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UY1: Phasors & Alternating Currents

Represent sinusoidal AC signals with phasors, combine phase-shifted signals cleanly, and compute RMS values correctly.

  • University Physics Year 1
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Learning objectives

  • Analyse magnetic forces, induction, inductance, and alternating-current systems with consistent signs.
Why this matters + quick links

This page gives the UY1 working model/result for Phasors & Alternating Currents. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

  • Phasors convert sinusoidal time functions into rotating vectors/complex amplitudes.
  • Use phasors only when signals have the same angular frequency ω.
  • Choose and stick to a convention, e.g. x(t) = Re {X tilde e^(iω t)} (this page uses cosine reference).
  • Differentiation and integration become algebra:
d/dt ⟷ iω, ∫ dt ⟷ 1/iω.
  • RMS values for pure sinusoids:
Iᵣₘₛ = I₀/(square root of 2), Vᵣₘₛ = V₀/(square root of 2).

Prerequisites: Complex Numbers
Next uses: Resistors, Inductors & Capacitors In A.C. Circuits, L-R-C Series Circuit With A.C.

2) Setup

Write AC quantities as:

v(t) = V₀ cos(ω t + φᵥ), i(t) = I₀ cos(ω t + φᵢ).

Define phasors (complex amplitudes):

V tilde = V₀e^iφᵥ, I tilde = I₀e^iφᵢ,

with physical signal as real part of V tilde e^(iω t) or I tilde e^(iω t).

Common traps (phasor conventions)
  • Don’t mix conventions. If you define x(t) = Re {X tilde e^(iω t)}, then your derivative rule is d/dt → iω.
  • Don’t add phasors of different frequencies: a 50 Hz term and a 100 Hz term do not have a single fixed phase relation.
  • Be consistent with angle units. Phasor arithmetic is clean, but sin/cos arguments in calculus must be in radians.
  • Keep track of peak vs RMS: phasors can be defined using either, but don’t mix them in one calculation.

3) Core derivation/explanation

Why phasors help:

  • Differentiation/integration in time become multiplication/division by iω.
  • Addition of sinusoids becomes vector addition of complex numbers.

If

v = v₁ + v₂,

then

V tilde = V tilde ₁ + V tilde ₂.

RMS derivation for current:

Iᵣₘₛ = square root of (1/T∫₀^T i²(t) dt) = square root of (1/T∫₀^T I₀² cos² (ω t) dt) = I₀/(square root of 2).

Same for voltage.

Checks (sanity)

  • If two equal-amplitude phasors are 180° apart, their sum is zero.
  • The magnitude of a sum is bounded: ||V tilde ₁|-|V tilde ₂|| ≤ |V tilde ₁ + V tilde ₂| ≤ |V tilde ₁| + |V tilde ₂|.

4) Worked example(s)

Add two voltages of same frequency:

v₁ = 10 cos ω t, v₂ = 10 cos(ω t + 60°).

Phasors:

V tilde ₁ = 10, V tilde ₂ = 10e^(i60°) = 5 + i8.66.

Total:

V tilde = 15 + i8.66.

Magnitude/phase:

|V tilde | = square root of (15² + 8.66²) = 17.3, ∠V tilde = tan⁻¹ (8.66/15) = 30°.

So:

v(t) = 17.3 cos(ω t + 30°).

5) Practice set (with hints + answers)

  1. For i(t) = 6 cos(ω t-20°) A, find Iᵣₘₛ. Hint: divide amplitude by square root of 2. Answer: 4.24 A.

  2. Add 5 cos ω t and 5 cos(ω t + 180°). Hint: opposite phasors. Answer: zero.

  3. Why is phasor addition invalid if one term has ω and another has 2ω? Hint: rotating vectors have different speeds. Answer: no fixed phase relation; single-phasor representation fails.

6) Summary + next steps

  • Phasors simplify AC algebra and make phase relations visual.
  • RMS gives DC-equivalent heating values.
  • Next lessons apply this to R, L, C elements and full AC series circuits.

Next: Resistors, Inductors & Capacitors In A.C. Circuits Previous: R-L Circuit Back To Electromagnetism (UY1)