Complex RLC Circuit Problem (System of diff eqs)

AI Thread Summary
The discussion revolves around solving a complex RLC circuit problem involving multiple currents and differential equations. Initial current values were calculated, assuming the capacitor acts as a perfect conductor, leading to final values for the currents. Participants suggest using Laplace transforms to simplify the process of solving the differential equations, as they involve numerous time-dependent variables. There is also a query about the capabilities of Python packages for solving these equations, whether numerically or analytically. The conversation emphasizes the need for additional differential equations to fully determine the six currents in the circuit.
milkism
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Homework Statement
I need to find the six currents I_i(t) via a system of diff eqs.
Relevant Equations
V=IR, Q=CV, etc
The following circuit is:
1701517638743.png

Going clockwise the current ##I_1## goes through resistor ##R_1## and voltage ##V_11##. Current ##I_2## goes through capacitator ##C_1## and ##R_2##.
Current ##I_3## goes through resistors ##R_3## and ##R_4##. Current ##I_4## goes through resistor ##R_5##, but from left to right.
Current ##I_5## goes through the inductor ##L_1## from right to left. And current ##I_6## goes through the voltage ##V_2## from under to above.
I have found the initial values of the currents, when the capacitator acts like a perfect a conductor.
$$I_1 = 0.57$$, $$I_2 = 1.20$$, $$I_3 = 0.57$$, $$I_4 = -0.63$$, $$I_5 = 0$$ and $$I_6 = 0.63$$.
I have gotten these results through these equations:
$$I_2=I_1+ I_6$$, $$V_1 = I_1 ( R_1 + R_3 + R_4) + I_2 * R_2$$, $$V_2 = I_2 * R_2 + I_6 * R_5$$, $$I_1 = I_3$$, $$I_5 = 0$$, $$I_6 = -I_4$$.
For final I have:
$$I_1 = 0.35$$, $$I_2 = 0$$, $$I_3 = 0$$, $$I_4 = 0$$, $$I_5 = 0.35$$ and $$I_6 = -0.35$$.
I have gotten these results through these equations:
$$I_5 + I_3 = I_1$$, $$-I_5 -I_6 - I_4 = 0$$, $$V_1 - V_2 = I_1 * R_1$$, $$V_1 - V_2 = I_1 * R_1 + I_4 * (R_3 + R_4 + R_2)$$, $$I_2 = 0$, $I_4 = I_3$$.
Now the question is, to find the currents at any time. For the top-left branch I got differential equation:
$$V_1 - V_2 = I_1 * R_1 + L \frac{dI_5}{dt}$$ or $$V_1 - V_2 = \frac{dQ_1}{dt} * R_1 + L \frac{d^2Q_5}{d^2t}$$
For top-right branch I got:
$$V_2 = \frac{1}{C} \int I_2 dt + I_2 * R_2 - I_4 * R_5$$ or $$V_2 = \frac{Q_2}{C} + \frac{dQ_2}{dt} R_2 - \frac{dQ_4}{dt} * R_5$$
For the rectangular branch below I got (going clockwise)
$$0=- L \frac{dI_5}{dt} + I_4 * R_5 + I_3 * (R_3 + R_4)$$ or $$0=- L \frac{d^2 Q_5}{d^2t} + \frac{dQ_4}{dt} * R_5 + \frac{dQ_3}{dt} * (R_3 + R_4)$$
I can make an another loop for the outer branch, but you can see each differential equation depends on too many time-dependant variables. So I really have no idea to do!
 
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Have you tried using Laplace Transforms? Are they saying that both voltage sources switch on at time zero?
 
scottdave said:
Have you tried using Laplace Transforms? Are they saying that both voltage sources switch on at time zero?
I can solve the differential equations with python. And yes both voltages switch on at time zero.
 
What do you need help with?

Does the python package solve it numerically or analytically?
 
Last edited:
scottdave said:
What do you need help with?

Does the python packages solve it numerically or analytically?
I need help to find enough differential equations to solve for the 6 currents.
 
Was this helpful @milkism ? Try making an eqauation for the bottom loop.

Which node equations are helpful?

I am curious, what python packages were you using?
 
Last edited:

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