Phase Locked Loop and FM Demodulation, Passive Filters.

Cuthbert Nyack
A block diagram of the phase Locked Loop is shown above. There are 3 main components in the loop. First is a Phase Detector shown as the subtractor and amplifier with gain K1. Second is a Loop Filter with transfer function F(s) and third is a Voltage Controlled Oscillator with transfer function K2/s. Here the feedback is a direct connection between the VCO and the Phase Detector. For other applications eg frequency synthesis, a divider is placed in the feedback path.
Filters are needed in the loop to attenuate high frequencies from the phase detector. However the transfer function of the filter not only affects high frequencies, it also affects the dynamic behaviour of the loop. Fortunately for most applications, the high frequencies are much higher than the desired frequency and relatively simple filters can be used. Several filter circuits are possible. Some possibilities using passive circuits are shown above. Fig 1 has 1 pole and attenuation of 20dB/decade at high frequencies. Fig 2 has 1 pole and 1 zero and its gain at high frequencies is constant at Tz/Tp1. The additional zero makes adjustment more flexible but is at the price of constant gain at high frequencies. Fig 3, with 1 zero and 2 poles retains the zero and has gain of 20dB/decade at high frequencies. Unfortunately the additional pole introduced may make the loop unstable and some care is required to locate the zero and poles to provide a satisfactory stable dynamic response.
The applet does not necessarily limit values to physically realistic ones. When changing Tz, Tp1 and Tp2 one should keep in mind that for Fig 2, Tz must be less than Tp1 since the gain of the passive circuit cannot be greater than 1. For Fig 3, Tz must be less than (Tp1 Tp2).

A comparison of the responses of the 3 filters is shown by Fn = 111 in the applet. Fn = 112 to 114 show special cases of Fn = 111.
The phase detector functions by multiplying the 2 inputs. Zero output is obtained when the 2 signals are out of phase by 90º. Inputs can be sinusoids or square waves.

Fn = 1 in the applet below shows the phase detector output when the input is 2 sinusoids out of phase by 90º.
Fn = 2 to 6 shows the output when the phase difference changes from 0º to 360º. The average output is (0.5 for phase = 0º), (0, 90º), (-0.5 180º), (0, 270º) and (0.5, 360º).
Output is proportional to cos(f)/2.0 and is nonlinear.
Fn = 7 shows the phase detector output when the input is 2 square waves out of phase by 90º.
Fn = 8 to 12 shows the output when the phase difference changes from 0º to 360º. The average output is (1.0, 0º), (0, 90º), (-1.0 180º), (0, 270º) and (1.0, 360º).
Output in this case is linear.

Fn = 13 is a simplified representation of the steady state situation in the loop. Input is in red. VCO output is in cyan. When the red and cyan signals are multiplied by the phase detector, the yellow signal results which is sent to the filter. Filter removes the high frequency components in the yellow signal leaving the green signal. Green signal is applied to the VCO which responds with the cyan signal. The result is the steady state situation indicated with the green signal being the demodulated FM output.

The unit step phase response is shown by Fn = 14. An example is shown in the image below.
In the applet, transfer functions for the open and closed loop system are shown in gray in the background and the closed loop poles in white.
The phase step response is in yellow and the corresponding frequency impulse response is in dark orange.
For the Bode plot, the transfer function for K/s is in light orange, for 1/(Tp1 s + 1) in blue magenta and the total open loop frequency response is in red. Combined open loop phase is in green. Light magenta x shows the frequency corresponding to 1/Tp1.
Closed loop frequency response is shown in pink for the magnitude and blue for the phase.
The first line of the blue text refers to the position of the white line which is moved by the WL scrollbar.
At the white line in the image, the frequency is ~0.0063rad/s for the Bode plot and the time is 1.6s for the transient plot. Magnitude and phase of open loop transfer function is 50dB and -90.1º and for the closed loop they are 8.6x10.0E-6dB and -0.18º. The peak in the step response is 1.07 at t = 1.6s.
The second blue line refers to the Bode plot. Phase Margin is 62.07º, open loop cross over frequency without the filter is 2rad/s and with the filter it is 1.767rad/s with a slope of -24.3dB/decade. At the crossover frequency, the closed loop gain is -0.266dB and the closed loop 3dB frequency is 2.8rad/s.
For this filter there is only 1 variable Tp1 which affects both the frequency and the damping. The closed loop transfer shows that as Tp1 is reduced K/Tp1 increases which reduces the risetime and 1/Tp1 also increases which damps out the transient oscillations.
Fn = 15 to 19 show special cases of Fn = 14.
For Fn = 15, the open loop cross over frequency without the filter is 1rad/s. This is usually used to define the lock frequency range for the PLL.
With the filter the crossover frequency is 0.8677rad/s. This is used to define the capture frequency range for the PLL.
If the center frequency of the VCO is w, then the lock range is w ± 1rad/s and the capture range is w ± 0.8677rad/s.
In principle, the loop with this filter cannot become unstable. Unfortunately nonlinear effects could cause problems. The applet assumes there are no nonlinear effects.
Fn = 16 is an example of the undesirable response that can arise when the filter pole is at too low a frequency.
The large overshoot in this case can easily cause the loop to become unlocked. The phase margin is 4.675º, the slope at crossover = -39.8dB/dec and the CL gain at crossover is 21.7dB. For Fn = 15 which has a much better response the corresponding numbers are :- 60.19º, -24.8dB/dec and -0.026dB.

One might be tempted to think that the small numbers used here for K may make the numbers obtained from the applet useless. The numbers obtained here can easily be scaled to more realistic values. If K = 10,000 instead of 1 then the time constants must be reduced by 10,000, the frequency scale for the Bode plot increased by 10,000 and the time scale for the transient response reduced by 10,000.
Fn = 18 and 19 are examples of this.
For Fn = 18, K = 1, Tp = 1.0, CL3db Freq = 1.271rad/s and peak response occurs at 3.6s.
For Fn = 19, K = 2, Tp = 0.5, CL3db Freq = 2.542rad/s and peak response occurs at 1.8s.
Note that the step response has the same shape in both cases.

Fn = 20 also shows the step response when the filter of Fig 1 is used. In this case Both the Bode Plot and the Root Locus are shown. This plot makes it easy to see the connection between the step response and the roots in the complex s plane. An example is shown in the image below.
In the root locus plot, the open loop poles are shown as black x's and the closed loop poles as white double x's.
In this case 1/Tp1 = 20, the open loop poles are at 0 and -20 and the closed loop poles are at -10 ± j10.
Although the log frequency scale is not shown, the white line can be used to find the Bode frequency. Fn = 21 to 25 show special cases of Fn = 20.


Another view of the unit step phase response of the loop when the filter of Fig 1 is used is indicated by Fn = 26. An example is shown in the image below.
In this case. Nyquist and root locus plots are also shown. The semicircular plot shows the left half of the Nyquist plot. 2 Nyquist curves are shown, the heavy pink magenta line is the regular Nyquist plot while the multicolored line is a "compressed" version of the Nyquist plot which shows the behaviour over a wider frequency range. Both curves coincide at the point where the magnitude of the gain is 1. The magenta semicircle has radius = 1 and the magenta x corresponds to the point where the gain = -1. White x on the Nyquist curve shows the point where the frequency is wc.
In this case 1/Tp1 = 10, the open loop poles are at 0 and -10 and the closed loop poles are at -5 ± j5.
From the Bode plot the open loop crossover frequency with the filter is 4.551rad/s, on the Nyquist plot, the white x shows the open loop crossover frequency which is shown as 4.55rad/s by wc.
Fn = 27 to 28 show special cases of Fn = 26.
The unit ramp phase response of the loop when the filter of Fig 1 is used is indicated by Fn = 29.
The applied ramp is in magenta and ramp phase response is in yellow. The corresponding frequency step response is in dark orange.
The output phase follows the input but is displaced from it, this means that there is a constant phase error. The first line of blue text shows that at 32s, the input is 32 and the output is 31.5 an error of 0.5. The theoretical error is 1/K = 1/2 = 0.5.
Theoretically this is a type 1 second order system. Type 1 means that it has zero steady error for a step input but a finite constant error for a ramp input.
Fn = 30 to 31 show special cases of Fn = 29.
Fn = 32 shows the response to a quadratic(parabolic) input as shown in the image below.
In this case, the phase error is increasing linearly in time while the frequency error is remaining constant. In th applet at t = 64 the input is 64 x 64 = 4096 and the output is 3938, an error of 4096 - 3938 = 158.
At t = 32, the input is 1024 and the output is 946.1 an error of ~78 or approximately 1/2 of the error at t = 64s.
Fn = 33, 34 show special cases of Fn = 32.
Fn = 35 shows the FM response as illustrated below.
The FM input is shown in cyan at the top and the VCO output is in fainter blue at the bottom. Input phase and frequency are shown as dark orange and blue cyan. Output phase and frequency is in yellow and blue magenta. Blue magenta is the demodulated FM out.
Fn = 36 show a special case of Fn = 35.
Fn = 37 shows the FM response with the FM being modulated by a ramp instead of a sine. Fn = 38 is a special case of Fn = 37.
Fn = 39 shows the FSK response with 2 frequencies. Fn = 40 is a special case of Fn = 39.

Fn = 41 shows the FSK response with 4 frequencies. Fn = 42 is a special case of Fn = 41.

Fn = 43 shows the PSK response with 2 phases. Fn = 44 is a special case of Fn = 43.

Fn = 45 to 77 show the corresponding cases for the Filter of Fig 2. Image below shows a case of the step response for this case.
For this case Tp1 can be adjusted to give the desired risetime then Tz can be used to damp out the transient behaviour.
NB The applet does not constrain the vaue of Tz but for the passive circuit here Tz = (R2C) must be less than Tp1 = (R1 + R2)C. One feature of this filter is the closed loop high frequency response only falls off at 20dB/decade.
Fn = 78 to 110 show the corresponding cases for the Filter of Fig 3. Since the open loop Transfer Function has 3 poles, then a bad choice of parameters can result in an unstable system with negative phase and gain margins as shown below.
The real part of the complex conjugate pole is ~+0.0333.
Because the zero is at too high a frequency, the phase drops to below -180º before the phase advance of the zero becomes significant.
With a different choice of parameters a positive phase margin of 58.24º can be obtained as shown below. Here the zero is between the 2 poles and stops the phase from dropping below -180º. Real part of the complex conjugate pole is now to the left of the imaginary axis.
A comparison of the responses of the 3 filters is shown by Fn = 111. Fn = 112 to 114 show special cases of Fn = 111.
Applet should appear below.


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COPYRIGHT © 2012 Cuthbert Nyack.