Effect of a potentiometer gap on the measured mean of a wind direction signal

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Effect of a potentiometer gap on the measured mean of a wind

direction signal

EFFECT OF A

POTENTIOMETER

ON

MEAN

OF

SIGNAL

by

J . T , Templin and W . A . DalgXiesh

The use of a standard one-turn potentiometer to measure an angular

signal such as wind direction can crcatc difficulties. The problems

arise because of t h e discontinuity in the potentiometer signal when t h e sliding contact p a s s e s from the maximum o u t p u t level across t h e gap t o t h e _{lowest output l e v e l , or vice versa. This results in an} e r r o r in t h e

c a l c u l a t i o n o f t h e mean.

Several s o l u t i o n s have been proposed i n c l u d i n g special f e a t u r e poten-

tiometers. One method proposed by Camuffo and Denegri (1976) uses a

standard potentiemeter b u t adds o r subtracts one unit t o an i n d e x of

periodicity each t i m e t h e sliding contact crosses t h e gap in either

d i r e c t i o n .

This

T h i s _{n o t e discusses a method of determining the e f f e c t o f t h e gap} by examining the measured mean and standard d e v i a t i o n of the wind d i r e c t i o n

signal. In some applications it is n o t important t h a t information be obtained f o r 3 1 1 directions. In t h e s e cases it is sufficient to know when t h c measured mean is i n crror so that i t can be rejected. The t e c h n i q u e s dcscribcd h e r e a l l o w an e s t i n i n t e to bc made of the error in

t h e measured mean so t h a t inaccurate data may be rejected. In some cases,

t h e r e f o r e , an ordinary potentiometer and simple averaging techniques can

be used to o b t a i n wind direction d a t a which can later be examined to

r e j e c t signals strongly affected by t h e gap.

The following a n a l y s i s assumes a s i g n a l w i t h a _{normal distribution} and a gap of some known l e n g t h . It i s a l s o assumed that when the s l i d i n g c o n t a c t is in t h e gap, t h e output drops to zero.

It is _{assumed that t h e t r u e standard d e v i a t i o n of t h e signal,} 0 , is known (from t y p i c a l signals measured t h a t are not near the g a p ) . The

full ranye o f t h e potentiometer, d , is expressed in units of the standard

then t h e full 36Q0 has a l e n g t h , d = 60. Similarly the gap s i z e i s

expressed

i n units

a srandard deviation o f the t r u e s i g n a l of 1.

F i g u r e I shows t h e probability density of a typical s i g n a l , x. The mean is _{located a distance y f ~ o m the gap. The t o t a l length of t h e} potentiometer is d (expressed in units of s t a n d a r d d e v i a t i o n ) . The

l e n g t h o f t h e gap i s " g " . The distribution is assumed n a n a l with the

form

where y = x-p, except in t h e range from x = d-g to x = d where t h e s i g n a l and t h e r e f o r e t h e contribution to t h e area i s _{0. The total area under} the curve i s

where t h e cumuPative distribution i s given by

-

To calculate t h e mean value of t h e distribution, x , t h e centre o f gravity of the d i s t r i b u t i o n must be found. The c e n t ~ e of g r a v i t y of the two

s h a d e d s e g m e n t s is found b y symmetry and weighted with t h e i r combined a r e a . The first moment about x = O is a l s o calculated f a r t h e unshaded

area.

area of c e n t e r of gravity o f

shaded shaded regions

T h i s y i e l d s

Substituting E q . [Z) i n t o E q . (5) and s a l v i n g f o r

_x

This is t h e calculated mean. S i n c e t h e actual mean is

u ,

is given by

The measured standard deviation, s, can be found by determining the second

moment of t h e area a b o u t x =

x.

the v a r i a b l e y to allow the result t o be expressed in terms of the tabulated functions f (y) and FEY).

'I'hc f i r s t ;ind second integral dctcsnnine the second n~omcnt for t h e r e g i o n

0 $ x < o r -11 .< y < TIIC t h i r d inacgrnl dctcrmincs t h e moment o f t l ~ c

region a t tlkc oxtrcnic r i g h t o f Fijir~rc 1 . I'hc mamcrlt am1 is corrected by

t h e f a c t o r d . ' T h i s cxprcssion assumcs t h a t t h e tails of t h c d i s t t r i h u t i o n

functions are cffcctivcly zero by t h c tirne t h c y reach t h c cnds of t h e

p o t c n t i o m c t c r r a n g e .

By making t h e variable change y = -y in t h e second and t h i r d integrals

and

(-YI-~*d)Z

I the following expression is obtained

Although t h e s e i n t e g r a l s cannot be evaluated in closed form, the s o l u t i o n

can be expressed in terms o f t h e cumulative distribution f u n c t i o n , E(y), t h e v a l u e s o f which a r e tabulased in mathematical t a b l e s . The resultant e q u a t i o n i s _:

This expression can bc evaluated and solved for s . The e f f e c t of the

discontinuity i n the potentiometer i s to increase t h e measured standard d e v i a t i o n whcr~ tflc

slid in^

t h e signal distribution movcs f a r from thc gap, thc v a l u c of s , t h e

mcasurcd st;lnrlarJ devi;rtion, appsoncl~t..s t l ~ c t r u c v a l u e of 1. Thc crror

i n thc measurcd mean i s always i n a d i r r c t ion t c l % h i t't it f:trtf~cr T I . ~ I ~ I \ the gap than t h e tme v a l u e .

To illustrate t h e e f f e c t s of t h e discontinuity, a s p e c i f i c case is

considered. Wind speed and direction have been measured by Dalgllesh

(1971, 1975) as p a ~ t of a program to measure wind pressures on full-scale buildings. Because of s h e dependence of p r e s s u r e s on t h e wind direction,

each pressure that is measured. The approach taken was to measure the

mean and standard deviation of the wind direction signal and r e j e c t all data associated w i t h large standard deviations

of

A standard potentiometer

with

to measure wind direction. By examination of wind direction signals w i t h

means away f r o m the gap, it was determined that t h e typical standard deviation of a 5-minute s i g n a l was D = 6". Therefore in normalized n o t a t i o n d = 3 6 0 / 6 = 60,

g = 5/6 = .833

The error in the measured mean and the corresponding measured standard

deviation are shown for several s e p a r a t i o n s of t h e true mean from the end

o f the gap in Table I. Results are a l s o shown for o t h e r values of a but

with

A s u i t a b l e value o f t h e measured standard deviation can be chosen to reduce t h e error in the mean to a desired level. F o r example, Dalgliesh

C1971, 1975) has chosen to r e j e c t d a t a w i t h standard deviations in t h e wind direction greater than 20'. This ensures t h a t the gap has

not

closer than 10 or I I degrees

ro

In some cases, t h e loss of a large sector of information may make she

preceding technique undesirable. Then, the o n l y alternative is to use

s p e c i a l equipment or to use a more complicated computing technique such

as that proposed by Camuffo and Denegri (1976) which requires additional data reduction before t h e mean wind direction i s calculated. I f , however,

t h e actual standard deviation of the measured s i g n a l is small and the

potentiometer gap can be placed in a sector where loss of information is

not critical, then na special equipment i s necessary to obtain accurate

L i s t of Variables

A area

under

in

d normalized l e n g t h of potentiometer range

f ( y ] normal probability f u n c t i o n d e f i n e d in Eq. (1)

F [ y ] c m l a t i v e probability d e f i n e d in E q . ( 3 )

g normalized width of gap

in

s measured standard d e v i a t i o n of normalized wind direction signal

x normalized wind d i r e c t i o n s i g n a l

-

x measured mean o f variable x

Y transformed variable, y = x-u

1 transformed variable, y 1 = -y - E e r r o r

in

P actual mean af normalized wind d i r e c t i o n s i g n a l

0 actual standard d e v i a t i o n of wind d i r e c t i o n signal

( i n

used to normalize a l l o t h e r parameters.

Rc f erences

Camuffo, I3. and D e n e ~ ~ i , A . (19761, '"A Method for Measurement of Mean Wind

i l i r c c t i o n w i t h t h c U s e of Standard Potentiomet~ic Transducers"

Atmospheric Environment, V o l . 1 0 , p . 415.

I ) n l g l i csll, _W .A. (19751, f7Compari son of Modcl / l : r r E 1-Sc:l l c Wind FErcssurcs on

TABLE I

EFFECTS OF GAB ON MEASURED MEAN AND STANDARD DEVIATION OF WIND D I R E C T I O N

t r u e standard d e v i a t i o n

u

6 "

distance of true

I mean from gap

v . 0

[X-ll).

l o 0

"

F I G U R E 1