SINUSOIDS AND PHASORS

SINUSOIDS AND PHASORS

Sinusoids

- A Sinusoid is a signal that has the form of the sine and cosine function. Sinusoidal variables are of special importance in electrical and electronic systems, not only because the occur frequently in such systems but also because any periodical signal can be represented as a linear combination of a set of sinusoidal signals of different frequencies, amplitudes, and phase angles.





Sinusoidal variable can be written as:




v(t) = Vm sin(ωt + Ф)

Where;

Vm = the amplitude of the sinusoid

ω = the angular frequency in radians/s

Ф = the phase









Example 1:

Given a sinusoid, i=3sin(3t + 70) , calculate its amplitude, phase, angular frequency

Solution:

Amplitude = 3, phase =70,angular frequency = 3




Example 2:

A sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20A at . Give the expression of this current as a function of time .
Solution:
We have , , , . As cosine function reaches maximum when (or ), the phase angle should satisfy where and , i.e.,














The current is














Alternatively, the phase angle can be found as shown below:














Solving this we get , same as above.

Example 3:

Find the phase angle between i1=-4sin(377t + 25) and i2=5cos(377t – 40), does i1 lead or lag i2?

Solution:

Add 180 @ i1:

i1=-4sin(377t + 25 + 180)

i1=4cos(377t + 205)

Change: cos to sin (add 90)

Since sin(ωt + 90) = cosωt

i2=5cos(377t – 40+90)

i2=5cos(377t +50)

therefore, i1 leads i2 155o(Phase Angle)
Complex Numbers

The mathematics used in Electrical Engineering to add together resistances, currents or DC voltages uses what are called “real numbers”. But real numbers are not the only kind of numbers we need to use especially when dealing with frequency dependent sinusoidal sources and vectors. As well as using normal or real numbers, Complex Numbers were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, √-1.

In electrical engineering this type of number is called an “imaginary number” and to distinguish an imaginary number from a real number the letter “ j ” known commonly in electrical engineering as the j-operator, is used. The letter j is placed in front of a real number to signify its imaginary number operation. Examples of imaginary numbers are: j3, j12, j100 etc. Then a complex number consists of two distinct but very much related parts, a “ Real Number ” plus an “ Imaginary Number ”.