06.10.2011. A sinusoids is signal that has the form of the sine or cosine function. Consider the sinusoidal voltage.

*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. Sinusoids

as 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. Sinusoids

Consider a more general expression for the sinusoids.

Phase (in radian or degrees)

2.2. Sinusoids

Let us consider two sinusoids.

İn this case, lags by

İf

İf

2.2. Sinusoids

A 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. Sinusoids

The graphical technique can be also used to add two sinusoids of the same frequency.

2.2. Sinusoids

For example;

?

sin

cos +3

-4

5

53.10

Example 2.1.

Example 2.2.

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

Solution:

Sam

e fo

rm

Example 2.2.

2.3. Phasors

A 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 part

Real part

2.3. Phasors

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

magnitude phase

z can be expressed in three forms;

2.3. Phasors

Relationship between polar and rectangular form;

2.3. Phasors

Following operations are important;

2.3. Phasors

2.3. Phasors

İn 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.