LTI Systems with Periodic Inputs

The frequency response at the harmonics, a square wave through an RC filter, total harmonic distortion, and why linear systems never create new frequencies

Currents that are not sinusoids

The mains voltage is a clean sinusoid, but much of what we plug into it does not draw a sinusoidal current. A rectifier feeding a capacitor — the front end of almost every electronic power supply — takes current only near the peaks of the voltage, in short pulses once or twice per cycle. The current is still periodic at 50 or 60 Hz, so it is a Fourier series: a fundamental plus a stack of harmonics. In a three-phase, four-wire building supply, the three phase currents are shifted by a third of a cycle, so their fundamentals cancel in the shared neutral wire. Their third harmonics do not: shifted by a third of a cycle at three times the frequency, they line up and add, driving a current at 150 or 180 Hz through a neutral that was sized on the assumption that it would carry almost nothing (Harmonics (electrical power)). IEEE Standard 519 sets limits on the harmonic currents a facility may inject into the grid. This lesson shows how to follow each harmonic through a system — and why a linear system can never be the source of them.

Lesson 3-1 promised that an LTI system acts on a Fourier series term by term. The frequency response

H(j\omega)

of a stable LTI system is the transform of its impulse response,

H(j\omega) = \int_{-\infty}^{\infty} h(t)\,e^{-j\omega t}\,dt,

and for systems described by differential equations it can be read straight off the equation, as the derivation below shows for the RC circuit.

💡Periodic in, periodic out

x(t) = \sum_k a_k e^{jk\omega_0 t}

passes through an LTI system, the output is periodic with the same period, and its Fourier coefficients are

b_k = a_k\,H(jk\omega_0).

Each harmonic is scaled by

|H(jk\omega_0)|

and shifted in phase by

\angle H(jk\omega_0)

. Nothing else happens: no harmonic appears in the output that was not in the input.

Derivation

The RC circuit's frequency response

The RC low-pass obeys

RC\,\dot y + y = x

(Lesson 2-3). Try

x = e^{j\omega t}

and, by the eigenfunction property,

y = H\,e^{j\omega t}

. Then

\dot y = j\omega H e^{j\omega t}

, and

RC\,j\omega H\,e^{j\omega t} + H\,e^{j\omega t} = e^{j\omega t} \quad\Longrightarrow\quad H(j\omega) = \frac{1}{1 + j\omega RC}.

Its magnitude and phase are

|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega/\omega_c)^2}}, \qquad \angle H(j\omega) = -\arctan(\omega/\omega_c), \qquad \omega_c = \frac{1}{RC}.

At the cut-off frequency

\omega_c

the magnitude is

1/\sqrt2

(−3 dB) and the phase lag 45°. Far above it the magnitude falls as

\omega_c/\omega

: the circuit halves each doubling of frequency. Every differential equation turns into an algebraic one this way.

📐A square wave through an RC filter

\pm1

square wave at

f_0 = 100

Hz enters an RC low-pass with cut-off

f_c = 200

Hz. Its harmonic amplitudes are

4/(\pi k)

for odd

k

, and the filter's gain at the

k

th harmonic is

1/\sqrt{1 + (k f_0/f_c)^2}

: | Harmonic

k

| Input amplitude | Filter gain | Output amplitude | |---|---|---|---| | 1 (100 Hz) | 1.273 | 0.894 | 1.139 | | 3 (300 Hz) | 0.424 | 0.555 | 0.235 | | 5 (500 Hz) | 0.255 | 0.371 | 0.095 | The high harmonics are cut hardest, so the output's corners round off: the square wave becomes the charge–discharge curve of Lesson 2-2. The table is the output, written in the frequency domain.

💡Total harmonic distortion

How far is a periodic signal from a pure sinusoid? The total harmonic distortion compares the power in all the harmonics with the power in the fundamental:

\mathrm{THD} = \frac{\sqrt{\sum_{k \ge 2} |c_k|^2}}{|c_1|},

where

c_k

is the amplitude of the

k

th harmonic. A pure sine has 0 %. A square wave, by Parseval, has

\sqrt{\pi^2/8 - 1} = 48.3\,\%

. Audio amplifiers are specified at small fractions of a per cent; power-quality limits in standards such as IEEE 519 are a few per cent.

Linear systems never create frequencies. Because

b_k = a_k H(jk\omega_0)

, a harmonic absent from the input (

a_k = 0

) is absent from the output, whatever

H

is. Harmonics are born in nonlinear devices — a rectifier that conducts only near the voltage peaks, an amplifier that clips, a transformer core that saturates — and then carried by the linear parts of the system: cables, filters, transformers. That division of labour is why power engineers place filters near the nonlinear loads that produce the harmonics.

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

– Attributed to Pythagoras, as quoted in Louise B. Young (ed.), The Mystery of Matter (1965), p. 113, on Wikiquote

⚠A new frequency means a nonlinearity

If a spectrum analyser shows a frequency at the output that was not at the input — a harmonic, or a sum or difference of two input tones — something in the chain is nonlinear: an amplifier near clipping, a saturating sensor, a diode. No amount of linear filtering can create it. (Time-varying systems, such as a mixer driven by a local oscillator, are the deliberate exception — Unit 10 uses them.)

💡Why delta windings help

Third harmonics, and the other multiples of three, are the same in all three phases, so they cannot flow out along the three lines of a delta-connected winding. They circulate inside the delta instead of travelling on through the system, which is why transformers with a delta winding are used to keep these triplen harmonics out of the upstream network.

💡Where you will use this

The frequency response is the subject of Unit 4, where Bode plots draw it and filters shape it. In discrete time it is the DTFT of Lesson 6-1. THD and harmonic spectra are everyday measurements in audio, power electronics and vibration analysis, and Unit 10 puts harmonic thinking to work in modulation.

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 3.8–3.9 — Fourier series and LTI systems, and filtering.
Lathi and Green, Linear Systems and Signals, section 6.4 — LTI system response to periodic inputs.
Harmonics (electrical power) on Wikipedia — sources, effects and the neutral-current problem.

↓But there is a problem

Periodic signals live on a comb of harmonics, and LTI systems act on each tooth of the comb separately. But a single clap, a lightning strike or a radar pulse never repeats. Make the period longer and longer: the harmonics crowd together, the comb's teeth merge, and in the limit every frequency is present. The sum becomes an integral — the Fourier transform — and it turns out to describe almost everything a prism does to light.