- Email: [email protected]
Resonance based DC current sensor
Measurement 45 (2012) 369–374
Contents lists available at SciVerse ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Resonance based DC current sensor B.V.M.P. Santhosh Kumar ⇑, K. Suresh, U. Varun Kumar, G. Uma, M. Umapathy Department of Instrumentation and Control Engineering, National Institute of Technology, Tiruchirappalli 620 015, India
a r t i c l e
i n f o
Article history: Received 31 March 2010 Received in revised form 22 July 2011 Accepted 15 November 2011 Available online 27 November 2011 Keywords: Current sensor Resonant sensor Cantilever beam Proximity sensor Piezoelectric
a b s t r a c t The paper presents design, development and testing of a resonant proximity DC current sensor to measure current in the range of 0–20 mA. The sensor is built using cantilever structure with piezoelectric excitation, sensing and closed loop electronics. The sensor measures the DC current by measuring the shift in resonance frequency of the cantilever beam. The proposed measurement system is novel, simple and accuracy is found to be 1.1% of full scale deflection. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Information regarding current flow is required in a wide range of electrical and electronics applications. Each application has different performance requirements in terms of cost, isolation, precision, bandwidth, measurement range, or size and many different current measurement methods have been developed to satisfy these demands. Current sensing principles based on either established techniques such as Hall-effect sensors and fluxgate principles or emerging technologies such as magneto resistance effect and fiber-optical techniques provide attractive alternatives for current sensing, although generally at a higher price than traditional techniques like Ohm’s law of resistance, trace resistance technique, Rogowski coil, current transformer, polarimeter detection method, interferometer detection method and magnetic induction method [1,2]. Today, current information increasingly needs to be available in digital form for digital control or monitoring purposes.
⇑ Corresponding author. E-mail addresses: [email protected] (B.V.M.P. Santhosh Kumar), [email protected] (G. Uma), [email protected] (M. Umapathy). 0263-2241/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.11.008
A resonant sensor is a device with an element vibrating at resonance, which changes its output frequency i.e. mechanical resonance frequency, as a function of a physical parameter. The frequency domain sensor output is digital in the sense that it is basically independent of analog levels and can be connected to digital circuitry. The conversion from the measurand to the resonance frequency of the vibrating element can be accomplished by means of a change in stress, mass, or shape of the resonator. Advantages of the resonance sensor are its high stability, high resolution and quasi digital output. The mechanical resonator structure has to be brought into vibration and the vibration has to be detected, excitation and detection technique can be seen in [3–5]. Among the excitation and detection methods used in resonant sensors, piezoelectric excitation and detection is gaining importance in recent years as the piezoelectric excitation offers the advantages like strong forces, low voltage, high energy efficiency, linear behavior, high acoustic quality, high speed and high frequency. The integration of piezoelectric materials in macro, meso, micro and nanosystems has made considerable progress in the last two decade [6,7]. Resonant sensors have been used in a wide range of sensing applications such as to measure force, acceleration and fluid flow characteristics [8–10]. In this paper, a new kind of resonant
370
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
based DC current sensor using cantilever beam with piezoelectric excitation and detection is proposed.
R Br
Bx
2. Measurement system The measurement system consists of a flexible aluminum beam clamped at one end is shown in Fig. 1. Two piezoceramic patches are surface bonded at a distance of lx from the fixed end of the beam. The piezoceramic patch bonded on the bottom surface acts as a sensor and the one on the top surface acts as an actuator or vice versa. A cylindrical disk type permanent magnet of flux density Br is mounted at the bottom surface of the free end of the cantilever beam and a current carrying coil having N turns is placed under the magnet at a distance of x.
L
x
Fig. 2. Magnet flux density at point x.
F current ¼ NIABx =Lc
ð2Þ
where N is the number of turns in the coil, I is the DC current through the coil, A is the cross sectional area of core and Lc is the length of the coil under magnetic field. From, Eqs. (1) and (2)
2
3
NIABr 6 Lþx x 7 ffi5 ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Lc 2 2 R þ x2 R þ ðL þ xÞ
3. Measurement principle
F current
To measure the DC current, the cantilever structure is placed close to the coil carrying DC current. The force between the permanent magnet and the current carrying coil induces additional stiffness (positive for repulsive force and negative for attractive force) on the structure and hence the resonant frequency of the structure gets altered. The closed loop electronics shown in Fig. 1 adapts to the changes and makes the structure to vibrate at its new resonance frequency [11]. This change in resonant frequency is the measure of the current through the coil.
The magnetic force between the permanent magnet and the coil induces an additional stiffness on the vibrating cantilever beam which in turn alters the resonant frequency of the beam. The additional stiffness induced from the magnetic force is positive for repulsive force and negative for attractive force; the attractive force can be represented as a spring under tension and the repulsive force as a spring under compression [12]. Thus, the stiffness due to magnetic force is
3.1. Theory
K current ¼
Considering a cylindrical permanent magnet in Fig. 2, the flux density of the permanent magnet at a distance x is given as
0
dF current dx NIABr ¼ 2Lc 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ ðL þ xÞ2 ðL þ xÞ2 ðR2 þ ðL þ xÞ2 Þ R2 þ x2 x2 ðR2 þ x2 ÞA @ ðR2 þ x2 Þ R2 þ ðL þ xÞ2
1
ð4Þ
Br B Lþx x C Bx ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2 2 R þ x2 R þ ðL þ xÞ
ð1Þ
where L is the length of the permanent magnet, R is the radius of the permanent magnet and Br is the flux density of the permanent magnet. Now, the force acting on the current carrying coil placed in the magnetic field is,
The lumped parameter model of the measurement system is shown in Fig. 3. The stiffness involved in the measurement system is the stiffness of the structure and the stiffness due to magnetic force, which depends on the current flow through the coil. The effective stiffness of the system for a given current flow would be smaller or greater than the beam stiffness.
Rf R1 Micro controller
+
LCD Display
R Piezoelectric Actuator lb
x z
Magnet
lx
x
Core
Piezoelectric Sensor C
ð3Þ
Lc
Fig. 1. Current measurement system using piezo actuated cantilever.
Current Input
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B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
The equation of motion for the measurement system is
y Kbeam
f
€ þ C y_ þ K eff y ¼ f M eff y
ð5Þ
Where,
K eff ¼ K beam K current
ð6Þ
33 ms 140
ð7Þ
meff
Kcurrent
M eff ¼ mtip þ
C Fig. 3. Lumped parameter model of the system.
with K beam ¼ 3EIl3m is the stiffness of the clamped-free beam, E is the Young’s modulus of the beam, Im is the moment of inertia of the beam, l is the length of the beam, ms is the mass of the cantilever beam, mtip is the mass of the permanent magnet placed at the tip of the beam, y is the displacement of the effective mass and f is the force acting
Table 1 Properties and dimensions of the measurement system. Symbol
Description
Value
Units
Br x L R Lc N
Residual flux density Distance between magnet and coil Length of the magnet Radius of the magnet and coil Length of the coil Number of turns in coil Permeability of intervening medium (air) Young’s modulus of the beam Length of the beam Width of the beam Thickness of the beam Location piezo from the fixed end Length of the piezo patch Width of the piezo patch Thickness of piezo patch Piezoelectric strain constant Piezoelectric stress constant Dielectric constant
1.2 5 2 3.25 16 2000 1.256 106 71 200 13 1.27 10 70.5 13 0.5 247 1012 9 103 3100
T mm mm mm mm – H m1 Gpa mm mm mm mm mm mm mm m V1 Vm N1 –
l0 E lb b h lx lp bp tp d31 g31 K T3
Cantilever beam
Piezoelectric actuator/sensor
Permanent magnet Electromagnetic coil
Closed loop electronics
Microcontroller with display
Fig. 4. Photograph of the measurement system.
372
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
(a)
4
Magnitude (abs)
0 mA 4 mA
3
8 mA 12 mA
2
16 mA 20 mA
1
Phase (deg)
0 0 -45 -90 -135 -180 22
23
24
25
26
27
28
29
30
31
32
Frequency (Hz)
(b)
30 Analytical
29.5
Experimental
29
Frequency (Hz)
28.5 28 27.5 27 26.5 26 25.5 25 0
2
4
6
8
10
12
14
16
18
20
current (mA) Fig. 5. Variation in natural frequency with current. (a) Frequency response of the measurement system and (b) comparison of experimental and analytical result.
on the cantilever beam due to current. The state space model of the measurement system is derived to be
X_ ¼ AX þ Bu;
Y ¼ CX
ð8Þ
Where, A is the system matrix, B is the input matrix, C is the output matrix, X is the state vector, u is the input to the actuator and Y is the measurement system output. The matrices, the state vector, and the input to the actuator are given as follows:
" A¼
0 K Meff eff
#
1 MC
eff
" ;B ¼
0 1 M eff
# ; C ¼ ½ 1 0 ; X ¼
y1 y2
Where the state variables are y1 ¼ y; y2 ¼ y_ and u = f. Hence the natural frequency of the measurement system (x) with current flow is defined as,
x¼
sffiffiffiffiffiffiffiffiffi K eff Meff
ð9Þ
4. Evaluation of measurement system The measurement system is designed and developed in the laboratory. The properties and dimensions of the measurement system components are given in Table 1. The
373
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
ARBITRARY WAVEFORM GENERATOR
Piezoelectric Actuator lb
x z
Magnet
lx
Core
x Piezoelectric Sensor
Current Input
Lc
FREQUENCY MEASUREMENT (OSCILLOSCOPE)
Fig. 6. Measurement system in open loop configuration.
29 Measured value Ture value
Frequency (Hz)
28.8
28.6
28.4
28.2
28
27.8
0
2
4
6
8
10
12
14
16
18
20
Current (mA) Fig. 7. Input–output characteristics showing accuracy.
29 Increasing order Decreasing order
28.8
Frequency (Hz)
performance of the measurement system is evaluated through experimentation and compared with the analytical results. The permanent magnet made of NdFeB having the radius and length of 3.25 mm and 2 mm is fixed at free end of the cantilever beam. The distance between the current carrying coil and the magnet is fixed as 5 mm. The piezoceramic patches (PZT-5H) are bonded at a distance of 10 mm from the fixed end. The patch bonded on the bottom surface acts as a sensor and the one on the top surface acts as an actuator. The photograph of the measurement system is shown in Fig. 4. The cantilever with piezo patch is excited using a closed loop electronics which makes the measurement system to vibrate at the natural frequency. When the current (I) is made to flow through the coil, the stiffness of the measurement system changes from Kbeam to Keff due to the addition of the stiffness Kcurrent, which in turns alter the natural frequency of the measurement system. The closed loop electronics acts on this change and vibrates the measurement system with the new natural frequency. Microcontroller is programmed to display the measured current from the frequency output of the closed loop electronics. Experiments are conducted to measure the current in the range of 0–20 mA. The theoretical result showing the variation of the natural frequency with the current is shown in Fig. 5a. The change in the natural frequency of the measurement system experimentally is found to vary linearly with current and close to the analytical results as shown in Fig. 5b. The zero bias in the experimental result is due to the non-inclusion of mass and stiffness of piezoelectric sensor and actuator in the theoretical analysis. To substantiate the experimental results shown in Fig. 5b, the current (0–20 mA) is measured by exciting the cantilever beam to resonance through a piezo actuator from the arbitrary waveform generator (measurement scheme in open loop) as shown in Fig. 6. Then the shift in natural frequency is measured for the variation in the input current flowing through the coil. The input–output characteristic obtained in open loop configuration is compared with the characteristics obtained with proposed measurement system in Fig. 7 and accuracy is found to be ±1.1% of full scale deflection.
28.6
28.4
28.2
28
27.8
0
2
4
6
8
10
12
14
16
18
20
Current (mA) Fig. 8. Input–output characteristics showing hysteresis.
The measurement system is also tested for hysteresis and repeatability the characteristics obtained are shown in Figs. 8 and 9 respectively. The maximum hysteresis is found to be
374
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
29
Frequency (Hz)
28.17
28.8
Frequency (Hz)
28.6
28.16
28.15
28.14 5.8
5.9
6
28.4
6.1
6.2
6.3
Current (mA)
28.2
Set 1 Set 2 Set 3
28
Set 4 Set 5
27.8 0
5
10
15
20
Current (mA) Fig. 9. Input–output characteristics showing repeatability.
approximately 5.69% of the full scale deflection. To evaluate the error in the measurement, repeated measurements are carried out over the measurement range, the standard deviation (r) and probable error (r) is found to vary from 0.3112 to 0.3126 Hz and 0.2099–0.2108 Hz respectively.
[3] [4]
[5]
5. Conclusion A novel resonant proximity sensor to measure the DC current using piezoelectric bonded cantilever beam is designed. The measurement system is experimentally tested and validated with theoretical results. The sensor is found be linear and the sensitivity is found to be 0.05 Hz/mA. The proposed system is simple in design and can be easily extended to design micro-DC current sensor. References [1] Silvio Ziegler, Robert C. Woodward, Herbert Ho-Ching Iu, Current sensing techniques: a review, IEEE Sensors Journal 9 (4) (2009) 354– 376. [2] Chucheng Xiao, Lingyin Zhao, Tadashi Asada, W.G. Odendaal, J.D. van Wyk, An overview of integratable current sensor technologies, in:
[6] [7] [8] [9]
[10] [11]
[12]
Industry Applications Conference, 38th IAS Annual Meeting, vol. 2, 2003, pp. 1251–1258. Goran Stemme, Resonant silicon sensors, Journal of Micromechanics and Microengineering 1 (1991) 113–125. K. Suresh, G. Uma, B.V.M.P. Santhosh Kumar, U. Varun kumar, M. Umapathy, Piezoelectric resonant mass sensor using phase measurement, Measurement 44 (2010) 320–325. G. Uma, M. Umapathy, A. Maria Josephine, S. Aishwarya, K. Suresh, Design of microcontroller based resonant sensor with piezoelectric excitation and detection, Instrumentation Science and Technology 36 (4) (2008) 367–374. E. Benes, M. Groschl, W. Burger, M. Schmid, Sensors based on piezoelectric resonators, Sensors and Actuators A 48 (1995) 1–21. Paul Muralt, Recent progress in materials issues for piezoelectric MEMS, Journal of America Ceramic Society 91 (5) (2008) 1385–1396. G. Murali Krishna, K. Rajanna, Tactile sensor based on piezoelectric resonance, IEEE Sensors Journal 4 (5) (2004) 691–697. V. Ferrari, A. Ghisla, D. Marioli, A. Taroni, Silicon resonant accelerometer with electronic compensation of input–output crosstalk, Sensors and Actuators A 123–124 (2005) 258–266. M. Shanmugavalli, M. Umapathy, G. Uma, Smart coriolis mass flowmeter, Measurement 43 (4) (2010) 549–555. G. Uma, K. Suresh, M. Umapathy, M. Shanmugavalli, Closed loop electronics for resonant sensors, in: Proceedings of ISSS 2008 International Conference on Smart Materials Structures and Systems, India, 2008, p. 90. Vinod R. Challa, M.G. Prasad, Yong Shi, Frank T. Fisher, A vibration energy harvesting device with bidirectional resonance frequency tunability, Smart Materials and Structures 17 (2008) 015035. pp. 10.
Contents lists available at SciVerse ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Resonance based DC current sensor B.V.M.P. Santhosh Kumar ⇑, K. Suresh, U. Varun Kumar, G. Uma, M. Umapathy Department of Instrumentation and Control Engineering, National Institute of Technology, Tiruchirappalli 620 015, India
a r t i c l e
i n f o
Article history: Received 31 March 2010 Received in revised form 22 July 2011 Accepted 15 November 2011 Available online 27 November 2011 Keywords: Current sensor Resonant sensor Cantilever beam Proximity sensor Piezoelectric
a b s t r a c t The paper presents design, development and testing of a resonant proximity DC current sensor to measure current in the range of 0–20 mA. The sensor is built using cantilever structure with piezoelectric excitation, sensing and closed loop electronics. The sensor measures the DC current by measuring the shift in resonance frequency of the cantilever beam. The proposed measurement system is novel, simple and accuracy is found to be 1.1% of full scale deflection. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Information regarding current flow is required in a wide range of electrical and electronics applications. Each application has different performance requirements in terms of cost, isolation, precision, bandwidth, measurement range, or size and many different current measurement methods have been developed to satisfy these demands. Current sensing principles based on either established techniques such as Hall-effect sensors and fluxgate principles or emerging technologies such as magneto resistance effect and fiber-optical techniques provide attractive alternatives for current sensing, although generally at a higher price than traditional techniques like Ohm’s law of resistance, trace resistance technique, Rogowski coil, current transformer, polarimeter detection method, interferometer detection method and magnetic induction method [1,2]. Today, current information increasingly needs to be available in digital form for digital control or monitoring purposes.
⇑ Corresponding author. E-mail addresses: [email protected] (B.V.M.P. Santhosh Kumar), [email protected] (G. Uma), [email protected] (M. Umapathy). 0263-2241/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.11.008
A resonant sensor is a device with an element vibrating at resonance, which changes its output frequency i.e. mechanical resonance frequency, as a function of a physical parameter. The frequency domain sensor output is digital in the sense that it is basically independent of analog levels and can be connected to digital circuitry. The conversion from the measurand to the resonance frequency of the vibrating element can be accomplished by means of a change in stress, mass, or shape of the resonator. Advantages of the resonance sensor are its high stability, high resolution and quasi digital output. The mechanical resonator structure has to be brought into vibration and the vibration has to be detected, excitation and detection technique can be seen in [3–5]. Among the excitation and detection methods used in resonant sensors, piezoelectric excitation and detection is gaining importance in recent years as the piezoelectric excitation offers the advantages like strong forces, low voltage, high energy efficiency, linear behavior, high acoustic quality, high speed and high frequency. The integration of piezoelectric materials in macro, meso, micro and nanosystems has made considerable progress in the last two decade [6,7]. Resonant sensors have been used in a wide range of sensing applications such as to measure force, acceleration and fluid flow characteristics [8–10]. In this paper, a new kind of resonant
370
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
based DC current sensor using cantilever beam with piezoelectric excitation and detection is proposed.
R Br
Bx
2. Measurement system The measurement system consists of a flexible aluminum beam clamped at one end is shown in Fig. 1. Two piezoceramic patches are surface bonded at a distance of lx from the fixed end of the beam. The piezoceramic patch bonded on the bottom surface acts as a sensor and the one on the top surface acts as an actuator or vice versa. A cylindrical disk type permanent magnet of flux density Br is mounted at the bottom surface of the free end of the cantilever beam and a current carrying coil having N turns is placed under the magnet at a distance of x.
L
x
Fig. 2. Magnet flux density at point x.
F current ¼ NIABx =Lc
ð2Þ
where N is the number of turns in the coil, I is the DC current through the coil, A is the cross sectional area of core and Lc is the length of the coil under magnetic field. From, Eqs. (1) and (2)
2
3
NIABr 6 Lþx x 7 ffi5 ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Lc 2 2 R þ x2 R þ ðL þ xÞ
3. Measurement principle
F current
To measure the DC current, the cantilever structure is placed close to the coil carrying DC current. The force between the permanent magnet and the current carrying coil induces additional stiffness (positive for repulsive force and negative for attractive force) on the structure and hence the resonant frequency of the structure gets altered. The closed loop electronics shown in Fig. 1 adapts to the changes and makes the structure to vibrate at its new resonance frequency [11]. This change in resonant frequency is the measure of the current through the coil.
The magnetic force between the permanent magnet and the coil induces an additional stiffness on the vibrating cantilever beam which in turn alters the resonant frequency of the beam. The additional stiffness induced from the magnetic force is positive for repulsive force and negative for attractive force; the attractive force can be represented as a spring under tension and the repulsive force as a spring under compression [12]. Thus, the stiffness due to magnetic force is
3.1. Theory
K current ¼
Considering a cylindrical permanent magnet in Fig. 2, the flux density of the permanent magnet at a distance x is given as
0
dF current dx NIABr ¼ 2Lc 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ ðL þ xÞ2 ðL þ xÞ2 ðR2 þ ðL þ xÞ2 Þ R2 þ x2 x2 ðR2 þ x2 ÞA @ ðR2 þ x2 Þ R2 þ ðL þ xÞ2
1
ð4Þ
Br B Lþx x C Bx ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2 2 R þ x2 R þ ðL þ xÞ
ð1Þ
where L is the length of the permanent magnet, R is the radius of the permanent magnet and Br is the flux density of the permanent magnet. Now, the force acting on the current carrying coil placed in the magnetic field is,
The lumped parameter model of the measurement system is shown in Fig. 3. The stiffness involved in the measurement system is the stiffness of the structure and the stiffness due to magnetic force, which depends on the current flow through the coil. The effective stiffness of the system for a given current flow would be smaller or greater than the beam stiffness.
Rf R1 Micro controller
+
LCD Display
R Piezoelectric Actuator lb
x z
Magnet
lx
x
Core
Piezoelectric Sensor C
ð3Þ
Lc
Fig. 1. Current measurement system using piezo actuated cantilever.
Current Input
371
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
The equation of motion for the measurement system is
y Kbeam
f
€ þ C y_ þ K eff y ¼ f M eff y
ð5Þ
Where,
K eff ¼ K beam K current
ð6Þ
33 ms 140
ð7Þ
meff
Kcurrent
M eff ¼ mtip þ
C Fig. 3. Lumped parameter model of the system.
with K beam ¼ 3EIl3m is the stiffness of the clamped-free beam, E is the Young’s modulus of the beam, Im is the moment of inertia of the beam, l is the length of the beam, ms is the mass of the cantilever beam, mtip is the mass of the permanent magnet placed at the tip of the beam, y is the displacement of the effective mass and f is the force acting
Table 1 Properties and dimensions of the measurement system. Symbol
Description
Value
Units
Br x L R Lc N
Residual flux density Distance between magnet and coil Length of the magnet Radius of the magnet and coil Length of the coil Number of turns in coil Permeability of intervening medium (air) Young’s modulus of the beam Length of the beam Width of the beam Thickness of the beam Location piezo from the fixed end Length of the piezo patch Width of the piezo patch Thickness of piezo patch Piezoelectric strain constant Piezoelectric stress constant Dielectric constant
1.2 5 2 3.25 16 2000 1.256 106 71 200 13 1.27 10 70.5 13 0.5 247 1012 9 103 3100
T mm mm mm mm – H m1 Gpa mm mm mm mm mm mm mm m V1 Vm N1 –
l0 E lb b h lx lp bp tp d31 g31 K T3
Cantilever beam
Piezoelectric actuator/sensor
Permanent magnet Electromagnetic coil
Closed loop electronics
Microcontroller with display
Fig. 4. Photograph of the measurement system.
372
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
(a)
4
Magnitude (abs)
0 mA 4 mA
3
8 mA 12 mA
2
16 mA 20 mA
1
Phase (deg)
0 0 -45 -90 -135 -180 22
23
24
25
26
27
28
29
30
31
32
Frequency (Hz)
(b)
30 Analytical
29.5
Experimental
29
Frequency (Hz)
28.5 28 27.5 27 26.5 26 25.5 25 0
2
4
6
8
10
12
14
16
18
20
current (mA) Fig. 5. Variation in natural frequency with current. (a) Frequency response of the measurement system and (b) comparison of experimental and analytical result.
on the cantilever beam due to current. The state space model of the measurement system is derived to be
X_ ¼ AX þ Bu;
Y ¼ CX
ð8Þ
Where, A is the system matrix, B is the input matrix, C is the output matrix, X is the state vector, u is the input to the actuator and Y is the measurement system output. The matrices, the state vector, and the input to the actuator are given as follows:
" A¼
0 K Meff eff
#
1 MC
eff
" ;B ¼
0 1 M eff
# ; C ¼ ½ 1 0 ; X ¼
y1 y2
Where the state variables are y1 ¼ y; y2 ¼ y_ and u = f. Hence the natural frequency of the measurement system (x) with current flow is defined as,
x¼
sffiffiffiffiffiffiffiffiffi K eff Meff
ð9Þ
4. Evaluation of measurement system The measurement system is designed and developed in the laboratory. The properties and dimensions of the measurement system components are given in Table 1. The
373
B.V.M.P. Santhosh Kumar et al. / Measurement 45 (2012) 369–374
ARBITRARY WAVEFORM GENERATOR
Piezoelectric Actuator lb
x z
Magnet
lx
Core
x Piezoelectric Sensor
Current Input
Lc
FREQUENCY MEASUREMENT (OSCILLOSCOPE)
Fig. 6. Measurement system in open loop configuration.
29 Measured value Ture value
Frequency (Hz)
28.8
28.6
28.4
28.2
28
27.8
0
2
4
6
8
10
12
14
16
18
20
Current (mA) Fig. 7. Input–output characteristics showing accuracy.
29 Increasing order Decreasing order
28.8
Frequency (Hz)
performance of the measurement system is evaluated through experimentation and compared with the analytical results. The permanent magnet made of NdFeB having the radius and length of 3.25 mm and 2 mm is fixed at free end of the cantilever beam. The distance between the current carrying coil and the magnet is fixed as 5 mm. The piezoceramic patches (PZT-5H) are bonded at a distance of 10 mm from the fixed end. The patch bonded on the bottom surface acts as a sensor and the one on the top surface acts as an actuator. The photograph of the measurement system is shown in Fig. 4. The cantilever with piezo patch is excited using a closed loop electronics which makes the measurement system to vibrate at the natural frequency. When the current (I) is made to flow through the coil, the stiffness of the measurement system changes from Kbeam to Keff due to the addition of the stiffness Kcurrent, which in turns alter the natural frequency of the measurement system. The closed loop electronics acts on this change and vibrates the measurement system with the new natural frequency. Microcontroller is programmed to display the measured current from the frequency output of the closed loop electronics. Experiments are conducted to measure the current in the range of 0–20 mA. The theoretical result showing the variation of the natural frequency with the current is shown in Fig. 5a. The change in the natural frequency of the measurement system experimentally is found to vary linearly with current and close to the analytical results as shown in Fig. 5b. The zero bias in the experimental result is due to the non-inclusion of mass and stiffness of piezoelectric sensor and actuator in the theoretical analysis. To substantiate the experimental results shown in Fig. 5b, the current (0–20 mA) is measured by exciting the cantilever beam to resonance through a piezo actuator from the arbitrary waveform generator (measurement scheme in open loop) as shown in Fig. 6. Then the shift in natural frequency is measured for the variation in the input current flowing through the coil. The input–output characteristic obtained in open loop configuration is compared with the characteristics obtained with proposed measurement system in Fig. 7 and accuracy is found to be ±1.1% of full scale deflection.
28.6
28.4
28.2
28
27.8
0
2
4
6
8
10
12
14
16
18
20
Current (mA) Fig. 8. Input–output characteristics showing hysteresis.
The measurement system is also tested for hysteresis and repeatability the characteristics obtained are shown in Figs. 8 and 9 respectively. The maximum hysteresis is found to be
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Frequency (Hz)
28.17
28.8
Frequency (Hz)
28.6
28.16
28.15
28.14 5.8
5.9
6
28.4
6.1
6.2
6.3
Current (mA)
28.2
Set 1 Set 2 Set 3
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Set 4 Set 5
27.8 0
5
10
15
20
Current (mA) Fig. 9. Input–output characteristics showing repeatability.
approximately 5.69% of the full scale deflection. To evaluate the error in the measurement, repeated measurements are carried out over the measurement range, the standard deviation (r) and probable error (r) is found to vary from 0.3112 to 0.3126 Hz and 0.2099–0.2108 Hz respectively.
[3] [4]
[5]
5. Conclusion A novel resonant proximity sensor to measure the DC current using piezoelectric bonded cantilever beam is designed. The measurement system is experimentally tested and validated with theoretical results. The sensor is found be linear and the sensitivity is found to be 0.05 Hz/mA. The proposed system is simple in design and can be easily extended to design micro-DC current sensor. References [1] Silvio Ziegler, Robert C. Woodward, Herbert Ho-Ching Iu, Current sensing techniques: a review, IEEE Sensors Journal 9 (4) (2009) 354– 376. [2] Chucheng Xiao, Lingyin Zhao, Tadashi Asada, W.G. Odendaal, J.D. van Wyk, An overview of integratable current sensor technologies, in:
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