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ng at general electric in the late 19th century 10 11 he got his inspiration from oliver heaviside heaviside s operational calculus was modified so that the variable p becomes jω the complex number j has simple meaning phase shift 12 glossing over some mathematical details the phasor transform can also be seen as a particular case of the laplace transform limited to a single frequency which in contrast to phasor representation 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 a e i ω t θ displaystyle a cdot e i omega t theta 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 2 π ω θ is the angle it forms with the positive real axis at t 0 and at t n 2 π ω for all integer values of n notation edit see also vector notation phasor notation also known as angle notation is a mathematical notation used in electronics engineering and electrical engineering a vector whose polar coordinates are magnitude a displaystyle a and angle θ displaystyle theta is written a θ displaystyle a angle theta 13 1 θ displaystyle 1 angle theta can represent either the vector cos θ sin θ displaystyle cos theta sin theta or the complex number cos θ i sin θ e i θ displaystyle cos theta i sin theta e i theta according to euler s formula with i 2 1 displaystyle i 2 1 both of which have magnitudes of 1 the angle may be stated in degrees with an implied conversion from degrees to radians for example 1 90 displaystyle 1 angle 90 would be assumed to be 1 90 displaystyle 1 angle 90 circ which is the vector 0 1 displaystyle 0 1 or the number e i π 2 i displaystyle e i pi 2 i multiplication and division of complex numbers become straight forward through the phasor notation given the vectors v 1 a 1 θ 1 displaystyle v_ 1 a_ 1 angle theta _ 1 and v 2 a 2 θ 2 displaystyle v_ 2 a_ 2 angle theta _ 2 the following is true 14 v 1 v 2 a 1 a 2 θ 1 θ 2 displaystyle v_ 1 cdot v_ 2 a_ 1 cdot a_ 2 angle theta _ 1 theta _ 2 v 1 v 2 a 1 a 2 θ 1 θ 2 displaystyle frac v_ 1 v_ 2 frac a_ 1 a_ 2 angle theta _ 1 theta _ 2 definition edit for a sinusoid with fixed angular frequency ω amplitude a and phase angle θ the time varying signal is expressed as v t a cos ω t θ displaystyle v t a cos omega t theta a phasor is a complex number representation that captures the amplitude and phase of this sinusoid the phasor v corresponding to the above sinusoid is defined as v a e i θ displaystyle mathbf v ae i theta using euler s formula e i θ cos θ i sin θ this can be expanded to v a cos θ i a sin θ displaystyle mathbf v a cos theta i cdot a sin theta alternatively the phasor can be expressed in polar form as v a θ displaystyle mathbf v a angle theta which reads as amplitude a at phase angle θ 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 a e i θ displaystyle ae i theta and the common factor e i ω t displaystyle e i omega t is reinserted prior to the real part of the result the function a e i ω t θ displaystyle ae i omega t theta is an analytic representation of a cos ω t θ displaystyle a cos omega t theta 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 15 as we do in the next section arithmetic edit see also complex number relations and operations multiplication by a constant scalar edit multiplication of the phasor a e i θ e i ω t displaystyle ae i theta e i omega t by a complex constant b e i ϕ displaystyle be i phi produces another phasor that means its only effect is to change the amplitude and phase of the underlying sinusoid re a e i θ b e i ϕ e i ω t re a b e i θ ϕ e i ω t a b cos ω t θ ϕ displaystyle begin aligned operatorname re left left ae i theta cdot be i phi right cdot e i omega t right operatorname re left left abe i theta phi right cdot e i omega t right ab cos omega t theta phi end aligned in electronics b e i ϕ displaystyle be i phi 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 addition edit 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 a 1 cos ω t θ 1 a 2 cos ω t θ 2 re a 1 e i θ 1 e i ω t re a 2 e i θ 2 e i ω t re a 1 e i θ 1 e i ω t a 2 e i θ 2 e i ω t re a 1 e i θ 1 a 2 e i θ 2 e i ω t re a 3 e i θ 3 e i ω t a 3 cos ω t θ 3 displaystyle begin aligned a_ 1 cos omega t theta _ 1 a_ 2 cos omega t theta _ 2 3pt operatorname re left a_ 1 e i theta _ 1 e i omega t right operatorname re left a_ 2 e i theta _ 2 e i omega t right 3pt operatorname re left a_ 1 e i theta _ 1 e i omega t a_ 2 e i theta _ 2 e i omega t right 3pt operatorname re left left a_ 1 e i theta _ 1 a_ 2 e i theta _ 2 right e i omega t right 3pt operatorname re left left a_ 3 e i theta _ 3 right e i omega t right 3pt a_ 3 cos omega t theta _ 3 end aligned where a 3 2 a 1 cos θ 1 a 2 cos θ 2 2 a 1 sin θ 1 a 2 sin θ 2 2 displaystyle a_ 3 2 a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 2 a_ 1 sin theta _ 1 a_ 2 sin theta _ 2 2 and if we take θ 3 π 2 3 π 2 textstyle theta _ 3 in left frac pi 2 frac 3 pi 2 right then θ 3 displaystyle theta _ 3 is sgn a 1 sin θ 1 a 2 sin θ 2 π 2 textstyle operatorname sgn a_ 1 sin theta _ 1 a_ 2 sin theta _ 2 cdot frac pi 2 if a 1 cos θ 1 a 2 cos θ 2 0 displaystyle a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 0 with sgn displaystyle operatorname sgn the signum function arctan a 1 sin θ 1 a 2 sin θ 2 a 1 cos θ 1 a 2 cos θ 2 displaystyle arctan left frac a_ 1 sin theta _ 1 a_ 2 sin theta _ 2 a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 right if a 1 cos θ 1 a 2 cos θ 2 0 displaystyle a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 0 π arctan a 1 sin θ 1 a 2 sin θ 2 a 1 cos θ 1 a 2 cos θ 2 displaystyle pi arctan left frac a_ 1 sin theta _ 1 a_ 2 sin theta _ 2 a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 right if a 1 cos θ 1 a 2 cos θ 2 0 displaystyle a_ 1 cos theta _ 1 a_ 2 cos theta _ 2 0 or via the law of cosines on the complex plane or the trigonometric identity for angle differences a 3 2 a 1 2 a 2 2 2 a 1 a 2 cos 180 δ θ a 1 2 a 2 2 2 a 1 a 2 cos δ θ displaystyle a_ 3 2 a_ 1 2 a_ 2 2 2a_ 1 a_ 2 cos 180 circ delta theta a_ 1 2 a_ 2 2 2a_ 1 a_ 2 cos delta theta where δ θ θ 1 θ 2 displaystyle delta theta theta _ 1 theta _ 2 a key point is that a 3 and θ 3 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 a 1 θ 1 a 2 θ 2 a 3 θ 3 displaystyle a_ 1 angle theta _ 1 a_ 2 angle theta _ 2 a_ 3 angle theta _ 3 another way to view addition is that two vectors with coordinates a 1 cos ωt θ 1 a 1 sin ωt θ 1 and a 2 cos ωt θ 2 a 2 sin ωt θ 2 are added vectorially to produce a resultant vector with coordinates a 3 cos ωt θ 3 a 3 sin ωt θ 3 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 2 π 3 radians or one third of a wavelength λ 3 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 cos ω t cos ω t 2 π 3 cos ω t 2 π 3 0 displaystyle cos omega t cos left omega t frac 2 pi 3 right cos left omega t frac 2 pi 3 right 0 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 λ displaystyle lambda 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 2 π 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 a max at 90 or π 2 and the negative peak value a max at 270 or 3 π 2 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 integration edit the time derivative or integral of a phasor produces another phasor b for example re d d t a e i θ e i ω t re a e i θ i ω e i ω t re a e i θ e i π 2 ω e i ω t re ω a e i θ π 2 e i ω t ω a cos ω t θ π 2 displaystyle begin aligned operatorname re left frac mathrm d mathrm d t mathord left ae i theta cdot e i omega t right right operatorname re left ae i theta cdot i omega e i omega t right operatorname re left ae i theta cdot e i pi 2 omega e i omega t right operatorname re left omega ae i theta pi 2 cdot e i omega t right omega a cdot cos left omega t theta frac pi 2 right end aligned therefore in phasor representation the time derivative of a sinusoid becomes just multiplication by the constant i ω e i π 2 ω textstyle i omega e i pi 2 cdot omega similarly integrating a phasor corresponds to multiplication by 1 i ω e i π 2 ω textstyle frac 1 i omega frac e i pi 2 omega the time dependent factor e i ω t displaystyle e i omega t is unaffected when we solve a linear differential equation with phasor arithmetic we are merely factoring e i ω t displaystyle e i omega t 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 d v c t d t 1 r c v c t 1 r c v s t displaystyle frac mathrm d v_ text c t mathrm d t frac 1 rc v_ text c t frac 1 rc v_ text s t when the voltage source in this circuit is sinusoidal v s t v p cos ω t θ displaystyle v_ text s t v_ text p cdot cos omega t theta we may substitute v s t re v s e i ω t displaystyle v_ text s t operatorname re left v_ text s cdot e i omega t right v c t re v c e i ω t displaystyle v_ text c t operatorname re left v_ text c cdot e i omega t right where phasor v s v p e i θ displaystyle v_ text s v_ text p e i theta and phasor v c displaystyle v_ text c is the unknown quantity to be determined in the phasor shorthand notation the differential equation reduces to i ω v c 1 r c v c 1 r c v s displaystyle i omega v_ text c frac 1 rc v_ text c frac 1 rc v_ text s derivation d d t re v c e i ω t 1 r c re v c e i ω t 1 r c re v s e i ω t displaystyle frac mathrm d mathrm d t operatorname re left v_ text c cdot e i omega t right frac 1 rc operatorname re v_ text c cdot e i omega t frac 1 rc operatorname re left v_ text s cdot e i omega t right eq 1 since this must hold for all t displaystyle t specifically t π 2 ω textstyle t frac pi 2 omega it follows that d d t im v c e i ω t 1 r c im v c e i ω t 1 r c im v s e i ω t displaystyle frac mathrm d mathrm d t operatorname im left v_ text c cdot e i omega t right frac 1 rc operatorname im left v_ text c cdot e i omega t right frac 1 rc operatorname im left v_ text s cdot e i omega t right eq 2 it is also readily seen that d d t re v c e i ω t re d d t v c e i ω t re i ω v c e i ω t d d t im v c e i ω t im d d t v c e i ω t im i ω v c e i ω t displaystyle begin aligned frac mathrm d mathrm d t operatorname re left v_ text c cdot e i omega t right operatorname re left frac mathrm d mathrm d t mathord left v_ text c cdot e i omega t right right operatorname re left i omega v_ text c cdot e i omega t right frac mathrm d mathrm d t operatorname im left v_ text c cdot e i omega t right operatorname im left frac mathrm d mathrm d t mathord left v_ text c cdot e i omega t right right operatorname im left i omega v_ text c cdot e i omega t right end aligned substituting these into eq 1 and eq 2 multiplying eq 2 by i displaystyle i and adding both equations gives i ω v c e i ω t 1 r c v c e i ω t 1 r c v s e i ω t i ω v c 1 r c v c e i ω t 1 r c v s e i ω t i ω v c 1 r c v c 1 r c v s displaystyle begin aligned i omega v_ text c cdot e i omega t frac 1 rc v_ text c cdot e i omega t frac 1 rc v_ text s cdot e i omega t lef...
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