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Encyclopedia > Phasor (electronics)

See wikibooks' book on Phasors

A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. 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. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = âˆ’1. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

1 Introduction
2 Phasor Calculus
3 Circuit laws
4 Phasor transform
5 Phasor arithmetic
6 See also

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)

$y=Acos{(omega t+phi)},!$

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 $ω = 2π f$ where f is frequency.
t is time.

This can be expressed as The radian (symbol: rad, or a superscript c ( half circle)) is the SI unit of plane angle. ...

$y=Re Big(Abig(cos{(omega{}t+phi)}+jsin{(omega t+phi)}big)Big),!$

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, In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... In electricity, current refers to electric current, which is the flow of electric charge. ... Error creating thumbnail: convert: unable to open image `/mnt/upload3/wikipedia/commons/e/eb/Eulers_formula. ...

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

$y=Re(Ae^{jphi}e^{jomega{}t}),!$

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

$Y = Ae^{j phi},$

such that

$y=Re(Ye^{jomega{}t}),!$

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 the form:

$Y = A angle phi ,$

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. This article treats electronics engineering as a subfield of electrical engineering, though this is not universally accepted. ... A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized Â°, is a measurement of plane angle, representing 1ï¼360 of a full rotation. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...

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: In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Algebraic chess notation is the method used today by all competition chess organizations and most books, magazines, and newspapers to record and describe the play of chess games. ... Eulers formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ...

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

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: A resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue infinite impulse response electronic filters. ...

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

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

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

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. Direct current (DC or continuous current) is the continuous flow of electricity through a conductor such as a wire from high to low potential. ...

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. In electrical engineering, Impedance is a measure of opposition to a sinusoidal electric current. ... In alternating current power transmission and distribution, complex power is a complex quantity which captures information about both the magnitude and phase of power consumed by a load. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ... A resistive circuit is a circuit containing only resistors, perfect current sources, and perfect voltage sources. ...

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.

eh?

Phasor arithmetic

As with other complex quantities the exponential (polar) form $A e j φ$ simplifies multiplication and division, while the Cartesian (rectangular) form $a + j b$ simplifies addition and subtraction. This article describes some of the common coordinate systems that appear in elementary mathematics. ... Cartesian means of or relating to the French philosopher and mathematician René Descartes. ...

Contents

Introduction GA_googleFillSlot("encyclopedia_square");

Phasor Calculus

Circuit laws

Phasor transform

Phasor arithmetic

See also

Introduction