1. Applied Electromagnetics EEE 161
Intro to Transmission Lines

2. LECTURE 1 – INTRO TO CLASS, Getting to know each other

3. Complex Numbers/Phasors at home reading
View the video about complex numbers:
View the phasor simulation:
Read chapter 1, sections on Complex numbers and phasors.
Post at least 1 “muddiest point” (most unclear) in this reading on the SacCT discussion Complex Numbers and Phasors. These are graded as extra credit.
Due Week 2, day before first class.

4. Objective
In order to understand transmission lines we have to be able to describe sinusoidal signals first!

5. Motivation Here is an equation of a wave
Here is an equation of a sinusoidal signal
How are they the same and how are they different? Use vocabulary to name green and yellow highlights.

6. Sinusoidal Signal
Describe this graph using at least 5 words!

8. Time Delay
Signal is given: Can you sketch the signal that is lagging or leading?

9. Time Delay
Which one is leading and which one is lagging?

10. Time Delay
Which one is leading and which one is lagging?

11. Time Delay – mathematical description
We replace t in the above equation with t-T and we get:

12. Sinusoidal signals – time delay

13. Phase Lead & Lag

14. Time Domain vs. Frequency Domain
Write an equation for a sin signal in time-domain
Write an equation for a sin signal in frequency-domain

15. Time Domain vs. Frequency Domain
Write an equation for a sin signal in time-domain
Write an equation for a sin signal in frequency-domain

16. Time Domain vs. Frequency Domain
Draw a sinusoidal signal in time-domain
Draw a sinusoidal signal in frequency-domain

17. Time Domain vs. Frequency Domain
Draw a sinusoidal signal in time-domain
Draw a sinusoidal signal in frequency-domain

18. Time vs. Frequency Domain
How is frequency domain better than time domain?

20. Motivation
Where have we seen complex numbers and how we used them?

21. Motivation Where have we seen complex numbers and how we used them?
Circuits with capacitors and inductors for addition of impedances/admittances
Bode Plots
Free body diagrams
Wave propagation
Complex power
Application problems in diff eq. pendulum swing
Euler’s Equation

22. Application to Electrical Engineering
Let’s look at two simple circuits

25. Complex Numbers

26. Objective
To solve simple circuits we have to be able to apply basic arithmetic operations with complex numbers.

27. Complex numbers Cartesian Coordinate System Polar Coordinate System
Euler’s Identity

28. Quiz Complex number Z=3+j2 is given.
Sketch the number in Cartesian coordinate system
Find the magnitude and phase
Explain what is magnitude and what is phase on the diagram

30. Some concepts from complex #s
Conjugate
Division
Addition

31. Which equation do we use to go from Polar to Rectangular coordinates?

32. Euler’s equation

33. Quiz

34. Quiz

35. Some more concepts
Power and Square Root.

36. Quiz
Calculate and sketch the magnitude and phase of the following three complex numbers:

37. Quiz
Find the magnitude and phase of the following complex numbers:

38. Division of complex numbers
Where have we seen division of complex numbers?

39. Division of complex numbers example in EE

41. Socrative Quiz
Find the voltage on the mystery circuit if the current through it was measured to be
The complex equation to find the voltage is

44. Homework 1 will be due soon!

45. Objective
Students will be able to solve simple circuits using phasors

46. ac Phasor Analysis: General Procedure

47. ac Phasor Analysis: General Procedure

48. Example 1-4 (p.37): RL Circuit
Cont.

49. Example 1-4: RL Circuit cont.

50. Graded Quiz TL#1
A series R-L circuit is given as shown in Figure below.
Derive equations for magnitude and phase of current and voltages on resistor and inductor in the phasor domain. Assume that the resistance of the resistor is R, inductance of the inductor is L, magnitude of the sourse voltage is Vm and phase of the source voltage is θ. Note that you don’t have numbers in this step, so to find the magnitude and phase for current I and voltages VR and VL you must first derive both numerator and denominator in polar form using variables R, omega, L, Vm, Vphase (do not use numbers). The solutions should look like equations in slide 62!
In this step, assume that R=3Ω, L=0.1mH, Calculate magnitude and phase of I, VL and VR by plugging in the numbers in the equations for VL, VR and I you found above.
Sketch the phasor diagram.
Finally, transfer the solution to the time domain.
See slide #59-60

52. Write phasor expression of the generator’s voltage if you know:
What would be the Phasor of this signal?
Phasor -> Time Domain
This is why we need to know complex numbers!

53. Write phasor expression of the generator’s voltage if you know:
What would be the Phasor of this signal?
This is a complex number in polar coordinates. This is why we need to know complex numbers!

54. Objective
Students will be able to derive the relationship between sinusoidal signals and phasors
If a sinusoidal signal is given in time domain, they will be able to recognize and derive the phasor expression
If a phasor of a signal is given, they will be able to recognize and derive the time-domain expression

55. How do we transform time-domain function to frequency domain?

56. How do we transform time-domain function to frequency domain?

57. How do we transform time-domain function to frequency domain?

58. Transformation
What is the transformation that we are using to transfer signals from time to frequency domain?

59. Phasor Domain
Phasor counterpart of

60. How do you find current in Frequency Domain if that current is given in the time domain?

61. Closer look: How do we transform the current from TD to FD

62. Closer look: How do we transform the signal from TD to FD
BUT i(t) and I are not the equal! When will they be equal?

63. To go from TD to FD
This is what we call a “phasor”

64. Quiz: How do we go from FD to TD

65. How do you find voltage on an inductor in the Frequency Domain if the current through the inductor is known in Time Domain?

66. Inductor in Time domain

67. Inductor in Time domain

68. If we know that the transformation of current from TD to FD is

69. Transformation from time-domain to frequency-domain

70. Transformation from time-domain to frequency-domain

71. Transformation from time-domain to frequency-domain

72. Inductor TD-> FD

73. Phasor Relation for Inductors
Time domain
Phasor Domain
Time Domain

74. Phasor Relation for Capacitors
Time domain
Time Domain
Phasor Domain

75. Introduction to Phasors
Students will be able to explain how and why was the phasor transformation introduced

76. Superposition If we have two generators in a linear circuit:
Remove one, find currents and voltages
Remove the other, find currents and voltages
The total voltage or current is equal to the sum of responses from the two cases above.

77. Backward Superposition
We have an existing generator in the circuit
We add the second generator in a linear circuit
Make this additional generator a purely imaginary complex number
Find the currents and voltages in the circuit for the combined generator
Keep only the real part of the response

78. Wouldn’t the response of the circuit change if we add this additional complex voltage? + Why are we adding an additional complex voltage?

79. Wouldn’t the response of the circuit change if we add this additional complex voltage? YES, but we are adding the second voltage as an imaginary part of complex number, so that we can keep track which part of voltage makes which response Why are we adding an additional complex voltage? TO SIMPLIFY DIFFERENTIAL EQ.

80. How are we simplifying these differential equations?
We start with an RL circuit
The original differential equation is
Now we add an additional source

81. How are we simplifying these differential equations?
Equations for this circuit with each individual generator are
With both generators equations are:

82. How are we simplifying these differential equations?
Using Euler’s Equation and re-writing complex current in polar coordinates:
We get:

83. How are we simplifying these differential equations?
Taking derivative of current in polar coord:

84. Does KVL work in the FD for a simple RL circuit?
We don’t know the answer yet, so don’t start with KVL in FD. This is what we want to prove.
Write KVL in time domain.
Write the transformation for i(t), v(t) from time to frequency domain
Plug in i(t) and v(t) in TD KVL
Can you cancel some common terms?
Example solution for a similar problem to TL#1

90. From FD to TD

91. Phasor Relation for Inductors
Time domain
Phasor Domain
Time Domain

92. Graded Quiz #TL2
Show that the impedance of a capacitor is 1/(j*omega*c).
Prove that KCL works in FD for the circuit below
Write kcl in time domain
Relate currents to R and C in time domain
Write phasor def of I,V
Plug 3 in to TD eq.

94. Socrative Quiz
To use voltage divider eq the same current has to flow through both impedances.

95. How are these two the same and how are they different?
You can use voltage divider in both
What is the difference between Xc=120Ohms and R=120Ohms
What is the difference between Xc(reactance) and Zc (impedance)?
To use voltage divider eq the same current has to flow through both impedances.

96. How are these two the same and how are they different?
You can use voltage divider in both
What is the difference between Xc=120Ohms and R=120Ohms
What is the difference between Xc(reactance) and Zc (impedance)?
To use voltage divider eq the same current has to flow through both impedances.

97. Homework 1 is due!
We have completed all lectures covered by Homework 1.
Submit HW 1 as soon as you are done with it!