Superposition of two sinusoids calculator

When two pure tones, AC voltages, or forced oscillations share the same frequency, their sum is always another sinusoid at that frequency—only the amplitude and phase change. Students, audio engineers, and circuit designers need that combined amplitude and phase without expanding product-to-sum identities by hand for every pair of phasors. This superposition of two sinusoids calculator accepts the two amplitudes and phases, converts them to rectangular phasor components, adds those components, and returns the magnitude and atan2 phase of the resultant. The common angular frequency never has to be supplied: it factors out of the phasor picture and cancels from the amplitude and phase of the sum.

What same-frequency superposition means in phasor form

A real cosinusoid of fixed angular frequency ω can be written A cos(ωt + φ). In the phasor convention used throughout AC circuit analysis and introductory waves courses, that wave is represented by the complex number A e to the jφ, or equivalently by the rectangular pair (A cos φ, A sin φ). Adding two such waves that share the same ω is then exactly the vector sum of their phasors: the real parts add and the imaginary parts add.

If the first wave has amplitude A1 and phase φ1 and the second has amplitude A2 and phase φ2, the rectangular sum is Rx equals A1 cos φ1 plus A2 cos φ2 and Ry equals A1 sin φ1 plus A2 sin φ2. The combined amplitude R is the Euclidean length hypot(Rx, Ry), and the combined phase φ is the two-argument arctangent atan2(Ry, Rx), which places the result in the correct quadrant instead of the principal half-range of a naïve arctan of Ry over Rx.

Reconstructing the time-domain signal is immediate: the sum is R cos(ωt + φ) with the same ω you started with. Signed amplitudes are allowed; a negative A is equivalent to a positive magnitude with an extra π (or one hundred eighty degrees) of phase, matching textbook algebra. Phases may be any finite real number in degrees or radians and are not reduced to a principal interval on input, so a full-turn equivalent such as three hundred sixty degrees behaves like zero after conversion.

Because frequency is shared, beats and amplitude modulation that require two different frequencies are outside this landing and belong on separate tools.

Where adding two waves of one frequency helps

Introductory physics and AC circuits courses assign dozens of two-tone or two-voltage phasor drills every term; a live calculator that returns both combined amplitude and quadrant-correct phase lets learners check work without flipping cosine signs by hand. Exam generators and LMS plugins need a deterministic oracle that rejects non-finite junk and never invents a second frequency when the problem states pure mono-frequency superposition. Electrical engineering homework on series AC sources and Kirchhoff voltage loops reduces to the same phasor sum before impedance enters the model.

Content sites that rank for superposition of two sinusoids calculator benefit from a free browser path that matches the paid automation path on the identical pure module. Failed validation does not consume balance. Keep requests small, pin the rounded fields in tests, and you get reproducible combined amplitudes and phases for homework, lab keys, and edge workers without standing up complex-number libraries of your own.

Different-frequency beats, Fourier series of many harmonics, and full SPICE phasor nets remain separate search intents and stay on their own landings so this page stays focused on the two-wave same-frequency case.