Harmonic addition theorem

The formula

The two-term workhorse:

a·sin(ωt) + b·cos(ωt) = R·sin(ωt + φ)

R = √(a² + b²)          φ = atan2(b, a)

Reading: a sine and a cosine of the same frequency collapse into one sine whose amplitude is the hypotenuse of the two coefficients and whose phase is the angle of the point (a, b).

Two sinusoids with arbitrary phases:

A₁·cos(ωt + φ₁) + A₂·cos(ωt + φ₂) = A·cos(ωt + φ)

A² = A₁² + A₂² + 2·A₁·A₂·cos(φ₁ − φ₂)
φ  = atan2( A₁·sin φ₁ + A₂·sin φ₂ ,  A₁·cos φ₁ + A₂·cos φ₂ )

Reading: the amplitude formula is the law of cosines — the two signals add like two sides of a triangle, with the phase difference as the included angle. Everything about the result hinges on φ₁ − φ₂.

The general case, any number of terms ^[1]^[2]:

Σ Aᵢ·cos(ωt + φᵢ) = A·cos(ωt + φ)

A·cos φ = Σ Aᵢ·cos φᵢ        (in-phase part)
A·sin φ = Σ Aᵢ·sin φᵢ        (quadrature part)

Reading: sum the in-phase components, sum the quadrature components, then convert the pair back to amplitude and phase. That is vector addition, done componentwise — which is why engineers draw these as arrows and call them phasors ^[3].

Where you meet it

The vibration test stand. A sine sweep on the shaker, and the analyzer reports each accelerometer channel as one amplitude and one phase per frequency. Under the hood it correlated the signal against a sine and a cosine reference, got the pair (a, b), and ran this theorem. Every DFT bin you have ever read as "magnitude and phase" is the amplitude-phase form of exactly this identity.
The RF bench. IQ demodulation is the theorem run in reverse: the receiver splits an incoming carrier into in-phase and quadrature channels, and √(I² + Q²) with atan2(Q, I) recovers the envelope and phase. Lock-in amplifiers pulling a microvolt signal out of noise do the identical arithmetic.
The power review board. Someone asks what the neutral current is when the three phases are unbalanced, or why two paralleled sources fight each other. Phasor addition is the answer both times. Two 120 V sources 120° apart don't make 240 V — the theorem says their sum has amplitude 120 V. Run the numbers before the meeting.
The structures desk. Combining a forced response with a reflected or coupled response at the same excitation frequency — two transmission paths into one gauge, multipath into one antenna, two pumps beating into one manifold at line frequency. Same-frequency contributions never add as amplitudes. They add as phasors.

How it works

The reason this works at all is that sin and cos at one frequency span everything that can happen at that frequency. A sinusoid has exactly two degrees of freedom — amplitude and phase, or equivalently an in-phase coefficient and a quadrature coefficient. Add any number of them and you still have just two numbers to determine. The cleanest way to see it is Euler's formula, e^(jωt) = cos ωt + j·sin ωt: each sinusoid becomes a complex number Aᵢ·e^(jφᵢ) riding a common e^(jωt), the time factor pulls out of the sum, and adding sinusoids reduces to adding complex numbers ^[3]^[8]. The geometry-of-arrows picture and the trig identity are the same fact in two costumes.

A worked number, since a formula without one is decoration: 3·sin(ωt) + 4·cos(ωt) = 5·sin(ωt + 53.13°). And the case that surprises people: equal parts sine and cosine give 10·sin + 10·cos = 14.14 at 45° — a factor of √2, not 2. In-phase components add; quadrature components combine root-sum-square.

The bounds are worth memorizing. Two signals of amplitudes A₁ and A₂ sum to somewhere between |A₁ − A₂| (opposed, 180° apart) and A₁ + A₂ (aligned, 0° apart). If a colleague adds two 5 g responses and reports 10 g, they have silently assumed perfect phase alignment — usually the worst case, sometimes flat wrong, never automatic.

Near cancellation is where the theorem gets treacherous. Two equal-amplitude signals 180° − δ apart leave a residual of 2·sin(δ/2) times either component: at δ = 1°, about 1.7% survives. This is why active noise cancellation, balanced mixers, and CMRR specs live or die on fractions of a degree — and why a small phase drift in a cancellation scheme turns a 40 dB null into a 20 dB one overnight.

The mistakes, in the order they actually happen. One: using atan(b/a) instead of atan2(b, a). Plain arctangent can't see the quadrant; with a = −3, b = 4 it returns −53.1° when the true phase is +126.9°, and the reconstructed signal comes out inverted while every amplitude check still passes. Two: mixing conventions. The sine-referenced form above and the cosine-referenced form differ in where the signs land — MathWorld's cosine version carries δ = tan⁻¹(−b/a) ^[1] — and half the sign errors in phase data trace to two tools disagreeing about which reference they used. State the reference. Three: applying the theorem across frequencies. Two sinusoids at different frequencies do not combine into one sinusoid at all; close frequencies produce beats, and no single amplitude-phase pair describes them. Same ω is not a footnote — it is the theorem's entire jurisdiction. Four: adding amplitudes of uncorrelated broadband signals with it. Random vibration from independent sources adds in power (RSS), not as phasors; the theorem is for deterministic, phase-locked, single-frequency content.

History

The pieces of this theorem are old trigonometry, but its modern form — complex numbers doing the bookkeeping — has a traceable birthday. Roger Cotes, Newton's young editor, published the germ in his one lifetime paper, Logometria (1714), a logarithm–trigonometry link equivalent to ln(cos θ + i·sin θ) = iθ ^[8]^[9]. Euler derived the exponential form around 1740 by comparing series expansions and published e^(ix) = cos x + i·sin x in his 1748 Introductio in analysin infinitorum ^[8]. That formula is the entire machinery of phasor addition; it just sat waiting a century and a half for someone with an AC circuit problem.

That someone arrived twice in the same year. In April 1893, Arthur Kennelly — then at Edison's West Orange laboratory — read his paper "Impedance" to the American Institute of Electrical Engineers, showing that resistance and reactance combine as a single complex number ^[6]^[7]. Months later, at the International Electrical Congress in Chicago, Charles Proteus Steinmetz presented "Complex Quantities and Their Use in Electrical Engineering," laying out the full symbolic method for AC analysis ^[4]^[5]. Steinmetz had fled Germany in 1889, one step ahead of prosecution for socialist activities, and landed at the company that became part of General Electric ^[4]. His 1897 textbook with Ernst Berg, Theory and Calculation of Alternating Current Phenomena, put the method in front of a generation of engineers ^[4]^[5]. The pitch, in his own framing, was that it reduced the calculus of AC circuits to "a simple problem of algebra" ^[4] — which is precisely what this theorem does every time you trade a trig-identity grind for one complex addition. Neither man called the arrows "phasors"; both said "vectors," and the newer name came later ^[3].

Related tools

/tools/rms-peak
/tools/series-rlc-impedance
/tools/lc-resonance
/tools/reactance
/tools/vibration-natural-freq
/tools/frequency-period
/tools/decibel-power-sum

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →