FM Wave Generation

Frequency Modulation

Another means of encoding a wave of information is by producing a complex
wave whose frequency is varied in proportion to the instantaneous amplitude of the
information wave. Such an encoding is frequency modulation. The result of this
encoding or modulation process is a complex modulated wave whose instantaneous
frequency is a function of the amplitude of the modulating wave and differs from the
frequency of the carrier from instant to instant as the amplitude of the modulating wave
varies.
The following equation provides the equivalent formula for FM:

v(t) Asin( t M sin( t)) c f i = w + w

where,
v(t) is the instantaneous voltage
A is the peak value of the carrier
wc is the carrier angular velocity
Mf is the modulation index
wi is the modulating signal angular velocity

The FM formula is really complex. In figure 1 is the waveform of a FM signal. To solve
for the frequency components of an FM wave requires the use of the Bessel functions.
They show that frequency-modulating carrier with a pure sine actually generates an
infinite number of sidebands spaced at multiples of the intelligence frequency, fm, above
and below the carrier. Fortunately, the amplitude of these sidebands approaches a
negligible level the farther away they are from the carrier, which allow FM transmission
within finite bandwidths.

                                                  Figure 1: FM Signal Representation
The Bessel functions solution to the FM equation is

where
1 = carrier component
2 = component at ± fi around the carrier
3 = component at ± 2fi around the carrier
4 = component at ± 3fi around the carrier
To solve for the amplitude of any side-frequency component, Jn is equal to

In figure 2 is an example of a FM spectrum

                                                                 Figure 2: FM Spectrum
Center Frequency
The center frequency is that frequency assigned to the carrier, but during
modulation the carrier is not always present in the complex modulated wave and the
instantaneous frequency of the complex modulated wave varies above and below the
frequency assigned to the carrier. This assigned frequency, then, is the “center” about
which the instantaneous frequencies of the modulated wave vary.

Frequency Deviation
In FM the shift in frequency is proportional to the amplitude of the modulating
wave. A weak modulating wave will have a small peak amplitude, which will produce a
small peak frequency variation. A strong modulating wave whose peak amplitude is the
maximum that the modulating system is designed to handle will produce the maximum
peak frequency variation. Any of these peak variations from the center frequency is
called the frequency deviation (fd).

The maximum value of the frequency deviation (fD) is a system constant, and
when it is intended that the modulated wave be radiated, its magnitude is established by
law.

Frequency Swing
The overall extreme of the excursion of instantaneous frequencies from maximum
negative to maximum positive is called the frequency swing. Frequency swing,
therefore, is equal to twice the maximum design frequency deviation. In FM the
frequency swing is a system constant and is usually expressed in terms of the maximum
frequency deviation as ±fD.

Deviation Ratio
Any modulating system is intended to accommodate some specific band of
modulating frequencies. Therefore the lower limit and especially the upper limit of this
band are of importance in the design of the equipment. The upper frequency is the most
important because it determines the maximum bandwidth requirements. In FM, the
highest modulating frequency (fM) is a system constant. The ratio of the maximum
frequency deviation (fD) to the highest modulating frequency (fM) is called the deviation
ratio (D).

The deviation ratio is strictly an equipment characteristic and is a quantity used to set the
circuit bandwidth of a modulating system.

Modulation Index
There is a very important signal characteristic that resembles the deviation ratio
and under certain circumstances is equal to it. It is the ratio,

Notice, however, that fd is any frequency deviation, not necessarily maximum and fm is
any modulating frequency, not necessarily the highest. There are system restrictions that
apply; for example fd cannot exceed fD and fm cannot be greater than fM.

Bandwidth Calculations
Since the sidebands are separated by multiples of the modulating frequency, the
frequency of the highest important sidebands is given by

The overall bandwidth from the highest to the lowest side frequency whose amplitude is
15 % (or greater) of the unmodulated carrier is twice this value. In figure 3 is the
commercial FM bandwidth allocation for two adjacent stations

                         Figure 3: Commercial FM bandwidth allocation for two adjacent stations

FM Generation Circuit (Reactance Modulator)
The reactance modulator is a very popular means of FM generation and is shown
in figure 4. The reactance modulator is an amplifier designed so that its input impedance
has a reactance that varies as a function of the amplitude of the applied input voltagemodulating
signal).

                                                       Figure 4: FM Reactance Modulator