- Email: [email protected]
Enhancing The Response Speed And Reducing The Mechanical Mounting Tolerances of Impulse Tachometric Systems
ENHANCING THE RESPONSE SPEED AND REDUCING THE MECHANICAL MOUNTING TOLERANCES OF IMPULSE TACHOMETRIC SYSTEMS AUTOMATION
L. Natens, Instrumentation Dept. , Agfa-Gevaert N. V., Mortsel, Belgium
PAPER 1.3
SUMMARY
DESCRIPTION OF THE FREQUENCY MEASURING METHOD
Speed monito r ing and control is very important to the P.R .P. -industries. More and more digital systems become to be used, and impulse tachometers seem to be the appropriate instruments to this purpose . This type of instrument presents the highest static accuracy, when used with a supervisory computer control. Mostly the dynamic control is still performed by more classical analog means, and a particular difficulty arises by converting the series of impulses into a continuous signal, the converter generally introducing considerable time constants . I n order to avoid this, one can measure instantaneous· Iy the true interval of one period . Afterwards an inversion is produced to obtain a speed proportional signal. A method, performing this with a maximum delay of one pe riod is dealed with . At the same time some considerations are made to redu · ce the very tight tolerances on mounting impulse generating systems . A double detection system is proposed, making the detector reading almost independent on the mechanical precision . Practical realisations are shown, the results of which can be compared to the performances of the prec ision d-c tachygenerator. Some thoughts are also given to the manufactur ing of coding discs, meeting the particular requirements of the method, by Cl photographical process of gene ration .
Pr inciple of operation The beam from a light source is interrupted by light· absorbing windows, moving in a plane normal to this beam . This causes the photodetector to generate a square wave current, the frequency of which is directly proportional to the speed of the moving gear. ERROR ANALYSIS OF THE METHOD Imperfections of the square wave Due to the imperfections of the square wave, frequency can change instantaneously, without any real change of speed (fig. 1).
Fig . 1. 1Nl RODUCTION Whenever highly accurate speed·control of rotation or translation is required, digital measuring means are often preferred to analog means . The most adequate solution appears to be the genera· tion of an impulse train whose frequency is proportional to the desired measured value. Generators for these purposes can be classified as : - mechanical . magnetodynamic, where a magnetic gear moves in front of a fixed coil or a moving contact, . magnetostatic, where a metallic gear moves in front of an oscillator coil , stopping oscillation due to the additional damping, - optical, where the light beam from source to receiver is period ically interrupted by means of an optically dense medium . The last method is especially preferred for high precision applications, because a very high pulse density can be obtained with engraving or photographic techn iques.
15
a) with the first window, the electrical impulse will arrive at the correct time: frequency will be un· affected, b) with the second one, the transition from I ight to dark is somewhat blurred , the impulse will arrive too late, resulting in an apparently lower frequency, c) in the third case the transition is irregular (particularly with engraving), causing the same effect as in the second case, d) in the fourth case the transition shows some defects (typical for photographic materials). An apparently lower instantaneous frequency will be generated, followed by a higher one, as an undesired impu Ise has been generated. Averaging the mean frequency over a relatively long period , gives a measurement of much better stability than can be obtained with analog techniques. It has the disadvantage, however, of masking real sudden speed changes. Therefore this method cannot be considered for precision drive systems. A much better
directly proportional to their order. The second factor produces a phase shift, also p r oportional to the harmonic order, which leads to a sawtooth output signa l. By increasing the number of windows, measured at the same time by optical averaging, the hyperbolic enveloping tends to decrease more rapidly. By narrow ing the sensing slit, the sinusoidal envelopings will broaden, meaning an increase of bandwidth, or · a mo re perfect reproduction of the ideal square wave. An infinitely small slit would reproduce this square perfectly , and would therefore del iver a signal with exactly the same amplitude and frequency spectrum as the original signal S. Instead of well -defined theoretical spectral lines, the Fourier spectrum of S will contain a number of probability density functions , due to the imperfections of the manufactured code-discs. The I inear sum of N delayed versions of S over 1, 2, 3 . . . N periods will influence the width of these distribution functions ( J of P[S(jw)] . Let Pn [S (t) ] be the probability density function of one slit signal , with
averaging method uses a complementary grating (fig. 2) which optically averages over a number of windows. In this way individual errors are divided by the number of windows being used at that moment .
Fig . 2
This method is widely used and in the paper some more details are dealt with . Let S be the signal generated by a square wave window profile , and sensed by a slit 6 , of exactly equal width as that of the windows. The Fourier transform of S will be : , F [S(6 1)* H(~) ]
= F~w +00
F'('J'w. )
= F(-r w ) J
h(6)
). H (tu) )
= F(tu})
PI = P2 = P3 = . . . Pn
e- rw6 d6
+00
_00
p[S(t)]= _
where h = 1 for 0 ~ 6 ~ 6 1 6 0 h = 0 for 6 1
J00
P1.P2 . . . PndS
and
< <
P[SU w )]=P1(iw)
0
P2(jw) ... Pn (jw)
also
F' (Ow ) = F (rQ) )
+00
J
pIt) dS= 1
_ 00
= F(tw)
sin .w 61
6 __ __2_ 1 W 01
0
e
-2-
The enveloping of the frequency spectrum of the window profile detection transformed by a slit of equal width , is given by an expression of the form sin x
x
(fig . 3)
21T , a spatial frequency deWith a first zero at 61 termined by the slit 61 at the second harmonic of
Fig . 4
Each delayed version of S can be obtained by the delay distribution function . Assuming the amplitude is constant, this will only give riseto a different phase shift of the Fourier components. The linear sum of an increasing number of delayed signal versions will result in a narrowing of the main frequency distribution . But this process also results in broadening the higher harmon ics distribution to a noise spectrum, which means a relatively higher certitude about the main frequency. The signal reproduction, however, will differ more and more from the original one, tending towards a sinusoidal wave.
27T 26 1 . I
2n
x
Fig. 3
a) Errors by excentricity With disc generators - the ones mostly encountered in practice - another er ror arises, when the generating
The same happens to each even harmon ic, being suppressed from S. The odd harmonics are attenuated,
16
averaging, even when this will cause a considerable change of amplitude of the ouput signal.
center of the impulses does not coincide with the center of the rotation. The number of pulses n, per unit time at an angular velocity and linear impulse density d at the circumference of the disc with radius R is given by
c) Errors by axial and/or radial clearance The errors due to axial and/or radial clearance can be treated in the same way as those caused by excentrical mounting and/or parallax. Resuming th is survey we can conclude that the best method for measuring frequency is the optical averag· ing method . Two major drawbacks however exist, namely the sensitivity to excentrical mounting and the difficulty of electrical processing of the signals.
n= GtJ. R. d
Excentric mounting of a disc by a valuE.' t results (fig. 5) in a change of distance between the pulse crown and the center of rotation. Therefore: n' =t.l (R ±e)d Resulting in an error S=
n - n' e n- "" - + - -R
Advantages and disadvantages of optical averaging
as long ast: is kept small.
The errors due to excentricity are transmitted directly when using optical averaging . Moreover, great care has to be taken when using this method, to mount the setup properly, especially when a not exactly complementary grating is used (figure 7), but rather a narrow slit grating, in order to obtain better rise and fall times of the measured impulses. This leads to less uncertainty in determining the exact frequency as there will be less phase and frequency modulation, resulting from an imperfect detection.
This error is equal to zero when averaged over one revolution .
,
~
..
. , '
~ : ~ ~
~
~
~
~ Moster GratinQ
"""" ""'"'"
-=-.".... == """"
==
~ ~
~w.......""""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ~ ~
'" . I'
CorT-plementcry Grotn9 CorT-plementary Slit GrOlinQ
Fig. 7
Fig. 5
Electrical aspects of frequency measurement The electrical processing of speed data collected by impulse tachometers is mostly done by converting each impulse into a square of constant height and width . Time integration of such an impulse train results in an analog value, directly proportional to the desired speed value. I n practical servosystem techniques, measured values must have a ripple content of less than a few percent, in order to keep the control amplifiers linear and unsaturated. As the measured value has 100 % ripple content, the ripple will have to be filtered out. This is usually done by R·e-filters, thereby introducing considerable time constants into the signal path, especially since the filtering has to be adequate even for the lowest input frequencies. This results in lowering the frequency-response of the servosystem, unless very high frequencies are chosen, which results in difficulties of generating code discs, working out the mechanical set-up, and processing of the electrical data with high speed electronics under industrial circumstances of RFI and EMI.
b) Errors by parallax Parallax errors arise when the light beam is not perfect· Iy parallel. Figure 6 shows how the image of a slit can be distorted, depending on the direction of the light beam.
---
~-------- . - . ~. -.~ ----,...,.--- ...:.~ . -
--- ----
------
PHOTOCELL
-~-=-:--,------<-'-- ' - - '
----
-------
-------
I
GRATING Fig . 6
This error can be particularly disturbing, when the cocje-disc is not mounted mechanically with sufficient care in the normal plane to the axis of rotation . However, this error will be greatly suppressed by optical
PERIOD MEASURING SYSTEMS Much more interesting than frequency measurements
17
a) Simple non par allel cut-off
are measurements of periods, because this gives instantaneous results , with a maximum delay equal to the measured period . This means that the time constant can be kept very small .
The case Of non parallel cut-off is completely similar to that of excentricity and these two will therefore be treated together.
Optical opportunities
b) Errors due to defects of the windows at parallel intersect ion
The method appears still more appealing considering the optical opportunities to work out this system . Instead of measuring the time elapsed between two successive impulses, it is also possible to measure the time between the actuation of two detection systems by one impulse of the code disc. In this case an absolute guarantee is present for measuring the exact tirr.e inte rval that an imaginary point of the periphery of the code disc, located at a constant distance of the real rotation center, needs to t ravel over a given trajectory, in other words, the exact linear speed of a point at the periphery of the disc, at a constant distan· ce of the rotation center, from which it is easy to calcul· ate the angular speed . Two important advantages result : . independence of the centering of the code disc, - independence of the impulse density tolerance along the periphery of the code disc. It can be shown that the measured time depends only on the distance between the two measuring elements and the linear speed V of an imaginary point P, located on a circle with radius R , equal to the distance between the detection systems and the rotation center (figure 8).
The errors generated by defects of the windows on the code disc (figure 9) as they result on photographic or engraved plates, will only cause a higher rise time of the measured impulses. This will delay the electrical impulse, without however affecting the measuring accuracy, because the same delay will result for both the detection systems. More detail is given further in this paper. It is clear that the error affects only the rise time of the electrical impulse.
Fig . 9
A
---r- - :
~-- -J
I
f
I
/ '
1
'
/
I
I
I
I
,
,I
I
I
I
'
I
:I
:1
1/
x I I
a)J __J_ 0-T---t-1
Fig. 10
I
Three phases (see figure 10) can be considered :
I
1. the opening has entered , but is not still complete with in the window 1 x b .
-----7------t----:----1r-
A=
f
o
(l-d)dx
whereas A = transmitting area x = jvdt dx = vdt
Fig . 8
2. the opening lies completely within the window 1 xb Error analysis A=
Let IJ S first consider the errors due to an imperfect ge neration of i mpulses on the code disc.
18
fd
1.dx
3. the opening no longer lies completely within the window 1 x b A=
b+d
f
infinitely small, and of length I, the rise time becomes to a first approximation (figure 12) \
JY R \ ~ -1.- ,AB , , /r - - - - -__ 8 - --. -_ - :-:.--:::--. 6 " I ,
d . dx
~-----=-=-
b The successive results are represented in fig . 10, where it appears clearly that the leading edge of the impulse has been delayed, in such a way that even the halfvalue no longer coincides with the ideal position.
........ '---' ---'" "
/
/
----------------;~S
---Jt----\
--- - -
,
-
I
~~---,/
c) Defects of windows w ith non parallel intersection
Fig. 12
Defects of windows with non parallel intersection can be considered as a superposition of non parallel intersection with the foregoing case . The area A grows differently, depending on how d is situated against 5, in other words, if the defect will fall in phase I , 2 or 3. Here again, because of the symmetry of the two detection systems, the measurement will not be affected . From this it can be concluded that errors due to defects of window edges have insignificant influence on the measurement, because of the symmetry of the two identical sensors.
and '
= 1
tB =
10 R2
(figure 13)
d) Errors by excentricity and non parallel intersection
,r---r---------------I ,,I , ,, ,,
The errors caused by excentricity are most difficult to eliminate from a mechanical system. Suppose 0 to be the generation center of the code disc, and 0' the center of rotation (figure 11). will describe a circular path around 0' during the movement, with radius 6 . Let A and B be the locations of the two sensors with slits SA and SB '
o
, I
, ts I
---1
t--
t
Fig. 13
If 0 =1= 0, SA and SB are finite, with length 1 and width b, the rise time can be calculated as follows (dividing this period in three phases) :
1) A 1 =
K
I
2) A2 =
/
Jl
v
t adt _ co g -
2
Co)
2 R 2 cotg at 2 2
o
0
J2 v.b.cotg adt =
(6
R b cotg at
o
Fig. 11 3) A3 Le! 13 be the angular displacement window W has to effect to reach SB from SA. The angles a: and 13 will be equal, so that 13 will be exactly the same as when cS would be equal to zero . Because T =
a:R
-;:i=f
=
= =
}3 (bv - v2 t) cotg adt
o
vt (b -
vt 2" ) cotg a Co)
a: w
/ o
t3
Rt
= l>iRt (b - - ) cotg a /
2
T will be independent of the excentricity 8.lf SA and Ss are infinitely small then the transition time for passing from zero to one for the optical impulse is also infinitely small if 8 = O. I f cS =1= 0, SA and Ss are
If
19
/
= constant A = Al + A2
t3
0
lA!
+ A3 ' (figure
14)
t1
A
r-----
T"'" - - - ; - -
--~.,....---------
formances for this system, and we therefore deliberately centered our attention on the simple detection system. f) Errors by parallax The principle itself eliminates errors due to parallax, because both the sensors are subjected to it. The actual error results from differences between the detectors, and more particularly from the asymmetry of the light projection system, because generally on Iy one such system is used . Much care has therefore to be taken to obtain a perfectly parallel light beam.
Fig . 14 g) Errors by axial and radial clearance In all these cases the ideal moment coincides with the half-value of the impulse, the value of which has to be detected with the greatest care. e) Errors by irregular distribution of the windows For each measuring cycle, only one window edge has to be used and therefore a certain amount of irregularity in distribution of these edges (figure 15) cannot in any way influence the measured value. However, should the distribution be very irregular, the impulse sequence at the two sensors could be disturbed. The actual error on the distribution of the windows is adsorbed within the dead zones between two successive measuring periods. Usi ng this method, optical averaging becomes very difficult, because the perfect matching of the earlier mentioned sensors has to be carried out here for optical gratings. Even supposing a suitable technique existed for doing this, there would still remain the difficulty of detecting the same edges, or at least, the same portion of each series of edges by both systems, because an essential requirement is that the sensors be spaced less than one period from each other .
The errors caused by axial clearance can be considered identically to the errors due to parallax, in so far as they are slow. Sudden displacements, appearing within one measuring period can cause considerable errors, because they disturb the identical character of the two sensors. They can be eliminated by the use of parallel light beams. Errors by radial clearance can be serious also if they occur very rapidly. They will be measured as differences in distance between the two detection systems and are proportional to ratio of the clearance value to the distance between the two sensors. Radial clearance can be eliminated by USing precision bearings, eventually air bearings. Electrical opportunities Measuring a period of time can be done electrically in a very simple way, by feeding a constant current into an integrator for the time of the measured period. The output voltage of such a circuit is directly proportional to this time interval, and is available immediately after this period. The resulting value contains no ripple, thus needs no filters . The only time delay is the measuring period itself. Unfortunately the output value is inversely proportional to the value usually asked for, nl. speed. DESCRIPTION OF A PRACTICAL SYSTEM Optical part
Fig. 15
This can be done by means of fibre optics. It would also be possible to separate both gratings, if the exact number of periods between them is known, and the second measurement is delayed by the same number of periods. However, this results in a loss of the response speed of the system, and complicates the electronics, requiring a shift register for each measurement. Optical averaging does not result in much better per-
The optics of the system were kept as simple as possible. The distance between two successive window edges was therefore the upper limit of the distance between the two sensors. This distance is of course limited to the physical diameter of photodetectors. In practice the process is reversed, as this distance will determine the distances between the window edges. Photocells are photoduodiodes, e. g. LS-400 or 600 from Texas Instruments, MRD 200 from Motorola. They measure 2,5 mm. This limits the impulse density to 400 per meter. In order to keep the number of impulsions per revolution reasonably high, the diameter, of the code disc was chosen to 500 mm. The impulsions were recorded photographically on a graphic-arts emulsion, coated on a dimensionally stable glass disc with sufficient mechanical strength. The impulsions were generated by succesive exposures of a slit by means of a flash lamp.
20
In front of the sensors diaphragms of 0 . 1 mm width are placed. They were al so produced photographically (figure 16) . I f a higher pulse density would be required, readout optics will have to be arranged in order to magnify the distance between two successive impulsions to a value slightly larger than the distance between the two sensors, as mentioned earlier . However, optics will have to be first class projection optics presenting no deformation of the magnified image (figure 17) . Another way to read out small impulsions would be fibre optics, leading to sensitive photocells , e. g. photomultipliers (fig . 18). A brief description of the electronic process has already been given . We will now describe some details . During the measuring period (figure 19) an integrator runs with constant current. Its output voltage at the end of the given interval is read out into an analog memory -cell, after wh ich the integrator is reset .
Fig . 16
GRATING
InteQrator
(J
~!
\
Cl
<
~-+--T-,Anol09 memory cell
PC . Fig . 19
<
This value is a measure of the period , whereas generally speed is of more direct interest. Should only one speed value be desired, the hyperbola can be approximated by its tangent . Should this not be sufficient, various methods are available to perform an inverse relation. One of them is the s. c. diode approximation method, straightening the hyperbola to a straight line by means of suc cessive approximations. However, this method is rather cumbersome, and therefore mostly a real division is performed . Here also many techniques can be useful, e. g. Hall-effect multiplication, logarithmic converters based on semiconductor junc tion properties, time division methods, etc . The logarithmic converters are used very frequently, because they are cheap and fast. Their static accuracy is mostly not good enough for this kind of application . A good compromise can be reached by using a digital technique, even a computer for static and slow frequency components of the measured value, and using the logarithmic conversion for higher frequencies (figure 20) .
Fig . 17
GRATING
Fiber Optics
AdS
Fig . 18
~ Over-Oll
Choroct
CharacteristiC
et
LOQOrithmic '" Divider Digital ioteQl'otar
The photographic way was chosen for its high resolution,and very sharp edges, due to small thickness of tne layer.
21
-f---~~----',""------------ log w
Fig. 20
This modulator is built around an operational amplifier , containing a positive and a negative feedback path . The posit ive feedback makes the output voltage switch between two distinct levels, the negative feedback is delayed, and assures the proportional ity between mean output voltage and input voltage . The mean current into the capacitor has to be equal to zero, since otherwise the voltage across its terminals woukt keep rising and no steady state would be attained ; the mean value of the input current must be equal to the mean value of the feedback current, assuring thereby the exact proportional ity between in- and output .
Description of the period measuring The output signals of the two sensors are amplified and applied to triggers T1 and T2 (figure 21) . These circuits control a flip-flop, which in turn controls the integrato r .
Controlling a perfect switch in series with a resistor by means of this square wave results in modulating the mean current through this resistor when a voltage is applied to this circuit. . ei I =~. ~
I
a = on/off ratio
Fig . 21
Here again averaging of the current can be done by connecting a capacitor of sufficient value across the circuit. Inserting this circuit into the feedback path of an operational amplifier makes a divider out of this circuit because
After the measuring period, one fl ip and flop is first started controlling the analog memory, after which a second flip and flop is started to reset the integrator. The reset of the integrator is done by means of a photo-electric coupled switch . The output voltage of this circuit is proportional to the measured period, and has to be inverted to be a measure of speed .
ii = if
=a . if
= -a ' .
eo
Rf
e I'
li = Rj Divider circuit An electronic divider circuit has been developed, based upon the principle of time division of a resistor value (figure 22) . This time-division is controlled by a pulse depth modulator.
Choosing the modulation frequency sufficiently high, the time constant CfR f
a can also be kept small, in order to keep the response speed of the divider circuit reasonably high.
eo
€i
Fig . 22
22
L. Natens, Instrumentation Dept. , Agfa-Gevaert N. V., Mortsel, Belgium
PAPER 1.3
SUMMARY
DESCRIPTION OF THE FREQUENCY MEASURING METHOD
Speed monito r ing and control is very important to the P.R .P. -industries. More and more digital systems become to be used, and impulse tachometers seem to be the appropriate instruments to this purpose . This type of instrument presents the highest static accuracy, when used with a supervisory computer control. Mostly the dynamic control is still performed by more classical analog means, and a particular difficulty arises by converting the series of impulses into a continuous signal, the converter generally introducing considerable time constants . I n order to avoid this, one can measure instantaneous· Iy the true interval of one period . Afterwards an inversion is produced to obtain a speed proportional signal. A method, performing this with a maximum delay of one pe riod is dealed with . At the same time some considerations are made to redu · ce the very tight tolerances on mounting impulse generating systems . A double detection system is proposed, making the detector reading almost independent on the mechanical precision . Practical realisations are shown, the results of which can be compared to the performances of the prec ision d-c tachygenerator. Some thoughts are also given to the manufactur ing of coding discs, meeting the particular requirements of the method, by Cl photographical process of gene ration .
Pr inciple of operation The beam from a light source is interrupted by light· absorbing windows, moving in a plane normal to this beam . This causes the photodetector to generate a square wave current, the frequency of which is directly proportional to the speed of the moving gear. ERROR ANALYSIS OF THE METHOD Imperfections of the square wave Due to the imperfections of the square wave, frequency can change instantaneously, without any real change of speed (fig. 1).
Fig . 1. 1Nl RODUCTION Whenever highly accurate speed·control of rotation or translation is required, digital measuring means are often preferred to analog means . The most adequate solution appears to be the genera· tion of an impulse train whose frequency is proportional to the desired measured value. Generators for these purposes can be classified as : - mechanical . magnetodynamic, where a magnetic gear moves in front of a fixed coil or a moving contact, . magnetostatic, where a metallic gear moves in front of an oscillator coil , stopping oscillation due to the additional damping, - optical, where the light beam from source to receiver is period ically interrupted by means of an optically dense medium . The last method is especially preferred for high precision applications, because a very high pulse density can be obtained with engraving or photographic techn iques.
15
a) with the first window, the electrical impulse will arrive at the correct time: frequency will be un· affected, b) with the second one, the transition from I ight to dark is somewhat blurred , the impulse will arrive too late, resulting in an apparently lower frequency, c) in the third case the transition is irregular (particularly with engraving), causing the same effect as in the second case, d) in the fourth case the transition shows some defects (typical for photographic materials). An apparently lower instantaneous frequency will be generated, followed by a higher one, as an undesired impu Ise has been generated. Averaging the mean frequency over a relatively long period , gives a measurement of much better stability than can be obtained with analog techniques. It has the disadvantage, however, of masking real sudden speed changes. Therefore this method cannot be considered for precision drive systems. A much better
directly proportional to their order. The second factor produces a phase shift, also p r oportional to the harmonic order, which leads to a sawtooth output signa l. By increasing the number of windows, measured at the same time by optical averaging, the hyperbolic enveloping tends to decrease more rapidly. By narrow ing the sensing slit, the sinusoidal envelopings will broaden, meaning an increase of bandwidth, or · a mo re perfect reproduction of the ideal square wave. An infinitely small slit would reproduce this square perfectly , and would therefore del iver a signal with exactly the same amplitude and frequency spectrum as the original signal S. Instead of well -defined theoretical spectral lines, the Fourier spectrum of S will contain a number of probability density functions , due to the imperfections of the manufactured code-discs. The I inear sum of N delayed versions of S over 1, 2, 3 . . . N periods will influence the width of these distribution functions ( J of P[S(jw)] . Let Pn [S (t) ] be the probability density function of one slit signal , with
averaging method uses a complementary grating (fig. 2) which optically averages over a number of windows. In this way individual errors are divided by the number of windows being used at that moment .
Fig . 2
This method is widely used and in the paper some more details are dealt with . Let S be the signal generated by a square wave window profile , and sensed by a slit 6 , of exactly equal width as that of the windows. The Fourier transform of S will be : , F [S(6 1)* H(~) ]
= F~w +00
F'('J'w. )
= F(-r w ) J
h(6)
). H (tu) )
= F(tu})
PI = P2 = P3 = . . . Pn
e- rw6 d6
+00
_00
p[S(t)]= _
where h = 1 for 0 ~ 6 ~ 6 1 6 0 h = 0 for 6 1
J00
P1.P2 . . . PndS
and
< <
P[SU w )]=P1(iw)
0
P2(jw) ... Pn (jw)
also
F' (Ow ) = F (rQ) )
+00
J
pIt) dS= 1
_ 00
= F(tw)
sin .w 61
6 __ __2_ 1 W 01
0
e
-2-
The enveloping of the frequency spectrum of the window profile detection transformed by a slit of equal width , is given by an expression of the form sin x
x
(fig . 3)
21T , a spatial frequency deWith a first zero at 61 termined by the slit 61 at the second harmonic of
Fig . 4
Each delayed version of S can be obtained by the delay distribution function . Assuming the amplitude is constant, this will only give riseto a different phase shift of the Fourier components. The linear sum of an increasing number of delayed signal versions will result in a narrowing of the main frequency distribution . But this process also results in broadening the higher harmon ics distribution to a noise spectrum, which means a relatively higher certitude about the main frequency. The signal reproduction, however, will differ more and more from the original one, tending towards a sinusoidal wave.
27T 26 1 . I
2n
x
Fig. 3
a) Errors by excentricity With disc generators - the ones mostly encountered in practice - another er ror arises, when the generating
The same happens to each even harmon ic, being suppressed from S. The odd harmonics are attenuated,
16
averaging, even when this will cause a considerable change of amplitude of the ouput signal.
center of the impulses does not coincide with the center of the rotation. The number of pulses n, per unit time at an angular velocity and linear impulse density d at the circumference of the disc with radius R is given by
c) Errors by axial and/or radial clearance The errors due to axial and/or radial clearance can be treated in the same way as those caused by excentrical mounting and/or parallax. Resuming th is survey we can conclude that the best method for measuring frequency is the optical averag· ing method . Two major drawbacks however exist, namely the sensitivity to excentrical mounting and the difficulty of electrical processing of the signals.
n= GtJ. R. d
Excentric mounting of a disc by a valuE.' t results (fig. 5) in a change of distance between the pulse crown and the center of rotation. Therefore: n' =t.l (R ±e)d Resulting in an error S=
n - n' e n- "" - + - -R
Advantages and disadvantages of optical averaging
as long ast: is kept small.
The errors due to excentricity are transmitted directly when using optical averaging . Moreover, great care has to be taken when using this method, to mount the setup properly, especially when a not exactly complementary grating is used (figure 7), but rather a narrow slit grating, in order to obtain better rise and fall times of the measured impulses. This leads to less uncertainty in determining the exact frequency as there will be less phase and frequency modulation, resulting from an imperfect detection.
This error is equal to zero when averaged over one revolution .
,
~
..
. , '
~ : ~ ~
~
~
~
~ Moster GratinQ
"""" ""'"'"
-=-.".... == """"
==
~ ~
~w.......""""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ~ ~
'" . I'
CorT-plementcry Grotn9 CorT-plementary Slit GrOlinQ
Fig. 7
Fig. 5
Electrical aspects of frequency measurement The electrical processing of speed data collected by impulse tachometers is mostly done by converting each impulse into a square of constant height and width . Time integration of such an impulse train results in an analog value, directly proportional to the desired speed value. I n practical servosystem techniques, measured values must have a ripple content of less than a few percent, in order to keep the control amplifiers linear and unsaturated. As the measured value has 100 % ripple content, the ripple will have to be filtered out. This is usually done by R·e-filters, thereby introducing considerable time constants into the signal path, especially since the filtering has to be adequate even for the lowest input frequencies. This results in lowering the frequency-response of the servosystem, unless very high frequencies are chosen, which results in difficulties of generating code discs, working out the mechanical set-up, and processing of the electrical data with high speed electronics under industrial circumstances of RFI and EMI.
b) Errors by parallax Parallax errors arise when the light beam is not perfect· Iy parallel. Figure 6 shows how the image of a slit can be distorted, depending on the direction of the light beam.
---
~-------- . - . ~. -.~ ----,...,.--- ...:.~ . -
--- ----
------
PHOTOCELL
-~-=-:--,------<-'-- ' - - '
----
-------
-------
I
GRATING Fig . 6
This error can be particularly disturbing, when the cocje-disc is not mounted mechanically with sufficient care in the normal plane to the axis of rotation . However, this error will be greatly suppressed by optical
PERIOD MEASURING SYSTEMS Much more interesting than frequency measurements
17
a) Simple non par allel cut-off
are measurements of periods, because this gives instantaneous results , with a maximum delay equal to the measured period . This means that the time constant can be kept very small .
The case Of non parallel cut-off is completely similar to that of excentricity and these two will therefore be treated together.
Optical opportunities
b) Errors due to defects of the windows at parallel intersect ion
The method appears still more appealing considering the optical opportunities to work out this system . Instead of measuring the time elapsed between two successive impulses, it is also possible to measure the time between the actuation of two detection systems by one impulse of the code disc. In this case an absolute guarantee is present for measuring the exact tirr.e inte rval that an imaginary point of the periphery of the code disc, located at a constant distance of the real rotation center, needs to t ravel over a given trajectory, in other words, the exact linear speed of a point at the periphery of the disc, at a constant distan· ce of the rotation center, from which it is easy to calcul· ate the angular speed . Two important advantages result : . independence of the centering of the code disc, - independence of the impulse density tolerance along the periphery of the code disc. It can be shown that the measured time depends only on the distance between the two measuring elements and the linear speed V of an imaginary point P, located on a circle with radius R , equal to the distance between the detection systems and the rotation center (figure 8).
The errors generated by defects of the windows on the code disc (figure 9) as they result on photographic or engraved plates, will only cause a higher rise time of the measured impulses. This will delay the electrical impulse, without however affecting the measuring accuracy, because the same delay will result for both the detection systems. More detail is given further in this paper. It is clear that the error affects only the rise time of the electrical impulse.
Fig . 9
A
---r- - :
~-- -J
I
f
I
/ '
1
'
/
I
I
I
I
,
,I
I
I
I
'
I
:I
:1
1/
x I I
a)J __J_ 0-T---t-1
Fig. 10
I
Three phases (see figure 10) can be considered :
I
1. the opening has entered , but is not still complete with in the window 1 x b .
-----7------t----:----1r-
A=
f
o
(l-d)dx
whereas A = transmitting area x = jvdt dx = vdt
Fig . 8
2. the opening lies completely within the window 1 xb Error analysis A=
Let IJ S first consider the errors due to an imperfect ge neration of i mpulses on the code disc.
18
fd
1.dx
3. the opening no longer lies completely within the window 1 x b A=
b+d
f
infinitely small, and of length I, the rise time becomes to a first approximation (figure 12) \
JY R \ ~ -1.- ,AB , , /r - - - - -__ 8 - --. -_ - :-:.--:::--. 6 " I ,
d . dx
~-----=-=-
b The successive results are represented in fig . 10, where it appears clearly that the leading edge of the impulse has been delayed, in such a way that even the halfvalue no longer coincides with the ideal position.
........ '---' ---'" "
/
/
----------------;~S
---Jt----\
--- - -
,
-
I
~~---,/
c) Defects of windows w ith non parallel intersection
Fig. 12
Defects of windows with non parallel intersection can be considered as a superposition of non parallel intersection with the foregoing case . The area A grows differently, depending on how d is situated against 5, in other words, if the defect will fall in phase I , 2 or 3. Here again, because of the symmetry of the two detection systems, the measurement will not be affected . From this it can be concluded that errors due to defects of window edges have insignificant influence on the measurement, because of the symmetry of the two identical sensors.
and '
= 1
tB =
10 R2
(figure 13)
d) Errors by excentricity and non parallel intersection
,r---r---------------I ,,I , ,, ,,
The errors caused by excentricity are most difficult to eliminate from a mechanical system. Suppose 0 to be the generation center of the code disc, and 0' the center of rotation (figure 11). will describe a circular path around 0' during the movement, with radius 6 . Let A and B be the locations of the two sensors with slits SA and SB '
o
, I
, ts I
---1
t--
t
Fig. 13
If 0 =1= 0, SA and SB are finite, with length 1 and width b, the rise time can be calculated as follows (dividing this period in three phases) :
1) A 1 =
K
I
2) A2 =
/
Jl
v
t adt _ co g -
2
Co)
2 R 2 cotg at 2 2
o
0
J2 v.b.cotg adt =
(6
R b cotg at
o
Fig. 11 3) A3 Le! 13 be the angular displacement window W has to effect to reach SB from SA. The angles a: and 13 will be equal, so that 13 will be exactly the same as when cS would be equal to zero . Because T =
a:R
-;:i=f
=
= =
}3 (bv - v2 t) cotg adt
o
vt (b -
vt 2" ) cotg a Co)
a: w
/ o
t3
Rt
= l>iRt (b - - ) cotg a /
2
T will be independent of the excentricity 8.lf SA and Ss are infinitely small then the transition time for passing from zero to one for the optical impulse is also infinitely small if 8 = O. I f cS =1= 0, SA and Ss are
If
19
/
= constant A = Al + A2
t3
0
lA!
+ A3 ' (figure
14)
t1
A
r-----
T"'" - - - ; - -
--~.,....---------
formances for this system, and we therefore deliberately centered our attention on the simple detection system. f) Errors by parallax The principle itself eliminates errors due to parallax, because both the sensors are subjected to it. The actual error results from differences between the detectors, and more particularly from the asymmetry of the light projection system, because generally on Iy one such system is used . Much care has therefore to be taken to obtain a perfectly parallel light beam.
Fig . 14 g) Errors by axial and radial clearance In all these cases the ideal moment coincides with the half-value of the impulse, the value of which has to be detected with the greatest care. e) Errors by irregular distribution of the windows For each measuring cycle, only one window edge has to be used and therefore a certain amount of irregularity in distribution of these edges (figure 15) cannot in any way influence the measured value. However, should the distribution be very irregular, the impulse sequence at the two sensors could be disturbed. The actual error on the distribution of the windows is adsorbed within the dead zones between two successive measuring periods. Usi ng this method, optical averaging becomes very difficult, because the perfect matching of the earlier mentioned sensors has to be carried out here for optical gratings. Even supposing a suitable technique existed for doing this, there would still remain the difficulty of detecting the same edges, or at least, the same portion of each series of edges by both systems, because an essential requirement is that the sensors be spaced less than one period from each other .
The errors caused by axial clearance can be considered identically to the errors due to parallax, in so far as they are slow. Sudden displacements, appearing within one measuring period can cause considerable errors, because they disturb the identical character of the two sensors. They can be eliminated by the use of parallel light beams. Errors by radial clearance can be serious also if they occur very rapidly. They will be measured as differences in distance between the two detection systems and are proportional to ratio of the clearance value to the distance between the two sensors. Radial clearance can be eliminated by USing precision bearings, eventually air bearings. Electrical opportunities Measuring a period of time can be done electrically in a very simple way, by feeding a constant current into an integrator for the time of the measured period. The output voltage of such a circuit is directly proportional to this time interval, and is available immediately after this period. The resulting value contains no ripple, thus needs no filters . The only time delay is the measuring period itself. Unfortunately the output value is inversely proportional to the value usually asked for, nl. speed. DESCRIPTION OF A PRACTICAL SYSTEM Optical part
Fig. 15
This can be done by means of fibre optics. It would also be possible to separate both gratings, if the exact number of periods between them is known, and the second measurement is delayed by the same number of periods. However, this results in a loss of the response speed of the system, and complicates the electronics, requiring a shift register for each measurement. Optical averaging does not result in much better per-
The optics of the system were kept as simple as possible. The distance between two successive window edges was therefore the upper limit of the distance between the two sensors. This distance is of course limited to the physical diameter of photodetectors. In practice the process is reversed, as this distance will determine the distances between the window edges. Photocells are photoduodiodes, e. g. LS-400 or 600 from Texas Instruments, MRD 200 from Motorola. They measure 2,5 mm. This limits the impulse density to 400 per meter. In order to keep the number of impulsions per revolution reasonably high, the diameter, of the code disc was chosen to 500 mm. The impulsions were recorded photographically on a graphic-arts emulsion, coated on a dimensionally stable glass disc with sufficient mechanical strength. The impulsions were generated by succesive exposures of a slit by means of a flash lamp.
20
In front of the sensors diaphragms of 0 . 1 mm width are placed. They were al so produced photographically (figure 16) . I f a higher pulse density would be required, readout optics will have to be arranged in order to magnify the distance between two successive impulsions to a value slightly larger than the distance between the two sensors, as mentioned earlier . However, optics will have to be first class projection optics presenting no deformation of the magnified image (figure 17) . Another way to read out small impulsions would be fibre optics, leading to sensitive photocells , e. g. photomultipliers (fig . 18). A brief description of the electronic process has already been given . We will now describe some details . During the measuring period (figure 19) an integrator runs with constant current. Its output voltage at the end of the given interval is read out into an analog memory -cell, after wh ich the integrator is reset .
Fig . 16
GRATING
InteQrator
(J
~!
\
Cl
<
~-+--T-,Anol09 memory cell
PC . Fig . 19
<
This value is a measure of the period , whereas generally speed is of more direct interest. Should only one speed value be desired, the hyperbola can be approximated by its tangent . Should this not be sufficient, various methods are available to perform an inverse relation. One of them is the s. c. diode approximation method, straightening the hyperbola to a straight line by means of suc cessive approximations. However, this method is rather cumbersome, and therefore mostly a real division is performed . Here also many techniques can be useful, e. g. Hall-effect multiplication, logarithmic converters based on semiconductor junc tion properties, time division methods, etc . The logarithmic converters are used very frequently, because they are cheap and fast. Their static accuracy is mostly not good enough for this kind of application . A good compromise can be reached by using a digital technique, even a computer for static and slow frequency components of the measured value, and using the logarithmic conversion for higher frequencies (figure 20) .
Fig . 17
GRATING
Fiber Optics
AdS
Fig . 18
~ Over-Oll
Choroct
CharacteristiC
et
LOQOrithmic '" Divider Digital ioteQl'otar
The photographic way was chosen for its high resolution,and very sharp edges, due to small thickness of tne layer.
21
-f---~~----',""------------ log w
Fig. 20
This modulator is built around an operational amplifier , containing a positive and a negative feedback path . The posit ive feedback makes the output voltage switch between two distinct levels, the negative feedback is delayed, and assures the proportional ity between mean output voltage and input voltage . The mean current into the capacitor has to be equal to zero, since otherwise the voltage across its terminals woukt keep rising and no steady state would be attained ; the mean value of the input current must be equal to the mean value of the feedback current, assuring thereby the exact proportional ity between in- and output .
Description of the period measuring The output signals of the two sensors are amplified and applied to triggers T1 and T2 (figure 21) . These circuits control a flip-flop, which in turn controls the integrato r .
Controlling a perfect switch in series with a resistor by means of this square wave results in modulating the mean current through this resistor when a voltage is applied to this circuit. . ei I =~. ~
I
a = on/off ratio
Fig . 21
Here again averaging of the current can be done by connecting a capacitor of sufficient value across the circuit. Inserting this circuit into the feedback path of an operational amplifier makes a divider out of this circuit because
After the measuring period, one fl ip and flop is first started controlling the analog memory, after which a second flip and flop is started to reset the integrator. The reset of the integrator is done by means of a photo-electric coupled switch . The output voltage of this circuit is proportional to the measured period, and has to be inverted to be a measure of speed .
ii = if
=a . if
= -a ' .
eo
Rf
e I'
li = Rj Divider circuit An electronic divider circuit has been developed, based upon the principle of time division of a resistor value (figure 22) . This time-division is controlled by a pulse depth modulator.
Choosing the modulation frequency sufficiently high, the time constant CfR f
a can also be kept small, in order to keep the response speed of the divider circuit reasonably high.
eo
€i
Fig . 22
22