There are two ways of decomposing the complete response into two components:
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First is to break it into a natural response and a forced response.
\(\boxed{\textrm{Complete response } v = \underset{\textrm{(stored energy)}}{\textrm{natural response } v_n} + \underset{\textrm{(independent source)}}{\textrm{forced response } v_f}}\)
For RC and RL circuits, the natural response \(v_n\) is
\(\displaystyle v_n = V_0\,e^{\displaystyle -t/\tau}\)
and forced response \(v_f\) is
\(\displaystyle v_f = V_s \left(1 - e^{\displaystyle -t/\tau}\right)\)
The forced response \(v_f\) is produced by the circuit when an external force (a voltage source in this case) is applied. The natural response \(v_n\) eventually dies out along with the transient component of the forced reponse \(v_f\), leaving only the steady-state component of the forced response \(v_f\).
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Second is to break it into a transient response and a steady-state response.
\(\boxed{\textrm{Complete response } v = \underset{\textrm{(temporary)}}{\textrm{transient response } v_t} + \underset{\textrm{(permanent)}}{\textrm{steady-state response } v_{ss}}}\)
For RC and RL circuits, the transient response \(v_t\) is
\(\displaystyle v_t = (V_0 - V_s) e^{\displaystyle -t/\tau}\)
and steady-state response \(v_{ss}\) is
\(\displaystyle v_{ss} = V_s\)
The transient response \(v_t\) is the circuit’s temporary response that will die out with time. The steady-state response \(v_{ss}\) is the behavior of the circuit a long time after an external excitation is applied.
The first decomposition of the complete response is in terms of the source of the responses, while the second decomposition is in terms of the permanency of the responses.