Microsoft Word - Article 10 - 81-488-1-Galley finale.docx ACTA IMEKO December 2013, Volume 2, Number 2, 56 – 60 www.imeko.org ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 56 The influence of source impedance on charge amplifiers Henrik Volkers, Thomas Bruns Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Section: RESEARCH PAPER Keywords: charge amplifier, calibration Citation: Henrik Volkers, Thomas Bruns, The influence of source impedance on charge amplifiers, Acta IMEKO, vol. 2, no. 2, article 10, December 2013, identifier: IMEKO-ACTA-02 (2013)-02-10 Editor: Paolo Carbone, University of Perugia Received February 15th, 2013; In final form November 13th, 2013; Published December 2013 Copyright: © 2013 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Funding:This work has received funding from the European Union on the basis of Decision No 912/2009/EC Corresponding author:Henrik Volkers, e-mail: henrik.volkers@ptb.de 1. INTRODUCTION Since its invention in 1950 by Kistler [1], the calibration of charge amplifiers for piezoelectric sensors is usually performed with a setup similar to Figure 1. The sensor is replaced by a voltage source generating ug and a standard capacitor of a well known value Cc. By assuming ui to be negligible, paralleled (cable) capacities Cp are ignored and the input charge is qc = qi = ug·Cc, resulting in the following formula for the complex transfer function: cg a uq )( Cu u S  (1) Calibrations at PTB and accredited laboratories of charge amplifiers performed with different standard capacitors Cc in a range from 10 pF to 2000 pF showed up significant systematic differences of Suq with increasing frequency. This led to the general conclusion that at higher frequencies, the burden voltage ui could no longer be ignored and the total source impedance of the sensor or calibration setup including cables as seen by the charge amplifier has to be regarded. While this effect is known for field measurements mainly caused by extended cable length capacity [2], the impact to laboratory calibrations had been overseen and has led to the following investigation. 2. MEASURED DEVIATIONS The sensitivity Suq of six laboratory grade charge amplifiers of different types were measured with varying source impedances. For these amplifiers, Figure 2 shows the relative amplitude and absolute phase deviation taking a calibration with a 100 pF standard capacitor as a reference and sourcing with an impedance of 1920 pF, quite well representing an Endevco Type 2270 transducer including the connecting cable. By using a 1000 pF standard capacitor for calibration, this systematic deviation still will be about halved. Figure 1. Schematic of a charge amplifier calibration setup. Cc Cp qc qp qi ug ui ua Hf(ω) Rfb A(ω) ABSTRACT This contribution discusses the influence of the source impedance on the complex sensitivity of a charge amplifier (CA). During calibration of a CA with varying source impedances, deviations at higher frequencies were observed, which if not properly taken into account may generate systematic errors beyond the limits of the measurement uncertainty budget. The contribution discusses a model to describe the effect as well as an extension to the CA calibration procedures which allow to quantify and correct the effect. ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 57 The sensitivity to the source impedance of a CA is mainly determined by the first amplifier stage, its feedback network and the surrounding input protection circuit. 3. MODELING THE CHARGE AMPLIFIER For further analyses the charge amplifier with its unknown circuit details is regarded as a black box with a complex, frequency dependent input impedance Zi(ω) and charge coupled base sensitivity S0(ω) as shown in Figure 3. S0(ω) can be interpreted as the transfer function of the charge amplifier when driven by an ideal charge source. The output voltage is ioa )( qSu   (2) and the input burden voltage iiiii )(j)( qZqZu    (3) for i j ii ˆ   teqq , Kirchhoff’s laws leads to pic qqq  , (4) cigc )( Cuuq  , (5) pip Cuq  , (6) with (5),(6) in (4): )( pcicgi CCuCuq  . (7) From the ”traditional” calibration equation (1) we know: )(uq a cg S u Cu  . (8) With the total source impedance Cs = Cc+Cp and including (2), (3) and (8) in (7) this results in si 0 uq j)(1 )( )( CZ S S      . (9) For an ideal charge amplifier with an input impedance of Zi = 0 Ω, the source impedance Cs would have no influence on the resulting sensitivity Suq. One important conclusion from (9) is that for different calibrations or sensor setups with equal total source impedance Cs, the resulting transfer functions Suq will be equal, too. For a given sensor cable impedance a charge amplifier calibration setup can be matched by adding an appropriate Cp for a given (smaller) calibration capacitor Cc. 4. COMPENSATION RESULTS To determine the characteristic values of the base sensitivity S0(ω) and input impedance Zi(ω) of a charge amplifier, formula (9) is rewritten in the form s 0 i 0uq )( )( j )( 1 )( 1 C S Z SS      , (10) which allows a linear complex fit for the measured Suq with varying Cs. Cs was composed of three different calibrated standard capacitors Cc1,2,3 of 10 pF, 100 pF and 1000 pF (GenRad Type 1404-A,B,C) and a variable capacitor GenRad Type 1422-D connected in parallel to the cable providing an adjustable capacitance Cp in the range of about 270 pF to 1400 pF. The resulting total source impedance was measured by shortcutting ug and measuring Cs at the connector to the amplifier using an HP4274A LCR meter. Suq was measured using a PXI-System with an NI-PXI 5422 16 bit signal generator applying sine signals and a two channel NI-PXI-5922 24 bit digitizer with an NI-PXI-5900 differential Figure 2. Systematic relative amplitude error and absolute phase error of the sensitivity Suq(ω) for different charge amplifiers if calibrated with CsCal = 100 pF+120 pF and sourced with an impedance of CsSen = 1920pF. The two lower graphs show the same data with different scaling. The legend indicates brand and type of the investigated CA, however the results represent individual devices and might not be generally representative for these types of CAs. Figure 3. Simplified charge amplifier model. -1,0% -0,5% 0,0% 0,5% 1,0% re la tiv e am p lit ud e d ev ia tio n 0,1 kHz 1 kHz 10 kHz 100 kHz -2,0 -1,5 -1,0 -0,5 0,0 BK2525 BK2650 EN2710 EN2775 PCB443 BK2635 Frequency P ha se d ev ia tio n in d eg re e -5,0% -3,0% -1,0% 1,0% 3,0% 5,0% re la tiv e am pl itu de d ev ia tio n 0kHz 10kHz 20kHz 30kHz 40kHz 50kHz -14,0 -12,0 -10,0 -8,0 -6,0 -4,0 -2,0 0,0 Frequency P h as e d ev ia tio n in d eg re e Cc Cp qc qp qi ug ui ua Zi qi·S0 ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 58 preamplifier to simultaneously capture ug and ua. Each measurement was taken twice with swapped input channels to cancel differences in the channel amplification and group delay. The major remaining uncertainties in this setup are the nonlinearities of the PXI-5922/5900 at a single frequency for ratio measurements u(ua/ug) < 60 ppm ([3],[4]), the uncertainty of the standard capacitors u(Cc1) < 50 ppm, u(Cc2,3) < 20 ppm and the charge amplifier noise. The overall expanded uncertainty is estimated to be less than 200 ppm. The uncertainty of the total source impedance measurement u(Cs) is less critical and the impact to the sensitivity uncertainty is about 2 orders smaller than u(Cc). An uncertainty of u(Cs) ≤ 0.5% is still sufficient for an U(Suq) ≤ 200ppm. For each frequency, the reciprocal measured complex Suq is split into a real and an imaginary part and a linear least square fit with the total source impedance Cs as the independent variable was applied. Figure 4 shows the reciprocal real and imaginary parts of Suq(ω) for one single amplifier. Each line represents 8 measurements at one frequency. The relative mean squared errors of the fits are smaller than 10-5 indicating the validity of the proposed model. Two transfer functions Suq(ω) were measured with nearly the same total source impedance of Cs=1300 pF, but one used the Cc1=10 pF and the second used the Cc3=1000 pF. Figure 5 shows the resulting Suq differences of the amplifier most sensitive to source impedance variations. The increase to 50 kHz indicates a slight mismatch of Cs of about 3 pF. For the BK2635 and PCB443 amplifiers, no differences larger than the standard deviation of the measurements (s ≤ 5·10-5) were observed and are another proof of equation (9). The amplitude and phase of the complex input impedance Zi(ω) in Ω for the CAs investigated are shown in Figure 6. While the ideal charge amplifier would have an input impedance of Zi = 0 Ω, the real-world amplifiers investigated have input impedances ranging from 45 Ω up to 500 Ω. Figure 5. Deviation of two calibrations with Cc1=10 pF and Cc3=1000 pF where the total source impedance is matched to Cs=1300(3) pF. Figure 6. Amplitude and phase of the complex input impedance Zi(ω) of six different charge amplifiers. Figure 7. Amplitude and phase deviations of Suq(ω) after compensation, all CAs investigated, 280 pF ≤ CS ≤ 2300 pF. Figure 4. The real and imaginary inverse of Suq(ω) for various source impedances Cs. 0,1 kHz 1 kHz 10 kHz 100 kHz -100 -50 0 50 100 150 200 250 Frequency re la tiv e de lta S in 1 e- 6 10 100 1000 |Z | i n O hm 0,1 kHz 1 kHz 10 kHz 100 kHz -80 -60 -40 -20 0 20 40 BK2525 BK2650 EN2710 EN2775 PCB443 BK2635 Frequency P ha se Z in d eg re e -150 -100 -50 0 50 100 150 re l a m pl itu de d ev ia tio n af te r co m pe ns at io n in 1 e- 6 0,1 kHz 1 kHz 10 kHz 100 kHz -0,04 -0,03 -0,02 -0,01 0,00 0,01 0,02 0,03 0,04 Frequency P ha se d ev ia tio n af te r co m pe ns at io n in d eg re e 0,09 0,092 0,094 0,096 0,098 0,1 R E {1 /S _u q} in p f/ m V 0 500 1000 1500 2000 2500 -0,01 0 0,01 0,02 0,03 0,04 0,05 0,06 100 Hz 160 Hz 330 Hz 560 Hz 820 Hz 1 k Hz 1,6 k Hz 3,3 k Hz 5,6 k Hz 8,2 k Hz 10 k Hz 11 k Hz 15 k Hz 20 k Hz 25 k Hz 30 k Hz 40 k Hz 50 k Hz C_s in pF IM {1 /S _u q} in p f/ m V ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 59 With S0(ω), Zi(ω) and the known source impedance, the influence of the source impedance to the transfer function Suq(ω) can be compensated by applying formula (9). Figure 7 shows the remaining deviations after compensation of all 6 CAs with source impedances from 280 pF to 2300 pF in a frequency range from 100 Hz to 50 kHz, 648 measurements in total. The values underlay the conservative estimation of a U(Suq(ω)) < 2·10-4. 5. IMPACT ON SENSOR CALIBRATION To validate the model approach of the charge amplifier for sensor calibration, we measured an Endevco 2270 BB accelerometer on a SE09 shaker with four different CAs. The acceleration was measured with two heterodyne interferometers. Acceleration, output voltage amplitude and phase were determined by applying a sine approximation method (method 3 in ISO 16063-11). The frequency ranged from 100 Hz to 40 kHz at an acceleration amplitude of â = 100 m/s². The accelerometer mounting and the interferometer alignments were only performed once at the beginning of the measurement campaign. The only variable was the exchange of the charge amplifiers. The four CAs were calibrated with source impedances built of a CC = 100 pF and four CP = {148, 806, 1585, 2143} pF to determine Zi and S0. The transfer functions Suq(ω) were then calculated according to formula (9) for a total source capacity of CSsen = 1740(2) pF measured at 10 kHz with 1 V excitation. The complex mean of all four measured sensor sensitivities Sqa1,2,3,4(ω) was taken as the reference and Figure 8 shows the resulting deviations. In addition, Figure 9 shows the systematic deviations that would occur without compensation and a ‘common’ CC = 100 pF charge amplifier calibration. At higher frequencies this systematic deviation, if not properly taken into account, may exceed the limits of the measurement uncertainty budget. Note the different scaling in the vertical axes. The uncertainties marked are determined as follows:     2222120 ² xxxxx suuukU   , (11) u0â = 3.62×10-4, u1â = 2.23×10-9, u2â = 2.39×10-13, u0P = 2.10×10-2, u1P = 3.18×10-7, u2P = 1.91×10-11, coverage factor k = 2 and the standard deviation sx of the measurement. 6. CONCLUSIONS The influence of source impedance on charge amplifiers can be measured and explained with high confidence by the model shown in Figure 3 and the methods proposed. By characterising the charge amplifier with its base sensitivity S0(ω) and input impedance Zi(ω) it is now possible to compensate this influence. For calibrations of charged-based sensors with lowest possible uncertainty, the charge amplifier as the key linking element to the data acquisition system is usually calibrated before and after the sensor calibration. To avoid systematic deviations, these charge amplifier calibrations should be performed with a source impedance matching the sensor impedance. Now, this procedure has become a common practice in our laboratory at PTB. ACKNOWLEDGEMENT The research leading to these results has received funding from the European Union on the basis of Decision No 912/2009/EC. Figure 8. Amplitude and phase deviations of Sqa(ω) after CA compensation, taking the compensated complex mean as reference. Figure 9. Amplitude and phase deviations of Sqa(ω) with a Cs=248 pF (Cc=100 pF) CA calibration without compensation, taking the compensated complex mean as reference. 100 1000 10000 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 S_qa_PCB S_qa_2635 S_qa_2525 S_qa_2650 +u_PCB +u_2635 +u_2525 +u_2650 -u_PCB -u_2635 -u_2525 -u_2650 Frequency in Hz P h as e de vi at io n in d eg re e -0,6% -0,4% -0,2% 0,0% 0,2% 0,4% 0,6% re l. am p lit u de d ev ia tio n 100 1000 10000 -12 -10 -8 -6 -4 -2 0 2 S_qa_PCB S_qa_2635 S_qa_2525 S_qa_2650 +u_PCB +u_2635 +u_2525 +u_2650 -u_PCB -u_2635 -u_2525 -u_2650 Frequency in Hz P ha se d ev ia tio n in d eg re e -4% -3% -2% -1% 0% 1% 2% re l. am pl itu de d ev ia tio n ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 60 REFERENCES [1] Schweizerische Eidgenossenschaft, Eidgenössisches Amt für Geistiges Eigentum, CH267431 (A) 1950-03-31, Messverstärker zur Messung elektrischer Ladungen, http://worldwide.espacenet.com/publicationDetails/biblio?FT =D&date=19500331&DB=EPODOC&locale=de_EP&CC=C H&NR=267431A&KC=A&ND=4. 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