cu loss = 23.1 kW and new shaft power calculation
• (d) AM modulation: m = A_m/A_c = 0.2; sideband amplitude = mA_c/2 = 5 V; bandwidth = 2f_m = 1 kHz; total power = P_c(1 + m²/2) with P_c = A_c²/(2R) = 2.5 W giving 2.55 W; frequency spectrum showing carrier at 100 kHz and sidebands at 99.5 kHz and 100.5 kHz
• (e) Two-port parameters: From given Y-parameters (y11=2, y12=-1, y21=-1, y22=2 S), derive ABCD parameters [A=-2, B=-1Ω, C=-3S, D=-2] and h-parameters [h11=0.5Ω, h12=0.5, h21=-0.5, h22=1.5S] using standard conversion formulas

Solve this multi-part numerical problem by allocating time proportionally to marks: approximately 40% for part (a) on wave reflection, 40% for part (b) on buck converter design, and 20% for part (c) on superposition theorem. Begin each part with clear identification of given parameters, apply relevant formulas with proper unit handling, show all intermediate calculations, and conclude with boxed final answers. For parts (a) and (b), include neatly labeled diagrams showing wave propagation and converter topology respectively.

• Part (a)(i): Calculate intrinsic impedances η₁ = 377Ω (air) and η₂ = 188.5Ω (dielectric), then Γ = (η₂-η₁)/(η₂+η₁) = -0.333, τ = 1+Γ = 0.667, and S = (1+|Γ|)/(1-|Γ|) = 2
• Part (a)(ii): Derive reflected field Eᵣ = -10 cos(ωt+z) aₓ V/m and Hᵣ = Eᵣ/η₁ = -26.53 cos(ωt+z) aᵧ mA/m; transmitted field Eₜ = 20 cos(ωt-2z) aₓ V/m with wavenumber change
• Part (b): Calculate duty cycle D = Vₒ/Vᵢₙ = 4/16 = 0.25; inductance L = Vₒ(1-D)/(fₛ×ΔIₗ) = 0.5 mH; capacitance C = ΔVₒ/(8L×fₛ²×ΔVₒ) or using C = (1-D)/(8L×fₛ²×(ΔVₒ/Vₒ)) yielding approximately 31.25 μF
• Part (c): Apply superposition by considering 10V source alone (with 5A open) then 5A source alone (with 10V short), calculate contributions through 2Ω and 3Ω resistances, sum to find V = 10V contribution + 5A contribution across 5Ω
• Verify boundary conditions at z=0 for part (a): tangential E and H continuity; verify ripple current and voltage specifications are met in part (b) design

Calculate the required quantities for all three parts systematically. For part (a), apply the cylindrical rotor synchronous motor power equations and phasor diagram; allocate ~40% time (20 marks). For part (b), use probability normalization for A, then convolution/characteristic function for Z = 3X + 4Y; allocate ~40% time (20 marks). For part (c), verify Stokes theorem by computing both line integral and surface integral in cylindrical coordinates; allocate ~20% time (10 marks). Present clear final answers with units.

• Part (a)(i): Correct application of power equation P = (3VE_f/X_s)sinδ to find torque angle δ at 0.8 pf lagging, with proper phasor diagram construction
• Part (a)(ii): Calculation of pull-out torque using T_max = (3VE_f)/(X_s·ω_s) with synchronous speed ω_s = 4πf/P rad/s
• Part (a)(iii): Determination of new excitation voltage E_f at half torque, then solving for armature current and power factor using modified power equation
• Part (b)(i): Normalization of f_Y(y) to find A = 3, verifying ∫f_Y(y)dy = 1 over 0 to ∞
• Part (b)(ii): Derivation of PDF of Z = 3X + 4Y using convolution of transformed variables or characteristic functions, with correct limits for -6 ≤ 3X ≤ 6 and 0 ≤ 4Y < ∞
• Part (c): Verification of Stokes theorem with correct curl computation in cylindrical coordinates, proper surface integral over z=1, 0<ρ<2, 0°<φ<45°, and matching line integral around the boundary contour

Solve all four sub-parts systematically, allocating approximately 25% time to each part given equal 10-mark weighting. Begin with clear conceptual definitions for (a)(i), followed by diagram construction for (a)(ii). For (b), apply communication theory formulas with careful PSD integration. For (c), compute energy losses over the loading cycle methodically. Present derivations stepwise with units at each stage.

• Clear distinction: Decade counter counts 0-9 then resets (MOD-10), while BCD counter outputs valid 4-bit BCD codes (0000-1001) for each decimal digit; example showing 7490 vs 4518 IC applications
• Cascaded BCD adder diagram showing three 4-bit BCD adder blocks with carry propagation between stages; example computation like 456 + 789 = 1245 demonstrating decimal correction
• DSB-SC SNR calculation: noise power integration over 196-204 kHz band, output SNR = (Si/ni) × (2/1) for coherent detection, correct handling of triangular PSD shape
• Coherent detector analysis: pre-filter noise power = N₀W, post-filter signal power = A²/4, SNR₀ = A²/(2N₀W) with proper bandwidth considerations
• All-day efficiency: energy output = Σ(kVA×time×pf) × 2400/240, energy losses = 153W×24h + 224W×[(1.25)²×2 + 1²×6 + 0.5²×8 + 0.25²×4] hours, final percentage calculation
• Proper unit handling throughout (W, kWh, dB where applicable) and verification of numerical reasonableness

Solve all five sub-parts systematically, allocating approximately 20% time to each part given equal 12-mark weighting. Begin with clear circuit diagrams and waveforms for (a), proceed through control system analysis for (b), derive mechanical time constant expression for (c), apply Kirchhoff's laws for traction problem (d), and apply CTFT properties for (e). Present derivations stepwise with final boxed answers for each sub-part.

• Part (a): Correct firing angle calculation using P = (V²/2πR)(2π - α + sin2α/2) = 1000W, yielding α ≈ 90°; power factor = √(P/VA) = √(1000×26.45/230²); circuit diagram showing SCR pair with resistive load; voltage and current waveforms showing conduction from α to π
• Part (b): Characteristic equation 1 + G(s)K(s) = 0 → τs³ + s² + KTd s + K = 0; Routh-Hurwitz criterion application; oscillation condition: auxiliary equation roots at s = ±j√2 yielding K = 2, Td = τ = 1/√2 or equivalent valid ranges
• Part (c): Mechanical time constant τm = J/f seconds; experimental method: run machine at no-load, disconnect supply, measure speed decay time to 36.8% of initial value using tachometer and stopwatch; or use retardation test plotting ω vs t on semilog paper
• Part (d): Set up voltage distribution V(x) = VA - I·r·x - (VA-VB-I·r·L)·x/L for x from A; find dV/dx = 0 for minimum potential point; calculate currents IA and IB using current continuity at minimum potential point
• Part (e): Express y(t) as 2[u(t)-2u(t-1)+u(t-2)] using time-shift and linearity; apply scaling and shifting to get Y(ω) = 2e^(-jω/2)[sin(ω/2)/(ω/2)][1-e^(-jω)]; for z(t) as triangular pulse, use convolution of rectangular pulses or differentiation property

Calculate all numerical quantities demanded across the three parts, allocating approximately 35% time to part (a) given its multi-step PWM analysis and comparison requirement, 30% to part (b) for induction generator power flow calculations, and 35% to part (c) for sequence network construction and fault analysis. Begin each part with the appropriate formula statement, show systematic substitution, and conclude with physical interpretation of results.

• Part (a): Correct application of bipolar PWM fundamental voltage formula V₁ = mₐ × Vdc and harmonic voltage calculation using normalized Fourier coefficients; impedance calculation at dominant harmonic frequency (mf × f₁ = 1050 Hz) and side frequencies
• Part (a): THD calculation using Iₕ/I₁ ratio summation for identified harmonics, and explicit comparison table showing V₁(50Hz) for PWM (304V), square wave (4Vdc/π = 484V), and quasi-square wave with pulse width δ
• Part (b): Correct determination of slip s = (Ns - N)/Ns = -0.01 for generator operation; accurate calculation of Thevenin equivalent or direct impedance method for rotor circuit; separation of air-gap power into active and reactive components
• Part (b): Proper accounting for core loss resistance Rsh and magnetizing branch in power calculations; correct sign convention for generator operation (P delivered positive, Q absorbed positive)
• Part (c)(i): Clear statement that single line-to-ground fault exceeds 3-phase fault severity when X₀ < X₁ (typically with solidly grounded neutral or low X₀/X₁ ratio), making I(LG) = 3E/(2X₁+X₀) > I(3φ) = E/X₁
• Part (c)(ii): Correct sequence network interconnection for single line-to-ground fault (series connection of positive, negative, zero sequence); proper base conversion (15 MVA, 6.6 kV base) and per-unit calculations for transformer, line impedances; final fault current in amperes at 33 kV bus B

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 → ∞

This is a multi-part numerical and theoretical problem requiring systematic solving. Begin with part (a) on cable grading: define grading, derive the condition for minimum gradient using calculus on the electric field expression, then calculate economic diameter using the optimal radius ratio condition. For part (b), apply variable frequency control principles: first determine rated torque and slip, then use constant V/f control below synchronous speed to find inverter frequency at 500 rpm, and calculate stator current using the equivalent circuit with modified frequency parameters. For part (c), explain core saturation physics, derive the inrush current expression from flux balance equation considering residual flux and switching angle, and sketch the Φ-i curve showing non-linear magnetization. Allocate approximately 35% time to (a), 40% to (b), and 25% to (c) based on computational complexity.

• Part (a)(i): Definition of cable grading as equalizing dielectric stress; identification of capacitance grading (using multiple dielectrics) and intersheath grading (using intermediate conducting layers) methods
• Part (a)(ii): Derivation showing E_max is minimized when ln(R/r) = 1, i.e., R/r = e ≈ 2.718, giving optimal conductor-to-sheath radius ratio
• Part (a)(iii): Calculation of economic diameter using g_max = V/(r·ln(R/r)) with ln(R/r)=1, yielding overall diameter ≈ 4.06 cm for 75 kV working voltage
• Part (b): Determination of rated torque from synchronous speed (1000 rpm) and rated slip; application of constant flux (V/f) control to find inverter frequency ≈ 25 Hz at 500 rpm; calculation of stator current ≈ 12-14 A using modified equivalent circuit parameters
• Part (c): Explanation of non-linear magnetizing current due to saturation in B-H curve; derivation of inrush current i = (V_m/ωL)(cosα - cos(ωt+α)) + Φ_r/L showing DC offset; graph showing peaked waveform with harmonic content

Solve all five numerical sub-parts systematically, allocating approximately 20% time to each part given equal 12-mark weighting. Begin with clear identification of given data and required unknowns for each sub-part. Present derivations first where asked (AM power expression in c-i, control system parameters in d-i and d-ii), followed by numerical substitutions and final answers with proper units. For relay coordination in part (e), clearly show PSM calculations and time grading logic.

• Part (a): Apply full converter voltage equation V_a = (2V_m/π)cosα for upward motion; use torque-speed relation T = KφI_a and back EMF E_b = Kφω to find speed; for downward motion with regenerative braking, determine α for same speed magnitude; calculate holding angle when E_b = 0
• Part (b): Calculate secondary voltages using turns ratio; determine currents in each secondary section using S = VI; apply ampere-turn balance equation N_1I_1 = N_2I_2 + N_3I_3 with proper phase consideration for inductive load; compute primary current magnitude and power factor
• Part (c): Derive AM wave expression v(t) = V_c[1 + m sin(ω_m t)]sin(ω_c t); expand and identify carrier, upper and lower sideband components; integrate over period to obtain average power P_avg = P_c(1 + m²/2); substitute m = 0.8 to find radiated power
• Part (d): Use standard second-order system relations: K_v = K/a, ζ from overshoot formula %OS = exp(-ζπ/√(1-ζ²)) × 100, settling time t_s = 4/(ζω_n); solve simultaneous equations for K and a in both cases
• Part (e): Calculate PSM for both relays as I_fault/I_pickup; interpolate operating time from given table for TMS=1; apply actual TMS for R_1; ensure time grading margin of 0.5s between R_2 and R_1; back-calculate TMS for R_2

This is a calculation-heavy question demanding precise numerical work across three distinct domains. Begin with part (a) by first computing full-load current and losses, then establishing the current limits for starter design, and systematically deriving the four resistance sections. For part (b), first differentiate the impedance concepts conceptually, then apply the long-line ABCD parameters (or nominal π approximation) for the 400 km line to find sending-end current and receiving-end voltage under no-load, commenting on the Ferranti effect. For part (c), design the boost converter by calculating duty ratios for both input extremes, determining inductor value for worst-case ripple condition, and sizing capacitor for output voltage ripple. Allocate approximately 35% time to (a), 35% to (b), and 30% to (c), ensuring all derivations are shown stepwise with proper units.

• Part (a): Calculation of full-load current (40 A), total losses (2222.22 W), armature copper loss (888.89 W), armature resistance (0.556 Ω), and starter resistances R1-R4 (2.083 Ω, 1.25 Ω, 0.75 Ω, 0.45 Ω approximately) with current limits 48-80 A
• Part (b)(i): Clear distinction that characteristic impedance Zc = √(z/y) is frequency-dependent complex quantity while surge impedance Zs = √(L/C) is real and lossless; SIL = V²/Zs where natural power flow occurs at unity power factor
• Part (b)(ii): Application of long transmission line equations or nominal π method with ABCD parameters for 400 km line; calculation of sending-end current (approximately 180-200 A) and receiving-end voltage (approximately 245-260 kV, higher than sending end due to Ferranti effect)
• Part (c): Duty ratio calculation for Vin=12V (D=0.75) and Vin=24V (D=0.5); inductor selection based on worst-case ripple at D=0.5 or D=0.75 with ΔIL ≤ 40% of IL; L ≈ 45-60 μH; capacitor for 2% ripple (≈960V peak-to-peak at 48V means 0.96V ripple) yielding C ≈ 10-15 μF
• Practical interpretation: Comment on starter step transition currents, Ferranti effect significance for 400 km Indian transmission corridors (e.g., HVDC back-to-back stations), and converter continuous conduction mode verification

Solve this multi-part numerical problem by allocating approximately 35% time to part (a) due to its computational complexity involving harmonic analysis, 30% to part (b) for deriving instantaneous electromagnetic quantities, and 35% to part (c) covering circuit breaker theory and transient calculations. Begin with clear problem statements for each part, show complete derivations with intermediate steps, and conclude with physically meaningful interpretations of results.

• Part (a): Calculate extinction angle β using transcendental equation for R-L load, verify continuity condition (β > π+α), compute dominant harmonic (2nd harmonic) RMS current using Fourier analysis, determine power absorbed, and derive voltage ripple factor
• Part (a): Apply correct formula for ripple factor considering only dominant harmonic component in the output voltage
• Part (b): Derive instantaneous coil voltage using Faraday's law (e = N dΦ/dt), determine current considering coil resistance/inductance, and compute magnetic force using Maxwell stress tensor or energy method (F = ½ i² dL/dx or B²A/2μ₀)
• Part (c)(i): Define restriking voltage as transient voltage across breaker contacts post-current zero, RRRV as rate of rise of restriking voltage (kV/μs), and express V_restrike(max) = 2V_m and RRRV_max = ω₀V_m where ω₀ = 1/√(LC)
• Part (c)(ii): Identify SF6 circuit breaker as preferred for 132 kV and above due to superior dielectric strength and arc quenching capability
• Part (c)(iii): Calculate L = 5/314 = 15.92 mH, C = 0.03 μF, natural frequency f₀ = 1/(2π√(LC)), V_restrike(max) = 2√2 × 132/√3 kV, and RRRV_max = V_restrike(max) × ω₀

Calculate all numerical results with systematic derivations, allocating approximately 35% time to part (a) on DTFS periodicity and harmonic analysis, 30% to part (b) on time-delay stability using Nyquist/Bode criteria, and 35% to part (c) on state-space design including observability test, matrix exponential computation, and pole placement via Ackermann's formula or direct method. Present each sub-part with clear headings, show all intermediate steps, and conclude with boxed final answers.

• For (a)(i): Compute digital frequencies ω₁=28π/90, ω₂=40π/90, ω₃=70π/90; find periods N₁=45, N₂=9, N₃=9; determine fundamental period N₀=LCM(45,9,9)=45
• For (a)(ii): Identify harmonic indices m₁=14, m₂=20, m₃=35 (or equivalently m₃=-10 mod 45 = 35) where Dₘ is non-zero within 0≤m<45
• For (a)(iii): State |D₁₄|=0.5, |D₂₀|=1, |D₃₅|=1.5 with correct conjugate symmetry |D₄₅₋ₘ|=|Dₘ|
• For (b)(i): Apply Nyquist stability criterion; find phase crossover frequency ωₚc=√2 rad/s; compute maximum delay Tₘₐₓ=π/(2√2)≈1.11s for K=1
• For (b)(ii): At K=√5, find gain crossover frequency ωgc=1 rad/s; calculate phase margin without delay as 90°-arctan(0.5)=63.43°; subtract delay phase ωgc×T×180/π=28.65° to get PM≈34.8°
• For (c)(i)-(iii): Check observability rank[O]=rank[C;CA]=2 (observable); compute state transition matrix Φ(t)=[1 (1-e⁻³ᵗ)/3; 0 e⁻³ᵗ]; design state feedback K=[5 3] using desired characteristic equation s²+2s+5=0