# Can higher order (more than 2nd order) differential equation be formed with electrical network?

Is it possible to have more than 2nd order D.E. in electrical network? Because order of d.e = energy storage element (i.e. capacitor and inductor). Actually i have seen only 2nd order d.e. so, can there be other energy storing elements possible? Can we form higher order d.e. (more than 2nd order) only with capacitor and inductor in the circuit? Kindly explain with examples.

• Google "LC ladder filter". – The Photon Jul 16 '17 at 5:37
• The question is tricky because of slightly ambiguous terminology. Many circuits can be governed by a large set of coupled first- or second-order ODEs, which can then be reduced to a single high-order equation using standard methods (and vice versa). So the order of the equations of motion of the system is not as well-defined as you'd like. – Emilio Pisanty Jul 16 '17 at 13:01

Yes, you can form arbitrary orders of differential equations with electric circuits. These circuits are called filters, and we also talk about the order of the filter, which is equivalent to the order of the differential equation describing its response.

For each energy storage element in the circuit (inductor or capacitor) you'll add one order to the circuit, so long as the elements aren't connected in a way that produces a degeneracy (like two capacitors in parallel).

The simplest such circuit is an LC ladder: Practically, these type of circuits aren't used much above 3rd or 4th order, because it gets difficult to maintain a desired behavior while allowing for variations in the components' characteristics due to manufacturing differences and temperature.

For higher order filters, at least at modest frequencies, active filter techniques are used to provide better control over the filter behavior.

There are many such circuits and one group of such circuits are the basis of analogue computers.

Here is one such circuit which is designed to solve a second order differential equation as described here. Solving a linear third order differential equation is described here.

• This requires op-amps, though, and those can be nonlinear. Do they run in that regime in this kind of example? – Emilio Pisanty Jul 16 '17 at 12:53
• The op amps act like a "buffer" so that if you have a potential divider consisting of a capacitor and resistor (integrator) and the output is taken across the resistor there is no loading of the resistor ie the current through the resistor and the capacitor is the same. The difference in voltage between the non-inverting and inverting inputs multiplied by a very large number (amplification of op amp) gives the output voltage. – Farcher Jul 16 '17 at 12:55