Topics: Basic definitions, FM, narrow band FM, wide band FM, transmission bandwidth of FM waves, and generation of FM waves: indirect FM and direct FM.
Angle modulation is a method of analog modulation in which either the phase or frequency of the carrier wave is varied according to the message signal. In this method of modulation the amplitude of the carrier wave is maintained constant.
In general form, an angle modulated signal can represented as
Where Ac is the amplitude of the carrier wave and θ(t) is the angle of the modulated carrier and also the function of the message signal.
The instantaneous frequency of the angle modulated signal, s(t) is given by
The modulated signal, s(t) is normally considered as a rotating phasor of length Ac and angle θ(t). The angular velocity of such a phasor is dθ(t)/dt , measured in radians per second. An un-modulated carrier has the angle θ(t) defined as
Where fc is the carrier signal frequency and ¢c is the value of θ(t) at t = 0. The angle modulated signal has the angle, θ(t) defined by
There are two commonly used methods of angle modulation:
1. Frequency Modulation, and
2. Phase Modulation.
In phase modulation the angle is varied linearly with the message signal m(t) as :
where kp is the phase sensitivity of the modulator in radians per volt.
Thus the phase modulated signal is defined as
Frequency Modulation (FM):
In frequency modulation the instantaneous frequency fi(t) is varied linearly with message signal, m(t) as:
where kf is the frequency sensitivity of the modulator in hertz per volt. The instantaneous angle can now be defined as
and thus the frequency modulated signal is given by
The PM and FM waveforms for the sinusoidal message signal are shown in the fig-5.1.
Example 5.1:
Find the instantaneous frequency of the following waveforms:
(a) S1(t) = Ac Cos [100π t + 0.25 π ]
(b) S2(t) = Ac Cos [100π t + sin ( 20 π t) ]
(c) S3(t) = Ac Cos [100π t + ( π t2) ]
Solution: Using equations (5.1) and (5.2):
(a) fi(t) = 50 Hz; Instantaneous frequency is constant.
(b) fi(t) = 50 + 10 cos( 20 π t); Maximum value is 60 Hz and minimum value is 40 Hz.
Hence, instantaneous frequency oscillates between 40 Hz and 60 Hz.
(c) fi(t) = (50 + t)
The instantaneous frequency is 50 Hz at t=0 and varies linearly at 1 Hz/sec.
A frequency modulated signal can be generated using a phase modulator by first integrating m(t) and using it as an input to a phase modulator. This is possible by considering FM signal as phase modulated signal in which the modulating wave is integral of m(t) in place of m(t). This is shown in the fig-5.2(a). Similarly, a PM signal can be generated by first differentiating m(t) and then using the resultant signal as the input to a FM modulator, as shown in fig-5.2(b).
Single-Tone Frequency Modulation:
Consider a sinusoidal modulating signal defined as:
m(t) = Am Cos( 2π fm t) …. (5.10)
Substituting for m(t) in equation (5.9), the instantaneous frequency of the FM signal is
The frequency deviation factor indicates the amount of frequency change in the FM signal from the carrier frequency fc on either side of it. Thus FM signal will have the frequency components between (fc - ∆f ) to (fc +∆f ). The modulation index, β represents the phase deviation of the FM signal and is measured in radians. Depending on the value of β, FM signal can be classified into two types:
1. Narrow band FM (β << 1) and
2. Wide band FM (β >> 1).
Example-5.2: A sinusoidal wave of amplitude 10volts and frequency of 1 kHz is applied to an FM generator that has a frequency sensitivity constant of 40 Hz/volt. Determine the frequency deviation and modulating index.
Solution: Message signal amplitude, Am = 10 volts, Frequency fm = 1000 Hz and the frequency sensitivity, kf = 40 Hz/volt.
Frequency deviation, ∆f = kf Am = 400 Hz
Modulation index, β = ∆f / fm = 0.4, (indicates a narrow band FM).
Example-5.3: A modulating signal m(t) =10 Cos(10000pt) modulates a carrier signal, Ac Cos(2pfct). Find the frequency deviation and modulation index of the resulting FM signal. Use kf = 5kHz/volt.
Solution: Message signal amplitude, Am = 10 volts, Frequency fm = 5000 Hz and the frequency sensitivity, kf = 5 kHz/volt.
Frequency deviation, ∆f = kf Am = 50 kHz
Modulation index, β = ∆f / fm = 10, (indicates a wide band FM).
Expanding the equation (5.12) using trigonometric identities,
The above equation represents the NBFM signal. This representation is similar to an AM signal, except that the lower side frequency has negative sign. The magnitude spectrum of NBFM signal is shown in fig-5.3, which is similar to AM signal spectrum. The bandwidth of the NBFM signal is 2fm, which is same as AM signal.
Frequency Domain Representation of Wide-Band FM signals:
The FM wave for sinusoidal modulation is given by
The complex envelope is a periodic function of time, with a fundamental frequency equal to the modulation frequency fm. The complex envelope can be expanded in the form of complex
Substituting in (5.15), the FM signal can be written as
The above equation is the Fourier series representation of the single tone FM wave. Applying the Fourier transform to (5.21),
The spectrum S(f) is shown in fig-5.4. The above equation indicates the following:
(i) FM signal has infinite number of side bands at frequencies (fc + nfm).
(ii) Relative amplitudes of all the spectral lines depends on the value of Jn(β).
(iii) The number of significant side bands depends on the modulation index (β). With (β<<1), only J0(β) and J1(β) are significant. But for (β>>1), many sidebands exists.
(iv) The average power of an FM wave is P = 0.5Ac2 (based on Bessel function property).
Bessel’s Function:
Bessel function is an useful function to represent the FM wave spectrum. The general plots of Bessel functions are shown in fig-5.5 and table (5.1) gives the values for Bessel function coefficients. Some of the useful properties of Bessel functions are given below:
The Spectrum of FM signals for three different values of β are shown in the fig-5.6. In this spectrum the amplitude of the carrier component is kept as a unity constant. The variation in the amplitudes of all the frequency components is indicated.
For β = 1, the amplitude of the carrier component is more than the side band frequencies as shown in fig-5.6a. The amplitude level of the side band frequencies is decreasing. The dominant components are (fc + fm) and (fc + 2fm). The amplitude of the frequency components (fc + nfm) for n>2 are negligible.
For β = 2, the amplitude of the carrier component is considered as unity. The spectrum is shown in fig-5.6b. The amplitude level of the side band frequencies is varying. The amplitude levels of the components (fc + fm) and (fc + 2fm) are more than carrier frequency component; whereas the amplitude of the component (fc + 3fm) is lower than the carrier amplitude. The amplitude of frequency components (fc + nfm) for n>3 are negligible.
The spectrum for β = 5, is shown in fig-5.6c. The amplitude of the carrier component is considered as unity. The amplitude level of the side band frequencies is varying. The amplitude levels of the components (fc + fm), (fc + 3fm), (fc + 4fm) and (fc + 5fm), are more than carrier frequency component; whereas the amplitude of the component (fc + 2fm) is lower than the carrier amplitude. The amplitude of frequency components (fc + nfm) for n>8 are negligible.
Fig: 5.6 – Plots of Spectrum for different values of modulation index. (Amplitude of carrier component is constant at unity)
Example-5.4:
An FM transmitter has a power output of 10 W. If the index of modulation is 1.0, determine the power in the various frequency components of the signal.
Solution: The various frequency components of the FM signal are
Example-5.5:
A 100 MHz un-modulated carrier delivers 100 Watts of power to a load. The carrier is frequency modulated by a 2 kHz modulating signal causing a maximum frequency deviation of 8 kHz. This FM signal is coupled to a load through an ideal Band Pass filter with 100MHz as center frequency and a variable bandwidth. Determine the power delivered to the load when the filter bandwidth is:
(a) 2.2 kHz (b) 10.5 kHz (c) 15 kHz (d) 21 kHz
Example-5.6:
A carrier wave is frequency modulated using a sinusoidal signal of frequency fm and amplitude Am. In a certain experiment conducted with fm=1 kHz and increasing Am, starting from zero, it is found that the carrier component of the FM wave is reduced to
zero for the first time when Am=2 volts. What is the frequency sensitivity of the modulator? What is the value of Am for which the carrier component is reduced to zero for the second time?
Transmission Bandwidth of FM waves:
An FM wave consists of infinite number of side bands so that the bandwidth is theoretically infinite. But, in practice, the FM wave is effectively limited to a finite number of side band frequencies compatible with a small amount of distortion. There are many ways to find the bandwidth of the FM wave.
1.Carson’s Rule: In single–tone modulation, for the smaller values of modulation index the bandwidth is approximated as 2fm. For the higher values of modulation index, the bandwidth is considered as slightly greater than the total deviation 2∆f. Thus the Bandwidth for sinusoidal modulation is defined as:
For non-sinusoidal modulation, a factor called Deviation ratio (D) is considered. The deviation ratio is defined as the ratio of maximum frequency deviation to the bandwidth of message signal.
Deviation ratio , D = ( ∆f / W ), where W is the bandwidth of the message signal and the corresponding bandwidth of the FM signal is,
BT = 2(D + 1) W ... (5.25)
2.Universal Curve : An accurate method of bandwidth assessment is done by retaining the maximum number of significant side frequencies with amplitudes greater than 1% of the un- modulated carrier wave. Thus the bandwidth is defined as “the 99 percent bandwidth of an FM wave as the separation between the two frequencies beyond which none of the side-band frequencies is greater than 1% of the carrier amplitude obtained when the modulation is removed”.
Transmission Bandwidth - BW = 2 nmax fm , (5.26)
where fm is the modulation frequency and ‘n’ is the number of pairs of side-frequencies such that ½Jn(b)½> 0.01. The value of nmax varies with modulation index and can be determined from the Bessel coefficients. The table 5.2 shows the number of significant side frequencies for different values of modulation index.
The transmission bandwidth calculated using this method can be expressed in the form of a universal curve which is normalised with respect to the frequency deviation and plotted it versus the modulation index. (Refer fig-5.7).
From the universal curve, for a given message signal frequency and modulation index the ratio (B/ ∆f ) is obtained from the curve. Then the bandwidth is calculated as:
Example-5.7:
Find the bandwidth of a single tone modulated FM signal described by S(t)=10 cos[2p108t + 6 sin(2p103t)].
Solution: Comparing the given s(t) with equation-(5.12) we get
Modulation index, β = 6 and Message signal frequency, fm = 1000 Hz. By Carson’s rule (equation - 5.24),
Transmission Bandwidth, BT = 2(β + 1) fm
BT = 2(7)1000 = 14000 Hz = 14 kHz
Q. A carrier wave of frequency 91 MHz is frequency modulated by a sine wave of amplitude 10 Volts and 15 kHz. The frequency sensitivity of the modulator is 3 kHz/V.
(a) Determine the approximate bandwidth of FM wave using Carson’s Rule.
(b) Repeat part (a), assuming that the amplitude of the modulating wave is doubled.
(c) Repeat part (a), assuming that the frequency of the modulating wave is doubled.
Solution: (a) Modulation Index, β = ∆f / fm = kf Am / fm = 3x10/15 = 2
By Carson’s rule; Bandwidth, BT = 2(β + 1) fm = 90 kHz
(b) When the amplitude, Am is doubled,
New Modulation Index, β = ∆f / fm = kf Am / fm = 3x20/15 = 4 Bandwidth, BT = 2(β+1)fm = 150 kHz
(c) when the frequency of the message signal, fm is doubled New Modulation Index, β = 3x10/30 = 1
Bandwidth, BT = 2(β+1)fm = 120 kHz.
Q. Determine the bandwidth of an FM signal, if the maximum value of the frequency deviation ∆f is fixed at 75kHz for commercial FM broadcasting by radio and modulation frequency is W= 15 kHz.
Solution: Frequency deviation, D = ( ∆f / W ) = 5
Transmission Bandwidth, BT = 2(D + 1) W = 12x15 kHz = 180 kHz
Q. Consider an FM signal obtained from a modulating signal frequency of 2000 Hz and maximum Amplitude of 5 volts. The frequency sensitivity of modulator is 2 kHz/V. Find the bandwidth of the FM signal considering only the significant side band frequencies.
Solution: Frequency Deviation = 10 kHz
Modulation Index, β = ∆f / fm = kf Am / fm = 5; From table –(5.2) ; 2nmax = 16 for β =5,
Bandwidth, BT = 2 nmax fm = 16x2 kHz = 32 kHz.
Example-5.11: A carrier wave of frequency 91 MHz is frequency modulated by a sine wave of amplitude 10 Volts and 15 kHz. The frequency sensitivity of the modulator is 3 kHz/V. Determine the bandwidth by transmitting only those side frequencies with amplitudes that exceed 1% of the unmodulated carrier wave amplitude. Use universal curve for this calculation.
Solution:
Frequency Deviation, ∆f = 30 kHz
Modulation Index, β = 3x10/15 = 2
From the Universal curve; for β = 2; (B / ∆f) = 4.3
Bandwidth, B = 4.3 ∆f = 129 kHz
