Servo Mechanism Calculator

Theory Notes

2nd-order transfer function:

$$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$

Settling time and overshoot:

$$t_s \approx \frac{4}{\zeta\omega_n}, \quad \text{OS} = \exp\!\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right) \times 100\%$$

Resonance peak and bandwidth:

$$M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}} \quad (\zeta < 0.707), \quad \omega_{BW} = \omega_n\sqrt{1-2\zeta^2+\sqrt{4\zeta^4-4\zeta^2+2}}$$

Engineer Dialogue — "Why is 45° the phase margin target?"

🧑‍🎓 "My professor says to aim for 45° phase margin. Why that number specifically?"

🎓 "Phase margin tells you how far you are from -180° at the gain crossover frequency — the point where feedback flips from negative to positive and causes oscillation. 45° corresponds roughly to a damping ratio of 0.42, which gives about 20% overshoot but fast response. It's a practical engineering sweet spot."

🧑‍🎓 "So if I increase PM to 60°, what changes?"

🎓 "ζ increases to about 0.6 — less overshoot (around 9%), slower settling, more robust to model uncertainty. Below 30° PM the transient gets quite oscillatory, which industrial machines usually can't tolerate."

🧑‍🎓 "In real servo drives, how do engineers actually achieve the target PM?"

🎓 "Mainly by tuning the D-term to advance phase at high frequencies, or by reducing gain to lower the crossover frequency. More advanced drives use notch filters to suppress mechanical resonances, which then allows a higher bandwidth without violating PM requirements."