1. Empirical Damping Ratio ($\zeta$) Extraction from Oscilloscope Ringing:
For an underdamped second-order step response, consecutive peak voltage amplitudes $V_1$ and $V_2$ are related by the logarithmic decrement $\delta$: $$\delta = \ln\left(\frac{V_1}{V_2}\right) \implies \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}}$$

2. Phase Margin ($\Phi_m$) and Loop Bandwidth ($f_c$):
The closed-loop phase margin is derived directly from the extracted damping ratio: $$\Phi_m \approx \arctan\left(\frac{2\zeta}{\sqrt{\sqrt{1 + 4\zeta^4} - 2\zeta^2}}\right) \approx 100 \cdot \zeta \quad (\text{for } \zeta < 0.6)$$ The actual crossover frequency $f_c$ is determined from the ringing frequency $f_{ring}$ and natural frequency $\omega_n$: $$\omega_n = \frac{2\pi f_{ring}}{\sqrt{1 - \zeta^2}}, \quad f_c \approx f_{ring} \sqrt{1 - \zeta^2}$$

3. Type II Compensator Placement:
$$\text{Zero: } f_{z1} = \frac{1}{2\pi R_c C_c} \approx \frac{f_{c\_target}}{4}, \quad \text{High-Freq Pole: } f_{p1} = \frac{1}{2\pi R_c \frac{C_c C_p}{C_c + C_p}} \approx \min\left(\frac{f_{sw}}{2}, f_{esr}\right)$$ $$\text{Mid-Band Gain Scaling: } R_{c\_new} = R_{c\_old} \times \frac{f_{c\_target}}{f_{c\_est}}$$

4. Type III Compensator Placement:
Provides an additional lead zero-pole pair ($f_{z2}, f_{p2}$) to cancel double-pole LC peaking in Voltage Mode Control (VMC) and inject up to $+75^\circ$ of phase boost: $$f_{z2} = \frac{1}{2\pi (R_1 + R_3) C_3} \approx f_0, \quad f_{p2} = \frac{1}{2\pi R_3 C_3} \approx \frac{f_{sw}}{2}$$