The Mechanics of Modulation
October 1931 QST

Paul Huntsinger wrote a nice introductory article on amplitude modulation (AM) in a 1931 edition of the ARRL's QST magazine. At the time, frequency modulation (FM) was still a laboratory curiosity, and many "experts" believed that FM would not provide any advantage over AM broadcasting. You might be tempted to think that sources of electrical noise that would interfere with AM were less at the time, thus negating the need for noise-immune FM, but the fact is by 1931 there was a lot of static caused by brushed motors, lousy automobile ignition systems, and arcing transmission lines, along with natural sources like lightning. Figures 6, 7, and 8 were missing in the original article, but fortunately I was able to get them from the next month's issue of QST.

The Mechanics of Modulation

By Paul R. Huntsinger*

Modulation is the process of varying the amplitude of a carrier wave in accordance with a signal to be transmitted. For purposes of explanation it is customary to assume that both the carrier frequency and the signaling frequency are sinusoidal although this is not necessarily the case in practice. In discussing the qualities of a given transmitter one usually speaks of the modulation capability of the outfit. The modulation capability of a transmitter may be expressed either as a factor or as a percentage, and in either case means the extent to which the amplitude of the carrier may be varied at the signal frequency without serious distortion to the latter.

The modulation factor is defined as the difference between the signaling and non-signaling amplitudes of the carrier divided by the non-signaling amplitude. Expressed as an equation,

         (1)

where M is the modulation factor. The difference between the modulated and unmodulated carrier wave amplitudes is evidently equal to the amplitude of the signal frequency, so we may write the above equation in the form,

         (2)

The modulation capability may be expressed as a percentage by merely multiplying the modulation factor by 100. Fig. 1 shows a blank carrier and also one which is modulated completely or 100%. Inspection of Equation 1 shows that for 100% modulation the amplitude of the carrier will be twice the non-signaling value. In the ideal case, as shown in Fig. 1, p = n = A.

Unfortunately in practice the positive and negative peaks are not always equal as shown in the figure, so the modulation factor p/A on the positive peaks may not be equal to that on the negative peaks, n/A. In case the positive and negative peaks, p and n, are unequal the modulation factor is defined as the arithmetic mean of the positive and negative modulation factors; that is,

        (3)

Inspection of the figure for the modulated wave shows that the limiting position of the negative peaks is zero; hence the ratio n/A can never be greater than unity. In other words the percentage of modulation on negative peaks can never be greater than 100. If a transmitter is modulated over 100 percent the tops of the negative peaks are cut off and the signaling frequency is seriously distorted.

Now the power of any wave varies as the square of the amplitude. Knowing the amplitude of the modulated carrier to be 1 + M times the amplitude of the blank carrier we may square it and find the ratio of the peak power of the modulated wave to the blank carrier power. As an equation,

        (4)

Thus we find that in the case of 100 percent modulation the peak power will be 2 squared or four times the unmodulated carrier power. Incidentally, this shows that for complete modulation the tube complement of the transmitter stages handling modulated r.f. must be capable of handling power peaks four times as great as the carrier power.

The average power of a modulated wave is found as follows: The amplitude of the modulated carrier wave is equal to the quantity (1 +M sin qt). As was stated before, the power of a modulated wave varies as the square of the amplitude, so the instantaneous power may be represented by the equation,

    (5)

The average power of the modulated wave, that is the average ordinate of the power wave, is found by integrating the power wave over a complete cycle and dividing by 2π. Thus we find

             (6)

For complete modulation we have M equal to unity and so the average power is seen to be 1 + 1/2 = 3/2 = 1.5 times the unmodulated carrier power. The power wave is shown in Fig. 2.

Now as for the antenna or tank current of a transmitter under modulation, the amplitude of the current will evidently be 1 + M times the unmodulated amplitude. Since the power varies as the square of the current, the average current will necessarily be equal to the square root of the average power. Thus we have

            (7)

In case the modulation is complete, or 100 percent, the average power will be 1.5 times the non-signaling power and hence the antenna current will be equal to the square-root of 1.5 or 1.226 times the unmodulated antenna current. It should be borne in mind that this will only be the case if the modulating signal is a pure sine wave and then only after sufficient time has elapsed for the antenna ammeter to reach its maximum steady reading.¹

Methods of Measuring Modulation Percentage

One method of measuring the modulation factor of a transmitter is by the use of a current-squared galvanometer.² The hook-up used is shown in Fig. 3, the pickup coil being coupled to the tank of the output stage of the transmitter. Two changes must be made in equation (1) before it is suitable for use in this method. First, since the galvanometer is calibrated in current squared, it will be necessary to take the square-root of the right hand member. Second, since the instrument reads effective rather than peak values and since the factor is defined in terms of peak values, we must also multiply by the square-root of 2. Making the above changes in the equation and simplifying we have,

        (8)

where D represents the deflection of the instrument under the conditions shown. A convenient way of finding the factor is to couple the pickup coil to the tank sufficiently so that a reading of 40 scale divisions is obtained with no modulation. Complete or 100 percent modulation will deflect the pointer to 60. For any deflection noted upon modulation, the factor may be computed by substituting the readings in the equation. Here is a check upon the above method. The full-scale reading of the instrument represents a current of 115 milliamperes. The current for any deflection will then be,

        (9)

From the above equation we find the current for a deflection of 40 divisions to be 72.773 ma. while that for 60 divisions is 89.079 ma. The ratio of the second to the first is 1.226 which, as shown before, is the ratio of the antenna currents for complete and zero modulation of the carrier wave. The fault in this method of determining the factor is that nothing whatever can be told of the relative amplitudes of the positive and negative peaks. However, the average modulation, as defined before, can be determined very easily by this method.

The modulation factor may be determined also by means of a linear rectifier and vacuum tube voltmeter.³ This method is used in the General Radio modulation meter. Both positive and negative peaks may be measured as well as the resulting carrier shift if the peaks are not equal. It is readily seen that if the positive peaks are greater than the negative peaks the carrier power will shift upward under modulation and vice versa as shown in Fig. 4. A transmitter should be tuned so that the mean plate current of the modulated stage and that of the linear amplifiers does not shift either up or down on modulation. If so tuned there will be no carrier shift and consequently the positive and negative peaks will be equal in amplitude.

The relation between audio-frequency volts inputs to a transmitter and percentage modulation must be linear over the working range of the transmitter if no distortion of the modulated signal is to take place. This relation, both for positive and negative modulation, is easily determined if one has a good audio frequency oscillator and level indicator. The ideal case is shown in Fig. 5A while a possible relation is shown in Fig. 5B. The negative peaks drop off slightly, hence there will be some carrier shift upward at this level. This falling off of the negative peaks is due to curvature in the lower end of the dynamic characteristic of the tubes. A transmitter should be adjusted so that there is no carrier shift in modulation and then the curves for both positive and negative peaks will coincide. If this is done practically the only distortion encountered will be a slight flattening of the tops of the wave at high modulation levels. If the curve for negative peaks is swung around we have the modulation characteristic of the entire transmitter. By assuming a sine wave input to the transmitter and plotting the output wave as shown in Fig. 6 we may analyze it for harmonic distortion. If the curves for positive and negative peaks coincide, the transmitter operating point is midway between the upper and lower bends in the dynamic characteristic of the tubes and the undistorted output will be a maximum.

The oscillograph is the best method for checking the operation of a transmitter since it will show the waveform of the output as well as the value of the peaks. A typical oscillogram is shown in Fig. 7A and was obtained by exciting the oscillograph element from the output of a linear rectifier which was coupled to the tank circuit. Fig. 7B shows what happens to the waveform of a signal when one tries to modulate 100% a transmitter having but 30% modulation capability. The cutting off of the positive halves of the cycle is caused by the modulator grid going positive.

Still another method of measuring the percentage of modulation is shown in Fig. 8. A resistor of proper size is connected in the plate supply lead to the modulated tube. The alternating-current voltage drop in the resistor is rectified by a copper-oxide rectifier which drives a direct current microammeter. The direct current component of the plate current is kept out of the rectifier by the blocking condenser C whose impedance is low at audio frequency. The resistor is made of such size that the instrument reads full scale when the alternating-current component of the plate current is equal to the direct current component, which is the condition for 100% modulation. Another form of the instrument uses a current transformer and alternating-current ammeter. This type is inferior to the first mentioned since the scale of the instrument is not linear and the damping is necessarily high, the last mentioned preventing the instrument from following the peaks.

* WOI. Iowa State College, Ames, Iowa.

1. The increase in antenna current as indicated by the r.f. ammeter will be 22.6% only when the modulating signal is sinusoidal. Complex modulation frequencies such as those caused by speech or music will give an antenna current increase of greater than 22.6% for amplitude modulation of 100%. - Editor.

2. See also Experimenters' Section, May, 1930; and "The Neglected Current-Squared Galvanometer," Feb., 1931. - Editor.

3. This method is used in the modulometer, described in QST, Aug., 1929. Positive or negative peaks may be measured by reversing the input connections to the grid circuit of the voltmeter tube. - Editor.

Posted March 29, 2022
(updated from original post on 6/22/2016)