PROCESSING TIME SERIES DATA (VELOCITY, SHEAR STRESS) FROM THE
PULSE WIDTH MODULATED-CONSTANT TEMPERATURE ANEMOMETER
(PWM-CTA)
J.F. Foss, Professor
Michigan State University
Mechanical Engineering Department
East Lansing, MI 48824
T.J. Hicks, Chief Engineer
Digital Flow Technologies, Inc.
4731 Nakoma Drive
Okemos, MI 48864
ABSTRACT
The digital output from the PWM-CTA represents the
short duration (e.g., 20sec) true time average of the
analog velocity (addressed herein) or shear stress.
The developed techniques to extract the moments of
the velocity (or shear stress) distribution and to create
a regular time series are presented in this paper.
INTRODUCTION
The constant temperature anemometer (CTA) has
been an important instrument for fluid mechanics
research since its widespread availability in the
1960s. The use of the Wheatstone bridge and the
feedback amplifier – whose differential input signals
an increase/decrease in current to compensate for a
momentary increase/decrease in the cooling velocity
at the probe – provides a servo system that is capable
of following rapid fluctuations of the velocity. The
CTA has been the subject of numerous technical
descriptions. Recommended ones include Perry1,
Comte-Bellot2, Fingerson and Freymuth3.
The Pulse Width Modulated-Constant Temperature
Anemometer (PWM-CTA) provides a distinctly
different electronic architecture (in comparison with
the conventional (C)-CTA) with which to track high
frequency velocity fluctuations. The PWM-CTA was
introduced in the publication by Foss, et al.4 and
protected by the subsequently issued patent: Hicks,
et al.5. The PWM-CTA has been under continuous
development since these early disclosures with the
most recent description provided by Foss, et al.6 in
which comparative measurements – with a Disa 55M
system – in a slit-jet flow field were reported. Strong
agreement, in terms of the first four moments of the
velocity distribution as well as spectra, were shown
in these comparative measurements.
The present paper is to: i) further clarify the
extraction of moments of the velocity (or shear
stress) distribution from the PWM-CTA output, and
ii) to report on the regularizing strategy that is
employed to develop a periodic output for single and
multiple sensor measurements. As shown below, this
transformation of the data is required since the actual
measurement period is a function of the effective
cooling velocity that is experienced by a given
sensor.
The basic circuit configuration for the PWM-CTA is
presented in the next section and the data processing
observations and strategies are provided in the
following two sections.
BASIC FEATURES OF THE PWM-CTA
Circuit
A simplified block diagram of the PWM-CTA is
presented in Fig. 1. When the electronic switch (S) is
closed at the beginning of the next heating cycle, the
adjustable DC voltage: EA, is imposed across the
cable-impedance matching resistor (R1) and the hotwire (hot-film, shear stress, etc) sensor. The resulting
current flow heats the sensor (resistance RS) with a
corresponding increase in the temperature (and hence
the resistance) of the sensor. The voltage: EB, across
the probe itself, will exhibit a corresponding increase
during this heating portion of the cycle. By adjusting
the parameters of the circuit, the dual inputs to the
comparator can be made to trigger its output-state
change when RS(t)=RH=[1+OHR Rc] where OHR is
the designated overheat ratio for the wire whose
ambient temperature resistance is Rc. The high gain
(104), differential amplifier [E B  E Ref ] , ensures that
a steeply rising signal will “sharply” trigger the
comparator when the RS=RH condition is met.
The triggered comparator output is used to: i) open
the switch that connects EA to the sensor, ii) stop the
accumulation of the counts from the master clock,
and iii) present this digital value as the PWM-CTA
output for the present cycle to the multiplexer (MUX)
for storage in the host P.C. The accumulated counts
represent the duration of the heating period: k (for
the kth cycle) of a given sensor. As noted in Fig. 1,
the PWM-CTA common unit (master clock, etc.) can
serve multiple (up to 32) measurement channels.
it is important to realize that this
equilibration occurs in one cycle and that, in
principle, there is no implied upper
frequency limit. Specifically, the cycle time
(T) can be arbitrarily reduced for arbitrary
increases in the density-velocity product
(V). Practical limitations in terms of the
measurement resolution do, of course, exist.
These are related to the frequency of the
master clock.
With the cessation of the heating current at time
tK+K (for the Kth cycle), the following operations
complete the cycle: a) the sensor is cooled in
preparation for the next heating cycle, and b) the 
counter is reset.
The multiple channel, common clock timing, digital
output (no A/D required) operation is considered to
make the PWM-CTA an excellent anemometer for
multi-sensor (rakes, arrays) measurements.
The period of one cycle (T) is a free parameter for the
operation of the PWM-CTA. It was selected as
20sec (1/50 KHz) for the present data.
The ~Velocity Magnitude (Q) Transfer Function
The electrical attributes of Fig. 1 can be represented
in terms of the sensor’s RS(t) behavior for several
cycles: K1, K, K+1; see Fig. 2. The prescribed
energy state for the sensor: RH, is reached at
(tK1+K1) whereupon the wire cools in preparation
for the next cycle. This same energy state is achieved
in the next cycle at tK+K. The symbol: <T K>
designates the time between these equal energy states
as:
<TK>=tK+K(tK-1+K-1)=T+(KK-1)
The present unit, which operates at 50KHz, is
considered to be appropriate for typical laboratory
investigations. For example, at 50m/sec, the 20sec
time period will deliver a “streamwise length
averaged” measurement of 1mm. Since a typical
sensor length and diameter are 1mm and 5m, this
will provide the “spatially averaged velocity” within
a 1mm  1mm  5m domain in space.
It can also be noted that a 500KHz unit is in
development. The first unit of this kind will be
utilized by Prof. Wm. Saric (Arizona State Univ.) in
his studies of boundary layer transition at M=2.4
The energy loss effects (conduction to the supports
plus convection to the surrounding gas) can be
equated to the electrical power input in the same time
period. With reference to Fig. 2, the following
expressions define this equilibration condition.
(1)
In a statistically stationary flow, the time average of
the <TK> values will be equal to T. The (KK1)
term in (1) is not a fluctuation (i.e., it is not like
u=u(t) u ); rather, it is an indication of (u/t).
A detailed feature of the PWM-CTA operating
condition can be noted here: a nominal one percent
change in RS: 0.99RS/RH1.00 is targeted for the
anemometer’s operation. This is controlled by EA for
a given T. Also, the targeted operating range for the 
values is 0.25/T0.5. This avoids starting transient
effects (i.e., /T<0.25 for velocity = 0) and it permits
adequate cooling of the sensor in the period
tK+K<t<tK+T). Note that there are 4096 counts to
define T. Hence, the  value is resolved to nominally
one count in 1024.
Given that the key to the precise transfer function
between K, <TK> and the temporally averaged <VK>
velocity during the time period <T K>, is the return of
the sensor to the preset energy state (RS=RH) at
t=tK+K,

t K τK
t
electrical energy liberated in <T K> by the
sensor:
(I S2 /R S ) dt

K
t K  K
t

K
t K  K
t
2


EA

 R S (t) dt
 R 1  R S (t) 
g(R S ; E A , R 1 ) dt
(2)
K
where EA is an adjustable constant in the
weakly time dependent g function. The
circuit component: R1, is set at 50 for the
hot-wire sensor of this study. Its value
would be adjusted if, for example, RS1K
for a shear stress sensor.

Heat transfer from sensor to
surroundings in the relevant time span:
t K τK
t
K 1  τ K 1
[A  BQ n ] dt
 [A  B   VK  n ]  TK 
the
(3)
where <VK> is the time average of the
effective cooling velocity V(t) in the period
<TK>.
The preceding has clarified that (2) and (3) are equal
in magnitude; hence
t K  K
t
g[R S ; E A , R 1 ] dt
(4)
K
 [A  B  VK  n ]  TK 
The coefficients (A,B) will be evaluated in a direct
calibration. Since RS(tK)RS(t)RS(tK+K)=RH and
since RS(tK)/RH0.99, the left hand side integrand is
nominally constant and its time variable contribution
can be absorbed into the coefficients of the following
equation:
τK
 A  B  VK  n
 TK 
(5)
A representative calibration data set for a 5m
diameter, 1mm long tungsten sensor (with 30m
diameter, 1mm long copper plated stubs) is shown in
Fig. 3.
The right-hand side of this expression, with the
exception of its short time average characteristic
(<V>), is identical to that for the conventional CTA.
The latter would be written with E2 on the l.h.s. This
observation leads to a primary difference between the
PWM-CTA and the C-CTA; the former is more
sensitive since it has a (nominal) second power:
/<T> ~ <Vn>, instead of a fourth power: E2~Vn,
response. That is, given that n0.5, an incremental
change in /<T> can be expressed as (/<T> ~
1
(V1/2)<V> whereas  E ~ (V 1/ 2 ) V .
E
POST-ACQUISITION PROCESSING
ALGORITHMS
Stochastic Values – Single Sensor
If single sensor data are acquired for a sufficiently
long time period, the corresponding moments of the
velocity magnitude distribution can be readily
obtained as shown below.
Each K value can be used to obtain <VK> for the
corresponding <TK> time period. The first moment
of the distribution (i.e., the time mean value) is
V
N

K 1
 VK  TK  /
N

K 1
 TK 
(6)
The extraction of the time mean value is a most
efficient process using the PWM-CTA.
It is
instructive to compare this data processing to that of
a digitized analog signal. Specifically, if the analog
velocity signal were digitized at the same rate as the
PWM-CTA samples were made, then the random
deviations between the discrete sample and the short
time temporal average (per the PWM-CTA) would
have to be averaged to zero in order for the sampled
analog values to represent the true time-mean value.
The higher moments: =2,3,4, then follow as
described in the next sub-section.
The Higher Moments
It can be expected that the true velocity, V(t), at the
sensor will not be constant during the <T K> sample
period albeit the magnitude of the velocity
differences will approach zero as T0. Let [V(t)]
be defined as
[V(t)]K=V(t) - <VK>.
(7)
One can then show that, for the second moment of
the fluctuations:
~
1
V 2  lim
T T
T
0
(V(t)  V) 2 dt ,
(8)
the PWM-CTA will provide (for each time segment
<TK>):
[VK>+[V(t)]K V ]2=<VK>2+2[V(t)]K<VK>
2
+[V(t) ] K 2(VK+[V(t)]K) V + V 2.
(9)
This quantity can be integrated with respect to time
for the period tK-1+K-1tK+K to obtain
<[<VK>+[V(t)]K V ]2>
= <VK>22<VK> V + V 2+<[V(t)] 2K >
=[<VK> V ]2+<[V(t)]2>
(10)
where the first power: [V(t)]K, terms integrate to
zero. Hence, the second moment of the velocity
distribution would be:
~
V2 
N

[ VK   V]2  TK  /
K 1

N

K 1
N

 TK 
K 1
 [ V(t)] 2  TK  /
N

K 1
 TK  (11)
It is evident that the PWM-CTA will provide too low
a value of the second moment given that only the first
term on the rhs can be evaluated using the data
available from the PWM-CTA. However, if the
sample rate is sufficiently large, the second RHS term
can be made sufficiently small.
The following strategy can be recommended to assess
the influence of the [V(t) ]2K contributions. It is
made feasible by post-processing the calibration and
the flow field  values as shown below. This strategy
relies upon the observation that the error will be zero
in the limit as T0.
With reference to Fig. 2, the basic transfer function
could also be established for the time period.
tK-1+K-1tK+1+K+1.
This will be referred to as the (2T) series. Similarly,
the transfer function could be defined over the
period:
tK-1+K-1tK+2+K+2.
This will be identified as the 3T series.
It can be noted that these additional calibrations do
not need to be acquired. Specifically, since the
calibration velocity is constant, <T K>=T (for all K)
and the /<T> values for the 2T and the 3T series are
the same as /<T> for the 1T series.
The original calibration data will then serve for the
coefficients: A,B,n for each series. Next, the flow
field data can be processed to yield
~
V 2 for series n=1 
~
V 2 for series n=2 
~
V 2 for series n=3 
~
V 2 (1)
~
V 2 (2)
~
V 2 (3)
and the function:
~
V 2 (n)=a+bn+cn2
(12)
~
can be formed using the three known values: V 2 (1),
~
~
V 2 (2), V 2 (3).
~
The desired V (n=0) value can be evaluated as
~
 V
~
~
V 2 (0)= V 2 (1)  
 (1)
  n  n 1
(13)
where
V
 b 2 c
 n 1
(14)
~
~
The magnitude of the difference: [V 2 (0)  V 2 (1)],
will provide an evaluation of the “confidence level”
in the measured second moment.
Similar algebraic considerations will show the impact
on the third and fourth moments. These details are
not provided here.
The Recovery of a Regular (Registered To T And
Not <T>) Time Series Output
The preceding sections have emphasized that an
output value, from each sensor, is available with an
associated time increment: [<T K>]j for each sensor:
j.
Clearly, if one wishes to process data from an x-array
(j=1,2), a 3-sensor probe (j=1,2,3), etc., there must be
a technique to temporally align these signals. This
would also be true for single sensors that are to be
processed for cross-correlations, etc. It will also be
beneficial for spectral analyses to have access to a
regularized time series.
A regularized time series can be readily obtained in
the post-processing phase of the data reduction for a
single sensor as described in the next sub-section.
Processing for a multi-sensor probe is discussed in
the second sub-section.
A Regularized Output – Single Sensor
It is clear that a regularized output will be created at a
frequency of (1/T). Centering the regularized output
with respect to tK is, however, an arbitrary selection.
Given that the  values are to be limited to the range:
0.25/T0.5, it is suggested that /T=0.375 be
selected as the starting point for the regularized
output. With this condition, /T=0.875 will be the
center point (or the designated time) for the output
signal.
Figure 4 schematically shows four scenarios for the
possible  values that will dictate the “regularized
output” for the corresponding period:
tK1+0.375TtK+0.375T. Condition “a”: K-10.375T,
is mutually exclusive with respect to “b” for which
K-1>0.375T. Conditions “a” and “b” can be paired
with condition “c”: K0.375T or “d”: K>0.375T.
The symbol: <VK>, represents, as before, the
average velocity in the <TK> time period. This value,
and its adjacent values, are shared for the regularized
output in the time period tK1+3T/8<ttK+3T/8. The
following conditions show the time-weighted
velocity values that will contribute to the regularized
velocity-time product.
This net product will
represent the data for tK1+3T/8tK+3T/8.
if “a”:
if “b”:
if “c”:
if “d”:
<VK>5T/8
<VK-1>(K-1-3T/8) + <VK>(T-K-1)
<VK>K + <VK+1>(3T/8-K)
<VK>3T/8
The regularized value for the Kth time period is, then:
=if “a” or if “b” plus if “c” or if “d”.
involves the equivalent of plotting the 13 distinct Q
values that correspond to the measured  (or E2)
values from the two sensors and identifying their
intersection point. For the conventional anemometer,
the sampled voltages are “instantaneous” and the Q
and  values are similarly defined for this very short
time segment.
The need to register the two PWM-CTA channels to
each other presents a different issue. Namely, the 1
and 2 samples must be co-registered and then the Q,
 values can be evaluated. For this, the Q value
corresponding to =0 will be used for the 1 and the
2 time series. This will then provide
K-1, K}1  <QK(=0)>}1
(19)
and
K-1, K}2  <QK(=0)>}2
(15)
(20)
where these are regularized to a center point of
The regularized velocity value will be designated as:
VK=/T.
An Algorithm for an X-Array
Processing the  values (1, 2) from an x-array (or
other configurations for which the  cannot be
directly related to a velocity magnitude) requires a
computing algorithm that includes, but goes beyond,
the regularization noted above. One such algorithm
(attributed to Scott C. Morris in terms of processing
C-CTA signals) is provided here.
Consider that each K/<TK> value of an x-array
represents a combination of velocity magnitude (Q,
in the plane of the x) and the pitch angle with respect
to the probe shaft. The latter is designated as .
Namely
( K )1
 f1 (Q,  )
 TK 1
7 

t K  8 T


(16)
The above described calibration data can be further
processed to create a two-equation, two-unknown
solution as:
 Q(  ) 
 Q K (  )  [ Q K (0) ]1 

 Q( 0) 
 Q K (0)  g 1 (  )
(21)
 Q(  ) 
 Q K (  )  [ Q K (0).] 2 

 Q( 0)  2
 Q K (0)  g 2 (  )
(22)
and
(17)
for which there will be a unique
and
<QK>, <K>
( K ) 2
 f 2 (Q,  )
 TK  2
(18)
Consider a family of (Q) values for the discrete 
values (-36, -30, -24…0…24, 30, 36) of the full
calibration data set. The Morris strategy (developed
for the equivalent E 2Bridge (Q) for discrete  values)
combination. One can think of the calculation for
(Q,) as identifying the intersection point of two
curves in the (Q,) plane. If sensor 1 is oriented such
that =+45o is perpendicular to the sensor and if =45o is perpendicular to sensor 2, then
g1() is a monotonically decreasing function
g2() is a monotonically increasing function
and, plotting the curves (3.17) and (3.18) will lead to
a unique intersection point. Note that the smoothed
calibration data: /T=f[Q;], permit arbitrarily fine
steps in Q(0) to define the respective “f” functions.
For example
h[;Q(0)]=a+b()+c(2)+…
where a=a[Q(0)]. b=b[Q(0)], etc., and the
a,b,c…coefficients are obtained from the (for
example) 13  conditions: 0, 6o, 12o…36o.
SUMMARY
The PWM-CTA provides an alternative to the
conventional (C)-CTA which has well served the
experimental fluid dynamics community since the
early 1960’s. The PWM-CTA offers a more sensitive
transfer function (nominally a second vs. a fourth
power response), a digital output, ease of
simultaneous measurements and (not discussed
above) relatively low cost.
Its intrinsic attribute of providing a short (e.g., 20
microsec) time average leads to a rapid convergence
to the true time average value (velocity or shear
stress) and to higher moments that will (slightly)
lower than the true values. The magnitude of the
latter effects can be assessed using the strategy
shown herein.
The second intrinsic attribute of the PWM-CTA, to
deliver an irregular time series, can be addressed
using the schemes noted above. These will be
addressed by representative examples in the August
2001 presentation and this additional information can
be obtained from the author at <foss@egr.msu.edu>
or by telephone at 517-355-3337.
REFERENCES
1). Perry, A.E., 1982, “Hot-wire Anemometry,”
Oxford University Press, New York.
2). Comte-Bellot, G., 1976, “Hot-Wire Anemometry”,
Annu. Rev. Fluid Mech., vol. 8, pp. 209-231.
3). Fingerson, L.M. and Freymuth, P., 1996, “Thermal
Anemometers”, Fluid Mechanics Measurements 2nd
Edition, R.J. Goldstein, Ed., pp. 115-173.
4). Foss, J.F., Bohl, D.G. and Hicks, T.J. 1996, "The
Pulse Width Modulated-Constant Temperature
Anemometer," Meas. Sci. and Tech., vol. 7, pp. 13881395.
5). Hicks, T.J., Foss, J.F. and Bohl, D.G., 1997, "Pulse
Width
Modulated
Constant
Temperature
Anemometer," U.S. Patent No. 5,654,507.
6). Foss, J.F., Hicks, T.J. and Morris, S.C., 2001,
“Operating Experience With The Pulse Width
Modulated-Constant Temperature Anemometer”,
Sixth
International
Thermal
Anemometry
Symposium, Melbourne, Australia.
MUX
EA
16 bit out
To other
channels
From other
channels
Terminator Board
R1
R1
EB'
+
EB
RS
x11
(x6 optional)
Comp
FlipFlop
R
_
Q
x10,000
S
EREF
Clock
(163.84 MHz)
Count = 0
4096
Figure 1: PWM-CTA Schematic
Note: R1=50 for a typical (RC=3.5) hot-wire sensor
RS(t)
Figure 2: Timing Diagram for RS(t)
16bit
Counter
R
Common
Unit
Figure 3: A representative /T=f(Q) calibration for a single sensor.
RS(t)
t
a
c
b
d
tj
t j -1
{
*
<Vj - 1>
<Vj>
<Vj -1>
a
<Vj>
b
<Vj>
<Vj + 1>
<Vj>
c
<Vj + 1>
d
* Shows the applicable velocity for the indicated values
Figure 4: A representation of the algorithm which creates a regularized <V(t j)> time series.