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Some books talk about the natural response of the RLC circuit. This is when the voltage source is taken out from the circuit. Some other books talk about the transient response of an RLC circuit, which is the time it takes the circuit to reach to steady state. The equations seem similar. I googled and some people are saying that natural response and transient response are the same thing. I don't see how, when one depends on the voltage source and the are doesn't. Could anyone shed some light on the two concepts?

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Natural response refers to the zero-input response, where only initial conditions generate the system response. Transient response refers to the system response to a time domain input signal, such as an impulse or a step. The exponential terms in both types of response will be closely related - having the same exponential time constants, for example.

For an RLC circuit, the natural response could be obtained by having an initial charge on the capacitor when the circuit is closed (by a switch, e.g.), but with no other sources of EMF in the loop.

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I know this is a late answer, but I've just come across this question and wanted to add my point of view.

Suppose we have a generic dynamic (and linear) circuit with one or more external input sources and with some initial conditions at the dynamic elements. That implies that we have chosen an arbitrary time instant as our "initial" time. We don't have to know what happened to circuit before that point, because all gets "summarized" into the initial conditions.

Then, thanks to superposition theorem, the total time response \$r(t)\$ can be decomposed as a sum of the response to only the external inputs (the so-called forced response) and the response to only the initial conditions (the free, or natural, response).

If the circuit is stable, the natural response will decay to zero after some time, leaving only the forced response, which will stand forever (provided that we don't turn off the input sources and they don't decay to zero too).

So, for a while starting at initial time, we will have a response composed of natural + forced response. This is called the transient response. But after this while the natural response will have gone to zero and only the forced response will remain. This is called the steady state response.

The steady state response will last until we turn off the input sources. That will leave probably the circuit "energized" with some voltages and currents into capacitors and inductors, which will generate a brief natural decaying response until the total energy of the circuit is dissipated and the output reaches zero. This last response is another transient, this time containing only natural/free response.

Summarizing: the transient response of a circuit is the response part where the natural response is still present, whether merged or not with forced response. So, it equals the natural response only if there's not forced response.

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