A simple RLC circuit, so there is such a way?

Generally, the RLC network can directly use the phasor analysis method to give a steady-state solution, as shown in the following figure:

Obviously, the result of the phasor method above is far from the simulation in the first post. where is the problem?

In fact, the premise of phasor analysis is single-frequency steady state, which limits many degrees of freedom. The following will consider other factors and use Laplace transform for analysis, as shown in the figure below:

The biggest feature of Laplace analysis is that it considers the initial state of the system, such as V0 (the initial voltage of the capacitor) and I0 (the initial current of the Inductor) as shown in the figure.

Note the two formulas in the above figure. The above formula gives the solution of the RLC network with the initial state, while the following formula uses a partial decomposition. Comparing the corresponding coefficients, the following equations can be obtained:

From the output response formula of the RLC network, the so-called “zero state response” and “zero input response” can be directly seen, as shown in the following formula:

These formulas are too simple and are not the subject of this post. Let’s take a look at the situation of the simulation diagram first, as shown in the figure below:

This is a zero-state response, and the input is a cosine voltage signal:

Uin = U0 cos(ωt)

Note that the response is the superposition of two cosine signals of equal amplitude but different frequencies. That is the simulation result of the first post-beat.

Someone may ask, can the Laplace analysis method transition to the phasor analysis method?

Of course, it would be unreasonable otherwise. In the Laplace analysis, as long as the appropriate initial conditions are selected, the “natural characteristics” of the system itself can not be revealed, as shown in the figure below:

Finally, it needs to be particularly pointed out that the following formula:

It is given that the response can be decomposed into the sum of the system’s own characteristics and external excitation characteristics.

Obviously, if there is a resistance R (R≠∞), then the natural characteristics of the system itself will decay exponentially over time. Eventually approaching phasor analysis!