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321 A practical voltage source has an internal resistance of 12 and generates a 30-V output under open-circuit conditions What is the smallest load resistance we can connect to the source if we do not wish the load voltage to drop by more than 2 percent with respect to the source open-circuit voltage 322 A practical current source has an internal resistance of 12 k and generates a 200-mA output under short-circuit conditions What percent drop in load current will be experienced (with respect to the short-circuit condition) if a 200- load is connected to the current source 323 Repeat the derivation leading to equation 342 for the case where the load resistance is xed and the source resistance is variable That is, differentiate the expression for the load power, PL , with respect to RS instead of RL What is the value of RS that results in maximum power transfer to the load
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NONLINEAR CIRCUIT ELEMENTS
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Until now the focus of this chapter has been on linear circuits, containing ideal voltage and current sources, and linear resistors In effect, one reason for the simplicity of some of the techniques illustrated in the earlier part of this chapter is the ability to utilize Ohm s law as a simple, linear description of the i-v characteristic of an ideal resistor In many practical instances, however, the engineer is faced with elements exhibiting a nonlinear i-v characteristic This section explores two methods for analyzing nonlinear circuit elements Description of Nonlinear Elements
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There are a number of useful cases in which a simple functional relationship exists between voltage and current in a nonlinear circuit element For example, Figure 376 depicts an element with an exponential i-v characteristic, described by the following equations: i = I0 e v i = I0 v>0 v 0 (345)
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Figure 376 i-v characteristic of exponential resistor
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There exists, in fact, a circuit element (the semiconductor diode) that very nearly satis es this simple relationship The dif culty in the i-v relationship of equation 345 is that it is not possible, in general, to obtain a closed-form analytical solution, even for a very simple circuit
Part I
Circuits
With the knowledge of equivalent circuits you have just acquired, one approach to analyzing a circuit containing a nonlinear element might be to treat the nonlinear element as a load, and to compute the Th venin equivalent of the ree maining circuit, as shown in Figure 377 Applying KVL, the following equation may then be obtained: vT = RT ix + vx (346)
Nonlinear element as a load We wish to solve for vx and ix
RT + vT + _ vx ix Nonlinear element
To obtain the second equation needed to solve for both the unknown voltage, vx , and the unknown current, ix , it is necessary to resort to the i-v description of the nonlinear element, namely, equation 345 If, for the moment, only positive voltages are considered, the circuit is completely described by the following system: ix = I0 e
vx > 0
vT = RT ix + vx
(347)
Figure 377 Representation of nonlinear element in a linear circuit
The two parts of equation 347 represent a system of two equations in two unknowns; however, one of these equations is nonlinear If we solve for the load voltage and current, for example, by substituting the expression for ix in the linear equation, we obtain the following expression: vT = RT I0 e vx + vx or vx = vT RT I0 e vx (349) (348)
Equations 348 and 349 do not have a closed-form solution; that is, they are transcendental equations How can vx be found One possibility is to generate a solution numerically, by guessing an initial value (eg, vx = 0) and iterating until a suf ciently precise solution is found This solution is explored further in the homework problems Another method is based on a graphical analysis of the circuit and is described in the following section Graphical (Load-Line) Analysis of Nonlinear Circuits The nonlinear system of equations of the previous section may be analyzed in a different light, by considering the graphical representation of equation 346, which may also be written as follows: ix = 1 vT vx + RT RT (350)
We notice rst that equation 350 describes the behavior of any load, linear or nonlinear, since we have made no assumptions regarding the nature of the load voltage and current Second, it is the equation of a line in the ix -vx plane, with slope 1/RT and ix intercept VT /RT This equation is referred to as the load-line equation; its graphical interpretation is very useful and is shown in Figure 378 The load-line equation is but one of two i-v characteristics we have available, the other being the nonlinear-device characteristic of equation 345 The intersection of the two curves yields the solution of our nonlinear system of equations This result is depicted in Figure 379 Finally, another important point should be emphasized: the linear network reduction methods introduced in the preceding sections can always be employed to