# Phasor

A **phasor** is a complex number that represents the amplitude and phase of a sinusoidal signal of a fixed frequency, with the frequency itself left implicit. In electrical engineering it is the standard tool for analyzing the steady-state behavior of linear circuits driven by sinusoidal (AC) sources, replacing differential equations in time with algebraic equations in the frequency domain.<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | A compact frequency-domain representation of a signal in which only magnitude and phase are shown explicitly; the frequency is implicit in the representation<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup> |
| Mapping | A time-domain signal \( v(t) = A\cos(\omega t + \varphi) \) corresponds to the phasor \( V = A e^{j\varphi} = A\angle\varphi \)<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup> |
| Why it works | In a linear system with a sinusoidal forcing function, the steady-state voltages and currents are always sinusoids of the same frequency as the source<sup>[2](https://engineering.louisville.edu/raise/EE220/AllLessons.html)</sup> |
| Impedances | \( Z_R = R \), \( Z_C = 1/(j\omega C) \), \( Z_L = j\omega L \); the ratio of phasor voltage to phasor current<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup> |
| What it gives up | The phasor solution misses transient terms such as a current decay of the \( \exp(-(R/L)t) \) form; it yields only the steady state<sup>[3](https://ee.iitm.ac.in/~uday/notes/em/phasors.html)</sup> |
| Wave use | Two equal-amplitude waves differing in phase by \( \varphi \) superpose to a resultant of amplitude \( 2A\cos(\varphi/2) \), zero at a 180° phase difference<sup>[4](https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves)</sup> |

## The phasor transform

The method rests on a property of linear systems. When a sinusoidal forcing function drives a linear system of resistors, inductors, and capacitors, the steady-state response (what happens after waiting a long time) is a sinusoid of the same frequency; only its amplitude and phase angle must be calculated. A 120 V, 60 Hz source driving such a system produces a 60 Hz steady-state current.<sup>[2](https://engineering.louisville.edu/raise/EE220/AllLessons.html)</sup>

Because every voltage and current shares the frequency \( \omega \), that frequency can be divided out. A signal \( v(t) = A\cos(\omega t + \varphi) \) is written as the real part of a complex exponential, \( v(t) = \mathrm{Re}\{A e^{j\varphi} e^{j\omega t}\} \), and the constant factor \( A e^{j\varphi} \) is taken as the phasor \( V = A\angle\varphi \).<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup> The Berkeley EE 100/42 lecture notes describe the same idea: a phasor is a complex number representing the complex exponential form of a voltage or current with the time dependence divided out, and the factor \( e^{j\omega t} \) can be dropped from the equations because it cancels.<sup>[5](https://rfic.eecs.berkeley.edu/courses/ee42_100/pdf/lect16_ann.pdf)</sup> A MIT 6.007 slide presents this procedure as a modern version of the Steinmetz AC analysis method, in which a time-dependent problem is replaced by a complex one and the real solution is recovered at the end.<sup>[6](https://ocw.mit.edu/courses/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/7c18778ca0e6c11c2264aadaf42c2bd6_MIT6_007S11_lec19.pdf)</sup>

## Impedance and circuit analysis

**Differentiation becomes multiplication.** The payoff of the phasor transform is that derivatives turn into multiplication by \( j\omega \). For a capacitor, the time-domain relation \( i(t) = C\,dv/dt \) becomes the algebraic relation \( I = j\omega C V \); for an inductor, \( v(t) = L\,di/dt \) becomes \( V = j\omega L\,I \); and the resistor relation \( v = Ri \) keeps the same form. The ratio of phasor voltage to phasor current for each element is called its impedance \( Z \), with units of ohms: \( Z_R = R \), \( Z_C = 1/(j\omega C) \), and \( Z_L = j\omega L \).<sup>[1](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)</sup> Impedance is a frequency-dependent complex quantity that has meaning only in the frequency domain and cannot be transformed directly back to the time domain.<sup>[7](https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/96c80ce0d5513139310a9526fdadb419_sss_phsor_impdce.pdf)</sup>

Because phasor voltages and currents satisfy Kirchhoff's laws, an entire circuit can be transformed into the frequency domain and solved without writing differential equations.<sup>[8](https://ctools.ece.utah.edu/Phasors/PhasorTutor.htm)</sup> Impedances combine exactly as resistances do: series impedances add, and parallel impedances combine through reciprocal sums. Linear-circuit methods carry over unchanged, including superposition, Thevenin and Norton theorems, and source transformations, with resistance replaced by impedance.<sup>[7](https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/96c80ce0d5513139310a9526fdadb419_sss_phsor_impdce.pdf)</sup> The Louisville ECE 220 notes summarize the working procedure in three steps: transform the system from time functions into phasors, solve the resulting time-independent problem with complex numbers using Kirchhoff's laws, node voltage, mesh current, and series-parallel impedance combination, then transform back into the time domain.<sup>[2](https://engineering.louisville.edu/raise/EE220/AllLessons.html)</sup>

In a series [RC circuit](https://www.edgepedia.org/rc-circuit), for example, the resistor and capacitor voltage phasors add as vectors to give the series voltage, and the magnitude of the total follows from the Pythagorean relation \( V_{RC}^2 = V_R^2 + V_C^2 \), because the resistive and reactive components are 90° out of phase with each other.<sup>[9](https://animations.physics.unsw.edu.au/jw/AC.html)</sup> In power work, phasor diagrams are commonly drawn with RMS values rather than peak values; RMS values are peak values times 0.71 (that is, \( 1/\sqrt{2} \)), so RMS phasor diagrams have the same shape as amplitude diagrams scaled by that factor.<sup>[9](https://animations.physics.unsw.edu.au/jw/AC.html)</sup>

## Limits of the method

The phasor method applies only to the steady state of linear circuits with sinusoidal sources of a single common frequency. Comparing the phasor solution of a series R-L circuit with a full Fourier-transform solution shows that the two give identical steady-state results, except that the phasor approach misses a transient current decay term of the \( \exp(-(R/L)t) \) form; the notes call the phasor approach a "lazy" [Fourier transform](https://www.edgepedia.org/fourier-transform) for this reason, while observing that for steady-state questions it is simpler and faster.<sup>[3](https://ee.iitm.ac.in/~uday/notes/em/phasors.html)</sup> Circuits with non-sinusoidal sources, non-linear elements, or sources at different frequencies fall outside the method.<sup>[3](https://ee.iitm.ac.in/~uday/notes/em/phasors.html)</sup>

Phasor analysis can be seen as a special case of Laplace-transform analysis in which the stimulus is sinusoidal; Laplace methods are more involved but provide both transient and steady-state solutions for general stimuli.<sup>[10](https://gab.wallawalla.edu/~curt.nelson/engr228/lecture/mark_haun.pdf)</sup> For non-sinusoidal periodic inputs, the standard extension is to expand the input in a [Fourier series](https://www.edgepedia.org/fourier-series), solve the circuit in the phasor domain for each frequency component separately, and add the individual responses using superposition.<sup>[11](https://sites.google.com/view/uhm-ee-213/lectures/fourier-series-application)</sup>

## Phasors in wave interference

The same construction serves in physics for adding waves of a common frequency. A sinusoidal wave \( y = A\sin(kx - \omega t) \) can be represented by a vector of magnitude equal to the wave's amplitude, rotating with angular speed \( \omega \), whose projection on the vertical axis reproduces the wave; two such phasors separated by the phase angle \( \varphi \) are added as vectors to find the resultant amplitude and phase.<sup>[12](https://web.njit.edu/~taozhou/bbb/ch16.pdf)</sup> For two waves identical in amplitude, wave number, and angular frequency but differing by a phase shift \( \varphi \), the superposition is a wave of the same wave number and angular frequency with amplitude \( 2A\cos(\varphi/2) \) and a phase shift of \( \varphi/2 \); at zero phase difference the amplitude is \( 2A \) (pure constructive interference), and at 180° it is zero (pure destructive interference).<sup>[4](https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves)</sup> The vector method also handles waves of different amplitudes, which the simple trigonometric special cases do not.<sup>[12](https://web.njit.edu/~taozhou/bbb/ch16.pdf)</sup>

## References

1. [Sinusoidal Steady State Response of Linear Circuits, 6.071/22.071 Lecture Notes, MIT OpenCourseWare (Chaniotakis and Cory, Spring 2006)](https://mitocw.ups.edu.ec/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/sss_phsor_impdce.pdf)
2. [ECE 220 Network Analysis I, Lesson 11: AC Circuits, University of Louisville](https://engineering.louisville.edu/raise/EE220/AllLessons.html)
3. [Notes on phasors, IIT Madras](https://ee.iitm.ac.in/~uday/notes/em/phasors.html)
4. [16.5 Interference of Waves, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves)
5. [Lecture 16: Sinusoidal Steady-State, EE 100/42, University of California, Berkeley (A. M. Niknejad)](https://rfic.eecs.berkeley.edu/courses/ee42_100/pdf/lect16_ann.pdf)
6. [Complex Numbers and Phasors, 6.007 Electromagnetic Energy, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/7c18778ca0e6c11c2264aadaf42c2bd6_MIT6_007S11_lec19.pdf)
7. [Sinusoidal Steady State Response of Linear Circuits (impedance section), 6.071/22.071, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/96c80ce0d5513139310a9526fdadb419_sss_phsor_impdce.pdf)
8. [Phasors Tutorial, Conceptual Tools, University of Utah ECE (Carl H. Durney)](https://ctools.ece.utah.edu/Phasors/PhasorTutor.htm)
9. [AC circuits, alternating current electricity, UNSW Physics (Joe Wolfe)](https://animations.physics.unsw.edu.au/jw/AC.html)
10. [ENGR 228 Lecture Notes (Mark Haun), Walla Walla University](https://gab.wallawalla.edu/~curt.nelson/engr228/lecture/mark_haun.pdf)
11. [Fourier Series Application, EE 213 Basic Circuit Analysis, University of Hawaii](https://sites.google.com/view/uhm-ee-213/lectures/fourier-series-application)
12. [Chapter 16 slides: Waves (phasor addition), NJIT (Tao Zhou)](https://web.njit.edu/~taozhou/bbb/ch16.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Wave phenomena and acoustics › Interference and diffraction › Superposition mathematics and wave addition*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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