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6.1997-1915

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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.
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Downloaded by UNIVERSITY OF FLORIDA on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-1915
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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
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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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