Friday, October 5, 2012

On this page, an the Fourier Series is applied to a real world problem: determining the solution for an electric circuit. Particularly, we will look at the circuit shown in Figure 1:

An R-C circuit Figure 1. A series R-C circuit.
In Figure 1, there is a source voltage, Vs, in series with a resistor R, and a capacitor C. We are interested in finding the voltage across the capacitor, which we label as the output voltage. Side note: this simple circuit can be used as a low pass filter: high frequency noise can be elminated.
The source voltage Vs(t) will be a periodic square wave shown in Figure 1. The Fourier Series coefficients for this function have already been found on the complex coefficients page.

Fourier series example Figure 2. A periodic square waveform.
Electric circuits like that of Figure 1 are easily solved in the source voltage is sinusoidal (sine or cosine function). When this happens, the capacitor has an impedance that is easily calculable. The impedance is analogous to resistance: it is the ratio of the voltage across the capacitor to the current that flows through it. For capacitors and inductors, the impedance is a complex number (meaning the voltage and current are out of phase), that depends on the frequency of the sinusoidal source voltage. If the source voltage has frequency f, then the impedance of the capacitor (Zc) is:

impedance of a capacitor
[Equation 1]

The output voltage can be easily found with some application of Ohm's Law (V=I*Z):

solution for output of voltage across capacitor
[Equation 2]

From equation [2], we see that the output voltage can be easily calculated when the source voltage Vs is sinusoidal.
The question now is: how can we calculate the output voltage, Vo(t), when the input is not a sinusoidal function, but rather any periodic function f(t)?
If you don't think the answer has something to do with Fourier Series, you probably need to work on your reading comprehension skills. It just so happens that we know that any periodic function IS the sum of sinusoidal functions. And we know how to solve the circuit of Figure 1 for any sinusoidal input. Finally, electric circuits are simple linear systems: this means that if an input voltage V1 produces an output X1, and an input voltage V2 produces an output X2, then when the input V1+V2 is applied, the output is X1+X2.
The facts in the proceeding paragraph mean that with Fourier Series, the solution is very simple. We rewrite the square wave in terms of a sum of sinusoidal functions, calculate the output via each one, and then sum up the solutions for each sinusoidal component.

To do this, let's choose a random component of the Fourier Series, say the nth component, corresponding to coefficient cn. This is the coefficient that multiplies the complex exponential, with frequency given by f=n/T:

source voltage, component n
[Equation 3]

Using equation [3] in equation [2], the output voltage for just this sinusoid (or complex exponential, they are basically the same), is:

output voltage for component n of input
[Equation 4]

In equation [4], note that the frequency f has been substituted with n/T, because that is the frequency of the corresponding complex exponential that the cn multiplies. The cn values were already calculated on the square wave page.
Since the electric circuit is linear, the total output voltage is given by the sum of all the components of the waveform:

complete solution for output voltage
[Equation 5]

When n=0, then f=0, so the output voltage is easily found from equation [2]. In this case, the output and input are equivalent, so that constant (non-varying) term passes directly to the output.
The output or solution voltage for R=1 Ohm, C=0.1 Farads, T=1s, is given in Figure 3:

solution plot Figure 3. Output Voltage, Vo(t).
The beautiful thing about Fourier Series is that this method works for any periodic function, no matter how complicated. Once the Fourier Series coefficients are found, the output can be quickly calculated. It doesn't matter that the solution comes out to be an infinite sum - this is still very enlightening. From a pure math perspective, you can observe how the infinite sum varies with n, and get an idea of its frequency response via that analysis. From an engineering perspective, infinite sums can be easily calculated and coded up, so that the solution can be found quickly.
This is the power of the Fourier Series. Many, many problems in engineering and physics can be solved analytically for the case of a pure sinusoid input function. By using Fourier Series, the solution for all periodic functions can be quickly found. Hence, Fourier Series is a very useful tool.

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