where kp = Phase sensitivity of the modulator
FM is that type of angle modulation in which the instantaneous frequency ωi is varied linearly with a message or baseband signal x(t) about an unmodulated frequency ωc.
where, kf = Frequency sensitivity of the modulator
An angle modulated signal in general can be written as
S(t) = Ac cos [θ(t)]
where, θ (t) is the phase of the signal and its instantaneous frequency is given by
If m(t) is the message signal, then in PM system, we have
For FM system, we have
where, kp and kf are phase and frequency deviation constants respectively.
The maximum change in instantaneous frequency from the average frequency ωc is called frequency deviation which depends upon the magnitude and sign of kf m(t). Frequency deviation:
Carrier Swing: The total variation in frequency from the lowest to the highest point is called carrier swing.
The carrier swing = 2 x frequency deviation = 2 x Δω
Modulation Index: For FM, the modulation index is defined as the ratio of frequency deviation to the modulating frequency.
mf = frequency deviation / modulating frequency
for FM (sinusoidal message signal)
for PM (sinusoidal message signal)
Note: The mf may be greater than unity.
Per cent Modulation: The term 'per cent modulation as it is used in reference to FM' refers to the ratio of actual frequency deviation to the maximum allowable frequency deviation.
Per cent modulation :
is Actual frequency deviation,
is Maximum frequency deviation
In per cent modulation, the following features will be considered
The FM signal can be classified based on sensitivity constant as:
where β is modulation index. In (β) is known as Bessel function of first kind of order n. Total bandwidth requirement for angle modulated signal is infinite. Basically the effective bandwidth is the separation between the two extreme significant side frequencies on either side of the carrier.
Bandwidth of the Angle Modulated Signal: In general, the effective bandwidth of an angle modulated signal, which contains atleast 98% of the signal power is given by.
Bc = 2 (β + 1) fm is also known as Carson's formula
where, β is modulation index.
Number of harmonics in the bandwidth (including the carrier) is:
It is also possible to generate a PM wave using a frequency modulator as shown in figure 3. The modulating signal is first passed through a differentiator and then applied to a frequency modulator.
This discussion has revealed that it is possible to obtain all the properties of FM from a PM wave and vice versa. At the output of the differentiator, we get the differentiated version of modulating signal i.e. dm(t)/dt. Hence, output of the frequency modulator will be,
The amplitude of the FM wave is constant. It is thus independent of the modulation depth, whereas in AM, modulation depth governs the transmitted power.
Note: A much wider channel typically 200 kHz is required in FM as against only 10 kHz in AM broadcast. This form serious limitation of FM.
The FM modulator circuits used for generating FM signals may be put into two categories as under
This method is most widely used for generation of WBFM signal.
Drawbacks of Direct Method for FM Generation
Following are the drawbacks of the direct method:
Despite all above drawbacks, the direct method is utilized for high power FM generation in several applications.
[Note: A low-pass filter compensates for higher phase shift and frequency deviation at the higher modulating frequencies to produce indirect FM.]
DC component is eliminated using an amplifier of gain 2 then o/p = cos 4πfct. when one more square law device is connected frequency gets multiplied again by 2. Thus 2 square law devices in cascade multiplies the frequency by 4.
Assume that the message signal and carrier are applied to a NBFM modulate. The output signal is Ac cos [2πfct + βsin 2πfmt]. If the signal is passed through frequency multiplier the final output is Ac cos [n(2π fct + βsin 2πfmt)]. In a frequency multiplier, carrier frequency and β are increased by a factor of n. But the message frequency is same. (As multiplier changes the carrier frequency, it must be brought back to the original carrier frequency. This is done by using a mixer)
The demodulation process of FM waves is exactly opposite to that of the frequency modulation. The AM detector is nothing but a rectifier, but FM demodulator is a frequency to amplitude converter. It is necessary to convert the frequency variations in FM wave at its input into amplitude variations at its output to recover the original modulating signal.
The FM demodulator must satisfy the following requirements:
Schematic of an FM demodulator based of FM to AM conversion