One problem that has recently been seen by a netizen is that it cannot be a simple RLC network. A simulation in that post is shown in the figure below: In general, the RLC network can directly use the phasor analysis method to give a steady-state solution, as shown below: 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, using Laplace transform analysis, see below: The most important feature of Laplacian analysis is to consider the initial state of the system, V0 (the initial voltage of the capacitor) and I0 (the initial current of the inductor) in the figure. Note the two equations in the figure above. The equation above shows the solution of the RLC network with the initial state, while the following formula uses the partial decomposition. Corresponding to the corresponding coefficient, the following equations can be obtained: The response equations output by that RLC network can be directly seen as the so-called "zero-state response" and "zero-input response", see below: These formulas are too simple and not the subject of this post. Let's take a look at the simulation diagram below. See the figure below: This is a zero-state response. The input is a cosine voltage signal: Uin = U0 cos(ωt) Note that the response is a superposition of two cosine signals of equal amplitude but different frequencies. This is the result of the simulation of the first post-beat. Some people may ask, Laplacian analysis method can transition to the phasor analysis method? Of course you can, otherwise it's not reasonable. In Laplacian analysis, as long as the proper initial conditions are selected, the “natural characteristics†of the system itself will not be revealed, as shown in the following figure: Finally, it needs to be specifically pointed out that the following formula: The sum of the two parts of the response can be decomposed into the system's own characteristics and the external incentive features. Obviously, if there is a resistance R (R ≠∞), then the natural features of the system itself will decay exponentially over time. The final approach to phasor analysis!
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