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A New Approach for Calibration of Hot-Wires

for use in Uncertain Environments

Amber L Favaregh *

Lockheed Martin Missiles and Fire Control, Grand Prairie, TX, 75053-1046

Colin P Britcher †, Drew Landman

‡

Old Dominion University, Norfolk, VA 23529-0247

Hot-wire sensors are known to exhibit sensitivity to multiple fluctuating quantities,

particularly at transonic conditions, variation of calibration from sensor-to-sensor, and

calibration shifts due to environmental effects, such as changing cable resistance. This paper

addresses a new calibration technique using Response Surface Methods (RSM) to achieve

rapid calibration of sensors intended for use in uncertain environments. A TSI 1201 hot-

film sensor was set up in the exit plane of an ASME nozzle supplied with compressed air. A

LabVIEW program was developed which includes a driver for a TSI IFA-300 anemometer,

permitting rapid selection of different sensor overheats and operating velocity. Typical

calibration procedures involve acquisition of anemometer outputs at a minimum of five

levels of both velocity and overheat. Using least squares regression, a polynomial prediction

model is developed. Using the prediction model, a “response surface” of output voltage,

velocity, and overheat (sensor temperature) is then created. Exploiting the characteristic

form of the response surface, it is possible to identify changes in operating environments that

may have caused output voltage shifts, such as change in cable resistance.

Nomenclature

b = coded least squares estimator

d = diameter

E = current

E(ε) = mean error

E = residual

Er = temperature corrected voltage

Fo = F distribution value

I = current

k = number of regression coefficients

MSE = mean squared error

MSR = mean squared regression

n = number of observations

o = overheat setting

p = number of residuals

R = resistance

R2

= statistical value to account for model fit to variability

Rr = resistance at Tr

SSE = sum of squares error / residuala = cylinder diameter

SSR = sum of squares regression / model

SST = sum of squares total

T = average sensor temperature

Tm = measured total temperature in the flow

* Data Reduction Engineer, High Speed Wind Tunnel, P.O. Box 531046, AIAA Member

† Professor and Chair, Department of Aerospace Engineering, ECSB 1307, AIAA Associate Fellow

‡ Associate Professor, Department of Aerospace Engineering, ECSB 1311, AIAA Senior Member

25th AIAA Aerodynamic Measurement Technology and Ground Testing Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-2809

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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to = student’s t distribution values

Tr = reference temperature

U = feedback

v = velocity

x = input variable

X = input matrix

y = response variable

α = temperature coefficient of resistance

β = regression coefficient

ε = error term

σ2 = variance

I. Introduction

ld Dominion University has been involved in an ongoing effort to support the acquisition of flow quality

information for wind tunnels, with particular emphasis on cases where practical challenges exist; for example

transonic, pressurized, cryogenic wind tunnels, such as the National Transonic Facility (NTF). To obtain flow

quality measurements in tunnels of this type, efficient methods must first be developed to calibrate hotwires for use

in the highly variable environment, which includes large variations in local pressure and temperature, leading to

independent variation of test Mach number and Reynolds number. Traditional calibration techniques for hotwire

sensors inherently focus on sensor parameters that depend on heat transfer characteristics. Sensitivities to multiple

fluctuating quantities, including velocity, density, and temperature, are significantly influenced at transonic

conditions. The challenge of separating these sensitivity variations in the presence of large temperature variations

associated with cryogenics are magnified by changing cable resistance, making it difficult to accurately establish the

probe operating conditions. A new calibration technique using Response Surface Methods (RSM) was developed to

achieve rapid calibration of sensors for both velocity and overheat. This may permit minimization of the uncertainty

arising due to varying cable resistances. Calibrating probe output versus both velocity and overheat, then fitting a

surface for the calibration allows, in principle, the separation of sensor and cable resistance changes.

II. Background

Hot-Wire Anemometry (HWA), otherwise known as Thermal Anemometry, studies the convective heat transfer

from a heated wire or film element in a fluid flow. Changes in the fluid flow properties can be correlated to changes

in the heat transfer from the heated element. The hot-wire’s ability to respond to these changes at very high

frequencies, provides information related to velocity and temperature fluctuations in the flow, themselves arising

due to turbulence. Despite the fact that the technical approaches used in HWA system design have changed

relatively little over their long existence, they remain superior to any other known technique for detection of low

levels of fluctuation at high frequencies (Bruun, 1995).

The physical size of a typical hot-wire sensor is a diameter of about 5µm and approximately 1.25mm long,

making it capable of essentially point measurements being minimally obtrusive to the flow being studied. Hot-wire

probes are also available in multiple sensor configurations to allow for measurements of the second or third

components of the velocity vector at a particular location in the flow field. The velocity range for application of

HWA systems is wide, from low velocities to supersonic flows. HWA systems output a continuous analog signal

providing the opportunity for both time- and frequency-domain analysis.

Convective heat transfer from an electrically heated sensor is the basis of HWA. Common sensor configurations

are cylindrical hot-wires and hot-films deposited on cylindrical fibers. For this paper, the term hot-wire will be used

to cover both hot-wires and hot-films unless otherwise stated. It is usually adequate to assume a linear relationship

between temperature and resistance R of the sensor (Goldstein, 1996):

( )[ ]rr TTRR −+= α1 (1)

Tr= ref temperature; Rr= resistance at Tr; Tm= average sensor temp;

α= temperature coefficient of resistance

O

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The value of α is critical as the variation of resistance with temperature dictates the output signal from the

anemometer. The fluid static temperature Ta is often used for Tr for convenience. The reference temperature used

dictates the value for α. Today, the constant-temperature mode of operation is practically universal. Here, a

feedback loop maintains Tm practically constant and extends the upper frequency limit to more than 100 kHz

(Goldstein, 1996). It is necessary to establish the upper frequency limit for each given sensor, environment,

amplifier, and bridge as well as adjusting the system for the flattest possible amplitude response. The connecting

cable between the hot wire probe and the anemometer circuitry is typically quite long, resulting in significant

resistance and capacitance. This impedance must be accounted for in the anemometer bridges to avoid instability of

the feedback loop, typically by adding capacitance to opposite arms of the bridge. Since the bridge must be balanced

with the cable resistance in place, the resistance must be subtracted in some way from output measurements to

properly apply the sensor calibration. Modern anemometers, such as the TSI IFA 300 used here, perform some of

the bridge adjustments automatically (“Smartune” in the case of the TSI IFA-300).

In order for the anemometer output to be significant, a calibration must be made to relate the output to changes in

velocity or other flow parameters. Since sensors cannot be reproduced precisely enough to have identical calibration

curves for two sensors of the same type, a separate calibration must be done for each sensor. A typical calibration

setup involves measuring the anemometer output as a function of velocity. Velocity calibration curves can be plotted

in real terms of voltage and velocity, holding temperature and density constant as in Figure 1. Plotting in terms of

voltage squared and the square root of velocity makes the plot quasi-linear for low velocities. This form of plot is

referred to as a King’s Law plot, as shown in Figure 2.

Once a calibration curve is obtained it can be used to obtain good measurements in unknown environments

provided that the fluid temperature and density in the unknown environment match those of the environment used to

complete the calibration.

III. Calibration Technique

To calibrate hot-wires in a way more applicable to cases where operating conditions may be varying widely, or

uncertain, a new calibration technique was developed. By calibrating a single wire for both velocity and overheat

settings, it can be determined if variation in output voltage are due to changes in velocity or changes in cable

resistance. Overheat of hot-wires is the ratio of the operating resistance to the cold resistance of the wire. When the

setting of the operating resistance is made, the cable resistance is taken into account. If the cable resistance is

changing, then the actual overheat being set is not known. Changes in overheat equate to changes in output voltage

which, in a standard calibration, will appear to be changes in velocity. Using RSM to fit a surface for voltage

changes in both velocity and overheat will allow for investigation into the actual causes of output voltage changes.

A. Test Plan

Using the guidelines of RSM, a test plan was designed to minimize the required number of runs for the

calibration. To cover a reasonable range of overheats while staying in a safe range to maintain the integrity of the

film sensors, the overheat range was chosen to very between 1.2 to 1.8. The velocity range was chosen to be

Figure 1. Typical velocity calibration curve

Figure 2. Typical “King’s Law” plot

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approximately 0.03 to 0.3 Mach. Each of the two factors was varied over five evenly spaced levels. This created a

design space with a grid of 25 evenly spaced design points. Design points were collected in a randomized order. The

collected voltage data was temperature corrected to 21°C using equation (2).

m

r

cTT

TTEE

−

−= (2)

Ec=temperature corrected voltage; T= average wire temperature;

Tm= measured total temperature in flow; Tr= reference temperature

B. Test Setup

An easily characterized low turbulence flow was used to ease the collection of data for this calibration technique

The system used was designed for use at NASA Langley Research Center for the specific purpose of hot-wire

calibrations. A temperature-controlled high-pressure air supply is routed through a system of valves to decrease the

pressure, and a heater to heat the air. To ensure stable conditions at lower velocities, a Teledyne Hastings flow

controller was placed at the end of the sequence of valves in the air supply system. A maximum velocity

corresponding to a mach number of 0.3 was chosen to avoid compressible flow conditions that complicate hotwire

calibrations. The air flow was directed to an ASME nozzle with a 0.75 in exit plane diameter. The hot-wire was

placed at the exit plane of the nozzle. The flow controller was computer controlled using a LabView software

interface. The LabView interface was also used to communicate with the TSI IFA-300 constant temperature hot wire

anemometer. Air supply pressure and temperature measurements were obtained using a Ruska pressure transducer

and thermocouples, via a Hewlett-Packard 1314 data acquisition card and LabView software. Photos of the test

setup are shown in Figure 3.

1800 psi

air

Reduced

to 300 psi

heater

Reduced

to 200 psi

through flow

regulator

(not shown)

to nozzle

Figure 3 – Experimental Set-Up for Probe Calibration

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C. Results

Data was collected using a TSI 1201-20 film sensor. The data collected was plotted in the classic format to

ensure that the response was as expected; Figure 4 and 5. Both classic plots show that there is slight inaccuracies in

the 1.35 overheat measurement at a velocity of 175 ft/s. Using the least squares regression technique a quadratic

surface was fitted to the data. The input matrix X is as follows with n equal to the number of observations, 25:

=

)(

)(

)(

1

1

1

1

22

11

22

2

2

2

222

2

1

2

111

nnnnnnvo

vo

vo

vovo

vovo

vovo

XMMMMM

(3)

To ease the regression process, the overheat and velocity input variables were converted to coded variables.

Equation (4) was used to do this conversion with xi1 and xi2 equal to the coded overheat and velocity terms and ξi1

and ξi2 equal to the natural overheat and velocity terms for i = 1, 2, … , n. The coded variables will fall between –1

and 1.

[ ][ ]

[ ][ ] 2

22

222

1

11

111

2/)min()max(

2/)min()max(

2/)min()max(

2/)min()max(

i

ii

iii

i

ii

iii

x

x

=−

+−

=−

+−

ξξ

ξξξ

ξξ

ξξξ

(4)

In coded variables the X matrix is represented as:

X=

1.000 -1.000 -0.993 1.000 0.986 0.993

1.000 -1.000 -0.490 1.000 0.240 0.490

1.000 -1.000 0.024 1.000 0.001 -0.024

1.000 -1.000 0.524 1.000 0.274 -0.524

1.000 -1.000 0.963 1.000 0.927 -0.963

1.000 -0.500 -0.991 0.250 0.982 0.495

1.000 -0.500 -0.486 0.250 0.236 0.243

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300 350

Velocity (ft/s)

Vo

lta

ge

, E

1.2 ovh 1.35 ovh

1.5 ovh 1.65 ovh

1.8 ovh

Figure 4. Classic Calibration Curve

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16 18 20

sqrt(V)

E^

2

1.2 ovh

1.35 ovh

1.5 ovh

1.65 ovh

1.8 ovh

Linear

Figure 5. Classic King’s Law Calibration Plot

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1.000 -0.500 0.024 0.250 0.001 -0.012

1.000 -0.500 0.517 0.250 0.267 -0.259

1.000 -0.500 0.968 0.250 0.936 -0.484

1.000 0.000 -1.000 0.000 1.000 0.000

1.000 0.000 -0.496 0.000 0.246 0.000

1.000 0.000 0.007 0.000 0.000 0.000

1.000 0.000 0.490 0.000 0.240 0.000

1.000 0.000 1.000 0.000 1.000 0.000

1.000 0.500 -0.999 0.250 0.998 -0.500

1.000 0.500 -0.502 0.250 0.252 -0.251

1.000 0.500 0.007 0.250 0.000 0.003

1.000 0.500 0.532 0.250 0.283 0.266

1.000 0.500 0.991 0.250 0.981 0.495

1.000 1.000 -1.000 1.000 1.000 -1.000

1.000 1.000 -0.501 1.000 0.251 -0.501

1.000 1.000 -0.009 1.000 0.000 -0.009

1.000 1.000 0.507 1.000 0.257 0.507

1.000 1.000 0.984 1.000 0.968 0.984

The residual values were calculated for each data point collected and were found to be small and randomly

scattered about zero. This is a good indication of a properly fit model.

An investigation into the accuracy of the surface fit was now carried out. First, the sum of squares of the

residuals was calculated as:

0.0685

'__

)ˆ(

1

2

1

2

=⇒

=

=

−=

∑

∑

=

=

E

n

i

i

n

i

iiE

SS

eeformmatrixin

e

yySS

(5)

An unbiased estimator of the variance is directly calculated using the residual sum of squares, also known as the

error sum of squares. The variance is low which also indicates a good surface fit.

0.0036ˆ

ˆ

2

2

=⇒

−=

σ

σpn

SSE

(6)

The denominator in equation (6) represents the number of degrees of freedom that are associated with the

residuals; n equals the number of observations taken, 25; and p is the number of residuals, 6 for a quadratic model.

Next, analysis of variance was done in order to test for significance of the regression. This procedure breaks up the

total sum of squares into a sum of squares due to the model, or regression, and the error sum of squares that was

calculated in equation (5). This relationship is expressed in equation (7).

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13.7925

2

1

2

1

=

−′′=⇒

′′−′=

−′=

+=

∑

∑

=

=

n

y

yXbSS

yXbyySS

n

y

yySS

SSSSSS

n

i

i

R

E

n

i

i

T

ERT

(7)

The calculation for regression significance, F-test, is done as:

765.2673

)1/(

/

=

=−−

=

o

E

R

E

R

o

F

MS

MS

knSS

kSSF

(8)

The critical value for this F-test is Fα,k,n-k-1=2.74 with the α chosen as 0.05 for a 95 % confidence interval. Since

765.2673 is far larger than 2.74 the model is significant. To determine the amount of reduction in the variability of y

obtained by using the regression coefficients, the R2 and R

2adj values are calculated as:

9938.0)1/(

)/(

9951.0

2

2

=−

−=

==

nSS

pnSSR

SS

SSR

T

Eadj

T

R

(9)

This indicates that the model accounts for 99.51 % of the variability in voltage. The adjusted R2, R

2adj is a more

accurate value because it will not always increase with an increase in model terms. A value of 99.38 % for R2

adj is a

very good indication that the model is fit correctly.

Additionally, tests were done to determine the significance of the individual regression coefficients. These tests

were done using the student’s distribution.

[ ]8.4969 7.8662- 1.2219- 31.2789 52.1052 116.1400

ˆ 2

=

=

o

jj

j

o

t

C

bt

σ (10)

The denominator of equation (10) is called the standard error of the regression coefficient bj where Cjj is the

diagonal element of (X’X)-1

corresponding to bj. (Myers, 2002) The critical value of to is tα/2,n-k-1 equal to 2.093 with

α equal to 0.05 and 1.328 with α equal to 0.2, which is only a confidence level of 80 %. All of the absolute values of

the regression coefficients are far larger than both of the critical student’s t distribution values except the fourth

coefficient, which relates to the velocity-squared term. It is close to the value with an 80 % confidence but it is not

clear if the term should be left out of the model.

The velocity-squared term was left out of the model and a validation run was done to determine the accuracy of

the model predictions versus the observed values at randomly chosen points. This initial trial showed higher

accuracy at lower velocities with considerable inaccuracy at higher velocities. It was then determined that the

velocity-squared term should be included in the model for an accurate prediction of the response surface. With the

full model the observed voltages were compared to the predicted voltages to calculate their residuals. The model

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prediction was proven to be accurate within 3% of the observed voltage, with the exception of the 25th

and 3rd

runs.

Those two runs are within 9% of their observed voltages.

With the model proven to be a good predictor of voltage output, the regression coefficients were converted to

natural variables so that they may be directly used to estimate a voltage for a given overheat and velocity in natural

variables. The final equation in natural coefficients is equation (11). Figure 6 is a plot of the fitted surface.

))((10*703.4)(10*090.1

)(3897.0)(10*055.5)(2785.32846.2

325

24

VovhV

ovhVovhE

−−

−

+−

−++−= (11)

D. High Speed / Unsteady Data

In an attempt to validate the quality of this calibration technique a series of runs were carried out using high-

speed data collection. The data was collected on a separate calibration setup could not achieve a sufficiently high

sample rate, so a separate laptop was used to collect data at a rate of 40kHz for 20 seconds, allowing for 20

ensemble averages. It is expected that the spectra for different overheats for a constant velocity will collapse when

their respective sensitivities are accounted for. As seen in the classic calibration plots, as overheat increases the

sensitivity of the wire increases as well. The increase in wire sensitivity is shown as an increase in slope of the

curve. To calculate this sensitivity, the partial derivative, in terms of velocity, was taken of equation (11). This

calculation is:

)(703.4)(090.1*2055.5 354ovhEVEE

V

E −−− +−=∂

∂ (12)

Voltage data was then divided by the sensitivity for the given overheat and velocity settings. Power Spectral

graphs were calculated for all five overheat settings at four different velocities, 73 ft/s, 143 ft/s, 211 ft/s, and 279

ft/s. Twenty ensemble averages were taken in each case. Figures 7 through 14 represent all these spectral graphs.

Figure 6. Fitted Response Surface

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E. Validation

After running a full calibration for the film sensor, a wire sensor was run to validate that the technique was

applicable for any type of sensor. A platinum coated tungsten wire, TSI model number 1214, was placed in the

setup. Figures 15 and 16 show the collapse of the spectra, albeit with some anomalies, for a velocity of 105 ft/s.

Figure 7. Unsteady Data V=73 ft/s

Figure 8. V=73 ft/s Sensitivity Removed

Figure 9. Unsteady Data V=143 ft/s

Figure 10. V=143 ft/s Sensitivity Removed

Figure 11. Unsteady Data V=211 ft/s

Figure 12. V=211 ft/s Sensitivity Removed

Figure 13. Unsteady Data V=279 ft/s

Figure 14. V=279 ft/s Sensitivity Removed

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F. Determination of Actual Overheat This calibration technique was intended to allow for identification of changes in cable resistance. Data was

collected with an incorrect cable resistance for comparison with the calibration surface. The cable resistance was set

to 0.75 Ω when it was actually measured to be 0.59 Ω. Setting the cable resistance higher should have driven the

overheat to a lower setting. For ease of viewing, the calibration surface was plotted in two dimensions along with the

points collected with the incorrect cable resistance setting. Figure 17 shows the two-dimensional plot with the

incorrect cable resistance points falling slightly below the overheat curves they would otherwise have been thought

to lie on.

IV. Discussion and Conclusions

The calibration technique shows some promise. With the incorrect cable resistance setting the voltages appear to

fall on curves that would indicate a proportionally lower overheat setting. Mathematical modeling could be done to

determine the actual overheat setting numerically. This would lead to the ability of in-situ validation of sensor

calibration in uncertain environments.

Further investigation needs to be done into fitting higher order models. These models may allow for a higher

order partial derivative for the sensitivity; possibly allowing for full collapse of overheat spectra at higher velocities.

In addition to model fitting the hardware setup still need much more investigation as well. The response from the

HWA system often behaves strangely leaving room for questionability of its accuracy. There may be issues with the

cables or any number of other problems that would cause the unreliable behavior of the response.

Figure 15. Unsteady Wire Data v=105 ft/s

1.2

1.7

2.2

2.7

3.2

3.7

4.2

0 100 200 300

Velocity

Vo

lta

ge

1.2

1.35

1.5

1.65

1.8

1.65 ?

1.8 ?

1.2 ?

Figure 17. Determining Actual Overheat Setting

Figure 16. Wire Data v=105 ft/s Sensitivity Removed

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.

Acknowledgments

This work was partially supported by NASA Langley Research Center under Grant NAG-1-394. The authors

express their gratitude to Dr. Gregory Jones of the Flow Physics Branch of NASA Langley Research Center for

technical guidance and for allowing access to the hot wire calibration set-up.

References

1 Myers, Raymond H., and Douglas C. Montgomery. Response Surface Methodology: Process and Product

Optimization Using Designed Experiments. 2nd

Edition. New York: John Wiley & Sons, Inc., 2002. 2 Montgomery, Douglas C. Design and Analysis of Experiments. 5

th Edition. New York: John Wiley & Sons,

2001. 3 Bruun, H. H. Hot-Wire Anemometry: Principles and Signal Analysis. New York: Oxford University Press Inc.,

1995. 4 Goldstein, Richard J., ed. Fluid Mechanics Measurements. 2

nd Edition. Pennsylvania: Taylor & Francis, 1996.