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Q. What do you mean by Superposition and linearity?
Mathematically a function is said to be linear if it satisfies two properties: homogeneity (proportionality or scaling) and additivity (superposition),
f (Kx) = Kf (x) (homogeneity)
where K is a scalar constant, and
f (x1 + x2) = f (x1) + f (x2) (additivity)
Linearity requires both additivity and homogeneity. For a linear circuit or system in which excitations x1 and x2 produce responses y1 and y2, respectively, the application of K1x1 and K2x2 together (i.e., K1x1 + K2x2) results in a response of (K1y1 + K2y2), where K1 and K2 are constants. With the cause-and-effect relation between the excitation and the response, all linear systems satisfy the principle of superposition. A circuit consisting of independent sources, linear dependent sources, and linear elements is said to be a linear circuit. Note that a resistive element is linear. Capacitors and inductors are also circuit elements that have a linear input-output relationship provided that their initial stored energy is zero. Nonzero initial conditions are to be treated as independent sources.
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Consider the circuit shown in Figure (a) in which the switch S has been in position 1 for a long time. Let the switch be changed instantaneously to position 2 at t = 0. Obtain v(t)
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