In physics and engineering, a phasor (a portmanteau of phase vector[1][2]) is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It is related to a more general concept called analytic representation,[3] which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude,[4][5] and (in older texts) sinor[6] or even complexor.[6]

An example of series RLC circuit and respective phasor diagram for a specific ω. The arrows in the upper diagram are phasors, drawn in a phasor diagram (complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current.

A common situation in electrical networks powered by time varying current is the existence of multiple sinusoids all with the same frequency, but different amplitudes and phases. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be represented as a linear combination of phasors (known as phasor arithmetic or phasor algebra[7]: 53 ) and the time/frequency dependent factor that they all have in common.

The origin of the term phasor rightfully suggests that a (diagrammatic) calculus somewhat similar to that possible for vectors is possible for phasors as well.[6] An important additional feature of the phasor transform is that differentiation and integration of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple algebraic operations on the phasors; the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain.[8][9][a] The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century.[10][11]

Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform, which additionally can be used to (simultaneously) derive the transient response of an RLC circuit.[9][11] However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.[11]

Fig 2. When function is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its magnitude is A, and it completes one cycle every 2π/ω seconds. θ is the angle it forms with the positive real axis at t = 0 (and at t = n 2π/ω for all integer values of n).


Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering.   can represent either the vector   or the complex number  , with  , both of which have magnitudes of 1. A vector whose polar coordinates are magnitude   and angle   is written  [12]

The angle may be stated in degrees with an implied conversion from degrees to radians. For example   would be assumed to be   which is the vector   or the number  


A real-valued sinusoid with constant amplitude, frequency, and phase has the form:


where only parameter   is time-variant. The inclusion of an imaginary component:


gives it, in accordance with Euler's formula, the factoring property described in the lede paragraph:


whose real part is the original sinusoid. The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids. Furthermore, all the mathematics can be done with just the phasors   and the common factor   is reinserted prior to the real part of the result.

The function   is called the analytic representation of   Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a phasor,[13] as we do in the next section. But the term phasor usually implies just the static complex number  


Multiplication by a constant (scalar)Edit

Multiplication of the phasor   by a complex constant,  , produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid:


In electronics,   would represent an impedance, which is independent of time. In particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid.


The sum of phasors as addition of rotating vectors

The sum of multiple phasors produces another phasor. That is because the sum of sinusoids with the same frequency is also a sinusoid with that frequency:


and, if we take  , then   is:

  •   if   with   the signum function;
  •   if  ;
  •   if  .

or, via the law of cosines on the complex plane (or the trigonometric identity for angle differences):


A key point is that A3 and θ3 do not depend on ω or t, which is what makes phasor notation possible. The time and frequency dependence can be suppressed and re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In angle notation, the operation shown above is written:


Another way to view addition is that two vectors with coordinates [A1 cos(ωt + θ1), A1 sin(ωt + θ1)] and [A2 cos(ωt + θ2), A2 sin(ωt + θ2)] are added vectorially to produce a resultant vector with coordinates [A3 cos(ωt + θ3), A3 sin(ωt + θ3)] (see animation).

Phasor diagram of three waves in perfect destructive interference

In physics, this sort of addition occurs when sinusoids interfere with each other, constructively or destructively. The static vector concept provides useful insight into questions like this: "What phase difference would be required between three identical sinusoids for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral triangle, so the angle between each phasor to the next is 120° (2π3 radians), or one third of a wavelength λ3. So the phase difference between each wave must also be 120°, as is the case in three-phase power.

In other words, what this shows is that:


In the example of three waves, the phase difference between the first and the last wave was 240°, while for two waves destructive interference happens at 180°. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength  . This is why in single slit diffraction, the minima occur when light from the far edge travels a full wavelength further than the light from the near edge.

As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360° or 2π radians representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0°, 180°, and at 360°.

Likewise, when the tip of the vector is vertical it represents the positive peak value, (+Amax) at 90° or π2 and the negative peak value, (Amax) at 270° or 3π2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represents a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, (t) and in our example above, this is at an angle of 30°.

Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians.

But if a second waveform starts to the left or to the right of this zero point, or if we want to represent in phasor notation the relationship between the two waveforms, then we will need to take into account this phase difference, Φ of the waveform. Consider the diagram below from the previous Phase Difference tutorial.

Differentiation and integrationEdit

The time derivative or integral of a phasor produces another phasor.[b] For example:


Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant  .

Similarly, integrating a phasor corresponds to multiplication by   The time-dependent factor,   is unaffected.

When we solve a linear differential equation with phasor arithmetic, we are merely factoring   out of all terms of the equation, and reinserting it into the answer. For example, consider the following differential equation for the voltage across the capacitor in an RC circuit:


When the voltage source in this circuit is sinusoidal:


we may substitute  

where phasor   and phasor   is the unknown quantity to be determined.

In the phasor shorthand notation, the differential equation reduces to:







Since this must hold for all  , specifically:   it follows that:







It is also readily seen that:


Substituting these into Eq.1 and Eq.2, multiplying Eq.2 by   and adding both equations gives:


Solving for the phasor capacitor voltage gives:


As we have seen, the factor multiplying   represents differences of the amplitude and phase of   relative to   and  

In polar coordinate form, the first term of the last expression is:

where  .



Ratio of phasorsEdit

A quantity called complex impedance is the ratio of two phasors, which is not a phasor, because it does not correspond to a sinusoidally varying function.


Circuit lawsEdit

With phasors, the techniques for solving DC circuits can be applied to solve linear AC circuits.[a]

Ohm's law for resistors
A resistor has no time delays and therefore doesn't change the phase of a signal therefore V = IR remains valid.
Ohm's law for resistors, inductors, and capacitors
V = IZ where Z is the complex impedance.
Kirchhoff's circuit laws
Work with voltages and current as complex phasors.

In an AC circuit we have real power (P) which is a representation of the average power into the circuit and reactive power (Q) which indicates power flowing back and forth. We can also define the complex power S = P + jQ and the apparent power which is the magnitude of S. The power law for an AC circuit expressed in phasors is then S = VI* (where I* is the complex conjugate of I, and the magnitudes of the voltage and current phasors V and of I are the RMS values of the voltage and current, respectively).

Given this we can apply the techniques of analysis of resistive circuits with phasors to analyze single frequency linear AC circuits containing resistors, capacitors, and inductors. Multiple frequency linear AC circuits and AC circuits with different waveforms can be analyzed to find voltages and currents by transforming all waveforms to sine wave components (using Fourier series) with magnitude and phase then analyzing each frequency separately, as allowed by the superposition theorem. This solution method applies only to inputs that are sinusoidal and for solutions that are in steady state, i.e., after all transients have died out.[14]

The concept is frequently involved in representing an electrical impedance. In this case, the phase angle is the phase difference between the voltage applied to the impedance and the current driven through it.

Power engineeringEdit

In analysis of three phase AC power systems, usually a set of phasors is defined as the three complex cube roots of unity, graphically represented as unit magnitudes at angles of 0, 120 and 240 degrees. By treating polyphase AC circuit quantities as phasors, balanced circuits can be simplified and unbalanced circuits can be treated as an algebraic combination of symmetrical components. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents. In the context of power systems analysis, the phase angle is often given in degrees, and the magnitude in RMS value rather than the peak amplitude of the sinusoid.

The technique of synchrophasors uses digital instruments to measure the phasors representing transmission system voltages at widespread points in a transmission network. Differences among the phasors indicate power flow and system stability.

Telecommunications: analog modulationsEdit

A: phasor representation of amplitude modulation, B: alternate representation of amplitude modulation, C: phasor representation of frequency modulation, D: alternate representation of frequency modulation

The rotating frame picture using phasor can be a powerful tool to understand analog modulations such as amplitude modulation (and its variants[15]) and frequency modulation.

where the term in brackets is viewed as a rotating vector in the complex plane.

The phasor has length  , rotates anti-clockwise at a rate of   revolutions per second, and at time   makes an angle of   with respect to the positive real axis.

The waveform   can then be viewed as a projection of this vector onto the real axis. A modulated waveform is represented by this phasor (the carrier) and two additional phasors (the modulation phasors). If the modulating signal is a single tone of the form  , where   is the modulation depth and   is the frequency of the modulating signal, then for amplitude modulation the two modulation phasors are given by,

 , and


The two modulation phasors are phased such that their vector sum is always in phase with the carrier phasor. An alternative representation is two phasors counter rotating around the end of the carrier phasor at a rate   relative to the carrier phasor. That is,

 , and


Frequency modulation is a similar representation except that the modulating phasors are not in phase with the carrier. In this case the vector sum of the modulating phasors is shifted 90° from the carrier phase. Strictly, frequency modulation representation requires additional small modulation phasors at   etc, but for most practical purposes these are ignored because their effect is very small.

See alsoEdit


  1. ^ a b Including analysis of the AC circuits.[7]: 53 
  2. ^ This results from   which means that the complex exponential is the eigenfunction of the derivative operator.


  1. ^ Huw Fox; William Bolton (2002). Mathematics for Engineers and Technologists. Butterworth-Heinemann. p. 30. ISBN 978-0-08-051119-1.
  2. ^ Clay Rawlins (2000). Basic AC Circuits (2nd ed.). Newnes. p. 124. ISBN 978-0-08-049398-5.
  3. ^ Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 1965. p269
  4. ^ K. S. Suresh Kumar (2008). Electric Circuits and Networks. Pearson Education India. p. 272. ISBN 978-81-317-1390-7.
  5. ^ Kequian Zhang; Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics (2nd ed.). Springer Science & Business Media. p. 13. ISBN 978-3-540-74296-8.
  6. ^ a b c J. Hindmarsh (1984). Electrical Machines & their Applications (4th ed.). Elsevier. p. 58. ISBN 978-1-4832-9492-6.
  7. ^ a b Gross, Charles A. (2012). Fundamentals of electrical engineering. Thaddeus Adam Roppel. Boca Raton, FL: CRC Press. ISBN 978-1-4398-9807-9. OCLC 863646311.
  8. ^ William J. Eccles (2011). Pragmatic Electrical Engineering: Fundamentals. Morgan & Claypool Publishers. p. 51. ISBN 978-1-60845-668-0.
  9. ^ a b Richard C. Dorf; James A. Svoboda (2010). Introduction to Electric Circuits (8th ed.). John Wiley & Sons. p. 661. ISBN 978-0-470-52157-1.
  10. ^ Allan H. Robbins; Wilhelm Miller (2012). Circuit Analysis: Theory and Practice (5th ed.). Cengage Learning. p. 536. ISBN 978-1-285-40192-8.
  11. ^ a b c Won Y. Yang; Seung C. Lee (2008). Circuit Systems with MATLAB and PSpice. John Wiley & Sons. pp. 256–261. ISBN 978-0-470-82240-1.
  12. ^ Nilsson, James William; Riedel, Susan A. (2008). Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN 978-0-13-198925-2., Chapter 9, page 338
  13. ^ Singh, Ravish R (2009). "Section 4.5: Phasor Representation of Alternating Quantities". Electrical Networks. Mcgraw Hill Higher Education. p. 4.13. ISBN 978-0070260962.
  14. ^ Clayton, Paul (2008). Introduction to electromagnetic compatibility. Wiley. p. 861. ISBN 978-81-265-2875-2.
  15. ^ de Oliveira, H.M. and Nunes, F.D. About the Phasor Pathways in Analogical Amplitude Modulations. International Journal of Research in Engineering and Science (IJRES) Vol.2, N.1, Jan., pp.11-18, 2014. ISSN 2320-9364

Further readingEdit

  • Douglas C. Giancoli (1989). Physics for Scientists and Engineers. Prentice Hall. ISBN 0-13-666322-2.
  • Dorf, Richard C.; Tallarida, Ronald J. (1993-07-15). Pocket Book of Electrical Engineering Formulas (1 ed.). Boca Raton,FL: CRC Press. pp. 152–155. ISBN 0849344735.

External linksEdit

  • Phasor Phactory
  • Visual Representation of Phasors
  • Polar and Rectangular Notation
  • Phasor in Telecommunication