Analog implementation of imaginary roots in a transfer function
A real transfer function (from the reals into the reals) must have either real poles or sets of (2) complex conjugate poles. There are no other options.

You can always combine such sets of complex conjugate poles into a real second-order polynomial. Which can thus be implemented with real physical components. This does require that the chosen circuit topology does not assume a priori that the poles are real.

In your example: \$s(s+j18000)(s-j18000)\$ can be simply combined to obtain: \$s(s^2+18000^2)\$ note that this is not physically realizable not because of it being complex, but because it has infinite Q as the poles are on the imaginary axis. It has no energy loss thus it cannot be implemented in a circuit with resistors. At least not with actual physical positive-valued resistors, if you include negative-valued resistors (which can be implemented with active components) then the dissipative terms can be cancelled out.

However, if you have ideal inductors and ideal capacitors, this is just an LC circuit like this one:







simulate this circuit &ndash; Schematic created using CircuitLab

(Note that I only bothered implementing the poles, the (complex) zeros and scaling factors are wherever they happened to fall.)



Although it should be possible to find a passive element arrangement that would do what you want, the most straightforward route to implement generalized second order sections with somewhat arbitrary poles and zeros is to use either a biquad topology or a state-space filter. That is decompose the problem into a set of active integrators with appropriate feedbacks. In your case, a biquad might be the easiest route.





(Images taken from Dr. Elizabeth Tuttle's class website)