Phase Locked Loop and FM Demodulation,
Active 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.
Some possibilities for filters using
active circuits are shown above. Compared with their passive
counterparts, these filters are of type 1, ie their gain
becomes ~ 1/(Tp1 s) at low frequencies. These filters can increase
the gain of the loop and reduce loop rise time. Being Type 1, they
can eliminite errors for ramp phase inputs. Fig 1 has constant gain of
Tz/Tp at high frequencies. Fig 2 has gain of Tz/(Tp1 Tp2 s) at
high frequencies and can be used for higher attenuation of high
frequency components coming from the phase detector.
There are very limited restrictions on Tz, Tp1 and Tp2 in the
applet and CAUTION is required when using high values of Tz
and low values of Tp1 and Tp2. These should be related to
physically realistic values. Tp2 must be less than Tz.
Fn = 62 shows the frequency response of the filters.
Fn = 63 to 66 show special cases of Fn = 62.
Some of the properties of loops which use the filter of
Fig 1 are shown by Fn = 1 to 30.
Fn = 1 shows the phase step, frequency impulse response.
Fn = 2 to 5 show special cases of Fn = 1.
Image below shows a case of Fn = 1, Transient Phase
response is in yellow and frequency response is in
dark orange. In this case Tp1 changes both the
risetime and the damping and Tz can be used to
damp out the transient oscillations.
Fn = 6 shows the transient behaviour, Bode plot and
Root Locus plot. Fn = 7 to 12 show special cases
of Fn = 6.
Fn = 13 shows the Bode plot, Nyquist plot, Root Locus and
transient behaviour.
Fn = 14 and 15 show special cases of Fn = 13.
A example of Fn = 13 is shown in the image below.
The standard Nyquist plot is the heavier pink magenta line and
the multicolored line is a "compressed" version of the Nyquist
plot which shows the behaviour over a wider range of
frequencies. The white x is moved by wc
and shows the point where the magnitude of the gain is 1. This
is the point at which the phase margin is calculated.
Fn = 16 shows the ramp response for this system.
Fn = 17 and 18 are special cases of Fn = 16.
An example of the ramp response for this
system is shown in the image below.
Ramp input is in magenta, phase ramp output is in yellow
and frequency step output is in dark orange.
Since this is a Type 2 system, then the phase error to a ramp
input is zero. In the first blue line we see that at t = 64,
the input is 64 and the response is also 64.
Fn = 19 shows a parabolic phase input.
Fn = 20 and 21 show special cases of Fn = 19.
An example of the parabolic response is shown in the
image below. The phase parabolic error which is the difference between the
magenta and yellow lines is constant in time while the
frequency ramp error tends to zero.
Fn = 22 shows the FM response.
Fn = 23 and 24 show special cases of Fn = 22.
Fn = 25 shows the FSK response with 2 frequencies.
Fn = 26 and 27 show special cases of Fn = 25.
Fn = 28 shows the PSK response with 2 phases.
Fn = 29 and 30 show special cases of Fn = 28.
Note that the PSK response contains a frequency impulse
everytime the phase changes.
Image below show a case of the FM response.
Fn = 31 to 61 show the corresponding cases
for the Filter of Fig 3. Since this is an active filter, it can
increase the 3dB bandwidth and reduce the risetime as shown below.
An example showing the Nyquist plot is shown below. The
compressed version of the Nyquist plot is convenient
for adjusting the frequency response so the minimum phase
delay occurs at the crossover frequency.
Fn = 62 shows the frequency response of the filters.
Fn = 63 to 66 show special cases of Fn = 62.
Applet should appear below.
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COPYRIGHT © 2012 Cuthbert Nyack.