1. ELECTRIC CIRCUIT ANALYSIS - I
Chapter 14 – Basic Elements and Phasors
Lecture 18
by Moeen Ghiyas
19/04/2017

2. Chapter 14 – Basic Elements and Phasors
TODAY’S lesson

3. Today’s Lesson Contents
Math Operations with Complex Numbers
Phasors

4. Math Operations with Complex Numbers
Let us first revise the symbol j associated with imaginary numbers. By definition,

5. Math Operations with Complex Numbers
Multiplication (Rectangular)
Simple algebra – To multiply two complex numbers in rectangular form, multiply the real and imaginary parts of one in turn by the real and imaginary parts of other.

6. Math Operations with Complex Numbers
Multiplication Example (Rectangular)
or j

7. Math Operations with Complex Numbers
Division (Rectangular)
To divide two complex numbers in rectangular form, multiply the numerator and denominator by conjugate of the denominator and the resulting real and imaginary parts collected
In algebraic terms: Remove imaginary terms from denominator

8. Math Operations with Complex Numbers
Example Division (Rectangular)
We know

9. Math Operations with Complex Numbers
Multiplication (Polar)
In polar form, the magnitudes are multiplied and the angles added algebraically. For example, for

10. Math Operations with Complex Numbers
Multiplication Example (Polar)

11. Math Operations with Complex Numbers
Division (Polar)
In polar form, division is accomplished by dividing the magnitude of the numerator by the magnitude of denominator and subtracting the angle of the denominator from that of the numerator.

12. Math Operations with Complex Numbers
Example Division (Polar)

13. Math Operations with Complex Numbers
Reciprocal
The reciprocal is 1 divided by the complex number. For example, the reciprocal of is
and of is
We obtain the value of reciprocal in the rectangular form by multiplying numerator and denominator by complex conjugate of the denominator:
and reciprocal (Polar) is obtained as

14. Math Operations with Complex Numbers
Example – Perform the operation
Solution:

15. Math Operations with Complex Numbers
Example – Perform the operation
Solution:

16. Phasors
Addition of sinusoidal v and i is frequently required in analysis of ac ccts. One lengthy method is to place both sinusoidal waveforms on same set of axes and add algebraically their magnitudes at every point along abscissa.
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17. Phasors
A shorter method uses the rotating radius vector already discussed in Ch 13 for derivation of sinusoidal waveform.
This radius vector, having a constant magnitude (length) with one end fixed at the origin, is called a phasor in electric circuit analysis.
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18. The Sine Wave – Ch 13

19. The Sine Wave – Ch 13

20. Phasors
The phasor (radius vector) of the sine wave, at instant t = 0, will have the positions shown in Fig.(a) for each waveform in Fig.(b).
Fig (a)
Fig (b)

21. Phasors
Note the magnitudes and position of radius vectors (phasors) for waveform of v1 and v2 at t = 0, and their vector sum vT.
Fig (a)
Fig (b)

22. Phasors Using vector algebra, we have
In other words, if we convert v1 and v2 to the phasor form (polar form) and add them using complex number algebra, we can find the phasor form for vT .
The resultant vT can then be converted to the time domain and plotted on the same set of axes, as shown in Fig (b).
Fig (a)
Fig (b)

23. Phasors
Figure (a), showing the magnitudes and relative positions of the various phasors, is called a phasor diagram. (It is actually a “snapshot” of the rotating radius vectors at t = 0 s.)
Fig (a)
Fig (b)

24. Phasors
In future, therefore, if the addition of two sinusoids is required, they should first be converted to the phasor domain (polar form) and the sum found using complex algebra. The result can then be converted back to the time domain.
Fig (a)
Fig (b)

25. Phasors
Adding two sinusoidal currents with phase angles other than 90°.
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26. Phasors
Since the rms (effective), rather than the peak, values are used almost exclusively in the analysis of ac circuits.
The phasor will now be redefined as having a magnitude equal to the rms value of the sine wave it represents.
In general, for all of the analysis to follow, the phasor form (polar form) of a sinusoidal voltage (rms) or current (rms) will be
where V and I are rms (effective) values and θ is the phase angle

27. Phasors
It should be pointed out that in phasor notation, the sine wave is always the reference, and the frequency is not represented.
Also phasor algebra for sinusoidal quantities is applicable only for waveforms having the same frequency.

28. Phasors
Example – Convert the following from the time to the phasor domain:
Solution:

29. Phasors
Example – Write the sinusoidal expression for the following phasors if the frequency is 60 Hz:
Solution:
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30. Phasors Example - Determine the current i2 for the network of Fig
Solution:
Applying KCL,
Converting time to phasor (polar x 0.707) domain or

31. Phasors Converting polar to rectangular (for subtraction purpose)
Thus KCL equation yields
Converting from rectangular to polar (phasor) then to Vm (polar) form
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32. Phasors Converting phasor to time domain Plotting all three currents
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33. Summary / Conclusion
Math Operations with Complex Numbers
Phasors

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