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Problem 14-6 What is the phase angle in the preceding circuit if the frequency is raised to 710 MHz You need to find the new value for XC, because it will change as a result of the frequency change Calculating:

Here s a challenge!

The trick is to start with T large and then gradually reduce it to zero Thus initially, the local search can wander around quite freely, with only a mild preference for low-cost solutions As time goes on, this preference becomes stronger, and the system mostly sticks to the lower-cost region of the search space, with occasional excursions out of it to escape local optima Eventually, when the temperature drops further, the system converges on a solution Figure 911 shows this process schematically Simulated annealing is inspired by the physics of crystallization When a substance is to be crystallized, it starts in liquid state, with its particles in relatively unconstrained motion Then it is slowly cooled, and as this happens, the particles gradually move into more regular con gurations This regularity becomes more and more pronounced until nally a crystal lattice is formed The bene ts of simulated annealing come at a signi cant cost: because of the changing temperature and the initial freedom of movement, many more local moves are needed until convergence Moreover, it is quite an art to choose a good timetable by which to decrease the temperature, called an annealing schedule But in many cases where the quality of solutions improves signi cantly, the tradeoff is worthwhile

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First, let s figure out the equations for these inverse relations To do that, we must transpose the values of the variables without changing their names We must also transpose the domain and the range The three original relations are y=x+2 y = (x1/2) y = x1/2 In Chap 13, we were given the domains and ranges for these relations In the first case, the domain and range are both the entire set of reals In the second case, the domain is the set of nonnegative reals, and the range is the set of all reals In the third case, the domain and range are both the set of nonnegative reals Switching the values of the variables by reversing their positions in the original equation, we get x=y+2 x = (y1/2) x = y1/2 When we manipulate these equations to get x in terms of y, the results are y=x 2 y = x2 y = (+x)2 When we transpose the domain and the range from the original relations in the first case, they both remain the entire set of reals In the second case, the domain becomes the set of all reals, and the range becomes the set of nonnegative reals In the third case, the domain and range both remain the set of nonnegative reals The plus sign in the last equation means that we consider it only for nonnegative values of x, because negative values of x aren t part of the domain Now that we know the equations for the inverse relations, we re ready to graph them A simple trick makes it easy to graph the inverse of any relation We draw the line representing all points where the independent and dependent variables have the same value We can imagine this line, represented by the equation y = x in these examples, as a point reflector It works like the number reflector for generating negative numbers from positive numbers on the number line (We devised that gimmick in Chap 3 Now we re operating in two dimensions instead of one) For any point that s part of the graph of the original relation, we can find its counterpart in the graph of the inverse relation by going to the opposite side of the point reflector, exactly the same distance away Figure 14-7 shows how this works The line connecting a point in the original graph and its mate in the inverse graph is perpendicular to the point reflector In addition, the point reflector intersects every point-connecting line exactly in the middle Technically, we say that the point reflector is a perpendicular bisector of any line connecting a point with its inverse point.

X C = 1/(628 710 0000150) = 1/0006688 = 150 The ratio XC /R in this case is equal to 150/130, or 115 The phase angle is the arctangent of 115, which turns out to be 490

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