Phasors and impedance

Almost all electric power is generated, transmitted and used as sinusoidal alternating current, and sinusoids are also the building blocks of signals: any periodic waveform can be built from them. When a linear circuit is driven by a sinusoid, every voltage and current in it settles into a sinusoid of the same frequency, differing only in amplitude and phase. Solving for those amplitudes and phases with differential equations is laborious. Phasors replace the calculus with complex-number algebra, so that every technique from ELE-202, including Ohm's law, Kirchhoff's laws, nodal and mesh analysis and Thévenin equivalents, works unchanged for AC circuits.

Sinusoids

A sinusoidal voltage is written

where \(V_m\) is the amplitude (peak value), \(\omega\) the angular frequency in rad/s, and \(\phi\) the phase angle. The frequency in hertz is \(f = \omega/2\pi\) and the period \(T = 1/f\). North American power systems run at 60 Hz, so \(\omega = 2\pi \times 60 = 377\) rad/s and \(T = 16.7\) ms.

This course uses cosine as the reference. A sine is a cosine delayed by 90°: \(\sin\omega t = \cos(\omega t - 90°)\). When comparing two sinusoids of the same frequency, the one whose peaks occur earlier leads the other; \(\cos(\omega t + 30°)\) leads \(\cos\omega t\) by 30°, and \(\cos\omega t\) lags it by 30°.

RMS values

The effective, or root-mean-square (RMS), value of a periodic waveform is the DC value that would deliver the same average power to a resistor:

AC voltages and currents are almost always quoted as RMS values. The 120 V household supply has a peak of \(120\sqrt{2} = 170\) V, and a 600 V industrial system has line-to-line peaks of 849 V. Meters labelled "true RMS" compute the RMS of any waveform; cheaper meters assume a sinusoid and read incorrectly on distorted waveforms, such as the current drawn by electronic power supplies.

Complex numbers

Phasors use complex numbers, written in rectangular form \(z = x + jy\) (engineers use \(j\) for \(\sqrt{-1}\), because \(i\) denotes current) or polar form \(z = r\angle\theta\), where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\) (adjusted to the correct quadrant). The two are related by \(x = r\cos\theta\), \(y = r\sin\theta\), and Euler's formula connects them to the exponential:

Addition and subtraction are easiest in rectangular form; multiplication and division in polar form, where magnitudes multiply or divide and angles add or subtract. The complex conjugate \(z^* = x - jy = r\angle{-\theta}\). Most engineering calculators handle both forms directly, and it is worth learning to use that feature.

Phasors

By Euler's formula, \(V_m\cos(\omega t + \phi) = \text{Re}\{V_m e^{j\phi} e^{j\omega t}\}\). In a circuit where every quantity oscillates at the same \(\omega\), the factor \(e^{j\omega t}\) is common to everything and can be dropped. What remains, the complex number carrying amplitude and phase, is the phasor:

Geometrically, the phasor is an arrow of length \(V_m\) at angle \(\phi\); the real signal is the projection onto the real axis of that arrow rotating counterclockwise at \(\omega\). In power engineering, phasors are usually written with RMS magnitude (120∠0° V rather than 170∠0° V); either convention works if it is used consistently, and this course states which is used. Here, unless stated otherwise, phasor magnitudes are RMS.

The key property of phasors is that differentiation becomes multiplication by \(j\omega\), and integration division by \(j\omega\), because \(\frac{d}{dt}e^{j\omega t} = j\omega e^{j\omega t}\). That turns the differential equations of capacitors and inductors into algebraic equations.

Impedance

Applying the phasor rule to each element's equation gives a ratio of voltage phasor to current phasor called impedance, \(\mathbf{Z} = \mathbf{V}/\mathbf{I}\), measured in ohms:

The magnitudes \(X_L = \omega L\) and \(X_C = 1/(\omega C)\) are the inductive and capacitive reactances. An inductor's impedance grows with frequency (it is a short circuit at DC and an open circuit at very high frequency); a capacitor's falls with frequency (open at DC, short at very high frequency). A useful memory aid is "ELI the ICE man": in an inductor (L), voltage (E) leads current (I); in a capacitor (C), current leads voltage.

A general impedance is \(\mathbf{Z} = R + jX\): its real part is resistance and its imaginary part reactance, positive for inductive and negative for capacitive circuits. The reciprocal, admittance \(\mathbf{Y} = 1/\mathbf{Z} = G + jB\), measured in siemens, is convenient for parallel circuits.

Impedances combine exactly like resistances: in series they add, in parallel their admittances add, and \(\mathbf{Z}_1 \parallel \mathbf{Z}_2 = \mathbf{Z}_1\mathbf{Z}_2/(\mathbf{Z}_1 + \mathbf{Z}_2)\). Voltage and current dividers, delta-wye transformations, nodal and mesh analysis, superposition, and Thévenin and Norton equivalents all carry over with complex numbers in place of real ones.

Parallel circuits and admittance

AC circuit analysis

Nodal and mesh analysis proceed exactly as in ELE-202, producing simultaneous linear equations with complex coefficients. The steps are: convert every source to a phasor and every element to an impedance at the source frequency; solve with the usual method; convert the phasor answers back to time functions if needed. If sources of different frequencies are present, they must be analysed separately (each with its own impedances) and the time-domain results added by superposition; phasors of different frequencies cannot be added.

Lesson 4 applies phasors systematically, with nodal and mesh analysis, Thévenin equivalents, superposition of different frequencies and op-amp circuits.