# Thread: Magnitude of a complex function

1. ## Magnitude of a complex function

Hello,

I have a problem here that is for homework if you feel uncomfortable helping me can you at least explain the concept.

I have the equation:

1000K/[(jw)^3 + 110*(jw)^2 + 1000*(jw)]

I'm asked to find the magnitude of this equation. K is a constant so you treat it as such. j is the complex constant (-1)^2. w is omega the variable.

I have always had problems with finding the magnitude of complex equations such as this one. Any help is much appreciated.

2. I have a problem here that is for homework if you feel uncomfortable helping me can you at least explain the concept.
That depends. If the homework counts towards a grade, it is forum policy not to help at all. In that case, I'd recommend you dream up a similar problem that does not count towards a grade.

3. Yes it is for a grade. I had suggested some one either explaining the concept or possibly using different coefficients. the Coefficients can be changed to 1 5 and 10 in the denominator and 12 in the numerator. Whatever the solution is I just want an example that helps me understand the process of the complex magnitude for high order systems.

4. All right, then. You want to compute the magnitude of

$\displaystyle \dfrac{12K}{(j\omega)^{3}+5(j\omega)^{2}+10j\omega },$ right?

Ok, here's the tips and tricks you need to solve this problem:

1. $\displaystyle j=\sqrt{-1}, j^{2}=-1, j^{3}=-j, j^{4}=1.$ This pattern then repeats.

2. Once you've reduced the denominator to the form $\displaystyle a+jb,$ multiply top and bottom by the conjugate $\displaystyle a-jb$ of the denominator. This will result in a real denominator. Then, to find the magnitude of the complex number, you take $\displaystyle |z|^{2}=z\bar{z},$ where $\displaystyle z$ is the complex number, and $\displaystyle \bar{z}$ is the complex conjugate of $\displaystyle z.$

What do you get when you do that?

5. I just realized the coefficients I picked will introduce even more complexity because I would have complex roots of that equation. I'm just going to use the original numbers for an example of what I would get. If you want though I can do the other coefficents too.

after your first step I get:

-j1000k(-w^2-jw110+1000)/[(w(w^2+10000)(w^2+100)]

When you write:

Originally Posted by Ackbeet
Then, to find the magnitude of the complex number, you take $\displaystyle |z|^{2}=z\bar{z},$ where $\displaystyle z$ is the complex number, and $\displaystyle \bar{z}$ is the complex conjugate of $\displaystyle z.$
I need the magnitude of the entire expression. So is z the entire expression and I must multiply z by it's conjugate?

6. You don't need to find roots of anything in order to compute the magnitude, so I'm going to remain with the changed coefficients.

$\displaystyle \left|\dfrac{12K}{(j\omega)^{3}+5(j\omega)^{2}+10j \omega}\right|^{2} =\left|\dfrac{12K}{\omega}\right|^{2}\left|\dfrac{ 1}{(j\omega)^{2}+5j\omega+10}\right|^{2} =\left|\dfrac{12K}{\omega}\right|^{2}\left|\dfrac{ 1}{10-\omega^{2}+5j\omega}\right|^{2}$

$\displaystyle =\left|\dfrac{12K}{\omega}\right|^{2}\left(\dfrac{ 1}{10-\omega^{2}+5j\omega}\right)\left(\dfrac{10-\omega^{2}-5j\omega}{10-\omega^{2}-5j\omega}\right)$

How does this continue?

7. Oh, I thought that I would have to factor and then use the factored form to find the conjugate.

So then the conjugate will produce a denominator of:

w^5 + 5w^3 + 100w

the numerator is the conjugate its self.

Why did we drop the magnitude bars?

So do we take the result from the conjugating and then multiply it by it's own conjugate?

8. Originally Posted by ThomasH
I have the equation:

1000K/[(jw)^3 + 110*(jw)^2 + 1000*(jw)]

I'm asked to find the magnitude of this equation. K is a constant so you treat it as such. j is the complex constant (-1)^2. w is omega the variable.
You can save yourself some work here by using the fact that $\displaystyle \left|\dfrac zw\right| = \dfrac{|z|}{|w|}$ (where z, w are complex numbers).

Therefore $\displaystyle \left|\dfrac{1000K}{(jw)^3 + 110(jw)^2 + 1000(jw)}\right| = \dfrac{1000|K|}{|(jw)^3 + 110(jw)^2 + 1000(jw)|}$.

To complete the calculation, write the denominator in the form $\displaystyle x+jy$ and use the fact that $\displaystyle |x+jy| = \sqrt{x^2+y^2}$.

9. ## Magnitude of a complex expression

I see so the key to this thing all along is to treat the Imaginary terms as an axis and the real terms as an axis.

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# magnitude of complex expression

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