Magnitude of 1/(1 + jwRC) with complex denominator

[kelp]

Hello, I am trying to find out how my teacher got the magnitude of this expression. This is the original expression:
1/(1 + jwRC) j is an imaginary number

Then, he ends up with 1/(sqrt(1 + (wRC)^2)).

I get something nasty like:
sqrt(2/(1 + 2jwRC - wRC))

Thanks.

 

[danago]

How did you come to your answer?

 

[kelp]

I squared the original expression and multiplied it by 2, then took the square root of it.

 

[danago]

Why did you multiply by 2? Anyway, the first indication that something has gone wrong is that you still have the imaginary unit j in your answer -- the modulus should be real.

For any complex number, its modulus can be calculated from:

$$\left| {x + jy} \right| = \sqrt {x^2 + y^2 }$$

 

[danago]

You could also make use of some of the properties of the modulus listed on this page:

http://planetmath.org/encyclopedia/ModulusOfComplexNumber.html

 

[Mark44]

The usual technique to find the reciprocal of a complex number is to multiply by 1 in the form of the conjugate over itself. That gives you a complex number in rectangular form, making it easier to find the magnitude.

For example, let z = 1/(1 + 2i) (my i is your j)
$$\frac{1}{1 + 2i} = \frac{1}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i}$$
$$=\frac{1 - 2i}{1 - 4i^2} = \frac{1}{5}(1 - 2i)$$

Now we can find |z|, which is (1/5)sqrt(1 + 4) = sqrt(5)/5