This document discusses phasors and their use in representing sinusoidal functions. It provides background on phasors, including that they use complex numbers to represent sinusoids in the transform domain rather than the time domain. Phasors allow complicated problems involving sinusoids to be transformed into simpler problems using complex numbers. The document gives examples of how to derive and work with phasors for sinusoids, including how to transform between the time and phasor domains. It also provides an example problem of using phasors to solve an integrodifferential equation describing a circuit.

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

2. Background

3. A positive phase shift causes the function to lead of f
For example, –sin(t) = cos(t + 90°) leads cos(t) by 90°
Background
0 45 90 135 180 225 270 315 360
-1
-0.5
0
0.5
1
time(t)
cos(t)
sin(t)
+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
0 45 90 135 180 225 270 315 360
-1
-0.5
0
0.5
1
time(t)
cos(t)
sin(t)
-180°

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

7. Why we use Phasors?
Problem Solution
Complicated and difficult
solution process
Transformed
Problem
Transformed
Problem
Transformed
Solution
Transformed
Solution
Transform
Relatively simple
solution process, but
using complex numbers
Inverse
Transform
Solutions Using Transforms
Real, or time
domain
Complex or
transform domain

8. 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;
Phasors

9. Phasors
• If we use sine for the phasor instead of cosine,
• then v(t) = Vm sin (ωt + φ) = Im (Vm𝒆j(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. Find the sinusoids represent by phasors.
Solution:

13. Converting this to time domain gives
in time domain

14. 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. İn time domain
İn phasor domain
ω=2 so;

16. İn time domain

17.  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.  İf the phasor form of current;

19.  For inductor L;
 The current trough L is
 The voltage acrosss L;

20. Transform to the phasor

22.  For capacitor C;
 The voltage across C is
 The current trough C is

25. Solution:

29.  İf Z=R+jX inductive
İf Z=R-jX capasitive