Fig. 31-A

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We should rst explain that the term baseband spectrum refers to the original location of the spectrum of a signal. The baseband spectrum will normally extend from near f 0 to f fh , where fh is the highest frequency component in the signal. The shape of a possible baseband spectrum is illustrated in Fig. 31-A. The baseband of an audio signal, as produced by a microphone, is generally taken to extend from approximately 16 Hz to approximately 16,000 Hz.

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Appendix

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In a simple telephone system, the voice signal is transmitted in its original baseband form, over copper wires, from transmitter to receiver. Such a case does not involve modulation. If, however, the same voice signal is to be transmitted by wireless, then some kind of modulation of a high-frequency carrier wave is required. Regardless of the type of modulation used, the result is always the production of a band of side-band frequencies, clustered symmetrically about the carrier. Since the original signal information is contained in this band of frequencies, it s desirable that the entire band be passed through circuits having a reasonably good band-pass characteristic, such as is illustrated in Fig. 240.

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First, by note 6:

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Trigonometric Identity for (sin x sin y)

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cos x y cos x cos y sin x sin y

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In the above equation, replace y with y; then, since (section 5.3) cos y cos y and sin y sin y, eq. (I) becomes cos x y cos x cos y sin x sin y hence cos x y cos x cos y sin x sin y Lastly, addition of eqs. (I) and (II) gives the identity we are after, thus sin x sin y 1 cos x y cos x y 2 II

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Lt L L 2L

L Proportional to N2

16-A

Consider a coil of TWO TURNS in which each turn, considered by itself, has L henrys of inductance. If there were NO COUPLING WHATEVER between the two turns the total inductance Lt would simply be the sum of the inductances of the individual turns; thus If, however, some amount of coupling does exist between the two turns, then, as we found in problem 198, the total inductance would be equal to Lt L L 2M or, by eq. (371) Lt L L 2kL Hence, for the ideal case in which there is COMPLETE COUPLING between the two turns (k 1), eq. (16-A) becomes Lt L L 2L 17-A

Appendix

Now consider a coil of N turns in which each turn, considered by itself, has the same inductance of L henrys. If there were NO COUPLING WHATEVER between any turn and any other turn, the total inductance Lt would simply be the sum of the individual inductances of the individual turns; thus Lt L L L L L NL 18-A Now imagine an inductor coil of N turns, of L henrys each, in which 100% COUPLING EXISTS BETWEEN EACH TURN AND ALL THE OTHER TURNS; this would constitute a true ideal inductor and, in such a case, the amount of 2L would have to be added to eq. (18-A) FOR EVERY POSSIBLE COMBINATION OF TWO TURNS in the coil. Thus, for example, 2L would be added one time for N 2, three times for N 3, six times for N 4, ten times for N 5, and so on; this is illustrated in the gures below for N 2; N 3, and N 4.

Thus eq. (18-A) would become for N 2; for N 3; for N 4; for N 5; Lt L L L 2L 4L 2 2 L Lt L L L 2L 2L 2L 9L 3 2 L Lt L L L L 2L 2L 2L 2L 2L 2L 16L 4 2 L Lt 5L 10 2L 25L 5 2 L

and upon continuing on in this manner it soon becomes evident that an ideal inductor of N turns has a total inductance of Lt N 2 L kN 2 ; as stated; where L inductance/turn. (It should be mentioned that the above result can also be derived by direct application of the formula for the combination of N things taken two at a time.)