锁相环概念

• Notch Filter based PLL
  
  • Closed Loop Phase TF
    • $H_o(s) = \frac{\theta_{out}(s)}{\theta_{in}(s)} = \frac{LF(s)}{s+LF(s)} = \frac{v_{grid}(K_ps+\frac{Kp}{T_i})}{s^2+v_{grid}K_ps+v_{grid}\frac{K_p}{T_i}}$
  • Compaing with the generic second order system transfer function, we can get the natural frequency and the damping ration of the linearalized PLL
    • $H(s) = \frac{2\zeta w_ns+w_n^2}{s^2+2\zeta w_ns+w_n^2}$
    • $w_n =\sqrt{\frac{v_{grid}K_p}{T_i}}$, $\zeta = \sqrt{\frac{v_{grid}K_pT_i}{4}}$
  • low grid frequency (50Hz) make it hard to design PI, therefore, notch filter is added
    
  • Design PI coefficients
    • for a general second order system, the step response:
      • $H(s) = \frac{2\zeta w_ns+w_n^2}{s^2+2\zeta w_ns+w_n^2}$
      • $y(t) = 1 - ce^{-\sigma t}sin(w_d+\phi)$ where
      • $\sigma = \zeta w_n$ and $c = \frac{w_n}{w_d}$ and $w_d = \sqrt{1-\zeta^2}w_n$
      • for an error band $\delta$, $1-\delta = 1-ce^{-\sigma t_s} \rightarrow t_s = \frac{1}{\sigma}\cdot ln(\frac{c}{\sigma})$ , where $t_s$ is the settling time that the system needs to attain error within $\delta$
    • solve equations and obtain $w_n$
    • use $K_p = 2w_n\zeta$ and $K_i = w_n^2$, obtain appropriate PI controller values
  • Use Bi-linear transformation and obtain the difference equations