Power and Average Power in Fourier Series

Power Factor
Fourier Series of an Inverter Circuit
Fourier Series of Full-Wave Rectifier Circuit

In a power conversion sysytem, the major issue is the control of energy flow between a source and a load. The usual measure of energy flow in an electrical network is the average rate at which energy is transfered, or the average power P. This can be computed from the instantaneous power, p(t) = v(t)i(t), with the average integration

P = \frac{1}{T} \int_\tau^{\tau+T} v(t)i(t)dt

Average power can also be computed by averaging the products of the Fourier Series of the voltage and current. This approach reveals some interesting properties, and we need to examine it. First, assume that the voltage v(t) and the current i(t) are periodic waveforms with the same period T = 2\pi/ \omega. The have Fourier Series that can be written as

v(t) = \sum\limits_{n=0}^\infty c_n cos(n\omega t + \theta_n)

i(t) = \sum\limits_{m=0}^\infty d_m cos(m\omega t + \phi_m)

Average Power

A plot of the ideal case where voltage and current in the circuit are exactly in phase producing a constant average power.

Notice that the coefficients and summation indices differ so we can tell them apart. The average power is the integral of the product. With the distributive law, the product can be written as a nested sum so the the integral becomes

P_{avg} = \frac{\omega}{2\pi} \int_0^{2\pi / \omega} \sum\limits_{n=0}^\infty [ \sum\limits_{m=0}^\infty c_n d_m cos(n\omega t + \theta_n) cos(m\omega t + \phi_m)] dt

This expression is linear, so we can integrate each term in this double series and add the result. This is expressed as

P_{avg} = \sum\limits_{n=0}^\infty [ \sum\limits_{m=0}^\infty \frac{\omega}{2\pi} \int_0^{2\pi / \omega} c_n d_m cos(n\omega t + \theta_n) cos(m\omega t + \phi_m) dt]

This sum of integrals is simplified considerably y applying the orthogonality relations to the terms. For a given term, the integral will be

2_24

Only the terms in the summation for which n=m will contribute to the total. The total becomes

P = c_0 d_0 + \sum\limits_{n=0}^\infty \frac{c_n d_n}{2} cos(\theta_n - \phi_n)

But the RMS magnitude of each Fourier component is |v_{RMS}(n)| = c_n / \sqrt{2} and  |i_{RMS}(n)| = d_n / \sqrt{2} for n \neq 0. This means that the average power can be written

P_{avg} = \sum\limits_{n=0}^\infty |v_{RMS}(n)| |i_{RMS}(n)| cos(\theta_n - \phi_n)

if we define phases \theta_0 = 0 and \phi_0 = 0.

The average power contribution by each Fourier component is the RMS magnitude of the voltage times the RMS magnitude of the current times the cosine of the phase angle difference.

But it is also true that purely sinusoidal voltage and current will give this same result! None of the cross-terms, involving different frequency components of voltage and current, appear in the final expression. Two crucial facts emerge:

  1. The average power for a given periodic function is the sum of the average powers contributed by each Fourier component of that function.
  2. Only Fourier components which appear in both the current and the voltage can contribute to the average power. Cross-frequency terms do not contribute.

The second fact can also be written as the frequency matching constraint.

Only voltage and current Fourier components of equal frequency contribute to the power.

Both of the facts are extremely important. The first means that we can apply conservation of power and energy component-by-component for a signal with a Fourier Series. The second is a tight restriction on how to transfer energy. An AC-DC converter, for example, will have average output power at DC only if both the output voltage and the output current have a DC component. A DC-AC converter intended for 60 Hz output will produce average power at 60 Hz only if the output foltage and current both have a 60 Hz Fourier component.

The average power results are one of the foundations of power electronics and are the basis of all choices of switch operation. They can make analysis very direct.

Fourier Series of an Inverter Circuit
Fourier Series of Full-Wave Rectifier Circuit

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