06 - Closed Loop Response

\[C(t) = K_c\tau_D\frac{d\epsilon(t)}{dt}\] \[G_c = K_c\tau_D s\] where \(\tau_D\) = derivative time.

Closed loop response of a first order system with derivative control is given by \[\begin{aligned} Y(s) &= \frac{\dfrac{K_p}{\tau_ps+1}K_c\tau_Ds}{1+\dfrac{K_p}{\tau_ps+1}K_c\tau_Ds}Y_{sp}(s) \\ &= \frac{K_pK_c\tau_Ds}{(\tau_p+K_pK_c\tau_D)s+1}Y_{sp}(s) \end{aligned}\]

The derivative control does not change the order of the response.

The effective time constant of the closed-loop response is \((\tau_p+K_pK_c\tau_D)\), i.e., larger than \(\tau_p\). This means that the response of the controlled process is slower than that of the original first-order process.

For constant nonzero error, derivative mode does not give any output.

This mode is never used alone. It is always used in combination with proportional or proportional-plus-integral control action.

Derivative mode should not be used on noisy loops (e.g.: flow).