Circuit Theory/Fourier Transform



This course started with phasors. We learned how to transform forcing sinusodial functions such as voltage supplies into phasors. To handle more complex forcing functions we switched to complex frequencies. This enabled us to handle forcing functions of the form:
 * $$e^{st}\cos(\omega t + \phi)$$

where s is:
 * $$s=\sigma + j\omega$$

And the convolution integral can do anything.

Along the way "s" began to transform the calculus operators back into algebra. Within the complex domain, "s" could be re-attached to the inductors and capacitors rather than forcing functions. The transfer function helped us use "s" to capture circuit physical characteristics.

This is all good for designing a circuit to operate at a single frequency &omega;. But what about circuits that operate at a variety of frequencies? A RC car may operate at 27mhz, but when a control is pressed, the frequency might increase or decrease. Or the amplitude may increase or decrease. Or the phase may shift. All of these things happen in a cell phone call or wifi/blue tooth/xbee/AM/FM/over the air tv, etc.

How does a single circuit respond to these changes?

Fourier analysis
Fourier analysis says we don't have to answer all the above questions. Just one question has to be answered/designed to. Since any function can be turned into a series of sinusoidals added together, then sweeping the circuit through a variety of omegas can predict its response to any particular combination of them.

So to start this we get rid of the exponential term and go back to phasors.

Set &sigma; to 0:
 * $$s = j\omega$$

The variable &omega; is known as the "radial frequency" or just frequency. With this we can design circuits for cell phones that all share the air, for set-top cable TV boxes that pack multiple channels into one black cable. Every vocal or pixel change during transmission or reception can be designed for within this framework. All that is required is to sweep through all the frequencies that a sinusoidal voltage or current source can produce.

Analysis stays in the frequency domain. Because everything repeats over and over again in time, there is no point in going back to the time from a design point of view.

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 60 Hz, the resultant radial frequency is:
 * $$\omega = 2\pi f = 2\pi(60) = 120\pi$$

Fourier Domain
The Fourier domain then is broken up into two distinct parts: the magnitude graph, and the phase graph. The magnitude graph has j&omega; as the horizontal axis, and the magnitude of the transform as the vertical 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 j&omega; as the horizontal axis, and the phase value of the transform as the vertical axis. Remember, we can compute the phase of a complex value as such:


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

The phase and magnitude values of the Fourier transform can be considered independent 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.