Phasor:: Subject: Fundamentals Of Electrical Engineering

Phasor :

Subject: Fundamentals of Electrical Engineering.

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Phasor (Alexander) : A phasor is a complex representation of the magnitude and the phase of a sinusoid.

Basically complex numbers are represented by;

Z=x+jy=

(Rectangular form)

Z = r ∟Ǿ ° = (polar form) Z = r e jǾ

r = Magnitude

Ǿ ° = Phase

(Exponential form)

The idea of Phasor representation depends upon Euler’s identity: e ± j Ǿ = Cos Ǿ + j Sin Ǿ - - - - - - - - - - - - - (9.19) Which may be describe as

(Real part of complex no.) Cos Ǿ = R e (e j Ǿ) - - - - - - - - (9.20 a) (Imag: part of complex no.) Sin Ǿ = I m (e j Ǿ) - - - - - - - - - (9.20 b)

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Sinusoids :

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Sinusoid:

The fact that V (t) repeats itself every T sec is shown; replacing t by (t + T) in above equation, we get; V (t + T) = V m sin ω (t + T) = V m sin ω (t + 2 π / ω)

=>

V m sin (ω t + 2 π)

Hence,

V (t + T) = V (t)

=>

V m sin ω t = V (t)

(A periodic function is one that satisfies f (t) = f (t + n T), for all t and for all integers n). 4

Sinusoid : 

The period T of the periodic function is the time of one complete cycle or the number of seconds per cycle.



The reciprocal of this quantity is the number of cycles per seconds, known as the cyclic frequency of the sinusoid. Thus, f=1/T

=>

T=1/f

so that,

=>

ω=2πf

=>

2π/ω=1/f

Where, f = Frequency in Hz. ω = Angular frequency in rad / sec. 5

Let us now consider a more general expression for the sinusoid, V (t) = Vm sin (ω t + Ǿ) Where, (ω t + Ǿ) = Argument Ǿ = Phase (Both argument & phase can be expressed in radians or degrees). Let us examine the two sinusoids,

V 1 (t) = V m sin ω t

&

V 2 (t) = V m sin (ω t + Ǿ)

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Example 9.1:

Find the amplitude, phase, period and frequency of the sinusoid. V (t) = 12 cos (50 t + 10 °)

Here, The amplitude V m = 12 v The phase = Ǿ = 10 ° The angular frequency is = ω = 50 rad / sec.

The period = T = 2 π / ω = 2 π / 50 = 0.1257 sec The frequency = f = 1 / T = 1 / 0.1257 sec = 7.958 Hz. 7

Sinusoid :

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Resistive Load:

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Resistive Load:

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Inductive Load:

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Inductive Load:

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Inductive Load:

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V-I Characteristics in Inductive load:

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V-I Characteristics in Inductive load:

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Capacitive Load:

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Capacitive Load:

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V & I Characteristics:

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