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The response of the parallel RLC circuit shown in Fig. 8-7 will be similar to that of the series RLC circuit, since a second-order di erential equation can be expected. The node voltage method gives v 1 t dv 0 1 v dt C R L 0 dt Di erentiating and dividing by C yields d 2v 1 dv v 0 dt2 RC dt LC A solution is of the form

1) 5 .9512 .8607 .7788 .7047 .6376 .5770 .5220 .4724 .4274 .3867 6 .9418 .8521 .7711 .6977 .6313 .5712 .5169 .4677 .4232 .3829 7 .9324 .8437 .7634 .6907 .6250 .5655 .5117 .4630 .4190 .3791 8 .9231 .8353 .7558 .6839 .6188 .5599 .5066 .4584 .4148 .3753 9 .9139 .8270 .7483 .6771 .6126 .5543 .5016 .4538 .4107 .3716

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Develop a measure, BIG, for people that combines both weight and height. Empirically, if two people are the same height, the heavier is bigger, and if two people are the same weight, the taller is bigger. If we use this notion, we can have a partial order that most people would agree with. The only pair of persons that we would not order by this would be if one was heavier and the other was taller. Numerically, we can use a tuple, < height, weight >. Each part of the tuple would be a real number. Two tuples would be related if both parts were related in the same direction. That is, if x ; y are tuples, than x is less than or equal to y in bigness if x height < y height and x weight < y weight. This is also a partial order, and both versions of the representation condition are satisfied.

1, 2, 3, c, 10) 5 6 7 8 9 10

CHAP. 8]

.36788 .13534 .04979 .01832 .006738 .002479 .000912 .000335 .000123 .000045 TO obtain values of e Example: e

Fig. 8-6

3.48 l

NOTE:

v A 1 es 1 t A 2 es 2 t 2 s q 1 1 2 1 2 !2 where s1 0 2RC 2RC LC s q 1 1 2 1 2 !2 s2 0 2RC 2RC LC p where 1=2RC and !0 1= LC . Note that , the damping factor of the transient, di ers from in the series RLC circuit.

for other values of l, use the laws of exponents. (.04979)(.6188) .03081.

Fig. 8-7

An important characteristic of a measure is monotonicity. It means that the value of the measure of an attribute does not change direction as the attribute increases in the object. For example, the count of lines of code will not decrease as more code is added.

3.00)(e 0.48)

51772 24033 45939 30586 03585 64937 15630 09448 21631 91097 74640 23491 60173 02133 79353 03355 64759 56301 91157 17480 42331 83587 52078 75797 81938 95863 51135 57683 77331 29414 29044 06568 25424 45406 82322 20790 98527 30277 60710 06829 46621 21960 11645 31041 96799 65304 62586 94623 52290 87843 62898 21387 55870 86707 85659 55189 41889 85418 16835 28195 93582 76105 56974 12973 36081 00745 25439 68829 48653 27279 04186 10863 37428 17169 50884 65253 88036 06652 71590 47152 19640 97453 93507 88116 14070 11822 24034 41982 16159 35683 87056 90581 94271 42187 74950 15804 67283 49159 14676 47280

The transient response is easiest to visualize by assuming an initial charge Q0 on the capacitor and a switch that closes at t 0. However, a step function voltage applied to the circuit will initiate the same transient response. Overdamped Case 2 > !2 0 In this case, the solution (2) applies.

.

, C 0:167 mF, and L 1:0 H, has an initial voltage V0 50:0 V on the capacitor. Obtain the voltage v t when the switch is closed at t 0. We have 1 2994 2RC 2 8:96 106 !2 0 1 5:99 106 LC

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