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where V DOP quanti es the magni cation of URE in estimating altitude, HDOP quanti es the magni cation of URE in estimating the horizontal position [n, e] , etc Most GPS receivers calculate and can be con gured to output the various DOP factors and URE This allows the user (or real-time software) to monitor the expected instantaneous estimation accuracy GPS planning software is also available which allows users to analyze the DOP factors as a function of time and location to determine either satisfactory or optimal times for performing GPS related activities (see 7 in [76]) The matrix H is a function of the number of satellites in view and the geometry of the line-of-sight vectors hi from the satellites to the user The

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Problem 13-2 What will be the inductive reactance of the preceding coil if the supply is a battery that supplies pure dc Because dc has a frequency of zero, XL = 628 0 0500 = 0 That is, there will be no inductive reactance Inductance doesn t have any practical effect with pure dc Problem 13-3 If a coil has an inductive reactance of 100 at a frequency of 500 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 = 628 500 L = 314 L Because the frequency is given in megahertz, the inductance will come out in microhenrys You can divide both sides of the equation by 314, getting L = 100/314 = 318 H

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Figure 810: Two dimensional illustration of the relationship between satellite-to-user geometry and the resulting position accuracy

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 plotted horizontally, and inductive reactance is plotted vertically upward

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e ect of satellite-to-user geometry on position estimation accuracy is illustrated in Figure 810 using two dimensions This gure shows two possible user-satellite con gurations In each con guration, a receiver measures the range to two satellites at the indicated positions Due to measurement errors, the range measurement is not known exactly, but the receiver position is expected to lie between the two indicated concentric circles In the left half of the gure the receiver vectors h1 and h2 are nearly collinear; the intersection of the two concentric circles for the two satellites results in a long thin region of possible positions with the largest uncertainty direction orthogonal to the satellite vectors In this case, the DOP factor would be large In the right half of the gure the vectors h1 and h2 are nearly orthogonal; the intersection of the concentric circles for each satellite results in a much more equally proportioned uncertainty region The measurement matrix H is not constant in time because of the GPS satellite positions change as they orbit Earth Large DOP values result when the rows of H are nearly linearly dependent This could be measured by the condition of the matrix H H, but this approach is not often used or necessary There may be times when for a given set of four satellites the H matrix approaches singularity and GDOP approaches in nity This condition is called a GDOP chimney, see Figure 811 This was a di culty in the early days of GPS, when satellites had fewer tracking channels or a limited number of satellites were in view Fortunately, there are usually more than four satellites in view and modern receivers track more than the minimum number of satellites; hence, either all or an optimal set of the tracked satellites can be selected to minimize a desired DOP URE ampli cation factor.

inductive reactance (XL ) and resistance (R) This is also known as the RL quarter-plane, or simply as the RL plane

Example 84 Consider the following case for which six satellites are in view of the receiver antenna The rows of the H matrix for each satellite

are: SV 2 SV 24 SV 4 SV 5 SV 7 SV 9 0557466 +0469913 +0086117 +0661510 0337536 +0762094 +0829830 +0860463 +0936539 0318625 +0461389 +0267539 0024781 +0196942 0339823 0678884 0820482 0589606 +1000000 +1000000 +1000000 +1000000 +1000000 +1000000

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