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°

Background

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 °

Compare cos(377t) and cos(377t + 45°)

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 °

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

Why we use Phasors?

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;

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

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

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

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

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

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.

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

Example 2.7. İn time domain

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;

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

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

2.4. Phasor Relationships for Circuits Element Transform to the phasor

2.4. Phasor Relationships for Circuits Element

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

2.4. Phasor Relationships for Circuits Element

2.4. Phasor Relationships for Circuits Element

Example 2.8. Solution:

Example 2.8.

2.5. Impedance and Admitance

2.5. Impedance and Admitance

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

2.5. Impedance and Admitance

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.