Here comes AC to rattle the cage

Until now we have been discussing what is known as Direct Current, DC, circuits where the current travels in only one direction. The perfect example of a DC source is a battery where a chemical reaction takes place resulting in an electric field and voltage which forces charges to continuously flow through the circuit in one direction from the negative to positive terminal (other way for conventional current of course, but we're talking about how things actually happen for the moment).

There exists another type of current which is also ubiquitous in the real world known as Alternating Current, AC, where the current periodically changes direction as well as its rate, expressed mathematically it is a sinusoidal waveform. It's important to unpack these two fundamental differences, each of them require attention in their own right to really understand what is going on to develop intuition. But before that, let's look at a nice picture which shows us what a typical AC waveform looks like in time.

2Vpk-pk 1kHz sine wave

Feast your eyes on that beauty! That right there is a 2Vpk-pk 1kHz sine wave, that means the voltage measured from the minimum to maximum (i.e. bottom to top) is 2V. We could also say it is a 1V 1kHz sine wave which would be referring to the amplitude, i.e. the voltage measured from ground (0V) to the maximum or minimum (as they are both equal, it doesn't matter). 1kHz refers to the frequency, which we can actually work out from inspecting the above. From physics frequency is defined as the number of cycles per second, or the reciprocal of the period (time to repeat), and as the period is 1ms it follows the frequency is 1/1ms (1kHz).

How does current flow in an AC Circuit?

Simple AC Circuit

The above is as simple as it gets, so initially the electrons will travel in one direction say from the top of the voltage source back to the bottom for HALF of the period, and then when the voltage goes negative (measured at the top of the voltage source relative to the GND symbol at the bottom) current flows back the other way from the bottom back to the top. The electrons themselves accelerate for one quarter of the period, then decelerate for the next quarter then start accelerating in the other direction for the third quarter and finally decelerate for the final quarter and this then repeats indefinitely.

To see this in motion for a frequency of 60Hz, please check out the following link which uses the excellent Falstad Circuit Simulator.

AC Waveforms in the wild

So where do these AC waveforms originate from? Good question, many areas actually, but for the moment let's just consider the most prevalent in our daily lives which is the AC power that is delivered to our homes.

Electricity originates from AC Generators which work on the principle of Faraday's Law of Electromagnetic Induction where electricity is generated by exposing a conductor to a varying magnetic field. Put simply if we have a coil of wire, such as an inductor and we vary the magnetic flux which flows through the coils we will induce a voltage proportional to the number of turns and the rate of change of magnetic flux, i.e.

The negative sign is due to Lenz's Law where the induced voltage acts to oppose the change of magnetic flux.

If we then have a rectangular loop of wire then recall that the magnetic flux is proportional to the magnetic field strength and area of the loop. If we had two permanent magnets placed on opposite sides of this loop, a North and South pole, which cause a magnetic field to cut through the loop and we rotate the loop continuously then we will see the voltage vary in time like an AC waveform and this is why it is the way it is. We will discuss this in much greater detail as part of the Power Electronics course but for now this is sufficient knowledge to know.

Expressing AC Waveforms Mathematically

How then can we express AC waveforms? The pure mathematical approach is to explain it in terms of Amplitude, Frequency and Phase, i.e.

Where A is the amplitude, i.e. the distance from zero to the maximum or minimum in the vertical axis, f is the frequency in Hz and theta is the phase shift in the horizontal axis expressed in radians.


  1. Determine the formula for an AC wave with an amplitude of 10V, frequency 60Hz and zero phase shift.
For this we can directly substitute the numbers into the formula provided above:

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