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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 100 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:

p o , pi ^ )

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Figure 814: Relative GPS range For the physical description, consider the geometry depicted in Figure 814 The gure shows signals arriving at the base and rover locations The geometry of the gure assumes that (1) the satellite is su ciently far away for the wave fronts to appear as planes that are indicated by dashed lines; and, (2) the line-of-sight vectors for the rover and base are identical From the gure geometry it is clear that subject to the two assumptions, eqn (8125) is an accurate approximation The analysis below shows that the d2 accuracy of the approximations is approximately 2Ro which has magnitude 25cm for d = 1km and 25m at d = 10km The following mathematical derivation will allow quantitative analysis The derivation will start from the following facts: R(pr , pi ) = R(po , pi ) = hr pr pi ho po pi

100 = 1/(628 100 C ) Dividing through by 100, you get: 1 = 1/(628 100 C ) Multiply each side of this by C, and you obtain C = 1/(628 100) This can be worked out with a calculator You should find that C = 0000159 to three significant figures Because the frequency is given in megahertz, the capacitance comes out in microfarads That means C = 0000159 F You can also say it is 159 pF (Remember that 1 pF = 0000001 F)

p p (pr i ) (po i ) where hr = pr i and ho = (po i ) are represented as row vectors p p so that the superscript for transpose can be dropped To decrease the notation slightly, we will use the following symbols: Ro = R(po , pi ) and Rr = R(pr , pi ) The analysis is as follows:

hr pr pi ho po pi + hr (po po ) (8126)

Figure 815 shows the notation and geometry that will be useful for the analysis of the last two terms in eqn (8126) From the gure it is clear that hr po pi = ho po pi cos( ) (8127)

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