PHASORS

PHASORS

-A Phasor is a complex number that represents the amplitude and phase of a sinusoid.


Phasors provide a simple means of analyzing linear circuits excited by sinusoidal sources; solutions of such circuits would be intractable otherwise. The notion of solving ac circuits using phasors was first introduced by charles Steinmetz in 1893. Before we completely define phasors and apply them to circuit analysis, we need to be throughly familiar with complex numbers.

A  complex number z can be written in rectangular form as:

                                             z=x+jy

where j=√-1; x is the real part of z; y is the imaginary part of z. In this context, the variables x and y do not represent a location as in two-dimensional vector analysis but rather the real and imaginary parts of z in the complex plane. Nevertheless, we note that there are some resemblances between manipulating complex numbers and manipulating two-dimensional vectors.

The complex number z can also be written in polar or exponential form as:

                                         z=r∠Φ
where r is the magnitude of z, and Φ is the phase of z. We notice that z can be represented in three ways:

                z=x+jy                     z=r∠Φ               z=re^jΦ



the relationship between the rectangular form and the polar form is shown in fig, 9.6, where the x axis represents the real part and the y axis represents the imaginary part of a complex number. Given x and y, we can get r amd Φ as

                              r=√󠄈x^2+y^2,                Φ=tan^-1 (y/x)



On the other hand, if we know r and Φ, we can obtain x and y as

                            x=rcosΦ             y=rsinΦ

Mathematical operations of complex numbers:

ADDITION

Z1+Z2=(X1+X2) + J(y1+y2)

SUBTRACTION

z1-z2=(x1-x2)+(y1-y2)

MULTIPLICATION

z1z2= r1r2∠Φ1+Φ2

DIVISION

z1/z2=r1/r2∠Φ1-Φ2

RECIPROCAL

1/z=1/r∠-Φ

SQUARE ROOT

square root of z=square root of r∠Φ/2

COMPLEX CONJUGATE

z*=x-jy=r∠-Φ