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You should know how complex numbers are multiplied, to have a full understanding of their behavior. When you multiply these numbers, you only need to treat them as sums of number pairs, that is, as binomials. It s easier to give the general formula than to work with specifics here. The product of (a + jb) and (c jd) is equal to ac jad jbc jjbd. Simplifying, remember that jj 1, so you get the final formula: (a jb)(c jd) = (ac bd) j(ad bc)

Within 30 years, the Internet has grown from a Cold War concept for controlling the tattered remains of a post-nuclear society to the Information Superhighway. Just as the railroads of the 19th century enabled the Machine Age and revolutionized the society of the time, the Internet takes us into the Information Age, and profoundly affects the world in which we live.

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What about Reactance Things are simple when there is no reactance in an ac circuit using transformers. But often, especially in RF antenna systems, pure resistance doesn t occur naturally. It has to be obtained by using inductors and/or capacitors to cancel the reactance out. The presence of reactance in a load makes a perfect match impossible with an impedance-matching transformer alone. Recall that inductive and capacitive reactances are opposite in effect, and that their magnitudes can vary. If a load presents a complex impedance R + jX, it is possible to cancel the reactance X by deliberately introducing an equal and opposite reactance X. This can be, and often is, done by connecting an inductor or capacitor in series with a load that contains reactance as well as resistance. The result is a pure resistance with a value equal to (R + jX ) jX, or simply R. When wireless communications is contemplated over a wide band of frequencies, adjustable impedance-matching and reactance-canceling networks can be placed between the transmitter and the antenna system. Such a device is called a transmatch or an antenna tuner. These devices not only match the resistive portions of the transmitter and load impedances, but they can tune out reactances in the load. Transmatches are popular among amateur radio operators, who use equipment capable of operation from less than 2 MHz up to the highest known radio frequencies.

As with the addition and subtraction of complex numbers, you must be careful with signs (plus and minus). And also, as with addition and subtraction, you can get used to doing these problems with a little practice. Engineers sometimes (but not too often) have to multiply complex numbers.

Refer to the text in this chapter if necessary. A good score is 18 or more correct. Answers are in the back of the book. 1. In a step-up transformer, (a) the primary impedance is greater than the secondary impedance. (b) the secondary winding is right on top of the primary. (c) the primary voltage is less than the secondary voltage. (d) All of the above are true. 2. The capacitance between the primary and the secondary windings of a transformer can be minimized by (a) placing the windings on opposite sides of a toroidal core. (b) winding the secondary right on top of the primary. (c) using the highest possible frequency. (d) using a center tap on the balanced winding.

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Real and imaginary numbers can be thought of as points on a line. Complex numbers lend themselves to the notion of points on a plane. This 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 you use to make graphs of everyday things like bank-account balance versus time.

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