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