# Introduction to AC Circuits

concept

All through DC we dealt with voltage and current sources that were constant. We analysed our circuits and figured out what the one, single value for a voltage somewhere or a current somewhere else.
In AC we'll be dealing with voltage and current sources that not only change their values, but change their polarity as well. So a current that was 5A in one direction could shift to be 5A in the opposite direction a second later.
AC is how power is delivered to your home and is a fundamental part of all electronics that draw power from the power grid. Even if you want your circuits to work with DC, if you're powering them from a socket in your home you'll have to convert that AC to DC, which you can't do unless you know how to analyse AC circuits.
AC can seem a little daunting at first because it changes a lot of what we've been doing up until now. But the truth is that going from DC to AC is much less difficult than picking up DC was in the first place.
If you are already able to analyse DC circuits then you can learn to analyse AC circuits, just don't let the differences intimidate you.

fact

AC stands for "Alternating Current", because circuits driven by AC signals cause the current to change direction back and forth as they operate.

We'll primarily deal with AC voltage sources (since this is what you deal with most often out in the real world).

fact

The symbol for an AC voltage source is:

fact

AC sources don't just change their values randomly, they are periodic, which means that they give off a repeating pattern (most often a sine wave) like the image below:

Sine waves are, by far, the most common way to transmit AC power and they'll be the main type of AC source we deal with here.

A sine wave is defined by three things:

- Amplitude
- Frequency
- Phase

When dealing with AC we'll most often use radians rather than degrees.
If you're not familiar with radians go take a quick read, it's important you understand what it is, how it connects to the sine wave, and how to use it.

fact

A sine wave with amplitude A, frequency \(\omega\) and phase \(\theta\) is given mathematically by:
\(g(t) = A\sin(\omega t + \theta)\)

fact

The amplitude of a sine wave \(A\) is half the vertical distance between the top peak and the bottom peak.
We'll almost always have the sine wave centered on the zero line so the amplitude is the height of the peak.

The amplitude is also sometimes called the magnitude, as a historical reference to the sine wave's connection to circles and polar coordinates.

fact

When writing the amplitude of a specific quantity (like the voltage), we often write it like \(V_m\) rather than a generic \(A\).

fact

The period of a sine wave is the time it takes for the wave to complete one cycle.

fact

The frequency of a sine wave (\(f\)) is a measure of how quickly it repeats itself.
We call 1 period the time it takes for the sine wave to completely repeat itself, for instance going from the top peak, down to the bottom peak and then back to the top peak, is one period.
The frequency is given by \(\frac{1}{T}\)Hz
Where \(T\) is the period in seconds and Hz stands for "Hertz", the unit of the number of revolutions per second.

example

Find the frequency in Hz of the following sine wave:

The frequency can be found either by counting the number of times the wave repeats itself in 1 second (which is the definition of Hertz) or by finding the period of the wave (\(T\)) and using the formula: \(f = \frac{1}{T}\) For this example it seems easier to use the second method since we can clearly see that the wave repeats every 20ms. So \(f = \frac{1}{20\cdot 10^{-3}} = 50\)Hz Incidentally voltage travelling through power lines to homes is often either 50Hz or 60Hz depending on your country. We often use a different unit for frequency in some applications, called "radians per second" which is given the symbol \(\omega\) (that's a small Greek omega).

fact

To find the frequency of a sine wave in radians per second (often written rad/s or just rads) use the following formula:
\(\omega = \frac{2\pi}{T}\)rad/s
Where \(T\) is the period of the wave.

fact

Convert between Hertz and rad/s using the following identities:
\(f = \frac{\omega}{2\pi}\)Hz
\(\omega = 2\pi f\)rad/s

You're likely to see \(2\pi f\) quite frequently when working with AC electronics.

fact

The phase of a sine wave is a measure of which angle the sine wave "started" at.
It is given by the formula:
\(\theta = \sin^{-1}(g(0))\).
That is, the phase shift is the angle that the wave "starts" at assuming that it is the \(\sin\) function.

example

Find the phase of the following sine wave:

Here \(g(0) = \frac{1}{\sqrt{2}}\) So \(\theta = \frac{\pi}{4}\)fact

As you can see in the figure below the two sine waves are really the same wave but one has been shifted a little compared to the other. We measure this shift (\(\phi\)) by the following formula:
\(\phi = \sin^{-1}(g(0)) - \sin^{-1}(h(0))\)
If \(\phi \gt 0\) we say that h "leads" g (or that g "lags" h).
If \(\phi \lt 0\) we say that g "leads" h (or that h "lags" g).
Our selection for which one is "h" and which is "g" is arbitrary.

You can see that leading just means that the wave seems to have started earlier whereas lagging means the wave seems to have started later.
Of course since these waves are periodic we could say that the lagging wave is really just leading by a whole lot, we choose "leading" and "lagging" based on whichever results in the lowest difference between the two waves.
It's always important to know if you're working in degrees or radians, lost marks abound when students forget which one they're using; or which one their calculator is using.

practice problems