Here we analyze the time response of the circuit in Figure 3 which contains capacitive and resistive elements and a voltage source, VS. The voltage source is switched between two nodes in the circuit, and we observe the voltage response across C1. All answers in this problem are looking for expressions in terms of VS, C1, C2, R1, and R2.
Initially for t<0 the switch connects the voltage source, VS to node 1 and the circuit is allowed to reach steady-state. What is the steady-state voltage, v(t), across C1 for t<0?
At t=0, the switch is repositioned to connect the voltage source to node 2 and the circuit is allowed to reach steady-state again. What is the voltage, v(t), across C1 after the circuit reaches steady-state?
We can use differential equations to write an expression for the voltage, v(t), across C1 for t>0. The homogeneous and particular solutions can be used to express the voltage, v(t)=H∗e−t/τ+P. Give expressions for H, P and τ.
τ
P
H
To solve the problem, we will break it down step by step.
1. Steady-state voltage across C1 for t < 0:
When t < 0, the switch connects the voltage source, VS, to node 1. At steady state, the capacitor behaves as an open circuit, and the voltage across C1 is determined solely by the voltage source.
Therefore, the steady-state voltage, v(t) across C1 for t < 0 is simply equal to VS.
2. Steady-state voltage across C1 after the switch repositioned:
At t = 0, the switch is repositioned to connect the voltage source to node 2. In steady state, the capacitor charges or discharges to a voltage determined by the circuit parameters.
To find the steady-state voltage across C1 after the switch is repositioned, we need to calculate the voltage using the voltage divider rule.
Let VR1 be the voltage across R1 and VR2 be the voltage across R2. Then, we can write the following equations:
VR1 = VS * (R2 / (R1 + R2))
VR2 = VS * (R1 / (R1 + R2))
Since the voltage across C1 is the difference between VR1 and VR2, we have:
v(t) = VR1 - VR2
= VS * (R2 / (R1 + R2)) - VS * (R1 / (R1 + R2))
= VS * (R2 - R1) / (R1 + R2)
Therefore, the voltage, v(t), across C1 after the circuit reaches steady-state is given by:
v(t) = VS * (R2 - R1) / (R1 + R2)
3. Expressions for H, P, and τ in the differential equation solution:
To find the expression for the voltage, v(t), across C1 for t > 0, we need to write the differential equation governing the circuit behavior.
The differential equation for a series RC circuit is given by:
v'(t) + v(t) / (R1 * C1) = VS / (R1 * C1)
The homogeneous solution describes the response when the driving force (VS) is absent. It is given by:
v_h(t) = H * e^(-t/τ)
The particular solution describes the transient response due to the driving force. Here, the driving force is a constant voltage source, VS, so the particular solution is constant:
v_p(t) = P
The total response is the sum of the homogeneous and particular solutions:
v(t) = v_h(t) + v_p(t)
= H * e^(-t/τ) + P
Therefore, in terms of the circuit parameters, the expressions for τ, P, and H are as follows:
τ = R1 * C1
P = VS * (R2 - R1) / (R1 + R2)
H - This expression is not provided in the given information. It could be a constant determined by initial conditions or can be calculated if further context is provided.
To find the steady-state voltage across C1 for t<0, we need to consider that the circuit has reached steady-state. Since the switch connects the voltage source, VS, to node 1, the voltage across C1 will be equal to the voltage of the voltage source, VS.
Therefore, the steady-state voltage across C1 for t<0 is:
v(t) = VS
After the switch is repositioned at t=0, connecting the voltage source to node 2, the circuit again reaches a new steady-state. In this new steady-state, there will be no initial voltage across C1.
Therefore, the voltage across C1 after the circuit reaches steady-state at t=0 is:
v(t) = 0
To find the expression for the voltage, v(t), across C1 for t>0, we can use differential equations. The differential equation for the circuit can be written as:
C1 * dv(t)/dt + (v(t) - VS) / R1 = 0
This is a first-order linear ordinary differential equation. The solution to this type of equation consists of a homogeneous solution and a particular solution.
The homogeneous solution is given by:
v_h(t) = H * e^(-t/τ)
where H is the constant determined by the initial conditions, and τ is the time constant of the circuit given by τ = C1 * R1.
The particular solution, P, is a constant that satisfies the differential equation. To find P, we substitute the particular solution into the differential equation:
C1 * dP/dt + (P - VS) / R1 = 0
Solving this equation for P, we find:
P = VS
Therefore, P = VS.
The complete expression for the voltage, v(t), across C1 for t>0 is:
v(t) = H * e^(-t/τ) + VS
where τ = C1 * R1, P = VS, and H is determined by the initial conditions.