Finding B1 and B2 coefficients in underdamped RLC step response

[mushiman]

I've been working on this problem to review (see the attachments), and my circuit analysis skills are quite rusty. I've determined that the circuit should be underdamped, and it will have a step response. Based on this, I'm using the equation V0(t)=Vf+B1e^(αt)cos(ωdt)+B2e^(ω0t)sin(ωdt). As you can see from the third attachment, I'm not sure as to what the values of B1 and B2 should be. Vf should be 20v as I interpreted, correct?

Also, it would seem that 50mA source would loop between the resistor while the switch is open (since the inductor would effectively be a short and the capacitor would be open), and when the switch is closed, the current would pass through the switch and loop in that manner, correct?

As I stated earlier, assuming that all of my former work is correct, I'm not sure what values I should assign to B1 and B2. A push in the right direction from here would be great -- I have terrible handwriting but hopefully it is clear enough to read.

 

[SGT]

I've been working on this problem to review (see the attachments), and my circuit analysis skills are quite rusty. I've determined that the circuit should be underdamped, and it will have a step response. Based on this, I'm using the equation V0(t)=Vf+B1e^(αt)cos(ωdt)+B2e^(ω0t)sin(ωdt). As you can see from the third attachment, I'm not sure as to what the values of B1 and B2 should be. Vf should be 20v as I interpreted, correct?

Also, it would seem that 50mA source would loop between the resistor while the switch is open (since the inductor would effectively be a short and the capacitor would be open), and when the switch is closed, the current would pass through the switch and loop in that manner, correct?

As I stated earlier, assuming that all of my former work is correct, I'm not sure what values I should assign to B1 and B2. A push in the right direction from here would be great -- I have terrible handwriting but hopefully it is clear enough to read.

You mean $$V_0(t) = V_f + B_1e^{\alpha t}cos(\omega_dt) + B_2e^{\alpha t}sin(\omega_dt)$$ with $$\alpha$$ in both exponents.
The integration constants $$B_1$$ and $$B_2$$ are determined from the initial conditions $$V_o(0) = V_c(0)$$ and $$\frac{dV_o}{dt}(0)$$. Those initial conditions can be determined from your diagram for t<0.

 

[mushiman]

Right (forgot about special characters on this forum).

The problem is that I'm not exactly sure what $$V_o(0)$$ and $$\frac{dV_o}{dt}(0)$$ are.

Isn't it true that the 50mA source will loop through the resistor and then through the switch when it is closed? If so, then I believe my work up till the portion where $$B_1$$ and $$B_2$$ must be determined is correct. This is where I was hoping for a hint -- I'm not exactly sure what either of them ($$V_o(0)$$ and $$\frac{dV_o}{dt}(0)$$) should be interpreted as.

 

[SGT]

Right (forgot about special characters on this forum).

The problem is that I'm not exactly sure what $$V_o(0)$$ and $$\frac{dV_o}{dt}(0)$$ are.

Isn't it true that the 50mA source will loop through the resistor and then through the switch when it is closed? If so, then I believe my work up till the portion where $$B_1$$ and $$B_2$$ must be determined is correct. This is where I was hoping for a hint -- I'm not exactly sure what either of them ($$V_o(0)$$ and $$\frac{dV_o}{dt}(0)$$) should be interpreted as.

$$V_o(0)$$ is the voltage in the capacitor at the closing of the switch. It is the difference between the voltages on the resistors.
$$\frac{dV_o}{dt}(0)$$ is the derivative of the voltage in the capacitor at the same instant.
Remember that $$i_c(t)=C\frac{dV_C}{dt}$$

 

[mushiman]

Great, that's what I was thinking it would be.

Thanks for the help.