Steady state and spectra
A filter does not deform a signal in some mysterious way: it changes the proportion among its harmonics. With that single idea, everything learned for sinusoids is useful for any periodic signal.
01Any periodic signal is a sum of sinusoids
Everything learned about sinusoidal steady state would seem to be useful only for pure sinusoidal signals, which in practice hardly exist. Fourier solved the problem in 1807: every reasonable periodic signal is a sum of sinusoids whose frequencies are integer multiples of the fundamental. Since linear circuits obey superposition, it is enough to solve the circuit once for each of those sinusoids and add the results.
02The line spectrum
Plotting the amplitudes \( c_n \) against frequency gives the spectrum of the signal: a series of lines at \( f_0, 2f_0, 3f_0, \ldots \). Each waveform has its signature.
| Waveform, amplitude 1 | Harmonics present | Amplitude of harmonic \( n \) | How they decay |
|---|---|---|---|
| Symmetric square wave | Odd only | \( 4/(\pi n) \) | Like \( 1/n \): slowly |
| Symmetric triangle wave | Odd only | \( 8/(\pi^2 n^2) \) | Like \( 1/n^2 \): quickly |
| Sawtooth | All | \( 2/(\pi n) \) | Like \( 1/n \) |
The rule behind this is general: the more abrupt a signal is, the more slowly its harmonics decay. A sudden jump, like that of the square wave, needs very high frequencies to be built; a continuous ramp, like the triangle wave, is put together almost entirely from the first few terms.
03The response, harmonic by harmonic
If the input is a sum of sinusoids, the output of a linear circuit is the sum of the responses to each one. Each harmonic comes out multiplied by the magnitude of \( H \) at its own frequency and shifted by its phase.
On top, the spectrum: in gray the input harmonics and in color those that come out of the low-pass filter. Below, the output waveform reconstructed by summing the first 99 harmonics.
04RMS value and harmonic distortion
Since harmonics of different frequencies do not exchange average power with one another, the square of the RMS value of a signal is the sum of the squares of the RMS values of its harmonics: the Parseval identity.
05From the line spectrum to the continuous spectrum
If the period grows without bound, the lines of the spectrum move closer to one another until they form a continuum: the Fourier series becomes the Fourier transform, which describes the frequency content of a signal that does not repeat, such as a single pulse. The power signals of the signals unit have a line spectrum; the energy signals, a continuous spectrum. It is the same idea with two tools, and the Signals and Systems course develops them in detail.
06Where it shows up in electronics
| Case | What to look at in the spectrum |
|---|---|
| Audio amplifier | Its THD at rated power: below 0.1 % it is inaudible |
| Mains with many switched-mode power supplies | The third harmonic, at 150 Hz, which in three-phase systems adds up in the neutral and overheats it |
| Electromagnetic compatibility | The harmonics of a fast-edged digital clock reach hundreds of MHz and have to be kept under the emission limits |
| Spectrum analyzer | The instrument that directly displays the lines of a signal |
07In the lab
Apply a 1 kHz square wave to an oscilloscope with an FFT function and measure the amplitude of harmonics 1, 3, 5 and 7. Verify that they follow the ratio \( 1 : 1/3 : 1/5 : 1/7 \).
Using this page’s lab, observe what happens to the output waveform when the filter cutoff goes down from 100·f₀ to 0.3·f₀, and relate the result to the RC charging in the passive components topic.
Calculate the THD of a square wave taking only the harmonics up to the 9th and compare it with the exact value of 48.3 %. Explain why the partial sum gives a lower value.
08Common mistakes
- Applying harmonic superposition to a nonlinear circuit. A diode or a saturated transistor generates new harmonics; the recipe in this unit holds only for linear circuits.
- Forgetting that \( H \) changes from harmonic to harmonic. Each one is multiplied by \( |H(jn\omega_0)| \), not by a single value.
- Adding harmonic amplitudes to obtain the RMS value. What gets added are the squares of the RMS values, not the amplitudes.
- Expecting even harmonics in a wave with half-wave symmetry. The symmetric square and triangle waves have only odd ones.
- Confusing a line spectrum with a continuous spectrum. The first belongs to periodic signals; the second, to signals that do not repeat.
09Self-assessment
Which harmonics does a symmetric square wave have?
Only the odd ones, with amplitude \( 4/(\pi n) \).
Why do the harmonics of the triangle wave decay faster than those of the square wave?
Because the triangle wave has no jumps: it is continuous, and only its slope changes abruptly. The smoother the signal, the faster its spectrum decays.
How is the output of a linear circuit calculated for a periodic wave?
The input is decomposed into harmonics, each one is multiplied by \( H(jn\omega_0) \) and the responses are added.
What does the Parseval identity say?
That the square of the RMS value is the sum of the squares of the RMS values of each harmonic component, plus that of the average value.
What is the THD of a square wave?
48.3 %.
10Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The chapter on Fourier series applies exactly this procedure to RC and RLC circuits.
- Alan V. Oppenheim, Alan S. Willsky and S. Hamid Nawab. Signals and Systems. 2nd ed., Prentice Hall, 1996. The formal bridge between the Fourier series and the transform, for whoever continues on to Signals and Systems.
- Keysight Technologies. Spectrum Analysis Basics (Application Note 150). Keysight. How the instrument that displays the line spectrum in the lab works on the inside.