Why can I ignore the “imaginary” value from the gain of my system?

Question

I’m in the final year of my education, and for the exam training I had to find the K (or max gain) value of the given system when the desired damping ration equals 0,7.

What i have done to addres the problem

I designed for my self a 6 step plan to calculate these kind of systems in general.

1. Calculate the transfer function
2. Draw the poles and zero's into a plot containing the unity circle
3. Determin the 'z' point of the gain for a given overshoot. (or when the system wil get unstable)
4. Take the denomerator of the transfer function and equal it to '0'
5. enter the found 'z' into this formula
6. calculate the gain (K)

To answer the question I first determined the transfer function

\$ \LARGE \frac{C(z)}{R(z)}=\frac{K(z+1)}{(z-1)(z-0.5)+K(z+1)} \$

after which I draw the polar plot with unity circle (in the picture "eenheidscirkel")

The question stated I had to find the damping ratio of 0,7 which was used to draw the following graph.

From which the "z" value was 0,719+0,215j. This Z value i enterd into the denomerator of my transferfunction equal to zero. $$ (z-1)(z-0.5)+k(z+1)=0 \\ (0.719+0.215j-1) (0.719+0.215j-0.5)+k(0.719+0.215j+1)=0 $$

This function I rewrote to the final statement for K with the next steps

(-0.281+0.215j)(0.219+0.215j)+k(1.719+0.215j)=0

-0.062-0.060j+0.047j-0.046+k(1.719+0.215j)=0

-0.108-0.013j+k(1.719+0.215j)=0

k=(0.108+0.013j)/(1.719+0.215j)

And here my problem starts

My teacher states that have to devide the first part (ignoring the IM part) of the function to get my gain (K).

0.108/1.719 = 0.063

I could also use the second part for this.

0.013/0.215 = 0.061

Or the sum ignoring the J again

(0.108+0.013)/(1.719+0.215) = 0.063

resulting in K=0.063

This result is correct but I don't understand why.

Why can I ignore the imaginairy part of the final function.

With a complex devision I would expect atleast a K value of a+bj

I tryed this for the final step:

k=(0.108 +0.013j)/(1.719+0.215j)*(1.719-0.215j)/(1.719-0.215j)

k=(0.189-0.046j)/3

RE : k= 0.189/3 = 0.063

IM : k=0.046j/3 = 0.015j

what makes me think the result shoud be

k= 0.063-0.015j

I know this smells like a home work question and there systems that can calculate everything with the snap of a finger, im trying to understand why im allowed (if im allowed) to ignore the imaginary value of the gain of this system.

Answer 1

You can ignore it because it is small, and the error comes because the pole has not been that precisely determined.

With the pole value you give (0.719 + 0.215 j), the solution from the characteristic equation for k is 0.0626793 - 0.0000849631 j. As the precision in the pole value is increased the imaginary part will become even closer to zero.

[I should also add that you are not plotting the polar (Nyquist) plot but rather the root locus plot.]