ECE 1100: Introduction to Electrical and Computer Engineering Dr. Dave Shattuck Associate Professor, ECE Dept. Set #8 – Sinusoids, Paradigms and Electronics Phone: (713) Office: Room W326-D3

Introduction I want to talk about sinusoids. I want to talk about how engineers think about attacking problems. I want to introduce the area called “electronics”.

Goal of this lecture: Answer Some Questions Why does a guitar sound different from a violin? Why don’t I sound like Phil Collins? Why don’t I make as much $ as Phil Collins? Why do people talk about audio systems in terms of sinusoids?

Engineering Paradigms We are going to introduce a couple of major engineering paradigms. What are paradigms?

About 20 cents. Get it? “Pair a dimes?” Okay, so it is not very funny…

Engineering Paradigms A paradigm is a way of thinking about something. A paradigm shift is a change in the way we think about something. I want to introduce a couple of engineering paradigm shifts. I will use sinusoids as the basis for this.

Euler’s Relation Who knows Euler’s Relation? To Play = To Lose

NO Wait! To Play = To Lose That was Oiler’s Relation. They are in Tennessee now. They’re called the Titans. This is now an obsolete joke. What is Euler’s Relation?

Euler’s Relation Euler’s Relation is: e j  = cos  + j sin  This means that sinusoids are just complex exponentials. When we solve certain kinds of problems, the solutions turn out to be complex exponentials, or sine waves.

Euler’s Relation Dr. Dave, can you say that again, in English? OK …. Sine waves happen.

Fourier’s Theorem Now, there is a special rule that concerns sinusoids. First, we need to be able to pronounce this. Furrier’s theorem applies in animal husbandry So, pronounce it 4 - E - A !!!

Fourier’s Theorem Any physically realizable waveform can be represented by, and is equivalent to, a summation of sinusoids of different amplitude, frequency and phase. This represented a major paradigm shift in engineering.

Fourier’s Theorem In English this time, Dr. Dave? OK…. You can get any real function by adding up sine waves. This represented a major change in the way we think about problem solving in engineering. We could look at what happens to sine waves, and know what would happen when other signals are used.

Fourier’s Theorem: Answers Question: Why does a guitar sound different from a violin? Answer: Sine wave components in the two instruments are different.

Fourier’s Theorem: Answers Question: Why don’t I sound like Phil Collins? Answer: Sine wave components in the two voices are different.

Fourier’s Theorem: Answers Question: Why don’t I make as much $ as Phil Collins? Equivalent Answer: He can sing. I can’t. Answer: Sine wave components in the two voices are different.

Fourier’s Theorem: Answers Question: Why do people talk about audio systems in terms of sinusoids? Answer: Sine wave components tell us everything we need to know about any signal. Therefore, Fourier’s Theorem allows us to describe and analyze a system without knowing what signals we use it with.

Fourier’s Theorem: Answers Question: Why do we introduce complicated mathematical concepts like Fourier’s Theorem? Answer: To make life hard for engineering students.

Fourier’s Theorem: Answers Question: Why do we introduce complicated mathematical concepts like Fourier’s Theorem? Answer: To make life hard for engineering students. No, this is RONG!

Fourier’s Theorem: Answers Question: Why do we introduce complicated mathematical concepts like Fourier’s Theorem? Answer: These concepts help us to solve problems, and to think about how to solve problems.

Demonstration of Sinusoids Let’s look at a system to send signals around. (Telephone, Radio, etc.)

Application to Electronics Let’s look at a system to send signals around. The telephone: –Converts sound to voltage, with a microphone. –Sends the voltage to the location needed. –Converts voltage to sound, with a speaker.

Send the voltage? How does the voltage get to where we want it? –We need to amplify it, to make it louder. –We need to amplify it, to send it a long way. –We need to deal with noise. –We may need to modulate it to send it through some kinds of channels.

Did I say modulate? In English, please, Dr. Dave! First, the problem: –Sometimes it is kind of awkward to run a wire to the place I want the signal to go. Modulate?

Did I say modulate? In English, please, Dr. Dave! –The problem: Sometimes it is kind of awkward to run a wire to the place I want the signal to go. –The solution: If I stick the wire up in the air, it will send that signal through the air. Modulate?

If I stick a wire up in the air, it will send that signal through the air. –However, to work well, it must be at least 1/10th of a wavelength ( ) long. –At 15 Hertz, the low end of human hearing, that is 0.1 = (0.1)c/f = (0.1)(186,000[miles/s])/(15[s -1 ]) = 1,240 miles in length.

Modulate! Need antennas that are about the length of Texas. Rhode Island would be out of luck. Only one signal can be sent at a time. The strongest signal would win, if you were lucky. Solution: Multiply by sinusoid at some high frequency. Wavelengths are shorter, and I can have lots of signals at once, each with a different frequency sinusoid. The different frequencies are called “stations”.

Modulate!! For example, for radio 740[kHz] –At 740,000[Hertz], we have 0.1 = (0.1)c/f = (0.1)(186,000[miles/s])/(740,000[s -1 ]) = 130 feet in length

Solution: Electronics!! In electronics, we: –Amplify signals (make them bigger). –Deal with noise in signals. –Modulate and demodulate signals. –Fool around with signals, and the devices that allow us to do this fooling around.

Who cares about this stuff? I do, obviously. But that is not really your question. Your question is, why should you care about this? You should only care about this if you are going into electrical engineering. If you are, this is the kind of way you will learn to approach problems. I am showing you electronics as one example. There are many different kinds of problems that Electrical Engineers solve.

Review of Complex Numbers – 1 A complex number is a number that is a function of the square root of minus one. We use the symbol “j” to represent this, Remember that j does not exist. It is a figment of our imagination. It is just a tool we use to get solutions that do exist.

Complex numbers can be expressed as having a real part, and an imaginary part. The imaginary part is the coefficient of j. The real part is the part that is not a coefficient of j. Thus, in the example given here, for the complex number A, the real part is 3, and the imaginary part is 4. Review of Complex Numbers – 2 Remember that the both the real part and the imaginary part are themselves real numbers.

Review of Complex Numbers – 3 Complex numbers can also be expressed as having a magnitude, and a phase. For example, in the complex number A, the real part is 3, the imaginary part is 4, the magnitude is 5, and the phase is 53.13[degrees]. Remember that all four are real numbers.

Review of Complex Numbers – 4 It is easiest to think of this in terms of a plot, where the horizontal axis (abscissa) is the real component, and the vertical axis (ordinate) is the imaginary component. So, if we were to plot our complex number A in this complex plane, we would get

Review of Complex Numbers – 5 We can get the relationships between these values from our trigonometry courses, just looking at the right triangle given here. For review, they are all given here. This one requires some thought. The value of  depends on the signs of y and x.

Review of Complex Numbers – 6 We often use a short hand notation for complex numbers, using an angle symbol instead of the complex exponential. Specifically, we write

Review of Complex Numbers – 7 Generally, we want to be able to move between notations and perform addition, subtraction, multiplication and division, quickly and easily. The rules are: to add or subtract, we add or subtract the real parts and the imaginary parts; and to multiply or divide, we multiply or divide the magnitudes, and add or subtract the phases. You should have a calculator or computer that does this for you. If so, practice this, because it will come in handy.

How does all this fit together? This is a good question. At this point it must seem like a series of strange, literally unreal, and unrelated subjects. However, while the numbers are not all real, the subjects are related. We will use the magnitude and phase of a complex number to get the magnitude and phase of the sinusoid in the solution in certain kinds of problems. It doesn’t matter whether the method is real or not. The solution is real. Any method that gets us the correct solution quickly, is a good method. Woody Allen tells a joke about this.

How does all this fit together? This is a good question. At this point it must seem like a series of strange, literally unreal, and unrelated subjects. However, while the numbers are not all real, the subjects are related. We will use the magnitude and phase of a complex number to get the magnitude and phase of the sinusoid in the solution in certain kinds of problems. It doesn’t matter whether the method is real or not. The solution is real. Any method that gets us the correct solution quickly, is a good method. Woody Allen tells a joke about this. We need the eggs.

Example Problems Let’s solve the following problems:

Example Problems The solution for the first problem is:

Example Problems The solution for the second problem is:

Example Problems The solution for the third problem is: