Nishino

Laser-lock box function of Mokulabs is very useful to built a good loop for cavity locking. An error signal is splitted into two paths, fast and slow paths, and one can create two independent filters to two actuations, PZT and laser crystal temperature, for example.

Filters are configured in PID controller. Intergartion and Derivative have their saturation limits, denoted as IS and DS, and you can also set unity-gain frequency.

This log is about how to derive mathmatical forms of the PID filters. It will be useful when you want to reconstruct something in the controling loop, optical transfer function of cavities, for example.

Definition:

g_I, f_I : Intergrater saturation limit, unity-gain frequency (w_I = 2*pi*f_I)

g_D, f_D: Derivative saturation limit, unity gain frequency (w_D = 2*pi*f_D)

Actually, it takes an envelope of two functions: complete integration (derivative) and constant gain. The overall gain, G_I(s), G_D(D) can be written as:

G_I(s) = g_I / (1 + g_I *s/w_I)

G_D(s) = (s/w_D)/(1+s/g_D/w_D)

Of course propotional gain (denoted as P) is just a frequency-independent gain. You can constract the overall filter function as:

G_sum(s) = P + G_I(s) + G_D(s)

See this if you want to check derivation.