# RLC Circuit Differential Equation Help!!!

#### RazedX

Joined Oct 21, 2012
2
In the circuit system shown below, the voltage source f(t) acts as the input to the system. The voltage across capacitor C1 is the measured system output y(t). Derive the differential equation to describe this system. This equation should be in terms of R, C1, C2, L1, and L2 and include y(t) and f(t) (or their derivatives, if necessary).

I always get thrown off by the extra inductors/capacitors. :/ Help please!

#### blah2222

Joined May 3, 2010
582
In the circuit system shown below, the voltage source f(t) acts as the input to the system. The voltage across capacitor C1 is the measured system output y(t). Derive the differential equation to describe this system. This equation should be in terms of R, C1, C2, L1, and L2 and include y(t) and f(t) (or their derivatives, if necessary).

I always get thrown off by the extra inductors/capacitors. :/ Help please!
This circuit condenses into two meshes as the C1 and C2 are in parallel so the analysis becomes less involved. Being careful with signs and integration/differentiation I ended up with this DE:

$$L_{1}(C_{1} + C_{2})\frac{d^{3}y}{dt^{3}} + R(C_{1} + C_{2})\frac{d^{2}y}{dt^{2}} + (\frac{L_{1}}{L_{2}} + 1)\frac{dy}{dt} + \frac{R}{L_{2}}y = \frac{df}{dt}$$

It should be third order because there are really only three distinct reactive components acting on this circuit as C1 and C2 can be thought of as a combined capacitor.

Work through it and see if you can get to something like this.

#### RazedX

Joined Oct 21, 2012
2
I got something similar after I combined the capacitors but I was wondering if it was also possible to combine the inductors so that it becomes a 3 component system with r, ceq, leq as opposed to having ceq, l1 l2, r. the third derivative is really throwing off my matlab programming :/

This circuit condenses into two meshes as the C1 and C2 are in parallel so the analysis becomes less involved. Being careful with signs and integration/differentiation I ended up with this DE:

$$L_{1}(C_{1} + C_{2})\frac{d^{3}y}{dt^{3}} + R(C_{1} + C_{2})\frac{d^{2}y}{dt^{2}} + (\frac{L_{1}}{L_{2}} + 1)\frac{dy}{dt} + \frac{R}{L_{2}}y = \frac{df}{dt}$$

It should be third order because there are really only three distinct reactive components acting on this circuit as C1 and C2 can be thought of as a combined capacitor.

Work through it and see if you can get to something like this.

#### blah2222

Joined May 3, 2010
582
I got something similar after I combined the capacitors but I was wondering if it was also possible to combine the inductors so that it becomes a 3 component system with r, ceq, leq as opposed to having ceq, l1 l2, r. the third derivative is really throwing off my matlab programming :/
Nope, not for this circuit as it's laid out.