# Underdamped Parallel RLC Circuit

• Engineering
• Jayalk97
In summary, the conversation is about a person having trouble solving for initial value constants in a circuit analysis problem. They used KCL to create an equation and found the type of response to be under-damped. The equation obtained was v(t) = [A_1 cos(600t)+A_2 sin(600t)]e^(-800t), and the person is unsure of how to solve for A_1 and A_2. They are advised to write two equations from the initial conditions of v(0) = 0 and current through the inductor being 0.5A, and to consider evaluating the circuit in the frequency domain for easier work.
Jayalk97
Please post this type of questions in HW section using the template.
Hey guys, so I'm having trouble with this circuit analysis question. I need to find voltage across the capacitor as a function of time. I nearly finished analyzing but I'm having trouble solving for the initial value constants in the voltage equation. So first I used KCL to create the equation, then created a characteristic equation to find the type of response, which ended up being under-damped. I'm not trying to get any help with the acgtual numbers, but I end up with the equation v(t) = [A_1 cos(600t)+A_2 sin(600t)]e^(-800t). I'm fairly sure I'm correct so far, but how would I go about solving for A_1 and A_2? Thanks in advance!

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Jayalk97 said:
Hey guys, so I'm having trouble with this circuit analysis question. I need to find voltage across the capacitor as a function of time. I nearly finished analyzing but I'm having trouble solving for the initial value constants in the voltage equation. So first I used KCL to create the equation, then created a characteristic equation to find the type of response, which ended up being under-damped. I'm not trying to get any help with the acgtual numbers, but I end up with the equation v(t) = [A_1 cos(600t)+A_2 sin(600t)]e^(-800t). I'm fairly sure I'm correct so far, but how would I go about solving for A_1 and A_2? Thanks in advance!
View attachment 214062
You should be able to write 2 equations from the initial conditions that will help you solve for those 2 constants.

What is v(0)? What other initial equation can you write for t=0+ ?

berkeman said:
You should be able to write 2 equations from the initial conditions that will help you solve for those 2 constants.

What is v(0)? What other initial equation can you write for t=0+ ?

V(0) = 0V right? and Current across the inductor is 0.5A. I understand that I am supposed to use those to find the initial conditions, but how are those related to the equation I obtained?

Jayalk97 said:
Current across the inductor is 0.5A
current *through the inductor is 0.5A. I know it sounds like a trivial difference, but it is not. current goes through a component. Voltage goes across a component.
Jayalk97 said:
I understand that I am supposed to use those to find the initial conditions, but how are those related to the equation I obtained?
Well V(0)=0 is the initial condition. I already know the form of your equation in your first post is not correct, because at t=0+ it does not match expectations. Do you know how to evaluate circuits in the frequency domain, it would make the work easier. If not, why don't you post your KCL/kvl equations.

berkeman

## 1. What is an underdamped parallel RLC circuit?

An underdamped parallel RLC circuit is an electrical circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in parallel. It is called underdamped because the system is characterized by a low damping ratio, meaning that the oscillations in the circuit can take a long time to dissipate.

## 2. How does an underdamped parallel RLC circuit behave?

The behavior of an underdamped parallel RLC circuit is characterized by oscillations. When the circuit is initially disturbed, the energy stored in the inductor and capacitor will cause the current to flow back and forth between the two at a specific frequency determined by the values of R, L, and C. These oscillations will gradually decrease over time due to energy dissipation in the resistor.

## 3. What is the resonant frequency of an underdamped parallel RLC circuit?

The resonant frequency of an underdamped parallel RLC circuit is the frequency at which the circuit will oscillate with the maximum amplitude. It is given by the formula: fr = 1 / (2π√(LC)), where L is the inductance and C is the capacitance of the circuit.

## 4. What factors affect the behavior of an underdamped parallel RLC circuit?

The behavior of an underdamped parallel RLC circuit is affected by the values of the resistor, inductor, and capacitor, as well as the voltage and current sources connected to the circuit. The resistance, inductance, and capacitance determine the natural frequency of the circuit, while the voltage and current sources determine the amplitude and initial conditions of the oscillations.

## 5. How is an underdamped parallel RLC circuit used in practical applications?

An underdamped parallel RLC circuit has various practical applications, including in radio and communication systems, electronic filters, and audio equipment. It is also used in medical devices such as MRI machines and in power systems for energy storage and stabilization.

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