# Governing Equation for an electrical system

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1. Oct 30, 2014

### mintsnapple

1. The problem statement, all variables and given/known data

2. Relevant equations
Kirchoff's Voltage/Current Law

3. The attempt at a solution
I first started by summing up the currents at node 1, which is the intersection of 3 wires at the top, and node 2, which is between the capacitor and inductor.
So, at node 1: $$\sum i = -i(t) + \frac{1}{R}V_1 + C\frac{d}{dt}(V_1 - V_2) = 0$$
At node 2: $$\sum i = -C\frac{d}{dt}(V_2 - V_1) + \frac{1}{L}\int_{-\infty}^t V_2 dt = 0$$
I also noted that $$V_2 = V_L$$

So I have these two equations, but I am not sure how to easily get rid of the V_1 dependence. It's not as simple as just solving for V_1, is it?

2. Oct 31, 2014

### Staff: Mentor

Best not introduce more variables than is necessary. I think you'll find that your node 2 does not qualify for the exhaulted designation "node".

Express the voltage acoss the C + L combination in terms of their current, i2(t)

Express the voltage across R in terms of its current, i(t) - i2(t)

Equate the two expressions, and rearrange to give i2(t) in terms of i(t).

Good luck!

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