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18. In Fig. 14-13, note that the R and X C scale divisions are not the same size. What is the actual phase angle (a) 1.42 (b) About 60 , from the looks of it (c) 58.9 (d) 88.6 19. Suppose an RC circuit consists of a 150-pF capacitor and a 330- resistor in series. What is the phase angle at a frequency of 1.34 MHz (a) 67.4 (b) 22.6 (c) 24.4 (d) 65.6 20. Suppose an RC circuit has a capacitance of 0.015 F. The resistance is 52 . What is the phase angle at 90 kHz (a) 24 (b) 0.017 (c) 66 (d) None of the above

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As the resistance in an RC circuit gets large compared with the capacitive reactance, the angle of lead becomes smaller. The same thing happens if the value of XC gets small compared with the value of R. When you call XC large, you mean large negatively. When you say that XC is small, you mean that it is close to zero, or small negatively. When R is many times larger than XC, whatever their actual values, the vector in the RC plane will be almost right along the R axis. Then the RC phase angle will be nearly zero, that is, just a little bit negative. The voltage will come nearly into phase with the current. The plates of the capacitor will not come anywhere near getting fully charged with each cycle. The capacitor will be said to pass the ac with very little loss, as if it were shorted out. But it will still have an extremely high XC for any ac signals at much lower frequencies that might exist across it at the same time. (This property of capacitors can be put to use in electronic circuits, for example when an engineer wants to let radio signals get through while blocking audio frequencies.) Ultimately, if the capacitive reactance gets small enough, the circuit will act as a pure resistance, and the current will be in phase with the voltage.

also get acquainted with admittance, the extent to which an ac circuit allows (or admits) current flow, rather than impeding it. As we develop these concepts, let s review, and then expand on, some of the material presented in the previous couple of chapters.

If you know the ratio of capacitive reactance to resistance, or XC /R, in an RC circuit, then you can find the phase angle. Of course, you can find this angle of lead if you know the precise values.

6

Have you been wondering what j actually means in expressions of impedance Well, j is nothing but a number: the positive square root of 1. There s a negative square root of 1, too, and it is equal to j. When either j or j is multiplied by itself, the result is 1. (Pure mathematicians often denote these same numbers as i or i.) The positive square root of 1 is known as the unit imaginary number. The set of imaginary numbers is composed of real-number multiples of j or j. Some examples are j4, j35.79, j25.76, and j25,000. The square of an imaginary number is always negative. Some people have trouble grasping this, but when you think long and hard about it, all numbers are abstractions. Imaginary numbers are no more imaginary (and no less real) than so-called real numbers such as 4, 35.79, 25.76, or 25,000. The unit imaginary number j can be multiplied by any real number on a conventional real number line. If you do this for all the real numbers on the real number line, you get an imaginary number line (Fig. 15-1). The imaginary number line should be oriented at a right angle to the real number line when you want to graphically portray real and imaginary numbers at the same time. In electronics, real numbers represent resistances. Imaginary numbers represent reactances.

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