Unit 2: Fundamentals of A.C. circuits

Alternating current (A.C.) is current whose magnitude and direction vary periodically with time, in contrast to the steady direction of direct current. Commercial supply systems (50 Hz in India, 60 Hz in the Americas) are sinusoidal because sinusoids are generated naturally by rotating machines and are preserved in shape under differentiation and integration, which keeps circuit analysis tractable.

• Period and frequency: T (seconds) is the time of one cycle; f = 1/T (hertz) is cycles per second; ω = 2πf (rad/s) is angular frequency.
• Sinusoidal standard form: v(t) = Vm sin(ωt + φ), with Vm the peak value and φ the phase angle.
• Linearity assumption: R, L and C are treated as linear, so steady-state response to a sinusoid is a sinusoid of the same frequency, differing only in amplitude and phase.
• Passive sign convention: current enters the positive terminal of a load; power absorbed is positive.

II. Alternating Current and Voltage — the time-varying source

An alternating quantity reverses polarity each half cycle and averages zero over a full cycle, which is what distinguishes it from D.C.

A. alternating current and voltage

A.C. waveforms are described by instantaneous equations derived from a rotating phasor.

• Generation: a coil rotating in a magnetic field gives e = Em sin(ωt) by Faraday's law, since flux linkage varies as cos(ωt).
• Instantaneous values: v = Vm sin ωt, i = Im sin(ωt ± φ); the sign of φ records lead or lag.
• Why sinusoidal: derivative of a sine is a cosine of equal frequency, so inductor and capacitor relations do not distort the waveform.

B. concept of notations (i, v, I, V)

A strict notation separates time-varying quantities from fixed descriptors.

• Lower case i, v: instantaneous values, functions of time — e.g. v = 325 sin 314t.
• Upper case I, V: magnitudes that do not change with time, normally the RMS values used in phasor and power work.
• Peak subscript Im, Vm: maximum (amplitude) values.
• Bold or hatted V̄, Ī: phasors, carrying both magnitude and angle, e.g. V̄ = V∠φ.

III. Characteristics of an A.C. Signal — the defining measures

Every sinusoid is fixed by its size, its timing and the averages used to rate it.

A. definitions of amplitude, phase, phase difference

Amplitude sets how large the wave is; phase and phase difference set where it sits in time relative to a reference.

• Amplitude: the peak value Vm or Im, the maximum displacement from zero.
• Phase: the angle (ωt + φ) giving the instantaneous position within a cycle; φ is the phase at t = 0.
• Phase difference: the angle φ between two sinusoids of the same frequency.
  • Leading: current peaks before voltage (capacitive tendency), i = Im sin(ωt + φ).
  • Lagging: current peaks after voltage (inductive tendency), i = Im sin(ωt − φ).

B. RMS value and average value of an AC signal

These two averages rate the heating effect and the rectified mean of a waveform.

• RMS (effective) value: the equivalent D.C. producing the same I²R heating.
  
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    Vrms = Vm / √2 = 0.707 Vm   (for a sinusoid)

• Average value: mean over a half cycle (full-cycle mean is zero for a symmetric sine).
  
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    Vavg = 2 Vm / π = 0.637 Vm   (half-cycle sinusoid)

• Form factor: Vrms / Vavg = 1.11 for a sine; peak factor: Vm / Vrms = 1.414.
• Worked example: a 230 V (RMS) supply has Vm = 230 × 1.414 ≈ 325 V and Vavg = 0.637 × 325 ≈ 207 V.

IV. Complex Representation of Impedance — algebra in place of calculus

Impedance recasts the differential relations of L and C as complex multiplication, so A.C. circuits solve like resistive ones.

A. complex representation of impedance

Impedance Z is the complex ratio of phasor voltage to phasor current.

• Definition: Z̄ = V̄ / Ī = R + jX (ohms), where R is resistance and X reactance.
• Element impedances:
  
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    Resistor:  Z = R          (current in phase)
    Inductor:  Z = jωL = jXL  (current lags V by 90°)
    Capacitor: Z = 1/(jωC) = −jXC (current leads V by 90°)

• Magnitude and angle: |Z| = √(R² + X²), θ = tan⁻¹(X/R).
• Admittance: Ȳ = 1/Z̄ = G + jB (siemens), convenient for parallel branches.

V. Steady-State Analysis of A.C. Circuits — series combinations

In the sinusoidal steady state each series circuit is solved by adding complex impedances and applying a phasor Ohm's law Ī = V̄ / Z̄.

A. steady state analysis of ac circuits consisting of RL

A resistor and inductor in series give a lagging current.

• Impedance: Z = R + jXL, |Z| = √(R² + XL²), XL = ωL.
• Phase: current lags voltage by θ = tan⁻¹(XL/R), between 0° and 90°.
• Phasor: VR in phase with I, VL leads I by 90°; V = √(VR² + VL²).

B. steady state analysis of ac circuits consisting of RC

A resistor and capacitor in series give a leading current.

• Impedance: Z = R − jXC, |Z| = √(R² + XC²), XC = 1/(ωC).
• Phase: current leads voltage by θ = tan⁻¹(XC/R).
• Phasor: VC lags I by 90°; the voltage triangle mirrors the RL case.

C. steady state analysis of ac circuits consisting of RLC (series)

All three elements in series combine inductive and capacitive reactance in opposition.

• Net reactance: X = XL − XC; Z = R + j(XL − XC).
• Three regimes:
  1. Inductive, XL > XC: net lagging current, θ > 0.
  2. Capacitive, XC > XL: net leading current, θ < 0.
• Magnitude: |Z| = √(R² + (XL − XC)²), so I = V / |Z|.

VI. Resonance in Series RLC Circuit — the matched-reactance condition

Series resonance occurs when the inductive and capacitive reactances cancel, leaving a purely resistive, minimum-impedance circuit.

A. resonance in series RLC circuit

At resonance the circuit behaves as a pure resistor and draws maximum current.

• Condition: XL = XC, i.e. ωL = 1/(ωC).
• Resonant frequency:
  
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    f0 = 1 / (2π √(LC))   (hertz)

• At f0: Z = R (minimum), I = V/R (maximum), power factor = unity.
• Quality factor: Q = ω0L / R = (1/R)√(L/C); higher Q gives a sharper, narrower response.
• Bandwidth: BW = f0 / Q, the span between the two half-power frequencies.
• Significance: basis of tuning in radio and filter circuits, selecting one frequency from many.

VII. Power Factor and Power Calculation — real, reactive and apparent power

Power in A.C. splits into a component that does work and one that only oscillates, linked by the power factor.

A. power factor and power calculation in RL, RC and RLC circuits

Power factor is the cosine of the phase angle and sets how much apparent power becomes useful power.

• Power triangle:
  
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    Apparent power  S = V·I            (volt-ampere, VA)
    Real power      P = V·I·cos θ      (watt, W)
    Reactive power  Q = V·I·sin θ      (volt-ampere reactive, VAR)
    S² = P² + Q²

• Power factor: pf = cos θ = R / |Z| = P / S.
• RL circuit: lagging pf; θ = tan⁻¹(XL/R), inductor stores energy, Q positive.
• RC circuit: leading pf; θ = tan⁻¹(XC/R), capacitor returns energy, Q negative.
• RLC circuit: pf lagging or leading per sign of (XL − XC); unity at resonance where P = S.
• Worked example: V = 230 V, I = 5 A, θ = 53° lagging gives pf = 0.6, P = 230×5×0.6 = 690 W, Q = 230×5×0.8 = 920 VAR.

VIII. Three-Phase Circuits — balanced polyphase supply

A three-phase system uses three EMFs of equal magnitude displaced by 120°, giving constant total power and more efficient transmission than single phase.

A. three-phase circuits — numbering and interconnection (delta or mesh connection) of three phases

The three phases are labelled and joined so that their voltages sum to zero in balance.

• Numbering: phases R, Y, B (or 1, 2, 3) with voltages 120° apart: eR = Em sin ωt, eY = Em sin(ωt − 120°), eB = Em sin(ωt − 240°).
• Balanced sum: eR + eY + eB = 0 at every instant.
• Delta (mesh) connection: end of one phase joined to the start of the next, forming a closed loop with three line terminals and no neutral; used where a neutral is not required, as in motor windings.

B. relations in line and phase voltages and currents in star and delta networks

Line and phase quantities differ by a factor of √3 fixed by the geometry of the connection.

1. Star (Y) connection: three phase-ends joined at a common neutral.
   
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      VL = √3 · Vph        IL = Iph

   
   Line voltage leads its phase voltage by 30°; a neutral wire is available.
2. Delta (Δ) connection: phases in a closed mesh.
   
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      VL = Vph             IL = √3 · Iph

   
   Line current lags its phase current by 30°; no neutral.
   • Total power (either connection): P = √3 · VL · IL · cos θ, since the √3 and phase/line swaps cancel between the two arrangements.