Phase Locked Loop and FM Demodulation,
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
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
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
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
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
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
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
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.