# Solve Coupled RC Circuit: Find V1(t) & V2(t)

• Engineering
• no_alone
In summary, the voltage on the resistors R1 and R2 develops with time according to the following equation: VR1 = VC1, VR2 = VC2, and IR1 + IC1 + VRC = Iinj.
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## Homework Statement

Hello I want to solve the following circuit, I want to know how the voltage on R1 and R2 develop with time,
It start at t=0 when there is no voltage in the circuit ( all V=0). Its a symbolic question, I need to show the equation.
V1(t) = ? ,
V2(t) = ?
The current is a DC

## The Attempt at a Solution

Well, noting much.. I just wrote it in a better way:
http://i62.tinypic.com/5w9gxz.pngVR1 = VC1
VR1 + VRc + VC2 = 0
VR2 = VC2
IR1 + IC1 + VRC = Iinj
IRc = IR2 + VC2
$$V_1 = V_{C1} = V_{R1} \\ V_2 = V_{R2} = V_{C2}$$

$$\frac{V_1}{R_1} + \frac{dV_1}{dt}*C_1 + \frac{V_{Rc}}{R_c} = I_{inj} \\ \frac{V_{Rc}}{R_c} = \frac{V_2}{R_2} + C_2*\frac{dV_2}{dt}$$-->

$$\frac{dV_1}{dt}*C_1 = I_{inj} -\frac{V_{Rc}}{R_c} - \frac{V_1}{R_1} \\ \frac{dV_2}{dt}*C_2 = \frac{V_2}{R_2} -\frac{V_{Rc}}{R_c}$$

Last edited:
Do you know how to do nodal analysis?

evaluate the circuit in the frequency domain using nodal analysis.

In case you don't know nodal analysis, simply write a KCL at node V1 and V2 and solve.

You haven't defined the question sufficiently. Do you have component values or is this strictly a symbolic problem? What are the properties of the current source? Is it a DC source or an AC source? Any initial conditions?

Hi @gneill and @donpacino thank you for the replay, I added some info about the question.
I also tried to make some progress.

I would start by defining the node in the upper left as V1, the node in the upper right as V2, and the bottom node as ground. so VRc=V1-V2, V1=VR1=VC1, etc
Do a laplace transform (will allow you to solve the system with algebra) to get to the frequency domain.

write two kcls at the upper left and right nodes using V1,V2,Iinj,R1,R2,Rc,C1,C2
then do some algebra and do an inverse laplace transform to get to your time domain solution.

alternatively you can solve the second order differential equation.

just a note: I am operating under the assumption that at t=0 the current source changes values. If it does not then you can disregard the capacitors.

thank you donpacino but I do not know how to do laplace transform...

ok, then do what i told you, but solve the differential equation in the time domain. you will still do a KCL, but in the time domain similar to what you did above

## 1. What is a coupled RC circuit?

A coupled RC circuit is a circuit that contains two or more resistors and capacitors that are connected to each other. The components are connected in such a way that the voltage across one component affects the voltage across the other component.

## 2. How is V1(t) and V2(t) related in a coupled RC circuit?

In a coupled RC circuit, V1(t) and V2(t) are related through the coupling coefficient, k. The voltage across each component can be calculated using the following equations:

V1(t) = V0 * (1 - e^(-t/RC))

V2(t) = V0 * (1 - k * e^(-t/RC))

## 3. How do you solve a coupled RC circuit?

To solve a coupled RC circuit, you need to first write down the Kirchhoff's laws equations for the circuit. Then, you can use the equations to solve for the unknown variables, such as V1(t) and V2(t). You may need to use algebraic manipulation and calculus to solve the equations.

## 4. What is the significance of the time constant in a coupled RC circuit?

The time constant in a coupled RC circuit is the amount of time it takes for the voltage across the capacitor to reach 63.2% of its final value. It is calculated using the equation τ = RC, where R is the resistance and C is the capacitance. The time constant is important because it determines the rate at which the voltage across the capacitor changes.

## 5. How does the coupling coefficient affect the behavior of a coupled RC circuit?

The coupling coefficient, k, affects the behavior of a coupled RC circuit by determining the amount of voltage that is transferred from one component to another. If k is close to 1, the circuit is said to be strongly coupled, and there is a significant transfer of voltage between the components. If k is close to 0, the circuit is said to be weakly coupled, and there is minimal transfer of voltage between the components.

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