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LAB 6. FM Modulation ∫ 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 t St()==++ A cos[φωττθ ()] t A cos tkfd∫ ( ) (6.2) FM FM c f 0 where the instantaneous frequency (in radians per second) of the signal is dtφ () FM =+ω kft(). Observe that the frequency of this signal is directly proportional dt cf 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, Page 1 of 9 Revision C f(t) = a cos (ωm t). (6.4) This results in ∆ω = kf |f(t)|max = kf a, (6.5) which in turn yields t St()==++ A cos[φωττθ ()] t A cos tkfd∫ ( ) (6.6) FM FM c f 0 =+()ωβω + θ Atcoscm sin( t ) with β given by ∆ω β == ω Modulation index. (6.7) m An FM signal can be represented using the Bessel function ∞ =++βωωθ StFM() AJ∑ n ( )cos[(cm n ) t ] (6.8) n=−∞ th where Jn(.) is an n 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 ≈+βω BW 21()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 SFOUT Signal Errors LPF Reference Signal VCO Control Voltage Controlled Voltage Oscillator Figure 1. Phase Locked Loop Page 2 of 9 Revision C Prelab 6.1 Generation of the normalized FM signal 1) Create an FM signal using MatLab using the following parameters: Modulating frequency, fm 1 kHz Amplitude of fm 2.5 Vpp Carrier frequency, fc 10 kHz Amplitude of fc 2.5 Vpp Sampling rate 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 m- file 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 Page 3 of 9 Revision C Table 1 Modulation index β∆ω kf 0.01 1.0 2.4 10.0 50.0 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 4) Complete Table 2 when kf = 1.2 ∗10 ∗ 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 Amplitude of modulating signal, ∆ω Modulation index β A 1 volt 2 4 2.4 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. Page 4 of 9 Revision C 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 1 volts for VDD . 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. F min1 = 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. F min2 = Fmax2 = 1 The MC14046 requires pin 16 to be connected to VDD and pin 8 to be connected to ground.
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