Reader et8017 electronic instrumentation, chapter4

666 views
617 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
666
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
23
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Reader et8017 electronic instrumentation, chapter4

  1. 1. 83Chapter 4DETECTION LIMIT INDIFFERENTIALMEASUREMENTS• Why differential measurements?• How is the quality of a differential measurement specified?• What is the detection limiting signal in a differential measurement?• How is the instrumentation amplifier used?4.1 IntroductionThe sub-systems discussed so far are composed of components or circuits with asingle-ended input and/or output. Information is transferred over two wires, butwith one at ground potential. One of the terminals of each signal source (voltageor current) is at ground potential. Moreover, one of the input terminals of a read-out circuit is at ground potential. As a consequence, the information content isdetermined by the absolute value of the signal at the other wire (the non-grounded wire that is interconnecting the source and read-out). Many practicalsignal sources, however, use the two wires actively and supply a differential sig-nal that is superimposed on a common signal. Such a signal source has a differ-ential output. Each of the two output terminals carries a signal and theinformation content is determined by the difference between these signals.A suitable circuit for the read-out of a differential source should be equippedwith a differential input (i.e. two active terminals, none directly forced to groundpotential). This read-out circuit should be designed to be sensitive only to thedifference between the signals at the input terminals (= the differential signal),and should be immune to the average value relative to ground potential (= thecommon-mode signal). A practical amplifier satisfies this requirement up to a
  2. 2. 84 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS certain extent, which results in an additive error and thus in a detection limit. This limited suppression of the common-mode signal is specified in the Com- mon-Mode Rejection Ratio (CMRR). Common-mode rejection in a measurement system is particularly important in medical instrumentation. Recording an electro-cardiogram (ECG) involves the positioning of two signal electrodes plus one reference electrode on the skin of a patient, as shown schematically and highly simplified in Fig. 4.1a. + Cc Instr. ampl. Uic Uid -- (a) Un ~ 220V / 50Hz Uid 2 Rs1 -- + + Uic Instr. Rs2 ampl. Zc + -- -- Uid (b) + 2 Rp2 Rp1 Un -- Figure 4.1, Application of the instrumentation amplifier for measuring the ECG: (a) set-up and (b) equivalent circuit of the input. The series resistance between the electrode and the skin is considerable and poorly-defined due to the skin conductivity and its dependence on the patient’s condition. This contact resistance can be reduced using conductive and moistur- ising cream before placement of the electrode. The read-out has a parallel resist- ance to ground for biasing purposes. As is demonstrated in Section 4.3.2, a very suitable differential amplifier is the instrumentation amplifier. The equivalent electrical circuit of the measurement set-up is shown in Fig. 4.1b. The signals in an ECG are differential bio-potentials of typically Uid= 10 µV which are superimposed on a common-mode signal (typically Uic> 1 mV, due to capacitive coupling of the mains voltage). The challenge in differential read-out design is to obtain a sufficient CMRR for reproducible detection of the differen- tial signal, despite the fact that the magnitude is two orders of magnitude lower than that of the common-mode signal.
  3. 3. Section 4.2 85 Detection limit due to finite common-mode rejection4.2 Detection limit due to finite common-mode rejection4.2.1 Differential-mode and common-mode sensitivityThe Wheatstone bridge shown in Fig. 4.2 is an example of a signal source thatsupplies a relatively small differential signal, UoWd, superimposed on a rela-tively large common-mode signal, UoWc. A fully balanced bridge results in azero value for the differential signal, UoWd= 0 (∆R= 0; see also Section 2.5.5).This differential signal is superimposed on a non-zero common signal UoWc=(U1+U2)/2= Uexc/2. A bridge imbalance yields: R + ∆R U1 = U exc 2R (4-1) R − ∆R U2 = U exc 2RThus: UoWd= (∆R/R)Uexc and UoWc= Uexc/2. UoWc UoWd R+∆R R-∆R + UoWd -- Uexc U1 U2 R-∆R U2 U1 R+∆R (a) (b)Figure 4.2, Bridge circuit with differential-mode and common-mode signalcomponents at the output.The circuit for the read-out of the differential signal is shown schematically inFig. 4.3a. The relevant specifications are the differential gain, Gd= Uo/Uid, andthe common-mode rejection ratio, CMRR= H. Uid Gd, H Uo Uid Gdd, H, F Uod Uic Uic Uoc (a) (b)Figure 4.3, Definition of common-mode rejection in the case of: (a) a single-ended output and (b) a differential output.The read-out should only be sensitive to Uid= UoWd (Uo= GdUid). However, apractical circuit also exhibits a non-zero sensitivity to Uic. Therefore, the outputvoltage, Uo, is more adequately described by:
  4. 4. 86 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS U o = G d U id + G cU ic (4-2) The unwanted second term is due to the limited CMRR. The CMRR is defined as the ratio between the sensitivity to the dif- ferential input signal, Uid, as expressed by the differential gain, Gd, and the sensitivity to the common input signal, Uic, as expressed by the common-mode gain, Gc. When using Fig. 4.3 this definition yields: Uo G U U  H = d = id =  ic  (4-3) Gc U o  U id  U o = const . U ic This expression indicates that CMRR can also be interpreted as the differential input signal, Uid, which is required to have the same effect on the output (i.e. Uo= constant) as any given common-mode signal, Uic. Note that the CMRR should be maximised to yield a minimum additive error, ε:  G   U  Uo = GdUid + GcUic = Gd Uid + c Uic  = Gd Uid + ic  = Gd (Uid + ε ) , (4-4)  Gd   H  with H= Uic/ε. 4.2.2 Maximising the CMRR The common-mode behaviour of an electronic measurement system is largely determined by the input circuit. A simplified schematic of the input stage of a differential read-out circuit is shown in Fig. 4.4. Obviously, the CMRR is max- imised if Uic is unable to cause a differential voltage, U+-U-. A qualitative inspection of the circuit reveals two opportunities to provide such an advantage: • No common-mode voltage would be generated across the differential input in case the Voltage across Rd1 due to Uic is equal to that generated across Rd2, so that the effects are cancelled out (Rd1= Rd2). • No common-mode current would flow in the case of an extremely high impedance between Uic and ground (Rc→ ∞). These qualitative considerations translate into two important design targets: • symmetry by using a fully balanced input circuit and • isolation using a floating input with respect to the common ground potential.
  5. 5. Section 4.2 87 Detection limit due to finite common-mode rejection Ui+ + Uid Rs Rd1 -- 2 Ucmi Uo + Uid Rs Rd2 G(Ui+-Ui-) -- 2 Rc + Ui- Uic --Figure 4.4, Input circuit of a differential amplifier with finite isolation fromground potential and unequal differential input resistors.These conclusions are confirmed by calculating the common-mode signal at theinput stage, Ucmi, which results in: U cmi = Rc U ic + ( R + R d 1 )( R s + R d 2 ) + s Rc 2 R s + Rd 1 + Rd 2  Rc ( Rs + Rd 2 ) Rc ( R s + Rd 1 )  (4-5)    Rc + R s + Rd 2 Rc + R s + Rd 1  U id -  Rc ( R s + Rd 2 ) Rc ( R s + Rd 1 )  2  R + R + R + R s + Rd 1 R + R + R + R s + Rd 2   c s d2 c s d1 The differential signal between the non-inverting input (U+) and inverting input(U-) can be derived as: U ic - U cmi U - U cmi Rd 1 + Rd 2 U+ -U− = Rd 1 - ic Rd 2 + U id (4-6) R s + Rd 1 Rs + Rd 2 Rd 1 + Rd 2 + 2RsAssuming: Rd1= Rd+∆Rd/2, Rd2= Rd-∆Rd/2 and Rc Rd yields: Rc Ucmi = U ic +  ∆R  ∆R   Rs +Rd + d  Rs +Rd − d  2  2  Rc +  2 ( Rs +Rd ) (4-7) ∆Rd Uid = Rs +Rd +Rs (1 + Rs / Rc + Rd / Rc ) 2Rc Uic ∆R U 2Rc Uic + d id ≈ 2Rc +Rs +Rd 2Rs +Rd 2Rc +Rs +RdNote that at the detection limit: Uic [∆Rd/(2Rs+Rd)]Uid.
  6. 6. 88 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS Hence: U + -U − ∆Rd  Rc  ≈  1-  Uic Rs + Rd  Rc + ( Rs + Rd ) / 2  (4-8) U + -U − Rd ≈ Uid Rs + Rd The common-mode rejection follows via Eqn. 4.1 as: U + -U − U id R 2R + Rs + Rd CMRR = H = = d × c (4-9) U + -U − ∆Rd Rs + Rd U ic Equation 4-9 confirms that two approaches are available for maximising com- mon-mode rejection. The first is by maximising the impedance between Ucmi and the common ground: Rc Rs+ Rd (i.e. high isolation). The second contribu- tion involves good matching between the input components: ∆Rd= 0 (i.e. sym- metry). A practical operational amplifier can be designed for (Rc/Rd)max to exceed 103. Moreover, matching could be better than 1%. Hence, the maximum achievable common-mode rejection can be in the 105-106 range (100-120 dB). The read-out circuit shown in Fig. 4.3a provides a single-ended output directly at the output of the first gain stage. A fully differential system offers a superior sup- pression of noise and interference. A gain stage with differential input and differ- ential output should, therefore, be used, as is shown in Fig. 4.3b. However, such a gain stage is only fully specified if both the differential and the common level of the output signal are specified in terms of the differential and common input signals. The definition of the CMRR specification of a differential input/output gain stage does not fundamentally differ from the one provided for the gain stage with differential input and single-ended output. The obvious change is that in a fully differential system the CMRR relates the differential output signal to the sig- nals supplied at the input terminals. Hence, Gcd (Gain common-input-to-differ- ential-output) is used instead of Gc. The common output level is generally considered to be determined by the com- mon input signal only (Uoc/Uid= 0). The common-mode signal transfer function is specified by the discrimination factor, F.
  7. 7. Section 4.2 89 Detection limit due to finite common-mode rejectionThe discrimination factor, F, is specified in terms of the gain for thecommon-mode input signal to common-mode output signal relative tothe differential gain: F= Gdd/Gcc.In equation: Uod Uod G U U  G U H = dd = id =  ic  F = dd = id (4-10) Gcd Uod  Uid U =const. Gcc Uoc od Uic UicThe terms CMRR and F are used for a fully-differential system throughout thisbook. Similar to the CMRR, a high value for F is also preferred.The fundamental difference between the CMRR (H) and F is that a finitevalue for the CMRR results in mixing of the differential and common inputsignals at the output and thus directly imposes a detection limit on the sys-tem. A finite value for F does not result in an additional differential outputsignal in the case of a non-zero common-mode input, but rather fails toblock the transfer of this common-mode input level to the differential out-put.Most importantly, in the case of a finite F, the differential and common-modesignals remain distinguishable and the detectivity is not yet impaired. A rela-tively high value for F is nevertheless very important, as the (partial) removal ofany common-mode signal reduces the requirements to be imposed on the CMRRof the next stage, which contributes to a high value of the overall CMRR of thesystem.It should be noted that the English literature generally uses only the common-mode gain, Gcc, and has not adopted the more convenient dimensionless nota-tion in terms of the discrimination factor, F.4.2.3 Practical aspects of the CMRR in bridge readoutThe limited CMRR in a differential amplifier has consequences for the Wheat-stone bridge; Fig. 4.5a shows the circuit. The differential and common-mode sig-nals are: ∆R U Uid = Uexc , Uic = exc (4-11) Ro 2
  8. 8. 90 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS + Ro+∆R Ro-∆R Ro+∆R Ro-∆R (1+δ)Uexc + 2 -- + + Ui Gd, H Ui G , H -- Uexc -- Uo -- d Uo + (1-δ)Uexc Ro-∆R Ro+∆R 2 Ro-∆R Ro+∆R -- (a) (b) Figure 4.5, Read-out of the full Wheatstone bridge using a differential amplifier with a finite CMRR: (a) single-ended excitation and (b) differential excitation. An expression results from the definition of CMRR for the minimum CMRR value required to detect of (∆R/Ro)det. The inaccuracy specification, ε, is included by requiring the output voltage due to Uid(min.) to be equal to 1/ε times that output voltage due to Uic: Uexc  Uic  1 2 1 Hmin =   = × = (4-12)  Uid U =const. ε  ∆R   ∆R  o   Uexc 2ε    Ro det  Ro det For (∆R/Ro)det.= 2×10-6 (which is the result of 1 µstrain when measured using a metal film strain gauge with kε= 2) and ε= 1%, the result is Hmin= 25×106 (= 148 dB), which is an extremely demanding specification. The reason for this high CMRR requirement is the presence of a large common-mode input signal, Uic. Reducing this value would reduce the required read-out specifications. This objective is achieved in a differential driving scheme, as shown in Fig. 4.5b. The differential input signal remains unchanged, Uid= ∆R/Ro, whereas the com- mon-mode signal is at ground potential, Uic= 0, in the case of perfectly balanced excitation voltages (δ= 0). Hence, a common-mode rejection H=1 would in prin- ciple be sufficient. The reason for this somewhat unrealistic outcome is the assumed perfect differential drive (δ= 0). In practice the two excitation voltage sources are not perfectly matched and any mismatch results in a common-mode input signal. A mismatch δ results in:
  9. 9. Section 4.2 91 Detection limit due to finite common-mode rejection U exc R o + ∆ R  ∆ R  U exc U i + = − (1 − δ ) + U exc =  δ +  2 2 Ro  Ro  2 U exc R o − ∆ R  ∆ R  U exc U i − = − (1 − δ ) + U exc =  δ −  (4-13) 2 2 Ro  Ro  2 ∆R U + U i− U U id = U i + − U i− = U exc , U ic = i + = δ exc Ro 2 2The required CMRR of the read-out results in: U  Uic  1 δ exc δ Hmin =   = × 2 = (4-14)  Uid Uo =C ε  ∆R  U  ∆R    exc 2ε    Ro det  Ro detand is proportional to the degree of mismatch between the two excitation voltagesources. Limiting the mismatch to δ= 1% is well feasible and results in Hmin=25×104 (108 dB), which is an acceptable specification. As a general rule it canbe stated that an increased symmetry (in this case balancing of excitationsources) helps in reducing CMRR requirements. Example 4.1 A Wheatstone bridge is imbalanced due to the resistance values shown in Fig. 4.6. Calculate the differential bridge output signal, Uid, the common- mode signal, Uic, and the required CMRR of the differential amplifier used, to ensure that Uic has the same influence on the amplifier output signal as Uid for versions (a) and (b). + 1002Ω 997Ω Uexc/2 997Ω + -- + 1002Ω + Ui H Uo Ui H Uo -- Uexc -- 996Ω -- + 996Ω 1001Ω Uexc/2 1001Ω -- (a) (b) Figure 4.6, Bridge circuit with differential-mode and common-mode signal levels at the output (Uexc= 10V). Solution version (a): Ui+ = (1001/1998)×10V and Ui- = (996/1998)×10 V. Uid = Ui+-Ui- = [(1001-996)/1998]×10 = 50/1998 = 25 mV.
  10. 10. 92 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS Uic = (Ui++Ui-) /2 = (1001+996)/3996 = 4.9975 V. Uo = Gd(Uid+Uic/H) → Hmin= Uic/Uid = 4.9975/0.025 = 199.9. Solution version (b): Ui+ = (1001/1998- 997/1998)×5V and Ui- = (996/1998- 1002/1998)×5 V. Uid = Ui+-Ui-= [(4+6)/1998]5 = 50/1998 = 25 mV. Uic = (Ui++Ui-)/2 = (4- 6)/3996 = 2.5 mV. Uo = Gd(Uid+Uic/H) → Hmin = Uic/Uid <1. 4.2.4 Using the CMRR for noise reduction A differential sensor structure combined with a differential read-out system offers huge benefits in terms of the detection limit, as it enables the conversion of an interfering signal into a common-mode signal, which is subsequently sup- pressed using the CMRR of the differential read-out. Cc Zg + ui Zi ug um _ Figure 4.7, Capacitive injection of the mains voltage in a single-ended system. Figure 4.7 shows a single-ended circuit for the read-out of ug with capacitive coupling. The transfer function from the source to input, ui,g/ug, results in an expression for the signal at the input terminals: Ri  Rg  (4-15) ui , g = ug 1 −  ug Rg + Ri  Ri  Similarly, for Rg« Ri, the noise signal at the input terminals due to the mains volt- age, ui,m, is expressed as: Ri Rg jω Cc Z g // Z i Ri + Rg ui ,m = um = um jω Rg Cc um (4-16) 1 RR Z g // Z i + 1 + jω i g Cc jωCc Ri + Rg The Electro-Magnetic Interference (EMI-see Section 5.8) is directly coupled to the input and imposes an unacceptable detection limit.
  11. 11. Section 4.2 93 Detection limit due to finite common-mode rejection Cc2 Cc1 Zg1 + + Ug/2 -- + Ui Zi Um + _ Ug/2 Zg2 -- _Figure 4.8, Capacitive injection of the mains voltage on a differential input.Figure 4.8 shows the differential read-out of a differential voltage source, Ug,with a relatively large source impedance, Zg. Since also Zi is large enough to ena-ble voltage read-out without source loading, the mains voltage, Um, is coupled tothe signal wires. However, for fully balanced source impedances, differentialread-out and equal coupling capacitances (Zg1= Zg2 and Cc1= Cc2), the injectednoise voltages are equal and thus only contribute to the common-mode signal. Inthe case of a sufficiently high CMRR, these are eliminated in the differential sig-nal.However, any differential signal due to unequal source impedances or couplingcapacitances can in principle not be distinguished from the source signal (unlessfiltering or other signal conditioning is possible- see Chapter 6) and can affectthe detection limit. Example 4.2 The single-ended circuit in Fig. 4.7 uses the following component values: Zi= Ri = 1 MΩ and Cc = 1pF. Calculate the signal level, Ui,g and the level of the injected voltage, Ui,m, at the input of the readout where Zg = Rg = 50 kΩ and the mains voltage is the external source of error (Um= 230 V and fre- quency 50 Hz). Solution: Equation (4-15) yields: Ui,g = 0.95×Ug. Similarly, for Rg«Ri, Equation (4-16) yields Ui,m= 100π×50×103×10-12× 230= 3.6 mV. Assuming an inaccuracy specification at ε= 5% results in a detection limit at Ui,g(min.)= Ui,m/ε= 72 mV. Referring this voltage level to the input Ug yields: Ug(min.)= 72×10-3/0.95= 75.8 mV. Obviously, this is too high a detection limit.
  12. 12. 94 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS The differential system, as shown in Fig. 4.8 with CMRR= 80 dB (the effect of coupling to the output is reduced by a factor 104) yields Ug(min.)= (72×10-3×10-4)/0.95= 7.58 µV, which is much more acceptable. 4.3 OPAMP circuits for differential readout 4.3.1 The 1-OPAMP differential amplifier An operational amplifier (OPAMP) is equipped with a differential input, U+-U-, which should make the OPAMP suitable for the direct read-out of a floating sig- nal source. U1 R1 R2 -- Ud -- + 2 Uo -- + Ud + 2 R3 + Uc U2 -- R4 Figure 4.9, Elementary opamp differential amplifier. Figure 4.9 shows the basic differential amplifier. It is instructive to note that this circuit is a combination of an inverting amplifier, Uo/U1, and a non-inverting amplifier, Uo/U2. Recognising these topologies simplifies the calculation of the transfer function using superposition: R2 R  U   U 2 = 0; U o1 = - U1 = - 2  U c − d   R1 R1  2    R4 R + R2 R ( R + R2 )  Ud  U 1 = 0; U o 2 = . 1 U2 = 4 1 U c +  R3 + R 4 R1 ( R3 + R4 ) R1  2  (4-17) R R ( R + R2 ) U o = U o1 +U o 2 = - 2 U 1 + 4 1 U R1 ( R3 + R4 ) R1 2 R2 R For : R1 R4 = R2 R3 → U o = [U 2 - U 1 ] = 2 U d R1 R1 In the case of ideal OPAMP characteristics and a perfect resistor matching in pairs (R1R4= R2R3), a perfect differential amplifier results in a gain of: Gd= Uo/ (U2-U1)= Uo/Ud= R2/R1= R4/R3. However, this circuit shows three unfavourable properties:
  13. 13. Section 4.3 95 OPAMP circuits for differential readout• Practical resistors are matched within a certain tolerance.• The limitations due to the finite and frequency-dependent open-loop gain of the OPAMP, A(ω), do apply.• The two terminals of the floating signal source are unevenly loaded by the inputs of the differential amplifier. The non-inverting input (connected to U2) has a large input impedance, Zi1 = R3+ R4. The inverting input (connected to U1) a small input impedance, Zi2 = R1. Problems can be expected in the read- out of a differential source with relatively large source impedances.The most significant problem is the resistor mismatch. Assume the effect resistormismatch is expressed as: (R1+R2)/R2 = (1+δ)(R3+R4)/R4. In this case the trans-fer function of the differential amplifier is expressed as: R2 R ( R + R2 ) R R Uo = − U1 + 4 1 U 2 = − 2 U 1 + 2 (1 + δ )U 2 = R1 ( R3 + R4 ) R1 R1 R1  R2 δ  δ R2 U 1 + U 2 (U 1 + U 2 )  1 +  (U 2 − U 1 ) + = Add (U 2 − U 1 ) + Acd R1  2 R1 2 2 (4-18) R2  δ  δ Add 1 +  1 + →H = = R1  2 = 2 ≈ 1 Acd R2 δ δ δ R1Equation (4-18) confirms that the CMRR becomes infinitely large for perfectlymatched resistors. However, it also demonstrates that the CMRR deteriorateswith increasing component tolerance. Assume δ= 5% limits the CMRR to:Hmax= 1/0.05= 20. This is insufficient to detect a differential input signal morethan two orders of magnitude smaller than the common-mode level, which is thecase in many applications. Therefore, a more versatile circuit, the instrumenta-tion amplifier, is used.4.3.2 The 3-OPAMP instrumentation amplifierCombining a special 2-OPAMP differential-to-differential voltage pre-amplifierwith the 1-OPAMP differential amplifier shown in Fig. 4.9, yields the instrumen-tation amplifier shown in Fig. 4.10. This circuit features a superior CMRR per-formance.Both inputs, Ui1 and Ui2, are connected to the non-inverting input of an opampwith the output fed back to the inverting input. The input impedance is extremelyhigh due to the local feedback, which attempts to reproduce the input voltage atthe inverting input node. Therefore, the input signal voltage sources are notaffected, which is desirable in voltage read-out.
  14. 14. 96 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS Ui1 + Uo1 R1 R2 -- OA1 Ud -- -- + 2 Uo R5 + R6 R7 -- Ud + 2 -- + R3 Uc OA2 -- + Uo2 Ui2 R4 Figure 4.10, Instrumentation amplifier. The expressions for Gdd, CMRR and F of the differential-to-differential voltage pre-amplifier can be derived via the transfer functions Uo1/Ui1, Uo2/Ui1, Uo1/Ui2 and Uo2/Ui2. It is, again, instructive to identify the various inverting amplifiers and non- inverting amplifiers within this circuit. The transfer function Uo1/Ui1 can be derived using superposition (Ui2=0). The transfer function can basically be derived directly from Fig. 4.10. Since Ui2= 0, node R6, R7 is at (virtual) ground potential and thus OA1 and the associated local feedback circuit is a non-invert- ing amplifier with gain: Uo1/Ui1= (R5+ R6)/R6. A similar argument applies to the transfer function Uo2/Ui1. OA1 ensures that Ui1 is (virtually) reproduced at node R5, R6. With respect to this node OA2 and the associated local feedback circuit are an inverting amplifier with a gain of: Uo2/Ui1= -R7/R6. The transfer function Uo2/Ui2 is derived similar to the approach used for Uo1/Ui1 and Uo1/Ui2 is derived similar to Uo2/Ui1, by exchanging the roles of OA1 and OA2: Uo2/Ui2= (R7+ R6)/R6 and Uo1/Ui2= -R5/R6. The differential-mode gain is calculated using Gdd= (Uo1-Uo2)/(Ui1-Ui2), the dif- ferential-to-common gain using Gcd= (Uo1-Uo2)/[(Ui1+Ui2)/2] and the common- mode gain using Gcc= (Uo1+Uo2)/(Ui1+Ui2). Finally, the common-mode rejec- tion and discrimination factor of this pre-amplifier result from: H1= Gdd/Gcd and F1= Gdd/Gcc, respectively.
  15. 15. Section 4.3 97 OPAMP circuits for differential readoutCombining the four transfer functions results in Uo1 and Uo2 expressed as: R5 + R6 R U o1 = U i1 − 5 U i 2 R6 R6 (4-19) R R + R7 Uo2 = − 7 U i1 + 6 Ui 2 R6 R6The differential and common-mode components of the output signal result in: R5 + R6 + R7 R + R6 + R7 R + R6 + R7 U o1 − U o 2 = U i1 − 5 Ui2 = 5 Ud R6 R6 R6 R5 + R6 + R7 = Gdd U d + Gcd U c → Gdd = , Gcd = 0 R6 (4-20) U o1 + U o 2 R5 + R6 − R7 R + R7 − R5 R − R7 U d = U i1 + 6 Ui2 = Uc + 5 2 2 R6 2 R6 R6 2 = GccU c + GdcU d GccU c → Gcc = 1Since the finite CMRR is an issue related to the differential output voltage whenUd«Uc, the effect of Gdc can be disregarded. Usually R5= R7, while R6 is used tovary the differential gain, Gdd. The CMRR and discrimination factor of the pre-amplifier can be derived as: Gdd R5 + R6 + R7 H1 = = →∞ Gcd 0 × R6 (4-21) G R + R6 + R7 F1 = dd = 5 Gcc R6Equation (4-21) indicates that the pre-amplifier has a favourable effect on theCMRR. For deriving an expression for the overall gain and CMRR, the cascadedsystem composed of the differential voltage pre-amplifier and the 1-OPAMP dif-ferential amplifier has to be considered. + Ud Uod1 = Uid2 -- 2 Uid1 + H1, F1 H2 Uo Ud -- 2 + Uc Uic1 Uoc1=Uic2 --Figure 4.11, Common-mode rejection in a cascaded system.
  16. 16. 98 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS An expression for the overall CMRR of such a cascaded system as a function of H1 and F1 of the pre-amplifier and the common-mode rejection H2 of the 1- OPAMP differential amplifier is derived using Fig. 4.10:  A   U  U o = Gd 2U id 2 +Gc 2U ic 2 = Gd 2  U id 2 + c 2 .U ic 2  = Gd 2  U id 2 + ic 2   Ad 2   H2   G   U  U id 2 = Gdd 1U id 1 +Gcd 1U ic1 = Gdd 1  U id 1 + cd 1 U ic1  =Gdd 1  U id 1 + ic1  (4-22)  Gdd 1   H1  G U U ic 2 = Gcc1 U ic1 = Gdd 1 cc1 U ic1 = Gdd 1 × ic1 Gdd 1 F1 Inserting the expressions for Uid2 and Uic2 into that of Uo yields:  U U   U  U 0 = Gdd 1 × Gd 2  U id 1 + ic1 + ic1 = Gd  U id 1 + ic1   H 1 F1 H 2   H tot   Gd = Gdd 1 × Ad 2 (4-23)  with : 1 1 1 H = H + FH  tot 1 1 2 In the instrumentation amplifier H1→ ∞, which yields: Htot= F1H2 H2. There- fore, the poor common-mode performance of the 1-OPAMP differential ampli- fier is improved by the differential-mode signal gain in the pre-amplifier. The CMRR of the instrumentation amplifier in terms of the passive components fol- lows from equations (4-21) and (4-23) as: 1 1 1 1 R5 + R6 + R7 1 = + → H tot F1H 2 = × (4-24) H tot H 1 F1H 2 F1H 2 R6 δ In summary: The merit of the differential-to-differential voltage pre-ampli- fier is that it combines a large CMRR with a large discrimination factor. The combined effect is an overall common-mode rejection of the instrumen- tation amplifier that is equal to the differential-mode gain in the preampli- fier and the inverse of the component mismatch in the 1-OPAMP differential amplifier (Htot = F1×H2). It should be noted that the operation of the instrumentation amplifier strongly relies on the floating input circuit. Recognising that the inverting and non-invert- ing inputs of an OPAMP circuit in feedback are at the same potential (U+-U-=0), leads to the conclusion that the differential input voltage, Uid, is across R6 and no current loop is available to Uic, as shown in Fig. 4.12a. This isolation criterion is discussed in Section 4.2.2. This benefit is lost in the modified version of the dif-
  17. 17. Section 4.3 99 OPAMP circuits for differential readoutferential-to-differential input stage of the instrumentation amplifier, as shown inFig. 4.12b, in which resistor R6 has been split into two equal parts. Ui1 Ui1 + + -- OA1 -- OA1 Ud -- Uo1 Ud -- Uo1 + 2 + 2 Rf Rs/2 Rf Rs -- Rf Ud Rs/2 Rf -- + + 2 -- + Ud Uc OA2 Uc + 2 -- -- + Uo2 -- Ui2 OA2 + Uo2 Ui2 (a) (b)Figure 4.12, Two different versions of the differential-to-differential voltagepre-amplifier.The modified input stage of the instrumentation amplifier in Fig. 4.12b is in factcomposed of two separate non-inverting amplifiers with the gain Uo1/Ui1= Uo2/Ui2= (Rf+Rs/2)/(Rs/2)= (2Rf+Rs)/Rs. Therefore, the differential gain is equal tothat of the basic instrumentation amplifier, as expressed in equation (4-19) forRf= R5= R7. However, the transfer function is without the cross-coupling termsUo2/Ui1 and Uo1/Ui2.The mismatch between the upper Rs/2 and its lower counterpart should be con-sidered in the CMRR performance analysis. When assuming Rs,upper= Rs(1+ε)/2and Rs,lower= Rs(1-ε)/2, the result is a transfer function described by: 2R f + Rs (1 + ε )  Uo1 = Ui1  Rs (1 + ε )  → 2R f + Rs (1 − ε ) Uo 2 = Ui 2  Rs (1 − ε )   (4-25) 2R f (1 − ε ) + Rs (1 − ε 2 ) 2R f (1 + ε ) + Rs (1 − ε 2 ) Uo1 − Uo2 = Ui1 − Ui 2 ≈ Rs (1 − ε 2 ) Rs (1 − ε 2 )  2R f + Rs  4ε R f (Ui1 + Ui 2 )   (Ui1 − Ui 2 ) − = GddUd + GcdUc  Rs  Rs 2Hence Gdd= (2Rf+Rs)/Rs.and Gcd= 4εRf/Rs. The common-mode gain can bederived as:
  18. 18. 100 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS 2 Uo1 +Uo2 2Rf (1− ε ) + Rs (1− ε ) 2Rf (1+ ε ) + Rs (1− ε 2 ) = Ui1 + Ui2 ≈ 2 2Rs (1− ε 2 ) 2Rs (1− ε 2 ) (4-26)  2Rf + Rs  (Ui1 +Ui2 )   = GccUc  Rs  2 which results in Gcc= (2Rf+Rs)/Rs. Consequently, H and F can be expressed as: Gdd (2 R f + Rs ) / Rs R f + Rs / 2 G H= = = and F = dd = 1 (4-27) Gcd (4ε R f ) / Rs 2ε R f Gcc Although the modified version provides the same differential-to-differential gain, the CMRR performance is significantly reduced, which is primarily due to the loss of isolation. The only remaining parameter for a high CMRR is symme- try. However, mismatch is introduced by using two components. Forcing a (virtual) potential across the terminals of a sensor could be important in some applications, such as the read-out of an electro-chemical cell. However, modifying the basic instrumentation amplifier to the circuit shown in Fig. 4.12b to achieve this is not a good approach. Example 4.3 Figure 4.13 shows a 2-OPAMP differential amplifier. 1. Derive the expression for the differential gain Gdd of the 2-OPAMP dif- ferential amplifier and Dimension the components in the circuit for a differ- ential gain G= Uo/(Ui1-Ui2)= 20 and R2= R3= 1 kΩ. 2. Derive the expression for the CMRR when considering practical toler- ances in the resistors (assume R1= R4(1+δ), R2= R3(1+ε), G= 20 and ε= -δ= 5%). The OPAMP’s can in a first approximation be considered ideal (no offset or bias and infinitely large CMRR). R2 -- R1 R4 + R3 Ui2 -- + Uo Ui1 Figure 4.13, 2-OPAMP differential amplifier.
  19. 19. Section 4.3 101 OPAMP circuits for differential readout Solution: The transfer function can be derived using superposition: R3 + R4 R +R R R +R  R +R R  Uo = Ui1 + 1 2 ×( −) 4 Ui2 = 3 4 Ui1 − 1 2 × 4 Ui2  (4-28) R3 R1 R3 R3  R3 + R4 R1  For R2R4= R1R3 the result is: Uo= G(Ui1-Ui2) with G= (R3+R4)/R3= (R1+R2)/R2= 20. Hence, R1= R4= 19 kΩ. 2. The CMRR results from Gd and Gc:  R (1 + δ ) + R3 (1 + ε ) R4  Uo = G Ui1 − 4 × Ui2  =  R3 + R4 R4 (1 + δ )    (δ − ε ) R3     (δ − ε )   G Ui1 − Ui 2 +  Ui 2  G Ui1 − Ui 2 +  Ui 2  = (4-29)    ( R3 + R4 ) (1 + δ )       G (1 + δ )     (δ − ε )  U −U + G  (δ − ε )  Ui1 +Ui2 = G U + G U G 1 −  ( i1 i 2 )    2G (1 + δ )   G (1 + δ )  d id c ic 2 Via H= Gd/Gc the result is: H= [2G(1+δ)-δ+ε]/[2(δ-ε)]= (40+1.95-0.05)/ (0.2)= 209.5 (= 46.4 dB). Since there is no floating input, this circuit does not provide the benefits of the instrumentation amplifier.4.3.3 Using the instrumentation amplifier in medical applicationsIn the ECG measurement shown in Fig. 4.1b and presented in the introduction ofthis chapter, the common-mode signal is capacitively coupled to the input cir-cuit, and resistors Rp1 and Rp2 are required for biasing purposes. These resistorsbasically ‘spoil’ the floating input of the instrumentation amplifier, the conse-quences of which are discussed in this section.The transfer function of the differential ECG signal and the common-mode sig-nal to the non-inverted and inverted inputs is basically similar to that of a differ-ential-to-differential input stage of a read-out circuit and is expressed as (Fig.4.1b): R p1  U id  U+ =  U ic +  R p1 + Rs1  2  (4-30) Rp2  U id  U-=  U ic −  R p 2 + Rs 2  2 
  20. 20. 102 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS Which yields:  R p1 Rp2   R p1 R p 2  U id U + -U − = -  U ic +  +   R +R  R p 2 + Rs 2   R +R R p 2 + Rs 2  2  p1 s1  p1 s1  (4-31) H= (U + -U − ) / U id 1 2 R p1 R p 2 + R p1 Rs 2 + R p 2 Rs1 = × (U + -U − ) / U ic 2 R p1 Rs 2 − R p 2 Rs1 Introducing Rp1= Rp+∆Rp, Rp2= Rp-∆Rp, Rs1= Rs+∆Rs and Rs2= Rs-∆Rs, while disregarding second-order terms (∆R2), yields:  ∆R p   ∆ Rs   ∆R p   ∆ Rs  2Rp +  Rp + 2   Rs - 2  +  R p − 2   Rs + 2  1  2      H= . → 2  ∆R p   ∆ Rs   ∆R p   ∆ Rs   Rp +  Rs -  -  Rp −  Rs +   2   2   2  2  (4-32) R p + Rs R p ( R p + Rs ) Rs H= = Rs ∆ R p − R p ∆ Rs ∆ R p ∆ Rs − Rp Rs The expression for the discrimination factor, F, can be derived via (U++U-)/2 as a function of Uic and Uid using a similar approach:  R p1 Rp2   R p1 R p 2  U id U + +U − = + U ic +  −   R +R R p 2 + Rs 2   R +R R p 2 + Rs 2  2  p1 s1   p1 s1  R p1 Rp2 (4-33) U + -U − + U id R + Rs1 R p 2 + Rs 2 F= = p1 =1 (U + +U − ) / 2 R p1 + Rp2 U ic R p1 + Rs1 R p 2 + Rs 2 The combination of this differential-to-differential input attenuator and instru- mentation amplifier can be considered a cascaded system, and the overall differ- ential gain and CMRR can be calculated using equation (4-23). Since the common-mode rejection of a practical instrumentation amplifier exceeds 104 at low frequencies, it can be concluded that the common-mode rejection of the input resistive divider is a limiting factor to the overall common- mode rejection and not to the instrumentation amplifier (equation (4-23): 1/Ht= 1/H1+1/(F1H2)= 1/H1+1/H2∼1/H1). Also in this case the primary cause is the limited isolation combined with component mismatch.
  21. 21. Section 4.3 103 OPAMP circuits for differential readoutThe CMRR of the input circuit, therefore, fully determines CMRR performance.As shown in equation (4-32), the CMRR is determined by the factor Rp/Rs andcomponent tolerances, ∆Rp/Rp and ∆Rs/Rs. The uncertainties in contact resistiv-ity to the human body are much larger than those in bias resistors. Therefore,∆Rs/Rs is usually the most significant.The discrimination factor of the input circuit does not contribute to an increasedoverall common-mode rejection (F1= 1). However, in this particular case thevalue of F1 is insignificant due to H2 H1. In an instrumentation amplifier typi-cally CMRR(DC) > 80 dB.Assuming Rp=10 MΩ ± 1% and Rs= 10 kΩ ± 20% yields:Ht= [1/H1+1/H2]−1= [(0.01+0.2)×10-3 +10-4]-1= 3226 (about 70 dB), which con-firms that the overall CMRR critically depends on the contact resistance betweenelectrode and skin. Actually, this value for the common-mode rejection is notgood enough. The common-mode signal is the capacitively coupled mains volt-age and depends strongly on the coupling capacitance, Cc. Assuming Cc= 0.1 pFyields: Uic jωCc ( Rs + Rp ) /2 Uic ωCc ( Rs + Rp ) /2 → = = 157 ×10−6 1+ jωCc ( Rs + Rp ) /2 (4-34) Um Um ( ) 1+ ωCc ( Rs + Rp ) /4 2A mains voltage Um= 230 Vrms at 50 Hz yields Uic= 36 mV. Assume a differen-tial gain Gdd= 100 and a differential input signal Uid= 10 µV superimposed onthis common-mode level. The resulting output signal is described by:  U   36 mV  U o = G d  U id + ic  = 100  10 µ V +  = 1mV + 1.12 mV (4-35)  Ht   3226 Consequently, the signal-to-common-mode interference ratio is equal to: 20log(1/1.12)= -1 dB, which is clearly not acceptable in a practical application.It should be noted that the ECG information content is contained in a frequencyband between about DC and 100 Hz, thus including the mains frequency. Fre-quency filtering, therefore, cannot be applied. The coupling capacitance shouldbe minimised. Reproducible detection of the ECG signals at a given Cc, there-fore, relies solely on the CMRR performance, which should be maximized usingwell-matched components, whenever possible, and a high value for Rp/Rs.
  22. 22. 104 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS 4.4 Frequency dependence of the CMRR of the instr. ampl. A remarkable and not-so-credible result of equation (4-21) is the infinitely high common-mode rejection of the differential-to-differential input stage in the instrumentation amplifier, H1 → ∞. This conclusion is the direct result of equa- tion (4-20), which yields Gcd= 0. This rather unrealistic conclusion thus results from the implicit assumptions made with respect to the opamps applied. It was assumed that: 1. The common-mode impedance Zic → ∞, 2. The effect of the finite and frequency-dependent value of the opamp CMRR can be disregarded and 3. The opamp open-loop gain is ideal: A(ω)→ ∞. The common-mode impedance is in parallel to any biasing resistor. The effect of assumption 1 is discussed in Section 4.3.3. 4.4.1 Frequency dependence of the CMRR of an OPAMP The open-loop gain of an OPAMP cannot be considered infinitely large (assump- tion 2). A practical OPAMP should, therefore, be considered a first-order system with an open-loop gain described by: A(ω)= Ao/(1+jωτv), with Ao as the DC gain and 1/τv as the -3 dB cut-off frequency, as shown by the solid line in Fig. 4.14. Usually this frequency-dependent behaviour is specified in terms of the DC gain, Ao [dB], and the unity-gain frequency, fT=Ao/(2πτv) [Hz]. This property directly implies that any transfer function to be realised using feedback should have a fre- quency response within this open-loop modulus plot (unless resonance occurs). A(ω) H(ω) [dB] Ao Ho H(ωT) 0 1/τv 1/τr ωT=Ao/τv log ω Figure 4.14, Specification of CMRR in an opamp relative to the open-loop gain. Also the CMRR of an OPAMP cannot be considered infinitely large. As men- tioned above, the major properties that limit common-mode rejection are compo- nent mismatch and differential input isolation. This conclusion also applies to the internal circuit of the opamp. The mismatch between the transistors that com- prise the differential input stage and the finite output impedance of the biasing
  23. 23. Section 4.4 105 Frequency dependence of the CMRR of the instr. ampl.current source give rise to an opamp common-mode rejection HOA(ω), whichcan be described by a first-order system: HOA(ω)= Ho/(1+jωτr) with the modulustransfer function shown by the dashed line in Fig. 4.14. Typical opamp specifica-tions are: 80 dB < Ho< 100 dB and 50 Hz<(2πτr)-1< 500 Hz.In a well-designed opamp the common-mode rejection is larger than unity forfrequencies beyond unity-gain ωT= Ao/τv. Therefore, the rejection-bandwidth islarger than the gain bandwidth, while Ho< Ao. As a consequence the common-mode rejection at the unity gain is: CMRR(ωT)= 20 log [HOA(ωT)]= 20 log [Ho/(1+jωTτr)]∼ 20 log(Ho/(ωTτr))= 20 log(Hoτv/(Aoτr)) >1.4.4.2 Effect of OPAMP A(ω) on the CMRRThe open-loop gain and CMRR of a practical OPAMP used in a the differential-to-differential pre-amplifier circuit, H1 or H2, are very large but not infinitelylarge (assumptions 2 and 3), which results in two operational constraints:• The CMRR at DC has a finite value and• The CMRR is frequency-dependent.Firstly, the consequence of these practical constraints on the CMRR at very lowfrequencies (DC) is analysed, using Fig. 4.15. HOA1=H0A(1+γ) U1 + + Ud,eq1 OA1 Uo1 -- Uoc/H1 -- Rf Rs Rf -- Uoc/H2 + + -- Uc Ud,eq2 OA2 Uo2 -- + U2 HOA2=H0A(1-γ)Figure 4.15, CMRR of the differential-to-differential voltage pre-amplifier inthe case of mismatch in the CMRR of the opamps.The effect of the opamp common-mode rejection on the common-mode rejectionof the instrumentation amplifier depends on the nominal value of the opampCMRR, HOA(ω), and the matching between the CMRRs of the two OPAMPscomprising the differential-to-differential pre-amplifier. These have to be consid-ered for a realistic estimate of the common-mode rejection at low frequencies
  24. 24. 106 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS (ω< 1/τv). From the definition of the CMRR follows that the effect of a common- mode signal at the output of an OPAMP, Uoc, can be represented by an equiva- lent differential input signal Ud,eq= Uoc/H. Figure 4.15 shows that this signal is subtracted from the differential signal applied to the OPAMP. For OPAMP OA1 the effect is described by Ud,eq1= Uoc/ H1. Similarly, the effect on OA2 is described by Ud,eq2= Uc/H2. Consider two OPAMPs, OA1 and OA2 with CMRR’s that are not perfectly matched to the nominal value HOA: HOA1= HOA(1+γ) and HOA2= HOA(1-γ). Since the circuit composed of OA1 and OA2 is a differential amplifier, the overall equivalent dif- ferential input signal representing the common-mode voltage is described by: Ud,eq= Ud,eq1- Ud,eq2, thus: H eq = 1 = 1 = H OA 1 - γ 2 (H ≈ OA ) U1 -U 2 1 1 2γ 2γ (4-36) - Uc H OA2 H OA1 Therefore, it is not the finite values of the CMRR of the OPAMP’s OA1 and OA2 in the pre-amplifier of the instrumentation amplifier that are limiting, but rather their CMRR mismatch. The CMRR of the differential-to-differential voltage pre-amplifier in an instrumentation amplifier, H1, is determined by the matching in CMRR of the opamps used. Secondly, the consequence of the practical open-loop gain and CMRR of OA1 and OA2, as shown in Fig. 4.14, on the CMRR at higher frequencies is analysed using Fig. 4.16. Ui1 OA1 + -- A(ω) Uo1 Ud -- + 2 Rf Rs -- Rf Ud + + 2 -- Uc A(ω) Uo2 -- + Ui2 OA2 Figure 4.16, CMRR of the differential-to-differential voltage pre-amplifier in the case of a frequency-dependent opamp open-loop gain.
  25. 25. Section 4.4 107 Frequency dependence of the CMRR of the instr. ampl.Note that only a feedback function with a modulus transfer function that fitswithin the open-loop of the OPAMP used can be realised. This constraint alsoapplies to Gd and Gc, which makes the CMRR of the instrumentation amplifierfrequency-dependent. Hence, the open-loop gain of the operational amplifiers,A(ω)= Ao/(1+jωτv), used in the pre-amplifier has to be included in the calcula-tion of both the common-mode rejection and the discrimination factor of theinstrumentation amplifier.This frequency dependence of the common-mode rejection of the instrumenta-tion amplifier is derived by applying the same routine as used in the derivation ofequations (4-20) through (4-22), but in this case with the non-ideal opamp prop-erties. Hence, Uo= A(ω)(U+-U-), with A(ω)= Ao/(1+jωτv), rather than stating thatU+= U- (which was based on the assumption that A(ω)→ ∞). The following setof thee equations can be derived for the feedback circuit of OA1: U 2 − − U 1− U o 1 − U 1− (a ) + =0 R6 R5 ( b ) U o1 = A (ω ) [U 1+ − U 1− ]  U o1  → U 1− = U i 1 − (4-37) U 1+ = U i 1  A (ω ) U o 2 = A (ω ) [U 2 + − U 2 − ]  U o2 (c )  → U 2− = U i 2 − , U 2+ = U i 2  A (ω )where U1- denotes the inverting input of OA1, etc.Substituting (b) and (c) with (a) yields:  Uo2   U o1   U o1  U i 2 −  − U i1 −  U o1 − U i1 −   A (ω )   A (ω )   A (ω )  (4-38) + =0 R6 R5Similarly for the feedback circuit of OA2: U 2 − − U 1− U 2− − U o 2 + = 0→ R6 R7  Uo2   U o1   U o2  (4-39) U i 2 −  − U i1 −  U i 2 −  − Uo2  A (ω )   A (ω )   A (ω )  + =0 R6 R7Subtracting equation (4-39) from (4-38) with R5= R7= Rf yields for the differen-tial gain Gdd:
  26. 26. 108 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS Gdd = U o1 − U o 2 = A (ω )  R6 + 2 R f    = ( R6 + 2 R f ) Ao (4-40) U i1 − U i 2 2 R f + R6 1 + A (ω ) R 6 (1 + Ao ) + 2 R f + jωτ v ( R6 + 2 R f   ) Adding equations (4-39) and (4-38) with R5= R7= Rf yields for the common- mode gain Gcc: Ao U o1 + U o 2 A (ω ) 1 + Ao 1 Gcc = = = ≈ (4-41) U i1 + U i 2 1 + A (ω ) 1 + jω τ v τ 1 + jω v 1 + Ao Ao The discrimination factor, F1, is by definition the ratio between Gdd and Gcc and follows from equations (4-40) and (4-41) as: G dd  R6 + 2 R f  1 + A (ω )   = F= = G cc 2 R f + R6 1 + A (ω )    ( R6 + 2 R f ) (1 + Ao )   τ     1 + jω v  Addo  1 + jω τ v    (4-42)  R 6 (1 + Ao ) + 2 R f   Ao  Ao    ≈  R6 + 2 R f A 1 + jωτ v 1 + jωτ v ddo R 6 (1 + Ao ) + 2 R f Ao [dB] Ao Gddo Gdd Gcc Gcco F1 0 Ao/(Gddoτv) log ω 1/τv ωT=Ao/τv Figure 4.17, Modulus plot of the differential gain, Gdd(ω), and common-mode gain, Gcc(ω), in the differential-to-differential voltage pre-amplifier with reference to the opamp open-loop gain, A(ω). Figure 4.17 shows the modulus plots of both the differential gain, Gdd, (line with long dashes) and the common-mode gain, Gcc, (line with short dashes) of the dif- ferential-to-differential pre-amplifier. The figure demonstrates one of the main
  27. 27. Section 4.4 109 Frequency dependence of the CMRR of the instr. ampl.problems. The discrimination factor F1 is shown as a solid grey line. At low fre-quencies a high value of F1 is based on a high value for Gdd, while Gcc= 1.However, the gain in a feedback circuit cannot exceed the open-loop gain, whichimplies that Gcc= 1 up to the unity-gain frequency, ωT, while Gdd is forced to fol-low the open-loop gain for frequencies where it would otherwise cross A(ω),which is at ωdd= Ao/(τvGdd). Consequently, F1 is equal to Gddo up to ωdd= Ao/τvGddo. At larger frequencies F1 reduces with frequency and remains constantand equal to F1= 1 for frequencies beyond ωT.The advantage of the differential-to-differential pre-amplifier to the overallCMRR of the instrumentation amplifier (Htot∼ F1H2) is, therefore, limited forfrequencies up to ωdd. This property needs to be included in the system designby restricting the differential signal bandwidth to ωdd.4.4.3 Instrumentation amplifier for high-CMRR bridge readoutThe instrumentation amplifier is very suitable for the read-out of resistive sen-sors, such as the strain gauge, when included in a Wheatstone bridge. Integratedcircuits are available with all the components of the instrumentation amplifierincluded, with the exception of R6. Either two pins are made available to connectan external gain-setting resistor, or a limited number of resistors are included on-chip, each with a pin for external gain selection to one common pin. On-chipcomponent matching is sufficient to enable the fabrication of instrumentationamplifiers. The frequency dependence shown in Fig. 4.17 applies when CMRR >100 dB typically for frequencies up to 100 Hz and reduces at higher frequencies. + R1 R2 -- OA1 Ro+∆R Ro-∆R -- -- Uexc + 2 Uid R5 + Uo R6 R7 -- Uexc Ro-∆R Ro+∆R Uic -- 2 + R3 OA2 + R4Figure 4.18, Instrumentation amplifier for the read-out of a Wheatstone bridgewith strain gauges.The considerations regarding offset, as presented in Chapter 3, and those relatedto excitation source balancing for reducing CMRR requirements, do apply. As a
  28. 28. 110 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS consequence the balanced AC excitation with balanced excitation, as shown in Fig. 4.18, is to be used for the best possible detectivity. The next detection limit- ing factor is noise, which is discussed in the next chapter. 4.5 Exercises Figure 4.19 shows an instrumentation amplifier. + 10 kΩ 20 kΩ -- R3 R4 R2 + R1 -- 12 kΩ Uid 1 kΩ Uo -- 12 kΩ + R2 -- 10 kΩ R4 + R3 20 kΩ Figure 4.19, Instrumentation amplifier for a gain of 50. 4.1 What is the gain, Uo/Uid, in the case of the component values shown in the figure? Solution:. U o 2 R2 + R1 R4 24k Ω + 1k Ω 20 k Ω Gd = = × = × = 50 (4-43) U id R1 R3 1k Ω 10 k Ω Rs R Uos + -- + Instrumentation Uexc Amplifier -- Uo R R Figure 4.20, Instrumentation amplifier with offset and finite CMRR. Figure 4.20 shows the application of an instrumentation amplifier for the read- out of an impedance bridge.
  29. 29. Section 4.5 111 Exercises4.2 Calculate Uo for DC excitation with Uexc= 10 V, Rs= 1.0002×R, Uos=0.5 mV and CMRR= 80 dB. The gain of the instrumentation amplifier is set toGd= 1.Solution:.  ∆R U   2 ×10−4 5  Uo = Gd  Uexc + ic + Uos  = 1×  ×10 + 4 + 5 ×10−4  = 1.5 mV (4-44)  4Ro H   4 10 The circuit shown in Fig. 4.21 is used as a differential voltage amplifier. TheOPAMPs can in a first approximation be considered ideal (no offset or bias,A(ω)= Ao→ ∞, and infinitely large values for the CMRR). R2 -- R1 + R7 R5 Ui2 Rg2 -- R6 + Uo R4 -- Rg1 R3 + Ui1Figure 4.21, 3-OPAMP differential amplifier.4.3 Derive an expression for G= Uo/(Ui1-Ui2) and dimension the circuit for a dif-ferential gain G= 20 for R2= R4= R7= 10 kΩ (several solutions are possible).Assume at this stage that Rg1 and Rg2 can be disregarded.Solution: R4 R R + R2 R Uo = (− ) × ( − ) 7 U i1 + 1 × (− ) 7 U i2 = R3 R6 R1 R5 (4-45) R7 R 2  R 4 R6 ( R1 + R2 ) R3  ×  U i1 − × U i2  R6 R3  R 2 R5 R1 R2 For R1= 10 kΩ, R3= 5 kΩ and R5= R6= 1 kΩ the result is:Uo= (10 kΩ/1 kΩ)×(10 kΩ/5 kΩ)[(10 kΩ/10 kΩ)×Ui1-(1 kΩ/1 kΩ)×(10 kΩ+10kΩ)×5 kΩ)/(10 kΩ×10 kΩ)×Ui2]= 10×2[Ui1-Ui2]= 20(Ui1-Ui2).
  30. 30. 112 Electronic Instrumentation R.F. Wolffenbuttel Chapter 4: DETECTION LIMIT IN DIFFERENTIAL MEASUREMENTS 4.4 Is the circuit in Fig. 4.21 sensitive to significant differences in the imped- ances Rg1 and Rg2 of the signal sources? Explain your answer. Solution: Yes, Ui1 is connected to the input of an inverting amplifier with input impedance Zip= R3, whereas Ui2 is connected to the input of a non-inverting amplifier with a very high input impedance Zis→ ∞ (inverting input is driven to the same -albeit virtual- potential). Hence, the difference in scale error due to source loading would be significant. 4.5 Derive an expression for the common-mode rejection (CMRR) of this differ- ential amplifier when using the nominal component values calculated in problem 4.3 and considering tolerances in the resistors (assume: Rg1= Rg2= 0, R1= R3, R4= 2R2(1+δ), R5= R6(1+ε), G= 20 and |ε| = |δ|«1). In a first approximation the OPAMPs can be considered ideal (no offset or bias and an infinitely large CMRR). Solution: In the solution to 4.3: R1= R2 and R7= 10R6. G 2  Uo =  2 (1 + δ ) Ui1 − Ui 2  G ( (1 + δ ) Ui1 − (1 − ε ) Ui 2 ) 2 1+ ε  U + Ui 2 Worst − case :δ = ε → G (Ui1 − Ui 2 ) + 2δ G i1 = GdUid + GcUic → (4-46) 2 G G 1 H= d = = Gc 2δ G 2δ This circuit is outperformed by the instrumentation amplifier. Source loading is unbalanced. CMRR performance depends on component matching, which is due to the non-floating differential input circuit.

×