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Suppose you change the voltage source, connected across the coil, from dc to ac (Fig 13-3) Imagine that you can vary the frequency of the ac, from a few hertz to hundreds of hertz, then kilohertz, then megahertz At first, the current will be high, just as it is with dc But the coil has a certain amount of inductance, and it takes some time for current to establish itself in the coil Depending on how many turns there are and on whether the core is air or a ferromagnetic material, you ll reach a point, as the ac frequency increases, when the coil starts to get sluggish That is, the current won t have time to get established in the coil before the polarity of the ac voltage reverses At high ac frequencies, the current through the coil will have difficulty following the voltage placed across the coil This sluggishness in a coil for ac is, in effect, similar to dc resistance As the frequency is raised, the effect gets more pronounced Eventually, if you keep increasing the frequency of the ac source, the coil will not even come near establishing a current with each cycle Then the coil will act like a high resistance The opposition that the coil offers to ac is called inductive reactance It, like resistance, is measured in ohms It can vary, just as resistance does, from near zero (a short piece of wire) to a few ohms (a small coil) to kilohms or megohms (bigger and bigger coils) Like resistance, inductive reactance affects the current in an ac circuit But, unlike simple resistance, reactance changes with frequency This effect is not merely a decrease in the current, although in practice this does happen Inductive reactance produces a change in the way the current flows with respect to the voltage.

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Inductive reactance is one of two form of reactance (The other form, called capacitive reactance, will be discussed in the next chapter) Reactance in general is symbolized by the italic uppercase letter X Inductive reactance is symbolized XL If the frequency of an ac source is given, in hertz, as f, and the inductance of a coil in henrys is given as L, then the inductive reactance in ohms, XL, is calculated as follows: XL = 2 f L In this formula, the symbol stands for the mathematical constant pi, which is the number of diameters around the circumference of a circle It is equal to approximately 314 We can consider the value of 2 to be equal to 628 in most practical situations Therefore, the preceding formula can be written a little more simply as: XL = 628f L 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 versus f is a straight line when graphed Inductive reactance also increases linearly with inductance Therefore, the function of XL versus L also appears as a straight line on a graph The value of XL is directly proportional to f, and is also directly proportional to L These relationships are graphed, in relative form, in Fig 13-4

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(881)

This P matrix could be used with the methods of Section 491 to de ne 1 Based on eqn probability ellipses De ne the matrix G = HT H (881), P = G 2 which shows that the variance of the x estimation error is determined by the product of the URE variance 2 and the matrix G which is determined by the geometry of the satellites relative to the user Due to the structure of the covariance matrix, as discussed in eqn (424), the variance of the position estimates and various useful error metrics can be computed, for example: var( ) = G11 2 , n var(d) = G33 2 , var( ) = G22 2 , e var(c tr ) = G44 2 ,

Problem 13-1 Suppose a coil has an inductance of 0500 H, and the frequency of the ac passing through it is 600 Hz What is the inductive reactance

and E p p = (G11 + G22 + G33 ) 2 The matrix G de nes various dilution of precision (DOP) factors that are convenient for specifying the ampli cation of the URE in the estimation of speci c portions of x Typically used DOP factors are V DOP HDOP P DOP GDOP = = = = = G33 G11 + G22 G11 + G22 + G33 G11 + G22 + G33 + G44 trace (HT H)

Using the preceding formula, calculate XL = 628 600 0500 = 188 This is rounded to three significant figures

(882)

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