Click here to purchase the entire book in PDF format. RMS Time Constants There's just one small problem with this explanation. We're talking about an RMS value of an alternating voltage being determined in part by an average of the instantaneous voltages over a period of time called the time constant. In Figure 2.5, we're assuming that the signal is averaged for at least one half of one cycle for the sine wave. If the average is taken for anything more than one half of a cycle, then our math will work out fine. What if this wasn't the case, however? What if the time constant was shorter than one half of a cycle? Figure 2.6: An arbitrary voltage signal with a short spike. Take a look at the signal in Figure 2.6. This signal usually has a pretty low level, but there's a spike in the middle of it. This signal is comprised of a string of 1000 values, numbered from 1 to 1000. If we assume that this a voltage level, then it can be converted to a power value by squaring it (we'll keep assuming that the resistance is 1 $\Omega$). That power curve is shown in Figure 2.7. Figure 2.7: The power dissipation resulting from the signal in Figure 2.6 being sent through a 1 $\Omega$ resistor. Now, let's make a running average of the values in this signal. One way to do this would be to take all 1000 values that are plotted in Figure 2.7 and find the average. Instead, let's use an average of 100 values (the length of this window in time is our time constant). So, the first average will be the values 1 to 100. The second average will be 2 to 101 and so on until we get to the average of values 901 to 1000. If these averages are plotted, they'll look like the graph in Figure 2.8. Figure 2.8: The average power dissipation of 100 consecutive values from the curve in Figure 2.7. For example, value 1 in this plot is the average of values 1 to 100 in Figure 2.7, value 2 is the average of values 2 to 101 in Figure 2.7 and so on. There are a couple of things to note about this signal. Firstly, notice how the signal gradually ramps in at the beginning. This is because, as the time window that we're using for the average gradually ``slides'' over the transition from no signal to a low-level signal, the total average gradually increases. Also notice that what was a very short, very high level spike in the signal in the instantaneous power curve becomes a very wide (in fact, the width of the time constant), much lower-level signal (notice the scale of the y-axis). This is because the short spike is just getting thrown into an average with a lot of low-level signals, so the RMS value is much lower. Finally, the end ramps out just as the beginning ramped in for the same reasons. So, we can now see that the RMS value is potentially much smaller than the peak value, but that this relationship is highly dependent on the time constant of the RMS detection. The shorter the time constant, the closer the RMS value is to the instantaneous peak value (in fact, if the time constant was infinitely short, then the RMS would equal the peak...). The moral of the story is that it's not enough to just know that you're being given the RMS value, you'll also need to know what the time constant of that RMS value is.