1. Background
As v cos(wt + f + 2p) = v cos(wt + f), restrict –p < f ≤ p
Engineers throw an interesting twist into this formulation
The frequency term wt has units of radians
The phase shift f has units of degrees: –180° < f ≤ 180°

2. Background

3. Background A positive phase shift causes the function to lead of f
For example, –sin(t) = cos(t + 90°) leads cos(t) by 90°
+ 90 °

4. Compare cos(377t) and cos(377t + 45°)

5. Background If the phase shift is 180°, the functions are out of phase
E.g., –cos(t) = cos(t – 180°) and cos(t) are out of phase
- 180 °

6. Compare cos(377t) and cos(377t – 180°)

7. Why we use Phasors?

8. Phasors
The idea of phasor representation is based on Euler’s identity. In general,
we use this relation to express v(t). If v(t) defines as;

9. Phasors If we use sine for the phasor instead of cosine,
then v(t) = Vm sin (ωt + φ) = Im (Vm 𝒆 j(ωt + φ) )
and the corresponding phasor is the same as that

10. Phasors Differentiating a sinusoid:
This shows that the derivative v(t) is transformed to the phasor domain
as jωV

11. Phasors Integrating a sinusoid
Similarly, the integral of v(t) is equivalent to dividing its corresponding phasor by jω.

12. Example 2.6.
Find the sinusoids represent by phasors.
Solution:

13. Example 2.6.
Converting this to time domain gives
in time domain

14. Example 2.7.
Using the phasor approach, determine the curret i(t) in a circuit described by the integrodiffential equation.
Solution:
We transform each term in the equation from time domain to phasor domain.

15. Example 2.7.
İn time domain
İn phasor domain
ω=2 so;

16. Example 2.7.
İn time domain

17. 2.4. Phasor Relationships for Circuits Element
Begin with the resistor;
İf the current trough a resistor R is
With Ohm’s law voltage acrosss R;
The phasor form of this voltage;

18. 2.4. Phasor Relationships for Circuits Element
İf the phasor form of current;

19. 2.4. Phasor Relationships for Circuits Element
For inductor L;
The current trough L is
The voltage acrosss L;

20. 2.4. Phasor Relationships for Circuits Element
Transform to the phasor

21. 2.4. Phasor Relationships for Circuits Element

22. 2.4. Phasor Relationships for Circuits Element
For capacitor C;
The voltage across C is
The current trough C is

23. 2.4. Phasor Relationships for Circuits Element

24. 2.4. Phasor Relationships for Circuits Element

25. Example 2.8.
Solution:

26. Example 2.8.

27. 2.5. Impedance and Admitance

28. 2.5. Impedance and Admitance

29. 2.5. Impedance and Admitance
İf Z=R+jX inductive
İf Z=R-jX capasitive

30. 2.5. Impedance and Admitance

31. 2.5. Impedance and Admitance
Y is the admittance, measured in siemens.
G is the conductance, measured in siemens.
B is the susceptance, measured in siemens.