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

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

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)

• There are very general algorithms for this. One of the papers on the topic is "The modified nodal approach to network analysis" by Chung-Wen Ho ; Ruehli, Albert E. ; Brennan, Pierce A.. This modified nodal analysis algorithms and its improved versions are being used by software like SPICE that can simulate such circuits. In general one will not try to extract the differential equations, anyway, because they can only be solved numerically but the software will set up a set of numerical equations right away. LTSpice is a very useful free tool: linear.com/designtools/software/#LTspice
– CuriousOne
Jan 17, 2016 at 22:21
• @CuriousOne very interesting, but can we at least extract the differential equation? (without solving it) out of interest
– user153330
Jan 17, 2016 at 22:23
• why the downvote?
– user153330
Jan 17, 2016 at 22:23
• Yes, there are tools for that, too. Look for "symbolic circuit analysis". One online tool that gives you the symbolic equations for linear circuits in matrix form is here analog-electronics.eu/slicap/slicap.html. Another one is thishttp://www.ewh.ieee.org/soc/es/May2001/12/Begin.htm. I am sure there are more.
– CuriousOne
Jan 18, 2016 at 0:23

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.

• yes, but how to get the differential equation in the first place? [ sketchtoy.com/66481556 as an example?] (without having to solve it right) Jan 26, 2016 at 20:54
• in nodal analysis, use V=I.R for resistors, V=L.di/dt and V=V0+integral(I/C) Jan 27, 2016 at 3:05
• but how to find $R_{equivalent}$ and $L_{equivalent}$ ...etc? could you please apply that to the example i gave that would be extremely helpful Jan 27, 2016 at 12:58
• The 'r_equivalent' etc for a capacitor is 1/(j.w.C), and j.w.L for an inductor. However, then you don't use differential equations to solve -- you use standard KVL or KCL analysis. Jan 28, 2016 at 2:29