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TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

Rapport No.442

A PHASE LOCKED LOOP SERVOSYSTEM

M.Buitenhek and

H.00ms

Deift University of Technology

Ship Hydromechanics Laboratory

rIflZ

(2)

Contents Page nr.

Summary I

Introduction I

Principle I

Description of the servosystem 2

A closer examination of the blocks 3

The PLL as a feedback system 7

The capture time II

The results 12

Conclusion 12

IO. References 13

Figures 14

(3)

i . Summary

This report describes a servosystem intended to synchronise a rotating motion with an incoming signal from a wave-heightrneter. In particular. the phase

shift between the motion and the reference signal must be kept within very small boundaries. To realize this, the phase locked loop principle (1) was used and adapted to our specific requirements.

2. Introduction

At the Delft Shiphydromechanics Laboratory an electromechanical transfer

function analyser is used as an aid in measuring hydrodynamic forces, moments and

movements of shipmodels, both in amplitude and phase, related to the

excitating forces,

In reference (2) a description is given of a system in which the mechanical part of the analyser is mechanically coupled with the apparatus which produces

the forced motion.

In the situation that will be described, the waves in the modelbasin are generating the excitatingforces and the frequency of encounter is determined

bythepropagation speed of the waves and the speed of the model.

The necessary coupling with the measured waves has been realised by means of a servosystem which produces a rotating motion, of which the angular frequency and phase are synchronous with respectively the angular frequency and phase of the wave, which is measured by means of a waveheightmeter.

3. Principle

The phase locked loop (PLL) system is a nonlinear feedback system in which the errorsignal is a function of the phase difference between the inputsignal and the feedback signal.

Figure 1 shows the block diagram of the system, consisting of the following

parts.

the waveheightmeter (which is a separate, existing apparatus) the multiplier

the low-pass filter the amplifier

the voltage controlled oscillator (VCO) the steppermotor

two optical sensors (each consisting of a emitter and light-detector)

(4)

4. Description of the servosystem

The waveheightsignal is fed to the X-input of a multiplier which, in the locked condition, has on its Y-input the phase-synchronised feedback signal. It can be proved (5.1.) that a phase error between the two signals results in a DC (direct current) correction voltage at the output of the multiplier, which is then used to eliminate the phase error.

The multiplication of the X-and-Y-signals also generates unwanted AC(alternatíng

current) components at the output of the multiplier which are eliminated by

the low pass filter.

Bij means of a VCO (voltage controlled oscillator) a steppermotor is driven which delivers the desired rotating motion. In its free running state

) (lock-in circuit disabled) the frequency (rps) of the rotating axis is manually adjusted to a value that resembles the expected frequency of encounter. Small

frequency errors between expected- and actual frequency of encounter, and

other phase errors, for example during the capture (lock-in) period, are

compensated by the correction voltage that comes from the amplifier. following the low-pass filter. The correction voltage forces the VCO to track the phase

and hence the frequency of the incoming wave signal. The amplifier is necessary to get a sufficiently large open loop gain, that, as we will prove(6.3.), makes the static phase error negligible.

The feedback signal, coming from the rotating axis, is generated by means of

two optical sensors and a disk with a slot in it.

Aset-reset flip-flop converts the pulsesfrom the sensors into a symmetrical

square wave centered on zero volts, This signal is fed to the Y-input of the

)

multiplier. Figure 2 shows the time relationship between the signals in the system in the locked situation. It should be noted that a phase shift

(5)

5. A closer examination of the blocks

5.1. The multiplier

As already mentioned in (4) the wavesignal U (t) is applied to the X-input of the multiplier and the feedback signal Uf(t) of the rotating axis to the Y-input.

Assuming that the wavesignal is sinusoidal U(t) can be described by

U(t)=

sin(t+)

The feedback signal, which is a square wave centered on zero, can be represented by the following Fourierseries

4U 4 411

U (t) cos wt cos 3wt + cos wt - 2

f Tt 371 57T

For the multiplier the following relationship exists between output-and input

voltages:

U .0 U(t). Uf(t)

U =

xy

o IO

3

Substitution from equations I and 2 into 3 yields

U0 =

{sin

+sin(2wt+)+ sin(2wt-)- sin(4wt+)- sin(4wt-)

+ - sin (6wt+)+ 4

In this equation the first term is a DC voltage which is proportional to the sine of the phase shift between the wavesignal and the position of the rotating

axis minus radians. Thus, maxima appear for 4= - - and -- radians.

The other terms represent AC-voltages with frequencies that are even multiples of the wavesignal's frequency.

In figure 3, the schematic diagram of the multiplier circuit is shown. The amplifier preceeding the actual multiplier is used to scale up the relatively

(6)

The main purpose of the low pass filter is to remove the undesired

AC-components from the output signal of the multiplier. However, it is

obvious that the introduction of a filter in a loop has an influence on the

overall response of the system. In fact, the tracking speed of the system is primarily determined by the low pass filter, amplifier and VCO. Because the latter two have a fixed transfer function, the system must be optimised with the low pass filter.

The purpose of the amplifier is to get a large open-loop gain,which is

necessary to get the desired static accuracy.

In figure 4 the actual implementation of the circuit is shown. The functions of low pass filter and amplifier are combined.

For low frequencies the amplifier acts as an integrator, which gives, as weh

known a phase-hag of - radians. Because, as we will see, the VCO also gives

a phase hag of radians the system will be unstable when no further

precautions are made.

Therefore, R2 has been added to the circuit, giving a small phase-lead at the frequency at which the open-hoop gain of the whohe system crosses the 0dB hine. At higher frequencies C2 takes over again as an integrator and C3 gives additional attenuation of higher frequencies. For how frequencies (w<12 rad/sec) an additional capacitor is connected in parahleh with C3 to keep the modulation distortion (momentary phase deviation) as small as possible. The transfer function of the circuit can be cahcuhated with the following eauation In which o1=R1C1=O.72 sec.

c22C10.6

sec. o3=R2C2=O.2 sec. G = C R3R4 4 3 R3+R4 =0.0382 sec. (w >l2rad/sec =0.218 sec. (w <12 rad/sec)

Figure 5 shows the calculated transfer function and phase response of the

circuit.

H(jw) =

-0.128

iwo1

(1+wO.753)(1+jwc4)

)

(7)

Between two measuringruns the reset switch is switched on and the output of the amplifier goes to zero volts. In this position the frequency of the rotating axis can be manually adjusted to the expected frequency of encounter for the next run. By doing this both the phase error and the lock in time are

minimised.

5.3. The voltage controlled oscillator

The VCO has been implemented with a Wavetek function generator module, type 120-021. By means of a few resistors and a capacitor the desired frequency range can be selected. With the values shown in figure 6 the oscillator covers a range from O to 4.5 KHz. By feeding a DC voltage to the VCO-module

input the frequency can be set anywhere in this range. The input sensitivity

is 450 Hz/V.

An operational amplifier preceeding the module delivers the required DC voltage. With no voltage applied to the input of the amplifier, the frequency can be manually adjusted from 100 to 990 Hz. For a voltage at the VCO amplifier input,

the gain of the amplifier can be set anywhere between O and 0,67. This gives

an overall sensitivity from the VCO of 3OOHz/V.

The square-wave output from the oscillatormodule is sent to a simple transistor

stage which converts the signal to the required (TTL) levels This signal goes

to the steppermotorlogic.

5.4. The stepper motor

The stepper motor has been inserted into the loop to deliver the required

synchronised rotating movement which drives the resolvers. To obtain a

regular proceeding movement the speed of the motor must be kept independent of load fluctuations (for example bearing friction). A stepper motor was the natural choice since, within certain load limits, the number of steps and there-fore the rotating speed is solely determined by the number and frequency of the trigger pulses.

The motor used has the following characteristics

manufacturer : Philips

type : PD 20

maximum working torque : 160 mNm

holding torque : 190 mNm

(8)

maximum pull-out rate 6000 steps/sec

step- angle 3°45'

number of steps/revolution 96

Figure 7 shows the relationship between torque and stepping-rate.

In our application the free running speed (unlocked condition) can be varied from 100 steps/sec to 990 steps/sec.

5.5. The optical sensors and square wave shaper

As already mentioned in 4 a feedback signal is derived from the axis by

means of two optical sensors, type MCA 81 (Monsanto).

The pulses are converted to a square-wave signal by means of a set-reset flip-flop that consists of two NAND gates (see fig. 8). Because the output signal

of the flip flop is not centered on zero an operational amplifier is added to

the circuit which delivers the required voltage levels. The output signal is fed to the Y-input of the multiplier and thus, the loop is closed.

5.6. The resolvers

The resolvers do not belong to the servosystem itself and are described in reference (2).So, we will not discuss them here.

)

(9)

6. The PLL as a feedback system

6.1. Considering the PLL as a linear system

Because a multiplier is inserted in the feedback loop, the PLL is not a linear system. However, it is possible to consider it as a linear system when the system is locked in and the phase disturbances on the input signal are relatively small. This can be seen by taking a closer look at equation 4 from 5.1.

110=

U{sin

sin(2wt+p)+

sin(2wt-)-}

4

Because the lowpass filter eliminates the AC components the only term that

passes the filter is,

UU

, f

U = s1n o 5T1

When the system is locked and has a sufficiently large static open look gain,

is almostradians which gives a sign reversal. When a small phase error

v(t) exists, caused by for example, speed variations of the towing carriage,

equation 6 becomes,

-U U

, f

U = sin u(t) o 5ir

For small values of the phase error sin u(t) ii(t), where u(t) is expressed

in radians.Thus,for small values of u(t) equation 7 can be approximated by

o

-U fU

5_ u(t)

Two things can be seen from this equation

The output voltage U is proportional with (t),

and therefore the system can be considered as a linear one.

y

Uf11

11 is also determined by This coefficient can be interpreted

as a gain factor. Unfortunately, U is a signal whose amplitude

is proportional to the waveheight, which is a variable.

6

7

(10)

For the locked condition the blockdiagram of figure 1 can be superseeded by

figure 9. The transfer function from this circuit is

H H2H3

H

1+H1H2H3

This equation consists of the following terms

Hi represents the transfer function from the multiplier and is

described by :

--U U

H = (V/rad)

i 5'Tf

We can calculate that Uf4.57V and U8.6V, so H1 becomes

H1= -2.5 (V/rad)

H2 describes the low pass filter response and is already found in

5.2, so

-0.303 i+jO.6u

9

H2

jLk) (i+jO,15w)(i+jwc4)

04=0.0383 sec for the high frequency range 04=0,218 sec, for the low frequency range

113 gives the transfer function of the VCO and the stepper motor

and is expressed in radians/volt. With a given VCO sensitivity of

300 Hz/V96steps/revolution and a reduction gear with a 2:1

ratio the overall sensitivity becomes 9,82 radians/volt sec. and

5

So, the open ioop gain depends on the waveheight, and therefore the

)

capture time and bandwidth. Also, the stability is influenced. The

amplitude cannot be stabilized by conventional automatic gain control circuits because they require a time constant which is far too long for our purposes. However, by assuming that at low wave amplitudes the input voltage of the multiplier is sufficient, and by clamping the voltage at higher wave amplitudes to a maximum value

this problem was solved.

(11)

)

From equation 8 we see that the phases of both signals are compared at the summing point in figure 9. The phase of the VCO-steppermotor combination can be described as

T

=

Iw

(t)dt 11

osc ) osc

o

w(t) consists of two parts, namely a constant factor w' which is manually adjusted to the expected frequency of encounter, and a

correction component with a value K II. (t), in which K is the conversion

factor of the VCO (9,82 rad/V5) and U.(t) is the input error voltage, Thus, equation 11 can be rewritten, (with K set equal to 1.75 rad/V5), as:

T

=w T+.75

I u.

(t)dt 12

osc o ) in o

Now, H3 can be calculated by assuming U.(t) to be sinusotdal and so we

find for the statical condition

1.75

=wT+

. U

osc. o

Because wT is not part of H3, the transfer function is

H 1.75 (rad/V) 14

3 3W

With the formula's derived for H1,H2 and H3 we can calculate the open

loop transfer function H1 of the loop, as such

H =HHH

=

ol

123

2

(jw)

(1+jO,15w)(1+w4)

This transfer function is shown in figure 10 for both the low- and high

frequency range. Figure 11 shows the phase response of the loop.

These figures show that the system is stable. However, the phase margin is very

small, especially at the low frequency range. For the closed loop this means

that peaking will occur near the cutt off frequency, which can give overshoot

when the system is locking or trying to retain its locked condition if phase

deviations are developing. In practice however, the phase deviations are very slowly varying phenomena which are tracked without giving any noticeable overshoot. Only during the capture process can some overshoot be observed.

(U. =11 sin wt) 13 in

1.82

l+O,6w

(12)

In figures 12 and 13 the transfer function and phase response of the closed loop are given. They were calculated by substituting equation 15 into 9 which yields

)

t

cy4 =O.0383sec for the high frequency range 04 =0.218 sec for the low frequency range

From figure 12 it can be seen that the bandwidth of the closed loop is

approximately 3.2. radians ( 0.5Hz), which is sufficient for tracking

existing speed variations.

6.3. The error components

)

Basically, four types of error components can be distinguished They are

A static phase difference e between the incoming and outgoing phases, which

exists when the open loop has a finite gain and a difference exists between the (adjusted) expected frequency of encounter and the actual frequency.

Because of the very small differences (<1%, mean value 40/00) and the use

of an integrator in the loop which gives a nearly infinite static open 1oop

gain,e is almost zero and can be neglected.

The second phase error u(t) is generated when the frequency of encounter

is changing due to, for example, speed variations of the towing carriage. This error is a slowly varying phenomena falling well within the bandwidth

of the system and is therefore completely tracked and thus also negligible.

)

A third error signal consists of the sum of the AC components that are present

at the output of the multiplier. When the input signal has no DC component these AC components are even multiples of the frequency of encounter, so the

lowest frequency we have to worry about is 2. For a lowest We628 rad/sec. 2 We becomes 12,56 rad/sec. (f2Hz). This frequency dictates the bandwidth

of the low pass filter because these unwanted frequencies must be sufficient-ly attenuated to prevent phase distortion. As we can see in figure 9, the transfer function for these error signals is the same as for the signals of interest. Hence, for the lowest We these ripple-components are attenuated more than 35dB(see figure 12, low frequency range). The resulting phase distortion

H

(13)

4. The last error component is caused by the distortion of the incoming wave signal from the pure sine-wave we desire. However, the amplitude of the

disturbances is very small and, to a large extent , of much higher

frequencies than the wave itself. Therefore these components may be

ignored. Theoretically, an exception must be made for odd harmonics of the

frequency of encounter. These frequencies can, depending on their amplitudes,

and relative phases, generate small DC voltages after the multiplier which

shall be compensated in the system by means of a phase shift, and thus a

phase error at the output of the system. In practice, however the harmonic

distortion of the waves and thus the phase-error it generates is negligible.

7. The capture time

The capture process does not lend itself to a simple mathematical analysis because of its highly complex nature. But it can be seen quite easily that the

capture time is determined by two factors, namely

The frequency difference between input- and feedback-signal when the

capture process is iniated.

The initial phase shift between both signals when the process starts.

In our case factor (1) does not apply because we already adjust the free running frequency the the expected frequency of encounter. Because, when the capture process starts, the phaseshift between the two signals can be anywhere between O and 2îr radians, the capture time can vary from zero to some maximum. To get an impression of the average capture time, it was measured 66 times and

plotted as a probability density (fig.14), as well as a cumulative probability

density diagram (fig. 15). The same measurement was repeated for a second set of

conditions, of which the results are shown in figures 16 and 17.

From the measured data the mean value was calculated, which was 13 and 8.7

seconds respectively. For the standard deviations, values of 3 and 2 seconds

where found. From the cumulative probability density diagrams it can be seen

that the system is locked within 19 and 13 seconds, respectively. Of course

the limited number of experiments must be taken into account, but nevertheless

the figures give a fairly good idea of the time necessary to achieve the locked

condition. With these times, the system can be locked in and a run be made before the end of the model-basin (effective length 100, speed 2m/sec) is reached.

(14)

The results

Originally, the system was developed for use with springing experiments (3) which were being prepared. But prior to this, it was used for measurements on

a model in regular head waves (4) . During these tests the accuracy of the

system was checked. This was done by measuring the extreme values of the

absolute motion of the model forward by means of a potentiometer and a

Uy recorder. Also, this value was calculated using the heave, pitch and

wave amplitudes and phases which were obtained by means of the PJ.L

servo-V

system. This value was called . Of course, the amplitudes found must be

the same. Some of the results re shown in figure 18. The straight lines

indicate the ideal functions, whereas the dots gives the mean values that were found. From figure 18 it can be seen that the results are quite satisfactory. The same conclusion was made during the springing tests. Here too, some of the values with the PLL servosystem were checked with the old method, using aTJ\T recorder, which confirmed the validity of the results.

Unfortunately, it is rather difficult to give an estimation of the accuracy of the system due to the fact that the method used to check the results has an accuracy that's not much better than 5%. However, from the results that we

have seen and the good reproducebility (for example, the phase errors between

repeated runs have a standard deviation of 1.850) we get the impression that the

PLL servosystem has an accuracy of better than 4%.

Conclusion

The PLL servosystem works properly and has the required accuracy. It has been proved to be labour-saving as compared with the older method using a UV recorder. With a somewhat more optimised version it is perhaps possible to extend the usable frequency-range to a lower value than was originally

requested. This would increase the usefulness for tests with lower frequencies However,the maximum permissible capture time must be taken into account,

because this time increases with decreasing bandwidth.

(15)

lo. References

I Signetics databook "Lineaire Integrierte Schaltungen't, Signetics corporation, 1972

2 M. Buitenhek: "Een fase-component meetsysteem",

Technische Hogeschool Delft, Laboratorium voor Scheepsbouwkunde, Rapport no. 113, February 1964

3 R. Wereldsma and G. Moeyes: "Wave and structural load experiments for

elastic ships",

Proceedings 11th Symposium on Naval Hydrodynamics, London, March/April 1976

4 Ir. J.M.J. Journe: "Motions, resistance and propulsion of a ship

in longitudinal regular waves",

Delft University of Technology, Ship Hydromechanics Laboratory, Report 428, May 1976.

(16)

wave-height

me te r

slotted disk

optical

sensors

multiplier

Isquare-wave

shaper

f

J

demodulator

I

resolver

demodulator

strain-gage

me te r

low-pass

filter

steer motor

a=hold

b=compute

/1/ J

frequency adjus4

(manual) I

ampli

fier

voltage

control led

oscillator

s

Fig.1 Block diagram of phase locked loop servo system(dashed

area) as part of a measuring system.

n

integrators /reset

o

jwin

carriage

)

wave-signal

pick up

dynamometer

(17)

I

O

U X U y

U .0 /10

X

y

Output after low-pass filter

and amplifier.

Signal from VCO (extended

time-base)..

Output from sensor 1.

Output from sensor 2.

t

(18)

Signal from

wave-height

meter

50 U i n

(from multiplier)

74

4

Fig.3 .1ultiplier circuit.

470k

HUb

_Ub

Reset

i O O 100k

cl R2

I I

120k

C2 Rl

120k

Fig.4 Low-pass filter and amlifier.

X-adj. Y-adj. Output zero.

+Ub

1ÌJ!I

20k 20k

4k7 56k

11

t

o-lu

out

llu J j lt

2u35!C3TC3b

\

t I (to VCO)

(

low frequency range.

Uout=U .0 /10

X

y

)

)

from square-wave

shaper.

)

(19)

O

I

Ò

o

high frequency ranqe(w>12ad./SeC.)

+

low

,, 1

Fig.5 Frequency- and ohase resoonse of low-pass fitter

plus amplifier.

(20)

560

18k

5k to rq u e (mNrn)

From amplifier.

o

6k2 400 300 200 mo o lo 6,2S

82k

H H

L.

3k

Sensitivity

Ficj.6 Voltage controlled oscillator (VCO).

30 18,75 loo 62,5 12k

680k

680k

300 187,5 1000 625 Ref. in

VCG

120-021

3n3

3000 i 875 10000 6250 I

11k

LH

I.

BAW62

77668021 1O power piI- oui (W) 8 6 £6 2 o 30000 steps/s 18750 rev/mm

I

3k3

o

/

To stepper

motor

ClO7logic.

Fig.7 Torque and relevant outout power versus

stenoing rate

from steooer motor.

)

)

UI

P020

III

ii!

ii.'

a

W)

R

uhlAn:'

IIiiaF

iit

II

iiii

11!1!iIÈiI::

Rau

IIIIIIIIiI

HIRR.IIIft

1I11*IUIIII

liii

liii

IiluuIi

aiihi,A4iîIai

uuIIrmawÁIsuaIIiIII

¶ut

HIIIft

+IUIIIIl

ii

I I

II II

torque

jIU4HllhI

I

,jI!IIIaa

all

auÏ!!L

UIIIUIflhILRMIIlIIII

IIIIIL

I

ut! in

i

RUlli

709

(21)

J-From optical

.

sensor 1.

J-+5v

I

1k

+5

10k

LOk

-r

1k

o

I

_177I_

From optical

sensor 2.

1k

BC1O7

68 1W

o-

j

+5v

To optical sensors

(liqht emitters)

disk

ri

L_

'MC.81

-

ooticaL sensor .$

Fig.8 The optical sensors and sauare-wave

shaDer.

6k8

o

To y-input

of

multi-plier.

To counter

(neriod)

10k

10k

(22)

it+ (t)

o

-18

-200

-220

-240

-260

AC-comoonents

o

wt

Hl

Ficj.9 Liniarised model of

the servo system.

Fig.l0 Bode-diagram of open

loon.

Fiq.11 Phase response of open

loop.

H2

H3

o

hiqh frequency range.

+

low

I,

.

o

.4 1.6 6.4

25.6

102.4

w (rad. ¡sec.)

102.4

w

(rad./sec.)

)

)

J

)

(23)

H (dB) 20

180

-Ï 0,1

.

e

-90-0,1 0,4

-20

-40

-60

-80

-100

Fiq.12 Bode-diaqram of closed loop.

cl

1,6 6,4

=low frequency range.

=high frequency range

270

-Fiq.13 Phase-diagram of closed loop.

25,6

102,4

(i)

(24)

P(t) o, o, o, ,

n=66

x=i 3sec. SD=3 , 2sec.

e2

rad. ¡sec. o,: u.

=2V

in tt 0,0 o 4 8

12

16

20

-

t(sec.)

Fig.14

Propability density function of lock-in time(low

frequency ranqe)

1, P(t) s s '8 '

12

116 o -

t(sec.)

Fig.15 Cumulative oronability density function of lock-in time

(low frequency range).

)

(25)

p (t) tl,o

-I

n=66

-

x=8,7sec.

SD=2sec.

oI8

U.=2V

in tt 0,6 0,0

0,4-g i i i i i I I t O 4 8 12 16 20

t(sec.)

Fig.16 Prooability density function of lock-in time(high

frequency range)

1,0 P(t)

I

0,8

0,6_

0,4

-

0,2-0,0 I O 4 8 12 16 20

t(sec.)

Fiq.17 Cumulative oropability density function of lock-in time

(26)

Fn =.15

FULL LOAD

CONDITION

Fn = .25

Fig.18 Relation between measured extreme values of absolute

motion and the motion amplitude calculated from heave,

pitch and wave.

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