LINEARITY

By
Advertisement

LINEARITY

THE WORKING RANGE OF MOST OF THE INSTRUMENT PROVIDES A LINEAR RELATIONSHIP BETWEEN THE OUTPUT (READING TAKEN FROM THE SCALE OF THE INSTRUMENT) AND INPUT (MEASURED, SIGNAL PRESENTED TO THE MEASURING SYSTEM). THE ASPECT TENDS TO FACILITATE A MORE ACCURATE DATA REDUCTION. LINEARITY IS DEFINED AS THE ABILITY TO REPRODUCE THE INPUT CHARACTERISTICS SYMMETRICALLY, AND THIS CAN BE EXPRESSED BY THE STRAIGHT LINE EQUATION 

Y=MX+C

WHERE Y IS THE OUTPUT, X IS THE INPUT, M IS THE SLOPE AND C IS THE INTERCEPT.APPARENTLY, THE CLOSENESS OF THE CALIBRATION CURVE TO A SPECIFIED STRAIGHT LINE IS THE LINEARITY OR THE INSTRUMENT.

ANY DEPARTURE FROM THE STRAIGHT-LINE RELATIONSHIP IS NON-LINEARITY.

THE NON-LINEARITY MAY BE DUE TO:
* NON-LINEARITY ELEMENTS IN THE MECHANICAL DEVICE
* MECHANICAL HYSTERESIS
* VISCOUS FLOW OR CREEP, AND
* ELASTIC AFTER EFFECTS IN THE MECHANICAL SYSTEM
IN A NOMINALLY LINEAR MEASUREMENT DEVICE, THE NON-LINEARITY MAY TAKE DIFFERENT FORMS AS ILLUSTRATED IN FIG.
1.      THEORETICAL SLOPE LINEARITY:
MAXIMUM DEPARTURE "A" FROM THE THEORETICAL STRAIGHT LINE OA PASSING THROUGH THE ORIGIN. THE LINE OA REFERS TO THE STRAIGHT LINE BETWEEN THE THEORETICAL END POINTS, AND IT IS DRAWN WITHOUT REGARD TO ANY EXPERIMENTALLY DETERMINED VALUES.

2.      END POINT LINEARITY:
MAXIMUM DEPARTURE "B" FROM THE STRAIGHT LINE OB PASSING THROUGH THE TERMINAL READING (EXPERIMENTAL END POINT - ZERO AND FULL SCALE POSITION).

3.      LEAST SQUARE LINEARITY:
MAXIMUM DEPARTURE "C" FROM THE BEST FIT STRAIGHT LINE OC DETERMINED BY THE LEAST SQUARE TECHNIQUE.
IN MOST INSTRUMENTS, THE LINEARITY IS TAKEN TO BE THE MAXIMUM DEVIATION FROM A LINEAR RELATIONSHIP BETWEEN INPUT AND OUTPUT, I.E FROM A CONSTANT SENSITIVITY AND IS OFTEN EXPRESSED AS A PERCENTAGE OF FULL SCALE.
THE CALCULATION OF MEASUREMENT ERROR REQUIRES NUMERICAL VALUES OF ACCURACY, RESOLUTION, AND LINEARITY ETC. FOR THE INSTRUMENT BEING USED. FOR THE MAJORITY OF LABORATORY INSTRUMENTS, THIS DATA IS GIVEN IN A MANUFACTURER HAND BOOK. HOWEVER, FOR SOME INSTRUMENTS SUCH AS MICROMETERS, VERNIER CALLIPERS, THERMOMETERS AND TESTING EQUIPMENT, THE DATA IS GIVEN IN THE STANDARDS MAINTAINED BY THE COUNTRY. 

SENSITIVITY: SENSITIVITY OF AN INSTRUMENT OR AN INSTRUMENTATION SYSTEM IS THE RATIO OF THE MAGNITUDE OF THE RESPONSE (OUTPUT SIGNAL) TO THE MAGNITUDE OF THE QUANTITY BEING MEASURED (INPUT SIGNAL), I.E.,                  
STATIC SENSITIVITY, K= CHANGE OF OUTPUT SIGNAL/CHANGE OF INPUT SIGNAL.

SENSITIVITY IS REPRESENTED BY THE SLOPE OF THE CALIBRATION CURVE IF THE ORDINATE ARE EXPRESSED IN THE ACTUAL UNITS. WITH A LINEAR CALIBRATION CURVE, THE SENSITIVITY IS CONSTANT. HOWEVER, IF THE CALIBRATION CURVE IS NON- LINEAR THE STATIC SENSITIVITY IS NOT CONSTANT AND MUST BE SPECIFIED IN TERMS OF THE INPUT VALUE AS SHOWN IN FIG.

IN CERTAIN APPLICATION, USE IS MADE OF RECIPROCAL OF SENSITIVITY AND THAT IS TERMED AS INVERSE SENSITIVITY OR DEFLECTION FACTOR.
SENSITIVITY HAS A WIDE RANGE OF UNITS, AND THESE DEPEND UPON THE INSTRUMENT OR MEASUREMENT SYSTEM BEING INVESTIGATED.

LET A DIFFERENT ELEMENTS COMPRISING A MEASUREMENT SYSTEM HAVE STATIC SENSITIVITIES OF K1, K2, K3 ... ETC. WHEN THESE ELEMENTS ARE CONNECTED IN SERIES, THEN THE OVERALL SENSITIVITY IS WORKED OUT FROM THE FOLLOWING RELATIONS

THE ABOVE RELATION IS BASED UPON THE ASSUMPTION THAT NO VARIATION OCCURS IN THE VALUE OF INDIVIDUAL SENSITIVITY K1, K2, K3 ...ETC. DUE TO LOADING EFFECTS.


WHEN THE INPUT TO AND OUTPUT FROM THE MEASUREMENT SYSTEM USED WITH ELECTRICAL/ELECTRONIC EQUIPMENT HAVE THE SAME FORM, THE TERM GAIN IS USED RATHER THAN SENSITIVITY. LIKEWISE, AN INCREASE IN DISPLACEMENT WITH THE OPTICAL AND MECHANICAL INSTRUMENT IS DESCRIBED BY THE TERM AMPLIFICATION. APPARENTLY THE TERM SENSITIVITY, GAIN AND MAGNIFICATION ALL MEAN THE SAME AND THEY DESCRIBED THE RELATIONSHIP BETWEEN THE OUTPUT AND INPUT. FURTHER WHEN THE INPUT OR OUTPUT SIGNAL IS CLANGING WITH TIME, THE TERM TRANSFER FUNCTION OR TRANSFER OPERATOR IS USED RATHER THAN THE SENSITIVITY, GAIN OR AMPLIFICATION.

2 comments: