Relation between trigonometric functions and complex numbers

 

 

 

 

 

 

         In the end of an AC electrical circuit analysis class         

 

Today, we learned that complex numbers are very useful to analyze an AC circuit.
You can solve a differential equation regarding an AC circuit like this below by using complex numbers.

 

 

Oh, time is almost up. See you next week.

 

 

Can we always transform trigonometric functions into complex numbers?

 

Hmm…

 

You look like you have a question.

 

Yeah.. I feel like Mr. Owl tells a lie…

 

Why do you feel so?

 

If it were really okay to use complex numbers when we analyze an AC circuit, I could calculate electric power very easily but the calculation would be wrong, right?

 

Will you explain it a little more to me?

 

Sure. Firstly, let me show you the character of multiplication of complex numbers below.

 

 

You are right. If you multiply blue complex number by yellow complex number, the length of the result (red complex number) will be the multiplication of both of the lengths, which is A×B, and the angle will be the addition of both of the angles, which is α+β.

 

How are you going to apply the character of the multiplication of complex numbers to the calculation of electric power?

 

This is how to calculate electric power, right?

 

 

Correct.

 

It would be very easy to calculate it if I really could make use of complex numbers. I showed it to you below.

 

 

Hahaha. You seem like you’ve already known that the calculation is absolutely wrong.

 

Yeah, I know it. That’s why I’m confused. I sometimes can apply complex numbers to calculations in an AC circuit but I sometimes can NOT..

 

I’d like you to look at the figure below. Both of the calculation make use of multiplication of complex numbers. However, the calculation of the left side is absolutely wrong..

When is it possible to transform trigonometric functions into complex numbers?

 

 

 

The reason why we can transform trigonometric functions into complex numbers

 

Okay. Your question has become clear to me. I’ll answer the question.

Firstly, let’s think about the reason why we can make use of complex numbers in an AC circuit. Do you sometimes calculate the addition below?

 

 

That’s when I need to add two currents like the figure below.

 

 

Can you calculate it?

 

I think I can.. It’s okay if I make use of complex numbers, isn’t it? That was what Mr. Owl told us.

 

Yes. Will you show me the actual calculation using complex numbers?

 

Sure. Here it is.

 

 

Right, but can you tell me the reason why you can calculate it like that?

 

Well… No, I can’t.. I just calculated it by using complex numbers as Mr. Owl told us..

 

Hahaha. Okay. Please take a look at the figure below.

I showed the reason why we can use complex numbers. I didn’t use complex numbers at all in the calculation.

Did you understand that the addition of trigonometric functions acts like the addition of complex numbers?

 

 

 

The reason why we can apply complex numbers to a differential equation

 

Yeah, I understood the reason why we can transform trigonometric functions into complex numbers when we add trigonometric functions.

However, I still haven’t understood the reason why we can make use of complex numbers in solving a differential equation.

 

Actually, we can’t always make use of complex numbers in solving a differential equation. If we solve a differential equation regarding an AC circuit, complex numbers will become useful.

 

What? Can you explain it a little more?

 

Okay. So, please take a look at the calculation below a little. I am trying to solve a differential equation regarding an AC circuit.

 

 

An addition of trigonometric functions appeared in the last calculation.

 

 

Well noticed. Do you remember that we can make use of complex numbers when we add trigonometric functions?

 

 

Yes. Let me make use of complex numbers in the calculation. I showed it below. Is it right?

 

 

Good job. Strictly speaking, Φ=0 but it’s okay for now. The calculation tells us that the result will be the red complex number if we add the blue complex number and the yellow complex number.

What does the red complex number mean? When we look at the equation, we can tell that the red complex number means the voltage.

Now, we can come to know the relation between the current and the voltage. Can you describe the relation?

 

Sure. We consider the current “i” as “I sin(ωt+θ)” in the calculation, and so the complex number of the current is the purple one in the figure below.

 

 

Did you understand how we can apply complex numbers to a differential equation in an AC circuit?

This is why the trigonometric differential equation above can be treated as a complex number equation.

 

 

When can we transform trigonometric functions into complex numbers?

 

I see. I feel like I understood when I can transform trigonometric functions into complex numbers. If we calculate electric power, we need to multiply trigonometric functions but we can’t apply complex numbers to multiplication of trigonometric functions. That’s why we can’t make use of complex numbers in the calculation of electric power.

 

 

You are right. Simply put, you can make use of complex numbers when you do linear calculations in trigonometric functions.

 

Thank you for having your lecture with me.