ELECTRICAL CIRCUIT ET 201

1. ELECTRICAL CIRCUIT ET 201 Define and explain phasors, time and phasor domain, phasor diagram. Analyze circuit by using phasors and complex numbers arithmetics.

2. BASIC ELEMENTS AND PHASORS (CHAPTER 1.5)

3. 14.12 Phasors • The addition of sinusoidal voltages and currents will frequently be required in the analysis of ac circuits. • One lengthy but valid method of performing this operation is to place both sinusoidal waveforms on the same set of axes and add algebraically the magnitudes of each at every point along the abscissa. • Long and tedious process with limited accuracy.

4. 14.12 Phasors • A shorter method uses the rotating radius vector. • The radius vector, having a constant magnitude (length) with one end fixed at the origin, is called a phasor when applied to electric circuits. • Phasors algebra is for sinusoidal quantities and is applicable only for waveforms having the same frequency f or angular velocity ω.

5. 14.12 Phasors • Phasor is a complex number that represent magnitude and angle for a sine wave. • Phasor diagram is a vector line that represent magnitude and phase angle of a sine wave. • The magnitude of the phasor is equal to rms value. • For example, if given a sine wave waveform • It can be represent by a phasor diagram

6. 14.12 Phasors PHASOR DOMAIN TIME DOMAIN Conversion of time domain to phasor domain • Amplitude and phase difference are two principal concerns in the study of voltage and current sinusoidals. • Phasor will be defined from the sine function in all our proceeding study. If a voltage or current expression is in the form of a cosine, it will be changed to a sine by adding 90o.

7. 14.12 Phasors Example 14.27(a) Convert the time domain to the phasor domain. Solution TIME DOMAIN PHASOR DOMAIN

8. 14.12 Phasors Example 14.27(b) Convert the time domain to the phasor domain Solution TIME DOMAIN PHASOR DOMAIN

9. 14.12 Phasors Example 14.27(c) Convert the time domain to the phasor domain Solution TIME DOMAIN PHASOR DOMAIN

10. 14.12 Phasors Example 14.28(a) Convert the phasor domain to the time domain if the frequency is 60 Hz. Solution PHASOR DOMAIN TIME DOMAIN

11. 14.12 Phasors Example 14.28(b) Convert the phasor domain to the time domain if the frequency is 60 Hz Solution PHASOR DOMAIN TIME DOMAIN

12. 14.12 Phasors Phasor diagram TIME DOMAIN PHASOR DOMAIN

13. 14.12 Phasors Phasor diagram TIME DOMAIN PHASOR DOMAIN

14. 14.12 Phasors Phasor diagram TIME DOMAIN PHASOR DOMAIN

15. 14.12 Phasors Phasor diagram • V2 leading V for or V lagging V2 for • V leading V1 for or V1 lagging V for • V2 leading V1 for or V1 lagging V2 for

16. BASIC APPROACH Steps to Analyze AC Circuits: • Transform the circuit to the phasor domain. • Analyze the circuit by using circuit techniques and perform the calculations with complex number arithmetics. • Transform the resulting phasor to the time domain. Phasor to Time Time to Phasor Solve in Phasor

17. 14.12 Phasors Example 14.29 Find ein

18. 14.12 Phasors Example 14.29 – solution (KVL) Transforming va and vb into the phasor domain;

19. 14.12 Phasors Example 14.29 – solution (cont’d) Converting from polar to rectangular form;

20. 14.12 Phasors Example 14.29 – solution (cont’d) Adding; Converting from rectangular to polar form;

21. 14.12 Phasors Example 14.29 – solution (cont’d) Inverse-transforming to time domain; PHASOR DOMAIN TIME DOMAIN

22. 14.12 Phasors Example 14.29 – solution (cont’d) Phasor diagram;

23. 14.12 Phasors Example 14.29 – solution (cont’d) Time domain representation;

24. 14.12 Phasors Example 14.30 Determine i2 in the following network;

25. 14.12 Phasors Example 14.30 – solution (KCL) In phasor form; Or;

26. 14.12 Phasors Example 14.30 – solution (cont’d) Transforming iT and i1 into the phasor domain; and

27. 14.12 Phasors Example 14.30 – solution (cont’d) Converting from polar to rectangular form;

28. 14.12 Phasors Example 14.30 – solution (cont’d) Adding; Converting from rectangular to polar form;

29. 14.12 Phasors Example 14.30 – solution (cont’d) Inverse-transforming to time domain; PHASOR DOMAIN TIME DOMAIN

30. 14.12 Phasors Example 14.30 – solution (cont’d) Time domain representation;

31. 14.12 Phasors Example 14.30 – solution (cont’d) Phasor representation;