Electrical Circuits and Machines: Analysis, AC Power and Conversion

Section 3 of the GATE Instrumentation Engineering (IN) paper. This chapter builds a reusable decision process for networks, AC quantities, machines and converters. Start by identifying the model and the conserved quantity: KCL/KVL for circuits, impedance and phasors for sinusoidal steady state, flux and turns ratio for transformers, and duty ratio for switching converters. The goal is not to memorise isolated formulas but to know which representation makes the unknown easiest to find.

Sources, circuit laws and first-order transients

KCL follows conservation of charge at a node; KVL follows conservation of energy around a loop. Independent sources specify their own voltage or current, while dependent sources are controlled by another circuit variable. When a source is deactivated for superposition or Thevenin resistance, an independent voltage source becomes a short circuit and an independent current source becomes an open circuit; dependent sources remain active.

For a first-order RC or RL response, use the same template: x(t) = x(final) + [x(initial) − x(final)]e^(−t/τ). For a simple RC circuit τ = RC; for a simple series RL circuit τ = L/R. The capacitor voltage and inductor current are continuous at an ordinary switching instant because an instantaneous change would require an impulse of current or voltage respectively.

⚠️ Exam trap
Do not blindly write τ = RC or L/R before reducing the network seen by the storage element. With other resistors or sources present, first find the equivalent resistance seen by the capacitor or inductor.

Network theorems and one/two-port thinking

Thevenin and Norton express a linear two-terminal network in forms that make load calculations simple. Vth is the open-circuit terminal voltage and Rth is the resistance seen at the terminals with independent sources deactivated, unless a test source is required because dependent sources are present. Norton current is the short-circuit current and Rn = Rth.

Superposition is linear addition of responses caused by independent sources. It is valid for voltage and current, but power is nonlinear in voltage or current, so powers from individual source cases cannot simply be added to obtain total power. Maximum power transfer is a load-matching result: for a resistive Thevenin network RL = Rth, while for an AC network the conjugate-match condition is ZL = Zth*.

MethodWhat it givesBest use
TheveninVth in series with Rth/ZthArbitrary load
NortonIn in parallel with Rn/ZnParallel-load calculations
SuperpositionOne independent source contribution at a timeUnderstanding source contributions

Sinusoidal steady state, complex power and transformers

A sinusoidal steady-state circuit becomes easier when derivatives are represented by jω multiplication and integrals by division by jω. Resistors, inductors and capacitors have impedances R, jωL and 1/(jωC). Phasor notation removes repeated time-domain differentiation while preserving amplitude and phase information.

For RMS phasors, complex power is S = VI* = P + jQ. P is average real power, Q is reactive power, and |S| is apparent power. Power factor is P/|S|. A transformer ideally transfers AC power through mutual magnetic coupling: V1/V2 = N1/N2 and I1/I2 = N2/N1, while frequency remains unchanged. The ideal transformer does not create energy; it trades voltage level against current level.

⚠️ Common trap
Keep RMS and peak values consistent. If V and I are RMS phasors, S = VI* directly gives watts, vars and VA. Mixing a peak value with an RMS value introduces a factor-of-two error in power.

Electrical machines: energy conversion and operating ideas

An electrical machine converts energy through electromagnetic interaction. In a transformer the conversion is electrical-to-electrical through a changing magnetic field; in a motor electrical input becomes mechanical output; in a generator mechanical input becomes electrical output. The same field principles—flux linkage, induced emf and electromagnetic force or torque—appear in each case.

For rotating machines, speed, electrical frequency and pole count are linked. Synchronous speed is Ns = 120f/P rpm. In an induction machine the rotor runs below synchronous speed in motoring operation, creating slip that permits rotor currents and torque. In a synchronous machine the rotor field locks to the rotating magnetic field at synchronous speed; a permanent-magnet synchronous motor uses permanent magnets for rotor excitation.

MachineEnergy directionKey idea
MotorElectrical → mechanicalElectromagnetic torque
GeneratorMechanical → electricalInduced emf
TransformerElectrical → electricalMutual induction

Power converters, PWM and practical calculation strategy

A switching converter regulates energy by repeatedly connecting and disconnecting a source through semiconductor switches. For an ideal buck converter in continuous conduction, the average output voltage is Vo = DVs, where D is duty ratio. For an ideal boost converter, Vo = Vs/(1−D). These relations are steady-state results and should not be applied blindly during startup or discontinuous conduction.

PWM changes the fraction of each switching period for which a switch is on. It can therefore change average output voltage while the switching frequency remains fixed. A good exam workflow is: identify topology, define the switching states, apply volt-second balance or charge balance where appropriate, then check limiting cases such as D → 0 or D → 1.

🧠 Memory hook
Buck steps voltage down; boost steps it up. Duty ratio controls average energy transfer.

Key takeaways

  • Start circuit problems by choosing the simplest valid representation: KCL/KVL, equivalent circuit, or phasor model.
  • For first-order transients, determine initial value, final value and the resistance seen by the storage element before writing the exponential.
  • Thevenin, Norton and superposition simplify linear networks, but power cannot be superposed directly.
  • Keep RMS phasors consistent when using complex power: S = VI* = P + jQ.
  • Transformer turns ratio changes voltage and current levels but not frequency in the ideal model.
  • Induction-motor torque requires slip; synchronous operation occurs at synchronous speed.
  • Ideal buck and boost formulas describe steady-state continuous-conduction operation, not every transient condition.
  • When a result is unfamiliar, test its dimensions and limiting cases before trusting the algebra.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A capacitor in an ordinary switching circuit is initially at 4 V and finally reaches 10 V. Which value is continuous at the switching instant?

    1. Capacitor voltage
    2. Capacitor current
    3. Resistor power
    4. Source voltage
    Show answer

    Answer: A — Capacitor voltage

    An ordinary capacitor voltage cannot jump instantaneously without an impulse current.

  2. For a resistive Thevenin equivalent with Rth = 6 ohm, maximum DC load power occurs when RL is:

    1. 3 ohm
    2. 6 ohm
    3. 12 ohm
    4. 18 ohm
    Show answer

    Answer: B — 6 ohm

    Maximum power transfer in a resistive DC network occurs at RL = Rth.

  3. Which statements about ideal transformer operation are correct?

    1. Frequency is unchanged
    2. Voltage ratio equals turns ratio
    3. Input and output power are equal in the ideal model
    4. A step-up transformer must increase current
    Show answer

    Answer: A — Frequency is unchanged; B — Voltage ratio equals turns ratio; C — Input and output power are equal in the ideal model

    An ideal transformer changes voltage and current inversely while conserving power and frequency.

  4. An ideal buck converter has input 48 V and duty ratio D = 0.25 in continuous conduction. Enter the average output voltage in volts.

    Numerical answer — type the value.

    Show answer

    Answer: 12

    For an ideal buck converter, Vo = DVs = 0.25 × 48 = 12 V.

  5. For a sinusoidal circuit using RMS phasors, complex power is:

    1. S = VI*
    2. S = V/I
    3. S = V + I
    4. S = V − I
    Show answer

    Answer: A — S = VI*

    With RMS voltage and current phasors, S = VI* gives P + jQ.

  6. A 4-pole machine operates from a 50 Hz supply. Its synchronous speed is:

    1. 750 rpm
    2. 1000 rpm
    3. 1500 rpm
    4. 3000 rpm
    Show answer

    Answer: C — 1500 rpm

    Synchronous speed is Ns = 120f/P = 120 × 50/4 = 1500 rpm. The 3000 rpm option is the answer for a 2-pole machine and 750 rpm for an 8-pole one, so both come from misreading the pole count rather than the formula: at a fixed supply frequency the speed falls as the number of poles rises.

  7. Which statements about superposition in linear circuits are correct?

    1. Independent sources are considered one at a time
    2. Voltage responses can be added algebraically
    3. Current responses can be added algebraically
    4. Power from separate source cases can always be added directly
    Show answer

    Answer: A — Independent sources are considered one at a time; B — Voltage responses can be added algebraically; C — Current responses can be added algebraically

    Linearity permits voltage and current responses to be superposed, but power is nonlinear.

  8. In ideal continuous-conduction operation, which expression gives the boost-converter output voltage?

    1. Vo = DVs
    2. Vo = Vs/(1−D)
    3. Vo = Vs(1−D)
    4. Vo = Vs/D
    Show answer

    Answer: B — Vo = Vs/(1−D)

    For an ideal boost converter in continuous conduction, Vo = Vs/(1−D).