(a) Draw the sequence networks and calculate the load sequence impedances of a load circuit as shown in figure. The load circuit is connected to a balanced three phase supply. The value of z₁, z₂ and zₙ are (4 + j6) Ω, –j45 Ω and j4 Ω. (20 marks)

(b

Question

(a) Draw the sequence networks and calculate the load sequence impedances of a load circuit as shown in figure. The load circuit is connected to a balanced three phase supply. The value of z₁, z₂ and zₙ are (4 + j6) Ω, –j45 Ω and j4 Ω. (20 marks)

(b) For the network shown in figure, draw a block diagram representing each circuit element by a block. Use block diagram reduction technique to obtain the transfer function of the network. The voltage $V_i(t)$ is the applied input and the voltage across the capacitor $V_o(t)$ is the output. (20 marks)

(c) A convolutional code is described by
$$g_1 = [1\ 1\ 0],\ g_2 = [1\ 0\ 1],\ g_3 = [1\ 1\ 1].$$
Find the transfer function and the free distance for this code. Also verify whether or not this code is catastrophic. (10 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction acknowledging the three distinct domains covered: symmetrical components, control systems, and coding theory. For part (a), spend approximately 40% of effort (20 marks) drawing sequence networks and computing Z₀, Z₁, Z₂ with proper handling of the neutral impedance. For part (b), allocate 40% (20 marks) to converting the electrical network to a block diagram, applying reduction rules systematically to obtain Vₒ(s)/Vᵢ(s). For part (c), use remaining 20% (10 marks) to construct the state diagram, derive the transfer function matrix, compute free distance via minimum weight path, and apply Massey-Sain criterion for catastrophic property. Conclude with a summary table of results.
- For (a): Correct sequence network diagrams showing positive, negative, and zero sequence connections with proper treatment of neutral impedance (3Zₙ in zero sequence)
- For (a): Accurate calculation of Z₁ = Z₂ = z₁ + z₂ = (4+j6) + (-j45) = 4-j39 Ω and Z₀ = z₁ + 3zₙ + z₂ = (4+j6) + j12 + (-j45) = 4-j27 Ω
- For (b): Proper block diagram construction with integrators, summers, and gain blocks representing the RLC network dynamics in Laplace domain
- For (b): Systematic application of block diagram reduction rules (series, parallel, feedback) to arrive at final transfer function without algebraic errors
- For (c): Correct generator matrix G(D) = [1+D, 1+D², 1+D+D²] and state diagram with 4 states (memory m=2)
- For (c): Computation of free distance d_free = 5 by finding minimum weight non-zero codeword path through state diagram
- For (c): Application of catastrophic code test: checking if GCD of generator polynomials equals D^l, concluding this code is non-catastrophic since GCD(1+D, 1+D², 1+D+D²) = 1

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates flawless understanding: for (a) correctly identifies that zero-sequence network includes 3Zₙ due to neutral current tripling; for (b) properly models dynamic elements as integrators in Laplace domain; for (c) correctly interprets convolutional code parameters (constraint length, code rate 1/3) and catastrophic code definition per Massey-Sain	Shows basic familiarity with concepts but makes minor errors: may forget 3Zₙ factor in zero sequence, or confuse block diagram elements, or miscalculate constraint length	Fundamental misconceptions: treats sequence networks as simple parallel/series combinations without transformation, or draws circuit schematic instead of block diagram, or misunderstands free distance as Hamming distance of generators
Numerical accuracy	20%	10	All calculations precise: (a) Z₁=Z₂=4-j39 Ω, Z₀=4-j27 Ω with correct complex arithmetic; (b) transfer function poles and zeros algebraically correct; (c) d_free=5 verified by exhaustive path analysis, catastrophic test conclusive	Correct approach but arithmetic slips: sign errors in complex numbers (e.g., j6+j4=j10 instead of j10), or algebraic mistakes in block diagram reduction, or path weight calculation errors in state diagram	Major computational errors: incorrect formula application leading to vastly wrong impedances, or transfer function with wrong order denominator, or d_free computed as 3 or 7 instead of 5
Diagram quality	20%	10	Professional diagrams: (a) three separate sequence networks with clear labeling of sequence currents and voltages; (b) clean block diagram with standard symbols (integrator as 1/s, summer with signs, gain blocks); (c) complete state diagram showing all 4 states with branch labels (input/output) and D-transform weights	Diagrams present but cluttered: missing labels on sequence networks, or block diagram with ambiguous signal flow, or state diagram with incomplete transitions	Diagrams absent or unusable: hand-drawn sketches without scanning, or wrong type of diagram (circuit schematic for block diagram), or state diagram confused with trellis diagram
Step-by-step derivation	20%	10	Logical progression with explicit steps: (a) transformation equations → sequence network construction → impedance calculation; (b) KVL/KCL → Laplace transform → block diagram → reduction steps clearly numbered; (c) generator matrix → state table → state diagram → path enumeration for d_free → GCD computation with Euclidean algorithm shown	Derivation present but skips key steps: jumps from network to impedances without showing 3Zₙ derivation, or block diagram reduction with merged steps, or states d_free without showing path	No derivations shown: only final answers stated, or incorrect sequence of operations (e.g., reduces block diagram before drawing it completely), or asserts catastrophic property without test
Practical interpretation	20%	10	Connects theory to practice: (a) explains why zero-sequence impedance matters for ground fault analysis in Indian power systems (e.g., 400kV NLC lines); (b) discusses physical meaning of poles/zeros in filter response; (c) relates d_free=5 to coding gain and error correction capability, notes non-catastrophic property ensures finite error propagation	Brief mention of relevance: states that sequence analysis is used for fault studies, or that convolutional codes are used in satellite communications, without elaboration	No practical context provided: purely mathematical exercise with no mention of application to power system protection, control system design, or digital communication systems

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