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Reactance and frequency
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Inductive reactance is one of two kinds of reactance; the other, called capacitive reactance, will be discussed in the next chapter. Reactance in general is symbolized by the capital letter X. Inductive reactance is indicated by the letter X with a subscript L: XL. If the frequency of an ac source is given, in hertz, as f, and the inductance of a coil in henrys is L, then the inductive reactance is XL 6 .28fL
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This same formula applies if the frequency, f, is in kilohertz and the inductance, L, is in millihenrys. And it also applies if f is in megahertz and L is in microhenrys. Just remember that if frequency is in thousands, inductance must be in thousandths, and if frequency is in millions, inductance must be in millionths. Inductive reactance increases linearly with increasing ac frequency. This means that the function of XL vs f is a straight line when graphed. Inductive reactance also increases linearly with inductance. Therefore, the function of XL vs L also appears as a straight line on a graph. The value of XL is directly proportional to f ; XL is also directly proportional to L. These relationships are graphed, in relative form, in Fig. 13-4.
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Problem 13-1
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Suppose a coil has an inductance of 0. 50 H, and the frequency of the ac passing through it is 60 Hz. What is the inductive reactance Using the above formula, XL 6.28 60 0. 50 188.4 ohms. You should round this off to two significant digits, or 190 ohms, because the inductance and frequency are both expressed to only two significant digits.
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234 Inductive reactance
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13-4 Inductive reactance is directly proportional to inductance and also to frequency.
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Problem 13-2
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What will be the inductive reactance of the above coil if the supply is a battery that supplies pure dc Because dc has a frequency of zero, XL 6.28 0 0. 50 0. That is, there will be no inductive reactance. Inductance doesn t generally have any practical effect with pure dc.
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If a coil has an inductive reactance of 100 at a frequency of 5.00 MHz, what is its inductance In this case, you need to plug numbers into the formula and solve for the unknown L. Start out with the equation 100 6.28 5. 00 L 31.4 L. Then, recall that because the frequency is in megahertz, or millions of hertz, the inductance will come out in microhenrys, or millionths of a henry. You can divide both sides of the equation by 31.4, getting L 100/31.4 3.18 H.
Points in the RL plane
Inductive reactance can be plotted along a half line, just as can resistance. In a circuit containing both resistance and inductance, the characteristics become two-dimensional. You can orient the resistance and reactance half lines perpendicular to each other to make a quarter-plane coordinate system, as shown in Fig. 13-5. Resistance is usually plotted horizontally, and inductive reactance is plotted vertically, going upwards. In this scheme, RL combinations form impedances. You ll learn all about this in chapter 15. Each point on the RL plane corresponds to one unique impedance value. Conversely, each RL impedance value corresponds to one unique point on the plane. For reasons made clear in chapter 15, impedances on the RL plane are written in the form R jXL, where R is the resistance and XL is the inductive reactance. If you have a pure resistance, say R 5 , then the complex impedance is 5 j0, and is at the point (5,0) on the RL plane. If you have a pure inductive reactance, such as
Vectors in the RL Plane 235
13-5 The RL quarter-plane.
XL 3 , then the complex impedance is 0 j3, and is at the point (0, 3) on the RL plane. These points, and others, are shown in Fig. 13-6. In real life, all coils have some resistance, because no wire is a perfect conductor. All resistors have at least a tiny bit of inductive reactance, because they take up some physical space. So there is really no such thing as a mathematically perfect pure resistance like 5 j0, or a mathematically perfect pure reactance like 0 j3. Sometimes you can get pretty close, but absolutely pure resistances or reactances never exist, if you want to get really theoretical. Often, resistance and inductive reactance are both deliberately placed in a circuit. Then you get impedances values such as 2 j3 or 4 jl.5. These are shown in Fig. 13-6 as points on the RL plane. Remember that the values for XL are reactances, not the actual inductances. Therefore, they vary with the frequency in the RL circuit. Changing the frequency has the effect of making the points move in the RL plane. They move vertically, going upwards as the ac frequency increases, and downwards as the ac frequency decreases. If the ac frequency goes down to zero, the inductive reactance vanishes. Then XL 0, and the point is along the resistance axis of the RL plane.
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