+ v (t) _ T

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+ vT (t) _

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Parallel case (a) RT

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(536)

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iS (t) iC (t) iL (t) = 0 Further, KVL applied to the right-hand loop yields

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(537)

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vC (t) = vL (t)

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Series case (b)

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It should be apparent that we have all the equations we need (in fact, more) Using the de ning relationships for capacitor and inductor, we can express equation 537 as vT (t) vC (t) dvC iL (t) = 0 C RT dt and equation 538 becomes diL dt (539)

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Figure 539 Second-order circuits

RT iS (t) + iC (t) + iL(t) L

vC (t) = L

(540)

+ v (t) v (t) C _ T

C vL(t) _

Substituting equation 540 in equation 539, we can obtain a differential circuit equation in terms of the variable iL (t): 1 d 2 iL L diL = LC 2 + iL (t) vT (t) RT RT dt dt or d 2 iL 1 1 diL vT (t) + iL = + 2 dt RT C dt LC RT LC (542) (541)

Figure 540 Parallel case

The solution to this differential equation (which depends, as in the case of rst-order circuits, on the initial conditions and on the forcing function) completely determines the behavior of the circuit By now, two questions should have appeared in your mind: 1 Why is the differential equation expressed in terms of iL (t) (Why not vC (t) ) 2 Why did we not use equation 536 in deriving equation 542

Part I

Circuits

In response to the rst question, it is instructive to note that, knowing iL (t), we can certainly derive any one of the voltages and currents in the circuit For example, vC (t) = vL (t) = L iC (t) = C diL dt (543)

dvC d 2 iL = LC 2 dt dt

(544)

To answer the second question, note that equation 542 is not the only form the differential circuit equation can take By using equation 536 in conjunction with equation 537, one could obtain the following equation: vT (t) = RT [iC (t) + iL (t)] + vC (t) (545)

Upon differentiating both sides of the equation and appropriately substituting from equation 539, the following second-order differential equation in vC would be obtained: d 2 vC 1 dvC 1 dvT (t) 1 + + vC = dt 2 RT C dt LC RT C dt (546)

Note that the left-hand side of the equation is identical to equation 542, except that vC has been substituted for iL The right-hand side, however, differs substantially from equation 542, because the forcing function is the derivative of the equivalent voltage Since all of the desired circuit variables may be obtained either as a function of iL or as a function of vC , the choice of the preferred differential equation depends on the speci c circuit application, and we conclude that there is no unique method to arrive at the nal equation As a case in point, consider the two circuits depicted in Figure 541 If the objective of the analysis were to determine the output voltage, vout , then for the circuit in Figure 541(a), one would choose to write the differential equation in vC , since vC = vout In the case of Figure 541(b), however, the inductor current would be a better choice, since vout = RT iout Natural Response of Second-Order Circuits From the previous discussion, we can derive a general form for the governing equation of a second-order circuit: d 2 x(t) dx(t) a2 + a1 + a0 x(t) = f (t) 2 dt dt (547)

+ vT _

Rout

vout _

(a) RT L iout (t)

+ v _ T

+ vout _

Rout

It is now appropriate to derive a general form for the solution The same classi cation used for rst-order circuits is also valid for second-order circuits Therefore, the complete solution of the second-order equation is the sum of the natural and forced responses: x(t) = xN (t) + xF (t) Natural response Forced response (548)