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aystyle 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 left i omega v_ text c frac 1 rc v_ text c right cdot e i omega t left frac 1 rc v_ text s right cdot e i omega t i omega v_ text c frac 1 rc v_ text c frac 1 rc v_ text s end aligned solving for the phasor capacitor voltage gives v c 1 1 i ω r c v s 1 i ω r c 1 ω r c 2 v p e i θ displaystyle v_ text c frac 1 1 i omega rc cdot v_ text s frac 1 i omega rc 1 omega rc 2 cdot v_ text p e i theta as we have seen the factor multiplying v s displaystyle v_ text s represents differences of the amplitude and phase of v c t displaystyle v_ text c t relative to v p displaystyle v_ text p and θ displaystyle theta in polar coordinate form the first term of the last expression is 1 i ω r c 1 ω r c 2 1 1 ω r c 2 e i ϕ ω displaystyle frac 1 i omega rc 1 omega rc 2 frac 1 sqrt 1 omega rc 2 cdot e i phi omega where ϕ ω arctan ω r c displaystyle phi omega arctan omega rc therefore v c t re v c e i ω t 1 1 ω r c 2 v p cos ω t θ ϕ ω displaystyle v_ text c t operatorname re left v_ text c cdot e i omega t right frac 1 sqrt 1 omega rc 2 cdot v_ text p cos omega t theta phi omega ratio of phasors edit 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 applications edit circuit laws edit 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 16 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 engineering edit main article phasor measurement unit 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 modulations edit 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 17 and frequency modulation x t re a e i θ e i 2 π f 0 t displaystyle x t operatorname re left ae i theta cdot e i2 pi f_ 0 t right where the term in brackets is viewed as a rotating vector in the complex plane the phasor has length a displaystyle a rotates anti clockwise at a rate of f 0 displaystyle f_ 0 revolutions per second and at time t 0 displaystyle t 0 makes an angle of θ displaystyle theta with respect to the positive real axis the waveform x t displaystyle x t 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 a m cos 2 π f m t displaystyle am cos 2 pi f_ m t where m displaystyle m is the modulation depth and f m displaystyle f_ m is the frequency of the modulating signal then for amplitude modulation the two modulation phasors are given by 1 2 a m e i θ e i 2 π f 0 f m t displaystyle 1 over 2 ame i theta cdot e i2 pi f_ 0 f_ m t 1 2 a m e i θ e i 2 π f 0 f m t displaystyle 1 over 2 ame i theta cdot e i2 pi f_ 0 f_ m t 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 f m displaystyle f_ m relative to the carrier phasor that is 1 2 a m e i θ e i 2 π f m t displaystyle 1 over 2 ame i theta cdot e i2 pi f_ m t 1 2 a m e i θ e i 2 π f m t displaystyle 1 over 2 ame i theta cdot e i2 pi f_ m t 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 2 f m 3 f m displaystyle 2f_ m 3f_ m etc but for most practical purposes these are ignored because their effect is very small see also edit in phase and quadrature components constellation diagram analytic signal a generalization of phasors for time variant amplitude phase and frequency complex envelope phase factor a phasor of unit magnitude footnotes edit 1 2 including analysis of the ac circuits 7 53 this results from d d t e i ω t i ω e i ω t textstyle frac d dt e i omega t i omega e i omega t which means that the complex exponential is the eigenfunction of the derivative operator references edit huw fox william bolton 2002 mathematics for engineers and technologists butterworth heinemann p 30 isbn 978 0 08 051119 1 clay rawlins 2000 basic ac circuits 2nd ed newnes p 124 isbn 978 0 08 049398 5...
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