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. 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.

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

6. 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.

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

9. Sinusoidal signals can be represented using complex numbers!

10. Complex Numbers

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

12. Complex numbers Cartesian Coordinate System Polar Coordinate System
For example:
Euler’s Identity

13. Quiz Complex number Z=3+j2 is given.
Sketch the number in Cartesian coordinate system
Find the magnitude and phase
Write a complete sentence to explain magnitude and phase on the diagram

15. Some concepts from complex #s
Conjugate
Division
Addition
It is easier to divide/multiply complex numbers when they are in polar coordinates.
It is easier to add/subtract complex numbers when they are in _____________ coordinates.

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

17. Euler’s equation A A A

18. Quiz What is the real and imaginary part of the voltage whose amplitude is 2 and phase is 45 deg? (no calc)

19. Quiz

20. Some more concepts
Power and Square Root.

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

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

23. Where did we see division of complex numbers
Where did we see division of complex numbers? example in EE: Find magnitude and phase of the current in the circuit below
How can we find magnitude and phase of this current in terms of variables R, L, V and ω?

24. Separately transfer R and OmegaL to FD

25. Quiz
Find the magnitude and phase of the voltage on the mystery circuit if the current through it was measured to be
The complex equation to find the voltage is

28. Homework 1 will be due soon!

29. Interesting Case Study in an Electrical Engineering Laboratory
Let’s look at two simple circuits

30. Dr. M is measuring voltage in the laboratory using her handy-dandy Fluke. She measures voltages on the resistors and inductor to be as shown. She is shocked with the measurement on top and bottom circuit. Why? What do you think she expects, and what did she measure?

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

33. ac Phasor Analysis: General Procedure

34. ac Phasor Analysis: General Procedure

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

36. Example 1-4: RL Circuit cont.

37. 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 var
iables R, omega, L, Vm, Vphase (do not use numbers). The solutions should look like equations in slide 24/27!
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, write the two voltages and the current in the time domain.

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

39. 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.

40. 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.

42. Objectives
Students will be able to explain why is the relationship between sinusoidal signals and phasors valid in the previous table
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

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

44. Transformation
What is the transformation that we are using to transfer signals from time to frequency domain?
That’s great, but why??

45. Objective
Students will be able to explain how and why was the phasor transformation introduced

46. 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.

47. Backward Superposition
We have an existing cos(ωt) generator in the circuit
We add the second generator j*sin(ωt)
We find the currents and voltages in the circuit for the combined generator ejωt
Keep only the real part of the response

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

51. Guided Example solution similar to TL#2
What is the current in the series RL circuit below in Frequency Domain?* To find the current, we have to derive the phasor relation for KVL!
*You can’t say right away V=RI+jωLI, because you don’t know if it is true. Start from time-domain KVL equation. What would that be?

52. How do we find out Phasor relation for KVL?
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?

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

55. Graded Quiz #TL2
Derive the phasor relation for a capacitor (prove that the imped. Of capacitor is 1/(jomegaC)
Derive the phasor relation for KCL in 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.