**Principles of Amplitude Modulation**

Examining the basic equation for an AM signal, introduced in article 3, gives us several clues as to how AM can be generated. The equation is

**υAM = Vc sin 2πfc t + (Vm sin 2πfmt)(sin 2πfc t)**

where the first term is the sine wave carrier and second term is the product of the sine wave carrier and modulating signals. (Remember that υAM is the instantaneous value of the amplitude modulation voltage.) The modulation index m is the ratio of the modulating signal amplitude to the carrier amplitude, or m = Vm/Vc, and so Vm = mVc. Then substituting this for Vm in the basic equation yields υAM = Vc sin2πfct + (mVc sin 2πfmt)(sin 2πfct). Factoring gives υAM = Vc sin 2πfct(1 + m sin 2πfmt).

**AM in the Time Domain**

When we look at the expression for υAM, it is clear that we need a circuit that can multiply the carrier by the modulating signal and then add the carrier. A block diagram of such a circuit is shown in Fig. 4-1. One way to do this is to develop a circuit whose gain (or attenuation) is a function of 1 + m sin 2πfmt. If we call that gain A, the expression for the AM signal becomes

**υAM = A(υc)**

where A is the gain or attenuation factor. Fig. 4-2 shows simple circuits based on this expression. In Fig. 4-2(a), A is a gain greater than 1 provided by an amplifier. In Fig. 4-2(b), the carrier is attenuated by a voltage divider. The gain in this case is less than 1 and is therefore an attenuation factor. The carrier is multiplied by a fixed fraction A. Now, if the gain of the amplifier or the attenuation of the voltage divider can be varied in accordance with the modulating signal plus 1, AM will be produced. In Fig. 4-2(a) the modulating signal would be used to increase or decrease the gain of the amplifier as the amplitude of the intelligence changed. In Fig. 4-2(b), the modulating

signal could be made to vary one of the resistances in the voltage divider, creating a varying attenuation factor. A variety of popular circuits permit gain or attenuation to be varied dynamically with another signal, producing AM.

**AM in the Frequency Domain**

Another way to generate the product of the carrier and modulating signal is to apply both signals to a nonlinear component or circuit, ideally one that generates a square-law function. A linear component or circuit is one in which the current is a linear function of the voltage [see Fig. 4-3(a)]. A resistor or linearly biased transistor is an example of a linear device. The current in the device increases in direct proportion to increases in voltage. The steepness or slope of the line is determined by the coefficient a in the expression i 5 aυ. A nonlinear circuit is one in which the current is not directly proportional to the voltage. A common nonlinear component is a diode that has the nonlinear parabolic response shown in Fig. 4-3(b), where increasing the voltage increases the current but not in a straight line. Instead, the current variation is a square-law function. A square-law function is one that varies in proportion to the square of the input signals. A diode gives a good approximation of a square-law response. Bipolar and field- effect transistors (FETs) can also be biased to give a square-law response. An FET gives a near-perfect square-law response, whereas diodes and bipolar transistors, which contain higher-order components, only approximate the square-law function. The current variation in a typical semiconductor diode can be approximated by the equation

**i = aυ + bυ2**

where aυ is a linear component of the current equal to the applied voltage multiplied by the coefficient a (usually a dc bias) and bυ2 is second-order or square-law component of the current. Diodes and transistors also have higher-order terms, such as cυ3 and dυ4 ; however, these are smaller and often negligible and so are neglected in an analysis. To produce AM, the carrier and modulating signals are added and applied to the nonlinear device. A simple way to do this is to connect the carrier and modulating sources in series and apply them to the diode circuit, as in Fig. 4-4. The voltage applied to the diode is then

The first term is the carrier sine wave, which is a key part of the AM wave; the second term is the modulating signal sine wave. Normally, this is not part of the AM wave. It is substantially lower in frequency than the carrier, so it is easily filtered out. The fourth term, the product of the carrier and modulating signal sine waves, defines the AM wave. If we make the trigonometric substitutions explained in Chap. 3, we obtain two additional terms—the sum and difference frequency sine waves, which are, of course, the upper and lower sidebands. The third term cos 2ωct is a sine wave at two times the frequency of the carrier, i.e., the second harmonic of the carrier. The term cos 2ωmt is the second harmonic of the modulating sine wave. These components are undesirable, but are relatively easy to filter out. Diodes and transistors whose function is not a pure square-law function produce third-, fourth-, and higher-order harmonics, which are sometimes referred to as intermodulation products and which are also easy to filter out. Fig. 4-4 shows both the circuit and the output spectrum for a simple diode modulator. The output waveform is shown in Fig. 4-5. This waveform is a normal AM wave to which the modulating signal has been added. If a parallel resonant circuit is substituted for the resistor in Fig. 4-4, the modulator circuit shown in Fig. 4-6 results. This circuit is resonant at the carrier frequency and has a bandwidth wide enough to pass the sidebands but narrow enough to filter out the modulating signal as well as the second- and higher-order harmonics of the carrier. The result is an AM wave across the tuned circuit. This analysis applies not only to AM but also to frequency translation devices such as mixers, product detectors, phase detectors, balanced modulators, and other heterodyning circuits. In fact, it applies to any device or circuit that has a square-law function. It explains how sum and difference frequencies are formed and also explains why most mixing and modulation in any nonlinear circuit are accompanied by undesirable components such as harmonics and intermodulation products.

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