AC Signal Representation

Alternating current (AC) is the form of electricity that powers virtually every home, school, and factory on Earth. Unlike direct current (DC), where charge flows steadily in one direction, AC reverses direction periodically — the voltage and current oscillate back and forth in a smooth, repeating wave called a sinusoid. The standard household supply in most countries delivers voltage that swings from a positive peak to a negative peak and back again 50 or 60 times every second. The mathematical description of an AC signal uses the sine (or cosine) function from trigonometry. A voltage signal can be written as v(t) = V_m sin(ωt + φ), where V_m is the peak amplitude, ω is the angular frequency, t is time, and φ is the phase angle. Understanding what each of these quantities means — and how they combine — is the foundation for analysing every circuit that runs on AC power, from the simplest light bulb to complex radio transmitters. Engineers and physicists use a powerful shorthand called a phasor to represent AC signals without writing out the full sine function every time. A phasor is essentially a rotating arrow in a mathematical plane: its length represents the amplitude of the signal and its angle represents the phase. Phasors transform the algebra of oscillating waves into the geometry of rotating vectors, making circuit calculations far more tractable. Mastering AC signal representation unlocks the study of impedance, resonance, filters, and power transmission.

Key takeaways

The Sinusoidal Waveform
Frequency and Angular Frequency
Phase Angle and Phase Difference
Phasor Representation
RMS Values in Practice

The Sinusoidal Waveform

An AC voltage waveform is described by v(t) = V_m sin(ωt + φ). The amplitude V_m (in volts) is the maximum value the signal reaches. The argument (ωt + φ) is an angle measured in radians that increases with time. Because sin repeats every 2π radians, the signal repeats every time ωt increases by 2π — that is, every period T = 2π/ω seconds. Think of it like a wheel rolling along a flat road: one full rotation of the wheel equals one complete cycle of the wave.

Frequency and Angular Frequency

The ordinary frequency f tells you how many complete cycles occur per second, measured in hertz (Hz). The angular frequency ω = 2πf converts cycles per second into radians per second, because each cycle sweeps through 2π radians. For UK mains electricity, f = 50 Hz, so ω = 2π × 50 ≈ 314 rad s⁻¹. The factor of 2π appears constantly in AC analysis, so using ω simplifies most formulas — just as astronomers prefer using radians for angular calculations rather than degrees.

Phase Angle and Phase Difference

The phase angle φ (measured in radians or degrees) shifts the waveform left or right in time. If φ = 0, the wave starts at zero and climbs upward at t = 0. If φ = π/2, the wave is already at its peak when t = 0 — it is a cosine. When comparing two signals in the same circuit (say, voltage and current through a capacitor), we talk about the phase difference between them. A current that reaches its peak before the voltage is said to lead; one that reaches its peak after lags. This lead/lag behaviour is the key signature of capacitors and inductors.

Phasor Representation

A phasor is a complex number (or equivalently, a 2D vector) that encodes both the amplitude and the phase of an AC signal in a single object. For a signal v(t) = V_m cos(ωt + φ), the phasor is written as V = V_m∠φ, or in complex form V = V_m e^{jφ} = V_m(cos φ + j sin φ). The actual time-domain signal is recovered by multiplying by e^{jωt} and taking the real part. The genius of phasors is that adding or subtracting AC signals becomes vector addition, and differentiating a signal (needed for capacitors and inductors) becomes multiplication by jω — turning differential equations into simple algebra.

RMS Values in Practice

The peak voltage V_m is not what appears on a power rating. Instead, engineers use the root-mean-square (RMS) value, V_rms = V_m / √2 ≈ 0.707 V_m. The RMS value is defined so that an AC supply of V_rms delivers the same average power to a resistor as a DC supply of the same numerical value. The UK mains is rated at 230 V RMS, meaning its peak voltage is actually 230√2 ≈ 325 V. Always distinguish between peak and RMS values when solving problems.

The Key Equation

v(t) = V_m \sin(\omega t + \varphi)

v(t) — Instantaneous voltage (V): The voltage at a specific moment in time; it oscillates continuously between +V_m and −V_m.
V_m — Peak (amplitude) voltage (V): The maximum voltage reached by the signal; equals √2 times the RMS voltage.
\omega — Angular frequency (rad s⁻¹): Equals 2πf; the rate at which the phase angle increases with time.
t — Time (s): Independent variable; the signal repeats with period T = 2π/ω.
\varphi — Phase angle (rad): Sets the starting position of the wave at t = 0; positive φ shifts the wave earlier in time (leads).

Worked Example

Problem. An AC generator produces a voltage given by v(t) = 340 sin(100πt + π/6) volts. Find: (a) the peak voltage, (b) the angular frequency, (c) the ordinary frequency, (d) the period, (e) the RMS voltage, and (f) the phasor representation.

V_{\rm rms} = \frac{340}{\sqrt{2}} \approx 240\,\text{V},\quad \mathbf{V} = 340\angle 30^\circ\,\text{V}

Answer. Peak voltage 340 V; ω = 100π rad/s; f = 50 Hz; T = 20 ms; V_rms ≈ 240 V; phasor V = 340∠30° V.

Figure

Diagram

Origins · 1 minCharles Proteus Steinmetz and how this was discovered

Charles Proteus Steinmetz

1865–1923 · German-American

Steinmetz introduced the use of complex numbers and phasor notation to analyse AC circuits in the 1890s while working at General Electric. His 1893 paper showed engineers how to replace messy differential equations with simple complex-number algebra — a transformation so powerful that it is still the standard method taught today. He also developed practical theories of magnetic hysteresis and AC machinery that made long-distance power transmission commercially viable. Steinmetz was barely 1.5 m tall due to a spinal condition, yet he became one of GE's most celebrated scientists. He reportedly once billed GE $10,000 for a consulting job (equivalent to hundreds of thousands today), writing on the invoice: 'Making chalk mark on generator:$ 1. Knowing where to make it: $9,999.'

Impedance and Reactance — Impedance is the AC generalisation of resistance, expressed using phasors; understanding AC signal representation is prerequisite for defining and calculating impedance in resistors, capacitors, and inductors.
Resonance in LC Circuits — Resonance occurs when the phasors of inductive and capacitive reactances cancel; phasor analysis makes the resonance condition immediately visible as a phase alignment.
Faraday's Law of Electromagnetic Induction — AC generators produce sinusoidal voltages precisely because of Faraday's law applied to a rotating coil in a magnetic field — the physical origin of the sinusoidal waveform.
Trigonometric Functions and Waves — The sine and cosine functions are the mathematical backbone of every AC waveform; fluency with their graphs, identities, and properties is essential for AC analysis.