6.4 Crank diagram

We actually still have some options open to us. One of thenicest, at least in terms of getting some insight,is call a crank diagram . Note that this equation is a complex equation, which requires us to take a the ratio of two complexnumbers; $1+{}_{}e^{-(i\times 2s)}$ and $1-{}_{}e^{-(i\times 2s)}$ .

Let's plot these two quantities on the complex plane, starting at $s=0$ (the load end of the line). We can represent ${}_{}$ , the reflection coefficient, by its magnitude and its phase, $\left|{}_{}\right|$ and ${}_{}$ . For the numerator we plot a 1, and then add the complex vector $$ which has a length $\left|\right|$ and sits at an angle ${}_{}$ with respect to the real axis . The denominator is just the same thing, except the $$ vector points in the opposite direction .

Now let's find the magnitude of $V(s)$ . To do this we need to square the real and imaginary parts, add them, and then take the square root.