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Complex Numbers in Electrical Engineering

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Today, we're going to talk about complex numbers in electrical engineering. So we're going to start by talking about how electrical engineers and scientists use complex numbers, and then, we'll do some examples.

So let's start by reviewing complex numbers. Complex numbers are in the form a plus bi, where a is the real part of a complex number, b is the imaginary part of the complex number, and i is the imaginary unit. We defined i to be equal to the square root of negative 1, and we say that it's an imaginary number. And the reason that we say it's an imaginary number is because no real number squared can equal a negative number.

So let's start by reviewing two topics needed when multiplying or dividing complex numbers. So the first concept is FOIL, which we'll use to when we're multiplying two complex numbers, because complex numbers are binomials. And so we'll use FOIL, which is the acronym for multiplying two binomials together. FOIL stands for First, Outside, Inside, Last. So we'll multiply our first two terms together, our outside two terms, our inside two terms, and our last two terms to expand the expression.

We also need to know about conjugates, which we'll use when we're dividing complex numbers, because conjugates are used so that the denominator of the fraction does not have any imaginary numbers. And the conjugate of a complex number, or of any binomial, is just another binomial with the opposite sign. So a plus bi and a minus bi are conjugates of each other.

So let's look at a formula that is used to represent electrical currents. The formula is V equals I times R, where the V is the voltage, measured in volts; I is the current, measured in amps; and R is the resistance, measured in ohms. And so electrical engineers will use complex numbers when using this formula. Another thing to note is that engineers and scientists will often use a lowercase letter j for the imaginary number i, and that is because if they use a lowercase letter i, that could be confused with the upper letter I, which in this equation is the current. So if you see a lowercase letter j with this formula, know that that represents the imaginary number.

So let's do a real world example. Let's say I have an electrical circuit. The current is 2 plus j amps, and the resistance is 1 minus 3j ohms. I want to find the voltage, or the number of volts, for this electric circuit. So let's start by using our formula, V equals I times R. I know that I want to find the voltage, and so I need to have values for the current and the resistance.

So I know that the current is 2 plus j amps, and the resistance is 1 minus 3j ohms. So because these are two binomials, I can use FOIL to multiply. I'm just going to ignore the units while I multiply. So I'm going to start by multiplying 2 times 1, which will give me 2. 2 times negative 3j will give me negative 6j. j times 1 will give me positive 1j, or just j. And j times negative 3j will give me a negative 3j squared. Multiplying my units, amps times ohms is going to give me volts.

I can simplify this by combining my two middle terms. Negative 6j plus j will give me negative 5j. And then, substituting negative 1 in for my j squared, I have minus 3 times negative 1, which is the same as adding 3. So finally, I can simplify 2 plus 3 will give me five. So I found that the volts, or the voltage, for my electrical circuit is going to be 5 minus 5j volts.

So let's do another example. Let's say I have an electrical circuit. And this time, I know that the voltage is 1 plus j volts, and I know that the resistance is 3 minus 2j ohms. I want to find the current in amps of the electrical circuit. So again, I'm going to start with the formula, V equals I times R. Now this time, I know that my voltage is 1 plus j volts. That will be equal to the current-- which I don't know, so I'm going to leave that as I-- multiplied by my resistance, which is 3 minus 2j ohms.

So if I want to find the current, I'm going to have to isolate this variable, which means I'll need to divide by 3 minus 2j on both sides. And now, I see that I have a fraction with a complex number in the denominator. So I know that I'm going to need to multiply by the conjugate of this complex number so that the denominator is not an imaginary number.

So I'm going to start by multiplying by the conjugate of 3 minus 2j, which will be 3 plus 2j, in the denominator and in the numerator of my fraction. Simplifying the numerator by FOIL, I have 3 times 1, which will give me 3; 3 times j, which would give me 3j; 2j times 1, which will give me 2j; and 2j times j, which will give me 2j squared. And this is in volts.

Then, I have my denominator. I'm going to also multiply using FOIL. 3 times 3 will give me 9. 3 times minus 2j will give me minus 6j. Positive 2j times 3 will give me positive 6j. And 2j times negative 2j will give me minus 4j squared. And that's in ohms. So now, I can simplify by combining my two middle terms. That's going to give me 3 plus 5j. And I know that j squared is equal to negative 1. So 2 times negative 1 will give me negative 2. So I've got plus negative 2.

In my numerator, my two middle terms are going to cancel, so I'll just be left with 9 minus 4j squared. Again, I know that j squared is equal to negative 1, so I have minus 4 times negative 1, which is just negative 4. So my denominator, I have 9 minus negative 4. My units, volts over ohms, is going to give me amps. And so I just have one last step to simplify. 3 plus negative 2 is going to give me 1. And then, I've got my 5j. And 9 minus negative 4 is 13. So I can write this as 1 over 13 plus 5 over 13 j amps, which is the current of my electric circuit.

So let's go over our key points from today. Electrical engineers often use complex numbers when working with the equation relating voltage, current, and existence. A complex number, a plus bi, contains a real part, a, and an imaginary part, b, and the imaginary unit, i. The conjugate of a binomial is a binomial with the opposite signs between its terms.

Conjugates are used when dividing complex numbers, so that the denominator has no imaginary numbers. And engineers and scientists often use the letter j to refer to the imaginary number i, so as not to confuse lowercase i with uppercase i, which is the variable for current.

So I hope that these key points and examples helped you understand a little bit more about using complex numbers in electrical engineering. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.