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Problem 16-20 Suppose a series RX circuit (shown by the generic block diagram of Fig 16-11) has a resistance of R = 500 and a capacitive reactance of X = 500 Suppose 100-V rms ac is applied to this circuit What is the current First, calculate Z 2 = R 2 + X 2 = 5002 + ( 500)2 = 2500 + 2500 = 5000 Then Z is the square root of 5000, or 707 Therefore, I = E/Z = 100/707 = 141 A rms Problem 16-21 What are the rms ac voltages across the resistance and the reactance, respectively, in the circuit described in Problem 16-20 The Ohm s Law formulas for dc will work here Because the current is I = 141 A rms, the voltage drop across the resistance is equal to ER = IR = 141 500 = 705 V rms The voltage drop across the reactance is the product of the current and the reactance: EX = IX = 141 ( 500) = 705 V rms This is an rms ac voltage of equal magnitude to that across the resistance But the phase is different Note that voltages across the resistance and the reactance a capacitive reactance in this case, because it s negative don t add up to 100 V rms, which is placed across the whole circuit This is because, in an RX ac circuit, there is always a difference in phase between the voltage across the resistance and the voltage across the reactance The voltages across the components always add up to the applied voltage vectorially, but not always arithmetically Problem 16-22 Suppose a series RX circuit (Fig 16-11) has R = 100 and X = 400 The applied voltage is 100V rms ac What is the current Calculate Z 2 = R 2 + X 2 = 100 + 1600 = 1700 This means that Z is the square root of 1700, or 412 Therefore, I = E/Z = 100/412 = 243 A rms Problem 16-23 What are the rms ac voltages across the resistance and the reactance, respectively, in the circuit described in Problem 16-22 Knowing the current, calculate ER = IR = 243 100 = 243 V rms Also, EX = IX = 243 400 = 972 V rms If you add ER + EX arithmetically, you get 243 + 972 = 1215 V as the total.
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an x n + an-1x n-1 + an-2x n-2 + + a1x + b = 0 where a1, a2, a3, an are nonzero rational-number coefficients of the variable x, b is a nonzero rational constant, and n is a positive integer greater than 3 If b = 0, then you can t use the process, but you will at least know that 0 is a root If b 0, then you can go through the following steps Make certain that all the numbers a1, a2, a3, an, and b are integers If that is not the case, multiply the equation through by the smallest constant that will turn all the numbers a1, a2, a3, an, and b into integers Find all the positive and negative integer factors of b, the stand-alone constant Call these by the general name m Find all the positive and negative integer factors of an, the coefficient of xn (sometimes called the leading coefficient) Call these by the general name n Write down all the possible ratios m /n Call them by the general name r
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Notice that each of these functions has signature ha1 ,a2 : [m]2 [m], that is, it maps a pair of integers in [m] to a single integer in [m] (b) H is as before, except that now m = 2k is some xed power of 2 130 The grade-school algorithm for multiplying two n-bit binary numbers x and y consists of adding together n copies of x, each appropriately left-shifted Each copy, when shifted, is at most 2n bits long In this problem, we will examine a scheme for adding n binary numbers, each m bits long, using a circuit or a parallel architecture The main parameter of interest in this question is therefore (c) H is the set of all functions f : [m] [m 1]
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With synthetic division, check every r to see if it is a root of the original polynomial equation If you find a root, you ll get a remainder of 0 at the end of the synthetic division process If none of the numbers r produces a remainder of 0, then the original polynomial equation has no rational roots If one or more of the ratios r produces a remainder of 0, then every one of those numbers is a rational root of the equation List all of the rational roots found after carrying out the preceding steps Call them r1, r2, r3, and so on Create binomials of the form (x r1), (x r2), (x r3), and so on Each of these binomials is a factor of the original equation If you re lucky, you ll end up with an equation in binomial to the nth form, or an equation in binomial factor form If you re less lucky, you ll end up with one or more binomial factors and a quadratic factor That factor can be set equal to 0, and then the quadratic formula can be used to find its roots Neither of those roots will be rational They might even be complex If you re unlucky, you ll be stuck with one or more binomial factors and a cubic or higher-degree factor If you set that factor equal to 0 to form a polynomial equation, you ll know that none of the roots of that equation are rational Some might even be complex You can try to solve it, but you should not expect the task to be easy
containing resistance and reactance Illustration for Problems 16-20 through 16-23
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