Quantization Effects in the Resoiution of the Measurand in a Measurement System

'.z3Sebastian Y. C. Catunda. 'Jean-Francois Naviner, 3Gupdip S. Deqp and, 'Raimundo C. S. Freire

Ecole Nationale Suptrieure des Ttlhmmunications, ComElec, 46, me Eiarrault 75634 Paris cedex 13 France Universidade Federal da Pami% DEE, AV. Aprigio Veloso, 882, Bodocongb, Campina Grande, PB, Brazil

Universidade Federal do Marahlo , DEE, S o Luis, MA, Brazil E-Mail: catunda@enst.fr, naviner@enst.fir, deep@dee.ufpbbr, kire@,dee.ufpb.br

Abstract: An analysis of the quantization errors propagation in a generic measurement system and a methodology to estimate the best resolution of A/D converters to obtain the desired resolution of the system, are presented in this paper. As application of the methodology, analysis of measurement systems using different sensors are presented.

k INTRODUCTION A measurement system is an important functional block

when employed for monitoring and conirolling industrial processes and laboratory experiments. By virtue of the evolution in the microelectronics and specially the digital signal processors of varying complexities, the above system, more often, happens to be a digital data acquisition system.

A typical measurement system for such applications is presented in Fig. 1. It is constituted of a sensor (transducer) for converting the physical quantities into an equivalenl electrical signal, a signal conditioning block to make this electrical signal compatible with the input characteristics of the analog to digital converter used for quantizing the signal. This signal-conditioning block may be an amplifier? filter, precision rectifier, etc. In modem instnunentation systems, an additional functional block, called a signal reconstruction block with a specific algorithm, is employed to recover the desired characteristic of the original physical quantity being monitored [l]. In this reconstruction block, undesired effect of the other variables like temperature and other parameters on the main measurand (output of the sensor), can be taken into account

The sensor employed in the measurement system may be active or passive. An active sensor is a self-generating element and OII contrary, the passive sensor requires an auxiliary power source for providing an electrical signal as a function of the physical quantity being monitored. As mentioned before, for compensating the undesired effect of other variables on the operational performance of the principal sensor, additional sensors (secondary) might be needed.

The final measurement resolution of the main measurand will not be necessarily determined solely by the resolution of the A/D converter used for digitizing this measurand. This resolution will depend upon the characteristics of the reconstruction algorithm referred to before as well as the sensitivity of the main measurand to the secondary variables Physical quai&

and the measurement resolution of these variables. In this paper, iZ methodology is presented to determine the

minimum resolution required of an analog to digital converter, to obtain the desired specified final resolution of the reconstructed signal corresponding to the principaI measurand. For this, an analysis of the error propagation in the measurement system with or without the secondary variables which interfere with the measurement accuracy of the measurand is carried out. Three specific cases of measurement system with varying complexities are also presented to illustrate the application of the proposed methodology. a &ALJ'SIS OF A TYPICAL MEASUREMENT SYSTEM

In Fig. 2, a model for the analysis of quantization error in a generic measurement system is presented, and is based on [ 11. In this model only those blocks which play major role in causing error in the measurement system have been included. These blocks are: sensor, analog to digital converters and the signal reconstruction blocks.

The main task of the measurement system is to estimate the principal mc:asurand x as accurately as possible. This measurand x acts directly on an appropriate sensor and produces an electrical output y, and the transfer characteristics is represented by some function f( ). The analog to digital converter quantifies the signal y at the output of the signal conditioner (Fig 1) as an N-bit word. The reconstruction block may be visualized as an estimation block to estimate the most precise value of the measurand from the sensor's output signal. The reconstruction function may be formed as R( ) which should ideally be the inverse of the fimctionf( ).

When the value of the measurand is affected or influenced by other n secclndary quantities, denoted as vi, it becomes necessary to measure the value of these secondary quantities using appropriate sensors to be able to estimate the principal physical quantiiy of interest and compensate the effect of these secondary quantities in the reconstruction block. The outputs of the semndary sensors denoted wi, are also digitized as M-bit words. In case, only one analog to digital converter, together with a time division multiplexer, is employed, then

n Intafenng CIUanbE45

Figure 1. A typical measurement system

0-7803-5491-5/99/$10.00 0 1999 IEEE

Figure 2. Measurzment system model for quantization error analysis

'700

hf= N. Finally. the measured values of the interfering secondary quantities, represented as sensors' transfer functions g,( 1, are estimated in reconstruction blocks using reconstruction functors S,{ ) and these should be ideally the inverses of the hct ions gd ), with i varying from 1 to n.

For the analysis and evaluation of the effect of quantization error and its propagation in the measurement system, we consider that the output voltage range of the signal conditioning blocks is so adjusted that it equals the total input range of the A./D converter

Some characteristics of the model of the measurement system are defined in the Table I.

In this generic measurement system, the principal measurand x varies from x- to x,, and the secondary quantity v, varies from v,,, to v,-:

x c [xmyl; x J e vr c [v,-; vt-l The signals y and w, are available at the outputs of the

corresponding sensors, having characteristic functionsfl) and gl( ), respectively and these signals extend over the ranges ym andy- and w,, and w,, respectively. Thus:

JJ

Y c Bm; ymaxl and ~1 c [wr-; w m 4

VI, . . . vn) and W, = g,<v,)

The signals y and w, are digtized in the A D converters and are available as and Gz with N and A4 bit resolution, respectively Considering that the measurement system is duly calibrated and that the analog signal values corresponding to the fuIl range of the measurand input always span the input range of the A/D converter with $5 LSB resolution, the worst case absolute values of the quantization error may be written as:

The quantization errors will thus range over by and ki and from this it is possible to analyze the propagation of the quantization error, in the measurement system. Considering

Table I. Parameter definitions Main Secondary Description

quantity quantities X vi Quantity to be measured

Y Wi Output signal of the sensor AXYl,. . ,VI) &vi> Transfer function of the sensor

Quantized values

secondary quantities Maximum and minimum values of the sensor output voltage within the measurement range respectively.

Reconstruction function

secondary quantities Loss of resolution, effective final

resolution and desired f d resolution in terms of the number of

bits, respectively

- 7 w, A X q. Estimated values of main and

y- , ymh U'- , wi-

Ev &Vi Worst case quantization mor

N M ADC resalutions for main and R( ) q - I ( ) S,( ) z gj-'( )

NL, Aks ND

always the worst case situation, the reconstructed or the estimated values of the principal measurand and the secondary interfering quantities are:

Gi =S,(IY~+E,~) ad_i-=RG,+sy,$,,C2 ,..., Cn). Thus, for a system with one principal measurand and n

secondary quantities, affecting the principal measurand, the total number of possible calculated values of f , using the positive and negative values of the A/D quantization errors, is 2*'. The worst case estimation error in the principal measurand may be written as:

(1)

For a given worst case error E ~ , the effective find resolution NE may be interpreted as the equivalent A D resolution for measuring a particular value of the main meamrand without the influence of the reconstruction block and considering the sensor with a linear transfer function Thus, the equivalent quantization mor for a particular value x of the measurand can be expressed as:

E, =

E, = maxlx - 21 -

IIEK(X) - &(X)

2NE+'

This consideration can be extended to whole measurement range and the effective resolution can be calculated in number of bits using the values of the worst case estimation error 8 as in eq. (l), as:

The loss of resolution in terns of number of bits NL, can be easily calculated from the measurement system effective resolution NE and the resolution N of the given A/D converter employed N, as: NL= N-NE. (3)

Defining EOX as the maximum quantization error in the case of ideal measurement system, limited only by the H&SB resolution of the N-bit A/D converter:

D (4) max (x ) - min (x) Eo+ = 2 N t '

then, the loss of resolution can be calculated as:

If the desired resolution for the main measurand is specified to be ND, then it is required that: min(NE) = No

and

Ex 5 mx(x> - min(x) . Z N D

d . Determination ofADC resolution The minimum number of bits required for the AID

70 I

converter to obtain the desired final resolution ND, in spite of some loss of resolution due to the quantization of the secondary quantities or non-linearity of the main sensor transfer characteristic, may be written as: N = No + maX(NL) (61

But the determination of the values of N may not be so straight forward and possibly one may not come across an analytical expression for the desired value of N. For some relatively complex systems one may employ an iterative procedure with rapid convergence. An iterative procedure may be of the following type: I. Define an initial value of N; II. Calculate the loss of resolution NL considering the whole

measurement range; III. Calculate the new value of N, as N = ND + ~ ( N L ) ; W.Repeat the steps 2 and 3 until N is obtained with a

specified convergence tolerance; V. The minimum number of bits of the A/D converter should

the higher whole integer number to the calculated value of N .

ILL SOME CASE STUDIES For the measurement of the physical quantity to be

monitored, we can distinguish different cases by the number of the secondary quantities that influence the measurement result and type of sensor transfer characteristic. For the later, we can define the sensitivity of the principal sensor as being: 4&,1JJ = mW)l~ (7)

Some typical measurement systems are included in the Table 11, with the number of the secondary quantities that influence the pMcipal measurand and the linear or non-linear transfer characteristics of the sersor.

With the measurement system without any secondary influence and the sensor output signal linearly dependent on the input measurand x, the reconstruction function R( ) is also a linear function. Thus there is no loss of resolution in this case and thus NL = 0 and NE = N = NO. In the present report, we have not included the case for two or more secondary quantities.

B. Temperature Memirement Using a Thermistor A thermistor-based temperature measurement system may

grossly be classified as one having no secondary influence quantities using a sensor with non-linear transfer characteristics. The temperature dependence of the thermistor resistance may be described by [2]:

2 = R @ ) = - P h(,y)-ln(I.r*)

y c i0.078; 0.759 V]

& R @ ) = P 1 8 - 1

y c [13.2 ; 127.3 pA]

IT of secondary Type of transfer quantities , characteristic

One Linear +AV) I 3.2 Non-linear

$&,Vi) I -

Sensitivity Case study

linear j I

Zero

where r& is the. thermistor resistance at temperature T in Kelvin and ri, and p are the thermistor‘s characteristic parameters.

Let us wnsiaier a temperature measurement system over a 0 to 50 O C (273 to 323 K) range, having a 10 bit A/D converter and employing a thermistor with: p = 4000 K and ri,=0.033 R [3]. We analyze two possible conventional configurations: I) Constant current (Fig. 3.A), where the thermistor is biased with a constant current source; II) Constaut voltage (Fig. 3.B), where the thermistor is biased with a constant voltage, the current variation contains the information about the temperature variation.

For these configurations, one can write the following equations in ternns of variables defined before, i.e. consider x as the measurattd (i.e. temperature 0, and y is the sensor output electrical variable.

Iy = f ( x ) = Z.rmeg’s (Io

Iy = f ( x ) = Y/r*eP’= (1)

Linear I Constant 1 -

Non-Linear I $I xo I 3.1

IN = 10 + E,, = 3.32 x IO-4 V IN = 10 + E,, = 5.57 x lo-’ pA

The worst case error r; in estimating the value of the measurand x mid the effective measurement resolution are shown as function of the temperature itself in Fig. 4(A) e (El), respectively, when the thermistor is current biased (I) and when it is voltage biased @I). It is interesting to observe that the error increases in the case of the current biased thermistor and decreases iri the case of voltage biased thermistor, as the value of the meamand (i.e. temperature) increases.

The curves for the effective resolution, as expected, exhiiit a tendenicy opposite to that in Fig. 4(A), for the two cases. We also observe that for a temperature below 19 O C

one gains in effixtive obtainable resolution in the case of the curve I (current biasing). In the case of m e II (voltage biasing) for the temperature between 28 and 50 “C, there is gain in the effective obtainable resolution. This behavior of the resolution cilwe is clearly due to the non-linearity of the thermistor R - l characteristic curve, where it produces expansion over a part of the temperature range compression over the other pxt and M e r depends upon the bias mode. It is also interesting to observe that the voltage bias mode produces smaller variation in resolution Over the temperature

v -i

(1) (W Figure 3. Temperature measurement configurations: (I) Current biased. (11)

Voltage biased

7’02

o.l 5(A) Temperature m o r ("c)

0.1

0.05

' 0 5 10 15 20 25 30 35 40 45 50

7 G - F 2 = R R , ( y , i t ) = f - ' ( x , v ) = - k$

x c [0;141 y c [-518;518 mv]

Temperature ('C) (9) Effsdive resolution (bit)

12

11

10

9

8 I t s .--I-. .: ....-- "-2 .....-.-..J. ....__... : ...-I-- *-"--A

, ! I ; ! ! ! .

*v=v v c [273; 373 K] E, = 0.0488 oc

I - I I I I I J . . . .

0 5 10 I5 20 25 30 35 40 45 50 Temperature ('C)

Figure 4. (A) Worst cas temperature estimation error .F+. and (B) Temperature measurement effective Resolution.

m g e considered as compared to that obtained with the current mode.

C. pH Measurenrent The properly calibrated pH electrode yields an output

voltage directly propoational to the pH of the solution in which it is immersed But fhis output voltage depends upon die temperame of the liquid and this dependence may be described as [4]:

where Tis the temperature in Kelvin. The electrode output voltage as a function of the pH value

is plotted in Fig. 5 for three Merent temperatures. In order to correct estimate the pH value, one needs to measure also the temperature. The pH of a solution may vary over a range of 0 to 14 and we consider that the operating temperature may vary over a range of 0 to 100 "C. Once again adopting the notation defined before. x should represent the pH, y the electrode output voltage and v the operating temperature T as a secondary measurand. Assuming that the temperature sensor has a linear R-T characteristic so that w = v, and thus with 10 bit A/Q converter, we have the following:

V =0.19847(7-pH)T

1~ = f ( ~ , V ) = kv(7 - x), k = 0.1984

Isy = 0.5059 mV

In Fig. 6, the worst case measurement error c; (for pH value) for two AfD converters with N = M=10 bits is plotted over the operating temperature range of 0 to 100 "C. The effective resolution obtained in the pH measurement with a 10 bit A/D converter is also plotted for the A/D converter for the temperature quantization with different resolutions e.g. Ad = 7, 8, 9, 10, CO. The loss in the effective resolution may be attributed to lower resolution of the A/D converter for

Output temion (mv) 600 I I I I I I I

-600' I I I I 1 L 0 2 4 6 pH 8 10 12 14

Figure 5. pH Electrode output voltage as a hnction of pH for three different temperatures

temperature measurement and to the reduction of the range of the pH sensor output voltage in the lower ranges of the operating temperature.

D. Dissolved w e n (DO) Mewrenrent The concentration of the dissolved oxygen (DO) is an

important parameter in wastewater treatment plants that use activated sludge. The membrane type electrode is by far the most used in the DO measurement in liquids. This electrode is constituted of two metallic solid electrodes in contact with an electrolytic solution and separated fiom the surrounding by an ion selective membrane. The output current at the electrode terminals is linearly proportional to the concentration of the molecular oxygen. The sensitivity of the membrane type electrode for DO is afkted by the operating temperature, the salinity of the liquid under observation, etc. This dependence on the salinity is important when we deal with estuary water and the other factors may be ignored. By fa, the most important interference factor for measurement of DO is the temperature.

The dependence of the electrode current on the operating temperature may be described by [5 ] : i(t) = exp(m.T + b).DO(t) (9) where rn and b are the characteristic parameters of the electrode.

The maximum or saturation concentration of DO in liquids also varies with the operating temperature and this affects the range of DO values that one may need to measure in a given situation. The DO saturation values are well defined and can be found usually in most commercial electrode manuals. For a commercial electrode used in our laboratory [6], the experimentally measured values are: rn = 3,839~10-~e b = -24,49 [71. Considering that we wish to measure the DO concentration in the liquids with temperature varying over a range of 0 to 50 "C, the variation of the

ERective malution IN = l!ll OH error. I: 1 i x ~ t i 3

. , . - , -,. 10

3.5 10

9 9

8.5 8

7

o i o ZG 30 40 50 60 70 80 90 io0 Temperature ('Q

Figure 6. Effective resolution for different values of M and worst case as a fisndion of the temperature for p€I measurement estimation error

703

-0 5 10 15 20 25 30 35 40 45 So’ Temperature (”C)

Figure 7. Maximum DO concentration and electrode output current as function of the temperature

maximum DO concentration x,, with temperature and its correspondent electrode output current y are shown in Fig. 7.

In order to reconstruct the estimate of the DO value, one needs to measure the temperature. There different temperatures measurement schemes were considered to estimate the effect of temperature quantization, on the final estimate of the DO: (i) Use of a sensor with linear R-T characteristic curve as employed in the case of pH measurement; (ii) Use of a current biased thermistor and (iii) Use of a voltage biased thermistor. Denoting DO by x, and temperature by v and with a 10 bit A173 converter, we have: y = J(X,V) = xexpGnv +b) P = R z b d ,C) c f -‘(x,v) = yd log(mC + h) x c [O; 14.55 mg/l] y t [@31.4 pA] E, = 15.3 nA

The worst case DO concentration error and the effective measurement resolution plotted as a function of operating temperature are shown in Fig. 8 (A) and (B}, for each of the three temperature measurement configurations referred to above. The use of a current biased thermistor produces the best results in the sense that its effective resolution has the l a s t variation over the full range of the operating temperature.

N. CONCLUSIONS A procedure to formulate the propagation of the

quantization error in a typicd digital measurement system ha!; been done, to estimate its effect on the effective resolution OC principal measurand This formulation permits the determination of the required number of bits in MI converter, to obtain the desired final resolution for the measurand. Three different cases with varying degrees of complexity have been discussed.

It is observed that the final resolution is degraded when some compensation scheme has to be employed to cancel t h e effect of secondary quantities which influence the principal measurand or to cancel some non-linearity of the sensor. The measuring technique employed for this secondary quantity also affects the final obtainable resolution. From the pIotting of the effective resolution against the interfering quantity its also possible to determine the minimal required resolution for the Ax) converter to achieve the desired final resolution to the user for all the three Mses.

This study establishes a procedure for evaluating the final

” 0 5 10 15 20 25 30 35 40 45 50

Temperature (“C)

0 5 10 15 20 25 30 35 40 45 50 Temperature (“C)

Figure 8. DO cclnccntration (A) ma?rimum measurement error and (B) effective resolution. Both as hnction of the temperature for temperaturr

msasurement usini:: (I) linear transducer, (II) current biased thermistor and flu) voltage biased thermistor.

resolution in a given measurement scheme and provides a tool to design a system for the desired final resolution of the measmand.

’V. ACKNOWLEDGMENTS The author:; acknowledge the CNPq‘and the CAPES-

COFECUB, for the support in tlie form of fellowship, during the period of thxs research

[l] MORAWSKI, R Z. “Unified approach to measurand reconstrocti~on”, IEEE Transactions on Instrumentation and Measurements, Vol. 43, No. 2, pp.226-231. April 1994.

[2] DORF, R. C. The Electrical Engineering Handbook. IEEE Press. New York, USA. 1993.2661 p,

[3] CATUNDA, S. Y. C. “Sistema p m deteslo de elementos toxicos em fsistemas de tratamento de hguas residuirias de lodo ativado” (System for detection of toxic elements in activated sludge wastewater treament systems). Master’s Dissertation, COPELE-UFPB, 1996.

[LF] Anonymous, Cole Parmer Catalog 1999-2000, pp. 598- 645.

[5] STANDARD METHODS for the examination of water and wastewater. APHA - AWWA - UIPCF. 4500-0 Oxygen (Diissolved). 1989,17th edition. pg. 4-149.

[6]YSI 5700 Series Dissolved Oxygen Probes. Yellow Springs Instrument Co., Inc. Yellow Springs. Ohio. USA

171 CATUNDA, S. Y. C., Deep, G. S. e Freire, R C. S., “Compens@o da temperatura M medi@o de oxighio dissolvido” (Temperature compensation in dissolved oxygen measurements), Congress0 Braileiro de Autunzuticcr - CBA ’98, 14-18 de September, Volume I, pp

VI. m R E N C E S

9-14,1998

704