AC power

In a DC circuit, power is simply voltage times current. In an AC circuit, voltage and current oscillate and are generally out of phase, so the power flowing into a load surges and falls twice each cycle and can even reverse briefly. Part of that flow does useful work, heating, turning motors, running computers, and part merely sloshes back and forth between the source and the magnetic and electric fields of inductors and capacitors. Separating the two is essential for sizing equipment, for understanding electricity bills, and for designing power systems that do not waste capacity carrying current that does no work.

This lesson derives average (real) power, introduces reactive, apparent and complex power and the power triangle, explains power factor and how to correct it, covers maximum power transfer for AC circuits, and briefly treats the non-sinusoidal currents drawn by electronic loads.

Instantaneous and average power

Let a load have voltage \(v = \sqrt2\,V\cos(\omega t + \theta_v)\) and current \(i = \sqrt2\,I\cos(\omega t + \theta_i)\), where \(V\) and \(I\) are RMS values. The instantaneous power, using the identity \(\cos A\cos B = \tfrac12[\cos(A - B) + \cos(A + B)]\), is

The first term is constant; the second oscillates at twice the supply frequency and averages to zero. The average, or real, power is therefore

measured in watts, where \(\theta\) is the angle by which voltage leads current, equal to the angle of the load impedance. For a resistor \(\theta = 0\) and \(P = VI = I^2R = V^2/R\), as in DC, provided RMS values are used. For a pure inductor or capacitor \(\theta = \pm 90°\) and \(P = 0\): energy flows in during one quarter-cycle and returns during the next.

Reactive, apparent and complex power

The product \(VI\) is the apparent power \(S\), measured in volt-amperes (VA). It determines the current a source, cable or transformer must carry, and therefore its size, regardless of how much of that current does work. The reactive power

measured in volt-amperes reactive (var), is the amplitude of the energy that oscillates between the source and the load's fields. By convention, inductive loads absorb positive reactive power (\(\theta > 0\), current lagging) and capacitive loads absorb negative reactive power, which is the same as supplying it.

All three are combined in the complex power, computed from phasors as

where \(\mathbf{I}^*\) is the complex conjugate of the current phasor (both phasors with RMS magnitudes). For a load impedance \(\mathbf{Z} = R + jX\), \(\mathbf{S} = I^2\mathbf{Z}\), so \(P = I^2R\) and \(Q = I^2X\): real power is dissipated in resistance and reactive power is associated with reactance.

The three quantities form the power triangle, with \(P\) along the horizontal, \(Q\) vertical and \(S\) as the hypotenuse at angle \(\theta\).

Complex power is conserved: in any circuit, the total complex power delivered by the sources equals the sum of the complex powers absorbed by all the elements. Real powers add, and reactive powers add (with signs). This makes it easy to combine loads without finding every current.

Power factor

The power factor is the ratio of real to apparent power:

It is described as lagging (current lags voltage, inductive load) or leading (capacitive load). Induction motors, transformers and fluorescent ballasts are inductive, with power factors from about 0.7 to 0.9 depending on load; resistive heaters have a power factor of 1; lightly loaded induction motors can fall below 0.5.

A low power factor matters because the current, and hence the losses and voltage drop in every conductor, depend on apparent power. For the same real power, a load at power factor 0.7 draws \(1/0.7 = 1.43\) times the current of a unity-power-factor load and causes \(1.43^2 = 2.04\) times the \(I^2R\) loss in the supply cables. Generators, transformers and cables are rated in VA or amperes, so reactive current uses up capacity that could otherwise carry real power.

Utilities therefore charge large customers for poor power factor. Canadian utilities typically do so through the demand charge or a surcharge: Hydro-Québec, for example, bases the billing demand of many business rates on the greater of the real power demand and 90% of the apparent power demand, which effectively penalizes power factors below 0.9, and other provincial utilities apply power factor surcharges below a similar threshold. Rates differ by utility and change over time, so check the applicable rate schedule.

Power factor correction

Because inductive loads absorb reactive power and capacitors supply it, a capacitor connected in parallel with an inductive load can supply some or all of the load's reactive power locally. The real power is unchanged, but the reactive power, and hence the current, drawn from the supply falls. To raise the power factor of a load of real power \(P\) from \(\cos\theta_1\) to \(\cos\theta_2\), the capacitor must supply

where \(V\) is the RMS voltage across the capacitor.

Over-correction is a hazard: a capacitor bank larger than the load's reactive power makes the power factor leading and can raise the voltage, especially when the load switches off and the capacitors remain connected. Large installations use automatic banks that switch capacitor steps in and out as the load changes, and where harmonics are present (below), capacitors are combined with reactors to avoid resonance.

Maximum power transfer

In ELE-202, a resistive load received maximum power from a source with Thévenin resistance \(R_{th}\) when \(R_L = R_{th}\). For an AC source with Thévenin impedance \(\mathbf{Z}_{th} = R_{th} + jX_{th}\), the load receives maximum average power when its impedance is the complex conjugate:

The load reactance cancels the source reactance, leaving a purely resistive circuit, and the load resistance matches the source resistance. If the load is restricted to a pure resistance, the best choice is \(R_L = |\mathbf{Z}_{th}|\), which delivers less power.

Non-sinusoidal currents and harmonics

Computers, LED drivers, variable-frequency drives and battery chargers draw current in short pulses rather than as a smooth sinusoid. Such a current can be decomposed (by Fourier analysis, studied in ELE-250) into a fundamental at 60 Hz plus harmonics at integer multiples: 180 Hz (3rd), 300 Hz (5th) and so on. Its RMS value combines them:

where THD is the total harmonic distortion. With a sinusoidal voltage, only the fundamental current delivers average power, so harmonics add current and losses without adding real power. The true power factor then has two parts: the displacement power factor \(\cos\theta_1\) of the fundamental, and the distortion power factor \(I_1/I_{\text{rms}}\).

Lesson 11 develops harmonics fully with the Fourier series, including triplen neutral currents and true power factor.