DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS

Transcription

1 83 Chapter 4 DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS Why differential meaurement? How i the quality of a differential meaurement pecified? What i the detection limiting ignal in a differential meaurement? How i the intrumentation amplifier ued? 4.1 Introduction The ub-ytem dicued o far are compoed of component or circuit with a ingle-ended input and/or output. Information i tranferred over two wire, but with one at ground potential. One of the terminal of each ignal ource (voltage or current) i at ground potential. Moreover, one of the input terminal of a readout circuit i at ground potential. A a conequence, the information content i determined by the abolute value of the ignal at the other wire (the nongrounded wire that i interconnecting the ource and read-out). Many practical ignal ource, however, ue the two wire actively and upply a differential ignal that i uperimpoed on a common ignal. Such a ignal ource ha a differential output. Each of the two output terminal carrie a ignal and the information content i determined by the difference between thee ignal. A uitable circuit for the read-out of a differential ource hould be equipped with a differential input (i.e. two active terminal, none directly forced to ground potential). Thi read-out circuit hould be deigned to be enitive only to the difference between the ignal at the input terminal (= the differential ignal), and hould be immune to the average value relative to ground potential (= the common-mode ignal). A practical amplifier atifie thi requirement up to a

2 84 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS certain extent, which reult in an additive error and thu in a detection limit. Thi limited uppreion of the common-mode ignal i pecified in the Common-Mode ejection atio (CM). Common-mode rejection in a meaurement ytem i particularly important in medical intrumentation. ecording an electro-cardiogram (ECG) involve the poitioning of two ignal electrode plu one reference electrode on the kin of a patient, a hown chematically and highly implified in Fig. 4.1a. C c ~ n 0V / 50Hz ic id Intr. ampl. (a) ic Z c n id id 1 p p1 Intr. ampl. (b) Figure 4.1, Application of the intrumentation amplifier for meauring the ECG: (a) et-up and (b) equivalent circuit of the input. The erie reitance between the electrode and the kin i coniderable and poorly-defined due to the kin conductivity and it dependence on the patient condition. Thi contact reitance can be reduced uing conductive and moituriing cream before placement of the electrode. The read-out ha a parallel reitance to ground for biaing purpoe. A i demontrated in Section 4.3., a very uitable differential amplifier i the intrumentation amplifier. The equivalent electrical circuit of the meaurement et-up i hown in Fig. 4.1b. The ignal in an ECG are differential bio-potential of typically id = 10 µv which are uperimpoed on a common-mode ignal (typically ic > 1 mv, due to capacitive coupling of the main voltage). The challenge in differential read-out deign i to obtain a ufficient CM for reproducible detection of the differential ignal, depite the fact that the magnitude i two order of magnitude lower than that of the common-mode ignal.

3 Section 4. Detection limit due to finite common-mode rejection Detection limit due to finite common-mode rejection 4..1 Differential-mode and common-mode enitivity The Wheattone bridge hown in Fig. 4. i an example of a ignal ource that upplie a relatively mall differential ignal, owd, uperimpoed on a relatively large common-mode ignal, owc. A fully balanced bridge reult in a zero value for the differential ignal, owd = 0 ( = 0; ee alo Section.5.5). Thi differential ignal i uperimpoed on a non-zero common ignal owc = ( 1 )/= exc /. A bridge imbalance yield: Thu: owd = ( /) exc and owc = exc /. 1 = = exc exc (4-1) owd - owc owd exc (a) (b) Figure 4., Bridge circuit with differential-mode and common-mode ignal component at the output. The circuit for the read-out of the differential ignal i hown chematically in Fig. 4.3a. The relevant pecification are the differential gain, G d = o / id, and the common-mode rejection ratio, CM= H. id G d, H o id G dd, H, F od ic (a) Figure 4.3, Definition of common-mode rejection in the cae of: (a) a ingleended output and (b) a differential output. The read-out hould only be enitive to id = owd ( o = G d id ). However, a practical circuit alo exhibit a non-zero enitivity to ic. Therefore, the output voltage, o, i more adequately decribed by: ic (b) oc

4 86 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS o = Gd id Gc ic The unwanted econd term i due to the limited CM. (4-) The CM i defined a the ratio between the enitivity to the differential input ignal, id, a expreed by the differential gain, G d, and the enitivity to the common input ignal, ic, a expreed by the common-mode gain, G c. When uing Fig. 4.3 thi definition yield: H G o d id ic = = = Gc o id o = cont. ic (4-3) Thi expreion indicate that CM can alo be interpreted a the differential input ignal, id, which i required to have the ame effect on the output (i.e. o = contant) a any given common-mode ignal, ic. Note that the CM hould be maximied to yield a minimum additive error, ε: G c ic o = Gdid G c ic = Gd id ic = Gd id = Gd ( id ε ), Gd H (4-4) with H= ic /ε. 4.. Maximiing the CM The common-mode behaviour of an electronic meaurement ytem i largely determined by the input circuit. A implified chematic of the input tage of a differential read-out circuit i hown in Fig Obviouly, the CM i maximied if ic i unable to caue a differential voltage, - -. A qualitative inpection of the circuit reveal two opportunitie to provide uch an advantage: No common-mode voltage would be generated acro the differential input in cae the Voltage acro d1 due to ic i equal to that generated acro d, o that the effect are cancelled out ( d1 = d ). No common-mode current would flow in the cae of an extremely high impedance between ic and ground ( c ). Thee qualitative conideration tranlate into two important deign target: ymmetry by uing a fully balanced input circuit and iolation uing a floating input with repect to the common ground potential.

5 Section 4. Detection limit due to finite common-mode rejection 87 ic id id i d1 cmi d c i- G( i - i- ) o Figure 4.4, Input circuit of a differential amplifier with finite iolation from ground potential and unequal differential input reitor. Thee concluion are confirmed by calculating the common-mode ignal at the input tage, cmi, which reult in: c cmi = ic ( d1)( d ) c d1 d ( ) ( ) (4-5) c d c d1 c d c d1 id - c( d ) c( d1) d1 d c d c d1 The differential ignal between the non-inverting input ( ) and inverting input ( - ) can be derived a: = - ic cmi ic cmi d1 d d1 d d1 d d1 d id (4-6) Auming: d1 = d d /, d = d - d / and c d yield: c cmi = ic d d d d c ( ) cic did cic c d d c d ( 1 / / ) Note that at the detection limit: ic [ d /( d )] id. d d d c d c id = (4-7)

6 88 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS Hence: - d c 1- ic d c ( d )/ - d id d The common-mode rejection follow via Eqn. 4.1 a: (4-8) id CM = H = = - - d d ic d c d (4-9) Equation 4-9 confirm that two approache are available for maximiing common-mode rejection. The firt i by maximiing the impedance between cmi and the common ground: c d (i.e. high iolation). The econd contribution involve good matching between the input component: d = 0 (i.e. ymmetry). A practical operational amplifier can be deigned for ( c / d ) max to exceed Moreover, matching could be better than 1%. Hence, the maximum achievable common-mode rejection can be in the range ( db). The read-out circuit hown in Fig. 4.3a provide a ingle-ended output directly at the output of the firt gain tage. A fully differential ytem offer a uperior uppreion of noie and interference. A gain tage with differential input and differential output hould, therefore, be ued, a i hown in Fig. 4.3b. However, uch a gain tage i only fully pecified if both the differential and the common level of the output ignal are pecified in term of the differential and common input ignal. The definition of the CM pecification of a differential input/output gain tage doe not fundamentally differ from the one provided for the gain tage with differential input and ingle-ended output. The obviou change i that in a fully differential ytem the CM relate the differential output ignal to the ignal upplied at the input terminal. Hence, G cd (Gain common-input-to-differential-output) i ued intead of G c. The common output level i generally conidered to be determined by the common input ignal only ( oc / id = 0). The common-mode ignal tranfer function i pecified by the dicrimination factor, F.

7 Section 4. Detection limit due to finite common-mode rejection 89 The dicrimination factor, F, i pecified in term of the gain for the common-mode input ignal to common-mode output ignal relative to the differential gain: F= G dd /G cc. In equation: H od Gdd id ic Gdd = = = F = = Gcd od id G od = cont. cc ic (4-10) The term CM and F are ued for a fully-differential ytem throughout thi book. Similar to the CM, a high value for F i alo preferred. The fundamental difference between the CM (H) and F i that a finite value for the CM reult in mixing of the differential and common input ignal at the output and thu directly impoe a detection limit on the ytem. A finite value for F doe not reult in an additional differential output ignal in the cae of a non-zero common-mode input, but rather fail to block the tranfer of thi common-mode input level to the differential output. Mot importantly, in the cae of a finite F, the differential and common-mode ignal remain ditinguihable and the detectivity i not yet impaired. A relatively high value for F i neverthele very important, a the (partial) removal of any common-mode ignal reduce the requirement to be impoed on the CM of the next tage, which contribute to a high value of the overall CM of the ytem. It hould be noted that the Englih literature generally ue only the commonmode gain, G cc, and ha not adopted the more convenient dimenionle notation in term of the dicrimination factor, F Practical apect of the CM in bridge readout The limited CM in a differential amplifier ha conequence for the Wheattone bridge; Fig. 4.5a how the circuit. The differential and common-mode ignal are: id = exc, ic = o exc od id oc ic (4-11)

8 90 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS o o - o o - (1δ) exc i G exc d, H i G o d, H o (1-δ) exc o - o o - o (a) (b) Figure 4.5, ead-out of the full Wheattone bridge uing a differential amplifier with a finite CM: (a) ingle-ended excitation and (b) differential excitation. An expreion reult from the definition of CM for the minimum CM value required to detect of ( / o ) det. The inaccuracy pecification, ε, i included by requiring the output voltage due to id (min.) to be equal to 1/ε time that output voltage due to ic : H min exc ic 1 1 = = = id ε o = cont. exc ε o o (4-1) For ( / o ) det.= 10-6 (which i the reult of 1 µtrain when meaured uing a metal film train gauge with k ε = ) and ε= 1%, the reult i H min = (= 148 db), which i an extremely demanding pecification. The reaon for thi high CM requirement i the preence of a large common-mode input ignal, ic. educing thi value would reduce the required read-out pecification. Thi objective i achieved in a differential driving cheme, a hown in Fig. 4.5b. The differential input ignal remain unchanged, id = / o, wherea the common-mode ignal i at ground potential, ic = 0, in the cae of perfectly balanced excitation voltage (δ= 0). Hence, a common-mode rejection H=1 would in principle be ufficient. The reaon for thi omewhat unrealitic outcome i the aumed perfect differential drive (δ= 0). In practice the two excitation voltage ource are not perfectly matched and any mimatch reult in a common-mode input ignal. A mimatch δ reult in: det det

9 Section 4. Detection limit due to finite common-mode rejection 91 i = ( 1 δ ) exc = δ o o exc o exc i = ( 1 δ ) exc = δ o o id = i i = exc, ic = = δ exc o exc The required CM of the read-out reult in: o i i exc (4-13) H min exc δ ic 1 δ = = = id ε o = C exc ε o o (4-14) and i proportional to the degree of mimatch between the two excitation voltage ource. Limiting the mimatch to δ= 1% i well feaible and reult in H min = (108 db), which i an acceptable pecification. A a general rule it can be tated that an increaed ymmetry (in thi cae balancing of excitation ource) help in reducing CM requirement. det Example 4.1 A Wheattone bridge i imbalanced due to the reitance value hown in Fig Calculate the differential bridge output ignal, id, the commonmode ignal, ic, and the required CM of the differential amplifier ued, to enure that ic ha the ame influence on the amplifier output ignal a id for verion (a) and (b). det 100Ω 997Ω exc / 997Ω 100Ω i H o i H o exc 996Ω 996Ω 1001Ω exc / 1001Ω (a) Figure 4.6, Bridge circuit with differential-mode and common-mode ignal level at the output ( exc = 10V). (b) Solution verion (a): i = (1001/1998) 10V and i- = (996/1998) 10 V. id = i - i- = [( )/1998] 10 = 50/1998 = 5 mv.

10 9 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS ic = ( i i- ) / = ( )/3996 = V. o = G d ( id ic /H) H min = ic / id = /0.05 = Solution verion (b): i = (1001/ /1998) 5V and i- = (996/ /1998) 5 V. id = i - i- = [(46)/1998]5 = 50/1998 = 5 mv. ic = ( i i- )/ = (4-6)/3996 =.5 mv. o = G d ( id ic /H) H min = ic / id < ing the CM for noie reduction A differential enor tructure combined with a differential read-out ytem offer huge benefit in term of the detection limit, a it enable the converion of an interfering ignal into a common-mode ignal, which i ubequently uppreed uing the CM of the differential read-out. Z g C c u g u i Z i um _ Figure 4.7, Capacitive injection of the main voltage in a ingle-ended ytem. Figure 4.7 how a ingle-ended circuit for the read-out of u g with capacitive coupling. The tranfer function from the ource to input, u i,g /u g, reult in an expreion for the ignal at the input terminal: u = u u i g ig, g 1 g g i i (4-15) Similarly, for g «i, the noie ignal at the input terminal due to the main voltage, u i,m, i expreed a: jω i g c Zg // Z i i g im, = m = m ω g c m 1 i g Zg // Z 1 jω C i c jωcc i g u u u j C u C (4-16) The Electro-Magnetic Interference (EMI-ee Section 5.8) i directly coupled to the input and impoe an unacceptable detection limit.

11 Section 4. Detection limit due to finite common-mode rejection 93 C c g / g / Z g1 Z g i _ Z i C c1 m _ Figure 4.8, Capacitive injection of the main voltage on a differential input. Figure 4.8 how the differential read-out of a differential voltage ource, g, with a relatively large ource impedance, Z g. Since alo Z i i large enough to enable voltage read-out without ource loading, the main voltage, m, i coupled to the ignal wire. However, for fully balanced ource impedance, differential read-out and equal coupling capacitance (Z g1 = Z g and C c1 = C c ), the injected noie voltage are equal and thu only contribute to the common-mode ignal. In the cae of a ufficiently high CM, thee are eliminated in the differential ignal. However, any differential ignal due to unequal ource impedance or coupling capacitance can in principle not be ditinguihed from the ource ignal (unle filtering or other ignal conditioning i poible- ee Chapter 6) and can affect the detection limit. Example 4. The ingle-ended circuit in Fig. 4.7 ue the following component value: Z i = i = 1 MΩ and C c = 1pF. Calculate the ignal level, i,g and the level of the injected voltage, i,m, at the input of the readout where Z g = g = 50 kω and the main voltage i the external ource of error ( m = 30 V and frequency 50 Hz). Solution: Equation (4-15) yield: i,g = 0.95 g. Similarly, for g «i, Equation (4-16) yield i,m = 100π = 3.6 mv. Auming an inaccuracy pecification at ε= 5% reult in a detection limit at i,g (min.)= i,m /ε= 7 mv. eferring thi voltage level to the input g yield: g (min.)= /0.95= 75.8 mv. Obviouly, thi i too high a detection limit.

12 94 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS The differential ytem, a hown in Fig. 4.8 with CM= 80 db (the effect of coupling to the output i reduced by a factor 10 4 ) yield g (min.)= ( )/0.95= 7.58 µv, which i much more acceptable. 4.3 OPAMP circuit for differential readout The 1-OPAMP differential amplifier An operational amplifier (OPAMP) i equipped with a differential input, - -, which hould make the OPAMP uitable for the direct read-out of a floating ignal ource. 1 1 d o c d 3 4 Figure 4.9, Elementary opamp differential amplifier. Figure 4.9 how the baic differential amplifier. It i intructive to note that thi circuit i a combination of an inverting amplifier, o / 1, and a non-inverting amplifier, o /. ecogniing thee topologie implifie the calculation of the tranfer function uing uperpoition: d =0; o1= - 1= - c d 1=0; o =. = c ( 1) ( ) 4( 1) ( ) = = - o o1 o For : = = - = [ ] o (4-17) In the cae of ideal OPAMP characteritic and a perfect reitor matching in pair ( 1 4 = 3 ), a perfect differential amplifier reult in a gain of: G d = o / ( - 1 )= o / d = / 1 = 4 / 3. However, thi circuit how three unfavourable propertie: d

13 Section 4.3 OPAMP circuit for differential readout 95 Practical reitor are matched within a certain tolerance. The limitation due to the finite and frequency-dependent open-loop gain of the OPAMP, A(ω), do apply. The two terminal of the floating ignal ource are unevenly loaded by the input of the differential amplifier. The non-inverting input (connected to ) ha a large input impedance, Z i1 = 3 4. The inverting input (connected to 1 ) a mall input impedance, Z i = 1. Problem can be expected in the readout of a differential ource with relatively large ource impedance. The mot ignificant problem i the reitor mimatch. Aume the effect reitor mimatch i expreed a: ( 1 )/ = (1δ)( 3 4 )/ 4. In thi cae the tranfer function of the differential amplifier i expreed a: 4( 1 ) ( ) o = = ( 1 δ ) = δ δ ( ) ( ) ( ) = Add 1 Acd 1 1 A δ 1 δ 1 dd 1 H = = = A cd δ 1 δ 1 δ (4-18) Equation (4-18) confirm that the CM become infinitely large for perfectly matched reitor. However, it alo demontrate that the CM deteriorate with increaing component tolerance. Aume δ= 5% limit the CM to: H max = 1/0.05= 0. Thi i inufficient to detect a differential input ignal more than two order of magnitude maller than the common-mode level, which i the cae in many application. Therefore, a more veratile circuit, the intrumentation amplifier, i ued The 3-OPAMP intrumentation amplifier Combining a pecial -OPAMP differential-to-differential voltage pre-amplifier with the 1-OPAMP differential amplifier hown in Fig. 4.9, yield the intrumentation amplifier hown in Fig Thi circuit feature a uperior CM performance. Both input, i1 and i, are connected to the non-inverting input of an opamp with the output fed back to the inverting input. The input impedance i extremely high due to the local feedback, which attempt to reproduce the input voltage at the inverting input node. Therefore, the input ignal voltage ource are not affected, which i deirable in voltage read-out.

14 96 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS i1 d OA o1 1 o c i d OA o 3 4 Figure 4.10, Intrumentation amplifier. The expreion for G dd, CM and F of the differential-to-differential voltage pre-amplifier can be derived via the tranfer function o1 / i1, o / i1, o1 / i and o / i. It i, again, intructive to identify the variou inverting amplifier and noninverting amplifier within thi circuit. The tranfer function o1 / i1 can be derived uing uperpoition ( i =0). The tranfer function can baically be derived directly from Fig Since i = 0, node 6, 7 i at (virtual) ground potential and thu OA1 and the aociated local feedback circuit i a non-inverting amplifier with gain: o1 / i1 = ( 5 6 )/ 6. A imilar argument applie to the tranfer function o / i1. OA1 enure that i1 i (virtually) reproduced at node 5, 6. With repect to thi node OA and the aociated local feedback circuit are an inverting amplifier with a gain of: o / i1 = - 7 / 6. The tranfer function o / i i derived imilar to the approach ued for o1 / i1 and o1 / i i derived imilar to o / i1, by exchanging the role of OA1 and OA: o / i = ( 7 6 )/ 6 and o1 / i = - 5 / 6. The differential-mode gain i calculated uing G dd = ( o1 - o )/( i1 - i ), the differential-to-common gain uing G cd = ( o1 - o )/[( i1 i )/] and the commonmode gain uing G cc = ( o1 o )/( i1 i ). Finally, the common-mode rejection and dicrimination factor of thi pre-amplifier reult from: H 1 = G dd /G cd and F 1 = G dd /G cc, repectively.

15 Section 4.3 OPAMP circuit for differential readout 97 Combining the four tranfer function reult in o1 and o expreed a: o1 = i1 i o = i1 i 6 6 The differential and common-mode component of the output ignal reult in: o1 o = i1 i = d = G G G =, G = dd d cd c dd cd 6 o1 o d = i1 i = c = G G G G = 1 cc c dc d cc c cc (4-19) (4-0) Since the finite CM i an iue related to the differential output voltage when d «c, the effect of G dc can be diregarded. ually 5 = 7, while 6 i ued to vary the differential gain, G dd. The CM and dicrimination factor of the preamplifier can be derived a: H 1 Gdd = = G 0 cd Gdd F1 = = G cc 6 6 (4-1) Equation (4-1) indicate that the pre-amplifier ha a favourable effect on the CM. For deriving an expreion for the overall gain and CM, the cacaded ytem compoed of the differential voltage pre-amplifier and the 1-OPAMP differential amplifier ha to be conidered. c d d ic1 id1 H 1, F 1 od1 = id oc1 = ic H o Figure 4.11, Common-mode rejection in a cacaded ytem.

16 98 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS An expreion for the overall CM of uch a cacaded ytem a a function of H 1 and F 1 of the pre-amplifier and the common-mode rejection H of the 1- OPAMP differential amplifier i derived uing Fig. 4.10: A c ic o =Gd id Gc ic=gd id. ic =Gd id Ad H G cd1 ic1 id =Gdd1 id1gcd1 ic1=gdd1 id1 ic1 =Gdd 1 id1 Gdd1 H1 G =G =G =G cc1 ic1 ic cc1 ic1 dd1 ic1 dd1 Gdd1 F1 Inerting the expreion for id and ic into that of o yield: (4-) ic1 ic1 ic1 0=Gdd1 Gd id1 =Gd id1 H1 FH 1 Htot G d =Gdd1 Ad with : = Htot H1 FH 1 (4-3) In the intrumentation amplifier H 1, which yield: H tot = F 1 H H. Therefore, the poor common-mode performance of the 1-OPAMP differential amplifier i improved by the differential-mode ignal gain in the pre-amplifier. The CM of the intrumentation amplifier in term of the paive component follow from equation (4-1) and (4-3) a: = = H H FH FH δ tot Htot F1H (4-4) In ummary: The merit of the differential-to-differential voltage pre-amplifier i that it combine a large CM with a large dicrimination factor. The combined effect i an overall common-mode rejection of the intrumentation amplifier that i equal to the differential-mode gain in the preamplifier and the invere of the component mimatch in the 1-OPAMP differential amplifier (H tot = F 1 H ). It hould be noted that the operation of the intrumentation amplifier trongly relie on the floating input circuit. ecogniing that the inverting and non-inverting input of an OPAMP circuit in feedback are at the ame potential ( - - =0), lead to the concluion that the differential input voltage, id, i acro 6 and no current loop i available to ic, a hown in Fig. 4.1a. Thi iolation criterion i dicued in Section 4... Thi benefit i lot in the modified verion of the dif-

17 Section 4.3 OPAMP circuit for differential readout 99 ferential-to-differential input tage of the intrumentation amplifier, a hown in Fig. 4.1b, in which reitor 6 ha been plit into two equal part. i1 d OA1 o1 i1 d OA1 o1 c i d f f OA (a) o c i d / / f f OA (b) o Figure 4.1, Two different verion of the differential-to-differential voltage pre-amplifier. The modified input tage of the intrumentation amplifier in Fig. 4.1b i in fact compoed of two eparate non-inverting amplifier with the gain o1 / i1 = o / i = ( f /)/( /)= ( f )/. Therefore, the differential gain i equal to that of the baic intrumentation amplifier, a expreed in equation (4-19) for f = 5 = 7. However, the tranfer function i without the cro-coupling term o / i1 and o1 / i. The mimatch between the upper / and it lower counterpart hould be conidered in the CM performance analyi. When auming,upper = (1ε)/ and,lower = (1-ε)/, the reult i a tranfer function decribed by: (1 ε) f o1= i1 (1 ε) f (1 ε) o = i (1 ε) f(1 ε) (1 ε ) f(1 ε) (1 ε ) o1 o i1 i (1 ε ) (1 ε ) = (4-5) f 4 εf ( i1 i) ( i1 i) = Gddd Gcdc Hence G dd = ( f )/.and G cd = 4ε f /. The common-mode gain can be derived a:

18 100 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS o1 o f ε ε f ε ε i1 i ε ε (1 ) (1 ) (1 ) (1 ) = (1 ) (1 ) f ( i1 i) = G cc c which reult in G cc = ( f )/. Conequently, H and F can be expreed a: G ( f )/ f / dd G and dd H = = = F = = 1 G (4 ε )/ ε G cd f f cc Although the modified verion provide the ame differential-to-differential gain, the CM performance i ignificantly reduced, which i primarily due to the lo of iolation. The only remaining parameter for a high CM i ymmetry. However, mimatch i introduced by uing two component. Forcing a (virtual) potential acro the terminal of a enor could be important in ome application, uch a the read-out of an electro-chemical cell. However, modifying the baic intrumentation amplifier to the circuit hown in Fig. 4.1b to achieve thi i not a good approach. Example 4.3 Figure 4.13 how a -OPAMP differential amplifier. (4-6) (4-7) 1. Derive the expreion for the differential gain G dd of the -OPAMP differential amplifier and Dimenion the component in the circuit for a differential gain G= o /( i1 - i )= 0 and = 3 = 1 kω.. Derive the expreion for the CM when conidering practical tolerance in the reitor (aume 1 = 4 (1δ), = 3 (1ε), G= 0 and ε= -δ= 5%). The OPAMP can in a firt approximation be conidered ideal (no offet or bia and infinitely large CM). 1 4 i 3 o i1 Figure 4.13, -OPAMP differential amplifier.

19 Section 4.3 OPAMP circuit for differential readout 101 Solution: The tranfer function can be derived uing uperpoition: ( ) o= i1 i= i1 i For 4 = 1 3 the reult i: o = G( i1 - i ) with G= ( 3 4 )/ 3 = ( 1 )/ = 0. Hence, 1 = 4 = 19 kω.. The CM reult from G d and G c : ( 1 δ) ( 1 ε) o = G i1 i = 3 4 4( 1 δ ) ( ) ( ) δ ε δ ε i 1 i G 1 i1 i G = G d id G c ic G 1 δ G( 1 δ) Via H= G d /G c the reult i: H= [G(1δ)-δε]/[(δ-ε)]= ( )/ (0.)= 09.5 (= 46.4 db). Since there i no floating input, thi circuit doe not provide the benefit of the intrumentation amplifier ing the intrumentation amplifier in medical application (4-8) δ ε 3 δ ε G (4-9) i1 i i G i1 i i = ( 3 4)( 1 δ) G( 1 δ) ( ) ( ) ( ) ( ) In the ECG meaurement hown in Fig. 4.1b and preented in the introduction of thi chapter, the common-mode ignal i capacitively coupled to the input circuit, and reitor p1 and p are required for biaing purpoe. Thee reitor baically poil the floating input of the intrumentation amplifier, the conequence of which are dicued in thi ection. The tranfer function of the differential ECG ignal and the common-mode ignal to the non-inverted and inverted input i baically imilar to that of a differential-to-differential input tage of a read-out circuit and i expreed a (Fig. 4.1b): id p1 = ic p1 1 id = p - ic p (4-30)

20 10 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS Which yield: - = - - / id 1 p1p p1 p1 H= = - / ( ) ( ) p1 p p1 p id ic p1 1 p p1 1 p ic p1 p 1 (4-31) Introducing p1 = p p, p = p - p, 1 = and = -, while diregarding econd-order term ( ), yield: p p p p - p 1 H=. p p p - - p p p( p ) H= = p p p p (4-3) The expreion for the dicrimination factor, F, can be derived via ( - )/ a a function of ic and id uing a imilar approach: = - p1 p p1 p id ic p1 1 p p1 1 p p1 p p11 p id F= = =1 ( )/ p1 p ic p1 1 p (4-33) The combination of thi differential-to-differential input attenuator and intrumentation amplifier can be conidered a cacaded ytem, and the overall differential gain and CM can be calculated uing equation (4-3). Since the common-mode rejection of a practical intrumentation amplifier exceed 10 4 at low frequencie, it can be concluded that the common-mode rejection of the input reitive divider i a limiting factor to the overall commonmode rejection and not to the intrumentation amplifier (equation (4-3): 1/H t = 1/H 1 1/(F 1 H )= 1/H 1 1/H 1/H 1 ). Alo in thi cae the primary caue i the limited iolation combined with component mimatch.

21 Section 4.3 OPAMP circuit for differential readout 103 The CM of the input circuit, therefore, fully determine CM performance. A hown in equation (4-3), the CM i determined by the factor p / and component tolerance, p / p and /. The uncertaintie in contact reitivity to the human body are much larger than thoe in bia reitor. Therefore, / i uually the mot ignificant. The dicrimination factor of the input circuit doe not contribute to an increaed overall common-mode rejection (F 1 = 1). However, in thi particular cae the value of F 1 i inignificant due to H H 1. In an intrumentation amplifier typically CM(DC) > 80 db. Auming p =10 MΩ ± 1% and = 10 kω ± 0% yield: H t = [1/H 1 1/H ] 1 = [(0.010.) ] -1 = 36 (about 70 db), which confirm that the overall CM critically depend on the contact reitance between electrode and kin. Actually, thi value for the common-mode rejection i not good enough. The common-mode ignal i the capacitively coupled main voltage and depend trongly on the coupling capacitance, C c. Auming C c = 0.1 pf yield: ( ) c( p) ( ) ( ω c( p) ) jωcc ic p / ωc ic c p / = = m 1 jωc / m 1 C /4 (4-34) A main voltage m = 30 V rm at 50 Hz yield ic = 36 mv. Aume a differential gain G dd = 100 and a differential input ignal id = 10 µv uperimpoed on thi common-mode level. The reulting output ignal i decribed by: ic 36mV o = Gd id = µ V = 1mV 1.1 mv H t 36 (4-35) Conequently, the ignal-to-common-mode interference ratio i equal to: 0 log(1/1.1)= -1 db, which i clearly not acceptable in a practical application. It hould be noted that the ECG information content i contained in a frequency band between about DC and 100 Hz, thu including the main frequency. Frequency filtering, therefore, cannot be applied. The coupling capacitance hould be minimied. eproducible detection of the ECG ignal at a given C c, therefore, relie olely on the CM performance, which hould be maximized uing well-matched component, whenever poible, and a high value for p /. 6

22 104 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS 4.4 Frequency dependence of the CM of the intr. ampl. A remarkable and not-o-credible reult of equation (4-1) i the infinitely high common-mode rejection of the differential-to-differential input tage in the intrumentation amplifier, H 1. Thi concluion i the direct reult of equation (4-0), which yield G cd = 0. Thi rather unrealitic concluion thu reult from the implicit aumption made with repect to the opamp applied. It wa aumed that: 1. The common-mode impedance Z ic,. The effect of the finite and frequency-dependent value of the opamp CM can be diregarded and 3. The opamp open-loop gain i ideal: A(ω). The common-mode impedance i in parallel to any biaing reitor. The effect of aumption 1 i dicued in Section Frequency dependence of the CM of an OPAMP The open-loop gain of an OPAMP cannot be conidered infinitely large (aumption ). A practical OPAMP hould, therefore, be conidered a firt-order ytem with an open-loop gain decribed by: A(ω)= A o /(1jωτ v ), with A o a the DC gain and 1/τ v a the -3 db cut-off frequency, a hown by the olid line in Fig ually thi frequency-dependent behaviour i pecified in term of the DC gain, A o [db], and the unity-gain frequency, f T =A o /(πτ v ) [Hz]. Thi property directly implie that any tranfer function to be realied uing feedback hould have a frequency repone within thi open-loop modulu plot (unle reonance occur). A(ω) H(ω) [db] A o H o H(ω T ) 0 1/τ v 1/τr ω T =A o /τ v log ω Figure 4.14, Specification of CM in an opamp relative to the open-loop gain. Alo the CM of an OPAMP cannot be conidered infinitely large. A mentioned above, the major propertie that limit common-mode rejection are component mimatch and differential input iolation. Thi concluion alo applie to the internal circuit of the opamp. The mimatch between the tranitor that comprie the differential input tage and the finite output impedance of the biaing

23 Section 4.4 Frequency dependence of the CM of the intr. ampl. 105 current ource give rie to an opamp common-mode rejection H OA (ω), which can be decribed by a firt-order ytem: H OA (ω)= H o /(1jωτ r ) with the modulu tranfer function hown by the dahed line in Fig Typical opamp pecification are: 80 db < H o < 100 db and 50 Hz<(πτ r ) -1 < 500 Hz. In a well-deigned opamp the common-mode rejection i larger than unity for frequencie beyond unity-gain ω T = A o /τ v. Therefore, the rejection-bandwidth i larger than the gain bandwidth, while H o < A o. A a conequence the commonmode rejection at the unity gain i: CM(ω T )= 0 log [H OA (ω T )]= 0 log [H o /(1jω T τ r )] 0 log(h o /(ω T τ r ))= 0 log(h o τ v /(A o τ r )) > Effect of OPAMP A(ω) on the CM The open-loop gain and CM of a practical OPAMP ued in a the differentialto-differential pre-amplifier circuit, H 1 or H, are very large but not infinitely large (aumption and 3), which reult in two operational contraint: The CM at DC ha a finite value and The CM i frequency-dependent. Firtly, the conequence of thee practical contraint on the CM at very low frequencie (DC) i analyed, uing Fig oc /H 1 d,eq1 H OA1 =H 0A (1γ) OA1 o1 f f c oc /H d,eq OA H OA =H 0A (1-γ) o Figure 4.15, CM of the differential-to-differential voltage pre-amplifier in the cae of mimatch in the CM of the opamp. The effect of the opamp common-mode rejection on the common-mode rejection of the intrumentation amplifier depend on the nominal value of the opamp CM, H OA (ω), and the matching between the CM of the two OPAMP compriing the differential-to-differential pre-amplifier. Thee have to be conidered for a realitic etimate of the common-mode rejection at low frequencie

24 106 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS (ω< 1/τ v ). From the definition of the CM follow that the effect of a commonmode ignal at the output of an OPAMP, oc, can be repreented by an equivalent differential input ignal d,eq = oc /H. Figure 4.15 how that thi ignal i ubtracted from the differential ignal applied to the OPAMP. For OPAMP OA1 the effect i decribed by d,eq1 = oc / H 1. Similarly, the effect on OA i decribed by d,eq = c /H. Conider two OPAMP, OA1 and OA with CM that are not perfectly matched to the nominal value H OA : H OA1 = H OA (1γ) and H OA = H OA (1-γ). Since the circuit compoed of OA1 and OA i a differential amplifier, the overall equivalent differential input ignal repreenting the common-mode voltage i decribed by: d,eq = d,eq1 - d,eq, thu: c OA OA1 ( γ ) 1 1 HOA 1- HOA H eq = = = γ γ H H (4-36) Therefore, it i not the finite value of the CM of the OPAMP OA1 and OA in the pre-amplifier of the intrumentation amplifier that are limiting, but rather their CM mimatch. The CM of the differential-to-differential voltage pre-amplifier in an intrumentation amplifier, H 1, i determined by the matching in CM of the opamp ued. Secondly, the conequence of the practical open-loop gain and CM of OA1 and OA, a hown in Fig. 4.14, on the CM at higher frequencie i analyed uing Fig i1 d OA1 A(ω) o1 f f c i d A(ω) OA Figure 4.16, CM of the differential-to-differential voltage pre-amplifier in the cae of a frequency-dependent opamp open-loop gain. o

25 Section 4.4 Frequency dependence of the CM of the intr. ampl. 107 Note that only a feedback function with a modulu tranfer function that fit within the open-loop of the OPAMP ued can be realied. Thi contraint alo applie to G d and G c, which make the CM of the intrumentation amplifier frequency-dependent. Hence, the open-loop gain of the operational amplifier, A(ω)= A o /(1jωτ v ), ued in the pre-amplifier ha to be included in the calculation of both the common-mode rejection and the dicrimination factor of the intrumentation amplifier. Thi frequency dependence of the common-mode rejection of the intrumentation amplifier i derived by applying the ame routine a ued in the derivation of equation (4-0) through (4-), but in thi cae with the non-ideal opamp propertie. Hence, o = A(ω)( - - ), with A(ω)= A o /(1jωτ v ), rather than tating that = - (which wa baed on the aumption that A(ω) ). The following et of thee equation can be derived for the feedback circuit of OA1: ( ) 1 o1 1 a = ( ) ( ω )[ ] b = A = = o1 1 1 o1 1 i1 1 i1 A ( )[ ] ( ω ) o = A ω o ( c) = i, = i A ( ω ) where 1- denote the inverting input of OA1, etc. (4-37) Subtituting (b) and (c) with (a) yield: o o1 o1 i i1 o1 i1 A( ω) A( ω) A( ω) = Similarly for the feedback circuit of OA: (4-38) 1 o = o o1 o i i1 i o A( ω) A( ω) A( ω) = (4-39) Subtracting equation (4-39) from (4-38) with 5 = 7 = f yield for the differential gain G dd :

26 108 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS G dd = ( ω) ( ) ( ) ωτ ( ) o A f o f Ao = = 1 A( ω) 1 A j i1 i f 6 6 o f v 6 f (4-40) Adding equation (4-39) and (4-38) with 5 = 7 = f yield for the commonmode gain G cc : Ao o1 o A( ω) 1 Ao 1 Gcc = = = (4-41) i1 i 1 A( ω) τ v τ v 1 jω 1 jω 1 A A The dicrimination factor, F 1, i by definition the ratio between G dd and G cc and follow from equation (4-40) and (4-41) a: ( ω ) ( ω ) G 6 f 1 A dd F = = = Gcc f 6 1 A ( 6 f )( 1 Ao) τ v 1 jω τ v A 1 ( 1 ) ddo jω 6 Ao f A o Ao 6 f Addo 1 jωτ 1 jωτ v v ( 1 A ) Ao 6 o f o o (4-4) [db] A o G dd G ddo G cc 0 G cco 1/τ v A o /(G ddo τ v ) ω T =A o /τ v log ω Figure 4.17, Modulu plot of the differential gain, G dd (ω), and common-mode gain, G cc (ω), in the differential-to-differential voltage pre-amplifier with reference to the opamp open-loop gain, A(ω). Figure 4.17 how the modulu plot of both the differential gain, G dd, (line with long dahe) and the common-mode gain, G cc, (line with hort dahe) of the differential-to-differential pre-amplifier. The figure demontrate one of the main F1

27 Section 4.4 Frequency dependence of the CM of the intr. ampl. 109 problem. The dicrimination factor F 1 i hown a a olid grey line. At low frequencie a high value of F 1 i baed on a high value for G dd, while G cc = 1. However, the gain in a feedback circuit cannot exceed the open-loop gain, which implie that G cc = 1 up to the unity-gain frequency, ω T, while G dd i forced to follow the open-loop gain for frequencie where it would otherwie cro A(ω), which i at ω dd = A o /(τ v G dd ). Conequently, F 1 i equal to G ddo up to ω dd = A o / τ v G ddo. At larger frequencie F 1 reduce with frequency and remain contant and equal to F 1 = 1 for frequencie beyond ω T. The advantage of the differential-to-differential pre-amplifier to the overall CM of the intrumentation amplifier (H tot F 1 H ) i, therefore, limited for frequencie up to ω dd. Thi property need to be included in the ytem deign by retricting the differential ignal bandwidth to ω dd Intrumentation amplifier for high-cm bridge readout The intrumentation amplifier i very uitable for the read-out of reitive enor, uch a the train gauge, when included in a Wheattone bridge. Integrated circuit are available with all the component of the intrumentation amplifier included, with the exception of 6. Either two pin are made available to connect an external gain-etting reitor, or a limited number of reitor are included onchip, each with a pin for external gain election to one common pin. On-chip component matching i ufficient to enable the fabrication of intrumentation amplifier. The frequency dependence hown in Fig applie when CM > 100 db typically for frequencie up to 100 Hz and reduce at higher frequencie. exc o o - id OA o exc o - o ic OA 3 4 Figure 4.18, Intrumentation amplifier for the read-out of a Wheattone bridge with train gauge. The conideration regarding offet, a preented in Chapter 3, and thoe related to excitation ource balancing for reducing CM requirement, do apply. A a

28 110 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS conequence the balanced AC excitation with balanced excitation, a hown in Fig. 4.18, i to be ued for the bet poible detectivity. The next detection limiting factor i noie, which i dicued in the next chapter. 4.5 Exercie Figure 4.19 how an intrumentation amplifier. 10 kω 0 kω id 1 kω 1 kω 1 kω o 10 kω kω Figure 4.19, Intrumentation amplifier for a gain of What i the gain, o / id, in the cae of the component value hown in the figure? Solution:. G d o 1 4 4kΩ 1kΩ 0kΩ = = = = 50 1kΩ 10kΩ id 1 3 (4-43) exc o Intrumentation Amplifier o Figure 4.0, Intrumentation amplifier with offet and finite CM. Figure 4.0 how the application of an intrumentation amplifier for the readout of an impedance bridge.

29 Section 4.5 Exercie Calculate o for DC excitation with exc = 10 V, = 1.000, o = 0.5 mv and CM= 80 db. The gain of the intrumentation amplifier i et to G d = 1. Solution:. 4 ic o d exc o 4 4o H 4 10 = G = = 1.5mV (4-44) The circuit hown in Fig. 4.1 i ued a a differential voltage amplifier. The OPAMP can in a firt approximation be conidered ideal (no offet or bia, A(ω)= A o, and infinitely large value for the CM). i 1 g o 4 g1 3 i1 Figure 4.1, 3-OPAMP differential amplifier. 4.3 Derive an expreion for G= o /( i1 - i ) and dimenion the circuit for a differential gain G= 0 for = 4 = 7 = 10 kω (everal olution are poible). Aume at thi tage that g1 and g can be diregarded. Solution: o = ( ) ( ) i1 ( ) i = (4-45) ( 1 ) 3 i1 i For 1 = 10 kω, 3 = 5 kω and 5 = 6 = 1 kω the reult i: o = (10 kω/1 kω) (10 kω/5 kω)[(10 kω/10 kω) i1 -(1 kω/1 kω) (10 kω10 kω) 5 kω)/(10 kω 10 kω) i ]= 10 [ i1 - i ]= 0( i1 - i ).

30 11 Electronic Intrumentation.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFEENTIAL MEASEMENTS 4.4 I the circuit in Fig. 4.1 enitive to ignificant difference in the impedance g1 and g of the ignal ource? Explain your anwer. Solution: Ye, i1 i connected to the input of an inverting amplifier with input impedance Z ip = 3, wherea i i connected to the input of a non-inverting amplifier with a very high input impedance Z i (inverting input i driven to the ame -albeit virtual- potential). Hence, the difference in cale error due to ource loading would be ignificant. 4.5 Derive an expreion for the common-mode rejection (CM) of thi differential amplifier when uing the nominal component value calculated in problem 4.3 and conidering tolerance in the reitor (aume: g1 = g = 0, 1 = 3, 4 = (1δ), 5 = 6 (1ε), G= 0 and ε = δ «1). In a firt approximation the OPAMP can be conidered ideal (no offet or bia and an infinitely large CM). Solution: In the olution to 4.3: 1 = and 7 = G o = 1 ( δ) i1 i G( ( 1 δ) i1 ( 1 ε) i) 1 ε i1 i Wort cae : δ = ε G ( i1 i) δg = Gdid Gc ic H Gd G 1 = = = G δ G δ c (4-46) Thi circuit i outperformed by the intrumentation amplifier. Source loading i unbalanced. CM performance depend on component matching, which i due to the non-floating differential input circuit.