Sinusoids and phasors
Mains electricity and most signals are, or can be built from, sine waves. Phasors turn calculus on sinusoids into simple arithmetic with complex numbers.
Sinusoidal signals
A sinusoidal voltage is \(v(t) = V_m \cos(\omega t + \phi)\), with amplitude \(V_m\), angular frequency \(\omega = 2\pi f\) in radians per second, frequency f in hertz, and phase \(\phi\). The period is \(T = \frac{1}{f}\). Mains electricity runs at 50 Hz in most of the world, including Europe, Africa, most of Asia, and Australia, and at 60 Hz in North America, parts of South America, and some other countries.
Why sinusoids matter
Sinusoids are natural outputs of rotating generators. Linear circuits driven by a sinusoid respond with sinusoids of the same frequency, changing only amplitude and phase. And by Fourier analysis, almost any periodic signal can be built from sinusoids of different frequencies, so understanding sinusoidal response explains how circuits treat any signal.
RMS values
The root-mean-square (RMS) value of a periodic signal is the DC value that would deliver the same average power to a resistor. For a sinusoid, \(V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707V_m\). Mains voltages are quoted as RMS: 230 V RMS mains has a peak of about 325 V, and 120 V RMS has a peak of about 170 V. Meters usually read RMS.
Phase relationships
Two sinusoids of the same frequency may be shifted in time. If one reaches its peak before the other, it leads; the other lags. Phase differences are measured in degrees or radians: 90 degrees is a quarter cycle. Phase relationships between voltage and current determine how much power is actually delivered.
Phasors
A phasor represents a sinusoid by a complex number holding its amplitude and phase: \(V_m \cos(\omega t + \phi)\) corresponds to \(\mathbf{V} = V_m \angle \phi\). The frequency is understood. This works because of Euler's identity, \(e^{j\theta} = \cos\theta + j\sin\theta\); engineers write j for the imaginary unit since i denotes current.
Complex number arithmetic
Phasors can be written in rectangular form, \(a + jb\), convenient for adding, or polar form, \(r \angle \theta\), convenient for multiplying and dividing. Converting: \(r = \sqrt{a^2 + b^2}\), \(\theta = \tan^{-1}(b/a)\) (adjusting for quadrant), \(a = r\cos\theta\), \(b = r\sin\theta\). Multiplying phasors multiplies magnitudes and adds angles.
Calculus becomes algebra
Differentiating a sinusoid multiplies its phasor by \(j\omega\), and integrating divides by \(j\omega\). So the differential equations of circuits with capacitors and inductors become algebraic equations in phasors. All the DC techniques, Kirchhoff's laws, nodal and mesh analysis, Thevenin equivalents, then apply directly using complex arithmetic.
Worked example
Add \(v_1 = 10\cos(\omega t)\) and \(v_2 = 10\cos(\omega t + 90^\circ)\). As phasors: \(10\angle 0^\circ = 10 + j0\) and \(10\angle 90^\circ = 0 + j10\). Their sum is \(10 + j10 = 14.14\angle 45^\circ\). So \(v_1 + v_2 = 14.14\cos(\omega t + 45^\circ)\): adding two equal sinusoids a quarter-cycle apart gives a larger sinusoid midway in phase.
Key takeaways
Sinusoids have amplitude, angular frequency omega = 2 pi f, and phase; mains runs at 50 Hz in most of the world and 60 Hz in North America. Linear circuits keep frequency and change only amplitude and phase, and Fourier analysis builds signals from sinusoids. RMS is the power-equivalent DC value: V_m divided by root 2, so 230 V RMS peaks near 325 V. Leading and lagging describe phase shifts. Phasors encode amplitude and phase as complex numbers via Euler's identity, using j. Rectangular form suits addition, polar form multiplication. Phasors turn differentiation into multiplication by j omega, so DC methods apply.
Flashcard
omega = 2 pi f
Check your understanding
A sinusoidal voltage has a peak of 170 V. What is its RMS value approximately?