1. Chapter 10 Sinusoidal Steady-State Analysis

2. Charles P. Steinmetz (1865-1923), the developer of the
mathematical analytical tools for studying ac circuits.
Courtesy of General Electric Co.

3. Heinrich R. Hertz ( ).
Courtesy of the Institution of Electrical Engineers.
cycles/second
Hertz, Hz

4. Sinusoidal voltage source vs  Vm sin(t  ).
Sinusoidal Sources
Amplitude
Period = 1/f
Phase angle
Angular or radian
frequency = 2pf = 2p/T
Sinusoidal voltage source
vs  Vm sin(t  ).
Sinusoidal current source
is  Im sin(t  ).

5. Voltage and current of a circuit element.
Example v i
circuit
element + v _ i
Voltage and current of a circuit element.
The current leads the voltage by  radians OR
The voltage lags the current by  radians

6. Example
Find their phase relationship
and
Therefore the current leads the voltage by

7. Recall
Triangle for A and B of Eq ,
where C 

8. Example A B B A

9. An RL circuit. Steady-State Response of an RL circuit
From #8&#9
Substitute the assumed solution into
Coeff. of cos
Coeff. of sin
Solve for A & B

10. Steady-State Response of an RL circuit (cont.)
Thus the forced (steady-state) response is of the form

11. Complex Exponential Forcing Function
Input
Response
magnitude
phase
frequency
Exponential Signal
Note

12. Complex Exponential Forcing Function (cont.)
try
We get
where

13. Complex Exponential Forcing Function (cont.)
Substituting for A
We expect

14. Example
We replace
Substituting ie

15. Example(cont.)
The desired answer for the steady-state current
interchangeable Or

16. Using Complex Exponential Excitation to Determine a
Circuit’s SS Response to a Sinusoidal Source
Write the excitation as a cosine waveform with
a phase angle
Introduce complex excitation
Use the assumed response
Determine the constant A

17. Obtain the solution
The desired response is
Example

18. Example (cont.)

19. Example (cont.)
The solution is
The actual response is

20. The Phasor Concept
A sinusoidal current or voltage at a given frequency
is characterized by its amplitude and phase angle.
Magnitude
Phase angle
Thus we may write
unchanged

21. The Phasor Concept(cont.)
A phasor is a complex number that represents the
magnitude and phase of a sinusoid.
phasor
The Phasor Concept may be used when the circuit is
linear , in steady state, and all independent sources are
sinusoidal and have the same frequency.
A real sinusoidal current
phasor notation

22. The Transformation
Time domain
Transformation
Frequency domain

23. The Transformation (cont.)
Time domain
Transformation
Frequency domain

24. Example
Substitute into
Suppress

25. Example (cont.)

26. Phasor Relationship for R, L, and C Elements
Time domain
Resistor
Frequency domain
Voltage and current are in phase

27. Inductor
Time domain
Frequency domain
Voltage leads current by

28. Capacitor
Time domain
Frequency domain
Voltage lags current by

29. Impedance and Admittance
Impedance is defined as the ratio of the phasor voltage
to the phasor current.
Ohm’s law in phasor notation
phase
magnitude or
polar
exponential
rectangular

30. Graphical representation of impedance
Resistor wL
Inductor
Capacitor
1/wC

31. Admittance is defined as the reciprocal of impedance.
conductance
In rectangular form
susceptance G
Resistor
1/wL
Inductor wC
Capacitor

32. Kirchhoff’s Law using Phasors
KVL
KCL
Both Kirchhoff’s Laws hold in the frequency domain.
and so all the techniques developed for resistive circuits hold
Superposition
Thevenin &Norton Equivalent Circuits
Source Transformation
Node & Mesh Analysis
etc.

33. Impedances in series
Admittances in parallel

34. Example 10.9-1 R = 9 W, L = 10 mH, C = 1 mF i = ?
KVL

35. Example v = ?
KCL

36. Node Voltage & Mesh Current using Phasors
va = ? vb = ?

37. KCL at node a
KCL at node b
Rearranging
Admittance matrix

38. If Im = 10 A and
Using Cramer’s rule to solve for Va
Therefore the steady state voltage va is

39. Example v = ?
use supernode concept
as in #4

40. Example (cont.)
KCL at supernode
Rearranging

41. Example (cont.)
Therefore the steady state voltage v is

42. Example i1 = ?

43. Example (cont.)
KVL at mesh 1 & 2
Using Cramer’s rule to solve for I1

44. Superposition, Thevenin & Norton Equivalents
and Source Transformations
Example i = ?
Consider the response to the voltage source acting alone = i1

45. Example (cont.)
Substitute

46. Example (cont.)
Consider the response to the current source acting alone = i2
Using the principle of superposition

47. Source Transformations

48. Example IS = ?

49. Example 10.11-3 Thevenin’s equivalent circuit?

50. Example 10.11-4 Thevenin’s equivalent circuit

51. Example 10.11-4 Norton’s equivalent circuit?

52. Phasor Diagrams
A Phasor Diagram is a graphical representation of phasors
and their relationship on the complex plane.
Take I as a reference phasor
The voltage phasors are

53. Phasor Diagrams (cont.)
KVL
For a given L and C there will be a frequency w that
Resonant frequency
Resonance

54. Summary Sinusoidal Sources
Steady-State Response of an RL Circuit for Sinusoidal Forcing Function
Complex Exponential Forcing Function
The Phasor Concept
Impedance and Admittance
Electrical Circuit Laws using Phasors