1. A sinusoidal current source (independent or dependent) produces a current
That varies sinusoidally with time

3. We define the Root Mean Square value of v(t) or rms as

4. The Root Mean Square value of
Expand using trigonometric identity

8. 9.2 The Sinusoidal Response
Solution for i(t) should be a sinusoidal of frequency 3
We notice that only the amplitude
and phase change
In this chapter, we develop a technique for calculating the response directly without solving the differential equation

9. Time Domain
Complex Domain
Deferential Equation
Algebraic Equation

10. 9.3 The phasor
The phasor is a complex number that carries the amplitude and phase angle information
of a sinusoidal function
The phasor concept is rooted in Euler’s identity
Euler’s identity relates the complex exponential function to the trigonometric function
We can think of the cosine function as the real part of the complex exponential and the
sine function as the imaginary part
Because we are going to use the cosine function on analyzing the sinusoidal steady-state
we can apply

11. We can move the coefficient Vm inside
is a complex number define to be the phasor
that carries the amplitude and phase angle
of a given sinusoidal function
The quantity
Phasor Transform
Were the notation
Is read “ the phasor transform of

12. Summation Property of Phasor
(can be shown)

14. Since
Next we derive y using phsor method

16. The V-I Relationship for a Resistor
Let the current through the resistor be a sinusoidal given as
Is also sinusoidal with
amplitude
And phase
The sinusoidal voltage and current in a resistor are in phase

17. Which is Ohm’s law on the phasor ( or complex ) domain
Now let us see the pharos domain representation or pharos transform of the current and voltage
Phasor Transform
Phasor Transform
Which is Ohm’s law on the phasor ( or complex ) domain

18. The voltage and the current are in phase
Imaginary
Real

19. The V-I Relationship for an Inductor
Let the current through the resistor be a sinusoidal given as
The sinusoidal voltage and current in an inductor are out of phase by 90o
The voltage lead the current by 90o or the current lagging the voltage by 90o
You can express the voltage leading the current by T/4 or 1/4f seconds were T is the period
and f is the frequency

20. Now we rewrite the sin function as a cosine function
( remember the phasor is defined in terms of a cosine function)
The pharos representation or transform of the current and voltage
But since
Therefore
and

21. and
The voltage lead the current by 90o or the current lagging the voltage by 90o
Imaginary
Real

22. The V-I Relationship for a Capacitor
Let the voltage across the capacitor be a sinusoidal given as
The sinusoidal voltage and current in an inductor are out of phase by 90o
The voltage lag the current by 90o or the current leading the voltage by 90o

23. The V-I Relationship for a Capacitor
The pharos representation or transform of the voltage and current
and

24. and
The voltage lag the current by 90o or the current lead the voltage by 90o
Imaginary
Real

25. Phasor ( Complex or Frequency) Domain
Time-Domain

26. Impedance and Reactance
The relation between the voltage and current on the phasor domain (complex or frequency) for the three elements R, L, and C we have
When we compare the relation between the voltage and current , we note that they are all of
form:
Which the state that the phasor voltage is some complex constant ( Z )
times the phasor current
This resemble ( شبه ) Ohm’s law were the complex constant ( Z ) is called “Impedance” (أعاقه )
Recall on Ohm’s law previously defined , the proportionality content R was real and
called “Resistant” (مقاومه )
Solving for ( Z ) we have
The Impedance of a resistor is
In all cases the impedance is measured
in Ohm’s W
The Impedance of an indictor is
The Impedance of a capacitor is

27. Impedance
The Impedance of a resistor is
In all cases the impedance is measured
in Ohm’s W
The Impedance of an indictor is
The Impedance of a capacitor is
The imaginary part of the impedance is called “reactance”
The reactance of a resistor is
We note the “reactance” is associated
with energy storage elements like the
inductor and capacitor
The reactance of an inductor is
The reactance of a capacitor is
Note that the impedance in general (exception is the resistor) is a function of frequency
At w = 0 (DC), we have the following
short
open

28. Time Domain
Phasor (Complex) Domain

31. Consider the following circuit
9.5 Kirchhoff’s Laws in the Frequency Domain ( Phasor or Complex Domain)
Consider the following circuit
Phasor Transformation
KVL
Using Euler Identity we have
Which can be written as
Factoring
Can not
be zero
Phasor
KVL on the phasor
domain
So in general

32. Kirchhoff’s Current Law
A similar derivation applies to a set of sinusoidal current summing at a node
Phasor Transformation
KCL
KCL on the phasor
domain

33. 9.6 Series, Parallel, Simplifications
and Ohm’s law in the phosor domain

34. Example 9.6 for the circuit shown below the source voltage is sinusoidal
(a) Construct the frequency-domain (phasor, complex) equivalent circuit ?
(b) Calculte the steady state current i(t) ?
The source voltage pahsor transformation or equivalent
The Impedance of the indictor is
The Impedance of the capacitor is

35. To Calculate the phasor current I

36. and Ohm’s law in the phosor domain

37. Example 9.7 Combining Impedances in series and in Parallel
(a) Construct the frequency-domain (phasor, complex) equivalent circuit ?
(b) Find the steady state expressions for v,i1, i2, and i3 ? ?
(a)

39. Delta-to Wye Transformations
D to Y
Y to D

40. Example 9.8

42. 9.7 Source Transformations and Thevenin-Norton Equivalent Circuits

43. Example 9.9

46. Next we find the Thevenin Impedance
Example 9.10
Source Transformation
KVL
Since
then
Next we find the Thevenin Impedance

47. Thevenin Impedance

48. 9.8 The Node-Voltage Method
Example 9.11
KCL at node 1
(1)
KCL at node 2
Since
(2)
Two Equations and Two Unknown , solving

49. To Check the work

50. 9.9 The Mesh-Current Method
Example 9.12
KVL at mesh 1
(1)
KVL at mesh 2
Since
(2)
Two Equations and Two Unknown , solving

52. 9.12 The Phasor Diagram