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*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.