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An RC circuit works at a frequency of 3.50 MHz. It has a resistance of 130 and a capacitance of 150 pF. What is the phase angle This problem is a little more involved. First, you must find the capacitive reactance for a capacitor of 150 pF. Convert this to microfarads, getting C 0.000150 F. Remember that microfarads go with megahertz (millionths go with minions to cancel each other out). Then 1/(6.28 3.50 0.000150) XC 1/0.003297 303 Now you can find the ratio XC/R 66.8 degrees. 303/130 2.33; the phase angle is arctan ( 2.33)

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The Complex Number Plane A complete complex number plane is made by taking the real and imaginary number lines and placing them together, at right angles, so that they intersect at the zero points, 0 and j0. This is shown in Fig. 15-2. The result is a Cartesian coordinate plane, just like the ones people use to make graphs of everyday things such as stock price versus time.

Problem 14-6

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Complex Number Vectors Complex numbers can also be represented as vectors This gives each complex number a unique magnitude and a unique direction The magnitude is the distance of the point a + jb from the origin 0 + j0 The direction is the angle of the vector, expressed counterclockwise from the positive realnumber axis This is shown in Fig 15-3 Absolute Value The absolute value of a complex number a + jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b) In the case of a pure real number a + j0, the absolute value is simply the real number itself, a, if a is positive If a is negative, then the absolute value of a + j0 is equal to a.

What is the phase angle in the above circuit if the frequency is raised to 7.10 MHz You need to find the new value for XC, because it will change as a result of the frequency change. Calculating, 1/(6.28 7.10 0.000150) XC 1/0.006688 150 The ratio Xc /R 49.0 degrees. 150/130 1.15; the phase angle is therefore arctan ( 1.15)

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Refer to the text in this chapter if necessary. A good score is at least 18 correct. Answers are in the back of the book. 1. As the size of the plates in a capacitor increases, all other things being equal: A. The value of XC increases negatively. B. The value of XC decreases negatively. C. The value of XC does not change. D. You can t say what happens to XC without more data. 2. If the dielectric material between the plates of a capacitor is changed, all other things being equal: A. The value of XC increases negatively.

In the case of a pure imaginary number 0 + jb, the absolute value is equal to b, if b (a real number) is positive If b is negative, the absolute value of 0 + jb is equal to b If the number a + jb is neither pure real or pure imaginary, the absolute value must be found by using a formula First, square both a and b Then add them Finally, take the square root This is the length, c, of the vector a + jb The situation is illustrated in Fig 15-4 Problem 15-1 Find the absolute value of the complex number 22 j0 This is a pure real number Actually, it is the same as 22 + j0, because j0 = 0 Therefore, the absolute value of this complex number is ( 22) = 22 Problem 15-2 Find the absolute value of 0 j34.

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