Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. Determination of the Frequency Response of a Constant-Voltage Hot-Wire Anemometer Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Michael A. Kegerise* Eric F. Spina* Syracuse University Center for Hypersonics Syracuse University Syracuse, NY 13244 Abstract The dynamic response of the CVA system was investigated both analytically and experimentally. The frequency response functions of the CVA system for a number of different circuit parameters and operating conditions (Re, r) were determined via laser-based radiative heating of the hot-wire sensor. A 2nd-order linear systems model of the CVA was developed to provide insight to the dynamic response and to help interpret the experimental results. With the use of properly selected circuit components, a bandwidth in excess of 100 kHz can be achieved (« 350 kHz in this study). The qualitative variations in the frequency response function with changes in circuit parameters are in agreement with the 2nrf-order model. The frequency-response functions of the CVA systems used in this study were found to have little dependence on the operating conditions of the wire. ' 1. Introduction The constant-voltage hot-wire anemometer (CVA) is a new anemometry system that can achieve a high static sensitivity and large bandwidth (> 100fc/f z), making it suitable for high-speed boundary-layer stability and transition measurements1'2'3. In the CVA operating mode, the voltage across the probe network (sensor and cable) is maintained at a constant value by an op-amp circuit, and fluctuations in the wire resistance are sensed as fluctuations in the wire current, similar to the constant-temperature anemometer (CTA) system. Like the constant-current anemometer (CCA) system, the CVA operates in an open loop with respect to the wire sensor and therefore in-line (feed-for ward) compensation is required. * Research Assistant. Student Member AIAA. t Associate Professor. Member AIAA. 0 "Copyright ©1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved." Previous investigations of the CVA system have focused primarily on "proof of concept" type studies including a first look at signal-to-noise ratio (SNR) 2 , theoretical modeling of the dynamic response3, analysis of the static response in subsonic4 and supersonic5 flow and measurements of the static response in a supersonic flow5. Applications of the CVA system include experiments on hypersonic boundary-layer stability6. What is missing, though, is an experimental characterization of the CVA frequency response function. As with any dynamic measurement system, characterization of the frequency response function is needed to bound the uncertainty in the frequency spectra and correlation functions of the measured flow properties. Therefore, it was a priority to perform an experimental study to determine the frequency response function of the CVA system. Of particular interest was the dependence of the frequency response on various circuit parameters and sensor operating conditions (Re = Ud/v, T = (Tw — Tg)/Tg). In this experiment, the frequency response function of the CVA was determined via laserbased radiative heating. To provide additional insight to the dynamic response, a linear systems model of the CVA was developed. This analysis extends that of Sarma3 to include the effects of operating conditions on the system response. It should be noted that the current study is an independent investigation of the CVA system. While the CVA developers (Tao Systems, Inc.) provided access to several anemometers, the circuit details remained proprietary. In the next section, a theoretical analysis of the CVA system is presented. In section 3, the experimental setup is discussed, and the experimental results are given in section 4. The conclusions are presented in section 5. 2. Analysis of CVA System In this section, a model of the CVA system is presented and discussed. The classical method of analysis, which involves the linearization of the governing nonlinear system equations, is employed so that tools from classical control theory can be used to provide insight to the dynamic response of the system. Specifically, an expression for the transfer function of the CVA system is derived. This approach has been used in similar analyses of the CTA and CCA systems7'8'9'10. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 A schematic diagram of the canonical CVA circuit is shown in Figure 1. The input voltage to the circuit, £",-, is a constant. The cable and lead resistances of the hot-wire probe are lumped into the resistance RL. The capacitor, C, is connected to the center-tap position of the potentiometer resistance R. The parameter x represents the position of the center tap along the potentiometer, varying from 0 to 1. 2.1 Governing Equations filament in cross flow is given by11: dRw ' dt ' + YUn)- Ei = Rw(Rw-Rg)(X (6) where X, Y, and c are constants for a given wire, Prandtl number, and constant flow properties and Rg denotes the resistance of the wire at the gas temperature. 2.2 The Linearized Dynamic Equations The linearized dynamic equations describing perturbations in the circuit variables with changes in the input variables, z.e. ; flow velocity and gas temperature, are obtained by expressing the governing equations in a Taylor series about their static values1 and truncating after the linear terms. The resulting equations are given by: 1 The central component of the CVA circuit is the operational amplifier (op-amp). In this analysis, it is assumed that the input impedance of the op-amp is infinite and the output impedance is zero. The inverting terminal of the op-amp is at voltage E\ and the noninverting terminal is connected to ground. Assuming that the frequency dependence of the open-loop op-amp gain can be modeled as a simple pole, the equation governing the output voltage of the op-amp is: de0(t) ,.-. „ ,., ,_. (8) Ri ' (9) ew(t) = - RLiw(t), (10) and (1) dt where G0 denotes the zero-frequency gain and w/, is the open-loop bandwidth of the op-amp. Applying Kirchoff's laws to the model circuit yields several other system equations given by: R *R 5?.- - El (2) (3) dt where the lower-case letters denote perturbation quantities and the explicit time dependence has been added for clarity. Combining these relations and taking the Laplace transform yields: A?s + I and where 2 fCR x(l-x)d V R + Rd CR(l-x) , (11) (cRxtl -x} — ( } R + Rd( _ CR2x(l-x) 1- Tt+l R+Rd ' Rd RRd (5) In addition to the governing circuit equations, a model equation for the hot wire is needed. The instantaneous relation for the heat balance of a hot-wire 1 The static values are obtained from the governing equations by assuming that all perturbations are zero. •ou| 'souneuojisv pue sonneuojev jo ein^sui ueoueaiv 'Z66I. © wBuA Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. the gas temperature, the system transfer function for velocity fluctuations is given by: (R+Rd)R{ The linearized dynamic equation describing the response of the hot-wire to fluctuations in the flow velocity and gas temperature is given as (in Laplace transformed variables)12: -U(s) TWS Do DO = G<,BG 4- B$, KT, Di = G0B4 + B3 D2 = G0B2+Bl Ew(a), (22) , TWS + 1 TWS + (13) and Th = 1/w/i. The coefficients Bn, n = 1, 2, ... ,6, are given by: where Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Eo(s) U(s) (14) is _ 2 v w ____y / ~ (2^-^"""' Bi = rw [A2Ki (Ri + Ri) + flt-Ai (1 + <HL#I)] , £2 = rwRiAi(l + . (15) B3 = rw [RtAs (1 + RLKi) -h A3Kl (Ri + RI)] + and (16) The static sensitivities to velocity and gas temperature fluctuations are given by: 1-1 Ku= (2^-^^ (17) and —R In deriving this relation, it was assumed that the wire resistance varies with the wire temperature according to: w^Tr)]t B6 = Ri (1 + RL (K, - A'2)) . (18) An identical transfer function holds for gastemperature fluctuations, except that the static sensitivity to velocity, KU, is replaced by the gastemperature sensitivity, Krg- (19) where the subscript r denotes a reference condition and a is the first temperature coefficient of resistance. Completing the set of equations, the Laplace transform of Equations 7 and 10 yields: The large zero-frequency gain of the op-amp (G0 > 105) allows for the approximation of several terms in the system transfer function. Assuming typical values for the circuit components, the coefficients of the polynomial in the denominator of the system transfer function can be approximated as: (20) +1 and Ew(s) = /it - RLIw(s). (21) 2.3 The Transfer Function of the CVA Equations 12, 13, 20, and 21 constitute a system of four equations in four unknowns (E\, Iw, E0, Ew). The solution of these equations for the output voltage yields an expression for the overall system transfer function of the CVA. Assuming there are no perturbations in and These approximations are valid for a wide range of operating conditions except when the parameter x is very close to 0 or 1. However, these extreme values of x are not of interest in practice since they correspond to a frequency response that is dominated by the thermal inertia of the wire, resulting in a bandwidth that is less than 1 kHz. The time constant of thejvire is a function of the mean-flow conditions ({/, Tg, Rw) as observed in Equation 16. Therefore, if the mean-flow conditions under which the wire is operating change, the time constant of the compensation network must be readjusted to match that of the wire (just as in CCA operation). Failure to do so would result in an over- or under-compensated system, which produces an increase or attenuation in the gain and phase factors of the system frequencyresponse function at a frequency on the order of 1 kHz. Such deviations from a flat frequency response function will lead to significant errors in measured spectra and correlations, particularly when the energy-containing region of measured turbulence overlaps the frequency range in which the deviations occur. Making use of these approximations and assuming the cable resistance is negligible, the system transfer function can be written as: (23) U(s) where un and £ denote the natural frequency and damping ratio of the system and are given by: u2 = (24) and (25) Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 The function $ is given by: _ Rd Rx(l-x) Several conclusions concerning the dependence of the CVA frequency response on circuit parameters can be drawn from Equation 23. The natural frequency and damping ratio of the system are seen to increase as the square root of the gain-bandwidth product, G0Uh, of the op-amp. The bandwidth of a system is defined as the frequency at which the gain factor has decreased 3d£ from the zero-frequency gain13. For a 2nd-order system, the bandwidth is given by: RRd (RdRx + (Rx + + 2.4 Discussion of the CVA Dynamic Response The block diagram representation of the CVA transfer function is shown in Figure 2. In this figure, H(s) represents the response of the hot-wire sensor under const ant-volt age conditions, l/(rws + l), and Hc(s) is equal to RCx(\ — x)s + 1. The CVA system operates in an open loop with respect to the hot-wire sensor; i.e., there is no closed-loop feedback that automatically compensates for the finite thermal inertia of the wire as there is in the CTA system. Instead, in-line or feedforward compensation is provided by the RC network in the CVA circuit, represented by Hc(s). By adjusting the parameters in the RC network such that rw = rc = RCx(l - x), BW = wn [(1 - 2C2) 1/2 (27) Therefore an increase in the gain-bandwidth product of the op-amp tends to increase the system bandwidth. However, the corresponding increase in C tends to drive the bandwidth down, and in some cases, may outweigh the increase in bandwidth due to increasing u;n. The function ^ in the system transfer function has a weak dependence on the value of the damping resistance, Rd, since Rd is typically an order of magnitude larger than the potentiometer resistance, R. Therefore, the natural frequency of the system transfer function has a weak dependence on Rd- The damping ratio of the system, however, is nearly inversely proportional to the damping resistance and therefore the flatness of the gain factor can be controlled by Rd with little effect on the natural frequency. (26) the pole introduced by the finite thermal inertia of the wire is cancelled by the zero introduced by the RC network. In the current prototype design, this adjustment amounts to varying the center-tap position of the potentiometer (represented by the parameter x). Both the natural frequency and the damping ratio have a strong dependence on the value of the potentiometer resistance, /J, with un decreasing and C increasing as R increases. The zero-frequency gain of the CVA system is given by: The overall response of the model system will then behave as that of a 2nd-order system with natural frequency and damping ratio as given in Equations 24 and 25. With proper selection of the circuit parameters (G0Uh, Rd, R, C), a natural frequency in excess of 100 kHz can be achieved, as can a damping ratio of 0.707 (which corresponds to a maximally flat frequency response up to the -3dB point). In this case, the dynamic sensitivity (or gain factor) is constant up to the -3dJB point and equal to the static sensitivity. The static sensitivity can then be used with confidence to convert the raw anemometer voltage signal to velocity or gastemperature fluctuations. 5=0 RRd Ku « RKu Rd (28) since Rd > R. Thus, to increase the static sensitivity of the system, it is desirable to maximize the resistance R. However, the result of achieving a higher static sensitivity is a loss in bandwidth, since the changes in un and £ with increasing R both act to decrease the bandwidth. 4 Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 The natural frequency, damping ratio, and bandwidth of the system are also functions of the capacitance, C, and resistances Ri and RI. The natural frequency and damping ratio decrease and increase, respectively, as C is increased; the dependence on the values of the resistors is rather weak. This discussion indicates that a significant increase in the bandwidth of the system is not possible with the adjustment of a single circuit parameter. However, a two parameter adjustment can result in an increased bandwidth. The potentiometer resistance will be fixed in a design by the desired static sensitivity, leaving adjustments of the gain-bandwidth product and damping resistance to extend and optimize the frequency response. Increasing G0Uh leads to increases in both (jjn and £. The natural frequency is invariant to changes in the damping resistance, so Rj can then be increased to reduce ( such that a maximally-flat frequency-response function is achieved. Based on similar results, Sarma3 has suggested the use of a two-stage cascade amplifier. The open-loop gain of a multi-stage amplifier can be much larger than that of a single-stage op-amp, thereby resulting in a larger system bandwidth through an increased cj n , provided the damping resistance is adjusted accordingly to reduce <;. The frequency response function of the CVA system is also dependent on the mean operating conditionsJJI, Tg> Rw). Both cjn and C depend explicitly on Rw, with both values increasing as Rw increases. However, un and C also depend implicitly on the wire resistance through the parameter x, since x must be changed according to Equation 26 as Rw changes, to provide a properly compensated system. Figure 3 shows the compensated frequency-response functions for three different resistance ratios (Rw/Rg) and a single flow velocity2. An increase in the bandwidth with an increasing resistance ratio is noted, while the damping ratio is observed to be nearly constant. The natural frequency and damping ratio of the model system are also implicitly dependent on the velocity and gas temperature through x. Compensated frequency-response functions for three different flow velocities and a single wire operating resistance are shown in Figure 4. A clear increase in the bandwidth and decrease in the damping ratio of the system is observed with increasing velocity. For an increase in the gas temperature, un decreases and C increases, resulting in a lower system bandwidth. Previously, it has been assumed that the cable resistance is negligible. For a finite cable resistance, the 2 The model transfer function (Equation 22) was used in these calculations. Circuit component values typical of the current prototype design were used. The gain factors have been normalized by the zero-frequency gain. low frequency behavior of the CVA system is still governed by a simple pole-zero pair, however the time constant of the simple pole will be slightly modified. The qualitative variations in un and C with circuit parameters and operating conditions will remain as discussed above. 3. The Experiment 3.1 Previous Experimental Methods In the past, a variety of experimental techniques have been used to determine the frequency response function of hot-wire anemometer systems: mechanical shaking of the hot-wire probe in an airstream14, placement of the hot-wire in the Karman vortex street of a cylinder14'15, electronic signal injection, and radiative based heating methods16'17'18. The most prevalent method for frequency response determination and optimization in CCA and CTA systems has been electronic signal injection. However, there is currently no signal injection method available for the CVA system, although one is under development19. Therefore, an alternative to electronic signal injection was sought. In the current study, the experimental method of Bonnet and de Roquefort18, which involves laser-based radiative heating of the hot-wire filament, was adopted to characterize the frequency response function of the CVA system. In this method, the intensity of the incident laser beam is modulated sinusoidally over a range of frequencies with an external acousto-optic modulator (or Bragg cell). This method is particularly attractive since the input signal and anemometer output can be measured simultaneously to give both gain and phase factors of the frequency response function. Furthermore, modulation frequencies on the order of 1 MHz can be achieved, providing a very wide bandwidth for frequency response determination. In the next section, an analysis of the hot-wire filament subjected to a fluctuating radiant heat source is presented. The analysis provides the justification of laser-based heating for determination of the CVA frequency response function. 3.2 Justification for Laser-Heating Method Consider the heat-transfer balance for a hot-wire filament with a radiant heat-flux, F, applied to the wire,20. (Rw-R})(X —aw irdlF, dt (29) where aw is the absorptivity of the wire, d is the wire diameter and / is the active length of the wire. In de- Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. ui S4jns9J xnij-:vB9q 9q^ jo ptre su9j viva QOQ 9q4 qSnojqi} passed U9q4 S-BM 4U9uu qi UI 98^9 J09p 9q^ 9JIM-40q 9q4 IIIOJJ 04 4X9U pg-j-eooi podiJ4 t? uo p94unotu J9S-BJ ^U9ppui 9q4 pUB iosu9S 9JIM uopotn 9Ai^i2j9J XU-B ^U9A9jd 04 04 p9^unoui si2M J9pjoq 9qojd 9JiM-4o •X4ISU94UI 9ou9inqjn4 MOJ q^iM X4poj9A MOU unojiun -B s9ptAOJd 4^q4 49f jre oiuosqns U-BUIS « jo gu^jd 41x9 9q4 •qou9q S-BM Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 4"B P94T20OJ S-BM 9qOld 9JlM-4Oq 9qj^ "JOSU9S 9q4 uo urB9q 9q4 S9snooj ^q^ SU9| tutu 'JOJJTUI si q^ pire p9ddip ST p9SS-Bd ST UTB9q •B ui 9joq -B Xq 0Q6 uioij si -U9duioo X|J9dojd 9q !|o J9pun 9JIM 9q^ jo ^-Bq^ 04 ^9S si S9i{duii si *p90UnOUOJd SS9J SI S^U-B^SUOO 9UII4 UI 90U9J9JJIp 9q(4 'p9STB9JOUI SI X^pO|9A MO^ JO 90UB4SIS9J 9 JIM 9q^ Sy •Sui4-B9q J9S-BJ ^noq^iM ^six9 pjnoM qoiqM ^q^ J9AO 9JIM 9q'4 JO ^UB^SUOO 9UII4 9H^ 9S129JDUI O^ pU9 m { Xq jo q-4og ' y 9DU'B^sis9J 9JiM 9q^ 9s-B9Joui HIM ^U-BIp-BJ UB9UI 9q^ 'X^pOpA MOJJ pU"B 99-B^jOA 9JIM ^ joj 'uoi^ipp-B uj -^g uoi^nbg ui l°By S^9 9q^ 4*B 90UB4SIS9J 9JIM 9q^ UI 9S-B9JDUI 9Al^D9JJ9 UB SB °iii<B9q stq-4 p9ss9jdx9 si P9JJ9 siq JL ' (MOU 9q^ ui 9JIM p9^'B9qun U-B jo z) 9JIM 9q4 jo 9Jn4<BJ9dui94 uinuqi| I xng ^i29q ^u^ip'BJ UB9UI -B jo uopipp-B ui *S90U9J9JJIp ^U-B^JOdUJI I-BJ9A9S 9J^ 9J9q^ 'J9A9MOJJ J9IS-B9 SUI'B9q UIOJJ 9JIM 'B^S UI S9DU9J9JJIp 9q^ JOJ ^d90X9 9UI'BS JO 9q pjnoqs QC PU'B %\ suo^nbg <Xy«9pi 'suop-Bn; n^J9dui9'4 SB£ pu« X^popA O4'9suods9j 9q^ 04 U-B i U-B 04 p9ijdd^ U9q^ S-BM *"/ uoipunj ^ uioij 9A«M 9UTS y jo 9uo 'ipo 9q-4 uioaj p9iiBUiBUJ9 suu<B9q I9AIIp si suoi^-Bnpng xng-4*B9q ^UBip'Bj 04 9JiM 9q^ jo 9 ^ uo 9suods9j 9q4 4"Bq4 SMoqs gj uoi^ribg q^iM uosu-BduioQ si _ p9^snf * -p« SBM IJ90 SS^jg < 9q^ JO ^U9UI4SUfp B U-B JOJ P9MOJJ-B ^-Bq^ JOSS90Old B Xq p9piAoid ^SB^IOA {-Bpiosnuis zffffl S-BM 9D 99-Bi 9 ^ ui si 08 -B Xq U9Aiip X^IAI^ISUSS xny-^aq = "Zf q^ S9piA -oid i^q^ ||90 9S-Bjg ^ o^ui p9p9iip U9 UTB9q 9q^ p9S'B9JDUl 9ss^d S-BM 04 punoj S-BM^ ui IlnJ 9T1^ I9AO siq^^snfp-B 'pVJLtU JO I9MOd (£9) J° l'\ pU« UttU gg-Q JO 9DU9§J9A (38) ui-B9q -B q^iM J9S^i uoi-uo^iy A\ ui -ip pu-B M J M 9l JO 9Djnos : <B9 S^ 3 J f^ J > H 1U9UIUOJIAU9 9UI3JJOM 99JJ-UOl^iqiA "B (IE) 9J9M S^U9UOdlIJOD II V *S 9J^!J u! UMoqs si 9JiM-^oq 9q^ jo joj dn-^ £'£ -ui-BJ-Bd p pu-B suoi^ipuoo 9suods9j Xou9nb9jj VAO 4 p9sn J9pun CI 9J-B -9q JOJ ui s9§u<Bqo s U-BO B 9tirBs pu-B *T\J ( f f U (OS) >r '= = »3T I 9J9qM I -1- smj, 1 - >I + smj, 4 ui suoi^npng 04 -Xp 9q4 joj uoiss9jdx9 9 4 uo 9suods9J -B SUIUIJOJJ9J 'JOSU9S y-B uuojiun 9q 04 xnu -^j 9q4 pu-B p9409|5?9U U99q 9A^q 9JIM 9q4 uiojj pu-e uoi4onpuoo pu9 jo S409JJ9 9q^ 'uotss9Jdx9 Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 redirected at 90° toward a 250 mm lens. This lens focuses the scattered light on a Mamamatsu Corp. photomultiplier tube, which provides a measure of the radiative heating to the hot-wire. The photomultiplier signal was measured during the experimental set-up to monitor the quality of the system perturbation, with particular attention to two requirements: that the intensity modulation of the radiative heating input is in the form of a sine wave, and that the sinusoidal heating input has an amplitude that is independent of frequency. A standard deviation in the RMS of the photomultiplier signal of only 0.1 dB (re RMS 100 Hz) was observed over the frequency range tested (100 Hz to 500 kHz). To determine the frequency response function (gain and phase factors) of the CVA, the modulation frequency, fm, was swept over a range of discrete values from 100 Hz to a frequency where the anemometer RMS output voltage had decreased 20 dB from the value at 100 Hz. At each value of the frequency / m , the photomultiplier and CVA output voltage signals were measured simultaneously. The gain factor was determined by calculating the ratio of the CVA signal RMS to the photomultiplier RMS. The phase factor was determined by calculating the cross-spectral density between the CVA and photomultiplier signals. Three different CVA units, provided by Tao Systems, were used in the current study. Unit 1 had a single-stage op-amp and a variable compensation setting (adjustment of x). This unit was used to investigate the dependence of the frequency response on G0Uh, Rd, and operating conditions (Re, T). Units 2 and 3 have two op-amps in cascade and differed only in the value of the potentiometer resistance. The compensation setting in 'both units was fixed at the same value. These units were used to investigate the effects of potentiometer resistance on the frequency response as well as to demonstrate that the CVA can achieve high bandwidths with multi-stage op-amps. The frequency response of a CTA system was also measured via laserbased heating to provide a benchmark to which to compare the CVA system. The CTA measurements were made with a Dantec 55M12 symmetric bridge with 50 0 top resistors. The hot-wire probe design used in this study was identical to that of Spina & McGinley21. Copper plated 5 /ira tungsten wires were used with the active portion etched to a length greater than 1.0 mm (l/d > 200). For the unit 1 CVA and CTA measurements, the anemometer and photomultiplier signals were sampled simultaneously with a NefF System 490 A/D with 12-bit resolution over 640 mV. Both signals were first high-pass filtered at 20 Hz and low-pass filtered at 100 kHz. The pairs of low-pass and high-pass filters were checked to ensure that no phase differences between the two signals were introduced. CVA units 2 and 3 had bandwidths in excess of 100 kHz, requiring an A/D with a higher sampling rate. Therefore, a Lecroy 9310AM digital oscilloscope with 8-bit resolution over 400 m V was used to measure the frequency response of these units. 4. Experimental Results In this section, the experimentally determined frequency response functions of the CVA system are presented. Curve fits to the experimental data in the form of the model transfer function (Equation 23) were performed to determine whether the model adequately describes the system. In view of Equation 23, a curve fit to the experimental data of the form: H(s) = (36) should properly characterize the system when the values of TC, rw, cj n , and £ are chosen to give the best curve fit. Note that all gain factors presented in this section have been normalized by the gain at a frequency of 100 Hz. 4.1 Effect of Compensation Setting Figure 6 presents experimental data from CVA unit 1 in three different states: properly compensated (rw = rc), over compensated (rw < rc), and under compensated (rw > rc). To achieve proper compensation of the system, the RMS output voltage at frequencies above (1 kHz) and below (100 Hz) the corner frequency of the wire were compared. The center-tap position of the capacitor was then varied until the two values were equal. For the current experimental conditions, the wire corner frequency was about 675 Hz. For the over- and under-compensated cases, the corner frequency of the compensator was set at 454 Hz and 1136 Hz respectively. The success of the curve fits shown in Figure 6 suggest that the CVA system can be characterized by a simple pole-zero pair at low frequencies as indicated in the model transfer function. The compensator time constants as determined from the curve fits were found to be within 6% of the actual values. The disagreement between experimental data and the curve fits at higher frequencies is most likely due to the presence of higherorder poles in the system. The curve fit is based on the assumption that the system is of 2nd-order and therefore cannot account for the additional roll-off in the phase factor past 180°. 4.2 Effect of Gain-Bandwidth Product here is representative of both data sets. Figure 7 presents the frequency-response function of CVA unit 1 for two different gain-bandwidth products. All other circuit component values and the operating conditions remained the same. The values of un and C as obtained from the curve fits increase with increasing gain-bandwidth product. These results are in qualitative agreement with the model transfer function of Equation 23. The gain and phase factors exhibit only a weak dependence on the wire Reynolds number (see Figure 10). This is in contrast to the calculations presented in Figure 4 of § 2.4, which show a clear increase in the natural frequency and decrease in the damping ratio with increasing Re. In Figure 11, the gain factor of the system is also observed to have a weak dependence on the overheat ratio, although the phase factor does exhibit a consistent increase in phase lag with decreasing overheat ratio, particularly at higher frequencies. Again, this behavior is in contrast to the calculations of § 2.4 that suggest a significant decrease in the natural frequency with decreasing overheat ratio. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 4.3 Effect of Damping Resistance Measurements of the frequency-response function of CVA unit 1 for several different damping resistances are shown in Figure 8. As before, all other circuit component values and the operating conditions remained the same. The decrease in the system damping ratio with increasing damping resistance is in qualitative agreement with the theory. The curve-fit values of ujn also indicate a near invariance with damping resistance; a characteristic that was observed in the model transfer function. To allay fears that the experimental method was causing this discrepancy, measurements of the CTA frequency response function were also made via laser-based radiative heating to ensure that the characteristics of the frequency response are captured properly by this method. A great deal of theoretical and experimental work has been focused on the characterization of the CTA frequency response, and the changes in the frequency response with operating conditions are well documented. 4.4 Effect of Potentiometer Resistance The frequency-response functions of CVA units 2 and 3 were measured at the same flow conditions (Re = 16, r = 0.43), with the only difference being the value of the potentiometer resistance, R (Figure 9). Both systems had a fixed compensation setting, resulting in the slight roll-off in the gain and phase factors that occur at a frequency of w 1 kHz. It appears that an increase in the potentiometer resistance causes a decrease in natural frequency and an increase in the The frequency response function of the CTA system is presented in Figure 12 for overheat ratios ranging from 0.28 to 0.63. Data are presented in Figure 13 for wire Reynolds number ranging from 8.0 to 16.0. For both data sets, the system parameters were adjusted to optimize the anemometer response (via square-wave injection) at the highest overheat ratio or wire Reynolds number. According to Wood9, a well-timed CTA sys- tem behaves as a 2nd-order system and therefore the curve fits to the experimental data in Figures 12 and 13 are of the form: damping ratio, resulting in a decrease in the system bandwidth. For an increase in # by a factor of four, the bandwidth'of the system decreases by nearly the same factor. This behavior is in qualitative agreement with the theory. (37) The experimental data in Figure 9 also demonstrate that the bandwidth of the CVA system can be extended well beyond 100 kHz with the use of a multistage amplifier (f-3db » SSOfc/fz). The large deviations between the 2nd-order system curve fit and the experimental data in Figure 9 are most likely due to the higher-order pole introduced by the multi-stage amplifier circuit. Overall, the curve fits of this form are seen to characterize the data reasonably well. As indicated in Figures 12 and 13, there is a clear increase in the bandwidth of the system with increasing overheat ratio and Reynolds number. This behavior is in agreement with the theory of Wood9 and Perry11, giving support to the method by which the frequency-response functions were determined in this study. 4.5 Effect of Operating Conditions In the preliminary measurements of the CVA frequency-response function, auxiliary circuit components, such as a wire current limiter, were found to have a dramatic effect on the frequency response. Figure 14 shows a comparison between the measured frequency response of CVA unit 1 with and without the current limiter. Since the "canonical" CVA circuit was the focus of this research, the current limiter was re- The frequency-response functions of CVA unit 1 for several different wire Reynolds numbers and over- heat ratios are presented in Figures 10 and 1.1. At each operating point, the system was properly compensated. Two different 5//m tungsten wires were used for the measurements to show repeatability and the data shown 8 •ou| 'souneucujsv pue sonneuojev jo e i ueouewv 'Z66I. © IMBuAdo Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 moved during further measurements. This clear sensitivity to auxiliary circuit components is important, however, since they are not included in the model of the CVA circuit. Thus, any unknown (and proprietary) circuit components could be partially responsible for the discrepancy between the model and the experimentally determined frequency-response functions as the operating conditions are varied. From an operational standpoint, the observed invariance of the frequency response with operating conditions is beneficial. The bandwidth of the system would remain the same as the operating conditions change, provided the compensation was properly adjusted at each operating point. This is in contrast to the CTA system, in which the bandwidth decreases with decreasing velocity and overheat ratio. It must be cautioned, however, that the range of the operating conditions tested is rather limited and measurements over a wider range would be useful to determine the universality of this behavior. In addition, a more representative model of the CVA system must be developed to identify the cause for the system's frequency response invariance with operating conditions before relying upon this behavior as a useful characteristic of the anemometer. 5. Conclusions In this paper, the dynamic response of the CVA system was investigated both analytically and experimentally. The frequency response functions of the CVA system for a number of different circuit parameters and operating conditions were determined via laser-based radiative heating of the hot-wire sensor. A 2nd-order linear systems model of the CVA was developed to provide insight to the dynamic response and to help interpret the experirhental results. Overall, the experimental results indicate that the CVA system displays the characteristics of the model 2nrf-order system. With the use of properly selected circuit components and a multi-stage op-amp circuit, a bandwidth in excess of 100 kHz can be achieved (350 kHz in this study). The qualitative variations in the frequency response function with changes in the gain-bandwidth product, damping resistance and potentiometer resistance are in accordance with the 2ndorder model of the CVA. The frequency response functions of the CVA systems used in this study were found to have little dependence on the operating conditions of the wire. This behavior is not in agreement with the model and may be the result of an unknown auxiliary circuit component that was not included in the system model. Further study is needed to explain this invariance as it would be a desirable characteristic for experimental measurements. Perhaps most importantly, future work should fo- cus on the development of a method for frequency response function determination and setting. A possible scheme for optimization of the frequency response function that involves the adjustment of the gain-bandwidth product and damping resistance was suggested in the analysis of § 2.4. However, there is currently no practical method for adjusting these system parameters and monitoring the resulting response. Laser-based radiative heating is an effective tool for a qualitative investigation of the CVA dynamic response but is not suitable for exact compensation of the system. Therefore, development of an alternative technique is needed. Preferably, this technique will involve some form of electronic signal injection. Acknowledgements Primary funding for this research was provided by the NASA Center for Hypersonics grant to Syracuse University (NAGW-3713), monitored by Dr. Isaiah Blankson. The support and assistance of NASA Langley's Flow Modeling and Control Branch is gratefully acknowledged, particularly Catherine McGinley and Stephen Wilkinson. The CVA anemometers were provided by Tao Systems, Inc., and special thanks are extended to Drs. Mangalam and Sarma for their cooperation and assistance. References 1 2 3 4 Sarma, G.R. "Flow-Rate Measuring Apparatus," US Patent 5074147, 1991. Mangalam, S.M., Sarma, G.R., Kuppa, S., & Kubendran, L.R. "A New Approach to High-Speed Flow Measurements Using Constant-Voltage Anemometry," AIAA Paper 92-3957, 1992. Sarma, G.R. "Analysis of a Constant-Voltage Anemometer Circuit," Presented at the IEEE/IMTC Conference, Irvine, CA., 1993. Comte-Bellot, G. "Hot-Wire Anemometry," Handbook of Fluid Dynamics. Oxford University Press, New York, 1995. 5 Kegerise, M.A. &; Spina, E.F. "A Comparative Study of Constant-Voltage and Constant-Temperature Hot-Wire Anemometers in Supersonic Flow," Presented at the Fourth International Symposium on Thermal Anemometry, ASME, San Diego, CA., 1996. 6 Lachowicz, J.T. "Hypersonic Boundary-Layer Stability Experiments in a Quiet Tunnel with Bluntness Effects," Ph.D. Thesis, North Carolina State University, North Carolina, 1995. 7 20 Freymuth, P. "Feedback Control Theory for Constant Temperature Hot-Wire Anemometers," Rev. Sci. Instrum.. Vol. 38, No. 5, 1967, pp. 677681. 8 21 Freymuth, P. "Frequency Response and Electronic Testing for Constant Temperature Hot-Wire Anemometers," J. Phvs. E: Sci. Instrum.. Vol. Smits, A.J. "Further Developments of Hot-Wire and Laser Methods in Fluid Mechanics," Ph.D. Thesis, University of Melbourne, 1974. Spina, E.F. fc McGinley, C.B. "Constant Temperature Anemometry in Hypersonic Flow: Critical Issues and Sample Results," Exp. in Fluids. Vol. 17, 1994, pp. 365-374. 10, 1977, pp. 705-710. 9 Wood, N.B. "A Method for Determination and Control of the Frequency Response of the Constant Temperature Hot-Wire Anemometer," J. Fluid Mech., Vol. 67, 1975, pp. 769-786. Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 10 Perry, A.E. & Morrison, G.L. "A Study of the Constant Temperature Hot-Wire Anemometer," J. Fluid Mech.. Vol. 47, No. 3, 1971, pp. 577599. 11 Perry, A.E. Hot-Wire Anemometrv, Oxford University Press, New York, 1982. 12 Kegerise, M.A. "A Study of the Constant-Voltage Hot-Wire Anemometer," M.S. Thesis, Syracuse University, Syracuse, NY, 1996. 13 Kuo, B.C. Automatic Control Systems. 7th Ed., Prentice Hall, Englewood Cliffs, New Jersey, 1995. 14 Perry, A.E. & Morrison, G.L. "Static and Dynamic Calibrations of Constant-Temperature Hot-Wire Systems," J. Fluid Mech.. Vol. 47, No. 4, 1971, pp. 765-777. 15 Grant, H.P. "Measuring the Frequency Response of Constant Current Hot-Wire Systems," Advances in Hot-Wire Anemometrv. Ed. Melnik & Weske, University of Maryland, 1968. 16 Smits, A.J., Perry, A.E., & Hoffman, P.H. "The Response to Temperature Fluctuations of a Constant Current Hot-Wire Anemometer," J. Phvs. E: Sci. Instrum.. Vol. 11, 1978, pp. 909914. 17 Kidron, I. "Application of Modulated Electromagnetic Waves for Measurement of the Frequency Response of Heat-Transfer Transducers," PISA Inf.. Vol. 4, 1966, pp. 25-29. 18 Bonnet J.P. & de Roquefort, T.A. "Determination and Optimization of Frequency Response of Constant Temperature Hot-Wire Anemometers in Supersonic Flow," Rev. Sci. Instrum.. Vol. 51, No. 2, 1980, pp. 234-239. 19 Sarma, G.R., Private Communication, 1996. 10 •ou| 'soijneuojisv pue sojjneuojev jo ejnwsui ueouewv Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. Figure 1: Schematic diagram of the canonical CVA circuit. H(s) —+> HC(S) ——> «n2 «2 + 2(,<0n8 + <Dn2 Figure 2: Block diagram representation of the constant-voltage anemometer. 11 Eo(S) ———+• 5 0 m TJ o R./R, = 2.20 R./R, = 1 -50 R./R, = 1.07 -5 -10 -15 -20 10* 10° Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Frequency (Hz) Frequency (Hz) Figure 3: Frequency response functions for different wire operating resistances (U = 25 m/s, Rg = 3.0Q). The natural frequency, damping ratio and potentiometer position for each case are: Rw/Rg = 1.07, un = 10.1 kHz, C = 0.64, x = 0.10; RW/R9 = 1.50, cjn = 12.3 kHz, C = 0.61, x = 0.08; Rw/Rg = 2.20, w n = 14.2 £#*, C = 0.61, x = 0.07. 5 0 U = 100 m/s U = 50 m/s U = 25 m/s "5 1-10 o -15 -20 10s" 1CT Frequency (Hz) Frequency (Hz) Figure 4: Compensated frequency-response functions for several different flow velocities (Rw = 6.5£2, Rg = 3.0f2). The natural frequency, damping ratio and potentiometer position for each case are: U = 25 m/s, un = 14.2, kHz C = 0.61, x = 0.07; U = 50 m/s, cjn = 15.1, fctf* C = 0.49, x = 0.05; [/ = 100 m/s, un = 16.0, itfz C = 0.39, x = 0.04; 12 •ou| 's ev ]o ain^sui UBOUSLUV 'Z66L © wBuAdoo Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. hot-wire probe constant-voltage Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 anemometer Bragg-diffracted beam 1 r intensity-modulated be 45 degree min amp litude-modu lated input signal to Bragg cell Figure 5: Experimental arrangement for laser-based radiative heating of the hot-wire sensor. 5 0 S -5 1-10 o -15 -20 10* Freq (Hz) 50 ? e Sf 0 -50 I-100 I -150 -200 10* 10" 10° Freq (Hz) Figure 6: Frequency response functions of CVA unit 1 for several different compensation settings: A overcompensated system, O compensated system, D under-compensated system (Re0 = 15.7, r = 0.54). 13 CD •o c -5 o -10 o -15 -20 10 Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Freq (Hz) Freq (Hz) Figure 7: Frequency response functions of CVA unit 1 for two different amplifier gain-bandwidth products: O Gofh = ", A Gofh = 80; (fle0 = 6.6, r = 0.43). 5 0 !s ~^ 1-10 O -15 -20 icr 1CT Freq (Hz) Freq (Hz) Figure 8: Frequency response functions of CVA unit 1 for several different damping resistors: O Rd —> oo, A fid = 2», O ^d = & (fle0 = 7.0, r = 0.36). 14 •ou| 'sojineuojisv pue sonneuojev jo ein^sui uBoueaiv 'Z66I. © wBuAdoo Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. •io -10 -15 -20 E. 102 10 Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Freq (Hz) 8 -50 ? 2, -100 I -150 -200 10 10° Freq (Hz) Figure 9: Frequency response functions of CVA units 2 and 3 with fixed compensation settings and two different values of potentiometer resistance: O R = K, A R = 0.253? (Re0 = 16.0, r = 0.43). : 5 0 1 ~ f ' ' B H •"••• B • iliigg 1 S 5E- 1-10 J j L -: o M;Q O 4 ft Q >. SZ -i- -15 -20 1C)Z J 4 10 10 1C Freq (Hz) 0 : f - • • ••• § ^ -50 V 3, -100 _ • - 8 • § : ": '•.,V =i fl 5 -150 D_ -200 t 10° 10° Freq (Hz) Figure 10: Frequency response functions of CVA unit 1 for several different Reynolds numbers and a single overheat ratio (r = 0.54). The symbols correspond to: O Re0 = 8.7, A Re0 = 13.5, D Re0 = 17.3. 15 5 : 0 f— • • • • • • • • * * A f t * ~ *"^> m •o v_^ -5 c E- * *'•. -10 — *0 o — °IH ^I-g -: . . . . < % . . - : -15 r -20 : 1C^ i *• 10J 104 1C Freq (Hz) • • • • • • • • * Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 0 -50 1 o A I- "•••• , - -100 o OT o -150 -C — 1 ^ °X x *~• ° ** : 0 A• 0 A»^ L. - _ D 1C)Z : °a*** -200 10J 104 nA » ." 1C Freq (Hz) Figure 11: Frequency response functions of CVA. unit 1 for several different overheat ratios and a single Reynolds number (Re0 = 16.7). The symbols correspond to: O r = 0.55, A r = 0.23, D r = 0.07. Freq (Hz) 10" Freq (Hz) 10U Figure 12: Frequency response functions of the CTA system for three different overheat ratios (Re0 = 11.4). The symbols correspond to: D r = 0.63, A r = 0.43, O r = 0.28. 16 •ou| 'sojineuojisv pue sonneucuev jo ein^sui ueoueaiv 'Z66I- ©iqBuAdoo Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. 5 0 T§ s—x —5 •i -10 O -15 -20 10" 10 10 To6 Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915 Freq (Hz) 10 Figure 13: Frequency response functions of the CTA system for three different wire Reynolds numbers (r = 0.6). The symbols correspond to: O Re0 = 16.0, A Re0 = 11.0, d .Rec, = 8.0. 5 0 TJ -5 j-,0 -15 -20 10 Freq (Hz) 50 ? •••"., o 8? «, -100 (0 I -150 -200 10 Freq (Hz) Figure 14: Frequency response functions of CVA unit 1 with and without the current limiter. The symbols correspond to: A no current limiter, O current limiter. 17

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