EX7-1 Express the following phasor in the polar...

EX7-1 Express the following phasor in the polar representation assuming that $V_1$ is a real positive number. $\frac{1}{1 + j\omega CR}V_1$ Hint With real numbers $r$ and $\phi$, put $re^{j\phi} = \frac{1}{1 + j\omega CR}V_1$. You are to find $r$ and $\phi$. Define $\theta$ with $\tan \theta = \omega CR$ and express $r$ and $\phi$ by using $\theta$ and $V_1$.

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00:01 So in this question, we're going to be looking at some argand diagrams.

00:05 So first of all, we've got z equals 1 plus i.

00:08 So this is the imaginary part and this is the real part.

00:13 So it's one on the imaginary axis and it's one on the imaginary axis and one on the real axis.

00:19 So here's our z.

00:22 So let's get the modulus.

00:24 So r is the modulus of z is the square root of 1 plus 1 squared, which is root 2.

00:30 And the angle here is going to be tan to the minus 1 of 1 over 1, which is at 45 degrees or pi divided by 4.

00:48 So that means that we can write the z is root to e to the i pi over 4.

00:55 Part b we've got z equals minus 1 plus i root 3.

01:04 So minus 1, root 3, got this point here, this is z.

01:12 Mod z is going to be the square root of 1 squared plus 3, which is root 4, which is 2.

01:20 And this angle here, theta, equals tan to the minus 1, root 3 over minus 1.

01:31 So the inverse tan of root 3, minus root 3, i should say.

01:43 Which is, well, this is giving me a value of minus a third pi, but this is going to be in this branch here.

01:53 So we want actually, this is going to be plus 2 pi divided by 3.

02:05 Okay.

02:07 So because tan has a, so this gives us minus pi by 3, but we know tan is periodic with period equals pi.

02:21 So we want to get the first positive solution here...

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