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PHASORS
Introduction
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)
where
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


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y is the quantity that is varying with time
φ 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
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.

gives the real part of the complex number z
Equivalently, by Euler's formula,
Y, the phasor representation of this sinusoid is defined as follows:
such that
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:
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.
Phasor Calculus
When sinusoids are represented as phasors, differential equations become algebraic equations.
This result follows from the fact that the complex exponential is the eigenfunction of the
derivative operation:
That is, only the complex amplitude is changed by the derivative operation. Taking the real part
of both sides of the above equation gives the familiar result:
Thus, a time derivative of a sinusoid becomes, in the phasor representation, multiplication by the
complex frequency. Similarly, integrating a phasor corresponds to division by the complex
frequency.
As an 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:
the differential equation (in phasor form) becomes:
where
Solving for the phasor capacitor voltage gives:
To convert the phasor capacitor voltage back to a sinusoid, we need to express all complex
numbers in polar form:
where
Then
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.
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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
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.
Phasor transform
The phasor transform or phasor representation allows transformation from complex form to
trigonometric form:
where the notation
is read "the phasor transform of ____."
The phasor transform transfers the sinusoidal function from the time domain to the complexnumber domain or frequency domain.
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Inverse phasor transform
The inverse phasor transform
domain.
allows one to move back from the phasor domain to the time
Phasor arithmetic
As with other complex quantities the exponential (polar) form Aejφsimplifies multiplication and
division, while the Cartesian (rectangular) form a + jb simplifies addition and subtraction.