r:ïfC4+fJ!SCHE HOGESCHOOL DELFT VLIEGTUIGBOUWKUNDE Michiel de RuyJerweg 10 - DELFT
3 r.;;i.1361
THE COLLEGE OF A E R O N A U T I C S
CRANFIELD
LOAD COMPENSATION TECHNIQUES USING
MULTI-TAPPED POTENTIOMETERS
by
T H E C O L L E G E OF A E R O N A U T I C S
C R A N F I E L D
Load Compensation Techniques Using Multi-tapped Potentiometers
b y
-K. C. Garner, B . S c . ( E n g . ) , A . M . I . E . E . , A . F . R . A e . S .
SUMMARY
The theory is developed for loading e r r o r compensation techniques utilising multi-tapped potentiometers. Two methods of compensation a r e derived, and their relative advantages discussed. Design formulae a r e given for calculating appropriate values of the shunt r e s i s t o r s to be used in association with the tapped potentiometers. Numerical examples of each method a r e given.
P a g e S t i m m a r y L i s t of Symbols Introduction 1 Method A: Self c o m p e n s a t e d p o t e n t i o m e t e r 1 Method B: Ganged p a i r c i r c u i t 5 R e f e r e n c e s 9 Appendix 1: Special Nonlinear
C o m p e n s a t i n g function for
Method A 10 Appendix 2: Application for o t h e r than
the Designed Load 12 Appendix 3: Method A: E x a m p l e D e s i g n 13
f ( l ) value of f(K) v/hen K = 1
f(K) a n o n l i n e a r function of p o t e n t i o m e t e r r e s i s t a n c e with K K P o t e n t i o m e t e r w i p e r d i s p l a c e m e n t
n = — = s e g m e n t length r a t i o ; i. e . for a ten s e g m e n t p o t e n t i o m e t e r "^ n = 0.1
r unloaded and unshimted total r e s i s t a n c e of the p o t e n t i o m e t e r winding in the g e n e r a l c a s e of Method B
R unloaded and unshunted t o t a l r e s i s t a n c e of a p o t e n t i o m e t e r winding
R shunt r e s i s t o r value a c r o s s s e g m e n t 1 of a tapped p o t e n t i o m e t e r R. m i n i m u m input r e s i s t a n c e of the loaded p o t e n t i o m e t e r c i r c u i t
used in Method A
R. shunt r e s i s t o r value a c r o s s s e g m e n t j of a tapped p o t e n t i o m e t e r J
R load r e s i s t a n c e at output of p o t e n t i o m e t e r c i r c u i t
L
R shunt r e s i s t o r value a c r o s s s e g m e n t m of a tapped p o t e n t i o m e t e r R t o t a l r e s i s t a n c e of the s p e c i a l n o n l i n e a r p o t e n t i o m e t e r
d e t e r m i n e d in Appendix 1
V. voltage at the input of the p o t e n t i o m e t e r c i r c u i t V voltage at •'.he output of the p o t e n t i o m e t e r c i r c u i t V o V. a. J X X
voltage developed at the w i p e r output of the loaded p o t e n t i o m e t e r i ^ j c i r c u i t of Method A when the w i p e r i s at the j th tapping point
( y + y + +y .) 3 1 2 ' 3 a ( y +y + + y ) m 1 2 m R. y . = ^ 3 nR + R. 3 f(K) K f(l) R-L L ' f(l)
1. Introduction
The e r r o r in the relationship between voltage output and shaft position introduced by having a finite resistive load at the output of an otherwise linear potentiometer is a common difficulty in the design of servomechanisms and analogue computers. The use of tapoed potentio-m e t e r s with shunt r e s i s t o r s between tappings can provide a suitable means of compensating for this e r r o r and two methods are discussed, each with its special m e r i t s .
The first method, using only one tapped potentiometer, applies where an accurate full scale linear relation between the potentiometer angular setting and attenuation is required, and where the resistive load at the output is known and constant. This method is particularly suitable for helical potentiometers with geared dials so that the scale calibration is always valid in the specified loaded condition. Applications with loads less than that specified are described in Appendix 2.
The second method utilizes an accurately wound, linear potentiometer, ganged with a tapped potentiometer having specially chosen shunt r e s i s t o r s to provide the compensation for the load. This circuit enables a linear relationship to be obtained whatever the value of the load, although for any finite fixed load there is a fixed insertion attenuation. This latter effect may be turned to advantage when the potentiometer assembly is used as a position feedback pick-off i n a servomechanism., since the load may be varied to provide a stiffness control.
2. Method A: Self compensated potentiometer
Using an accurate linear tapped potentiometer, it is necessary to calculate the values for the shunt r e s i s t o r s to be inserted between taps using the following specified data
:-(a) the minimum input resistance to be presented to the source (b) the load resistance
It is of course necessary that item (a) should be rather less than (b). Figure 1 shows the basic circuit configuration with a multi- tapped potentiometer having an unloaded andunshunted resistance R. The wiper is loaded by the resistance R . There a r e m segments, and it is assumed that a c r o s s each segment, between adjacent tapping points, there is a
Such a n e t w o r k h a s been analysed in s o m e d e t a i l in a p r e v i o u s a r t i c l e w h e r e it w a s shown that it could b e used to g e n e r a t e a wide r a n g e of n o n l i n e a r functions. In t h i s c a s e a v e r y s p e c i a l n o n l i n e a r function i s r e q u i r e d in the unloaded condition, s o that with the load connected the final r e s u l t i s l i n e a r within the specified t o l e r a n c e . T h i s s p e c i a l non-l i n e a r function i s d e r i v e d in Appendix 1 and i s shown to be
(1) R
«K)
= f
KR o \ K R o + 4 R (1) w h e r e K i s the w i p e r d i s p l a c e m e n tand R i s the t o t a l r e s i s t a n c e of the shunted p o t e n t i o m e t e r which gives t h e d e s i r e d n o n l i n e a r l a w .
The p r o b l e m i s i n t r a c t a b l e in t h i s form h o w e v e r , s i n c e R i s not known until the d e s i g n i s c o m p l e t e . N e v e r t h e l e s s equation 1 i n d i c a t e s the f e a s i b i l i t y of the m e t h o d , and to overconae the above d i l e m m a the following a n a l y s i s h a s b e e n developed.
R e t u r n i n g to F i g . 1, c o n s i d e r the input-output voltage r a t i o s for the w i p e r s e t at each tapping point, with the load in c i r c u i t .
F o r the w i p e r at the junction of the f i r s t and second s e g m e n t s ,
V i ' •> nRR \ /^nRR, nR + R , / L / t nR + R^ + R, nR R R R nR + R. n R + R * ' • • # • • " ' ' m nR nRR + R "^ (nR+Ri j I^jnR+R (2) RR \ w h e r e y . = J R, y L 1 (nR y + R - ) ( y + y + + y ) + R ^ y 1 L 2 3 m L i R. 3 _ nR + R. , . / (3) (4)
S i m i l a r l y V o V. 1 - 2 R. ( y + y ) L, 1 2 rnR( y + y )-i- R, ~l(y +y +.,.. + y ) + R^ ( y + y )
]
m ( 5 ) V o V. 1 ' 3 RT ( y + y + y ) i-i 1 2 3rnR(y + y + y ) + RJ1( y+y + . . . . + y )+R. ( y+y + y)
L L m and by induction o r ( 6 ) RT (y +y + + >^) L 1 2 3 ' i'^3 nR(y + y + 1 2. . + y.) + R^ 3 •L' 3+1 3+2 m L 1 2 J ( 7 ) i / j n R. Rr
or-
3V - >'j
( 8 ) w h e r e and a ^ m ( y + y+ , . , . + y.) •! 2 3 = ( y + y + . . . . + y ) 1 2 n i (9) (10) T h i s i s the g e n e r a l f o r m u l a for the voltage r a t i o at any tapping point in the p r e s e n c e of the load R. . T h u s if t h e s e voltages a r e a r r a n g e d to l i e along a l i n e a r l a w , then atTLeast at t h e s e p o s i t i o n s of the w i p e r , the r e s u l t i n g network p e r f o r m s a s a l i n e a r p o t e n t i o m e t e r , a s shown in F i g . 2. However sonae p e r m i s s i b l e e r r o r will have been p r e s c r i b e d , and s i n c e it i s known that the loading effect will c a u s e e a c h s e g m e n t of the g e n e r a t e d c u r v e to droo{3, a m o r e a c c u r a t e d e s i g n will be achieved if t h e s e voltage r a t i o s a r e chosen to lie high by an amount equal to the p o s i t i v e t o l e r a n c e , a s indicated in F i g . 3 . T h i s method of optimizing the d e s i g n of t a p p e d - p o t e n t i o m e t e r function g e n e r a t o r s i s d i s c u s s e d fully in Ref. 1.A s s i u n i n g that the p o t e n t i o m e t e r r e s i s t a n c e R, and the load
r e s i s t a n c e R^ a r e specified, t o g e t h e r with n, the s e g m e n t length r a t i o ,
L
it i s now p o s s i b l e to w r i t e (m - 1) e q u a t i o n s , w h e r e y to y a r e the unknowns. One m o r e equation i s r e q u i r e d to m a k e the solution p o s s i b l e . T h i s i s obtained b y c o n s i d e r i n g the m i n i m u m r e s i s t a n c e , F . , allowed by the s o u r c e p o w e r r e q u i r e m e n t s . T h i s value will o c c u r with the w i p e r at the top of the p o t e n t i o m e t e r in the loaded condition.
Hence o r R. = 1 •^m n R R . a L m n R a + R m n R (R- - R.) L 1 (11) (12)
w h e r e a will be a constant for any given specification. Substituting t h i s value of a in equation 8 y i e l d s V (
v.,
1 3 n RL R. + R^ R. L 1 nR(R^ -R.) JLi 1 + 1 (13) Solving t h i s for a . g i v e s •If\
^M
R. 1 (R. -R.) 1-1 1H
- 1 n I 3 R ^L R. 1 ( R ^ - R . ) - 1l/"-' ""^
\V.). (Rj^-R.)Also it i s known that a. - a.
3 3-1 y .
(14) (15)
T h u s if all a t e r m s a r e computed, a l l y t e r m s can be d e r i v e d . T h e r e f o r e , s i n c e y . = / „ ^ .^r, ^ (16) J then R. 3 (nR + R.) J n R y j (1 - y.) J n R(a . - a . 3 3 -1 R. = - r ; V- (17) 3 Hence n R ( a . - a . , ) > . ; ^ 3 3-^ ^- - ^ ^ c , (1 - a. +a. , J 3 3-1 R = , . ^ ^ ^~\ ( } ^ ' (18) ^ , ( l - a. +a. . ) '
which i s the g e n e r a l f o r m u l a r e q u i r e d in o r d e r to c a l c u l a t e the shunt r e s i s t a n c e s for specified v a l u e s of R^ , R. and R.
L 1
3 . Method B: Ganged p a i r c i r c u i t
T h e second method p r o v i d e s l i n e a r o p e r a t i o n r e g a r d l e s s of the choice of R . A s s u m e the network shown in F i g . 4 , w h e r e the tapped p o t e n t i o m e t e r h a s a n o n l i n e a r r e s i s t a n c e c h a r a c t e r i s t i c f(K), and the l i n e a r p o t e n t i o m e t e r
i s gauged to i t . r The voltage r a t i o Is then
V K R^ ^ = fe (19) i R. + K R (1 - K) + f(K) w h e r e K i s the wiper d i s p l a c e m e n t . T h u s in o r d e r to g e n e r a t e the d e s i r e d l i n e a r law V ^ = K (20) i f r o m t h i s c i r c u i t , it would be n e c e s s a r y that f(K) = - K R(l - K) (21) which i s c l e a r l y i m p o s s i b l e with p a s s i v e c o m p o n e n t s .
R
However f(K) has a maximum negative resistance of - j at K = |^. Thus if it is a r b i t r a r i l y chosen that
f(K) = ^ - K R(l - K) (22) the network is passive realizable. Any other value of the positive
resistance greater than the chosen one of £• is admissible but as shown below does not produce such an efficient design. The overall performance of the ganged pair of potentiometers is now given by
V - 5 = K V. 1 R L ^ L ^ 4 J (23)
where the factor R / ( RT + T ) 1^ ^ constant for any fixed load.
Therefore the desired linear law is modified only by a constant multiplying factor and the result is shown gmphically in F i g . 5 . , i . e . the load
resistance merely changes the slope but does not affect the linearity. This configuration, unlike that of method A is not well suited for analogue computer applications, since for every change of R , (usually the input r e s i s t o r for the following amplifier), a new scale factor would have to be calculated. However the circuit is very suitable as a servo feedback potentiometer since the constant factor only affects the loop stiffness, and indeed can be used as the feedback gain control directly by varying R , without affecting the position linearity in any way.
The calculation for the r e s i s t o r values is simple, and derived as follows. Referring to Fig. 6, where there a r e m equal segments of fractional length n = 1_ where m is an even number. Only m / 2 segments need be
considered as the law f(K) is symmetrical. The potentiometer total resistance is r . F o r K = where 0
«K)o = f
R. 3 ' j n r + R . = n r ( y _ , , + + y ^ + . . . . + y +y ) (24) (25) m / 2 /2L n 2 1 ^2 'F o r K = n n R 4 f(K) = ^ - n R(l
"^ = "^<>'m/2 % V , , + . . . . + y + y )
(26) F o r K = 2n f(K) 2nf - 2nR(l - 2n) = nr(y^,2-^ y
^ 2 '
. . + y +y ) 4 3 (27) F o r K = 3n e t c . , o r R f(K)3^ = ^ - 3 n R ( l - 3 n ) = n r ( y ^ , 2 ^ y * 2 ^(y^/a"" ••••'" V ^ ^ = f / nr
. + Y +y ) 5 4 m / 2 ^yr^lo^ + y + y ) m / i 4 3<>'m/2''"--^3 ""^J
R 4 R 4 n r - 2nR(l n r 3nR(l -- 2n) • 3n) n r (28) (29) (30) (31) (32) e t c . Solving for y , y , e t c . 1 2 R V = 4J -nR(l
")]
1 n r ^ ( 1 - n ) r ^ ( 1 - 3 n ) r (33) (34) y R R (1 - 5n) y = - ( 1 - 7n) (35) (36)and by induction R
-]
1 - (2 j - l)n (37) and from equation 25
n r y .
R. = - ; ^ (38)
1
i-y^which from equation 37 gives
n R F I - ( 2 j - l)n1
R. = ^ i - (39)
.\ 1 - 7 [ 1 -<23 - l ) n ]
which is the general formula for the j th shunt r e s i s t o r which will provide the desired compensation.
A design simplification is to make both potentiometers of equal r e s i s t a n c e , i . e . r = R, so that for this special case
^ R T I - ( 2 j - l ) n ] ,
3 (23 - 1) ^ ^ If unequal valued potentionaeters a r e used it i s , of course, essential that the tapped potentiometer must have a resistance 3 R when K = 0
4
and K = 1. The condition for this is that the pair of segments having the maximtim incremental resistance can be achieved in the absence of shunt r e s i s t o r s a c r o s s these segments.
In the general case considered the maximtun slope occurs a c r o s s segments 1 and m, and it is necessary to consider the value of r wliich yields an infinite value for R^ in equation 39. This will be the minimum permissible value of r for a specified nvimber of segments, and linear potentiometer resistance R, Thus for R^ •* cc (41) from equation 39 - R < 1 " "^ •* „ (42) 1 - - ( 1 - n ) r or 1 - - ( 1 - n ) = 0 (43) r
Hence the minimum value of r is given by
4. References
1. Garner, K. C. Linear multi-tapped potentiometers with loaded outputs.
Electronic Engineering, v o l . 3 1 , 1959, pp 192 - 199.
.APPENDIX 1
SPECIAL NONLINEAR COMPENSATING FUNCTION FOR METHOD A
It was assimied that it would be possible to construct a special non-linear potentiometer which when loaded would yield an overall non-linear law. The purpose of this appendix is to derive the resistance law of this special potentiometer and to see if it could be constructed using only passive
components. The result obtained in this appendix fully justifies this assumption. Thus what is to be determined is this nonlinear law f(K), that the potentiometer must possess initially, so that when loaded the overall result will be linear.
Using the nomenclature given in Fig, 7 it can be written that
K = V c V. o R^^K) R^ + f(K) f(l) - f(K) + R L f(K) R^ + f(K) ( A l . l ) R^ f(K)
[f(l) - f(KJ
[RJ^+ f(K)] + R^ f(K)
(A1.2) L e t^<^> -K and
""L f(l) K f(l) (A1.3) so that K \ \ X +X L K K (A1.4) o r -K" A _±L . I j-k _ X = 0 K A K / K L (A1.5) Hence K K: - ^ ^ l ( ^ - - J . 4 K
K (A1.6)Now f(l) is the total resistance of the nonlinear potentiometer. Let this be equal to R so that o R and K f(K) R o (Al. 7)
Substituting these t e r m s in equation A1.6 and re-arranging gives R
f(K)
= f
R,
K R \ K R + 4 R (A1.8)
which is the desired nonlinear function. This function may be obtained using a shaped former technique etc, , or by using a tapped potentiometer
APPENDIX 2
METHOD A: APPLICATION FOR OTHER THAN THE DESIGNED LOAD
Particularly in analogue computer applications, the load will be changed from time to time as the problem set-up is renewed. This is a typical consideration with servomultipliers, and dial-set coefficient potentiometers. However this presents no particular obstacle if either of the following techniques are employed.
The l e s s attractive method is to provide pre-wired sets of com-pensating shunt r e s i s t o r s which when connected suit the load present. This would necessitate carrying a large stock of such networks, although connection could easily be arranged using standard plug-in c a r d s .
However, it is yet another task in an already tedious patching procedure. A second and I'ar better solution suggested by J . E . F i s h e r , is to design the compensated potentiometer for the lowest likely load. Thus whenever the computer set-up is changed, the potentiometer output can readily be shunted with a variable resistance until the correct loading is achieved. This is readily adjusted by setting the compensated potentio-meter wiper to say, m i d - s c a l e , so that it should indicate half the applied voltage. If this wiper voltage is then compared with a reference voltage of appropriate value, using a circuit similar to that shown in Fig. 8, then a null indication on the detecting instrument shov/s that the variable
resistance has been set to the correct value. This procedure can readily be mechanized for a large installation having several servomultipliers.
APPENDIX 3
METHOD A: E X A M P L E DESIGN
In o r d e r to i l l u s t r a t e the s i m p l i c i t y of the method a n u m e r i c a l c a s e i s included a s an e x a m p l e . T h e following t y p i c a l specification will be assvuned :
-M a x i m u m load on s o u r c e to be 50K F i x e d load on p o t e n t i o m e t e r to be lOOK Stock s i z e of p o t e n t i o m e t e r to be 3OOK
N u m b e r of s e g m e n t s between equally s p a c e d t a p s t o b e 10 F i g ' i r e 9 shows the c i r c u i t w h e r e the r e s i s t o r s R, to R a r e to be
° 1 10
d e t e r m i n e d . T h i s i s c a r r i e d out by computing the v a l u e s for a^ to a and substituting t h e s e v a l u e s in equation 14.
V/e know that
and R. 1 R, - R. L 1 n R « L = = 50 100 - 50 0.1 X 300 100 = l.C = 0.3
which r e m a i n constant throughout the c a l c u l a t i o n . Also we w r i t e down t h e voltage r a t i o s :
-X 3 1 2 3 4 5 6 7 8 9 10 * 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 * In t h i s e x a m p l e the d r o o p e r r o r optimizing p r o c e d u r e h a s been omitted for
s i m p l i c i t y , e . g . for a 0,1% V t o l e r a n c e o . . would b e 0.101 : ^ ( X ) i i n s t e a d of 0.1 a s used h e r e .
Hence ^ 1 and similarly a 2 a a 4 a 5 a 6 a 7 0 a = = = :: = = = 2 = = and a = 10 Substituting thes Rl R 2 R 3 R 4 ï^3 R g R 7 R R 9 R 10 = = = = = = = = = = "(0.1)(1)-0.366 0.781 1.233 1.667 2.050 2.402 2«694 2.942 3.153 3.333 e values in 17.3 K n 25.7 K n 24.2 KQ 23.0 KO 19.4 K n 15.6 K n 12.4 K n 9.90 K n 8.02 K n 6.58 K n l ] + ^| [(0.1)(1)- l ] ' + 4 ( 0 . 1 ) ' ( 1 ) 2(0.1) (0.3) equation 18 gives
Component Tolerances
A complete analysis to include the effect of the many component tolerances is possible but very tedious, and since the basic calculation is relatively rapid, it is easy enough to test one or two t r i a l designs using component values at each end of their tolerance limits. It is however, worth noting the significant factors influencing the final
accuracy. The nominal value of the manufactured potentiometer resistance is usually stated with a small production tolerance. Before commencing the above calculation it is worthwhile measuring the actual value, if only one unit is to be constructed. The same applies to the load resistance, which also has to be known to a fair degree of accuracy. Similarly, small e r r o r s will a r i s e if the tapping points a r e not precisely positioned.
The better the linearity of the unloaded, unshunted potentiometer, the better the final result v/ill be, although the effect of shunting sections of a potentiometer is to make it l e s s sensitive to linearity e r r o r s . Lastly, while not always n2cessary, it is preferable to use high-stability 1%
r e s i s t o r s to make up the shunt r e s i s t a n c e s . This is nearly always possible taking by suitable combinations of standard preferred values. If it is
desired to compute the entire performance curve for the completed design reference should be made to an e a r l i e r article by the author^l', where a complete analysis of such a network is discussed,
APPENDIX 4
METHOD B: EXAMPLE DESIGN
Figure 10 shows a final circuit where R to R are to be determined 1 10
to give the desired law. As a simplifying factor it is assumed that both the linear, and the tapped potentiometers have the same overall resistance R. Since the law is symmetrical it is apparent that R = R^^ R^ - Rg etc, Thus it is only n e c e s s a r y to consider R to R . By inspection n = 0.1.
Since the potentiometers a r e ganged we must consider the resistance of the tapped potentiometer at the tapping points at K = 0.1, 0.2, 0.3, 0.4 and 0,5, and make the resistance at each setting conform with the law
f(K)
= I -
KR(1
-K)
Using the relation (equation 40) R. = R [l - <23 - Dr] J R l = R = 2 R 3 = R 4 = Rs = 23 - 1 0.90R = R 1 0 0.23R = R 9 0.1 OR = Rg 0.043 R = R^ O.OllR = R^ 0
V i -Rm nR Rm-i< S n R K - I O nR nR •nR
T
-^VrFIG. I. GENERAL CIRCUIT FOR METHOD A
DESIRED LAW
ACHIEVED LAW
l O K-FIG. 2. METHOD A COMPENSATION
t o V,
l O
K-FIG.4 BASIC CIRCUIT FOR METHOD B
FIG, a METHOD A COMPENSATION UTIUZING TOLERANCE
FIG. 5. METyop B PERFORMANCE CURVES.
V i o
FIG. 6. GENERAL QRCUIT FOR METHOD B.
f(KK
V f (0 WHEN K- I
<3Vo
FIG. 7 GENERALIZED NONLINEAR FUNCTION POTENTIOMETER.
FIG. 8. METHOD A TRIMMING CIRCUIT FOR USE WITH OTHER THAN DESIGN - SPECIFIED LOAD
