SINUSOIDS AND PHASORS

A Sinusoid is a signal that has the form of the sine or cosine function.

                              There are two parts of sinusoid, the Sinusoidal Current and Sinusoidal Voltage. Sinusoidal current is usually referred to as alternating current. Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuit. 

Sinusoidal voltage,

v(t) = Vmsinωt

where;

Vm= the amplitude of the sinusoid

ω = the angular frequency in radian/s

ωt = the argument of the sinusoid

Sample equation to determine it's label;

  6cos(200t + 15° )

Amplitude- 6

Phase angle- 15°

Angular Frequency- 200t

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

2 PHASES

* IN PHASE

*OUT OF PHASE

IN PHASE,

The same;

*Time

*Period

*Frequency

OUT OF PHASE,

It's either have the same amplitude or not.

~Sinusoids are easily expressed in terms of phasors, in which are more convenient to work with than sine and cosine function.

Sinusoid-Phasor Transformation 

Time Domain representation	Phasor Domain Representation
Vmcos(ωt + ɸ )	Vm ∠ ɸ
Vmsin(ωt + ɸ )	Vm ∠ ɸ - 90 °
Imcos(ωt + 0 )	Im∠0
Imsin(ωt + 0 )	Im∠ 0 - 90 °

To transform the Time domain into the Phasor domain, the time domain is in the rectangular form of

 z = x + jy,

where in x is the real part of z and y is the imaginary part.

The equation going to polar form for the amplitude;

Square root of x squared plus y squared

For the Phase angle;

arctan(y divided by x) 

Example; 

5 + j2
= 5.39∠ 21.80°

Since all of these was all about the currents and voltages, in getting the value of each of them we go through graphing in a sinusoidal form. So between that two, there must be leading and lagging. 

Looking at the figure, the voltage leads the current since leading is when a sinusoid peaks first in time and it is closer to the reference axis. And the current here is lagging.