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Hafiz Zaheer Hussain

2. 04/03/2015 Hafiz Zaheer Hussain
Circuit Analysis-II Spring-2015 EE Instructor: Hafiz Zaheer Hussain Department of Electrical Engineering The University of Lahore Week 1 & 2
04/03/2015
Hafiz Zaheer Hussain

3. Fundamentals of Electric Circuits by Alexander-Sadiku
Chapter 9
Sinusoidal Steady-State Analysis
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Hafiz Zaheer Hussain

4. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors
9.4 Phasor relationships for circuit elements
9.5 Impedance and admittance
9.6 Kirchhoff’s laws in the frequency domain
9.7 Impedance combinations
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Hafiz Zaheer Hussain

5. 9.1 Introduction
We now begin the analysis of circuits in which the source voltage or current is time-varying. In this chapter, we are particularly interested in sinusoidally time-varying excitation, or simply, excitation by a sinusoid
A sinusoid is a signal that has the form of the sine or cosine function.
A sinusoidal current is usually referred to as alternating current (AC).
Such a current reverses at regular time intervals and has alternately positive and negative values.
Circuits driven by sinusoidal current or voltage sources are called ac circuits
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Hafiz Zaheer Hussain

6. 9.2 Sinusoids’ features
We are interested in sinusoids for a number of reasons.
nature itself is characteristically sinusoidal.
The motion of a pendulum
The vibration of a string
The ripples on the ocean surface etc.
Sinusoidal signal is easy to generate and transmit.
Sinusoidal play an important role in the analysis of periodic signals.
Sinusoid is easy to handle mathematically.
The derivative and integral of a sinusoid are themselves sinusoids.
For these the sinusoid is an extremely important function in circuit analysis.
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Hafiz Zaheer Hussain

7. 9.2 Sinusoids’ (2) where Vm = the amplitude of the sinusoid
A sinusoid is a signal that has the form of the sine or cosine function.
A general expression for the sinusoid,
where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
Ф = the phase
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Hafiz Zaheer Hussain

8. 9.2 Sinusoids’ (3)
A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n. If
Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference.
If phase difference is zero( ) , they are in phase;
If phase difference is not zero( ) , they are out of phase.
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Hafiz Zaheer Hussain

9. 9.2 Sinusoids’ (4)
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10. 9.2 Sinusoids’ (5) Q
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11. 9.2 Sinusoids’ (6)
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12. 9.2 Sinusoids’ Example 9.1
Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency.
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Hafiz Zaheer Hussain

13. 9.2 Sinusoids’ PP 9.2
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14. 9.2 Sinusoids’ (7) Trigonometric Identities (9.9) (9.9) 04/03/2015
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15. 9.2 Sinusoids’ (8)
A graphical approach may be used to relate or compare sinusoids as an alternative to using the trigonometric identities in Eqs. (9.9) and (9.10). Consider the set of axes shown in
Fig. 9.3(a).
The horizontal axis represents the magnitude of cosine, while the vertical axis (pointing down) denotes the magnitude of sine.
Angles are measured positively counterclockwise from the horizontal,
as usual in polar coordinates. This graphical technique can be used to relate two sinusoids.
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Hafiz Zaheer Hussain

16. 9.2 Sinusoids’ (9)
We obtain
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Hafiz Zaheer Hussain

17. 9.2 Sinusoids’ Example
Given v(t) = Vm sin (ωt +10o). Transform to Cosine
Solution
v(t) = Vm sin (ωt +10o)
v(t) = Vm cos (ωt + 10o - 90o)
v(t) = Vm cos (ωt – 80o)
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Hafiz Zaheer Hussain

18. 9.2 Sinusoids’ Example 9.2 Solution:
Let us calculate the phase in three ways. The first two methods use trigonometric identities, while the third method uses the graphical approach.
■ METHOD 1 In order to compare v1 and v2 we must express them in the same form. If we express them in cosine form with positive amplitudes,
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Hafiz Zaheer Hussain

19. 9.2 Sinusoids’ Example 9.2(count..)
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20. 9.2 Sinusoids’ PP 9.2
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Hafiz Zaheer Hussain

21. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors
9.4 Phasor relationships for circuit elements
9.5 Impedance and admittance
9.6 Kirchhoff’s laws in the frequency domain
9.7 Impedance combinations
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Hafiz Zaheer Hussain

22. 9.3 Phasors
A phasor is a complex number that represents the amplitude and phase of a sinusoid.
It can be represented in one of the following three forms:
Rectangular
Polar
Exponential
where
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Hafiz Zaheer Hussain

23. 9.3 Phasors , Example 9.3
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24. 9.3 Phasors , Example 9.3
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25. 9.3 Phasors , Example 9.3 (count..)
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26. 9.3 Phasors , PP 9.3 Evaluate the following complex numbers: a. b.
Solution:
a. – j13.67
b j2.2
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Hafiz Zaheer Hussain

27. 9.3 Phasors (2) Mathematic operation of complex number: Addition
Subtraction
Multiplication
Division
Reciprocal
Square root
Complex conjugate
Euler’s identity
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Hafiz Zaheer Hussain

28. 9.3 Phasors (3) (time domain) (phasor domain)
Transform a sinusoid to and from the time domain to the phasor domain:
(time domain) (phasor domain)
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.
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Hafiz Zaheer Hussain

29. Table 9.1 Sinusoid-phasor transformation
9.3 Phasors (3)
Table 9.1 Sinusoid-phasor transformation
Time domain representation
Phasor domain representation
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Hafiz Zaheer Hussain

30. Lead or Lag seen Via Phasors
Such a graphical representation of phasors is known as a phasor diagram.
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Hafiz Zaheer Hussain

31. 9.3 Phasors , Example 9.4
Transform the following sinusoids to phasors:
i = 6cos(50t – 40o) A
v = –4sin(30t + 50o) V
Solution:
I A b.
v(t) = -4 sin (30t +50o)V
v(t) = 4 cos (30t o)
v(t) = 4 cos (ωt o)
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Hafiz Zaheer Hussain

32. 9.3 Phasors , Example 9.5
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33. 9.3 Phasors , Example 9.5
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34. 9.3 Phasors , PP 9.5 Transform the sinusoids corresponding to phasors:
Solution:
v(t) = 10cos(wt + 210o) V
Since
i(t) = 13cos(wt o) A
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35. 9.3 Phasors , Example 9.6
Answer
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36. 9.3 Phasors , PP 9.6
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Hafiz Zaheer Hussain

37. 9.3 Phasors (4) The differences between v(t) and V:
v(t) is instantaneous or time-domain representation V is the frequency or phasor-domain representation.
v(t) is time dependent, V is not.
v(t) is always real with no complex term, V is generally complex.
Note: Phasor analysis applies only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency.
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Hafiz Zaheer Hussain

38. 9.3 Phasors (5)
Relationship between differential, integral operation in phasor listed as follow:
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Hafiz Zaheer Hussain

39. 9.3 Phasors , Example 9.7
Use phasor approach, determine the current i(t) in a circuit described by the integrodifferential equation.
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Hafiz Zaheer Hussain

40. 9.3 Phasors , PP 9.7
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Hafiz Zaheer Hussain

41. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors
9.4 Phasor relationships for circuit elements
9.5 Impedance and admittance
9.6 Kirchhoff’s laws in the frequency domain
9.7 Impedance combinations
04/03/2015
Hafiz Zaheer Hussain

42. 9.4 Phasor relationships for circuit elements
Transform the voltage-current relationship from the time domain to the
frequency domain for each element.
we will assume the passive sign convention.
Voltage and Current are in phase in resistance
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Hafiz Zaheer Hussain

43. 9.4 Phasor relationships for circuit elements
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44. 9.4 Phasor relationships for circuit elements
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45. 9.4 Phasor relationships for circuit elements
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46. The V-I Relationships for Capacitor
9.4 Phasor relationships for circuit elements
The V-I Relationships for Capacitor
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47. 9.4 Phasor relationships for circuit elements
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48. 9.4 Phasor relationships for circuit elements
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49. Summary of voltage-current relationship
9.4 Phasor relationships for circuit elements
Summary of voltage-current relationship
Resistor:
Inductor:
Capacitor:
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Hafiz Zaheer Hussain

50. Summary of voltage-current relationship
9.4 Phasor relationships for circuit elements
Summary of voltage-current relationship
Element
Time domain
Frequency domain R L C
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Hafiz Zaheer Hussain

51. 9.4 Phasor relationships for circuit elements Example 9.8
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Hafiz Zaheer Hussain

52. 9.4 Phasor relationships for circuit elements PP 9.8
If voltage v(t) = 6cos(100t + 30o) is applied to a 50 μF capacitor, calculate the current, i(t), through the capacitor.
Answer: i(t) = 30 cos(100t + 120o) mA
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