It is possible to write a mathematical expression for the output signal of this circuit:
The constant is the modulation index for FM. It is defined as follows:
The Greek letter δ represents the frequency deviation and fm represents the modulating frequency that causes the deviation. As in the case of AM, this time domain representation of the FM signal can be converted to an equivalent frequency-domain expression that includes the carrier and sidebands. Because the mathematics required for this conversion are quite complex, we will only consider the result:
The Jn(x) functions are known as Bessel Functions of the First Kind. Graphs of Jn(x) look like slowly decreasing sine and cosine functions. The Jn(x) functions are a closely related family of functions in the same way that sin(nx) and cos(nx) for a family of similar functions.
The zeroth order Bessel function, J0(m) determines the amplitude of the carrier. The nth Bessel function Jn(m) determines the amplitude of the nth pair of sidebands. There are two important concepts contained in the expression shown above:
The amplitude of the carrier depends on m. the modulation index. This is quite different from AM, where the amplitude of the carrier was independent of the value of m
There are an infinite number of sidebands. Thus the theoretical bandwidth of FM is infinite.
An infinite bandwidth signal would be very difficult to transmit. Fortunately, the higher order sidebands in FM have extremely low amplitude and may be ignored. For example: if the modulation index is 5, only the first 7 sidebands are significant in value.
There is a rule of thumb, known as Carson’s Rule, that predicts the bandwidth occupied by the significant sidebands of an FM signal, based on the maximum modulation frequency and its corresponding modulation index:
Parameter fm is the frequency of the modulating signal, δmax is the maximum deviation, and m is the corresponding modulation index. If a range of frequencies is used to modulate the carrier, the maximum modulating frequency and its corresponding modulation index are used.
Commercial FM broadcasting uses a maximum deviation of 75 KHz and a maximum modulating frequency of 15 KHz. Substituting these values into Carson’s Rule gives:
B = 2*(75+15) = 180 KHz.
The carrier amplitude of an FM signal is determined by the value of J0(m). There are certain values of m for which the carrier amplitude is zero. These permit a technician to measure FM modulator linearity. This is done as follows:
A sine wave of known frequency is applied to the FM modulator. The amplitude of the modulating signal is slowly increased while the FM carrier is observed on a spectrum analyzer. As the amplitude of the modulating signal is increased, the carrier will decrease to zero. The first zero carrier condition occurs when m = 2.4. At this point, the deviation δ is 2.4 * modulating frequency. The amplitude of the input signal is recorded. Then the amplitude of the input is increased more until the second zero carrier condition is reached. At this point, m = 5.5 and δ is 5.5 * modulating frequency. The amplitude of the input signal is recorded again. If the modulator is linear then:
(Amplitude at m = 5.5)/(Amplitude at m = 2.4) = 2.917
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