03.6 Pole Placement (Const Damping and Const Freq Line) s-plane


Fig : Constant Damping Line of $45\degree$ or $\zeta=0.707$ Red = $1 Hz$ Green = $3 Hz$ Blue = $10 Hz$	Fig : Constant Damping Line of $\zeta=0.3$ Red = $1 Hz$ Green = $3 Hz$ Blue = $10 Hz$
Meaning, across the same line, no matter how far you go, it is the same damping ratio, $\zeta$	Meaning, across the same line, no matter how far you go, it is the same damping ratio
Now, the time constant, $\tau$ depends on the frequency, $\omega$ because $\tau=\frac{1}{\zeta \omega}$ 03.4 Impulse Response Function Plotting	Now, the time constant, $\tau$ depends on the frequency, $\omega$ because $\tau=\frac{1}{\zeta \omega}$ 03.4 Impulse Response Function Plotting
That means, blue one is further from the imaginary axis = the most negative real part = faster convergence	In this case, the damping is much less, therefore, all the poles (same $1,2,3 Hz$) are closer to the imaginary axis = longer settling to converge
Overshoot depends on the damping ratio only	Overshoot depends on the damping ratio only. Here, we would see that the overshoot would be much higher since the damping ratio is less
Changing the radius = Changing the frequency


To increase the damping, I would rotate anti-clockwise because we are keeping the same frequency by making it a constant radius. And we are making it more damped by going away from the imaginary axis	Here, to keep the damping ratio same, i would go along the same radial line. And I would increase the radius, for more frequency, hence, more negative real part, faster response = faster convergence