Frequency Modulation Fundamentals
August 1939 QST Article
modulation (FM) was, is, and shall always be: x(t) = Xc·cos
[Ωct + β·sin (Ωmt)], where the carrier is
xc(t) = Xc·cos (Ωct), and the modulating
signal is xm(t) = β·sin (Ωmt). Yea verily,
Edwin H. Armstrong. Amen. The methods for generating and degenerating[sic]
FM might vary, but the fundamentals do not vary. Mr. Armstrong developed
and patented his system of frequency modulation in the late 1920s
and early 1930s, so when this article appeared in QST in
1939, FM was still fairly new - or even unknown - to most people.
Household radio dials still had only markings for the commercial
AM band (520 - 1720 kHz) and, in a few cases, a couple shortwave
bands (also AM). The information presented here is suitable for
study by anyone at any time.
August 1939 QST
articles are scanned and OCRed from old editions of the ARRL's QST magazine. Here is a list
of the QST articles I have already posted. All copyrights are hereby acknowledged.
Frequency Modulation Fundamentals
How Frequency Modulation Works; Its Advantages in Overcoming Noise
By Daniel E. Noble, W1CAS
Two 50,000-watt experimental transmitters and several lower-powered
transmitters will be placed in regular operation in the Fall using
the Armstrong frequency-modulation system. The marked noise suppression
which is the important characteristic of the system will make possible
a new standard of high-fidelity reception. The writer has been asked
to explain the action of this frequency-modulation system without
too much technical terminology. With all qualifications aside, the
picture looks something like this:
Fig. 1 - Illustrating amplitude and frequency modulation.
A, the motor-driven alternator used as an example; B, output
with constant field and constant speed (sine wave); C, output
with constant speed and variable field (amplitude modulation);
D, constant amplitude and variable speed (frequency modulation).
Fig. 2 - Vector Diagram of phase modulation. The modulator
vector reverses, producing a resultant Θ degrees ahead or
behind the carrier vector. This is equivalent to a sudden
change in the time axis, with the result that the frequency
changes. The vector will oscillate back and forth between
A and B at the modulating frequency. The more rapid the
oscillation the faster the change in the time axis, therefore
the greater the frequency deviation produced.
Every amateur knows what
frequency modulation is - it's something in his transmitter operation
that he doesn't want! To make the picture a little more exact, we
shall make use of a pure sine wave alternator. A pure sine wave
is a single-frequency wave; that is, no side bands and no harmonics
will be associated with it. A perfect frequency meter could locate
only one frequency with such a wave. If our alternator is the usual
motor-driven type with an external field supply, we can vary the
voltage output of the alternator by varying the field current. Let's
vary the field current slowly up and down and observe the result.
First, the output voltage of the alternator will increase and decrease,
and we have a condition commonly referred to as amplitude modulation.
See Fig. 1 (A, B, and C). Second, the output wave is no longer a
pure sine wave, and if we examine the wave with our perfect frequency
meter we shall find several frequencies present, because only the
pure sine wave will be limited to a single frequency. So much for
Now regulate the field supply so that the amplitude of the alternator
output will not change while the driving motor is made to speed
up and slow down. The frequency of the alternator will be determined
by the speed of the motor; if we speed up the motor the output frequency
will increase, and it will decrease when the motor slows down. Assuming
that the amplitude of the output remains constant, we have produced
a frequency-modulated wave by the simple process of speeding up
and slowing down the motor. What has happened to the wave? First,
obviously the wave is no longer a pure sine wave, since the frequency
is changing. Second, since the wave is not a pure sine wave, several
frequencies will be present (theoretically, an infinite number).
When we neglect inertia and speed up and slow down the motor in
such a way that the change in speed is at the rate of ten cycles
per second, and the cycles are perfect sine-wave cycles, we will
produce a frequency series for a 1000-cycle generator something
like this: . . . 1000 - 30, 1000 - 20, 1000 - 10, 1000, 1000 + 10,
1000 + 20, 1000 + 30 . . . and so on to an infinite number of side
bands. Although frequency modulation will produce a composite wave
made up of the carrier, plus and minus a regular harmonic series
of the modulating-signal frequency and the carrier, we are fortunate
in the fact that the amplitudes of the side bands decrease rapidly
as the signal harmonic number increases.
To go back to our
motor-generator again, the motor was speeded up and slowed down
to produce our frequency modulation but we didn't say how much we
speeded it up or how much we slowed it down. We can change the motor
speed so that the frequency will vary instantaneously as follows:
1000 --> 1025 --> 1000 --> 975 --> 1000 cycles, and
make the entire excursion in one-tenth of a second for a modulating
frequency of ten cycles per second. Or we can go 1000 -->1050
--> 1000 --> 950 --> 1000 in one-tenth of a second for
a 10-cycle modulation frequency. The difference is found in the
more extended change in frequency in the second case. This change
is called the "deviation." For the first case the deviation is 25
cycles and for the second, 50 cycles. Deviation is then the maximum
instantaneous change in frequency. Just to increase the confusion,
we might add that we can't find the deviation with the frequency
meter since no continuous spectrum is produced but, rather, we produce
discrete side bands which may be detected and their physical existence
made evident by means of our frequency meter. These side bands may
be found far beyond the limits of the deviation. We might define
the maximum instantaneous frequency for our special case as the
frequency we would get from our alternator if we held the speed
constant when the maximum speed was reached. We do not actually
produce such a maximum frequency because the speed does not remain
constant. All this leads to conclusion that we can expect the band-width
of the frequency modulated wave to be greater than twice the deviation.
Fig. 3 - A practicable frequency-modulator
circuit, after Weir. The oscillator is frequency-modulated by the
a.f.c. tube (modulator) which causes a frequency deviation in proportion
to the amplitude of the audio voltage. A small part of the output
signal is fed to the converter tube, which is heterodyned by a stable
crystal oscillator to give a beat frequency at 1500 kc. The i.f.
output operates the rectifier (discriminator) and by providing the
modulator with a d.c. bias which varies when the mean oscillator
frequency tends to change (a.f.c. action) maintains the carrier
frequency constant. Deviations of approximately 30 to 40 kc. may
be obtained in the region of 20 Mc. using a 6L6 modulator and 6F6
oscillator. The stability of the system will be determined by the
discriminator circuit stability.
A frequency-modulated wave
may be produced much more readily with vacuum tube equipment than
with rotating machinery. Rotating a condenser back and forth to
change the capacity in an oscillator circuit will produce a frequency-modulated
wave. Placing a condenser microphone in an oscillator circuit in
such a way that changes in the microphone capacity will influence
the frequency of the oscillator is an obvious means of producing
a modulated wave. The circuit used in automatic frequency control
systems is an excellent frequency-modulation system.
The modulation method invented by Major Edwin Armstrong is very
stable since the carrier is controlled by a quartz crystal oscillator.
A 200-kc. oscillator supplies voltage to a phase-shift network from
which two components of the carrier are extracted, differing only
in phase. One component is 90° out of phase with the other. Mathematically,
the difference between the amplitude-modulated wave and the frequency-modulated
wave is the difference in the phase relations between side bands
and carrier. If the side bands of an amplitude-modulated wave could
be extracted from the carrier, shifted in phase 90°, and then recombined
with the carrier, a frequency- modulated wave would result. Major
Armstrong did not extract the side bands but he did arrange to produce
side bands without a carrier by means of a balanced modulator working
with one of the 200-kc. components mentioned above, and then to
combine the side bands with the second component in such a way that
the side bands were 90° out of phase with the normal arrangement
for carrier and side bands in the amplitude modulated wave. His
result was a frequency-modulated wave of the special type sometimes
referred to as a "phase-modulated" wave. Another way to describe
the action of Major Armstrong's modulator is to say that he combined
a carrier voltage with a side-band voltage which had been rotated
through 90°. This gives us the simple picture of two vectors 90°
out of phase combining to give the resultant voltage. Fig. 2 will
assist the reader to visualize the process. As the side-band voltage
is increased and decreased, the resultant of the two vectors is
caused to shift phase. The shift in phase corresponds to a frequency
change, and the amount of frequency change produced will depend
upon the magnitude of the phase shift and upon how rapidly the phase
shift is taking place. Since the magnitude and the speed of the
phase shift is determined by the side-band vector, the deviation
produced will be determined by the magnitude and the frequency of
the modulating signal. The only difference between pure frequency
modulation1 and phase modulation is the fact that the
deviation is a function of the amplitude only of the modulating
signal for pure frequency modulation while the frequency of the
signal also determines the deviation for phase modulation. A network
placed in the audio input amplifier making the output signal voltage
inversely proportional to frequency will make the overall response
independent of signal frequency, and thus the phase modulator will
produce a pure frequency-modulated wave. The actual deviation produced
at 200 kc. is small, something of the order of 15 to 20 cycles.
Therefore, a series of doublers must be introduced to increase the
maximum deviation to 100 kc. A total of twelve or thirteen doubler
stages is used to reach the required deviation.
Fig. 4 - Essentials of a superheat receiver suitable for
Fig 5. - Elementary detector circuit for frequency-modulated
waves. CD is tuned to the lower extremity of useful side
bands, AB to the upper extremity. The voltage appearing
across either circuit is determined by the amplitude of
the audio modulating voltage. A hybrid wave appears across
each circuit. Rectification recovers the audio component.
of modulation suggested by Murray C. Crosby and developed by Irving
R. Weir makes use of the automatic frequency control variable oscillator
for modulating the frequency, and of the a.f.c. discriminator circuit
for stabilizing the oscillator carrier. Fig. 3 illustrates the type
of circuit used. The modulator tube injects 90° out-of-phase current
into the oscillator tank circuit. The effect of changing the modulator
injector current is comparable to changing the tank capacity in
the oscillator circuit. The stabilizing circuit functions in the
usual a.f.c. manner.
The receiver requirements are not so complicated as one might suspect.
The usual super-heterodyne is used with a few additions and changes.
The pass band must be greater than twice the transmitter deviation.
A limiter precedes the detector, and this limiter has the very important
function of "wiping off" any amplitude modulation which may have
been introduced by noise voltages. The limiter passes on the frequency-modulated
wave with constant amplitude to the frequency detector, which changes
the frequency-modulated wave into a hybrid wave with both amplitude
and frequency modulation components. An ordinary detector then recovers
the signal from the amplitude component. Fig. 4 illustrates the
line-up. Figs. 5 and 6 show two frequency detectors.
Fig. 6 - The discriminator circuit combines the functions
of frequency detector and rectifier to recover the audio
Fig. 7 - A parallel-resonant circuit will act as a frequency
detector when tuned to carrier at points A or B. With the
circuit tuned so that the carrier is at A. The voltage across
the circuit will rise and fall in step with the deviation
produced by the modulating voltage; the result is an amplitude-
modulated wave which is also frequency-modulated. A rectifier
will recover the amplitude audio component.
Transmitter house and six-bay turnstile antenna at W1XPW,
a 1000-watt experimental frequency-modulated transmitter
located on top of West Peak, Meriden Mountain, near Meriden,
Conn. The transmitter operates on 43.4 megacycles. WDRC,
Inc., is the owner.
circuit of the type shown in Fig. 7 also will act as a frequency
detector. The carrier is tuned in on one side of the resonance curve.
A steady-state r.f. voltage will result from the unmodulated carrier,
and modulation will produce instantaneous frequency changes. Taking
A as the operating point, any change corresponding to an increase
in frequency of the signal will increase the amplitude of the voltage
across the parallel-resonant circuit, and an equivalent decrease
in frequency will decrease the voltage across the circuit. Therefore,
since the modulation produces magnitudes of frequency change or
deviation corresponding to the amplitude of the audio modulating
signal, and since the rate at which the changes or deviations take
place corresponds to the frequency of the audio signal, the voltage
appearing across the parallel-resonant circuit will be amplitude-modulated.
Frequency modulation will also be present but we are no longer interested
in that. Rectification will recover the audio signal. Any receiver
of the usual type can be made to receive frequency- modulated signals
after a fashion by detuning slightly, but the reader is assured
that the "fashion" is not very satisfactory.
Remarkable results in the suppression
of noise and interference are possible with the frequency modulation
system. Since the limiter wipes off all amplitude variations, noise
of this type must appear as frequency modulation produced by the
phase shift resulting from the combination of the signal and noise
voltages. For the case where the peak noise amplitude is half the
signal amplitude and the phase relation between signal and noise
is ninety degrees, the maximum phase shift would be approximately
26.5°. Very little frequency modulation will be produced if this
phase shift is the result of noise modulated by a low-frequency
audio component, but the frequency modulation will increase directly
with the frequency of the audio noise component. The receiver will
display greatest susceptibility to noise frequencies above audibility.
Logical design of the receiver would call for a sharp cut-off of
the audio amplifier response or, better still, a falling high-frequency
characteristic which will reduce the hiss response. A simple predistortion
network at the transmitter will present a compensating rising high-frequency
response so that the overall response of the system is flat. This
is the arrangement used in the stations now on the air.
very remarkable effect of the limiter action upon the suppression
of interference has been demonstrated by Weir.2 He reports
that with two stations operating on the same channel the stronger
station would prevail 100 per cent at the receiver whenever the
stronger station's signal was more than twice the strength of the
weaker signal. He also reports in the same paper that no interference
area of the usual kind existed where the signals were of nearly
the same amplitude. In this area the movement of the antenna a few
inches would throw one program out and bring in the other one. The
presence of standing waves accounts for the phenomenon since the
nodes would permit the selection of the required voltage radio.
Mathematically the action of the limiter is rather complicated,
but the results of the limiter action are an overall effect of cutting
the amplitude of the received voltage in such a way that the strong
signal component dominates while the weak signal is suppressed.
In other words, the strong signal will always take control of the
receiver. The frequency-modulation system permits2 as
much as 25 dB gain in signal-plus-noise- to-noise ratio over that
possible with an amplitude system of equal carrier strength.
this gain in equivalent power is due in part to the limiter action
it is also the result of the very interesting effect which makes
the magnitude of the recovered power at the receiver a function
of the modulation deviation. If a deviation of 50 kc. produces voltage
A at the receiver, then a deviation of 100 kc. will produce a voltage
two times A at the receiver. Here the received voltage has been
doubled without changing the carrier power at the transmitter. In
the amplitude case the peak carrier power must increase four times
when the modulation changes from zero to 100 per cent. Since without
the power change the received voltage increases for the frequency-modulated
system with an increase in deviation, it follows that the advantage
of the system over the amplitude system will increase as the deviation
is increased. The practicable limits must be determined by available
channel width. The Federal Communications Commission has assigned
200-kc. channels for the broadcast stations now in operation. For
this channel width the deviation will probably be restricted to
80 kc. or less. Present standards seem to point to a modulation
index (that is: ratio of deviation to audio frequency) of 80,000/15,000'
or approximately 5.3.
Necessarily this is a very sketchy
account of Major Armstrong's invention. The writer hopes that it
may serve as an introduction to the subject, and for those who are
interested in the more detailed and technical aspects a carefully
selected bibliography is appended.
1: The term "phase modulation" is something
of a pain. Actually there are as many types of frequency modulation
as there may be functions of X. Phase modulation is one type. The
type referred to as "pure" frequency modulation
is the unadulterated, holy, sweet, etc. variety in which the deviation
produced is a linear function of the modulating
signal amplitude only. "Phase modulation" is still "frequency modulation."
2: I.R. Weir, "Field Tests and Amplitude Modulation with
Ultra-High-Frequency Waves," Part I. General Electric Review,
Selected Frequency Modulation
John R. "Notes on the Theory of Modulation," Proceedings IRE, Vol.
10, No.1, February, 1922. First paper in which the frequency-modulated
wave is analyzed mathematically and the required "band width" determined.
2. Armstrong, Edwin H. "A Method of Reducing Disturbances in
Radio Signaling by a System of Frequency Modulation." Proc. I.R.E.,
Vol. 24, No.5, May, 1936. Undoubtedly the classical paper in the
field. The first published account of the wide-band frequency modulation
3. Crosby, Murray C. "Frequency Modulation Noise
Characteristics." Proc. I.R.E., Vol. 25, No.4, April, 1937. Mathematical
treatment and experimental verification of wide-band frequency modulation
vs. amplitude noise suppression.
4. Roder, Hans. "Frequency
Modulation." Electronics, Vol. 10, No.5, May, 1937. Mathematical
analysis of validity of noise-suppression effect in wide-band frequency
5. Carson, John R. and Fry, Thornton C. "Variable
Frequency Electric Circuit Theory with Application to the Theory
of Frequency Modulation." Bell System Technical Journal, Vol. 16,
No.4. Fundamental formulas for variable frequency electric circuit
theory are developed. Transmission, reception and detection of frequency
modulated waves are studied analytically.
6. Roder, Hans.
"Tuned Circuits and a Frequency Modulated Signal." Proc. I.R.E.,
Vol. 25, No. 12, Mathematical treatment of tuned circuits.
7. Weir, I. R. "Field Tests of Frequency-and-Amplitude-Modulation
with Ultra-High-Frequency Waves." General Electric Review, May,
1939, Part I; June, 1939, Part II. A very important paper of interest
to both the technical and non-technical readers. Describes a simplified
8. Day, John R . "A Receiver for Frequency Modulation."
Electronics, June, 1939. The first published constructional data
for a seven-tube frequency-modulation receiver.