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Signals and Circuits 2 Time responses The nature of the time response at the electrical system is the same as at the mechanical system. As example in the figure 1 are presented speed-time graph of the airplane is the same current time graph of electrical system with one energy-storage element (in this case kinetic). Such graphs are characteristics of a first-order system.

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Signals and Circuits 2 Time responses The energy storage elements in the electrical system are capacitance and inductance and response of circuits containing these elements may be determined using the relationships

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Signals and Circuits 2 Time responses Forced response is also called steady- state response, being that to which the capacitor voltage eventually settles. However, the forcing function is not in fact connected until the switch is closed and is unlikely that, at the point of switching, the initial condition of the energy storage element in the network. To bridge this mismatch an additional component of response arises. This is the natural response, existing for the period of adjustment only. The total response is the sum of the natural and forced components.

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Signals and Circuits 2 Time responses A special case occurs when forcing function is zero. Then, the forced response does not exist and the total response is also the natural response, which is dependent only on the network elements and initial conditions and known as the zero input response. An example in the freely swinging pendulum which is given and initial displacement and in then released.

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Signals and Circuits 2 Time responses The solution of this equation may be found by the standard procedure of forming an auxiliary or characteristic equation. is a circuit parameter gives

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Signals and Circuits 2 Time responses

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The standard method, known as the D-operator method could be used. In this method differentiation of a variable in this case, is represented by the operational notation The complete response is given by the sum of the two component responses: natural and forced. For the natural and forced responses:

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Signals and Circuits 2 Time responses

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