Finding the voltage of a capacitor

Question

I was given this problem by my prof and his answer is simply that it is a parallel RLC circuit and he uses $$s^2 +\frac{1}{RC}s+\frac{1}{LC} = 0$$ for the characteristic equation. It seems to me that it is invalid and when I try to find the solution using differential equations and either nodal analysis or mesh I can't seem to arrive at this solution.

Am I wrong on this "standard" characteristic equation is only valid for a circuit where the three elements are parallel and in this case the inductor and resistor are in series and thus it cannot be used?

Answer 1

I have a little different opinion from your prof. The characteristic equation of an RLC circuit will be either of the two given below. $$s^2 + \frac{R}{L}s +\frac{1}{LC} = 0\tag1$$ $$s^2 + \frac{1}{RC}s +\frac{1}{LC} = 0\tag2$$

Reason:
The characteristic equation of a second order system is $$s^2 + 2\alpha s + w_0^2 = 0$$

Where \$\alpha\$ is a damping related parameter and \$w_0\$ is the natural undamped frequency.

For an RLC circuit, \$w_0\$ will depend only on L and C and it will be \$\frac{1}{\sqrt{LC}}\$.

Coming to the dimension of the characteristic equation, each term in it should have a dimension of \$\mathrm{radian^2}\$. So \$2\alpha\$ must have dimension of \$\mathrm{radian}\$. The only possible options are \$\frac{1}{RC} \$ and \$\frac{R}{L}\$.

Which equation applies where?

Equation (1) applies to RLC circuits in which two more components are in series. The two possible connection are:

The R L and C can take any of these positions Z1, Z2 and Z3, but only one position at a time.

Equation (2) applies to RLC circuits in which two more components are in parallel. The two possible connection are:

The verification of this result is left to the reader. :-)

In the circuit given in the question, two components R and L are in series and hence the corresponding characteristic equation will be (1).