Handbook of Industrial Automationedited - Chapter 2 pot

The instrumentation ampli®er error budget tion ofTable 3 employs the parameters of Table 1 toobtain representative ampli®er error, expressed both as tabula-an input-amplitude-threshold u

Modern technology leans heavily on the science of

measurement The control of industrial processes and

automated systems would be very dif®cult without

accurate sensor measurements Signal-processing

func-tions increasingly are being integrated within sensors,

and digital sensor networks directly compatible with

computer inputs are emerging Nevertheless,

measure-ment is an inexact science requiring the use of reference

standards and an understanding of the energy

transla-tions involved more directly as the need for accuracy

increases Seven descriptive parameters follow:

Accuracy: the closeness with which a measurement

approaches the true value of a measurand,

usually expressed as a percent of full scale

Error: the deviation of a measurement from the true

value of a measurand, usually expressed as a

pre-cent of full scale

Tolerance: allowable error deviation about a

refer-ence of interest

Precision: an expression of a measurement over

some span described by the number of signi®cant

®gures available

Resolution: an expression of the smallest quantity to

which a quantity can be represented

Span: an expression of the extent of a measurement

between any two limits

A general convention is to provide sensor ments in terms of signal amplitudes as a percent of fullscale, or %FS, where minimum±maximum values cor-respond to 0 to 100%FS This range may correspond

measure-to analog signal levels between 0 and 10 V (unipolar)with full scale denoted as 10 VFS Alternatively, a sig-nal range may correspond to 50%FS with signallevels between 5 V (bipolar) and full scale denoted

at  5VFS

1.2 INSTRUMENTATION AMPLIFIERSAND ERROR BUDGETS

The acquisition of accurate measurement signals, cially low-level signals in the presence of interference,requires ampli®er performance beyond the typical cap-abilities of operational ampli®ers An instrumentationampli®er is usually the ®rst electronic device encoun-tered by a sensor in a signal-acquisition channel, and inlarge part it is responsible for the data accuracy attain-able Present instrumentation ampli®ers possess suf®-cient linearity, stability, and low noise for total error inthe microvolt range even when subjected to tempera-ture variations, and is on the order of the nominalthermocouple effects exhibited by input lead connec-tions High common-mode rejection ratio (CMRR) isessential for achieving the ampli®er performance ofinterest with regard to interference rejection, and forestablishing a signal ground reference at the ampli®er

espe-137

that can accommodate the presence of ground±return

potential differences High ampli®er input impedance

is also necessary to preclude input signal loading and

voltage divider effects from ®nite source impedances,

and to accommodate source-impedance imbalances

without degrading CMRR The precision gain values

possible with instrumentation ampli®ers, such as

1000.000, are equally important to obtain accurate

scaling and registration of measurement signals

The instrumentation ampli®er of Fig 1 has evolved

from earlier circuits to offer substantially improved

performance over subtractor instrumentation

ampli-®ers Very high input impedance to 109

with no resistors or their associated temperature

coef-®cients involved in the input signal path For example,

achieved with Avdiff values of 103with precision internal

resistance trimming

When conditions exist for large potentials between

circuits in a system an isolation ampli®er should be

considered Isolation ampli®ers permit a fully ¯oating

sensor loop because these devices provide their own

input bias current, and the accommodation of very

high input-to-input voltages between a sensor input

and the ampli®er output ground reference Off-ground

Vcm values to 10 V, such as induced by interference

coupled to signal leads, can be effectively rejected bythe CMRR of conventional operational and instru-mentation ampli®ers However, the safe and linearaccommodation of large potentials requires an isola-tion mechanism as illustrated by the transformercircuit of Fig 2 Light-emitting diode (LED)-photo-transistor optical coupling is an alternate isolationmethod which sacri®ces performance somewhat toeconomy Isolation ampli®ers are especially advanta-geous in very noisy and high voltage environments andfor breaking ground loops In addition, they providegalvanic isolation typically on the order of 2 mA input-to-output leakage

The front end of an isolation ampli®er is similar inperformance to the instrumentation ampli®er of Fig 1and is operated from an internal dc±dc isolated powerconverter to insure isolation integrity and for sensorexcitation purposes Most designs also include a

of catastrophic input fault conditions The typicalampli®er isolation barrier has an equivalent circuit of

respectively An input-to-output Viso rating of 2500

V peak is common, and is accompanied by an tion-mode rejection ratio (IMRR) with reference to theoutput Values of CMRR to 104 with reference to theinput common, and IMRR values of 108with reference

isola-Figure 1 High-performance instrumentation ampli®er

to the output are available at 60 Hz This dual rejection

capability makes possible the accommodation of two

sources of interference, Vcm and Viso, frequently

encountered in sensor applications The performance

of this connection is predicted by Eq (1), where

non-isolated instrumentation ampli®ers are absent the Viso/

tions are at low frequencies because of the limited

response of the physical processes from which

mea-surements are typically sought The selection of an

instrumentation ampli®er involves the evaluation of

ampli®er parameters that will minimize errors ciated with speci®c applications under anticipatedoperating conditions It is therefore useful to perform

asso-an error evaluation in order to identify signi®casso-ant errorsources and their contributions in speci®c applications

ampli®ers described in ®ve categories representative ofavailable contemporary devices These parametersconsist of input voltage and current errors, interferencerejection and noise speci®cations, and gain nonlinear-

de®nitions

The instrumentation ampli®er error budget tion ofTable 3 employs the parameters of Table 1 toobtain representative ampli®er error, expressed both as

tabula-an input-amplitude-threshold uncertainty in volts tabula-and

as a percent of the full-scale output signal These errortotals are combined from the individual device para-meter errors by

Figure 2 Isolation instrumentation ampli®er

The barred parameters denote mean values, and the

unbarred parameters drift and random values that are

combined as the root-sum-square (RSS) Examination

of these ampli®er error terms discloses that input offsetvoltage drift with temperature is a consistent error, andthe residual Vcmerror following upgrading by ampli®erCMRR is primarily signi®cant with the subtractorinstrumentation ampli®er Ampli®er referred-to-inputinternal rms noise Vn is converted to peak±peak at a3.3 con®dence (0.1% error) with multiplication by 6.6

to relate it to the other dc errors in accounting for itscrest factor The effects of both gain nonlinearity anddrift with temperature are also referenced to the ampli-

®er input, where the gain nonlinearity represents anaverage amplitude error over the dynamic range ofinput signals

The error budgets for the ®ve instrumentationampli®ers shown inTable 3include typical input con-ditions and consistent operating situations so that theirperformance may be compared The total errorsobtained for all of the ampli®ers are similar in magni-tude and represent typical in-circuit expectations.Signi®cant to the subtractor ampli®er is that Vcmmust be limited to about 1 V in order to maintain areasonable total error, whereas the three-ampli®erinstrumentation ampli®er can accommodate Vcmvalues to 10 V at the same or reduced total error.1.3 INSTRUMENTATION FILTERS

Lowpass ®lters are frequently required to bandlimitmeasurement signals in instrumentation applications

to achieve frequency-selective operation The tion of an arbitrary signal set to a lowpass ®lter canresult in a signi®cant attenuation of higher frequencycomponents, thereby de®ning a stopband whoseboundary is in¯uenced by the choice of ®lter cutoff

applica-Table 1 Example Ampli®er Parameters

Subtractor ampli®erOP-07 Three-ampli®erAD624 Isolation ampli®erBB3456 Low-bias ampli®erOPA 103 CAZ DC ampli®erICL 7605

5 pA/8C0.17 V=ms

600 Hz

105

10 nV=pHz0.01%

Rtempco1:2  1011

3  107

25 mV0.2 mV=8C

1 kHz

104…106†

7 nV=pHz0.01%

1 kHz

104

30 nV=pHz0.01%

Rtempco

1014

1013

2 mV0.05 mV=8C

150 pA

1 pA/8C0.5 V/ms

10 Hz

105

200 nV/pHz0.01%

Input-offset-current temperature driftDifferential input impedance

Common-mode input impedanceSlew rate

Input-referred noise voltageInput-referred noise currentOpen-loop gain

Common-mode gainClosed-loop differential gainGain nonlinearity

Gain temperature drift

3 dB bandwidthCommon-mode (isolation-mode)numerical rejection ratio

frequency, with the unattenuated frequency

compo-nents de®ning the ®lter passband For instrumentation

purposes, approximating the lowpass ®lter amplitude

responses described in Fig 3 is bene®cial in order to

achieve signal bandlimiting with minimum alteration

or addition of errors to a passband signal of interest

In fact, preserving the accuracy of measurement signals

is of suf®cient importance that consideration of ®lter

charcterizations that correspond to well-behaved

func-tions such as Butterworth and Bessel polynomials are

especially useful However, an ideal ®lter is physically

unrealizable because practical ®lters are represented by

ratios of polynomials that cannot possess the

disconti-nuities required for sharply de®ned ®lter boundaries

Figure 3 describes the Butterworth and Bessel

low-pass amplitude response where n denotes the ®lter

order or number of poles Butterworth ®lters are

char-acterized by a maximally ¯at amplitude response in the

vicinity of dc, which extends toward its 3 dB cutoff

frequency fcas n increases Butterworth attenuation is

rapid beyond fcas ®lter order increases with a slightly

nonlinear phase response that provides a good

approx-imation to an ideal lowpass ®lter Butterworth ®lters

are therefore preferred for bandlimiting measurement

signals

unity-gain networks tabulated according to the ber of ®lter poles Higher-order ®lters are formed by acascade of the second- and third-order networksshown Figure 4 illustrates the design procedurewith a 1 kHz-cutoff two-pole Butterworth lowpass ®l-ter including frequency and impedance scaling steps.The choice of resistor and capacitor tolerance deter-mines the accuracy of the ®lter implementation such

num-as its cutoff frequency and pnum-assband ¯atness Filterresponse is typically displaced inversely to passive-component tolerance, such as lowering of cutoff fre-quency for component values on the high side of theirtolerance

evaluated for their amplitude errors, by

over the speci®ed ®lter passband intervals One-pole

RC and three-pole Bessel ®lters exhibit comparableerrors of 0.3%FS and 0.2%FS, respectively, for signalbandwidths that do not exceed 10% of the ®lter cutofffrequency However, most applications are better

Table 3 Ampli®er Error Budgets (Avdiffˆ 103; VFSˆ 10 V;
T ˆ 208C; Rtolˆ 1%; Rtempcoˆ 50 ppm=8C†

Ampli®er parameters

Subtractorampli®erOP-07

ampli®erAD624

Three-Isolationampli®erBB3456

Low-biasampli®erOPA103

CAZ DCampli®erICL7605

served by the three-pole Butterworth ®lter which offers

an average amplitude error of 0.2%FS for signal

pass-band occupancy up to 50% of the ®lter cutoff, plus

good stopband attenuation While it may appear

inef-®cient not to utilize a ®lter passband up to its cutoff

frequency, the total bandwidth sacri®ced is usually

small Higher ®lter orders may also be evaluated

when greater stopband attenuation is of interest

with substitution of their amplitude response A… f † in

Eq (4)

1.4 MEASUREMENT SIGNALCONDITIONING

Signal conditioning is concerned with upgrading thequality of a signal of interest coincident with measure-ment acquisition, amplitude scaling, and signal band-limiting The unique design requirements of a typicalanalog data channel, plus economic constraints ofachieving necessary performance without incurringthe costs of overdesign, bene®t from the instrumenta-Figure 3 (a) Butterworth and (b) Bessel lowpass ®lters

tion error analysis presented.Figure 5describes a basic

signal-conditioning structure whose performance is

described by the following equations for coherent

and random interference:



‡"2 coherent

1=2

n 1=2

…7†Input signals Vdiff corrupted by either coherent orrandom interference Vcm can be suf®ciently enhanced

by the signal-conditioning functions of Eqs (5) and(6), based upon the selection of ampli®er and ®lterparameters, such that measurement error is principallydetermined by the hardware device residual errorsderived in previous sections As an option, averagedmeasurements offer the merit of sensor fusion wherebytotal measurement error may be further reduced by theTable 4 Unity-Gain Filter Network Capacitor Values (Farads)

0.7071.3920.9240.3831.3540.3090.9660.7070.2591.3360.6240.2230.9810.8310.5560.195

0.2020.421

0.488

0.9071.4230.7351.0121.0091.0410.6350.7231.0730.8530.7251.0980.5670.6090.7261.116

0.6800.9880.6750.3900.8710.3100.6100.4840.2560.7790.4150.2160.5540.4860.3590.186

0.2540.309

0.303

Figure 4 Butterworth lowpass ®lter design example.

Table 5 Filter Passband Errors

1.0000.9980.9880.9720.9510.9240.8910.8520.8080.7600.707

1.0001.0001.0001.0000.9980.9920.9770.9460.8900.8080.707

0%

0.30.91.93.34.76.38.09.711.513.3

0%

0.20.71.42.33.34.66.07.79.511.1

0%00000.20.71.42.64.46.9

factor n 1=2for n identical signal conditioning channels

combined Note that Vdiff and Vcm may be present in

any combination of dc or rms voltage magnitudes

External interference entering low-level

instrumen-tation circuits frequently is substantial, especially in

industrial environments, and techniques for its

attenuation or elimination are essential Noise coupled

to signal cables and input power buses, the primary

channels of external interference, has as its cause

local electric and magnetic ®eld sources For example,

unshielded signal cables will couple 1 mV of

interfer-ence per kilowatt of 60 Hz load for each lineal foot of

cable run on a 1 ft spacing from adjacent power cables

Most interference results from near-®eld sources,

pri-marily electric ®elds, whereby the effective attenuation

mechanism is re¯ection by a nonmagnetic material

such as copper or aluminum shielding Both

copper-foil and braided-shield twinax signal cables offer

attenuation on the order of 90 voltage dB to 60 Hz

interference However, this attenuation decreases by

20 dB per decade of increasing frequency

For magnetic ®elds, absorption is the effective

attenuation mechanism, and steel or mu-metal

shield-ing is required Magnetic-®eld interference is more

dif-®cult to shield against than electric-®eld interference,

and shielding effectiveness for a given thickness

diminishes with decreasing frequency For example,

steel at 60 Hz provides interference attenuation on

the order of 30 voltage dB per 100 mils of thickness

Magnetic shielding of applications is usually

imple-mented by the installation of signal cables in steel

con-duit of the necessary wall thickness Additional

magnetic-®eld cancellation can be achieved by periodictransposition of a twisted-pair cable, provided that thesignal return current is on one conductor of the pairand not on the shield Mutual coupling between cir-cuits of a computer input system, resulting from ®nitesignal-path and power-supply impedances, is an addi-tional source of interference This coupling is mini-mized by separating analog signal grounds fromnoisier digital and chassis grounds using separateground returns, all terminated at a single star-pointchassis ground

Single-point grounds are required below 1 MHz toprevent circulating currents induced by couplingeffects A sensor and its signal cable shield are usuallygrounded at a single point, either at the sensor or thesource of greatest intereference, where provision of thelowest impedance ground is most bene®cial This alsoprovides the input bias current required by all instru-mentation ampli®ers except isolation types, which fur-nish their own bias current For applications where thesensor is ¯oating, a bias-restoration path must be pro-vided for conventional ampli®ers This is achieved withbalanced differential Rbiasresistors of at least 103timesthe source resistance Rs to minimize sensor loading.between the ampli®er input and the single-pointground as shown in Fig 5

Consider the following application example.Resistance-thermometer devices (RTDs) offer com-platinum RTD For a 0±1008C measurement range theFigure 5 Signal-conditioning channel

stant-current excitation of 0.26 mA converts this

resis-tance to a voltage signal which may be differentially

sensed as Vdiff from 0 to 10 mV, following a 26 mV

ampli®er offset adjustment whose output is scaled 0±

10 V by an AD624 instrumentation ampli®er

differen-tial gain of 1000 A three-pole Butterworth lowpass

bandlimiting ®lter is also provided having a 3 Hz cutoff

frequency This signal-conditioning channel is

evalu-ated for RSS measurement error considering an input

Vcm of up to 10 V rms random and 60 Hz coherent

interference The following results are obtained:

"RTDˆtolerance ‡ nonlinearity  FS

ˆ0:18C ‡ 0:0028

8C8C 1008C

ˆ 0:48%FS

An RTD sensor error of 0.38%FS is determined for

this measurement range Also considered is a 1.5 Hz

signal bandwidth that does not exceed one-half of the

®lter passband, providing an average ®lter error

con-tribution of 0.2%FS fromTable 5 The representative

error of 0.22%FS fromTable 3for the AD624

instru-mentation ampli®er is employed for this evaluation,

and the output signal quality for coherent and random

input interference from Eqs (5) and (6), respectively, is

1:25  10 5%FS and 1:41  10 3%FS The

acquisi-tion of low-level analog signals in the presence of

appreciable intereference is a frequent requirement indata acquisition systems Measurement error of 0.5%

or less is shown to be readily available under thesecircumstances

1.5 DIGITAL-TO-ANALOG CONVERTERSDigital-to-analog (D/A) converters, or DACs, providereconstruction of discrete-time digital signals into con-tinuous-time analog signals for computer interfacingoutput data recovery purposes such as actuators, dis-plays, and signal synthesizers These converters areconsidered prior to analog-to-digital (A/D) convertersbecause some A/D circuits require DACs in theirimplementation A D/A converter may be considered

a digitally controlled potentiometer that provides anoutput voltage or current normalized to a full-scalereference value A descriptive way of indicating therelationship between analog and digital conversionquantities is a graphical representation Figure 6describes a 3-bit D/A converter transfer relationshiphaving eight analog output levels ranging betweenzero and seven-eighths of full scale Notice that aDAC full-scale digital input code produces an analogoutput equivalent to FS 1 LSB The basic structure

of a conventional D/A converter incudes a network ofswitched current sources having MSB to LSB valuesaccording to the resolution to be represented Eachswitch closure adds a binary-weighted current incre-ment to the output bus These current contributionsare then summed by a current-to-voltage converter

Figure 6 Three-bit D/A converter relationships

ampli®er in a manner appropriate to scale the output

signal Figure 7 illustrates such a structure for a 3-bit

DAC with unipolar straight binary coding

correspond-ing to the representation ofFig 6

In practice, the realization of the transfer

character-istic of a D/A converter is nonideal With reference to

Fig 6, the zero output may be nonzero because of

ampli®er offset errors, the total output range from

zero to FS 1 LSB may have an overall increasing or

decreasing departure from the true encoded values

resulting from gain error, and differences in the height

of the output bars may exhibit a curvature owing to

converter nonlinearity Gain and offset errors may be

compensated for leaving the residual temperature-drift

variations shown in Table 6, where gain temperature

coef®cient represents the converter voltage reference

error A voltage reference is necessary to establish a

basis for the DAC absolute output voltage The

major-ity of voltage references utilize the bandgap principle,

whereby the Vbe of a silicon transistor has a negative

temperature coef®cient of 2:5 mV=8C that can be

extrapolated to approximately 1.2 V at absolute zero

(the bandgap voltage of silicon)

Converter nonlinearity is minimized through

preci-sion components, because it is essentially distributed

throughout the converter network and cannot be

elimi-nated by adjustment as with gain and offset error

Differential nonlinearity and its variation with

tem-perature are prominent in data converters in that

they describe the difference between the true and actual

outputs for each of the 1-LSB code changes A DAC

with a 2-LSB output change for a 1-LSB input code

change exhibits 1 LSB of differential nonlinearity as

shown Nonlinearities greater than 1 LSB make theconverter output no longer single valued, in whichcase it is said to be nonmonotonic and to have missingcodes

1.6 ANALOG-TO-DIGITAL CONVERTERSThe conversion of continuous-time analog signals todiscrete-time digital signals is fundamental to obtain-ing a representative set of numbers which can be used

by a digital computer The three functions of sampling,quantizing, and encoding are involved in this processand implemented by all A/D converters as illustrated

devices and their functional operations as we were withthe previously described complementary D/A conver-ter devices In practice one conversion is performedeach period T, the inverse of sample rate fs, whereby

a numerical value derived from the converter ing levels is translated to an appropriate output code.The graph of Fig 9 describes A/D converter input±output relationships and quantization error for pre-vailing uniform quantization, where each of the levels

quantiz-q is of spacing 2 n…1 LSB† for a converter having ann-bit binary output wordlength Note that the maxi-mum output code does not correspond to a full-scaleinput value, but instead to …1 2 n†FS because thereexist only …2n 1† coding points as shown in Fig 9.Quantization of a sampled analog waveforminvolves the assignment of a ®nite number of ampli-tude levels corresponding to discrete values of inputsignal Vi between 0 and VFS The uniformly spacedquantization intervals 2 n represent the resolutionlimit for an n-bit converter, which may also beexpressed as the quantizing interval q equal to

VFS=…2n 1†V These relationships are described by

word-length in bits to a required analog input signal span

to be represented digitally For example, a 10

mV-to-10 V span (0.1%±mV-to-100%) requires a minimum converterwordlength n of 10 bits It will be shown that addi-tional considerations are involved in the conversionFigure 7 Three-bit D/A converter circuit

Table 6 Representative 12-Bit D/A ErrorsDifferential nonlinearity (1/2 LSB)

Linearity temp coeff (2 ppm/8C)(208C)Gain temp coeff (20 ppm/8C)(208C)Offset temp coeff (5 ppm/8C)(208C)

0:012%0:0040:0400:010

of an input signal to an n-bit accuracy other than the

choice of A/D converter wordlength, where the

dynamic range of a digitized signal may be represented

by an n-bit wordlength without achieving n-bit data

accuracy However, the choice of a long wordlength

A/D converter will bene®cially minimize both

quanti-zation noise and A/D device error and provide

increased converter linearity

The mechanization of all A/D converters is by either

the integrating method or the voltage-comparison

method The successive-approximation

voltage-com-parison technique is the most widely utilized A/D

con-verter for computer interfacing primarily because its

constant conversion period T is independent of input

signal amplitude, making its timing requirements veniently uniform This feedback converter operates bycomparing the output of an internal D/A converterwith the input signal at a comparator, where each bit

con-of the converter wordlength n is sequentially testedduring n equal time subperiods to develop an outputcode representative of the input signal amplitude Theconversion period T and sample/hold (S/H) acquisi-tion time tacqdetermine the maximum data conversionthroughput rate fs …T ‡ tacq† 1 shown in Fig 10

successive-approximation converter The internal elements arerepresented in the 12-bit converter errors of Table 8,where differential nonlinearity and gain temperaturecoef®cient are derived from the internal D/A converterand its reference, and quantizing noise as the 1/2 LSBuncertainty in the conversion process Linearity tem-perature coef®cient and offset terms are attributable tothe comparator, and long-term change is due to shiftsoccurring from component aging This evaluationreveals a two-binary-bit derating in realizable accuracybelow the converter wordlength High-speed, succes-sive-approximation A/D converters require high-gainfast comparators, particularly for accurate conversion

at extended wordlengths The comparator is thereforecritical to converter accuracy, where its performance isultimately limited by the in¯uence of internal andexternal noise effects on its decision threshold.Integrating converters provide noise rejection forthe input signal at an attenuation rate of 20 dB/decade of frequency Notice that this noise improve-ment capability requires integration of the signal plusnoise during the conversion period, and therefore isnot provided when a sample-hold device precedes theconverter A conversion period of 16 2/3 ms willprovide a useful null to the conversion of 60 Hzinterference, for example Only voltage-comparisonconverters actually need a S/H to satisfy the A/D-conversion process requirement for a constant inputsignal

Figure 8 Analog-to-digital converter functions

Figure 9 Three-bit A/D converter relationships

Dual-slope integrating converters perform A/D

conversion by the indirect method of converting an

input signal to a representative time period that is

totaled by a counter Features of this conversion

tech-nique include self-calibration that makes it immune to

component temperature drift, use of inexpensive

com-ponents in its mechanization, and the capability for

multiphasic integration yielding improved resolution

of the zero endpoint as shown in Fig 12 Operationoccurs in three phases The ®rst is the autozero phasethat stores the converter analog offsets on the inte-grator with the input grounded During the secondphase, the input signal is integrated for a constanttime T1 In the ®nal phase, the input is connected

to a reference of opposite polarity Integration thenproceeds to zero during a variable time T2 while clockpulses are totaled to represent the amplitude of theinput signal The representative errors of Table 8

show slightly better performance for dual-slope pared with successive-approximation converters, buttheir speed differences belie this advantage The self-calibration, variable conversion time, and lower costfeatures of dual-slope converters make them espe-cially attractive for instrumentation applications.Sample/hold component errors consist of contribu-tions from acquisition time, capacitor charge droopand dielectric absorption, offset voltage drift, andhold-mode feedthrough A representative S/H errorbudget is shown in Table 9 Hold-capacitor voltagedroop dV=dt is determined primarily by the outputampli®er bias-current requirements Capacitor values

com-in the 0.01± 0.001 mF range typically provide a balancefor reasonable droop and acquisition errors Capacitordielectric absorption error is evident as voltage creepfollowing repetitive changes in capacitor charging

Table 7 Decimal Equivalents of Binary Quantities

1234567891011121314151617181920

2481632641282565121,0242,0484,0968,19216,38432,76865,536131,072262,144524,2881,048,576

0.50.250.1250.06250.031250.0156250.00781250.003906250.0019531250.00097636250.000488281250.0002441406250.00012207031250.000061035156250.0000305175781250.00001525878906250.000007629394531250.0000038146972656250.00000190734863281250.00000095367431640625

50.025.012.56.253.121.560.780.390.190.0970.0490.0240.0120.0060.0030.00150.00080.00040.00020.0001

Figure 10 Timing relationships for S/H±A/D conversion

resulting from incomplete dielectric repolarization.Polycarbonate capacitors exhibit 50 ppm dielectricabsorption, polystyrene 20 ppm, and Te¯on 10 ppm.Hold-jump error is attributable to that fraction ofthe logic signal transferred by the capacitance of theswitch at turnoff Feedthrough is speci®ed for the holdmode as the percentage of an input sinusoidal signalthat appears at the output.

1.7 SIGNAL SAMPLING ANDRECONSTRUCTIONThe provisions of discrete-time systems include theexistence of a minimum sample rate for which theore-tically exact signal reconstruction is possible from asampled sequence This provision is signi®cant inthat signal sampling and recovery are considered

Figure 11 Successive-approximation A/D conversion

Table 8 Representative 12-Bit A/D Errors

12-bit successive approximationDifferential nonlinearity (1/2 LSB)

Quantizing uncertainty (1/2 LSB)

Linearity temp coeff (2 ppm/8C)(208C)

Gain temp coeff (20 ppm/8C)(208C)

simultaneously, correctly implying that the design of

real-time data conversion and recovery systems should

also be considered jointly The following interpolation

formula analytically describes this approximation ^x…t†

of a continuous time signal x…t† with a ®nite number ofsamples from the sequence x…nT† as illustrated byFig

Table 9 Representative Sample/Hold Errors

Figure 13 Ideal signal sampling and recovery.

Table 10 Signal Interpolation Functions

35

1=2

100%

D/A + 1-pole RC ‰1 ‡ …f =fc†2Š 1=2

V 2 FS

3 7 7 5

ing in a time-domain sinc amplitude response owing to

the rectangular characteristic of H… f † Due to the

orthogonal behavior of Eq (8), however, only one

nonzero term is provided at each sampling instant by

a summation of weighted samples Contributions of

samples other than the ones in the immediate

neigh-borhood of a speci®c sample, therefore, diminish

rapidly because the amplitude response of H… f † tends

to decrease Consequently, the interpolation formula

provides a useful relationship for describing recovered

bandlimited sampled-data signals of bandwidth BW

with the sampling period T chosen suf®ciently small

to prevent signal aliasing where sampling frequency

fsˆ 1=T

It is important to note that an ideal interpolation

function H… f † utilizes both phase and amplitude

infor-mation in reconstructing the recovered signal ^x…t†, and

is therefore more ef®cient than conventional

band-limiting functions However, this ideal interpolation

function cannot be physically realized because its

impulse response is noncausal, requiring an output

that anticipates its input As a result, practical

inter-polators for signal recovery utilize amplitude

informa-tion that can be made ef®cient, although not optimum,

by achieving appropriate weighting of the

recon-structed signal

Of key interest is to what accuracy can an original

continuous signal be reconstructed from its sampled

values

It can be appreciated that the determination of

sam-ple rate in discrete-time systems and the accuracy with

which digitized signals may be recovered requires the

simultaneous consideration of data conversion and

reconstruction parameters to achieve an ef®cient

allo-cation of system resources Signal to

mean-squared-error relationships accordingly represent sampled and

recovered data intersample error for practical

interpo-lar functions inTable 10 Consequently, an

intersam-ple error of interest may be achieved by substitution of

a selected interpolator function and solving for the

sampling frequency fs by iteration, where asymptotic

convergence to the performance provided by ideal

interpolation is obtained with higher-order practical

interpolators

The recovery of a continuous analog signal from a

discrete signal is required in many applications

Providing output signals for actuators in digital

con-trol systems, signal recovery for sensor acquisition

sys-tems, and reconstructing data in imaging systems are

but a few examples Signal recovery may be viewed

from either time-domain or frequency-domain

perspec-tives In time-domain terms, recovery is similar to

interpolation procedures in numerical analysis withthe criterion being the generation of a locus that recon-structs the true signal by some method of connectingthe discrete data samples In the frequency domain,signal recovery involves bandlimiting by a linear ®lter

to attenuate the repetitive sampled-data spectra abovebaseband in achieving an accurate replica of the truesignal

A common signal recovery technique is to follow aD/A converter by an active lowpass ®lter to achieve anoutput signal quality of interest, accountable by theconvergence of the sampled data and its true signalrepresentation Many signal power spectra have longtime-average properties such that linear ®lters are espe-cially effective in minimizing intersample error.Sampled-data signals may also be applied to controlactuator elements whose intrinsic bandlimited ampli-tude response assist with signal reconstruction Theseterminating elements often may be characterized by asingle-pole RC response as illustrated in the followingsection

An independent consideration associated with thesampling operation is the attenuation impressed uponthe signal spectrum owing to the duration of thesampled-signal representation x…nT† A useful criterion

is to consider the average baseband amplitude errorbetween dc and the full signal bandwidth BWexpressed as a percentage of departure from full-scaleresponse This average sinc amplitude error isexpressed by

A data-conversion system example is provided by asimpli®ed three-digit digital dc voltmeter (Fig 14) Adual-slope A/D conversion period T of 16 2/3 msprovides a null to potential 60 Hz interference,which is essential for industrial and ®eld use, owing

to sinc nulls occurring at multiples of the integrationperiod T A 12-bit converter is employed to achieve anominal data converter error, while only 10 bits arerequired for display excitation considering 3.33 binarybits per decimal digit The sampled-signal error eva-luation considers an input-signal rate of change up to

an equivalent bandwidth of 0.01 Hz, corresponding to

an fs=BW of 6000, and an intersample error mined by zero-order-hold (ZOH) data, where Vsequals VFS:

3 7 5

3 7 5

2

‡ sin  1 ‡

1 6000

3 7 5

3 7 7 7 5

The RSS error of 0.07/% exceeds 10 bits required for a

three-digit display with reference toTable 7

1.8 DIGITAL CONTROL SYSTEM ERROR

The design of discrete-time control loops can bene®t

from an understanding of the interaction of sample

rate and intersample error and their effect on system

performance The choice of sample rate in¯uences

sta-bility through positioning of the closed-loop transfer

function pole locations in the z-domain with respect to

the origin Separately, the decrease in intersample error

from output interpolation provided by the closed-loop

bandwidth of the control system reduces the

uncer-tainty of the controlled variable Since the choice of

sample rate also in¯uences intersample error, an

ana-lysis of a digital control loop is instructive to illustrate

these interrelationships

con-trol loop with a ®rst-order process and unity feedback.All of the process, controller, and actuator gains arerepresented by the single constant K with the compen-sator presently that of proportional control The D/Aconverter represents the in¯uence of the sampling per-iod T, which is z-transformed in the closed-loop trans-fer function of the following equations:

ˆZ 0:80:5Z ‡Z 10:5Z

C…n† ˆ ‰… 0:5†…0:8†n‡ …0:5†…1†nŠU…n†

ˆ 0:50 final value …n large†

The denominator of the transfer function de®nes thein¯uence of the gain K and sampling period T on thepole positions, and hence stability Values are substi-tuted to determine the boundary between stable andunstable regions for control loop performance evalu-ated at the z-plane unit circle stability boundary of

z ˆ 1 This relationship is plotted in Fig 15

Calculation of the 3dB closed-loop bandwidth

BW for both ®rst- and second-order processes is sary for the determination of interpolated intersampleerror of the controlled-variable C For ®rst-order pro-cesses, the closed-loop BW is obtained in terms of therise time tr between the 10% and 90% points of thecontrolled-variable amplitude response to a step inputFigure 14 Three-digit digital voltmeter example

neces-as de®ned in Table 11 The constant 0.35 de®nes the

ratio of 2.2 time constants, required for the response to

rise between 10% and 90% of the ®nal value, to 2

radians for normalization to frequency in Hertz

Validity for digital control loops is achieved by

acquir-ing tr from a discrete-time plot of the

controlled-vari-able amplitude response Tcontrolled-vari-able 11 also de®nes the

bandwidth for a second-order process which is

calcu-lated directly with knowledge of the natural frequency,

sampling period, and damping ratio

In the interest of minimizing sensor-to-actuator

variability in control systems the error of a controlled

variable of interest is divisible into an analog

measure-ment function and digital conversion and interpolation

functions Instrumentation error models provide a

uni-®ed basis for combining contributions from individual

devices The previous temperature measurement signal

conditioning associated withFig 5 is included in this

temperature control loop, shown by Fig 16, with the

averaging of two identical 0.48%FS error

measure-ment channels to effectively reduce that error by

n 1=2 or 2 1=2, from Eq (7), yielding 0.34%FS This

provides repeatable temperature measurements to

within an uncertainty of 0.348C, and a resolution of0.0248C provided by the 12-bit digital data buswordlength

The closed-loop bandwidth is evaluated at vative gain and sampling period values of K ˆ 1 and

conser-T ˆ 0:1 sec …fsˆ 10 Hz†, respectively, for unit-stepexcitation at r…t† The rise time of the controlled vari-able is evaluated from a discrete-time plot of C…n† to be1.1 sec Accordingly, the closed-loop bandwidth isfound from Table 11 to be 0.318 Hz The intersampleerror of the controlled variable is then determined to

be 0.143%FS with substitution of this bandwidth valueand the sampling period T…T ˆ 1=fs† into the one-poleprocess-equivalent interpolation function obtained

scaling signal amplitudes of less than full scale, but aretaken as VS equalling VFS for this example.Intersample error is therefore found to be directlyproportional to process closed-loop bandwidth andinversely proportional to sampling rate

The calculations are as follows:

3 7 5

3 7 5

2

1 ‡ 10 Hz ‡ 0:318 Hz0:318 Hz

2 6 6 6 6 6 6 6 6

3 7 7 7 7 7 7 7 7

1=2

 100%

ˆ 0:143%FS

" controlled variable ˆ …"measurement 21:2†2‡ "2S=H‡ "2A=D

‡" 2 D=A ‡ " 2 sinc ‡ " 2 intersample

ˆ 0:39%FS

Figure 15 Elementary digital control loop

Table 11 Process Closed-Loop Bandwidth

The addition of interpolation, sinc, and device

errors results in a total rss controlled-variable error

of 0.39%FS, corresponding to 8-bit binary accuracy

This 0.39%FS de®ned error describes the baseline

variability of the control loop and hence the process

quality capability It is notable that control-loop

track-ing cannot achieve less process disorder than this

de®ned-error value regardless of the performance

enabled by process identi®cation and tuning of the

PID compensator

BIBLIOGRAPHY

1 JW Gardner Microsensors New York: John Wiley,

1994

2 G Tobey, J Graeme, L Huelsman Operational

Ampli®ers: Design and Applications New York:

McGraw-Hill, 1971

3 J Graeme Applications of Operational Ampli®ers:

Third-Generation Techniques New York:

McGraw-Hill, 1973

4 PH Garrett Computer Interface Engineering for

Real-Time Systems Englewood Cliffs, NJ: Prentice-Hall,

9 M Budai Optimization of the signal conditioning nel Senior Design Project, Electrical EngineeringTechnology, University of Cincinnati, 1978

chan-10 LW Gardenshire Selecting sample rates ISA J April:1964

11 AJ Terri The Shannon sampling theorem ± its variousextensions and applications: a tutorial review Proc IEE

65 (11): 1977

12 N Weiner, Extrapolation, Interpolation, andSmoothing of Stationary Time Series with EngineeringApplications Cambridge, MA: MIT Press, 1949

13 E Zuch Data Acquisition and Conversion Handbook.Mans®eld, MA: Datel-Intersil, 1977

14 ER Hnatek A User's Handbook of D/A and A/DConverters New York: John Wiley, 1976

15 PH Garrett Advanced Instrumentation and ComputerI/O Design New York: IEEE Press, 1994

Figure 16 Process controlled-variable de®ned error

Chapter 2.2

Fundamentals of Digital Motion Control

Ernest L Hall, Krishnamohan Kola, and Ming Cao

University of Cincinnati, Cincinnati, Ohio

2.1 INTRODUCTION

Control theory is a foundation for many ®elds,

includ-ing industrial automation The concept of control

the-ory is so broad that it can be used in studying the

economy, human behavior, and spacecraft design as

well as the design of industrial robots and automated

guided vehicles Motion control systems often play a

vital part of product manufacturing, assembly, and

distribution Implementing a new system or upgrading

an existing motion control system may require

mechanical, electrical, computer, and industrial

engi-neering skills and expertise Multiple skills are required

to understand the tradeoffs for a systems approach to

the problem, including needs analysis, speci®cations,

component source selection, and subsystems

integra-tion Once a speci®c technology is selected, the

suppli-er's application engineers may act as members of the

design team to help ensure a successful implementation

that satis®es the production and cost requirements,

quality control, and safety

Motion control is de®ned [1] by the American

Institute of Motion Engineers as: ``The broad

applica-tion of various technologies to apply a controlled force

to achieve useful motion in ¯uid or solid

electromecha-nical systems.''

The ®eld of motion control can also be considered

as mechatronics [1]: ``Mechatronics is the synergistic

combination of mechanical and electrical engineering,

computer science, and information technology, which

includes control systems as well as numerical methodsused to design products with built-in intelligence.''Motion control applications include the industrialrobot [2] and automated guided vehicles [3±6].Because of the introductory nature of this chapter,

we will focus on digital position control; force controlwill not be discussed

2.2 MOTION CONTROL ARCHITECTURESMotion control systems may operate in an open loop,closed-loop nonservo, or closed-loop servo, as shown

approach, shown in Fig 1(a), has input and outputbut no measurement of the output for comparisonwith the desired response A nonservo, on±off, orbang±bang control approach is shown in Fig 1(b)

In this system, the input signal turns the system on,and when the output reaches a certain level, it closes

a switch that turns the system off A proportion, orservo, control approach is shown in Fig 1(c) In thiscase, a measurement is made of the actual outputsignal, which is fed back and compared to the desiredresponse The closed-loop servo control system will bestudied in this chapter

The components of a typical servo-controlledmotion control system may include an operator inter-face, motion control computer, control compensator,electronic drive ampli®ers, actuator, sensors and trans-ducers, and the necessary interconnections The actua-

157

tors may be powered by electromechanical, hydraulic,

or pneumatic power sources, or a combination

The operator interface may include a combination

of switches, indicators, and displays, including a

computer keyboard and a monitor or display The

motion control computer generates command signals

from a stored program for a real-time operation

The control compensator is a special prgram in the

motion control computer Selecting the compensator

parameters is often a critical element in the success

of the overall system The drive ampli®ers and

elec-tronics must convert the low power signals from the

computer to the higher power signals required to

drive the actuators The sensors and transducers

record the measurements of position or velocity

that are used for feedback to the controller The

actuators are the main drive devices that supply

the force or torque required to move the load All

of these subsystems must be properly interconnected

in order to function properly

2.3 MOTION CONTROL EXAMPLEConsider the simple pendulum shown in Fig 2 that hasbeen studied for more than 2000 years Aristotle ®rstobserved that a bob swinging on a string would come

to rest, seeking a lower state of energy Later, GalileoGalilei made a number of incredible, intuitive infer-ences from observing the pendulum Galileo's conclu-sions are even more impressive considering that hemade his discoveries before the invention of calculus.2.3.1 Flexible-Link Pendulum

The pendulum may be described as a bob with mass,

M, and weight given by W ˆ Mg, where g is the eration of gravity, attached to the end of a ¯exible cord

accel-of length, L as shown in Fig 2 When the bob is placed by an angle , the vertical weight componentcauses a restoring force to act on it Assuming thatviscous damping, from resistance in the medium,with a damping factor, D, causes a retarding forceproportional to its angular velocity, !, equal to D!.Since this is a homogeneous, unforced system, thestarting motion is set by the initial conditions Letthe angle at time …t ˆ 0† be 458 For de®niteness letthe weight, W ˆ 40 lb, the length, L ˆ 3 ft, D ˆ 0:1 lbsec and g ˆ 32:2 ft/s2

dis-The analysis is begun by drawing a free-body gram of the forces acting on the mass We will use thetangent and normal components to describe the forcesacting on the mass The free-body diagram shown inFig 2(b) and Newton's second law are then used toderive a differential equation describing the dynamicresponse of the system Forces may be balanced in anydirection; however, a particularly simple form of the

dia-Figure 1 Motion control systems may operate in several

ways such as (a) open loop, (b) closed-loop nonservo, or

(c) closed-loop servo

Figure 2 Pendulum as studied by Galileo Galilei

equation for pendulum motion can be developed by

balancing the forces in the tangential direction:

X

This gives the following equation:

The tangential acceleration is given in terms of the rate

of change of velocity or arc length by the equation

Note that the unit of each term is force In imperial

units, W is in lbf, g is in ft/sec2, D is in lb sec, L is in

feet,  is in radians, d=dt is in rad/sec and d2=dt2is in

rad/sec2 In SI units, M is in kg, g is in m/sec2, D is in

kg m/sec, L is in meters,  is in radians, d=dt is in rad/

sec, and d2=dt2 is in rad/sec2

This may be rewritten as

This equation may be said to describe a system While

there are many types of systems, systems with no

out-put are dif®cult to observe, and systems with no inout-put

are dif®cult to control To emphasize the importance

of position, we can describe a kinematic system, such as

y ˆ T…x† To emphasize time, we can describe a

dynamic system, such as g ˆ h… f …t†† Equation (7)

describes a dynamic response The differential

equa-tion is nonlinear because of the sin  term

For a linear system, y ˆ T…x†, two conditions must

be satis®ed:

1 If a constant, a, is multiplied by the input, x,

such that ax is applied as the input, then the

output must be multiplied by the same constant:

2 If the sum of two inputs is applied, the output

must be the sum of the individual outputs and

the principal of superposition must hold asdemonstrated by the following equations:

Invariance is an important concept for systems In

an optical system, such as reading glasses, positioninvariance is desired, whereas, for a dynamic systemtime invariance is very important

Since an arbitrary input function, f …t† may beexpressed as a weighted sum of impulse functionsusing the Dirac delta function, …t † This sum can

be expressed as

f …t† ˆ

1 1

f …† …t † d

24

Therefore, the response of the linear system is terized by the response to an impulse function Thisleads to the de®nition of the impulse response, h…t; †,as

Since the system response may vary with the timethe input is applied, the general computational formfor the output of a linear system is the superpositionintegral called the Fredholm integral equation [7,8]: