a controlled-temperature hot-wire anemometer with voltage feedback linearization
TRANSCRIPT
A controlled-temperature hot-wire anemometer
with voltage feedback linearization
L. V. Araujo, S. Y. C. Catunda, C. E. T. Dórea
Department of Computer Engineering and Automation,
UFRN
Natal, RN, Brazil
[email protected], [email protected],
R. C. S. Freire
Engineering Electronic Department, UFCG
Campina Grande, PB, Brazil
Abstract—This paper proposes a new architecture of a
controlled-temperature hot-wire anemometer using voltage
feedback linearization. The voltage feedback linearizes the sensor
input-output relationship and the controller is designed to
achieve null steady-state error and reduce the system response
time. Analysis of the behavior of the architecture modeled using
Simulink is presented for a NTC sensor. Simulation results are
presented and discussed, and the architecture is compared with
the classical constant-temperature anemometer (CTA) one.
Keywords—Hot-wire anemometer; thermoresistive sensor;
feedback linearization; constant temperature; controller
I. INTRODUCTION
Measurement systems that use thermoresistive sensors have the capability to exploit the variation of the electrical resistance as function of sensor temperature [1]. These systems can be used to measure temperature, incident radiation and fluid velocity [2].
Hot-wire anemometers are used for measuring fluid velocity and are usually implemented using architectures that maintain constant the sensor temperature (CTA, constant temperature anemometer) [3]. These anemometers can use thermoresistive sensors of PTC (Positive Temperature Coefficient) or NTC (Negative Temperature Coefficient) type.
In the CTA architecture, the sensor is heated by an electrical current, due to the Joule effect, up to a reference temperature [4]. The temperature is kept constant through the use of feedback control, which compensates the thermal variations and reduces the system response time. In the classical CTA architecture, the control function is implemented as a high gain proportional controller and it is carried on by an operational amplifier, using a Wheatstone bridge configuration, as shown in Fig. 1. The thermoresistive sensor is placed in one arm of the bridge and the other operates with fixed resistances. Any disturbance in the thermal balance causes a variation in sensor temperature, changing sensor resistance and making the bridge unbalanced. In this case, the operational amplifier acts varying its output voltage, compensating variations on the sensor temperature, making the temperature almost constant [5].
Fig. 1. Schematic of constant temperature anemometer architecture with
Wheatstone bridge.
One of main difficulties in designing controlled-temperature anemometers is that of the design of the feedback control, due to the nonlinear relationship between the applied voltage or current and the temperature response. This difficulty can be also observed for the CTA configuration, which, in some cases, can lead the system to take long time to stabilize or even become instable [6].
With the availability of low-cost microcontrollers, it is possible to digitally implement the anemometer temperature control system [7]. Contrasting with the analog architecture, with a digital system it is easier to explore diverse feedback control strategies. One strategy, to avoid the possible problems related to the sensor nonlinear behavior, is to employ linearization through voltage feedback [1].
This paper proposes an architecture of a hot-wire anemometer which performs the linearization of the thermoresistive sensor transfer function through output voltage feedback. Analysis of the behavior of the proposed architecture modeled using Simulink is presented for a NTC sensor. Simulation results are presented and discussed, and the architecture is compared with the classical constant temperature anemometer (CTA) one.
II. THEORETICAL REVIEW
Thermoresistive sensors can be classified as PTC or NTC. PTC sensors increase their resistance with the temperature. For a metallic PTC, the relation between these quantities is given by:
0 (1 )S SR R Tβ≅ + (1)
where R0 is the resistance of the sensor to 0 °C and β is the coefficient of temperature.
NTC sensors work inversely, i.e., decreasing their resistance with the increase of the temperature. The equation of this sensor is given by the simplified Steinhart-Hart model using only a temperature coefficient:
inf K
S
B
TR R e≅ (2)
where Rinf is the resistance of the sensor considering a temperature tending to infinite and TK is the temperature in Kelvin.
Apart from the temperature coefficient characteristic of the sensor, the first law of thermodynamics can be employed to express the sensor thermal balance. For applications in anemometry, the sensor should be in the form of a very thin wire to prevent the absorption of energy by incident radiation [2]. Excluding the effect of incident radiation, the sensor thermal balance can be written as:
( )2 Se S S S a
dTP R I hS T T mc
dt= = − + (3)
where Pe and IS are the electric power and current of the sensor, respectively, Ta is the ambient temperature and h is the heat transfer coefficient on the sensor surface. The sensor parameters are: the mass m, the specific heat c, and area S. In these instruments the value of h is not constant and is given by King’s equation:
nh a b= + υ (4)
where a, b e n are constants which can be experimentally
determined and υ is the fluid velocity.
The equations (2-4) constitute the NTC sensor model as it can be represented in Fig. 2. The inputs of this model are the current through to the sensor, the fluid velocity and the ambient temperature. The outputs are the sensor temperature and voltage.
Fig. 2. Block diagram model of the NTC sensor.
III. WB-CTA ARCHITECTURE
Traditional methods of measurement used in hot-wire anemometers are to maintain the voltage (CV), current (CCA) or temperature (CTA) constant [3]. The architecture proposed in this paper carries the control of the sensor temperature making it constant. Thus, it is important to show a comparison between the performance of the proposed and a consolidated architecture. To conduct this comparative analysis, the WB-CTA architecture was chosen because it is fairly consolidated and present advantages such as reduced time constant, good sensitivity and accuracy.
The WB-CTA architecture typically uses a thermoresistive sensor in a Wheatstone bridge driven by an operational amplifier. Fig. 3 shows the schematic diagram of the architecture using a NTC sensor placed in the Wheatstone bridge connected to the operational amplifier positive input. If the PTC sensor is to be used it would be placed in the other arm of the bridge.
Fig. 3. Esquematic diagram of the WB-CTA architecture using NTC
thermoresistive sensor.
The principle of operation of the architecture is as follows: The operational amplifier negative feedback aims to maintain the Wheatstone bridge balanced and, hence, the sensor temperature approximately constant. Any variations in the measurand, fluid velocity in this case, will tend to unbalance the bridge, causing the operational amplifier to vary its output to change the electric power dissipated by the thermoresistive sensor, returning the bridge to balance.
Analyzing the circuit, the output voltage of the operational amplifier, considering the input offset voltage, is given by:
( )o S OS
V A V V V−= + − (5)
where VS is the voltage on the sensor and V- is the voltage at
the inverting input. Moreover, in order to simplify the final equation, a parameter that relates two fixed resistors of the circuit is introduced. This parameter K is given by:
3 2
3
R R
KR
+= (6)
From equations (3-6) it is possible to obtain an expression of the output voltage of the operational amplifier:
( )( ) n SO S f S A OS
dTAKV R S a b T T mc V
A K dtυ
= + − + + +
(7)
It is possible to simplify (7) considering the gain, A, much greater than the parameter K. Furthermore, if the offset voltage, VOS, is small and can be disregarded, the simplified expression for reconstruction the fluid velocity can be found as:
12
( )
n
O
f
S S A
V
aK
bSR T T bυ
= − −
(8)
The WB-CTA architecture was simulated using Matlab and Simulink, using the parameters shown in Table 1.
TABLE I. PARAMETERS OF WB-CTA ARCHITECTURE
Parameters Values
A 100.000
VOS 1 mV
R1 500 Ω
K 3.5
Fig. 4 shows the behavior of the temperature sensor, in steady state, as a function of the fluid velocity. It can be seen that this architecture presents a small variation in temperature with increasing the fluid velocity.
Fig. 4. Temperature sensor as a function of fluid velocity.
Fig. 5 and 6 show the dynamic behavior of the output voltage, for two values of offset voltage 1 mV and 10 mV, respectively, considering a step variation on the fluid velocity, from 0 to 25 m/s, applied at time instant zero. It is possible to observe that the value of the operational amplifier input offset voltage influences the system behavior and performance strongly.
Fig. 5. WB-CTA output voltage as function of time for an step variation of
the fluid velocity, with VOS = 1 mV.
Fig. 6. WB-CTA output voltage as function of time for an step variation of
the fluid velocity, with VOS = 10 mV.
The fluid velocity was estimated from (8) and it is shown in Fig. 7, for the operational amplifier input offset voltage of 1 mV and for the same fluid velocity step. The settling time (using 2% criterion) obtained for this case was 42.73 ms and the percent overshoot 659.51%.
Fig. 7. Reconstruction of fluid velocity in WB-CTA architecture.
The proposed architecture aims to avoid these unwanted behaviors and obtain a better performance for estimating the fluid velocity.
IV. FEEDBACK LINEARIZATION
Equation (3) presented in section II shows that the relationship between the electric current and the temperature of the NTC sensor is nonlinear. On the other hand, the relationship between the electrical power and the temperature is linear. To make it possible to employ electrical power as the controlling variable, the architecture presented in Fig. 8 is considered.
Fig. 8. Feedback linearization configuration of the system with electrical
power as input.
Applying the Laplace transform in (3) the transfer function can be written as:
1
( )
( ) 1
th
e
GT s
P s sτ
−∆ =
+ (9)
where ( ) ( ) S a
T s T s T∆ = − , τ is the time constant which is given
by th
th
C
G, Cth and Gth are the thermal capacitance and the thermal
conductance, respectively. The thermal capacitance is calculated by the product mc and thermal conductance by the product hS.
This configuration makes linear the relationship between the input and output signal of the sensor system, which makes it possible to apply conventional control techniques in order to improve system performance.
V. PROPOSED ARCHITECTURE
The proposed architecture is shown in Fig. 9. For the sensor to operate at a heated temperature, the input signal is a reference temperature. A comparison between reference and sensor temperatures is made producing an error signal, which is the input of the controller. The controller generates a power signal output that is converted to a current signal by voltage feedback, making complete feedback control system linear.
Fig. 9. Proposed architecture.
In order simulate the proposed architecture, the parameters of a commercial sensor were used, and are shown in Table 2.
TABLE II. PARAMETERS OF COMMERCIAL SENSORS
Parameters Values
Rinf 0.0265 Ω
B 3068 K
Cth 7 J/K
Gth 4.284×10-4 W/K
a 2375 W/m².K
b 976 W.s0.5/m2.5.K
n 0.5
For the design of the controller, the requirements of simplicity, zero steady-state error and low system response time were considered. Thus, a comparison between proportional (P) and proportional-integral (PI) controllers is carried out to verify their adequacy to these requirements. The transfer function of P controller is given by:
( )P th PC s G K= (10)
The transfer function of the PI controller can be written as:
( ) IPI th P
KC s G K
s
= +
(11)
The transfer function in open loop is given by:
( )2
1
P I
P
P I
K Ks
KG s
K Ks s
τ
τ τ
+
=+
+ +
(12)
where 12P
n
Kξω
τ
+ =
and 2 I
n
Kω
τ= . The damping
coefficient ξ and natural frequency ωn were chosen to achieve a system time response of 5 ms, with 2% criterion. In Table 3 these values and gains of P and PI controller are presented.
TABLE III. PARAMETERS OF CONTROLLER AND SPECIFICATIONS CHOSEN
Parameters Values
1
800
25.08
10432
Similar to what was developed for the WB-CTA architecture is necessary to obtain an expression that estimates
the behavior of the fluid velocity to the proposed architecture. Through equation (3), the fluid velocity is given by:
1n
e
S A
f
Pa
T T
bυ
−
−
= (13)
To simulate the proposed architecture, the reference temperature of 100 °C was considered. The fluid velocity step of 0 m/s to 25 m/s was applied at time instant zero. Fig. 10 and 11 shows the power signal at the controller output and current signal at the input of the sensor, respectively, for both controllers.
Fig. 10. Power signal generated.
Fig. 11. Input current signal of the sensor .
Figure 12 shows the behavior of output temperature of the sensor for both controllers. The PI controller presents a zero steady-state error when the fluid velocity step is applied. The system response time, to reach 2% of the final value, was found about 2.46 ms as expected.
Fig. 12. Temperature response for both controllers.
From equation (14) it was possible to estimate the fluid velocity. In Figure 13 it is possible to observe this behavior, in which is again applied a step of 25 m/s. The settling time and the percent overshoot obtained were 7.95 ms and 28.71% respectively.
Fig. 13. Reconstruction of fluid velocity in proposed architecture.
VI. CONCLUSION
This paper presented an architecture of a controlled-temperature hot-wire anemometer using thermoresistive sensors that performs a linearization of the transfer function through a feedback output voltage. Furthermore, a control system was designed for the architecture to ensure that steady-state error is zero and low system response time. The proposed architecture was compared to the classical Wheatstone Bridge Constant Temperature Anemometer (WB-CTA).
Through simulations, it was possible to verify that the WB-CTA architecture has a rapid response but cannot keep the temperature constant as noted. Moreover, it was also found that it performance, in terms of response to a fast measurand variation, is strongly influenced by the operational amplifier input offset voltage.
The proposed architecture has shown it can maintain a constant temperature in the sensor due to use of a PI controller and it has a better response performance for a rapid variation of the fluid velocity when compared to the classical architecture.
ACKNOWLEDGMENT
The authors would like to thank the Federal University of Rio Grande do Norte (UFRN), CAPES e CNPq for the financial support and all colleagues for the technical support.
0 0.02 0.04 0.06 0.08 0.10
20
40
60
80
Time (s)
Pow
er (
mW
)
P controller
PI controller
0 0.02 0.04 0.06 0.08 0.110
15
20
25
Time (s)
Cu
rren
t (m
A)
P controller
PI controller
0 0.02 0.04 0.06 0.08 0.194
96
98
100
102
104
Time (s)
Tem
per
ature
(°C
)
P controller
PI controller
Reference
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