Periodic signals
Four numbers matter in a periodic signal, and two of them give the form factor: the 1.111 that a cheap multimeter applies blindly, and that makes it wrong for any wave that is not sinusoidal.
01Repeating exactly
A signal is periodic if there exists a positive \( T \) that maps it back onto itself:
If \( T \) works, so do \( 2T \), \( 3T \) and all its multiples. That is why we speak of the fundamental: it is the one that defines the fundamental frequency \( \omega_0 = 2\pi/T \), and everything else —the harmonics— are multiples of it.
A constant signal is a curious case: it satisfies the definition for any \( T \), so it has no fundamental period. It is considered periodic by convention, with zero frequency.
02Adding two periodic signals
The result is not always periodic. With periods \( T_1 \) and \( T_2 \), the sum repeats only if there exist integers \( m \) and \( n \) such that \( mT_1 = nT_2 \), that is:
A concrete case: 50 Hz and 60 Hz have periods of 20 ms and 16.67 ms, whose ratio is \( 6/5 \). The sum repeats every 100 ms, that is at 10 Hz, which is the greatest common divisor of the two frequencies. That is what is heard as a beat when two nearby tones sound together, and it is the same phenomenon that makes a motor and the power line produce a slow vibration when their frequencies do not quite coincide.
03The four measures taken from it
Four numbers matter for a periodic signal, all defined over one period:
And from those four come two ratios that describe the shape of the wave regardless of its amplitude:
For a sinusoid they equal \( 1.111 \) and \( 1.414 \). For a square wave, \( 1 \) and \( 1 \). For a narrow pulse, both shoot up. Those numbers are not curiosities: the first is the one that makes a cheap multimeter lie.
04Why a cheap multimeter lies
Truly measuring the RMS value requires squaring, averaging and taking a root, and that costs circuitry. An inexpensive multimeter does something much simpler: it rectifies, averages and multiplies by 1.111. The result is correct as long as the signal is sinusoidal, because that number is precisely its form factor.
With any other waveform, the error is direct and can be enormous. The lab computes it for several signals; the most brutal case is that of a narrow pulse, where the reading falls below half of the true value. It is exactly what happens when measuring the input current of a switched-mode power supply, which flows in brief peaks near the maximum of the line voltage.
Choose the waveform and look at the four values. The last row compares the true RMS value with what a calibrated average-responding multimeter would show.
05Average power
A periodic signal has infinite energy —it repeats forever— so what gets measured is the average power over one period:
That equivalence is the historical origin of the concept. A sinusoid of 311 V peak is 220 V RMS because it heats a resistor exactly the same as a 220 V battery. And that is why all power specifications are given as RMS values.
An amplifier that delivers 100 W RMS into 8 Ω needs to be able to supply 40 V peak if the signal is sinusoidal, but 56 V if the crest factor is 2, as in music. That is why power stages are designed with supply voltage headroom: the average power specification is not enough to know whether the signal is going to clip.
06In the lab
Generate a square wave of 1 V peak and measure it with an inexpensive multimeter and with a true-RMS one. Check that the first reads around 1.11 V and explain the 11 % of difference.
Add two sinusoids of 500 Hz and 505 Hz with an amplifier and observe the envelope on the oscilloscope. Measure the period of the repetition and verify that it matches what the greatest common divisor of the two frequencies predicts.
Measure with a clamp and with an oscilloscope the input current of a computer power supply without power factor correction. Calculate its crest factor and estimate how wrong a calibrated average-responding multimeter would be.
07Common mistakes
- Using 0.707 of the peak for any wave. That factor belongs to the sinusoid only.
- Confusing the mean value with the mean value of the rectified signal. In a sinusoid the first is zero and the second, 0.637 of the peak.
- Believing an ordinary multimeter measures RMS. Almost none does, unless it says true RMS.
- Adding two periodic signals without checking the ratio of periods. If it is irrational, the sum is not periodic.
- Sizing a power stage by the RMS value alone. You have to look at the crest factor.
08Self-assessment
What is the period of the sum of 50 Hz and 60 Hz?
100 ms, which corresponds to 10 Hz: the greatest common divisor of the two frequencies.
What is the form factor of a sinusoid?
\( \pi/(2\sqrt{2}) \approx 1.111 \). It is the number by which a calibrated average-responding multimeter multiplies.
What does that multimeter read for a square wave of 1 V peak?
1.11 V, when the true RMS value is 1 V. It is off by 11 %.
Why is the RMS value called that?
Because it is the DC voltage that would dissipate the same power in a resistor: \( P = X_{\text{rms}}^2/R \).
What crest factor does a pulse with duty cycle 0.1 have?
\( 1/\sqrt{0.1} \approx 3.16 \). The narrower, the higher.
Why does the crest factor matter in a power stage?
Because it defines the peak voltage that must be deliverable without clipping, which can be much larger than what the average power suggests.
09Further reading
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Prentice Hall, 1998. Periodicity, average power and the condition for a sum of periodic signals to stay periodic.
- Robert L. Boylestad. Introductory Circuit Analysis. 13th ed. (Spanish edition, “Introducción al análisis de circuitos”), Pearson, 2017. RMS value, form factor and how measuring instruments actually work.
- Muhammad H. Rashid. Power Electronics. 4th ed. (Spanish edition, “Electrónica de potencia”), Pearson, 2015. Non-sinusoidal waveforms, crest factor and power factor in real converters.