- Introduction to AC Theory
Introduction to AC Theory
What is AC?
So far we have only considered cases in which the current is only flowing in one direction around any particular loop within a circuit. That situation being known as Direct Current (DC). When monitored on an oscilloscope, which displays a plot of voltage on the ‘Y’ axis against time on the ‘X’ axis, looks like this:
Alternating Current (AC), on the other hand is when current constantly alternates the flow in one direction around the loop and then in the opposite direction around the loop. Again monitored on an oscilloscope displaying a plot of voltage (Y) against time (X) can look something like this (notice that the signal repeatedly crosses the zero volt horizontal axis line):
Just to confuse the issue, this alternating signal can be (and often is) superimposed upon a standing steady DC offset. This is known as the DC bias of the signal. If the AC signal is of a lower amplitude than the DC bias voltage, then overall current still only flows in one direction, but with a varying amplitude. This is still known as an AC signal because if you subtract the DC bias from the signal, then it would still be constantly alternating it’s direction of flow. Notice in the plot below that the signal is completely above the zero volt horizontal axis line which demonstrates the DC bias:
The traces shown above feature a sinusoidal waveform, but this is just one of an infinite variety of shapes, the majority of which are complex and have no name. Here are just a few simple example shapes for AC signals captured on my oscilloscope:
In all cases a single cycle is from any point on the repeating waveform to the next exact point on the waveform. It is just as valid to start counting from part way up (or down) a slope, so long as you end at the same point on the slope where you stop counting. There is no absolute starting point for measuring a cycle. Although many drawings will either start at zero or one of the positive or negative peaks, that is only for convenience. in fact on an oscilloscope you actually choose the trigger point (which will be the point from which it starts to display the trace).
The take away from all this should be that:
- AC signals alternate the direction of current flow at least once per cycle.
- The AC signal can be, and often is, superimposed on to a DC bias which can result in a varying current always flowing in the same direction.
- An AC signal can be any shape as long as it varies over time. It doesn’t have to be a classic repeating periodic waveform such as a sine, triangle or square wave.
- There is no absolute point that is always the start for a given signal shape. Any point on the shape is equally valid.
I realise that it may seem like I am labouring points that you may well feel that you already know, but they are very important. Pretty much all of the complications in analysing AC signals can be traced back in some way to these things.
Amplitude
Specifying, or measuring, the amplitude of a DC circuit is unambiguous. The voltage of a battery, or bench power supply, can be considered as constant. Even changes due to changes in load can still be considered as constant for that given load.
The situation for AC is much less straight forward since, by definition, it varies over time during each cycle. Depending upon when you choose to look at it, the value can be anywhere between some maximum negative value and some maximum positive value (including zero).
There are four commonly used methods of expressing the amplitude of an AC signal. Which one is most appropriate will generally depend upon the circumstances. The four common methods are:
- Peak value
- Peak to peak value
- Average value
- RMS value
Let’s look at each of these methods in turn to see how they are calculated and what their respective strengths and weaknesses are.
Peak Value
A seemingly simple quantity, being the height of the peak of the waveform above zero.
However, consider the square (yellow) and triangle (blue) waves shown here.
Both have the same peak, but clearly the square wave is at the peak value for much longer than the triangle wave and hence delivers more power in to the load. Notice also that the peak value says nothing about the negative part of the wave. The assumption being that the waveform is symmetrical about the zero axis.
Peak to Peak Value
Another seemingly simple quantity, being the vertical height between the tip of the positive peak and the bottom of the negative peak.
This method does take some account of the negative part of the cycle. However, like the peak value dissimilar shape waves of the same peak-peak value can impart wildly differing power in to their load.
Average Value
An intuitive solution may be to just take the algebraic average of the trace value. However, consider the effect of doing that for any symmetrical shape such as this sin wave. The amount of the waveform below the zero axis exactly matches that above the axis, thus resulting in an average of zero!
A practical solution for this is to take the mean of the absolute (rather than signed) values. That is ignore the sign and treat all values as positive.
Taking this approach would be the equivalent of taking the mean of this waveform i stead of the true sin.
The average calculated in this way gives a slightly more useful indicator for comparing signals. However, it does not give a useful indicator of comparative power of the signals.
RMS Value
It would be very useful to be able to relate the amplitude of an AC signal to that of an equivalent DC amplitude (in terms of the work that it can do). In practical terms that requires that the amplitude be based on the waveform’s ability to deliver power in to a load.
As you will recall from Joules Law:
\(P = I^2R = \frac{V^2}{R}\)
In other words, power is proportional to the square of the voltage. This is generally translated as taking the mean average of the square of each point on the waveform, and then taking the square root of that mean value to be the amplitude.
This is known as the “Root Mean Square” or RMS value. Sometimes the term “DC equivalent Value” is used instead, but it means the same thing.
OK. Having described how the RMS value is calculated, you can relax a little bit because there are simple short cuts for the common symmetrical periodic wave forms of sin, square and triangle waves. Which I will summarise here:
Sin Waves:
\(V_{rms} = \frac{V_{peak}}{\sqrt{2}} ~ \approx ~ v_{peak} \times 0.7071 \\\) \(\text{I will show where the } \frac{1}{\sqrt{2}} \text{ comes from later} \\\) \(\text{Similarly:} \\\) \(V_{peak} = V_{rms} \times \sqrt{2} ~ \approx ~ V_{rms} \times 1.4142 \)
Square Waves
\(V_{rms} = V_{peak} \)
Triangle Waves
\(V_{rms} = \frac{V_{peak}}{2} \\\) \(V_{peak} = V_{rms} \times 2 \)
Note that there is a limitation that the triangles must NOT have obtuse angles (which would in any case then be called a Saw Tooth and not a Triangle waveform).
Derivation of Sin Wave RMS Shortcut
You do not need to know this if you are prepared to take the above on faith. It is only shown here to satisfy the curious.
Assuming power delivery into a purely resistive load, then P = V x I
Looking to corresponding points on the traces for V, I & P:
When V & I = 0, then P = 0;
When V & I are positive, then P is positive;
When V & I are negative, then P is positive.
Since the power curve is a positive sin wave at double the frequency of the V & I curves.
\(P_{mean} = \frac{P_{peak}}{2} \)
\(\text{Since power can also be expressed as } P = \frac{V^2}{R} \text{ then substituting in above equation we get} \\\) \(\frac{\frac{{V_{peak}}^2}{R}}{2} = \frac{{V_{rms}}^2}{R} \\\) \(\text{Simplifying we get } \frac{{V_{peak}}^2}{2R} = \frac{{V_{rms}}^2}{R} \\\) \(\text{and multiplying both sides by R we get } R \times \frac{{V_{peak}}^2}{2R} = R \times \frac{{V_{rms}}^2}{R} \\\) \(\text{the R’s cancel out leaving us with } \frac{{V_{peak}}^2}{2} = {V_{rms}}^2 \\\) \(\text{taking the square root of both sides we get } \sqrt{\frac{{V_{peak}}^2}{2}} = \sqrt{{V_{rms}}^2} \\\) \(\frac{\sqrt{{V_{peak}^2}}}{\sqrt{2}} = V_{rms} \\\) \(\frac{V_{peak}}{\sqrt{2}} = V_{rms} \\\)