1. 1 EENG224 Chapter 9 Complex Numbers and Phasors Huseyin Bilgekul EENG224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University Chapter Objectives:  Understand the concepts of sinusoids and phasors.  Apply phasors to circuit elements.  Introduce the concepts of impedance and admittance.  Learn about impedance combinations.  Apply what is learnt to phase-shifters and AC bridges.

2. 2 EENG224 Complex Numbers  A complex number may be written in RECTANGULAR FORM as: x is the REAL part. y is the IMAGINARY part. r is the MAGNITUDE. φ is the ANGLE.  A second way of representing the complex number is by specifying the MAGNITUDE and r and the ANGLE θ in POLAR form.  The third way of representing the complex number is the EXPONENTIAL form.

3. 3 EENG224 Complex Numbers  A complex number may be written in RECTANGULAR FORM as: forms.

4. 4 EENG224 Complex Number Conversions  We need to convert COMPLEX numbers from one form to the other form.

5. 5 EENG224 Mathematical Operations of Complex Numbers  Mathematical operations on complex numbers may require conversions from one form to other form.

6. 6 EENG224

7. 7 Phasors  A phasor is a complex number that represents the amplitude and phase of a sinusoid.  Phasor is the mathematical equivalent of a sinusoid with time variable dropped.  Phasor representation is based on Euler’s identity.  Given a sinusoid v(t)=V m cos(ωt+φ).

8. 8 EENG224 Phasors  Given the sinusoids i(t)=I m cos(ωt+φ I ) and v(t)=V m cos(ωt+ φ V ) we can obtain the phasor forms as:

9. 9 EENG224 Phasors  Amplitude and phase difference are two principal concerns in the study of voltage and current sinusoids.  Phasor will be defined from the cosine function in all our proceeding study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase. Example Transform the following sinusoids to phasors: –i = 6cos(50t – 40 o ) A –v = –4sin(30t + 50 o ) V Solution: a. I A b. Since –sin(A) = cos(A+90 o ); v(t) = 4cos (30t+50 o +90 o ) = 4cos(30t+140 o ) V Transform to phasor => V V

10. 10 EENG224 Phasors Solution : a) v(t) = 10cos(t + 210 o ) V b) Since i(t) = 13cos(t + 22.62 o ) A Example 5: Transform the sinusoids corresponding to phasors: a) b)

11. 11 EENG224 Phasor as Rotating Vectors

12. 12 EENG224 Phasor Diagrams  The SINOR Rotates on a circle of radius V m at an angular velocity of ω in the counterclockwise direction

13. 13 EENG224 Phasor Diagrams

14. 14 EENG224 Time Domain Versus Phasor Domain

15. 15 EENG224 Differentiation and Integration in Phasor Domain  Differentiating a sinusoid is equivalent to multiplying its corresponding phasor by jω.  Integrating a sinusoid is equivalent to dividing its corresponding phasor by jω.

16. 16 EENG224 Adding Phasors Graphically  Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors. V=V 1 +V 2

17. 17 EENG224

18. 18 EENG224

19. 19 EENG224  We can derive the differential equations for the following circuit in order to solve for v o (t) in phase domain Vo.  However, the derivation may sometimes be very tedious. Is there any quicker and more systematic methods to do it?  Instead of first deriving the differential equation and then transforming it into phasor to solve for V o, we can transform all the RLC components into phasor first, then apply the KCL laws and other theorems to set up a phasor equation involving V o directly. Solving AC Circuits