Confusion about Laplace and frequency response of a electrical circuit

Question

I have a confusion between the Laplace transform of a circuit and the frequency domain AC analysis. Suppose we have a series RLC resonant circuit connected to an ideal sinusoidal voltage source.

We can write $$V_m \sin(\omega t)=i R + \frac{1}{C} \int i dt + L \frac{d i}{dt} $$

now if we take the Laplace transform (assume initial inductor current to be zero),

$$\frac{V_m \omega }{s^2+\omega^2} =I(s) \big(R + \frac{1}{s C} + s L \big) \Rightarrow I(s) = \frac{V_m \omega }{(s^2+\omega^2) \big(R + \frac{1}{s C} + s L \big)} \tag{A}$$

On the other hand, using frequency domain analysis we can derive current as, $$ I= \frac{V_m }{R +j \omega L + \frac{1}{j\omega C}} \tag{B}$$

My question is: How can we deduce (B) directly from (A) without inverse Laplace transform? just putting \$s=j \omega\$ will not simply reproduce (A).

To further elaborate my question: If we have a current of a branch as the form of $$I_n=\frac{ V_n (s \sin\theta+\omega_1 \cos\theta)}{s^2+\omega_1^2} \frac{1}{Z_1(s)} $$

can we directly deduce (without taking the inverse) that particular branch is effectively acting as it is connected to a voltage source of \$ V_n \sin(\omega_1 t + \theta) \$ and impedance of the branch is \$ Z_1(j \omega) \$?

Answer 1

$$I(s) = \frac{V_m \omega_0 }{(s^2+\omega_0^2) \big(R + \frac{1}{s C} + s L \big)} \tag{A}$$

For a linear system with sinusoidal voltage input, the current also will be sinusoidal. In which case, \$I(s)\$ can be written as,

$$I(s) = \frac{I_m \omega }{(s^2+\omega_0^2)}$$

Substituting this into \$(A)\$ and replacing \$s\$ with \$j\omega\$ should will result in \$(B)\$.

Answer 2

You need the \$ \frac{V(s)}{I(s)} = (R + \frac{1}{cs} + sL)\$ so the transfer function for \$ \omega \$ is :

\$ \frac{I( \omega)}{V(\omega)} = \frac{1}{(R + \frac{1}{cj\omega} + j \omega L)}\$

you insert the Laplace transform of the input source to transfer function.

Note: the Laplace can get any input function and you can solve the equation for time response of the system to any arbitrary input. But in frequency model, you know that the input is sinusoidal.