Solving Critical Damping Circuit: Find R, i, di/dt, v_C(t)

In summary, the circuit shown has a resistor adjusted for critical damping, with an initial capacitor voltage of 15 V and an initial inductor current of 6 mA. The numerical value of R is 1250 Ω. Immediately after the switch is closed, the current is 6.00 mA due to the continuity principle. To find di/dt immediately after the switch is closed, the general solution was used to get i(t) = 0.006 + K_2*t. However, without any other boundary conditions, it is not possible to solve for K_2. Similarly, for v_C(t) for t≥0, using the given boundary conditions and the homogeneous equation, the equation v_C(t) =
  • #1
zealeth
25
0

Homework Statement



In the circuit in the following figure, the resistor is adjusted for critical damping. The initial capacitor voltage is 15 V, and the initial inductor current is 6 mA.

Figure_P08.44.jpg


Find the numerical value of R.

Find the numerical values of i immediately after the switch is closed.

Find the numerical values of di/dt immediately after the switch is closed.

Find v_C(t) for t≥0.

Homework Equations



x_C(t) = K_1 e^(s_1*t)+K_2 * t * e^(s_1*t)
(General solution to 2nd order differential equation)
damping ratio = [itex]\zeta[/itex] = [itex]\alpha[/itex]/[itex]\omega[/itex]_0
[itex]\alpha[/itex] = R/(2L), [itex]\omega[/itex]_0 = 1/[itex]\sqrt{LC}[/itex]
s_1,2 = -[itex]\alpha[/itex] +/- ([itex]\alpha[/itex]2-[itex]\omega[/itex]_02)^(1/2)
KVL

The Attempt at a Solution



Find the numerical value of R.

Critical damping, so zeta = 1. Using the equation for damping ratio, I solved for R to be 1250 Ω (correct answer).

Find the numerical values of i immediately after the switch is closed.

Once the switch is closed, I found i to be 6.00 mA due to the continuity principle i_C(0-) = i_C(0+) (correct answer).

Find the numerical values of di/dt immediately after the switch is closed.

Here is where I'm having trouble. I started by using the general solution and evaluating at t=0 to find K_1 = 6 mA.

s_1 = 0, s_2 = -10000

So I now have:

i(t) = 0.006 + K_2 * t * e^(-10000*t)

EDIT: Just realized it's s_1 in the exponent of both terms, however I'm still not getting the correct answer. Is it possible I have the values for s_1 and s_2 mixed up? Assuming I have them right, the equation should be:

i(t) = 0.006 + K_2*t

Obviously I need to solve for K_2 here to be able to differentiate the equation and find di(0)/dt, but I'm not sure how I would go about doing that. I can't plug in t=0 because that would remove K_2 from the equation, and I don't know any other boundary conditions that I could make use of.

Find v_C(t) for t≥0. **where t is in milliseconds**

Currently working on this using a similar approach to above.

EDIT: Not getting this one either. I started with v_C(0) = 15V, which was given in the problem. I used that boundary condition in the homogeneous equation to solve for K1 = 15.

V_C(t) = 15*e^(0t) + K_2*t*e^(0t)

Now to solve for K_2, I needed dv_C(0)/dt which I calculated using i_C(0) = C * dv_C(0)/dt and got 18750. Differentiating the homogeneous equation and evaluating at t=0, I got:

0*K_1+K_2=18750

Therefore K_2 should = 18750. I'm not sure if I need to convert the units since t is in milliseconds, but the equation I got was:

15+18750*t, which is obviously incorrect.
 
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  • #2
Hi zealeth. So the circuit you've shown is not representative of the circuit in use here, because what you show does not indicate how the inductor can have an initial non-zero current?

Do you happen to know the correct answer for ##di/dt## at ##t=0^+## ?
 
  • #3
The circuit I've shown is what was given with the problem statement. And no, I do not know what di/dt is at t=0+.
 

FAQ: Solving Critical Damping Circuit: Find R, i, di/dt, v_C(t)

1. How do you calculate the value of R in a critical damping circuit?

To calculate the value of R in a critical damping circuit, you can use the formula R = 2 * sqrt(L/C), where L is the inductance and C is the capacitance of the circuit. This value of R will ensure that the circuit is critically damped, meaning that it will return to its equilibrium state without any oscillations.

2. What is the equation for current (i) in a critical damping circuit?

The equation for current (i) in a critical damping circuit is i = i_0 * e^(-Rt/2L), where i_0 is the initial current and t is the time. This equation shows that the current decreases exponentially over time due to the resistance in the circuit.

3. How do you find the rate of change of current (di/dt) in a critical damping circuit?

The rate of change of current (di/dt) in a critical damping circuit can be found by taking the derivative of the current equation, which is di/dt = -i_0 * R/2L * e^(-Rt/2L). This shows that the rate of change of current is also decreasing exponentially over time.

4. What is the formula for voltage (v_C(t)) in a critical damping circuit?

The formula for voltage (v_C(t)) in a critical damping circuit is v_C(t) = v_0 * (1 - e^(-Rt/L)), where v_0 is the initial voltage and t is the time. This equation shows that the voltage also decreases exponentially over time, but with a slower rate compared to the current.

5. How does a critical damping circuit differ from an over-damped or under-damped circuit?

A critical damping circuit is designed to have a damping factor that is equal to the critical damping ratio, which means it will return to equilibrium without any oscillations. In contrast, an over-damped circuit has a damping factor that is greater than the critical damping ratio, causing it to return to equilibrium slowly without any oscillations. An under-damped circuit, on the other hand, has a damping factor that is less than the critical damping ratio, resulting in oscillations before returning to equilibrium.

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