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, thus sayeth
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
of Contents]These 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 and
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
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 amplitude modulation.
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.
Producing Frequency Modulation
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
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 frequency-modulated
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.
A system 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 signal.
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
A simple 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.
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
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.
While 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
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
1. Carson, 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 system.
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
4. Roder, Hans. "Frequency Modulation." Electronics,
Vol. 10, No.5, May, 1937. Mathematical analysis of validity of noise-suppression
effect in wide-band frequency modulation.
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.
Hans. "Tuned Circuits and a Frequency Modulated Signal." Proc. I.R.E.,
Vol. 25, No. 12, Mathematical treatment of tuned circuits.
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 transmitter.
8. Day, John R . "A Receiver for Frequency Modulation." Electronics,
June, 1939. The first published constructional data for a seven-tube
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