Electrical Principles: Thevenin and RLC Worked Revision Questions

Electrical Principles: Worked Revision Questions

These independently prepared examples practise DC network analysis and AC series resonance. They are not an authenticated copy of a 2021 KNEC paper or an official marking scheme. Read the assumptions, attempt each calculation and use the worked answers to check your method.

Question 1: DC Network Theorems & Kirchhoff’s Laws

(a) State Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).

• Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering any electrical node or junction in a circuit is equal to zero (Sum of I_in = Sum of I_out).
• Kirchhoff’s Voltage Law (KVL): The algebraic sum of all electromotive forces (EMFs) and potential differences around any closed loop in an electric circuit is equal to zero (Sum of V = Sum of I*R).

(b) Two DC voltage sources E1 = 12 V (internal resistance r1 = 1 Ohm) and E2 = 10 V (internal resistance r2 = 2 Ohms) are represented by ideal voltage sources in series with their stated internal resistances. Their positive terminals feed the same load node, with a common negative reference, across a load resistor RL = 5 Ohms. Using Thevenin’s Theorem, determine:
1. The Thevenin open-circuit voltage (V_th).
2. The Thevenin equivalent resistance (R_th).
3. The load current (I_L) and power dissipated in the load resistor.

Model Solution & Derivations:

1. Calculate Thevenin Voltage (V_th):
   With load resistor RL removed from terminals A-B, circulating loop current between sources is:
   I_c = (E1 – E2) / (r1 + r2) = (12 – 10) / (1 + 2) = 2/3 A = 0.667 A.
   V_th = E1 – (I_c * r1) = 12 – (0.667 * 1) = 11.33 Volts.
2. Calculate Thevenin Resistance (R_th):
   Deactivating independent voltage sources (short circuits):
   R_th = (r1 * r2) / (r1 + r2) = (1 * 2) / (1 + 2) = 0.67 Ohms.
3. Calculate Load Current (I_L) & Power (P_L):
   I_L = V_th / (R_th + RL) = 11.33 / (0.67 + 5) = 11.33 / 5.67 = 2.00 Amperes.
   P_L = (I_L)^2 * RL = (2.0)^2 * 5 = 20.0 Watts.

Question 2: AC Series Resonance Calculations

An R-L-C series circuit with R = 15 Ohms, L = 0.2 H, and C = 50 uF is analysed with a 240 V RMS, 50 Hz sinusoidal AC supply. Assume ideal L and C and the stated series resistance. Calculate:
1. Inductive reactance (X_L) and capacitive reactance (X_C).
2. Total circuit impedance (Z) and phase angle.
3. Resonant frequency (f_0).

Model Solution:
– X_L = 2 * pi * f * L = 2 * pi * 50 * 0.2 = 62.83 Ohms.
– X_C = 1 / (2 * pi * f * C) = 1 / (2 * pi * 50 * 50*10^-6) = 63.66 Ohms.
– X_net = X_L – X_C = 62.83 – 63.66 = -0.83 Ohms (Capacitive).
– Z = sqrt(R^2 + (X_L – X_C)^2) = sqrt(15^2 + (-0.83)^2) = 15.02 Ohms.
– Resonant frequency f_0 = 1 / (2 * pi * sqrt(L * C)) = 1 / (2 * pi * sqrt(0.2 * 50*10^-6)) = 50.33 Hz.

Check the DC result another way

Keeping the fractions until the final step gives V_th = 34/3 V and R_th = 2/3 ohm. Therefore I_L = (34/3) / (17/3) = 2 A. The loaded terminal voltage is I_L × R_L = 10 V. In the stated ideal model, the 12 V branch supplies (12 – 10) / 1 = 2 A and the 10 V branch supplies (10 – 10) / 2 = 0 A. The branch currents add to the load current, providing a useful independent check.

Common revision errors

• When finding Thevenin resistance, deactivate only the independent sources according to the circuit model; do not remove the internal resistances.
• Convert 50 microfarads to 50 × 10^-6 F before calculating capacitive reactance.
• Do not add resistance and net reactance as ordinary positive numbers when finding the magnitude of AC impedance.
• Distinguish the 50 Hz supply frequency from the calculated 50.33 Hz resonance frequency. The given circuit is close to resonance, but not exactly at resonance.
• State the phase convention and whether current leads or lags, rather than giving an unexplained signed angle.

These are paper-based circuit models, not instructions to connect unequal real sources together or assemble a mains-voltage experiment. Practical work needs approved equipment and competent supervision.

Further study

OpenStax provides background on Ohm’s law and series RLC circuits. Use your institution’s current syllabus to decide which methods to practise. Browse Jemshah courses for related learning resources, or report a correction with the exact calculation step.