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The challenge is to find a line that has this particular Zo. Commercially manufactured lines come in standard Zo values, and a perfect match might not be obtainable. In that case, the closest obtainable Zo should be used. In this case, it would probably be 150 . If nothing is available anywhere near the characteristic impedance needed for a quarter-wave matching section, then a coil-type transformer will probably have to be used instead. A quarter-wave matching section should be made using unbalanced line if the load is unbalanced and balanced line if the load is balanced. The major disadvantage of quarter-wave sections is that they work only at specific frequencies. But this is often offset by the ease with which they are constructed, if radio equipment is to be used at only one frequency, or at odd-harmonic frequencies.

When two waves having different frequencies are combined in a nonlinear circuit, new waves are produced at frequencies equal to the sum and difference of the frequencies of the input waves. Diodes can provide this nonlinearity. Suppose there are two signals with frequencies f1 and f2. For mathematical convenience, let s assign f2 to the wave with the higher frequency, and f1 to the wave with the lower frequency. If these signals are combined in a nonlinear circuit, new waves result. One of them has a frequency of f2 + f1, and the other has a frequency of f2 f1. These sum and difference frequencies are known as beat frequencies. The signals themselves are called mixing products or heterodynes (Fig. 20-4). Figure 20-4, incidentally, is an illustration of a frequency domain display. The amplitude (on the vertical scale or axis) is shown as a function of the frequency (on the horizontal scale or axis). This sort of display is what engineers see when they look at the screen of a lab instrument known as a spectrum analyzer. In contrast, an ordinary oscilloscope displays amplitude (on the vertical scale or axis) as a function of time (on the horizontal scale or axis). The oscilloscope provides a time domain display.

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Things are simple when there is no reactance in an ac circuit using transformers. But often, especially in radio-frequency circuits, pure resistance doesn t occur naturally. It has to be obtained by using inductors and/or capacitors to cancel the reactance out.

First, calculate the reactance of the inductor at 10.0 MHz. Proceed as follows: jXL = j 6.28f L = j(6.28 10.0 5.00) = j314 Next, calculate the reactance of the capacitor at 10.00 MHz. Proceed as follows: jX C = j[1/(6.28f C )] = j[1/(6.28 10.0 0.000200)] = j 79.6 Finally, add the inductive and capacitive reactances to obtain the net reactance: jX = jXL + jX C = j314 + ( j 79.6) = j234 This is a pure inductive reactance. For series-connected components, the condition in which the capacitive and inductive reactances cancel is known as series resonance. We ll deal with this in more detail in the next chapter.

0.5 0 800

342 Transformers and impedance matching Reactance makes a perfect match impossible, no matter what the turns ratio or Zo of the transformer. A small amount of reactance can be tolerated at lower radio frequencies (below about 30 MHz). A near-perfect match becomes more important at higher frequencies. The behavior of reactance, as it is coupled through transformer windings, is too complicated for a thorough analysis here. But if you re interested in delving into it, there are plenty of good engineering texts that deal with it in all its mathematical glory. 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, with X not equal to zero, it is always possible to cancel out the reactance X by adding an equal and opposite reactance ( X) in the circuit. This can be done by connecting an inductor or capacitor in series with the load. For radio communications over a wide band, adjustable impedance-matching and reactance-canceling networks can be placed between a transmitter and an antenna system. Such a device is called a transmatch and is popular among radio hams, who use frequencies ranging from 1.8 MHz to the microwave spectrum.

Adding Impedance Vectors In the real world, there is resistance, as well as reactance, in an ac series circuit containing a coil and capacitor. This occurs because the coil wire has some resistance (it s never a perfect conductor). It can also be the case because a resistor is deliberately connected into the circuit. Whenever the resistance in a series circuit is significant, the impedance vectors no longer point straight up and straight down. Instead, they run off toward the northeast (for the inductive part of the circuit) and southeast (for the capacitive part). This is illustrated in Fig. 16-2.

1200 Wavelength (nm)

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