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I'm looking to find the transient response to a step impulse of a type-2 compensator.

Type II compensator

I already found the transfer function, but when it comes to finding the transient response by using the inverse Laplace transform I don't get a good result. Anyway, here is the transfer function:

$$ Hd(s)=\frac{1+R_2C_2 s}{s(R_1(C_1+C_2)+R_1R_2C_1C_2s)} $$ Reacting to a step function I multiply it by 1/s which gives: $$ Hd(s)=\frac{1+R_2C_2 s}{s^2(R_1(C_1+C_2)+R_1R_2C_1C_2s)} $$ Then using partial fraction I get: $$ \frac{1+R_2C_2 s}{s^2(R_1(C_1+C_2)+R_1R_2C_1C_2s)}=\frac{A}{s^2}+\frac{B}{s}+\frac{C}{R_1(C_1+C_2)+R_1R_2C_1C_2s} $$ When I perform the inverse Laplace transform on the result it gives an unexpected unreal time response. $$ Hd(t)=\frac{(R_1R_2C_1C_2)^2-(1+R_2C_2)(R_1(C_1+C_2))(R_1R_2C_1C_2)e^{\frac{-(R_1(C_1+C_2)t}{(R_1R_2C_1C_2)}}}{(R_1(C_1+C_2))^2(R_1R_2C_1C_2)}+\frac{(1+R_2C_2)(R_1(C_1+C_2))-R_1R_2C_1C_2}{(R_1(C_1+C_2))^2}+\frac{t}{R_1(C_1+C_2)} $$ It's quite a big formula, but what bother me is the last term because the response should not be linear.

Has anybody already calculated the transient response of a type-2 compensator?

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    \$\begingroup\$ Have you noted that this circuit is an "integrator"? \$\endgroup\$
    – Antonio51
    Commented Aug 11, 2022 at 8:44
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    \$\begingroup\$ You will make your life easier if, in the original formula for \$Hd(s)\$, you replace: \$R_2C_2=\tau_2,\;R_1(C_1+C_2)=\tau_1,\;R_1R_2C_1C_2=\tau_{12}\$, because then the step response will show up in a more readable form:$$s(t)=\dfrac{t}{\tau_1}+\dfrac{\tau_1\tau_2-\tau_{12}}{\tau_1^2}\left(1-\text{e}^{-\frac{\tau_1}{\tau_{12}}t}\right)$$The presence of the free term with \$t\$ tells you it's an integrator, if the fact that it has no DC path in the feedback, only capacitors, won't. \$\endgroup\$
    – a concerned citizen
    Commented Aug 11, 2022 at 10:09

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This circuit is "fundamentally" an "integrator".
So the last term shows that output is growing with time.
The last term seems thus ok.

enter image description here

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