LAB 6. FM Modulation
Introduction
In this lab, you will investigate frequency modulation (FM) and its properties. During
this lab you will
•
•
•
•
Create an FM signal by modulating an audio waveform onto a carrier,
Examine the spectrum of the modulated carrier,
Evaluate the modulated carrier when the modulation index is varied and
Demodulate the signal and recover the original modulating waveform using
two methods for demodulation.
In your report you will compare the properties of the FM signal you created
experimentally with those suggested by theory.
The IC used for this experiment is the MC14046. The data sheet for the MC14046 can be
obtained from Motorola at their web site:
http://mot-sps.com/books/dl131/pdf/mc14046brev4.pdf
In this modulation scheme, the frequency of carrier is varied in time based upon a
modulating signal. This modulation method is, therefore, referred to as FM or frequency
modulation. Since the frequency change of a sinusoid in time results in alternation of
phase in time, FM may also be referred to as phase modulation, or PM, as well.
Theory
Consider a carrier signal,
S(t) = A cos (ω c t + θ )
(6.1)
where A, ϖc, and θ denote the amplitude, frequency, and phase of the carrier signal
respectively.
Now consider a situation where the frequency of this signal changes in accordance with a
modulating signal, f(t). The resulting signal can be expressed as

S FM (t ) = A cos[φFM (t )] = A cos ω c t + k f

t

∫ f (τ )dτ + θ 
(6.2)
0
where the instantaneous frequency (in radians per second) of the signal is
dφFM (t )
= ω c + k f f (t ) . Observe that the frequency of this signal is directly proportional
dt
to the modulating signal. Also, kf denotes a scaling factor, limiting the maximum
frequency deviation of signal ∆ω,
∆ω = kf |f(t)|max
(6.3)
Because FM is a nonlinear modulation it is highly sensitive to the frequency content of
modulating signal. To see this, start with a sinusoidal modulating signal,
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f(t) = a cos (ωm t).
(6.4)
∆ω = kf |f(t)|max = kf a,
(6.5)
This results in
which in turn yields

S FM (t ) = A cos[φFM (t )] = A cos ω c t + k f

t
∫
0

f (τ )dτ + θ 

(6.6)
= A cos(ω c t + β sin(ω m t ) + θ )
with β given by
β=
∆ω
= Modulation index.
ωm
(6.7)
An FM signal can be represented using the Bessel function
∞
S FM (t ) = A ∑ J n ( β ) cos[(ω c + nω m )t + θ ]
(6.8)
n =−∞
where Jn(.) is an nth order Bessel function. Note that the spectrum of the FM signal in this
case consists of an infinite sum of delta functions. Realizing that Jn(β) ≈ 0 for n > β, the
bandwidth of the above FM signal may be shown to be
BW ≈ 2( β + 1)ω m radians per second,
(6.9)
based on Carson’s rule.
The MC14046 Integrated Circuit is a Phase Locked Loop, which can be used as both an
FM modulator and demodulator. In figure 1 below, when the input signal is the
modulated carrier, the error voltage at the output of the low-pass filter represents the
modulating signal.
Phase Comparator
Input
Phase
Signal
SFOUT
Errors
LPF
Reference
Signal
VCO Control
Voltage
Controlled
Oscillator
Voltage
Figure 1. Phase Locked Loop
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Prelab 6.1 Generation of the normalized FM signal
1)
Create an FM signal using MatLab using the following parameters:
Modulating frequency, fm
Amplitude of fm
Carrier frequency, fc
Amplitude of fc
Sampling rate
1 kHz
2.5 Vpp
10 kHz
2.5 Vpp
1 MHz
You can make use of the m-file Pre6_1.m.
2)
Generate plots of the FM signals created when the modulating signals are sine,
triangle, and square waves using the same settings for frequency and amplitude.
Make sure that the triangle and square waves are both antipodal (waveforms balanced
above and below 0 volts).
The FM output for a square wave looks like Frequency Shift Keying (FSK). The mfile pre6_1.m is designed to show the frequency variation more clearly in the time
domain. Note: The m-file uses the function cumsum of MatLab for FM modulation.
Cumsum models the integration operation for a digital signal, which means that the
output of cumsum is not exactly same as the mathematical modeling for the analog
signal. As a result, the value of the modulation index will differ somewhat from the
actual result of mathematical calculation. To correct this, use integral-based
equations to model FM.
Prelab 6.2 Narrow band and Wide band Modulation
FM can be categorized to narrow band and wideband. When the modulation index is
very small, it is usually called narrow band FM (NBFM). The m-file pre6_2.m is a
function file for this procedure. In the file, you only need to change the value of the
modulation index β. It generates the output for the time domain signal based on the
sampling frequency. Create plots of the FM output spectra using psdplot.m. The
parameters you will need to run the m-files are
>> [ y, fs ] = pre6_2 ( Beta ) ;
>> psdplot ( y, fs)
1) Select a value of the modulation index that satisfies the condition for the NBFM (β ,
0.2). Using that value, generate the plot of a NBFM signal using pre6_2.m. Plot the
FFT of the output signal using psdplot.m file.
2) Generate the FFTs of an FM modulated signal when β = 0.01, 1, 2.4, 10, and 50.
Observe the characteristic of the side bands using zoom for the FFT plots. In your
report, comment on the time and frequency outputs for varying modulation indices.
3) Determine the maximum frequency deviation and scaling factor based the value of the
modulation index. The m-file will show these values in the MatLab command
window. The specification for amplitude and frequency are given in the m-file.
Complete Table 1 and comment on the results on your plots for the outputs
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Table 1
Modulation index β
0.01
1.0
2.4
10.0
50.0
∆ω
kf
When β is 2.4, you will observe a carrier null in the frequency spectrum. Explain the
relationship of this carrier null and the value of the modulation index.
4) Complete Table 2 when kf = 1.2 ∗104 ∗ π radians per second per volt, the modulating
signal frequency is 6 kHz, and the amplitude of the modulating signal is as given in
the table.
Table 2
∆ω
Amplitude of modulating signal,
A
1 volt
2
4
2.4
Modulation index β
Prelab 6.3. The Modulation Index of an FM signal
You can observe the change of the spectral shape for various modulation indices, β. In
this step, you will generate a series of line spectra.
Generate two sets of plots, one for varying β when ωm is held constant, and one for
varying β when ∆ω is held constant. Use 1, 2, 5, 10, and 20 as values for β.
The m-file pre6_3 will generate the spectra for the values of modulation index. Once you
execute the m-script file, you have to press any key after you have one figure in order to
see the subsequent figures.
In the Lab Report, explain the patterns of the line spectra.
Prelab 6.4. Demodulation of FM signal
There are two predominant methods for demodulating an FM signal. One is direct
method that uses a linear frequency-to-voltage transfer characteristic. Such a system is
called a frequency discriminator. The simplest discriminator is a differentiator.
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Figure 2. FM Modulator
The second method, considered an indirect method, uses a Phase-Locked Loop.
In this procedure, a direct method is simulated, using a frequency differentiator and an
envelope detector.
1) Generate the plots for modeling the demodulation. The FM signal is generated and
demodulated in the m-file, pre6_4. The carrier is a sinusoid with 1 Vpp amplitude and 1
MHz. Modulating signal is a sinusoid with a 10 Vpp and 10 kHz. Try using different
values for β to see the impact on the displayed spectra. To see what your FM radio
receives, try using values such as are used for Commercial FM transmission (88 - 108
MHz).
Lab 6: Frequency Modulation
Parts required for Lab 6
Resistors:
1 – 150 Ω
Others as determined by circuit
Capacitors:
1 – 510 pF
1 – 4700 pF
Others as determined by circuit
Integrated circuits:
2 – MC14046 Phase Locked Loop
1. Construct the circuit shown in Figure 2 using 91 kΩ for R1 and 22 kΩ for R2. Use +5
volts for VDD1. Connect pin 9 (the VCO input) to ground and measure the frequency
of the signal at pin 4 (the VCO output). Record this frequency as Fmin1. Remove the
ground from pin 9 and connect it, instead to VDD. Again, measure the frequency at pin
4 and record it as Fmax1. Fmin1 and Fmax1 establish the dynamic range of the PLL.
Fmin1 =
Fmax1 =
2. Using a second MC14046, construct a second circuit based on Figure 2. This time
use 33 kΩ for R1 and 30 kΩ for R2. As you did for the first circuit, connect pin 9 (the
VCO input) to ground and measure the frequency of the signal at pin 4 (the VCO
output). Record this frequency as Fmin2. Remove the ground from pin 9 and connect
it, instead to VDD. Again, measure the frequency at pin 4 and record it as Fmax2.
Fmin2 =
Fmax2 =
1
The MC14046 requires pin 16 to be connected to VDD and pin 8 to be connected to ground. The
schematics given in Figure 2 and Figure 4 follow the convention of not showing these connections. The
omission is deliberate, not accidental.
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Figure 3. FM Modulator
1. Determine which of the two circuits to use as the modulator and which to use as the
demodulator by comparing the respective Fmin and Fmax values. The demodulator
must be able to lock onto any frequency deviation produced by the modulator.
Therefore, choose as the demodulator the circuit with the lowest frequency for Fmin
and the highest frequency for Fmax.
If neither circuit meets both criteria, modify one of them by changing the values for R11
and/or R12, using the formulas for VCO output frequency given in the data sheet on
page 4, Figure 2. Notice that Fmax depends on Fmin. Record the final values:
Fmin1 =
Fmax1 =
Fmin2 =
Fmax2 =
2. Once the demodulator stage has been selected, add the low pass filter and connect the
VCO output (pin 4) to the B input of the phase comparator (pin 3) as is shown in
Figure 4. Cascade the stages by connecting VCO output of modulator (pin 4) to the
input of the phase comparator of the demodulator (pin 14).
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Figure 4. FM Demodulator
1. Measure the output frequencies of both stages (at pin 4) when the VCO input of the
modulator is connected to VDD. Use channel 2 and 1 of the oscilloscope to observe
the output waveforms of modulator and demodulator respectively. Is the
demodulator’s VCO phase-locked to the modulator’s VCO? Measure the output
frequencies of both stages again when the VCO input is connected to ground. Is the
demodulator’s VCO still phase-locked to the modulator’s VCO? [If phase-lock did
not occur, return to step 3 and modify the demodulator so that its Fmin is at least 10%
below the modulator’s Fmin, and its Fmax is 10% above the modulator’s Fmax.]
VCO input to VDD:
Fmod =
Fdem =
VCO input to ground:
Fmod =
Fdem =
2. Connect a 51 Ω resistor between pin 9 and ground of the modulator
stage, then apply a sinusoid signal to the VCO input of the modulator with the
following parameters: amplitude = 2 Vpp, offset level =3 Vdc, and frequency = 1 kHz
(all displayed values). Use channel 2 of the oscilloscope to observe this signal and
channel 1 to observe the demodulated signal (at pin 10 of the demodulator).
3. If you observe clipping in the output signal, decrease the DC offset level of the input
signal. Observe that the output signal eventually becomes unclipped. Adjust the
amplitude and the offset level of the input signal to attain the maximum amplitude of
signal at the output without clipping. One at a time, apply square, triangle, and ramp
waveforms at the modulator's input. In your report, comment on the modulator’s
success at demodulating the waveform. Also explain the presence of any unexpected
frequencies in the output signal (seen by viewing the FFT of the output), or
unexpected amplitude variations.
4. Using the square wave as the input signal, change the frequency of the input signal to
30 mHz (millihertz) and measure the minimum and maximum frequencies (at pin 4)
attained by the modulator/demodulator circuit.
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Modulator
Demodulator
Fmax =
Fmin =
Fmax =
Fmin =
5. Compute the peak frequency deviation of the modulator using
∆ F = (Fmax-Fmin)/2
and the center frequency (the carrier frequency) by
fc = Fmin + ∆ F
6. Compute the frequency of the modulating signal (the signal being applied by the
function generator) to attain the following values of the modulation index:
β = ∆ F/fm
β=
0.2; fm =
β=
1.0; fm =
β=
2.4; fm =
For β = 0.2, set the corresponding fm frequency as the output of the function generator
Figure 5. Low Pass Filter
and use channel 1 to observe the FM signal (at pin 4 of the modulator) through the low
pass filter shown in Figure 5. The LP filter is required because the signal at this point is a
square wave. You may find it helpful to modify the RC time constant of the LP filter to
properly display the signal. To obtain a better oscilloscope display, either disconnect the
probe from the channel 2 input or ground the channel 2 probe. Use the FFT of the
oscilloscope to display the spectrum of the FM signal. Adjust the "Center Freq" to about
fc, and set the "Freq Span" about 244 kHz.
Using "Cursors" complete the following table:
Observed
Calculated
Frequency of main peak fc =
Frequency lower peak f1 =
Frequency upper peak f2 =
Bandwidth f2-f1 =
Amplitude of main peak =
Amplitude of lower peak =
Amplitude of upper peak =
fc =
fc-fm =
fc+fm =
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1. As β increases the number of significant spectral components also increases. Repeat
the measurements in (10) for the remainder values of β and fm, measuring
frequencies and amplitudes of the added components. Then compute the bandwidth.
If necessary, decrease the "Freq Span" to better display the spectrum for the highest
values of β. Pay special attention to the amplitude of the carrier frequency
component. Note: Do not change the "Center Freq" setting.
Lab Report
1. Compare the Fmin and Fmax frequencies produced by the two configurations built in
steps 1 and 2 of the lab with the theoretical values given for the MC14046. How
close were the calculated values to the theoretical values? The formulas for these
values may be located on page 4, Figure 2 of the data sheet as the VCO output
frequency.
2. How would the system composed of a modulator and a demodulator function if the
circuit you used as the demodulator was used as the modulator, and the circuit used as
the modulator was used as the demodulator? Why?
3. Compare the theoretical spectra of the FM signals with the results obtained in the
experiment for each value of Beta. Comment your results.
4. Verify Carson's rule for each value of Beta.
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