Applied Electromagnetics EEE 161 Intro to Transmission Lines

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

Complex Numbers/Phasors at home reading View the video about complex numbers: https://www.youtube.com/watch?v=T647CGsuOVU View the phasor simulation: https://en.wikipedia.org/wiki/File:Circle_cos_sin.gif 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.

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

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.

Sinusoidal Signal Describe this graph using at least 5 words!

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

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

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

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

Sinusoidal signals – time delay

Phase Lead & Lag

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

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

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

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

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

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

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

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

Complex Numbers

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

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

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

Some concepts from complex #s Conjugate Division Addition

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

Euler’s equation

Quiz

Quiz

Some more concepts Power and Square Root.

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

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

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

Division of complex numbers example in EE

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

Homework 1 will be due soon!

Objective Students will be able to solve simple circuits using phasors

ac Phasor Analysis: General Procedure

ac Phasor Analysis: General Procedure

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

Example 1-4: RL Circuit cont.

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

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!

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!

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

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

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

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

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

Phasor Domain Phasor counterpart of

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

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

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?

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

Quiz: How do we go from FD to TD

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

Inductor in Time domain

Inductor in Time domain

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

Transformation from time-domain to frequency-domain

Transformation from time-domain to frequency-domain

Transformation from time-domain to frequency-domain

Inductor TD-> FD

Phasor Relation for Inductors Time domain Phasor Domain Time Domain

Phasor Relation for Capacitors Time domain Time Domain Phasor Domain

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

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.

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

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

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.

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

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

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

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

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

From FD to TD

Phasor Relation for Inductors Time domain Phasor Domain Time Domain

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.

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

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.

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.

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