AC Circuit (Phasor Diagram)

Historical Context & Motivation

The story of phasor diagrams is inseparable from the electrification of the modern world. In the late nineteenth century, engineers faced a daunting challenge: how to design, predict, and troubleshoot circuits carrying currents that reverse direction dozens of times every second. Direct-current (DC) analysis—where voltage and current are constant—was already well understood, but alternating current (AC) introduced a new dimension: time-varying phase relationships between voltages and currents. The phasor method emerged as an ingenious shortcut, replacing calculus-heavy differential equations with simple vector geometry.

The central question that phasor diagrams answer is deceptively simple: when voltage and current both oscillate sinusoidally at the same frequency but may peak at different times, how do we keep track of their relative timing, add them, and compute power—without writing integrals for every calculation?

Core Principles & Definitions

A phasor is a rotating vector whose projection onto a reference axis generates a sinusoidal waveform. Instead of tracking the full time-domain function v(t) = V₀ sin(ωt + φ), we represent it as a vector of length V₀ pointing at angle φ in a two-dimensional plane. Because every quantity in a linear AC circuit oscillates at the same angular frequency ω, we can "freeze" the rotation and work entirely with magnitudes and angles. Below are the five foundational ideas you need.

Visual Explanation — The Phasor Diagram

The phasor diagram below shows the fundamental relationship between the voltage phasors across a resistor (V_R), an inductor (V_L), and a capacitor (V_C) in a series RLC circuit driven by a sinusoidal source. The current phasor I is chosen as the reference (pointing along the positive horizontal axis) because in a series circuit the same current flows through every element.

In this diagram the current phasor I lies along the positive real axis. Because current through a resistor is in phase with the voltage across it, V_R also points horizontally. An ideal inductor causes voltage to lead the current by 90°, so V_L points straight up. Conversely, a capacitor's voltage lags the current by 90°, so V_C points straight down. The total source voltage V_total is the vector sum of all three component voltages. Its angle with respect to I is the circuit's phase angle φ, which tells us whether the circuit is overall inductive (φ > 0, voltage leads current) or capacitive (φ < 0, voltage lags current).

Mathematical Framework

The phasor diagram is the visual face of an algebraic framework built on complex numbers. Below are the four key equations that govern series AC circuits. In every case, angular frequency ω = 2πf, where f is the frequency of the source in hertz.

The impedance Z is a complex number whose real part is the resistance and whose imaginary part is the net reactance X = X_L − X_C. When X > 0 the circuit behaves inductively; when X < 0 it behaves capacitively; when X = 0 the circuit is at resonance.

The phase angle φ appears directly in the phasor diagram as the angle between the total voltage phasor and the current reference. It also determines the power factor, cos φ, which quantifies how much of the apparent power is actually doing useful work.

Notice the remarkable parallel with DC circuits: Ohm's law V = IR is simply extended to V = IZ, where both V and I are now phasors (complex numbers) and Z is the complex impedance. Multiplication by a complex number simultaneously scales the amplitude and rotates the phase — exactly what inductors and capacitors do to the voltage relative to the current.

Detailed Breakdown — Impedance Triangle & Component Behaviour

The impedance of every two-terminal combination of R, L, and C can be represented as a right triangle known as the impedance triangle. This triangle is the backbone of the phasor diagram: its horizontal leg is R, its vertical leg is the net reactance X, its hypotenuse is |Z|, and its angle is the phase angle φ. Below is a detailed diagram showing both the impedance triangle and how voltage phasors map onto it.

The mnemonic "ELI the ICE man" is a time-tested way to remember phase relationships. In an inductor (L), the EMF (E) leads the current (I) — E-L-I. In a capacitor (C), the current (I) leads the EMF (E) — I-C-E. When both reactive elements are present, their vertical phasors partially cancel, and the remaining net reactance determines whether the overall circuit is inductive or capacitive.

Worked Example

A series RLC circuit has R = 40 Ω, L = 0.15 H, and C = 50 μF. It is connected to a 120 V_rms, 60 Hz AC source. Find the impedance, current, phase angle, component voltages, and power factor. Draw the phasor diagram description.

Strengths, Limitations & Comparisons

Phasor diagrams are extraordinarily powerful, but like any model they have boundaries. Understanding when the technique excels and when it falls short is essential for choosing the right analytical approach.

The phasor method's greatest strength is its ability to convert calculus into geometry. Engineers can see at a glance whether a circuit is inductive or capacitive, estimate the power factor from the angle, and predict the effect of changing a component value. However, it is limited to linear circuits at a single frequency in sinusoidal steady state. Circuits with switching transients, distorted waveforms, or nonlinear devices (diodes, transistors operating in saturation) require time-domain or more advanced frequency-domain techniques (Fourier analysis, Laplace transforms).

Connection to Advanced Theory

Phasor analysis is not a dead-end technique; it is the gateway to a family of increasingly powerful frequency-domain methods used throughout electrical engineering, signal processing, and physics.

When you evaluate a Laplace-domain transfer function at s = jω, you recover exactly the phasor result. In this sense, phasor analysis is a special case of the Laplace transform restricted to steady-state sinusoids. The Bode plot — a cornerstone of control theory and filter design — is simply the magnitude and phase of the transfer function plotted versus frequency, which is a continuous family of phasor diagrams, one for each frequency.

Three-phase power systems extend phasor diagrams into sets of three phasors separated by 120°, enabling compact analysis of industrial power distribution. In quantum mechanics and wave optics, phasors (often called "amplitudes" or "Jones vectors") describe the superposition of electromagnetic waves, linking this circuit technique to some of the deepest ideas in modern physics.

Practice Problems

Lesson Summary