How to find differential equation satisfied by \$u_{R_{equivalent}}\$, \$u_{L_{equivalent}}\$ or \$u_{C_{equivalent}}\$ in complicated RLC circuits?

Question

When i'm given simple RLC circuits i know how to find the expression for each voltage, for example the expression of \$u_R\$ the voltage between the borns of the resistor \$R\$, or the expression of \$u_C\$ the voltage between the borns of the capacitor \$C\$, or the expression of \$u_L\$ between the borns of the inductor \$L\$... it all starts by finding first the differential equation they satisfy usually using Kirchkov's law of currents. not only but also in complicated circuits like this one

http://sketchtoy.com/66481514 (someone please post image i can't do it (on ipad))

the basic idea is to find \$R_{equivalent}\$ and \$L_{equivalent}\$ and \$C_{equivalent}\$ and then to use same past technique, Kirchkov's law => differential equation => solution (expression desired). (what we would be searching then is an expression for the voltage between the borns of the equivalent resistor, and not each resistor)

but how to do when you have the resistors mixed up with capacitors in parallel and stuff like that ? (as below) how can we find the diff. equation satisfied by \$u_{R_{eq}}\$, \$u_{C_{eq}}\$, and \$u_{L_{eq}}\$, in that case?

http://sketchtoy.com/66481556

Edit: I'm not looking in particular for the solution of these differential equations, just to find the differential equation itself.

(sorry for writing and links to images, using ipad to write is difficult)

Answer 1

In linear circuits (basically R,L,C and controlled sources), you can analyze at each individual frequency. Each frequency is a pure sinusoid, and because the response of each element to a sinusoid is also a sinusoid (at the same frequency), but with a different amplitude and/or phase, it is possible to use complex numbers to represent the behavior. Thus a capacitor's impedance becomes 1/(j.2.pi.f.C) where j = sqrt(-1). When an input signal is repetitive, it is possible to break this down to a collection of sinusoids, analyze and then combine the results from each frequency.

This analysis can be performed symbolically, or numerically.

In non-linear circuits, this method does not apply, and in general it is not possible to solve the differential equations directly. Numerical approximation methods are used to solve them in simulators such as SPICE.