2. 2

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

7. The V-I Relationship for an Inductor
Let the current through the resistor
be a sinusoidal given as
Now we rewrite the sin function
as a cosine function
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

8. The V-I Relationship for a Capacitor
Let the voltage across the capacitor
be a sinusoidal given as
Now we rewrite the sin function
as a cosine function
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

9. The Sinusoidal Response
KVL
This is first order differential equations which has the following solution
We notice that the solution is also sinusoidal of the same frequency w
However they differ in
amplitude
and
phase

10. Complex Numbers
Rectangular Representation

11. Complex Numbers (Polar form)
Rectangular Representation

12. Euler’s Identity
Euler’s identity relates the complex exponential function to the trigonometric function
Adding
Subtracting

13. Euler’s Identity The left side is complex function
The right side is complex function
The left side is real function
The right side is real function

14. Complex Numbers (Polar form)
Rectangular Representation
Short notation

15. Real Numbers
Rectangular Representation
Polar Representation OR

16. Rectangular Representation
Polar Representation OR

17. Imaginary Numbers
Rectangular Representation
Polar Representation OR

18. Rectangular Representation
Polar Representation OR

19. Complex Conjugate
Complex Conjugate is defined as

20. Complex Numbers (Addition)

21. Complex Numbers (Subtraction)

22. Complex Numbers (Multiplication)
Multiplication in Rectangular Form
Multiplication in Polar Form
Multiplication in Polar Form is easier than in Rectangular form

23. Complex Numbers (Division)
Division in Rectangular Form

24. Complex Numbers (Division)
Division in Polar Form
Division in Polar Form is easier than in Rectangular form

25. Complex Conjugate Identities ( can be proven)OR
Other Complex Conjugate Identities ( can be proven)

26. Let the current through the resistor
be a sinusoidal given as
Let the current through the resistor
be a sinusoidal given as
From Linearity if
then

27. The solution which was found earlier

28. The V-I Relationship for an Inductor
Let the current through the resistor
be a sinusoidal given as
Now we rewrite the sin function
as a cosine function
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

29. From Linearity if
The solution which was found earlier

30. The solution is the real part of
This will bring us to the PHASOR method in solving sinusoidal excitation of linear circuit

32. The real part is the solution
Now if you pass a complex current
Phasor
You get a complex voltage
Phasor

33. 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
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

34. Moving the coefficient Vm inside
Phasor Transform
Were the notation
Is read “ the phasor transform of

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

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

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

38. 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

39. 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

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

41. 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

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

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

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

45. 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

46. 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

47. 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

48. 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

49. 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

50. To Calculate the phasor current I

51. 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)

53. Ex 6.4:Determine the voltage v(t) in the circuit
Replace: source with
desired voltage v(t) with
Impedance of capacitor is 53

54. A single-node pair circuit
Hence time-domain voltage becomes 54

55. Ex 6.5 Determine the current i(t) and voltage v(t)
Single loop phasor circuit
The current
By voltage division
The time-domain 55

56. Ex 6.6 Determine the current i(t)
The phasor circuit is
Combine resistor and inductor 56

57. Use current division to obtain capacitor current
Hence time-domain current is: 57

58. 9.7 Source Transformations and Thevenin-Norton Equivalent Circuits

59. Example 9.9

60. Ex 6.7 Determine i(t) using source transformation
Phasor circuit
Transformed source
Voltage of source:
Hence the current
In time-domain 60

61. Ex 6.9 Find voltage v(t) by reducing the phasor circuit at terminals a and b to a Thevenin equivalent
Phasor circuit 61

62. 62

63. The Thevenin impedance can be modeled as 1.19 resistor in series
with a capacitor with value or 63