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Capacitive Reactance and Frequency
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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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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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Capacitive Reactance and Frequency 217
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14-4 Capacitive reactance
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is negatively, and inversely, proportional to capacitance. Capacitive reactance is also negatively, and inversely, proportional to frequency.
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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.
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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:
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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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218 Capacitive Reactance
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Points in the RC Plane
In a circuit containing resistance and capacitive reactance, the characteristics are two-dimensional in a way that is analogous to the situation with the RL plane from the previous chapter. The resistance ray and the capacitive-reactance ray can be placed end to end at right angles to make a quarter plane called the RC plane (Fig. 14-5). Resistance is plotted horizontally, with increasing values toward the right. Capacitive reactance is plotted downward, with increasingly negative values as you go down. The combinations of R and X C in this RC plane form impedances. You ll learn about impedance in greater detail in the next chapter. Each point on the RC plane corresponds to one and only one impedance. Conversely, each specific impedance coincides with one and only one point on the plane. Any impedance that consists of a resistance R and a capacitive reactance X C can be written in the form R + jX C. Remember that X C is always negative or zero. Because of this, engineers will often write R jX C instead. If an impedance is a pure resistance R with no reactance, then the complex impedance is R j 0 (or R + j 0; it doesn t matter if j is multiplied by 0!). If R = 3 with no reactance, you get an impedance of 3 j 0, which corresponds to the point (3,j 0) on the RC plane. If you have a pure capacitive reactance, say X C = 4 , then the complex impedance is 0 j 4, and this is at the point (0, j 4) on the RC plane. Again, it s important, for completeness, to write the 0 and not just the j 4. The points for 3 j 0 and 0 j 4, and two others, are plotted on the RC plane in Fig. 14-6. In practical circuits, all capacitors have some leakage resistance. If the frequency goes to zero (pure dc), a tiny current always flows, because no capacitor has a perfect insulator between its plates. In addition to this, all resistors have a little capacitive reactance because they occupy a finite physical space. So there is no such thing as a mathematically perfect resistor, either. The points 3 j0 and