Complex RLC Circuit Problem (System of diff eqs)

• milkism
milkism
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:

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!

Last edited by a moderator:
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.

I am curious, what python packages were you using?

Last edited:
SammyS

What is a Complex RLC Circuit?

A complex RLC circuit consists of resistors (R), inductors (L), and capacitors (C) arranged in a network. These components can be connected in series or parallel, and the circuit can be driven by alternating current (AC) or direct current (DC) sources. The complexity arises from the interactions between the components, which can lead to differential equations that describe the circuit's behavior over time.

How do you derive the differential equations for a complex RLC circuit?

To derive the differential equations for a complex RLC circuit, you use Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL). KVL states that the sum of voltages around any closed loop in a circuit must be zero, while KCL states that the sum of currents entering a junction must equal the sum of currents leaving the junction. By applying these laws, you can write equations for the voltages and currents in the circuit, which often result in a system of differential equations.

What methods can be used to solve the system of differential equations in an RLC circuit?

Several methods can be used to solve the system of differential equations in an RLC circuit, including:1. Analytical methods such as the Laplace transform, which converts differential equations into algebraic equations.2. Numerical methods like the Runge-Kutta method, which approximate solutions at discrete points.3. Matrix methods, where the system of equations is written in matrix form and solved using linear algebra techniques.

What is the significance of the natural frequency and damping factor in an RLC circuit?

The natural frequency and damping factor are critical parameters that describe the behavior of an RLC circuit. The natural frequency is the frequency at which the circuit would oscillate if there were no damping (resistance). The damping factor indicates how quickly the oscillations decay over time due to the resistance in the circuit. Together, these parameters help determine whether the circuit is underdamped, overdamped, or critically damped, which affects the transient response of the circuit.

How do initial conditions affect the solution of the differential equations in an RLC circuit?

Initial conditions, such as the initial voltage across the capacitor and the initial current through the inductor, are essential for solving the differential equations in an RLC circuit. These conditions determine the specific solution to the differential equations, as they provide the necessary constants for the general solution. Without initial conditions, you would only have a family of possible solutions rather than a unique solution that describes the actual behavior of the circuit.

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