Making Modulation Easy to Understand
September 1966 Radio-Electronics

September 1966 Radio-Electronics [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.

Norman Crowhurst's 1966 Radio-Electronics article demonstrates how vector analysis simplifies understanding modulation in circuits. Building on his previous work, Mr. Crowhurst explains that amplitude modulation (AM) can be visualized using a stationary carrier vector and rotating sideband vectors, making waveform addition easier than point-by-point sine wave graphing. For frequency modulation (FM) and phase modulation (PM), vectors reveal why additional sidebands are necessary to maintain linearity and constant amplitude. The first-order sidebands introduce phase deviation, while higher-order sidebands (2nd, 3rd, etc.) correct amplitude fluctuations and improve phase linearity. He uses vector diagrams to show how FM/PM sidebands interact, contrasting them with AM's phase-locked resultant. The article highlights that while mathematical analysis (e.g., Bessel functions) is complex, vectors provide an intuitive grasp of modulation dynamics, especially in balancing amplitude stability and phase accuracy as deviation increases. Figures illustrate key concepts, such as mixed modulation effects and sideband contributions at different modulation levels. See Vectors Show How Circuits Work in the July 1966 issue.

Making Modulation Easy to Understand

This sequel to "Vectors Show How Circuits Work" proves the point.

By Norman H. Crowhurst

In the July issue, I showed you how vectors can help you understand circuit operation. In this followup article, I'm going to show you how to uncover some of the mysteries of modulation through the use of vectors. Take a quick look through the first article (page 58 July Radio-Electronics) if you've forgotten any of the fundamentals I brought out there. Then proceed, and see how vectors truly do help you understand circuit operation.

Amplitude Modulation

You may recall that amplitude modulation can be represented by a carrier and two sidebands. You can graph this by laboriously adding the sine waves that represent these three frequencies, point by point. Once you realize that vectors "work", you can do the same thing much more easily from a simple vector diagram.

You're probably familiar with the convention that a vector rotates counterclockwise at the frequency it represents. If we have three different frequencies, such as a carrier and two sidebands, then we shall have three vectors, each rotating counterclockwise at a rate determined by its frequency.

With vectors that represent a circuit working at only one frequency, we examine relationships in only one position of the vectors, to make the drawing easier. We imagine, in effect, that we're going around with the vectors, so they appear stationary to us. Or, if you imagine the vectors being driven around by some sort of motor, they could be viewed as stationary by using a strobe light operated at the same frequency. We "stop" them all at a particular point in their travel.

Now, when we imagine the amplitude-modulation vector diagram, we work the strobe light at the carrier frequency. The carrier vector will now appear stationary. The lower sideband will appear to rotate clockwise, and the upper sideband vector will appear to rotate counterclockwise, each at the frequency by which it differs from the carrier frequency.

Completing the analysis, we can show the vector diagrams at different points through the modulation cycle, and show that the resultant vector, made up by adding the carrier and its two sidebands, is always in phase with the carrier (never at an angle to it) and is of varying amplitude, to represent true amplitude modulation (Fig. I).

In this diagram, C is for carrier, L for lower sideband and U for upper sideband, with R for resultant. We also see a half cycle of all three frequencies at points marked 1, 2 and 3 on the modulation envelope, to demonstrate how the waves themselves add up at the corresponding points on the modulation wave.

Frequency or Phase Modulation

So far, so good. Now we turn to frequency or phase modulation. For anyone modulating frequency, these two types of modulation come out to the same thing. The distinction between them shows up when we consider impressing a complex wave, or a complete program on the carrier, and then determine the relationship between the modulating waveform and the carrier deviation it produces. Frequency modulation changes the frequency to match the instantaneous amplitude of the modulating waveform; phase modulation changes the carrier phase (Fig. 2).

For our analysis, at one modulating frequency the methods are identical: linear frequency modulation is also linear phase modulation, and vice versa.

Linear amplitude modulation makes the amplitude of the carrier fluctuate in faithful replica of the modulating waveform. It also keeps the carrier frequency or phase constant, as shown by the fact that the resultant wave is always in phase with the carrier. The object of frequency (or phase) modulation is to have the frequency or phase of the resultant vary in faithful replica of the modulating wave, while keeping its amplitude constant.

Frequency or phase modulation is developed by using circuits that modulate the carrier in one of these ways. This modulation must add sidebands to achieve the desired effect. For other reasons (connected with bandwidth and interference), the number and magnitude of the sidebands must be restricted.

What limitation on performance do these restrictions impose? Some discussions imply that nothing is really lost by clipping off some of the sidebands. Even the best explanations have left the matter somewhat hazy. To examine why sidebands are necessary and what they contribute to overall performance quality, we will examine them as if they are added consecutively, in pairs, as required. Then we will analyze just what can be done with successive limited numbers of side-band pairs.

The first step is to add the first pair of sidebands in a different phase relationship from that used for amplitude modulation (Fig. 3). These sidebands have equal amplitude, and start 180° out of phase with each other. Now the resultant swings forward and backward in phase, as well as changing amplitude much less than in Fig. 1 (and at twice the frequency, being minimum at zero and increased at both extremes of the modulating wave). As long as the phase deviation is small, this looks good: the resultant swings on alternate sides of the carrier, and its magnitude doesn't change too much.

But as soon as we step up the amount of modulation a little more, the amplitude starts to change more rapidly than the phase. Before the phase swing can reach 90° in each direction from center, amplitude goes up to infinity. This is why frequency or phase modulation must add further sidebands. First we add another pair, called second-order sidebands. Their frequency of contrarotation, relative to the carrier, is double that of the first pair (Fig. 4). These new sidebands "b" start out in phase with each other. Their amplitudes are equal.

If the second-order sidebands augment the carrier at the zero-modulation (center) point, they will neutralize one an-other at the half-phase (45°) points, where the first-order sidebands produce a resultant 0.707 of the maximum deviation. At maximum deviation (90°) of the modulating wave), the second-order sidebands will oppose the carrier by as much as they augmented it at the zero point.

Now we can examine how the resultant changes at these three positions (five positions, if we consider positive and negative phase excursions, which are identical). Each deviation point considered is analyzed into in-phase (zero-angle) and quadrature (right-angle) components. For the zero-phase point, the resultant is the carrier plus twice the individual second-order magnitude. For the maximum deviation point (90° and 270°), the in-phase component is carrier minus twice the second-order magnitude, while the quadrature component is twice the first-order magnitude.

Using C for carrier, a for first-order magnitude and b for second-order magnitude, we can write expressions for the resultant, R, using the Pythagorean theorem. We equate those expressions, to represent the condition that the" magnitude is the same at zero and maximum deviation points. (Actually, we equate R², to make the algebra simpler: if R² is equal at both points, R will be also). Then we derive an expression for second-order magnitude, in terms of carrier and first-order magnitude.

We want to know two more things: Does the amplitude stay constant, because we've made it the same at the middle and extremes. Is the phase movement linear with the modulating signal it represents?

To answer the first: making the extremes equal to the middle position magnitude required the value of the resultant at each to satisfy R = C + 2b, or R² = C² + 4Cb + 4b². Substituting the condition that yields this equality, the intermediate value of R² (at the 45° point) becomes R² = C² + 2a² (which it is regardless, but by substitution) = C² + 4Cb. In this last form, it is obviously less than R² at the middle and extremes, by 4b².

So the bigger b becomes, the more the amplitude fluctuates at the intermediate modulation point.

The phase-linearity test involves a little trig, but this is fairly simple if we use different reference angles. At the 30° phase-modulation points, deviation (measured as an angle) should be half the maximum, because sin 30° = 0.5. So now we have different relationships at the intermediate phase point, although the middle and extremes are the same as be-fore (Fig. 5).

The first-order sidebands will have half their maximum quadrature value in the resultant. The second-order side-bands will be half their maximum value, in-phase-augmenting the carrier (they augmented it by twice this in the zero or middle position).

To check what happens now, we make the half-phase deviation precisely half the maximum deviation, using the tangent formula

tan 2Φ = 2tanΦ/(1 - tan²Φ) 

Making the proper substitutions, tan Φ (the half-angle vector) is its quadrature component divided by its in-phase component, or a/(C+b), while tan 2Φ is the same ratio for the full-angle vector, or 2a/(C-2b) . Equating the two values for tan 2Φ (one from its vector and the other from the formula using tan Φ and simplifying by ordinary algebra (not shown in detail) yields the condition a² = 3 (Cb + b²).

As the expression for the zero-phase amplitude does not include any quadrature component (component with a in it) and those for half-phase and full-phase amplitudes both do, we can substitute for a² in each of the latter, to get the same form, so amplitude variation is more apparent. Now the expressions are

For zero phase, R² = C² + 4Cb + 4b²

For half phase, R² = C² + 5Cb + 4b²

For full phase, R² = C² + 8Cb + 16b²

 The value substituted for a², in terms of C and b, makes the half-phase point true. Maintaining linearity of phase modulation makes the amplitude vary even more than it does when zero and maximum-phase amplitudes were made equal, because half-phase and full-phase values of R² are both greater than the zero-phase value,

From this point, we can add the third-order sidebands and see what they yield. This has been done, qualitatively, in Fig. 6. The math to go with it gets a little more involved.

The point to emphasize is that, as each sideband is added, vector analysis provides a complete picture of what happens, which cannot readily be gained from the analysis using pure math involving Bessel functions. Adding this third sideband pair enables both the constant-amplitude and the phase-linearity conditions to be preserved more closely.

As deviation is increased, fourth-order sidebands are needed to keep phase linearity.

What FM Sidebands Do

In summary, the first-order sidebands change their amplitude before they show appreciable nonlinearity of phase modulation. The second-order sidebands primarily correct the variation in amplitude, but in doing so fail to maintain phase linearity. The third-order sidebands enable phase linearity to be preserved further, but again leave amplitude deviating a little more. The fourth-order sidebands correct this, and so on, back and forth, as wider overall deviations become possible.

We can see that the first difference between amplitude and phase modulation is in the phasing of the same first-order sidebands. From this it can be shown that a phase shift here will transfer some modulation energy from amplitude to phase, or vice versa (Fig. 7).

The successive magnitudes of the carrier and the different sidebands can be shown for different magnitudes of maximum deviation.

The carrier gets smaller and smaller and eventually passes through zero and reverses its phase, relative to the rest of the family. Further study would show that all the sideband amplitudes go through cyclic variation of magnitude and phase, as deviation is increased, to preserve the same constant resultant magnitude.

This is an interesting and enlightening way to study frequency modulation. It shows "what has to give." from a performance viewpoint, to maintain linear modulation and constant amplitude. We can use vectors, as we said, to show how various other electronic circuits work. In later articles, if we find you're interested, we can take up some of them.