THÔNG TIN TÀI LIỆU
Chapter 2.1 Measurement and Control Instrumentation Error-Modeled Performance Patrick H. Garrett University of Cincinnati, Cincinnati, Ohio 1.1 INTRODUCTION 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 measure- ments in terms of signal amplitudes as a percent of full scale, or %FS, where minimum±maximum values cor- respond to 0 to 100%FS. This range may correspond to analog signal levels between 0 and 10 V (unipolar) with full scale denoted as 10 V FS . Alternatively, a sig- nal range may correspond to Æ50%FS with signal levels between Æ5 V (bipolar) and full scale denoted at Æ5V FS . 1.2 INSTRUMENTATION AMPLIFIERS AND ERROR BUDGETS The acquisition of accurate measurement signals, espe- cially low-level signals in the presence of interference, requires ampli®er performance beyond the typical cap- abilities of operational ampli®ers. An instrumentation ampli®er is usually the ®rst electronic device encoun- tered by a sensor in a signal-acquisition channel, and in large 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 in the microvolt range even when subjected to tempera- ture variations, and is on the order of the nominal thermocouple effects exhibited by input lead connec- tions. High common-mode rejection ratio (CMRR) is essential for achieving the ampli®er performance of interest with regard to interference rejection, and for establishing a signal ground reference at the ampli®er 137 Copyright © 2000 Marcel Dekker, Inc. 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 10 9 is typical with no resistors or their associated temperature coef- ®cients involved in the input signal path. For example, this permits a 1 k source impedance imbalance with- out degrading CMRR. CMRR values to 10 6 are achieved with A v diff values of 10 3 with 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 V cm values to Æ10 V, such as induced by interference coupled to signal leads, can be effectively rejected by the CMRR of conventional operational and instru- mentation ampli®ers. However, the safe and linear accommodation of large potentials requires an isola- tion mechanism as illustrated by the transformer circuitofFig.2.Light-emittingdiode(LED)-photo- transistor optical coupling is an alternate isolation method which sacri®ces performance somewhat to economy. Isolation ampli®ers are especially advanta- geous in very noisy and high voltage environments and for breaking ground loops. In addition, they provide galvanic isolation typically on the order of 2 mA input- to-output leakage. The front end of an isolation ampli®er is similar in performance to the instrumentation ampli®er of Fig. 1 and is operated from an internal dc±dc isolated power converter to insure isolation integrity and for sensor excitation purposes. Most designs also include a 100 k series input resistor R to limit the consequences of catastrophic input fault conditions. The typical ampli®er isolation barrier has an equivalent circuit of 10 11 shunted by 10 pF representing R iso and C iso , respectively. An input-to-output V iso rating of Æ2500 V peak is common, and is accompanied by an isola- tion-mode rejection ratio (IMRR) with reference to the output. Values of CMRR to 10 4 with reference to the input common, and IMRR values of 10 8 with reference 138 Garrett Figure 1 High-performance instrumentation ampli®er. Copyright © 2000 Marcel Dekker, Inc. to the output are available at 60 Hz. This dual rejection capability makes possible the accommodation of two sources of interference, V cm and V iso , frequently encountered in sensor applications. The performance of this connection is predicted by Eq. (1), where non- isolated instrumentation ampli®ers are absent the V iso / IMRR term: V 0 A v diff V diff 1 1 CMRR V cm V diff V iso IMRR where (1) A v diff 1 2R 0 R G The majority of instrumentation-ampli®er applica- 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 asso- ciated with speci®c applications under anticipated operating conditions. It is therefore useful to perform an error evaluation in order to identify signi®cant error sources and their contributions in speci®c applications. Table1presentsparameterspeci®cationsforexample ampli®ers described in ®ve categories representative of available contemporary devices. These parameters consist of input voltage and current errors, interference rejection and noise speci®cations, and gain nonlinear- ity.Table2providesaglossaryofampli®erparameter de®nitions. The instrumentation ampli®er error budget tabula- tionofTable3employstheparametersofTable1to obtain representative ampli®er error, expressed both as an input-amplitude-threshold uncertainty in volts and as a percent of the full-scale output signal. These error totals are combined from the individual device para- meter errors by Measurement and Control Instrumentation 139 Figure 2 Isolation instrumentation ampli®er. Copyright © 2000 Marcel Dekker, Inc. " amp1RTI V os I os R s f A v V FS A v diff 2 4 dV os dT dT 2 V cm CMRR IMRR 2 6:6 V n f hi p 2 dA v dT dT V FS A v diff 2 5 1=2 " amp1%FS " amp1RTI A v diff V FS  100 3 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 offset voltage drift with temperature is a consistent error, and the residual V cm error following upgrading by ampli®er CMRR is primarily signi®cant with the subtractor instrumentation ampli®er. Ampli®er referred-to-input internal rms noise V n is converted to peak±peak at a 3.3 con®dence (0.1% error) with multiplication by 6.6 to relate it to the other dc errors in accounting for its crest factor. The effects of both gain nonlinearity and drift with temperature are also referenced to the ampli- ®er input, where the gain nonlinearity represents an average amplitude error over the dynamic range of input signals. The error budgets for the ®ve instrumentation ampli®ersshowninTable3includetypicalinputcon- ditions and consistent operating situations so that their performance may be compared. The total errors obtained 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 V cm must be limited to about 1 V in order to maintain a reasonable total error, whereas the three-ampli®er instrumentation ampli®er can accommodate V cm values to 10 V at the same or reduced total error. 1.3 INSTRUMENTATION FILTERS Lowpass ®lters are frequently required to bandlimit measurement signals in instrumentation applications to achieve frequency-selective operation. The applica- tion of an arbitrary signal set to a lowpass ®lter can result in a signi®cant attenuation of higher frequency components, thereby de®ning a stopband whose boundary is in¯uenced by the choice of ®lter cutoff 140 Garrett Table 1 Example Ampli®er Parameters Subtractor ampli®er OP-07 Three-ampli®er AD624 Isolation ampli®er BB3456 Low-bias ampli®er OPA 103 CAZ DC ampli®er ICL 7605 V os dV os =dT I os dI os =dT S r f hi A v diff 10 3 CMRR (IMRR) V n rms f A V dA V =dT R i cm R i diff 60 mV 0:2 mV=8C 0.8 nA 5 pA/8C 0.17 V=ms 600 Hz 10 5 10 nV= Hz p 0.01% R tempco 1:2  10 11 3  10 7 25 mV 0.2 mV=8C 10 nA 20 pA/8C 5V/ms 25 kHz 10 6 4nV= Hz p 0.001% 5 ppm/8C 10 9 10 9 0.25 mV 1 mV=8C 10 mA 0.3 nA/8C 0.5 mV=ms 1 kHz 10 4 10 6 7nV= Hz p 0.01% 10 ppm/8C 5  10 9 10 7 100 mV 1 mV=8C 0.2 pA 7%I os =8C 1.3 V/ms 1 kHz 10 4 30 nV= Hz p 0.01% R tempco 10 14 10 13 2 mV 0.05 mV=8C 150 pA 1 pA/8C 0.5 V/ms 10 Hz 10 5 200 nV/ Hz p 0.01% 15 ppm/8C 10 12 10 12 Table 2 Ampli®er Parameter Glossary V os dV os =dT I os dI os =dT R i diff R i cm S r V n I n A v o A v cm A v diff f A v dA v =dT f hi CMRR (IMRR) Input offset voltage Input-offset-voltage temperature drift Input offset current Input-offset-current temperature drift Differential input impedance Common-mode input impedance Slew rate Input-referred noise voltage Input-referred noise current Open-loop gain Common-mode gain Closed-loop differential gain Gain nonlinearity Gain temperature drift À3 dB bandwidth Common-mode (isolation-mode) numerical rejection ratio Copyright © 2000 Marcel Dekker, Inc. frequency, with the unattenuated frequency compo- nents de®ning the ®lter passband. For instrumentation purposes, approximating the lowpass ®lter amplitude responsesdescribedinFig.3isbene®cialinorderto 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 À3 dB cutoff frequency f c as n increases. Butterworth attenuation is rapid beyond f c as ®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. Table4providesthecapacitorvaluesinfaradsfor unity-gain networks tabulated according to the num- ber of ®lter poles. Higher-order ®lters are formed by a cascade of the second- and third-order networks shown.Figure4illustratesthedesignprocedure with 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 as its cutoff frequency and passband ¯atness. Filter response is typically displaced inversely to passive- component tolerance, such as lowering of cutoff fre- quency for component values on the high side of their tolerance. Table5presentsatabulationoftheexample®lters evaluated for their amplitude errors, by " filter%FS 0:1 f =f c f =0:1f c 0 1:0 À Af  100 4 over the speci®ed ®lter passband intervals. One-pole RC and three-pole Bessel ®lters exhibit comparable errors of 0.3%FS and 0.2%FS, respectively, for signal bandwidths that do not exceed 10% of the ®lter cutoff frequency. However, most applications are better Measurement and Control Instrumentation 141 Table 3 Ampli®er Error Budgets (A v diff 10 3 ; V FS 10 V; T 208C; R tol 1; R tempco 50 ppm=8C Ampli®er parameters Subtractor ampli®er OP-07 Three- ampli®er AD624 Isolation ampli®er BB3456 Low-bias ampli®er OPA103 CAZ DC ampli®er ICL7605 Input conditions V cm Æ1V Æ10 V Æ1000 V Æ100 mV Æ100 mV R s 1k 1k 1k 10 M 1k Offset group V os Nulled Nulled Nulled Nulled 2 mV dV os dT T 4 mV5mV20mV20mV1mV I os R s 0:8 mV 10 mV 10 mV 2 mV 0:15 mV Interference group V cm CMRR IMRR inckt 30 mV 10 mV 10 mV 12 mV1mV 6:6V n f h1 p 1:6 mV4:1 mV1:5 mV6:2 mV4:1 mV Linearity group f A v V FS A v diff 1 mV 0:1 mV 1 mV 1 mV 1 mV dA v DT T V FS A v diff 10 mV1mV2mV10mV3mV Combined error ampi RTI 34 mV22mV33mV29mV8mV ampi%FS 0:34 0:22 0:33 0:29 0:08 Copyright © 2000 Marcel Dekker, Inc. 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 Af in Eq. (4). 1.4 MEASUREMENT SIGNAL CONDITIONING Signal conditioning is concerned with upgrading the quality of a signal of interest coincident with measure- ment acquisition, amplitude scaling, and signal band- limiting. The unique design requirements of a typical analog data channel, plus economic constraints of achieving necessary performance without incurring the costs of overdesign, bene®t from the instrumenta- 142 Garrett Figure 3 (a) Butterworth and (b) Bessel lowpass ®lters. Copyright © 2000 Marcel Dekker, Inc. tionerroranalysispresented.Figure5describesabasic signal-conditioning structure whose performance is described by the following equations for coherent and random interference: " coherent V cm V diff R diff R cm ! 1=2 A v cm A v diff 1 f coh f c 2n 45 À1=2  100 5 " random V cm V diff R diff R cm ! 1=2 A v cm A v diff 2 p f c f hi ! 1=2 Â100 6 " measurement " 2 sensor " 2 amplifier " 2 filter " 2 random  " 2 coherent à 1=2 Án À1=2 7 Input signals V diff corrupted by either coherent or random interference V cm can be suf®ciently enhanced by the signal-conditioning functions of Eqs. (5) and (6), based upon the selection of ampli®er and ®lter parameters, such that measurement error is principally determined by the hardware device residual errors derived in previous sections. As an option, averaged measurements offer the merit of sensor fusion whereby total measurement error may be further reduced by the Measurement and Control Instrumentation 143 Table 4 Unity-Gain Filter Network Capacitor Values (Farads) Poles Butterworth Bessel C 1 C 2 C 3 C 1 C 2 C 3 2 3 4 5 6 7 8 1.414 3.546 1.082 2.613 1.753 3.235 1.035 1.414 3.863 1.531 1.604 4.493 1.091 1.202 1.800 5.125 0.707 1.392 0.924 0.383 1.354 0.309 0.966 0.707 0.259 1.336 0.624 0.223 0.981 0.831 0.556 0.195 0.202 0.421 0.488 0.907 1.423 0.735 1.012 1.009 1.041 0.635 0.723 1.073 0.853 0.725 1.098 0.567 0.609 0.726 1.116 0.680 0.988 0.675 0.390 0.871 0.310 0.610 0.484 0.256 0.779 0.415 0.216 0.554 0.486 0.359 0.186 0.254 0.309 0.303 Copyright © 2000 Marcel Dekker, Inc. 144 Garrett Figure 4 Butterworth lowpass ®lter design example. Table 5 Filter Passband Errors Frequency Amplitude response Af Average ®lter error " filter%FS f f c 1-pole RC 3-pole Bessel 3-pole Butterworth 1-pole RC 3-pole Bessel 3-pole Butterworth 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.000 0.997 0.985 0.958 0.928 0.894 0.857 0.819 0.781 0.743 0.707 1.000 0.998 0.988 0.972 0.951 0.924 0.891 0.852 0.808 0.760 0.707 1.000 1.000 1.000 1.000 0.998 0.992 0.977 0.946 0.890 0.808 0.707 0% 0.3 0.9 1.9 3.3 4.7 6.3 8.0 9.7 11.5 13.3 0% 0.2 0.7 1.4 2.3 3.3 4.6 6.0 7.7 9.5 11.1 0% 0 0 0 0 0.2 0.7 1.4 2.6 4.4 6.9 Copyright © 2000 Marcel Dekker, Inc. factor n À1=2 for n identical signal conditioning channels combined. Note that V diff and V cm 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 periodic transposition of a twisted-pair cable, provided that the signal return current is on one conductor of the pair and not on the shield. Mutual coupling between cir- cuits of a computer input system, resulting from ®nite signal-path and power-supply impedances, is an addi- tional source of interference. This coupling is mini- mized by separating analog signal grounds from noisier digital and chassis grounds using separate ground returns, all terminated at a single star-point chassis ground. Single-point grounds are required below 1 MHz to prevent circulating currents induced by coupling effects. A sensor and its signal cable shield are usually grounded at a single point, either at the sensor or the source of greatest intereference, where provision of the lowest impedance ground is most bene®cial. This also provides the input bias current required by all instru- mentation ampli®ers except isolation types, which fur- nish their own bias current. For applications where the sensor is ¯oating, a bias-restoration path must be pro- vided for conventional ampli®ers. This is achieved with balanced differential R bias resistors of at least 10 3 times the source resistance R s to minimize sensor loading. Resistors of 50 M, 0.1% tolerance, may be connected between the ampli®er input and the single-point ground as shown in Fig. 5. Consider the following application example. Resistance-thermometer devices (RTDs) offer com- mercial repeatability to 0.18C as provided by a 100 platinum RTD. For a 0±1008C measurement range the resistance of this device changes from 100.0 to Measurement and Control Instrumentation 145 Figure 5 Signal-conditioning channel. Copyright © 2000 Marcel Dekker, Inc. 138.5 with a nonlinearity of 0.00288C/8C. A con- stant-current excitation of 0.26 mA converts this resis- tance to a voltage signal which may be differentially sensed as V diff 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 V cm of up to 10 V rms random and 60 Hz coherent interference. The following results are obtained: " RTD tolerance nonlinearity  FS FS  100 0:18C 0:0028 8C 8C  1008C 1008C  100 0:38FS " ampl 0:22FS (Table 3) " filter 0:20FS (Table 5) " coherent 10 V 10 mV 10 9 10 9 45 1=2 Â10 À6  1 60 Hz 3Hz 6 45 À1=2 Â100 1:25  10 À5 FS " random 10 V 10 mV 10 9 10 9 45 1=2 Â10 À6  2 p 3Hz 25 kHz ! 1=2 Â100 1:41  10 À3 FS " measurement " 2 RTD " 2 ampl " 2 filter " coherent  " 2 random à 1=2 0:48FS 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- tributionof0.2%FSfromTable5.Therepresentative errorof0.22%FSfromTable3fortheAD624instru- 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 À5 %FS and 1:41  10 À3 %FS. The acquisi- tion of low-level analog signals in the presence of appreciable intereference is a frequent requirement in data acquisition systems. Measurement error of 0.5% or less is shown to be readily available under these circumstances. 1.5 DIGITAL-TO-ANALOG CONVERTERS Digital-to-analog (D/A) converters, or DACs, provide reconstruction of discrete-time digital signals into con- tinuous-time analog signals for computer interfacing output data recovery purposes such as actuators, dis- plays, and signal synthesizers. These converters are considered prior to analog-to-digital (A/D) converters because some A/D circuits require DACs in their implementation. A D/A converter may be considered a digitally controlled potentiometer that provides an output voltage or current normalized to a full-scale reference value. A descriptive way of indicating the relationship between analog and digital conversion quantities is a graphical representation. Figure 6 describes a 3-bit D/A converter transfer relationship having eight analog output levels ranging between zero and seven-eighths of full scale. Notice that a DAC full-scale digital input code produces an analog output equivalent to FS À 1 LSB. The basic structure of a conventional D/A converter incudes a network of switched current sources having MSB to LSB values according to the resolution to be represented. Each switch closure adds a binary-weighted current incre- ment to the output bus. These current contributions are then summed by a current-to-voltage converter 146 Garrett Figure 6 Three-bit D/A converter relationships. Copyright © 2000 Marcel Dekker, Inc. [...]... 16 32 64 128 25 6 5 12 1, 024 2, 048 4,096 8,1 92 16,384 32, 768 65,536 131,0 72 2 62, 144 524 ,28 8 1,048,576 LSB weight, 2 n 0.5 0 .25 0. 125 0.0 625 0.03 125 0.015 625 0.0078 125 0.00390 625 0.001953 125 0.0009763 625 0.00048 828 125 0.00 024 4140 625 0.000 122 0703 125 0.00006103515 625 0.000030517578 125 0.00001 525 87890 625 0.000007 629 39453 125 0.00000381469 726 5 625 0.0000019073486 328 125 0.00000095367431640 625 Dual-slope integrating... 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 Figure 7 Three-bit D/A converter circuit Copyright © 20 00 Marcel Dekker, Inc 147 Table 6 Representative 1 2- Bit D/A Errors Differential nonlinearity (1 /2 LSB) Linearity temp coeff (2 ppm/8C) (20 8C) Gain temp coeff (20 ppm/8C) (20 8C) Offset temp coeff (5 ppm/8C) (20 8C)... by the capacitance of the switch at turnoff Feedthrough is speci®ed for the hold mode as the percentage of an input sinusoidal signal that appears at the output 0.080%FS 1.7 1 2- bit dual slope Differential nonlinearity (1 /2 LSB) Quantizing uncertainty (1 /2 LSB) Gain temp coeff (25 ppm/8C) (20 8C) Offset temp.coeff (2 ppm/8C) (20 8C) 0:0 127 0.0 12 0.050 0.004 A=D 0.063%FS Copyright © 20 00 Marcel Dekker,... 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 determined by zero-order-hold (ZOH) data, where Vs equals VFS : 154 Garrett Figure 14 Three-digit digital voltmeter example QÀ1 =2 P T "intersample T R 2 VFS 2 1:644  VS sinc2 1 À BW 1 BW sinc2 fs fs U !U S Â1007 QÀ1 =2 P 1 VP T Q2 P Q2 WU T b... x
nT nÀx 2TBW x nÀx ej2BW
tÀnT À eÀj2BW
tÀnT j2
t À nT x
nT sin 2 BW
t À nT 2 BW
t À nT x
t is obtained from the inverse Fourier transform of the input sequence and a frequency-domain convolution with an ideal interpolation function H
f , result- 1 52 Table 10 Garrett Signal Interpolation Functions Interpolator A
f D/A + 1-pole RC 1
f =fc 2 À1 =2 D/A + Butterworth n-pole lowpass...
f =fc 2n À1 =2 P T T T T R Intersample error (%FS) QÀ1 =2 2 VFS R h iS Â1007 2 1:644VS sinc2 1 À BW sinc2 1 BW fs fs sinc
f =fs |{z} D/A P 2 VS @ h U 2 U VFS !À1 h !À1 U U i 2n i 2n S 1 fs ÀBW 1 fs BW sinc2 1 À BW sinc2 1 BW fs fs fc fc (fs Æ BW substituted for f in A
f Figure 13 Ideal signal sampling and recovery Copyright © 20 00 Marcel... (1 /2 LSB) Quantizing uncertainty (1 /2 LSB) Linearity temp coeff (2 ppm/8C) (20 8C) Gain temp coeff (20 ppm/8C) (20 8C) Offset (5 ppm/8C) (20 8C) Long-term change 0:0 127 0:0 12 0.004 0.040 0.010 0.050 A=D resulting from incomplete dielectric repolarization Polycarbonate capacitors exhibit 50 ppm dielectric absorption, polystyrene 20 ppm, and Te¯on 10 ppm Hold-jump error is attributable to that fraction of. .. QÀ1 =2 P 1 "intersample T P Q2 U U T 4 T sin 1 À 0:318 Hz 5À1 U U TT U 10 Hz À 0:318 Hz 2 10 Hz TT U 1 U U TR S 0:318 Hz 0:318 Hz U T 1À U T 10 Hz U T U TP 2 Q U T Hz 4 T sin 1 0:318 2 5À1U U TT U 10 Hz 0:318 Hz 10 Hz U U TT U TR S 1 0:318 Hz 0:318 Hz S R 1 10 Hz  1007 "ontrolled vrile 0:1437p 4 51 =2
"mesurement  2 1 :2 2 "2 "2 =r e=h 2 2 2. .. Closed-Loop Bandwidth Process First order Second order Copyright © 20 00 Marcel Dekker, Inc À3dB BW of controlled variable 0:35 Hz (tr from C
n) BW 1:1tr p1 =2 1 Hz where a 4 2 !2 4!3 T À 2! 2 À !4 T 2 BW 2 Àa 1 a2 4!4 n n n n n 2 (natural frequency !n , sample period T sec, damping ratio ) 156 Garrett Figure 16 Process controlled-variable de®ned error The addition of. .. Copyright © 20 00 Marcel Dekker, Inc T
x1 y1 2 The tangential acceleration is given in terms of the rate of change of velocity or arc length by the equation 2 T
x1 x2 y1 y2
9
10 and T
x2 y2
11 Equation (7) is nonlinear because the sine of the sum of two angles is not equal to the sum of the sines of the two angles For example, sin 458 0:707, while sin 908 1 Invariance is an important . LSB) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 4 8 16 32 64 128 25 6 5 12 1, 024 2, 048 4,096 8,1 92 16,384 32, 768 65,536 131,0 72 2 62, 144 524 ,28 8 1,048,576 0.5 0 .25 0. 125 0.0 625 0.03 125 0.015 625 0.0078 125 0.00390 625 0.001953 125 0.0009763 625 0.00048 828 125 0.00 024 4140 625 0.000 122 0703 125 0.00006103515 625 0.000030517578 125 0.00001 525 87890 625 0.000007 629 39453 125 0.00000381469 726 5 625 0.0000019073486 328 125 0.00000095367431640 625 50.0 25 .0 12. 5 6 .25 3. 12 1.56 0.78 0.39 0.19 0.097 0.049 0. 024 0.0 12 0.006 0.003 0.0015 0.0008 0.0004 0.00 02 0.0001 Figure. Equivalents of Binary Quantities Bits, n Levels, 2 n LSB weight, 2 Àn " %FS (1 LSB) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 4 8 16 32 64 128 25 6 5 12 1, 024 2, 048 4,096 8,1 92 16,384 32, 768 65,536 131,0 72 2 62, 144 524 ,28 8 1,048,576 0.5 0 .25 0. 125 0.0 625 0.03 125 0.015 625 0.0078 125 0.00390 625 0.001953 125 0.0009763 625 0.00048 828 125 0.00 024 4140 625 0.000 122 0703 125 0.00006103515 625 0.000030517578 125 0.00001 525 87890 625 0.000007 629 39453 125 0.00000381469 726 5 625 0.0000019073486 328 125 0.00000095367431640 625 50.0 25 .0 12. 5 6 .25 3. 12 1.56 0.78 0.39 0.19 0.097 0.049 0. 024 0.0 12 0.006 0.003 0.0015 0.0008 0.0004 0.00 02 0.0001 Figure. LSB) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 4 8 16 32 64 128 25 6 5 12 1, 024 2, 048 4,096 8,1 92 16,384 32, 768 65,536 131,0 72 2 62, 144 524 ,28 8 1,048,576 0.5 0 .25 0. 125 0.0 625 0.03 125 0.015 625 0.0078 125 0.00390 625 0.001953 125 0.0009763 625 0.00048 828 125 0.00 024 4140 625 0.000 122 0703 125 0.00006103515 625 0.000030517578 125 0.00001 525 87890 625 0.000007 629 39453 125 0.00000381469 726 5 625 0.0000019073486 328 125 0.00000095367431640 625 50.0 25 .0 12. 5 6 .25 3. 12 1.56 0.78 0.39 0.19 0.097 0.049 0. 024 0.0 12 0.006 0.003 0.0015 0.0008 0.0004 0.00 02 0.0001 Figure
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