Circuit Theory/Fourier Transform

Joseph Fourier, after whom the Fourier Transform is named, was a famous mathematician who worked for Napoleon.

Fourier Transform

The Fourier Transform is a specific case of the Laplace transform. If we separate s into its real and imaginary parts:

$s=\sigma +j\omega$

Where s is the complex laplace variable, $\sigma$ is the real part of s, and $\omega$ is the imaginary part of s. Remember, in Electrical Engineering, j is the imaginary number, not i.

Now, if we set $\sigma \to 0$, we can get the following:

$s=j\omega$

Plugging into the Laplace transform, we get the following formula:

$F(j\omega )=ℱ\{f(t)\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-j\omega t}dt$

The variable $\omega$ is known as the "radial frequency" of the circuit. This term refers to the frequency of the circuit. The Fourier transform, in the respect that it accounts only for the response of the circuit to a given frequency is very similar to phasor notation. However the Fourier Transform produces an equation that can be used to analyze the circuit for all frequencies, not just a single frequency like phasors are limited to.

As with the Laplace transform, there is an inverse Fourier transform:

${ℱ}^{-1}\{F(j\omega )\}=f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(j\omega )e^{j\omega t}d\omega$

However, there are extensive tables of Fourier transforms and their inverses available, so we need not waste time computing individual transforms.

Radial Frequency

In the Fourier transform, the value $\omega$ is known as the Radial Frequency, and has units of radians/second (rad/s). People might be more familiar with the variable f, which is called the "Frequency", and is measured in units called Hertz (Hz). The conversion is done as such:

$\omega =2\pi f$

For instance, if a given AC source has a frequency of 60Hz, the resultant radial frequency is:

$\omega =2\pi f=2\pi (60)=120\pi$

Fourier Domain

The Laplace transform converts functions from the time domain to the complex s domain. s has real and imaginary parts, and these parts form the axes of the s domain: the real part is the horizontal axis, and the imaginary part is the vertical axis. However, in the Fourier transform, we have the relationship:

$s\to j\omega$

And therefore we don't have a real part of s. The Fourier domain then is broken up into two distinct parts: the magnitude graph, and the phase graph. The magnitude graph has jω as the horizontal axis, and the magnitude of the transform as the horizontal axis. Remember, we can compute the magnitude of a complex value C as:

$C=A+jB$

$|C|=\sqrt{A^2+B^2}$

The Phase graph has jω as the horizontal axis, and the phase value of the transform as the horizontal axis. Remember, we can compute the phase of a complex value as such:

$C=A+jB$

$\mathrm{\angle }C={\tan }^{-1}\left(\frac{B}{A}\right)$

The phase and magnitude values of the Fourier transform can be considered independant values, although some abstract relationships do apply. Every fourier transform must include a phase value and a magnitude value, or it cannot be uniquely transformed back into the time domain.

The combination of graphs of the magnitude and phase responses of a circuit, along with some special types of formatting and interpretation are called Bode Plots, and are discussed in more detail in the next chapter.

Impedance and Reactance

In the Fourier domain, the concepts of capacitance, inductance, and resistance can be generalized into a single complex term called "Impedance." Impedance in this sense is exactly the same as the impedance quantities from the Laplace domain and the phasor domain. In the fourier domain however, the impedance of a circuit element is defined in terms of the voltage frequency across that element, as such:

$Z(j\omega )=R(j\omega )+jX(j\omega )$

Where R is the fourier transform of resistance, and X is the transform of reactance, that we discussed earlier.

Frequency Domain Analysis

Individual circuit elements can be transformed into the Fourier frequency domain according to a few simple rules. These transformed circuit elements can then be used to find the Frequency Response of the circuit.

Resistors

Resistors are not reactive elements, and their resistance is not a function of time. Therefore, when transformed, the fourier impedance value of a resistor is given as such:

$Z_{resistor}(j\omega )=r$

Resistors act equally on all frequencies of input.

Capacitors

Capacitors are reactive elements, and therefore they have reactance, but no resistance, as such:

$Z_{capacitor}(j\omega )=\frac{1}{j\omega C}=\frac{-j}{\omega C}$

Inductors

Inductors are also reactive elements, and have the following fourier transform:

$Z_{inductor}(j\omega )=j\omega L$

Current and Voltage Sources

The frequency representation of a source is simply the transform of that source's input function.

Frequency Response

If we set $s\to j\omega$, and plug this value into our transfer function:

$H(s){|}_{s\to j\omega }=H(j\omega )$

The function $H(j\omega )$ is called the "Frequency Response". The frequency response can be used to find the output of a circuit from the input, in exactly the same way that the Transfer function can be:

$Y(j\omega )=X(j\omega )H(j\omega )$

In addition, the Convolution Theorem holds for the $\omega$ domain the same way as it works for the S domain: