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 (CHAPTER 1.5) BASIC ELEMENTS AND PHASORS

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.

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.

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 Phasors

6 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 90 o. TIME DOMAIN PHASOR DOMAIN

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

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

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

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

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

Phasors TIME DOMAIN PHASOR DOMAIN Phasor diagram

Phasors TIME DOMAIN PHASOR DOMAIN Phasor diagram

Phasors TIME DOMAIN PHASOR DOMAIN Phasor diagram

i)V2 leading V for or V lagging V2 for ii)V leading V1 for or V1 lagging V for iii)V2 leading V1 for or V1 lagging V2 for Phasors Phasor diagram

16 BASIC APPROACH Time to Phasor Solve in Phasor Phasor to Time Steps to Analyze AC Circuits: 1.Transform the circuit to the phasor domain. 2.Analyze the circuit by using circuit techniques and perform the calculations with complex number arithmetics. 3.Transform the resulting phasor to the time domain.

Phasors Example Find e in

Phasors Example – solution Transforming v a and v b into the phasor domain; (KVL)

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

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

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

Phasors Phasor diagram; Example – solution (cont’d)

Phasors Example – solution (cont’d) Time domain representation;

Phasors Example Determine i 2 in the following network;

Phasors Example – solution In phasor form; Or; (KCL)

Transforming i T and i 1 into the phasor domain; and Phasors Example – solution (cont’d)

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

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

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

Phasors Time domain representation; Example – solution (cont’d)

Phasors Phasor representation; Example – solution (cont’d)