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In one sense, capacitive reactance behaves like a reflection of inductive reactance. But looked at another way, X C is an extension of XL into negative values. If the frequency of an ac source (in hertz) is given as f, and the capacitance (in farads) is given as C, then the capacitive reactance in ohms, X C, is calculated as follows: X C = 1/(2 f C ) Again, we meet our friend ! And again, for most practical purposes, we can take 2 to be equal to 6.28. Thus, the preceding formula can be expressed like this: XC = 1/(6.28f C ) This same formula applies if the frequency, f, is in megahertz and the capacitance, C, is in microfarads. Capacitive reactance varies inversely with the frequency. This means that the function X C versus f appears as a curve when graphed, and this curve blows up as the frequency gets close to zero. Capacitive reactance also varies inversely with the actual value of capacitance, given a fixed frequency. Therefore, the function of X C versus C also appears as a curve that blows up as the capacitance approaches zero. The negative of X C is inversely proportional to frequency, and also to capacitance. Relative graphs of these functions are shown in Fig. 14-4.

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The formula for calculating resonant frequency fo, in terms of the inductance L in henrys and the capacitance C in farads, is fo 0.159/(LC)1/2

Problem 14-1 Suppose a capacitor has a value of 0.00100 F at a frequency of 1.00 MHz. What is the capacitive reactance

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The 1/2 power is the square root. If you know L and C in henrys and farads, and you want to find fo, do these calculations in this order: First, find the product LC, then take the square root, then divide 0.159 by this value. The result is fo in hertz. The formula will also work to find fo in megahertz (MHz), when L is given in microhenrys ( H) and C is in microfarads ( F). These values are far more common than hertz, henrys, and farads in electronic circuits. Just remember that millions of hertz go with millionths of henrys and millionths of farads. This formula works for both series-resonant and parallel-resonant RLC circuits.

So when should Ethernet be used as opposed to Token Ring Both have their advantages and disadvantages, both have solid industry support, and both are manufactured by a number of respectable, well-known players. CSMA/CD (for all intents and purposes, Ethernet) is far and away the most widely deployed LAN technology because it is simple, inexpensive, and capable of offering very high bandwidth to any marketplace,

is negatively, and inversely, proportional to capacitance. Capacitive reactance is also negatively, and inversely, proportional to frequency.

Problem 17-10

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Use the formula and plug in the numbers. You can do this directly, because the data is specified in microfarads (millionths) and in megahertz (millions): X C = 1/(6.28 1.0 0.00100) = 1/(0.00628) = 159 This is rounded to three significant figures, because all the data is given to that many digits.

Find the resonant frequency of a series circuit with an inductance of 100 H and a capacitance of 100 pF. First, convert the capacitance to microfarads: 100 pF 0.000100 F. Then find the product LC 100 0.000100 0.0100. Take the square root of this, getting 0.100. Finally, divide 0.159 by 0.100, getting fo 1.59 MHz.

Problem 14-2 What is the capacitive reactance of the preceding capacitor if the frequency decreases to zero (that is, if the voltage source is pure dc) In this case, if you plug the numbers into the formula, you get a zero denominator. Mathematicians will tell you that such a quantity is undefined. But we can say that the reactance is negative infinity for all practical purposes. Problem 14-3 Suppose a capacitor has a reactance of 100 at a frequency of 10.0 MHz. What is its capacitance In this problem, you need to put the numbers in the formula and solve for the unknown C. Begin with this equation:

4

Problem 17-11

100 = 1/(6.28 10.0 C ) Dividing through by 100, you get: 1 = 1/(628 10.0 C ) Multiply each side of this by C, and you obtain C = 1/(628 10.0). This can be worked out with a calculator. You should find that C = 0.000159 to three significant figures. Because the frequency is given in megahertz, the capacitance comes out in microfarads. That means C = 0.000159 F. You can also say it is 159 pF. (Remember that 1 pF = 0.000001 F.)

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