Experiment 7: Frequency Modulation and Phase Locked Loops October 11, 2006

Transcription

1 Experient 7: Frequency Modulation and Phase ocked oops October 11, 2006 Frequency Modulation Norally, we consider a voltage wave for with a fixed frequency of the for v(t) = V sin(ω c t + θ), (1) where ω c is the fixed angular frequency and θ is the phase. When we have a voltage wave for with a variable frequency, this has the for v(t) = V sin φ(t) (2) where φ(t) is the total angular displaceent at tie t. Consistent with this viewpoint, the instantaneous frequency in radians/sec is ω i (t) = 2 π f i (t) = d( φ(t)) dt For exaple, a constant frequency, ω c, iplies φ(t) = ω c t + θ (4) in which case d( φ(t)) ω i = = ω (5) c dt In this anner, a frequency odulated wave with sinusoidal odulation has an instantaneous frequency f i (t) = f c + Δf cos (2πf t) (6) where f c is the average frequency of the carrier wave, and Δf is the axiu deviation of the instantaneous frequency fro the average frequency. Notice that Δf is proportional to the peak aplitude of the odulating signal and is independent of the odulating frequency, f. Fro equation (3) and equation (6) we see that d( φ(t)) dt = ω c + 2 π Δf cos ω t 2πΔf φ t) = ωct + sin ω ω ( t + θ Neglecting θ for siplicity, we then have the frequency odulated signal fro equation (2), v(t) 2πΔf = Vsin ωct + sin ωt ω 1 (3) (7) (8) (9)

2 Equation (9) is usually written in the for where v(t) = V sin (ω c t + f sin ω t) (10) f = 2πΔf 2πf = Δf f (11) the quantity f is known as the odulation index. As can be seen in equation (6), Δf is proportional to the axiu aplitude of the odulation signal, V, and Δf = (VCO Sensitivity) V (12) As will be discussed later, frequency odulation is ipleented with a voltage controlled oscillator (VCO) and the VCO sensitivity is the transfer characteristic of this device in Hz/V. Making use of the trigonoetric identity equation (10) becoes sin (A + B) = sin A cos B + cos A sin B (13) v(t) = V sin ω c t cos ( f sin ω t) + V cos ω c t sin ( f sin ω t). (14) We now use the identities: cos ( f sin ω t) = J 0 ( f ) + 2J 2 ( f ) cos 2ω t + 2J 4 ( f ) cos 4ω t +. (15) And sin ( f sin ω t) = 2J 1 ( f ) sin ω t + 2J 3 ( f ) sin 3ω t +. (16) Equation (14) can now be expanded in ters of Bessel functions of the first kind of order n and arguent f v(t) = Vsin ω c t [J 0 ( f ) + 2J 2 ( f ) cos 2ω t + ] + V cos ω c t [2J 1 ( f ) sin ω t + 2J 3 ( f ) sin 3ω t +.] (17) Using the trigonoetric identities for su and difference angles v(t)/v = J 0 ( f ) sin ω c t + J 1 ( f ) [sin (ω c + ω )t sin (ω c - ω )t] + J 2 ( f ) [sin (ω c + 2ω )t + sin (ω c - 2ω )t] + J 3 ( f ) [sin (ω c + 3ω )t sin (ω c - 3ω )t] +. (18) 2

3 So, a frequency odulated signal consists of a carrier and a large nuber of upper and lower side bands with aplitudes given by Bessel functions of the first kind with the odulation index, f, as its arguent. Soe of these values are shown in the able 1 below: f J 0 J 1 J 2 J 3 J 4 J 5 J 6 J 7 J able 1 Bessel functions specifying carrier (J 0 ) and side band values (J 1 to J 8 ) Notice that these Bessel functions have a daped oscillatory behavior. Voltage Controlled Oscillator V c Multiplier V O V c V o R C V R R R V 0 V =V Z +V D Figure 1 Voltage controlled oscillator Frequency odulation can be perfored using a voltage-controlled oscillator (VCO). Figure 1 shows a relatively siple but very practical VCO that consists of a ultiplier, an integrator, a coparator with hysteresis (or Schitt trigger circuit), and Zener diodes to liit the output voltage swing. As we will see, the frequency of the output voltage is deterined by the tie constant of the integrator and the control voltage input, V c. 3

4 Schitt rigger Circuit First, look at the Schitt trigger circuit on the right side of Figure 1. Because of the positive feedback voltage divider, the differential voltage into the operational aplifier is V + = (V 0 + V R )/2 (19) where V R is the integrator output. his circuit always operates in the non-linear ode. he output, V 0, is always saturated at either +V or V. If the output is at +V, then the input, V R, ust fall below V before the output will change. When the input drops below V, the output will go to V as quickly as the ap can slew because of the positive feed back through R. In order for the output to go to +V again V R ust now rise above +V. his produces the hysteresis in the transfer characteristic shown in Figure 2. V 0 +V -V +V V R -V Figure 2 Schitt circuit transfer characteristic Assue for the oent that V c is a constant (DC) voltage. Because of the charging capacitor in the integrator, the reference voltage, V R, changes fro +V to V in a half period of oscillation. his behavior and the transfer characteristic shown in Figure 2 produce the wavefors shown in Figure 3. V R V 0 +V +V /2 t /2 t -V -V Figure 3 Ideal wave fors for the reference voltage, V R, and output voltage, V 0. Integrator he integrator in Figure 1 perfors the function t 1 V out (t) Vout (0) = Vin (t) dt (20) 0 4

5 In ters of the paraeters of our circuit equation (20) becoes t 1 VR (t) VR (0) = Vc V0 (t) dt Evaluating equation (21) over the half period fro 0 to /2 gives V R ( ) 2 V R 2 1 (0) = Vc ( + V ) dt 0 o (21) (22) Or giving V V VcV = 2 (23) f = 1/ = V C /(4) (24) Notice that the control voltage, V c, ust be ultiplied instead of added. Otherwise we would just have a change in duty cycle with V c, and the period would be 2V = V + V c 2V + V V c (25) Phase ocked oop Sin[ω c t + θ] Input Signal Phase Sensitive Detector 0.5{cos[(ω c + ω v )t + θ] + cos[(ω c ω v )t + θ]} Ap. ow Pass Filter R 1 (R 12 ) 3.6 kω +V CC C 1 Output Signal Sin[(ω c - ω v )t + θ] Sin[ω v t] Voltage Controlled Oscillator Sin [(ω c -ω v )t + θ] θ(t) when locked Note: θ(t) = f sin(ω t) R 0 C 0 +V CC -V CC Figure 4 Scheatic diagra of the M565 phase locked loop Although useful in any applications, phase locked loops perfor FM deodulation in a straightforward anner. Referring to Figure 4 a frequency odulated input signal, sin[ω c t + θ], 5 (26)

6 is applied to a phase sensitive detector, which is essentially a ultiplier. Fro equations (10) and (11) the phase angle, indicated by θ in Figure 4, is Δf θ ( t) = sinωt f where we have assued sinusoidal, single frequency odulation. In the ultiplier this input signal is ultiplied by the output signal fro the voltage controlled oscillator, ω v t, to produce a sinusoida signal with su and difference frequencies. his signal containing the su and difference frequencies is aplified and low pass filtered to obtain only a signal with the difference frequency as the output. When ω v equals ω c, this output is just the phase angle, θ, which is fed back into the VCO to aintain the VCO output frequency, ω v, exactly equal to ω c. he syste is said to be in lock when ω v = ω c. Since the input phase, θ, varies with tie, the output will also vary in tie the sae way. If we copute the frequency of the output we find, as we expect, that f out 1 dθ = = Δf (cosωt) (27) 2π dt So, with an FM odulated input signal, the output, except for a scale factor, is just the original odulating signal. Preliinary Preparation Before you coe to the laboratory, you deterine the values for R 0 and C 0 that will give a voltage controlled oscillator (VCO) center frequency (f 0 ) of 30 khz. (See page 8 of Reference 1 posted on the Electronics aboratory Website for the necessary equations. Note that R o and C o in this note are the sae as R and C in Equation (24). ) R o should be in the range fro 2 k to 20 kω with an optiu value between 4 kω and 5 kω, so assue that R 0 = 4.7 kω. he filter bandwidth (the capture range of the loop) is deterined by the internal resistance R 12 (R 1 in Ref 1) of 3600 Ω and C 1. Deterine the value of C 1 that will give a capture range (f n ) of 8 khz. You will also need an additional low pass filter to eliinate the carrier frequency in the deodulated output. Deterine appropriate resistance and capacitance values for this filter. Experient Equipent ist 1 Integrated Circuit M 565 Phase ocked oop 1 Solderless Wiring Fixture 1 HP 33120A Function Generator 1 HP 3580A Spectru Analyzer Assorted Resistors and Capacitors 6

7 Procedure 1. Frequency Modulation. he HP 33120A function generator can only be frequency odulated internally. o set the 33120A carrier frequency, odulation shape, odulation frequency, and frequency deviation, proceed as follows- 1) Obtain the A:MOD enu by pressing SHIF, ENER. 2) Using the and > keys, go down and across the sub-enu to FM SHAPE. 3) Go down and across to SINE, SQUARE, OR RIANGE odulation wavefor. 4) Press ENER to set the selected odulation waveshape. 5) Set the carrier frequency and aplitude with the front panel controls in the usual way. 6) Press SHIF, FM to obtain a frequency odulated output. he front panel should display FM. 7) Press SHIF, FREQ to allow adjusting the odulation frequency to the desired value. 8) Press SHIF, EVE to allow adjusting the frequency deviation to the desired value. 1(a). Set the 33120A to a carrier frequency of 30 khz, 1 Vp-p output, 500 Hz sine wave odulation, with Δf = 125 Hz initially. Record the spectru. Adjust Δf to obtain zero carrier in the spectru. Record this value of Δf and the resulting spectru. Use the linear scale of the spectru analyzer and note the frequency scale calibration. 1(b). With, successively, a sine, square, and triangular odulation wavefor at 500 Hz, with Δf = 4 khz and 1 V p-p fro the 33120A, record the spectru. Note all pertinent data. 2. Phase ocked oop -V cc 1 Input 2 Phase Detector M565 AMP 6 Reference Output V cc 5 VCO Output 4 VCO 9 8 C 0 R 0 7 R 1 C 1 Deodulation Output V CC = 12V -V cc +V cc Figure 5 Phase locked loop circuit Construct the circuit shown in Figure 5 using your design paraeters for a VCO center frequency of 30 khz and a capture range of 8 khz. Add an additional low pass R-C filter to the 7

8 output of the deodulator to iniize the effect of the su frequency. Set the 33120A generator to provide a 30 khz, 1V pp sinusoidal output with no offset and no FM and apply this signal to the oscilloscope, the spectru analyzer, and the odulation input of the phase locked loop. he oscilloscope should be synchronized with this signal. Apply the VCO output to the other oscilloscope channel and the deodulation output to the DMM. For proper operation the VCO output should be a square wave varying fro about 0 to +12 V and the deodulation output should be a dc voltage of about +9 V. 2(a). Center Frequency. Adjust the generator frequency until the two oscilloscope signals are exactly 90 degrees out of phase. Record this frequency and the DC output voltage on the DMM. his frequency is the loop center frequency and the DC voltage is the noinal output voltage at zero signal input. 2(b). Upper ock Range. Slowly increase the signal generator frequency above center until the phase has shifted by 90 degrees fro its value at the loop center frequency. At this point the loop has dropped or is about to drop out of lock. If you have gone too far, return to the center and try again. Record the signal generator frequency and loop dc output voltage at this point. Beware of false locks. 2(c). ower ock Range. Next, slowly decrease the frequency below center until the phase has shifted by 90 degrees in the other direction. Again, the loop has dropped or is about to drop out of lock at this point. If you issed the values, return to the center and try again. Record the frequency and the loop dc output voltage. 2(d). Capture Range. Go up in frequency until the loop drops out of lock, then carefully coe back down until it regains lock. Avoid early false locks. In false lock the two wavefors on the oscilloscope will be of different frequency, whereas in true lock they will be of the sae frequency. A true lock will be at a lower frequency than a false lock. Record the true lock frequency. Repeat this procedure fro the low frequency side of center frequency. Repeat this the capture range easureent with C 0 equal to one-fourth its original value, four ties its original value, and sixteen ties its original value. 3. Deodulated Signal. Observe the low pass filtered output of the P FM deodulator for sine and square wave odulation. Set the 33120A frequency to the center frequency of the P. Also set 1 Vpp output, sine FM odulation, and Δf = 1 khz. Use C o consistent with a capture range of 8 khz. Observe the deodulated output on the scope. We ay regard the syste consisting of 33120A FM odulator and carrier generator and P as a linear syste and characterize it by its frequency response, uch as we would an aplifier. Accordingly, for a sinusoidal odulation signal, record the agnitude and phase (relative to the sync output of the 33120A) of the filtered deodulated output for a series of five odulation frequencies ranging fro the lower liit to the upper liit of the capture range. hen also record the wavefor of the filtered deodulated output for a square wave odulation with f = 1 khz. 8

9 Report 1(a) For the 500 Hz sine wave odulation data, copare the relative values of the frequency coponents in your recorded spectra with the table of Bessel functions given in the anual. Include the recorded spectra in this write up. Explain any discrepancies. 1(b) Refer to the recorded spectra of the sine, square, and triangle odulation with high odulation index. Include the spectra in this write up. Coent on the differences in the 3 different spectra. Why is the square wave spectru so sharply peaked? Also, relate the spectra widths to the odulation frequency and index. 2(a) Copare and discuss any differences between your experiental and calculated P center frequencies. 2(b) Copare and discuss any differences between your experiental and calculated upper lock range. 2(c) Copare and discuss any differences between your experiental and calculated lower lock range. 2(d) ranges. Copare and discuss any differences between your experiental and calculated capture 3 Plot the deodulated aplitude and phase data that you recorded vs. the odulation frequency. Are the results reasonable? Why? Is the wavefor of the deodulated output with square wave FM odulation consistent with the data? Why? References and Suggested Reading 1. M565/M565C Phase ocked oop Data Sheet, National Seiconductor, May eon W. Couch, Digital and Analog Counication Systes, 6th Edition, (Prentice Hall, Upper Saddle River, New Jersey, P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, (Wiley, New York, 1977). 4. F. M. Gardner, Phase-ock echniques, (Wiley, New York, 1966). 5. R. Kellejian, Applied Electronic Counication, (Science Research, Chicago, 1980). 9