Need some help on Circuit modeling(1 sample)

1. Feb 3, 2012

genxium

I'm learning circuit modeling recently, and got stuck by a simply serial circuit sample(shown in the attachment), here're equations I wrote for the sample, but I have no idea what initial conditions and algorithm I can use to solve it, could any one give me a hand or some tips?

$$i_{21}+{\int_0}^t \frac{V_3-V_1}{L} d\tau =0$$
$$i_{12}+\frac{GND-V_2}{R} =0$$
$${\int_0}^t \frac{V_1-V_3}{L} d\tau + C \cdot (\frac{dGND}{dt}-\frac{dV_3}{dt}) =0$$
$$C \cdot (\frac{dV_3}{dt}) + \frac{V_2-GND}{R}=0$$

where GND=0 V is constant, I see that $i_{12}=-i_{21}$ can be used to reduce the equations, but then the remaining equations are 2nd order diff equations, how do computers solve this?

$${\int_0}^t \frac{V_2-V_1}{L} d \tau = \frac{-V_2}{R}$$
$${\int_0}^t \frac{V_1-V_3}{L} d\tau + C \cdot -\frac{dV_3}{dt} =0$$
$$C \cdot (\frac{dV_3}{dt}) + \frac{V_2}{R}=0$$

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Last edited: Feb 3, 2012
2. Feb 4, 2012

genxium

I just read some tutorial of numerical solutions, but the easily found materials are all about ordinary diff equations of only 1 unknown function, could any one tell me some tutorial that shows the idea to multiple unknown functions' diff equation(numerical solution)?

3. Feb 4, 2012

4. Feb 4, 2012

genxium

Uhm..... I don't think this is the main issue here, take derivatives of these diff equations, then

$$\frac{V_2-V_1}{L}=-\frac{dV_2}{R \cdot dt}$$

$$\frac{V_1-V_3}{L}+C \cdot -\frac{d^2V_3}{dt^2}=0$$

$$C \cdot \frac{dV_3}{dt}+\frac{V_2}{R}=0$$

and wanna get numerical solutions for $$V_1(t),V_2(t),V_3(t)$$.

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