Experiment 7: Frequency Modulation and Phase Locked Loops Fall 2009

Transcription

1 Experiment 7: Frequency Modulation and Phase Locked Loops Fall 2009 Frequency Modulation Normally, we consider a voltage wave orm with a ixed requency o the orm v(t) = V sin(ω c t + θ), (1) where ω c is the ixed angular requency and θ is the phase. When we have a voltage wave orm with a variable requency, this has the orm v(t) = V sin φ(t) (2) where φ(t) is the total angular displacement at time t. Consistent with this viewpoint, the instantaneous requency in radians/sec is ω i (t) = 2 π i (t) = d (!(t)) dt d( "(t)) =! dt d (!(t)) dt For example, a constant requency, ω c, implies in which case φ(t) = ω c t + θ (4) ω i = c In this manner, a requency modulated wave with sinusoidal modulation has an instantaneous requency i (t) = c + Δ cos (2π m t) (6) where c is the average requency o the carrier wave, and Δ is the maximum deviation o the instantaneous requency rom the average requency. Notice that Δ is proportional to the peak amplitude o the modulating signal and is independent o the modulating requency, m. From equation (3) and equation (6) we see that = ω c + 2 π Δ cos ω m t 2$# %( t) = " ct + sin " mt +! (8) " m Neglecting θ or simplicity, we then have the requency modulated signal rom equation (2), v(t) & 2)( # = Vsin $ ' t + sin ' t! c m % ' m " 331exp7.rt 8/18/ (3) (5) (7) (9)

2 Equation (9) is usually written in the orm where v(t) = V sin (ω c t + m sin ω m t) (10) m = 2"! 2" m =! m (11) the quantity m is known as the modulation index. As can be seen in equation (6), Δ is proportional to the maximum amplitude o the modulation signal, V m, and Δ = (VCO Sensitivity) V m (12) As will be discussed later, requency modulation is implemented with a voltage controlled oscillator (VCO) and the VCO sensitivity is the transer characteristic o this device in Hz/V. Making use o the trigonometric identity equation (10) becomes sin (A + B) = sin A cos B + cos A sin B (13) v(t) = V sin ω c t cos (m sin ω m t) + V cos ω c t sin (m sin ω m t). (14) We now use the identities: cos (m sin ω m t) = J 0 (m ) + 2J 2 (m ) cos 2ω m t + 2J 4 (m ) cos 4ω m t +. (15) And sin (m sin ω m t) = 2J 1 (m ) sin ω m t + 2J 3 (m ) sin 3ω mt +. (16) Equation (14) can now be expanded in terms o Bessel unctions o the irst kind o order n and argument m v(t) = Vsin ω c t [J 0 (m ) + 2J 2 (m ) cos 2ω m t + ] + V cos ω c t [2J 1 (m ) sin ω m t + 2J 3 (m ) sin 3ω mt +.] (17) Using the trigonometric identities or sum and dierence angles v(t)/v = J 0 (m ) sin ω c t + J 1 (m ) [sin (ω c + ω m )t sin (ω c - ω m )t] + J 2 (m ) [sin (ω c + 2ω m )t + sin (ω c - 2ω m )t] + J 3 (m ) [sin (ω c + 3ω m )t sin (ω c - 3ω m )t] +. (18) 331exp7.rt 8/18/2009 2

3 So, a requency modulated signal consists o a carrier and a large number o upper and lower side bands with amplitudes given by Bessel unctions o the irst kind with the modulation index, m, as its argument. Some o these values are shown in the Table 1 below: m J 0 J 1 J 2 J 3 J 4 J 5 J 6 J 7 J Table 1 Bessel unctions speciying carrier (J 0 ) and side band values (J 1 to J 8 ) Notice that these Bessel unctions have a damped oscillatory behavior. Voltage Controlled Oscillator V c Multiplier V O V c V o R C V R R R V 0 V L =V Z +V D Figure 1 Voltage controlled oscillator Frequency modulation can be perormed using a voltage-controlled oscillator (VCO). Figure 1 shows a relatively simple but very practical VCO that consists o a multiplier, an integrator, a comparator with hysteresis (or Schmitt trigger circuit), and Zener diodes to limit the output voltage swing. As we will see, the requency o the output voltage is determined by the RC time constant o the integrator and the control voltage input, V c. 331exp7.rt 8/18/2009 3

4 Schmitt Trigger Circuit First, look at the Schmitt trigger circuit on the right side o Figure 1. Because o the positive eedback voltage divider, the dierential voltage into the operational ampliier is V + = (V 0 + V R )/2 (19) where V R is the integrator output. This circuit always operates in the non-linear mode. The output, V 0, is always saturated at either +V L or V L. I the output is at +V L, then the input, V R, must all below V L beore the output will change. When the input drops below V L, the output will go to V L as quickly as the amp can slew because o the positive eed back through R. In order or the output to go to +V L again V R must now rise above +V L. This produces the hysteresis in the transer characteristic shown in Figure 2. V 0 +V L -V L +V L V R -V L Figure 2 Schmitt circuit transer characteristic Assume or the moment that V c is a constant (DC) voltage. Because o the charging capacitor in the integrator, the reerence voltage, V R, changes rom +V L to V L in a hal period o oscillation. This behavior and the transer characteristic shown in Figure 2 produce the waveorms shown in Figure 3. V R V 0 +V L +V L T/2 T t T/2 T t -V L -V L Figure 3 Ideal wave orms or the reerence voltage, V R, and output voltage, V 0. Integrator The integrator in Figure 1 perorms the unction V (t) " V (0) = " 1 RC out out in 331exp7.rt 8/18/ t! V (t) dt (20)

5 In terms o the parameters o our circuit equation (20) becomes t 1 VR (t) " VR (0) = "! VcV0 (t) dt RC Evaluating equation (21) over the hal period rom 0 to T/2 gives V R ( T ) 2 " V R 2 1 (0) = "! Vc ( + VL ) dt RC T 0 o (21) (22) Or giving VcV! VL! VL =! RC L T 2 = 1/T = V C /(4RC) (23) (24) Notice that the control voltage, V c, must be multiplied instead o added. Otherwise we would just have a change in duty cycle with V c, and the period would be T 2V LRC = V + V L c 2V LRC + V! V L c (25) Phase Locked Loop 0.5{cos[(ω c + ω v )t + θ] + cos[(ω c ω v )t + θ]} Sin[ω c t + θ] Input Signal Phase Sensitive Detector Amp. Low 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) = m sin(ω m t) R 0 C 0 +V CC -V CC Figure 4 Schematic diagram o the LM565 phase locked loop 331exp7.rt 8/18/2009 5

6 Although useul in many applications, phase locked loops perorm FM demodulation in a straightorward manner. Reerring to Figure 4 a requency modulated input signal, sin[ω c t + θ], is applied to a phase sensitive detector, which is essentially a multiplier. From equations (10) and (11) the phase angle, indicated by θ in Figure 4, is # (26) " ( t) = sin! mt m where we have assumed sinusoidal, single requency modulation. In the multiplier this input signal is multiplied by the output signal rom the voltage controlled oscillator, ω v t, to produce a sinusoida signal with sum and dierence requencies. This signal containing the sum and dierence requencies is ampliied and low pass iltered to obtain only a signal with the dierence requency as the output. When ω v equals ω c, this output is just the phase angle, θ, which is ed back into the VCO to maintain the VCO output requency, ω v, exactly equal to ω c. The system is said to be in lock when ω v = ω c. Since the input phase, θ, varies with time, the output will also vary in time the same way. I we compute the requency o the output we ind, as we expect, that 1 d# = = " (cos mt) (27) 2$ dt out! So, with an FM modulated input signal, the output, except or a scale actor, is just the original modulating signal. Preliminary Preparation Beore you come to the laboratory, you determine the values or R 0 and C 0 that will give a voltage controlled oscillator (VCO) center requency ( 0 ) o 30 khz. (See page 8 o Reerence 1 posted on the Electronics Laboratory Website or the necessary equations. Note that R o and C o in this note are the same as R and C in Equation (24). ) R o should be in the range rom 2 k to 20 kω with an optimum value between 4 kω and 5 kω, so assume that R 0 = 4.7 kω. The ilter bandwidth (the capture range o the loop) is determined by the internal resistance R 12 (R 1 in Re 1) o 3600 Ω and C 1. Determine the value o C 1 that will give a capture range ( n ) o 8 khz. You will also need an additional low pass ilter to eliminate the carrier requency in the demodulated output. Determine appropriate resistance and capacitance values or this ilter. Experiment Equipment List 1 Integrated Circuit LM 565 Phase Locked Loop 1 Solderless Wiring Fixture 1 HP 33120A Function Generator 1 HP 3580A Spectrum Analyzer Assorted Resistors and Capacitors 331exp7.rt 8/18/2009 6

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8 Procedure 1. Frequency Modulation. The HP 33120A unction generator can only be requency modulated internally. To set the 33120A carrier requency, modulation shape, modulation requency, and requency deviation, proceed as ollows- 1) Obtain the A:MOD menu by pressing SHIFT, ENTER. 2) Using the and > keys, go down and across the sub-menu to FM SHAPE. 3) Go down and across to SINE, SQUARE, OR TRIANGLE modulation waveorm. 4) Press ENTER to set the selected modulation waveshape. 5) Set the carrier requency and amplitude with the ront panel controls in the usual way. 6) Press SHIFT, FM to obtain a requency modulated output. The ront panel should display FM. 7) Press SHIFT, FREQ to allow adjusting the modulation requency to the desired value. 8) Press SHIFT, LEVEL to allow adjusting the requency deviation to the desired value. 1(a). Set the 33120A to a carrier requency o 30 khz, 1 Vp-p output, 500 Hz sine wave modulation, with Δ = 125 Hz initially. Record the spectrum. Adjust Δ to obtain zero carrier in the spectrum. Record this value o Δ and the resulting spectrum. Use the linear scale o the spectrum analyzer and note the requency scale calibration. 1(b). With, successively, a sine, square, and triangular modulation waveorm at 500 Hz, with Δ = 4 khz and 1 V p-p rom the 33120A, record the spectrum. Note all pertinent data. 2. Phase Locked Loop -V cc 1 Input 2 Phase Detector LM565 AMP 6 Reerence Output V cc VCO Output 5 4 VCO 9 8 C 0 R 0 7 R 1 C 1 Demodulation Output V CC = 12V -V cc +V cc Figure 5 Phase locked loop circuit Construct the circuit shown in Figure 5 using your design parameters or a VCO center requency o 30 khz and a capture range o 8 khz. Add an additional low pass R-C ilter to the 331exp7.rt 8/18/2009 8

9 output o the demodulator to minimize the eect o the sum requency. Set the 33120A generator to provide a 30 khz, 1V pp sinusoidal output with no oset and no FM and apply this signal to the oscilloscope, the spectrum analyzer, and the modulation input o the phase locked loop. The oscilloscope should be synchronized with this signal. Apply the VCO output to the other oscilloscope channel and the demodulation output to the DMM. For proper operation the VCO output should be a square wave varying rom about 0 to +12 V and the demodulation output should be a dc voltage o about +9 V. 2(a). Center Frequency. Adjust the generator requency until the two oscilloscope signals are exactly 90 degrees out o phase. Record this requency and the DC output voltage on the DMM. This requency is the loop center requency and the DC voltage is the nominal output voltage at zero signal input. 2(b). Upper Lock Range. Slowly increase the signal generator requency above center until the phase has shited by 90 degrees rom its value at the loop center requency. At this point the loop has dropped or is about to drop out o lock. I you have gone too ar, return to the center and try again. Record the signal generator requency and loop dc output voltage at this point. Beware o alse locks. 2(c). Lower Lock Range. Next, slowly decrease the requency below center until the phase has shited by 90 degrees in the other direction. Again, the loop has dropped or is about to drop out o lock at this point. I you missed the values, return to the center and try again. Record the requency and the loop dc output voltage. 2(d). Capture Range. Go up in requency until the loop drops out o lock, then careully come back down until it regains lock. Avoid early alse locks. In alse lock the two waveorms on the oscilloscope will be o dierent requency, whereas in true lock they will be o the same requency. A true lock will be at a lower requency than a alse lock. Record the true lock requency. Repeat this procedure rom the low requency side o center requency. Repeat this the capture range measurement with C 0 equal to one-ourth its original value, our times its original value, and sixteen times its original value. 3. Demodulated Signal. Observe the low pass iltered output o the PLL FM demodulator or sine and square wave modulation. Set the 33120A requency to the center requency o the PLL. Also set 1 Vpp output, sine FM modulation, and Δ = 1 khz. Use C o consistent with a capture range o 8 khz. Observe the demodulated output on the scope. We may regard the system consisting o 33120A FM modulator and carrier generator and PLL as a linear system and characterize it by its requency response, much as we would an ampliier. Accordingly, or a sinusoidal modulation signal, record the magnitude and phase (relative to the sync output o the 33120A) o the iltered demodulated output or a series o ive modulation requencies ranging rom the lower limit to the upper limit o the capture range. Then also record the waveorm o the iltered demodulated output or a square wave modulation with m = 1 khz. 331exp7.rt 8/18/2009 9

10 Report 1(a) For the 500 Hz sine wave modulation data, compare the relative values o the requency components in your recorded spectra with the table o Bessel unctions given in the manual. Include the recorded spectra in this write up. Explain any discrepancies. 1(b) Reer to the recorded spectra o the sine, square, and triangle modulation with high modulation index. Include the spectra in this write up. Comment on the dierences in the 3 dierent spectra. Why is the square wave spectrum so sharply peaked? Also, relate the spectra widths to the modulation requency and index. 2(a) Compare and discuss any dierences between your experimental and calculated PLL center requencies. 2(b) Compare and discuss any dierences between your experimental and calculated upper lock range. 2(c) Compare and discuss any dierences between your experimental and calculated lower lock range. 2(d) ranges. Compare and discuss any dierences between your experimental and calculated capture 3 Plot the demodulated amplitude and phase data that you recorded vs. the modulation requency. Are the results reasonable? Why? Is the waveorm o the demodulated output with square wave FM modulation consistent with the data? Why? Reerences and Suggested Reading 1. LM565/LM565C Phase Locked Loop Data Sheet, National Semiconductor, May Leon W. Couch, Digital and Analog Communication Systems, 6th Edition, (Prentice Hall, Upper Saddle River, New Jersey, P. R. Gray and R. G. Meyer, Analysis and Design o Analog Integrated Circuits, (Wiley, New York, 1977). 4. F. M. Gardner, Phase-Lock Techniques, (Wiley, New York, 1966). 5. R. Kellejian, Applied Electronic Communication, (Science Research, Chicago, 1980). 331exp7.rt 8/18/