Phasor (electronics)

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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.

Contents

[edit] 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)

y=A\cos{(\omega t+\phi)}\,\!

where

  • 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

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

where

  • j is the imaginary unit \sqrt{-1}. 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 (no pun intended) 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.

[edit] 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:

\frac{d}{dt}(e^{j \omega t}) = j \omega e^{j \omega t}

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:

\frac{d}{dt} \cos{\omega t} = - \omega \sin{\omega t}\,

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:

\frac{dv_C}{dt} + \frac{1}{RC}v_C = \frac{1}{RC}v_S

When the voltage source in this circuit is sinusoidal:

v_S(t) = V_P \cos(\omega t + \phi)\,

the differential equation (in phasor form) becomes:

j \omega V_c + \frac{1}{RC} V_c = \frac{1}{RC}V_s

where

V_s = V_P e^{j \phi}\,

Solving for the phasor capacitor voltage gives:

V_c = \frac{1}{1 + j \omega RC} V_s

To convert the phasor capacitor voltage back to a sinusoid, we need to express all complex numbers in polar form:

V_c = \frac{1}{\sqrt{1 + (\omega RC)^2}}e^{j \theta(\omega)} V_s

where

\theta(\omega) = -\arctan(\omega RC)\,

Then

v_C(t) = \frac{1}{\sqrt{1 + (\omega RC)^2}} V_P \cos(\omega t + \phi + \theta(\omega))

[edit] 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.

[edit] Phasor transform

The phasor transform or phasor representation allows transformation from complex form to trigonometric form:

V_m e^{j \phi } = \mathcal{P} \{ V_m \cos( \omega t + \phi ) \}

where the notation \mathcal{P} \{ \} is read "the phasor transform of ____."

The phasor transform transfers the sinusoidal function from the time domain to the complex-number domain or frequency domain.

[edit] Inverse phasor transform

The inverse phasor transform \mathcal{P}^{-1} allows one to move back from the phasor domain to the time domain.

V_m \cos( \omega t + \phi ) = \mathcal{P}^{-1} \{ V_m e^{j \phi } \} = \Re \{ V_m e^{j \phi } e^{j \omega t } \}

[edit] 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.

[edit] 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 circuits. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents.

[edit] See also

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