Department of Aerospace Engineering Sciences, University of
Colorado, Boulder, Colorado 80309-0429, USA
1 The theory of third order by Grant and Kronauer (1962) was too
rudimentary for an analysis of frequency optimization.
Experiments in Fluids 23 (1997) 175—176 Springer-Verlag 1997
Second or third order control theory for constant-temperature hot-wire
It is shown that only a third order control theory is
capable of describing dynamically optimized modern constant-
temperature hot-wire anemometers. Second order theories or
approximations contain serious errors in judgment.
Points of view
The earliest theory of linear control of hot-wires to constant
temperature by Ziegler (1934) and Weske (1943) was of ﬁrst
order. It emphasized the role of ampliﬁer gain in overcoming
the thermal lag of the hot-wire for achieving high frequency
response. Subsequently Kovasznay (1948) realized the import-
ance of proper damping for optimizing frequency response,
within the framework of a linear theory of second order.
However, already the early anemometer design by Weske
(1943) and most subsequent designs allow for an optimization
of frequency response by means of at least two parameters, in
contemporary terms referred to as ﬁrst and second order
damping or ‘‘offset’’ and ‘‘trim’’. In order to include the effect
of second order damping on frequency optimization the linear
theory was extended further to third order 1by Berger, et al.
(1963) and in more detail by Freymuth (1967, 1977, 1997).
Recently a point of view has been championed from which
a second order control theory looks sufﬁcient for describing
the frequency response of contemporary constant-temperature
hot-wire anemometers, but this view is not without problems
Blackwelder (1981) argued that a cable inductance or
compensating reactances in the bridge introduce a time
constant which in comparison to the time constant of his
feedback ampliﬁer is negligible. Therefore the large second
order term in the theory caused by cannot be compensated
for by and this keeps the theory essentially second order.
The problem with this point of view is that enters into the
theory in the form /A
(Blackwelder 1981; Freymuth 1977),
is the ampliﬁer gain, while enters by itself
(Freymuth 1977). Therefore, the relevant time constant of the
ampliﬁer is /A
, not . For typical ampliﬁer gains of order 10
, the argument for a second order theory thus contains an
error in judgment by three to ﬁve orders of magnitude.
Previously, Perry and Morrison (1971) and Wood (1975)
expressed this view that a second order theory is sufﬁcient.
While they concede that the system has basically three poles
they suggest that one of these poles has a magnitude far beyond
the range of interest in anemometers. Therefore a second order
theory captures the essence.
The problem with this argument is its presentation without
proof. The most striking counter example for invalidating this
argument is the adjustment of the ﬁrst and second order
damping parameters for maximally ﬂat frequency response.
For this most desirable operating condition Freymuth
(1977, 1997) found that all three system poles have equal
magnitude and the system behaves as a Butterworth ﬁlter of
order three. The point of view that a second order theory is
sufﬁcient thus again contains an error in judgment.
Perry and Morrison (1971) furthermore asserted that
reactances in the bridge are of primary importance for
frequency response of the anemometer, while ampliﬁer lag has
only a secondary effect, and they developed their analysis
accordingly. This view was not shared by Wood (1975) and is
opposite to that of Blackwelder (1981) within the framework of
his second order theory. Moreover, Freymuth (1977, 1997)
showed within the framework of the more relevant third order
theory that, aside from thermal inertia of the hot-wire, the
primary inﬂuence is ampliﬁer lag, which may be expressed in
terms of gain-bandwidth products of the ampliﬁer. A cable
inductance may have a secondary inﬂuence.
Finally, electronic testing of a frequency optimized constant-
temperature hot-wire anemometer by Freymuth and Fingerson
(1977) strongly supported the point of view that a third order
theory is essential for the description of contemporary hot-
wire anemometers (see also the comments by Bruun 1995).
This support is based on the agreement between experimental
and predicted rolloff during sine wave testing and its relation
to square wave testing.
Berger E; Freymuth P; Froebel E (1963) Theory and design of
constant-temperature hot-wire anemometers (in German). Kon-
struktion 15: 495—497
Blackwelder RF (1981) Hot-wire and hot-ﬁlm anemometers. In:
Methods in experimental physics (ed. RJ Emrich) Vol. 8, pp 259—314,