*SINUSOIDS AND PHASORS

06.10.2011

2.2. Sinusoids

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

Consider the sinusoidal voltage.

2.2. Sinusoidsas a function of ωt

as a function of t

Sinusoids repeat itself every T seconds.

T is called the period of sinusoids.

2.2. Sinusoids

İf write t+T instead of t

2.2. Sinusoids The frequency f of the sinusoids

2.2. SinusoidsConsider a more general expression for the

sinusoids.

Phase (in radian or degrees)

2.2. SinusoidsLet us consider two sinusoids.

İn this case, lags by

İf

İf

2.2. SinusoidsA sinusoids can be expressed either in sine or

cosine function.

We can transform a sinusoids from sine to cosine or vice versa.

2.2. Sinusoids

2.2. SinusoidsThe graphical technique can be also used to add

two sinusoids of the same frequency.

2.2. SinusoidsFor example;

?

sin

cos +3

-45

53.10

Example 2.1.

Example 2.2. Calculate the phase angel between and . State which sinusoid is leading.

Solution:

Sam

e for

m

Example 2.2.

2.3. PhasorsA phasor is a complex number that represents the

amplitude and phase of a sinusoid.

Before we completely define phasors and apply them to circuit analysis, we need to be thoroughly familiar with complex numbers,

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

imaginary partReal part

2.3. PhasorsThe complex number z can be written in polar or

exponential form as;

magnitude phasez can be expressed in three forms;

2.3. PhasorsRelationship between polar and rectangular form;

2.3. PhasorsFollowing operations are important;

2.3. Phasors

2.3. Phasorsİn general;

Real part

imaginary part

Time-domain represantaion

Phasor-domain represantaion

2.3. Phasors

Sinusoid-Phasor Transformations

Time-domain represantaion

Phasor-domain represantaion

2.3. Phasors

Difference Between and V

Example 2.3.

Example 2.3.

Example 2.3.

Example 2.4.

Solution:

Example 2.4.

Example 2.5.

Solution: Here is an important use of phasors for summing

sinusoids of the same frequency.

Current is in standart form. Its phasor is;

Example 2.5. we need to express in cosine form. The rule for converting

sine to cosine is to substract .

if we let , then

Example 2.5.