Editing Phasor (section)

== Applications ==

=== Circuit laws ===

With phasors, the techniques for solving [[direct current|DC]] circuits can be applied to solve linear AC circuits.{{Efn|name="ac-circuits"}}

; Ohm's law for resistors: A [[resistor]] has no time delays and therefore doesn't change the phase of a signal therefore {{math|1=''V'' = ''IR''}} remains valid.
; Ohm's law for resistors, inductors, and capacitors: {{math|1=''V'' = ''IZ''}} where {{mvar|Z}} is the complex [[electrical impedance|impedance]].<!-- we probably want a justification of this somewhere-->
; [[Kirchhoff's circuit laws]]: Work with voltages and current as complex phasors.

In an AC circuit we have real power ({{mvar|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]] {{math|1=''S'' = ''P'' + ''jQ''}} and the apparent power which is the magnitude of {{mvar|S}}. The power law for an AC circuit expressed in phasors is then {{math|1=''S'' = ''VI''<sup>*</sup>}} (where {{math|1=''I''<sup>*</sup>}} is the [[complex conjugate]] of {{math|1=''I''}}, and the magnitudes of the voltage and current phasors {{math|1=''V''}} and of {{math|1=''I''}} are the [[Root mean square#Definition|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 [[inductor]]s. 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.<ref>{{Cite book|title=Introduction to electromagnetic compatibility| last=Clayton|first=Paul| publisher=Wiley|year=2008|isbn=978-81-265-2875-2|pages=861}}</ref>

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===
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 [[Degree (angle)|degree]]s, and the magnitude in [[Root Mean Square|RMS]] value rather than the peak amplitude of the sinusoid.

The technique of [[synchrophasor]]s 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 ===
[[File:Modulation phasors.svg|thumb|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<ref name=IJRES>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</ref>) and [[frequency modulation]].

<math display="block">x(t) = \operatorname{Re}\left( A e^{i \theta} \cdot e^{i 2\pi f_0 t} \right),</math>
where the term in brackets is viewed as a rotating vector in the complex plane.

The phasor has length <math>A</math>, rotates anti-clockwise at a rate of <math>f_0</math> revolutions per second, and at time <math>t = 0</math> makes an angle of <math>\theta</math> with respect to the positive real axis.

The waveform <math>x(t)</math> 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 <math>Am \cos{2\pi f_m t} </math>, where <math>m</math> is the modulation depth and <math>f_m</math> is the frequency of the modulating signal, then for amplitude modulation the two modulation phasors are given by,

<math display="block">{1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi (f_0+f_m) t},</math>
<math display="block">{1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi (f_0-f_m) t}.</math>

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 <math>f_m</math> relative to the carrier phasor.  That is,

<math display="block">{1 \over 2} Am e^{i \theta} \cdot e^{i 2\pi f_m t},</math>
<math display="block">{1 \over 2} Am e^{i \theta} \cdot e^{-i 2\pi f_m t}.</math>

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 <math>2f_m, 3f_m</math> etc, but for most practical purposes these are ignored because their effect is very small.