Phasor (electronics)

Introduction


A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. (In older texts, a phasor is alternatively referred to as a sinor.) It is usually expressed in exponential form. Phasors are used in engineering to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.


A sinusoid (or sine waveform) is defined to be a function of the form (the reason for using cosine rather than sine will become apparent later)


y=A\cos{(\omega t+\phi)}\,\! 
where

   – y is the quantity that is varying with time
   – Ф is a constant (in radians) known as the phase or phase angle of the sinusoid
   – A is a constant known as the amplitude of the sinusoid. It is the peak value of the function.
   – ω is the angular frequency given by ω = 2πf where f is frequency.
   – t is time.

This can be expressed as


y=\Re \Big(A\big(\cos{(\omega{}t+\phi)}+j\sin{(\omega t+\phi)}\big)\Big)\,\! 

where


   – j is the imaginary unit . Note that i is not used in electrical engineering as it is commonly used to represent the changing current.
   – \Re (z) gives the real part of the complex number z



Equivalently, by Euler’s formula,



y=\Re(Ae^{j(\omega{}t+\phi)})\,\!
y=\Re(Ae^{j\phi}e^{j\omega{}t})\,\!

Y, the phasor representation of this sinusoid is defined as follows:



Y = Ae^{j \phi}\,

such that



y=\Re(Ye^{j\omega{}t})\,\!

Thus, the phasor Y is the constant complex number that encodes the amplitude and phase of the sinusoid. To simplify the notation, phasors are often written in angle notation:



Y = A \angle \phi \,

Within Electrical Engineering, the phase angle is commonly specified in degrees rather than radians and the magnitude will often be the rms value rather than a peak value of the sinusoid.


The overarching conceptual motive behind phasor calculus is that it is generally far more convenient to manipulate complex numbers than to manipulate literal trigonometric functions. Noting that a trigonometric function can be represented as the real component of a complex quantity, it is efficacious to perform the required mathematical operations upon the complex quantity and, at the very end, take its real component to produce the desired answer. This is quite similar to the concept underlying complex potential in such fields as electromagnetic theory, where—instead of manipulating a real quantity, u—it is often more convenient to derive its harmonic conjugate, v, and then operate upon the complex quantity u + jv, again recovering the real component of the complex “result” as the last stage of computation to generate the true result.



Circuit laws


With phasors, the techniques for solving DC circuits can be applied to solve AC circuits. A list of the basic laws is given below.




  • 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.
  • 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 forward. 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).
  • Kirchhoff’s circuit laws work with phasors in complex form

Given this we can apply the techniques of analysis of resistive circuits with phasors to analyse single frequency AC circuits containing resistors, capacitors, and inductors. Multiple frequency linear AC circuits and AC circuits with different waveforms can be analysed to find voltages and currents by transforming all waveforms to sine wave components with magnitude frequency and phase then analysing each frequency separately. However this method does not work for power as power is based on voltage times current.

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