Fourier Series Representations of Switching Functions

Fourier Series of an Inverter Circuit

To take full advantage of Fourier analysis, we need to consider series representations of switching functions. The idea is to figure out the Fourier Series of a switching function in some general way, in order to avoid recomputing it multiple times in the future. Since any switching function q(t) is either 0 or 1, and is normally periodic, a plot of a given q(t)  will be a train of square pulses. Let us take a general pulse train and figure out its Fourier Series. A pulse train of arbitrary period T, with one pulse centered on the time t = t_0, is shown below. The pulses in the train each have duration DT , where D is defined as the duty cycle or duty ratio. Notice that 0 \leq D \leq 1. This pulse train has frequency f = 1/T and radian frequency \omega = 2\pi/T.

A periodic pulse train.

A periodic pulse train.

Let us compute the Fourier components of this q(t). The DC component a_0 is given by

a_0 = \frac{1}{T} \int^{t_0+T/2}_{t_0-T/2} q(t) dt

Only the portion of each cycle with q(t) = 1 will contribute to the integral. Since q(t) = 0 except between t_0-DT/2 and t_0+DT/2, the integral becomes

a_0 = \frac{1}{T} \int^{t_0+T/2}_{t_0-T/2} (1) dt = \frac{DT}{T} = D

An inspection of the waveform above confirms that it has a time average value of D, the duty cycle.

The values of a_0 and b_0, can be found and used to compare c_n and \theta_n. This is left as an exercise. The results are:

c_n = \frac{2}{\pi} \frac{sin(n\pi D)}{n} for n \neq 0

The value of phase is \theta_n = n \omega t_0, where \omega = 2\pi /T. It is conventional to define a reference phase \phi_0 = \omega t_0, so that \theta_n = -n \phi_0. Thus, the Fourier Series for this very general q(t) can be written

q(t) = D + \frac{2}{\pi} \sum \limits_{n=1}^\infty \frac{sin (n\pi D)}{n} cos(n \omega t - n \phi_0)

The Fourier representation shows that the function q(t) is determined completely by just three parameters: the duty cycle D, the radian frequency \omega = 2\pi f (or the period T), and the reference time t_0 (or the reference phase \phi_0). These three numbers fully define the switching function, and the switch action can always be interpreted in terms of one of more of them.

The switch action in a power converter will usually have to adjust over time to account for any changes in the environment, the input source, or the output load. The series representation makes a few possibilities salient:

  1. Duty cycle adjustment. The duty factor is related to the pulse width DT. Converters that operate by adjusting the duty cycle of their switches exhibit pulse-width modulation (PWM) action.
  2. Frequency adjustment. Frequency adjustment is unusual in power electronics for a very basic reason: The need to provide matching frequency components of voltage and current in a given source or load often places tight constrains on the switching frequencies. The most important exception is in DC-DC conversion. Since only the average values is of interest in the DC case, it is possible to adjust the frequency without causing trouble. This concept is called frequency control. Although true frequency control modulation (FM) is rare in power conversion, it is mathematically possible.
  3. Timing adjustment. One of the oldest ways to alter the behavior of a power converter is to adjust the timing of the switch action. Since the wave shapes are almost always more important than the specific frequencies in this technique, timing adjustmen is normally studied in the angular time frame \theta = \omega t. The term phase control describes the idea of adjusting the switch action in time. Some converters vary the phase in a regular way, which corresponds to phase modulation (PM).

Most of the converters we will study use either PWM or phase control to permit adjustment of their operation.

Fourier Series of an Inverter Circuit

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