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EC-305 · Network Analysis/Unsolved PYQ Paper

EC-305 Network Analysis - Jun 2024 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

  1. 7 Marks
    aIn the circuit of fig-1 the switch S is moved from position 1 to 2 at $t = 0$, having been in position 1 for a long time before $t = 0$. Capacitor $C_2$ is uncharged at $t = 0$. i) Find the particular solution for $i(t)$ for $t > 0$ ii) Find the particular solution for $v_2(t)$ for $t > 0$
    Figure-1: A circuit showing a DC source V_0 in series with resistor R connected to terminal 1 of switch S. In parallel across terminal 1 is capacitor C_1 to the common reference rail. Switch S can connect to terminal 1 or terminal 2. From terminal 2, a resistor R_1 connects in series to a parallel branch containing capacitor C_2, across which voltage v_2 is defined.
    How to answer
    • Find initial voltage on C1: v1(0) = V0, v2(0) = 0
    • Write KVL for t > 0 with switch at position 2: R1*i(t) + (1/C1) int i dt + (1/C2) int i dt = 0
    • Calculate equivalent capacitance Ceq = (C1*C2)/(C1+C2) and time constant tau = R1*Ceq
    • Derive expression for i(t)
    • Determine v2(t) by integrating i(t)/C2 with initial condition v2(0) = 0
  2. 7 Marks
    bThe circuit of fig. 2 reaches a steady state in position 2 and at $t = 0$ the switch S is moved to position 1. Find $i(t)$.
    Figure-2: Circuit with switch S having positions 1 and 2. Position 2 connects through a 10V DC source loop. Position 1 connects to a loop containing a 20V source and a 30 ohm resistor. Both loops connect via a 20 ohm resistor in series with an inductor L = 1/2 H carrying current i(t).
    How to answer
    • Find initial inductor current i(0^-) from steady state in position 2
    • Find final inductor current i(infinity) in position 1
    • Find Thevenin equivalent resistance Rth seen by the inductor for t > 0
    • Calculate time constant tau = L / Rth
    • Formulate complete response expression i(t) = i(infinity) + [i(0+) - i(infinity)] * exp(-t/tau)
  3. 7 Marks
    aIn the network shown in fig. 3, the switch S is closed at $t = 0$. Find the value of $i_1(t)$ and $i_2(t)$.
    Figure-3: Two-mesh circuit. A 100V DC source connects in series with switch S, a 10 ohm resistor, an inductor of 1 H, and meets a central branch having a 10 ohm resistor. Mesh 1 carries current i1(t). From the central node, a second 1 H inductor connects to a parallel 10 ohm resistor, forming mesh 2 carrying current i2(t).
    How to answer
    • State initial conditions: i1(0-) = 0, i2(0-) = 0
    • Convert circuit to s-domain with transform of 100V step input (100/s)
    • Formulate s-domain mesh equations for loop 1 and loop 2
    • Solve for I1(s) and I2(s) using Cramer's rule or substitution
    • Take inverse Laplace transform to obtain time-domain solutions i1(t) and i2(t)
  4. 7 Marks
    bState and prove RLC series resonance.
    How to answer
    • Define series resonance: condition where inductive reactance equals capacitive reactance and total impedance is purely resistive
    • Write series impedance expression: Z = R + j(omega*L - 1/(omega*C))
    • Set imaginary part to zero: omega*L - 1/(omega*C) = 0
    • Derive resonant frequency f0 = 1 / (2*pi*sqrt(L*C))
    • Sketch impedance and current response curves vs frequency, noting unity power factor and maximum current at resonance
  5. 14 Marks
    For the network shown in fig-4 write the tie set matrix for the twigs (4, 5, 6) and determine the loop currents and branch currents for the same.
    Figure-4: Planar oriented graph/network with 4 nodes forming a bridge/tetrahedral shape. Branch 1 has 5 ohm resistor and 10V source, branch 2 has 10 ohm resistor, branch 3 has 5 ohm resistor, branch 4 has 10 ohm resistor, branch 5 has 5 ohm resistor, branch 6 has 5 ohm resistor. Twigs are specified as branches 4, 5, 6.
    How to answer
    • Identify network graph, tree with twigs {4, 5, 6} and co-tree links {1, 2, 3}
    • Form fundamental loops corresponding to each link
    • Construct fundamental tie-set matrix [B] specifying orientations
    • Formulate network equilibrium equations using branch impedance matrix [Zb]: [B][Zb][B]^T [IL] = [B][Vs]
    • Solve for loop currents [IL] and determine branch currents [Ib] = [B]^T [IL]
  6. 7 Marks
    aDetermine the value of $R_L$ to be connected across AB in fig. 5, for maximum power transfer. Also calculate the maximum power absorbed by the $R_L$.
    Figure-5: Circuit with a 4V independent DC voltage source in series with a 2 kΩ resistor. Connected across this is a dependent current source directed upward with value v/4000. In series to terminal A is a 3 kΩ resistor, with terminals A and B as open terminals across which RL is to be connected.
    How to answer
    • Find open-circuit voltage Vth across terminals A-B
    • Find short-circuit current Isc or apply a test source to determine Rth with dependent source present
    • Set RL = Rth for maximum power transfer condition
    • Calculate maximum power using Pmax = Vth^2 / (4 * Rth)
  7. 7 Marks
    bFind the current in the $14\,\Omega$ resistor of the circuit of fig. 6 by Thevenin theorem.
    Figure-6: Circuit consisting of a 10V source in series with a 5 ohm resistor, in parallel with a dependent current source of 0.1*Vx pointing upward, an 8 ohm series resistor, and a load branch of 14 ohm across terminals A and B. The voltage across the 14 ohm resistor is designated as Vx.
    How to answer
    • Remove the 14 ohm load resistor across terminals A-B
    • Determine the open-circuit voltage Vth across terminals A-B, noting the control variable Vx
    • Determine Rth by applying a 1V test source or finding short-circuit current Isc
    • Draw the Thevenin equivalent circuit with Vth, Rth, and RL = 14 ohm
    • Calculate the load current IL = Vth / (Rth + 14)
  8. 7 Marks
    aSolve for $i(t)$ in the circuit as shown in fig. 7 in which 3F capacitor is initially charged to 20V and 6F capacitor to 10V, and the switch is closed at $t = 0$. Also draw the S-transformed circuit.
    Figure-7: Single loop circuit with a 3F capacitor initially charged to 20V, a 1 ohm resistor, switch closing at t = 0, another 1 ohm resistor, and a 6F capacitor charged initially to 10V, carrying clockwise loop current i(t).
    How to answer
    • Identify initial condition voltages with proper polarities: V1(0) = 20V, V2(0) = 10V
    • Draw s-transformed equivalent circuit showing capacitors as 1/(sC) in series with V(0)/s voltage sources
    • Write loop KVL equation in s-domain: I(s) * [1 + 1 + 1/(3s) + 1/(6s)] = (20/s - 10/s)
    • Simplify expression for I(s)
    • Perform inverse Laplace transform to find i(t) for t > 0
  9. 7 Marks
    bIn the circuit of fig-8, switch is in position 1 for long time and moved to position 2 at $t = 0$. Find the voltage across the capacitor $v_c(t)$ for $t > 0$.
    Figure-8: Circuit with a 6V source and 2 ohm resistor in position 1 of switch S. Position 2 connects to a parallel RLC network containing a branch with a 1 ohm resistor in series with a 0.5 H inductor, and a branch with a 1 F capacitor across which vc(t) is measured.
    How to answer
    • Determine initial conditions at t = 0-: vc(0-) and iL(0-) with switch in position 1
    • Write differential equation or s-domain equation for t > 0 with switch in position 2
    • Find characteristic roots to identify damping condition (overdamped, critically damped, or underdamped)
    • Apply boundary conditions vc(0+) and dv_c(0+)/dt = i_c(0+)/C
    • Write the final expression for vc(t) for t > 0
  10. 7 Marks
    aState and prove super position theorem.
    How to answer
    • State superposition theorem formally
    • State conditions of applicability (linear and bilateral elements)
    • Describe procedure: deactivate independent sources (short voltage sources, open current sources)
    • Provide mathematical proof using matrix formulation of network equations ([Z][I] = [V]) or an illustrative two-source network
    • Summarize limitations regarding power calculation (non-linear)
  11. 7 Marks
    bVerify reciprocity theorem for the circuit shown in fig. 9.
    Figure-9: An AC ladder network with a voltage source 10∠0° V in series with a 5 ohm resistor. A parallel branch has impedance 2 + j4 ohm. The output branch has impedance 4 - j8 ohm carrying current I.
    How to answer
    • State reciprocity theorem: ratio of response to excitation remains identical upon interchanging excitation and response locations
    • Case 1: Calculate current I in the (4 - j8) ohm branch with 10∠0° V source in the input branch
    • Case 2: Transfer the 10∠0° V source to the output branch and calculate response current I' in the original source branch
    • Compare numerical values of I and I' to show I = I', thus verifying reciprocity
  12. 14 Marks
    Obtain the Z and Y parameters of the network shown in fig-10.
    Figure-10: Two-port network ladder. From port 1, a 2 ohm resistor connects to a shunt 4 F capacitor branch, followed by a series 4 ohm resistor connecting to a shunt 2 ohm resistor, and then a series 2 H inductor to port 2.
    How to answer
    • Define Z-parameters: Z11, Z12, Z21, Z22 using open-circuit conditions (I2 = 0 and I1 = 0)
    • Calculate Z11 = V1/I1 |(I2=0) and Z21 = V2/I1 |(I2=0)
    • Calculate Z22 = V2/I2 |(I1=0) and Z12 = V1/I2 |(I1=0)
    • Formulate Z-matrix
    • Compute Y-matrix by inverting the Z-matrix ([Y] = [Z]^-1) or directly using short-circuit tests
  13. 14 Marks
    Write short notes on any two of the following: a) Tellegen's theorem b) Hybrid parameters c) Millman theorem
    How to answer
    • For Tellegen's theorem: state sum of products of branch voltages and currents is zero (sum v_k * i_k = 0); mention independence from element characteristics (linear, nonlinear, time-variant)
    • For Hybrid parameters: define h11, h12, h21, h22 equations (V1 = h11*I1 + h12*V2, I2 = h21*I1 + h22*V2), units, and equivalent circuit
    • For Millman's theorem: state theorem for finding common voltage of parallel voltage sources with series impedances Vm = sum(Ek*Yk) / sum(Yk), and draw equivalent circuit
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