Fractional Order Linear Variable Differential Transformer: Design and analysis

Accepted Manuscript Fractional order Linear Variable Differential Transformer: Design and Analysis Parthasarathi Veeraian, Uma Gandhi, Umapathy Mangalanathan PII: DOI: Reference:

S1434-8411(17)30217-0 http://dx.doi.org/10.1016/j.aeue.2017.05.037 AEUE 51908

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International Journal of Electronics and Communications

Received Date: Accepted Date:

30 January 2017 23 May 2017

Please cite this article as: P. Veeraian, U. Gandhi, U. Mangalanathan, Fractional order Linear Variable Differential Transformer: Design and Analysis, International Journal of Electronics and Communications (2017), doi: http:// dx.doi.org/10.1016/j.aeue.2017.05.037

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Fractional order Linear Variable Differential Transformer: Design and Analysis Parthasarathi Veeraian1, Uma Gandhi2 and Umapathy Mangalanathan3,* 1

Research Scholar, 2Associate Professor, 3Professor Department of Instrumentation and Control Engineering National Institute of Technology, Tiruchirappalli, India. E-mail: [email protected], [email protected], 3,*[email protected]. Abstract Linear Variable Differential Transformer (LVDT) is the most preferred displacement measurement transducer used in industries because of its simple design and proven reliability. The major challenge faced in LVDT is linearity in limited stroke range. In this paper, a fractional order model of LVDT is developed and its characteristics like sensitivity, nonlinearity and stroke range are analysed by considering fractional behaviour only in selfinductance, mutual inductance and both. The results presented in this paper shows that Fractional Order Linear Variable Differential Transformer (FOLVDT) provides high sensitivity with reduced nonlinearity compared to the conventional LVDT. The results also demonstrate that the FOLVDT can be operated for increased stroke range without compromising on sensitivity and linearity. Keywords: LVDT, self-inductance, mutual-inductance, fractional order, displacement measurement. 1. Introduction Fractional calculus has been extensively used in engineering design and modelling [1, 2]. Many researchers and scientists have contributed to the development and application of fractional calculus [3]. I. Podlubny [3] gave the geometrical and physical interpretation of fractional integration, fractional differentiation and the initial conditions of fractional

differential equations. Fractional differentiation with respect to time can be interpreted as an existence of memory effects which correspond to intrinsic dissipation in the system [4], as fractional derivatives are characterised by long term memory effects, it is preferred over classical calculus for describing many physical systems like distributed transmission lines [5], biological tissues [6], viscoelastic materials [7] and flexible structures [8]. Fractional order diff erential operators are used to describe real-world complex relaxation and diffusion phenomena [9-10]. The presence of extra fractional-order parameters adds more flexibility in modelling and analysis of many electrical systems like modelling of lossy capacitor [11], lossy coils [12], constant phase elements [13], fractal behaviour of a metal insulator solution interface [14], electromagnetics [15] and heat conduction problem [16]. Nidhi et al., [17] proposed selective image encryption in fractional wavelet transform domain. Conventional circuit theorem has been extended to fractional order systems and applied for designing fractional order sinusoidal oscillator [18], filters [19] and multivibrator [20]. A study of fractional order oscillators based on operational trans-resistance amplifiers is described in [21]. A generalized study of fractional order impedance analysis is carried out by Radwan and Salama [22-23]. In analog-to-digital converter to improve the design accuracy at high frequency region, fractional Euler transform based on the digital fractional delay operator has been proposed by Tahar and Abdelfatah [24]. Implementation of fractional order system is an emerging research area in recent years, a simple method for fabricating a constant phase element is addressed in [25, 26]. Switched capacitor technology is used for developing integrated circuit implementation of fractional order elements [27]. Hardware realization of fractional order operators using FPGA is proposed in [28]. Analog implementation technique for fractional-order controller using Operational Trans Conductance Amplifiers (OTCA) is proposed by Ilias Dimeas et al., [29]. Georgia Tsirimokou et al., constructed constant phase emulators using OTCA and capacitors and implemented in monolithic form through the

Analog and Mixed Signal (AMS) 0.35 μm CMOS process [30]. Prototypes for implementing fractional-order capacitors and inductors are presented in [31, 32]. Linear variable differential transformer (LVDT) has wide applications in high accuracy displacement measurement [33]. LVDT is the most preferred choice for the measurement of displacement, pressure, force, level, flow, and other physical quantities in engineering applications and in industries due to their ruggedness, long mechanical life and high resolution [34, 35]. The performance of the LVDT is subjective to the transducer geometry, the influence of physical parameters on linearity and sensitivity are examined in [36]. If full range of LVDT is used the accuracy of measurement gets affected due to the presence of nonlinearity in the input-output characteristics. The problem of restricted linear range is overcome by sophisticated precise adjustment of windings [37, 38] and applying signal conditioning [39, 40]. Artificial neural network is used to effectively compensate for the nonlinearity existing in the sensors [33]. In this work a Fractional Order Linear Variable Differential Transformer (FOLVDT) is proposed for improving the sensitivity, linearity and the stroke range. As LVDT works on the principle of mutual induction between the primary and two secondary windings, the relation between the input and output of the LVDT is derived using the concept of mutually coupled circuits. The conventional models used to describe the coupling between the coils doesn’t account for the losses in the network, hence they fail in accurately describing the real behaviour of such circuits. The concept of fractional-order mutual coupled circuits is proposed in [41], which could be used instead of the integer-order mutual inductance because it increases the design degree of freedom. The phase and magnitude response of the generalized mutual inductance is analysed, and an equivalent circuit for the fractional mutual inductance based on the differential voltage current controlled conveyor transconductance amplifier is presented to study the behaviour of the fractional-order mutual inductance for different fractional order.

In this work LVDT designed by considering fractional order self and mutual inductance, is first of its kind in the literature. Introduction of fractional order inductors in LVDT design modifies the conventional characteristics of the system. In the proposed system the extra independent fractional order terms, increases the design freedom in choosing the parameters of the transducer, which facilitates the multi objective design like obtaining higher sensitivity, reduced nonlinearity and enhanced stroke range. Modern studies demonstrated the presence of fractional behaviour in the circuit elements and this incites the chance of implementing fractional device in near future. The analysis and results of FOLVDT presented in this article will facilitate the instrumentation system designer a new approach of designing transducers using fractional calculus. 2. Review of fractional calculus concepts There are number of definitions in the literature for fractional differentiation of which Riemann–Liouville, Riesz, Weyl, Grünwald–Letnikov, and the Caputo derivative were extensively used. Unfortunately Riemann–Liouville approach leads to fractional order initial conditions which do not have a direct known physical meaning. A solution to this conflict was proposed by Caputo in 1967 [42], who defined a fractional derivative allowing the application of initial conditions with physical meaning which can be handled by using an analogy with the classical integer case. The Caputo derivative is defined as [3] C 0

Dt f (t ) 

t 1 f ( n ) ( ) d  (n   ) 0 (t  )  n 1

where C0 Dt 

(1)

d is the Caputo derivative of order   R with respect to time (t), () is gamma dt 

function and n is an integer such that n  1    n . Laplace transform of Caputo derivative is L



C  0 Dt



f (t )  s F ( s) 

m 1

 s k 0

 k 1

f ( k ) (0).

(2)

Existence of ideal inductance in reality is merely a hypothesis. Inductance for real time application is built by conductor loops or coils. The ideal relation between the current through and voltage across the inductor vL  L

di does not describe the real time inductance dt

accurately. Several methods were proposed in the literature to model an inductor taking into account the losses due to ohmic resistance, eddy-current and hysteresis. In the field of mechanics the behaviour of magnetic core coils is comparable to that of viscoelastic materials [43] and this motivated to use fractional calculus to model coils. In [31, 12] fractional derivatives are used for modelling coils and showed that such models provide clearly more realistic description than the conventional model. In [31] fractional order model of the coil having inductance L is given by vL  RCu i  L

d i dt 

(3)

vL is the voltage drop across the coil, i is the current flowing through it, RCu is the ohmic

resistance and  is the degree of derivation between 0 and 1. Since the conventional mutually coupled circuits are lossless networks, which is not practical, in [41] authors introduced the concept of mutual inductance in the fractional-order domain to model practical coupled circuits. 3. Description of LVDT and its fractional order model 3.1. Description Linear variable differential transformer (LVDT) measures the displacements as small as few millionths of an inch to several inches. Generally LVDT is designed for required operating stroke range and sensitivity which depends on its structural parameters [34]. The schematic of LVDT is shown in Fig. 1; it consists of a primary winding of length h1 arranged in line with two identical secondary windings of length h2 on either side. The windings have an inner diameter d and outer diameter D. A moveable core of length ha made of high-permeable

magnetic material is placed running through the centre of the three coils. Length of core extending into the secondary windings when the core is at the centre is t0 and x is displacement to be measured. The output of the LVDT is voltage across the secondary windings, which are connected in series opposition. The flux linkage between the primary and secondary windings changes with the core position. The output measure is based on the variation in mutual inductance between the primary winding and each of two secondary windings when the core moves due to subjected displacement. The displacement to be measured is proportional to the position of the core which is extracted from the output voltage.

h2

h1

h2 x

D d

-

0

Rm

Vo

+

t0 ha Fig. 1. Schematic diagram of LVDT construction

3.2. Fractional order model Rp

ip Vp

Rs 2

M1

C O R E

Lp

Ls 2

is

Rs 2 Ls 2

M2 x Fig. 2. Electrical equivalent circuit of LVDT

Electrical equivalent of LVDT shown in Fig. 2 is considered for deriving the mathematical model of the LVDT. The inductance of primary and secondary windings Lp and Ls are considered to exhibit fractional behaviour of order α. M1 and M2 are fractional order mutual inductance between the primary and each secondary respectively with an order β. Rp is the

resistance of the primary coil,

Rs is the resistance of each secondary coil, V p is primary 2

excitation voltage, Vo is output voltage measured across the load resistance Rm. Let the current through the primary and secondary coils be ip and is respectively, using equation (3) the mesh equations are v p  i p Rp  Lp

d i p dt 

  M 2  M1 

d is dt 

d i p d  is 0  is  Rs  Rm   Ls    M 2  M 1   dt dt

(4)

Applying Laplace transform to equation (4) with zero initial conditions,

V p  I p R p  s  Lp I p  I s s   M 2  M 1  0  I s  Rs  Rm  s  Ls   I p s  M 2  M1 

(5)

Output voltage of the Fractional Order Linear Variable Differential Transformer (FOLVDT) which is direct indication of the displacement to be measured is derived from the above equations and is given by Vo  Rm I s 

Vp Rm (m) s

Ls Lp s 2  (m)2 s 2   Lp  Rs  Rm   Ls Rp  s   Rp  Rs  Rm 

(6)

where m  M1  M 2 Mutual inductance between the primary and two secondary coils depends on the position of the core with respect to the secondary windings. When the core is displaced by x the change in mutual inductance m is given by [44]

 0 R 2 x gt x  m  M1  M 2  N1 N 2  2  0  2 2 h2   h2  2h2t0  t0  x

(7)

where N1 and N 2 are number of turns in primary and each secondary coils, 0  4107 Hm-1 is the permeability of free space, R is the flux effective radius determined by the magneticcore radius and its air gap (approximately equal to zero), g is the specific magnetic

conductance, for cylindrical surface it is given by g 

20  . The sensitivity of FOLVDT ln  D d 

is derived to be Ls Lp s 2  (m)2 s 2   Lp  Rs  Rm   Rp Ls  s   Rp  Rs  Rm   0 R 2  h22  2h2t0  t02  x 2  gto  dVo    N1 N 2Vp Rm s  2 2 2 2 2 d x h2  Ls Lp s 2  (m)2 s 2   Lp  Rs  Rm   Rp Ls  s   Rp  Rs  Rm    h2  2h2t0  t0  x  





(8) Sensitivity expression for conventional LVDT is obtained by substituting α=β=1 in the above equation. For a LVDT with known physical parameters, the sensitivity is a function of the core position or displacement of the core with respect to the null point ( x ). From the above expression it is clear that the relation between x and sensitivity is not linear. However for smaller displacement x << t0 , neglecting the higher order terms of x , the eq. (7) reduces to

m  M1  M 2  x , and thus the sensitivity

dVo will be constant, hence LVDT operation d x

is restricted to a smaller stroke range. In the full range of LVDT the accuracy of measurement gets affected due to the presence of nonlinearity. 4. Analysis of FOLVDT characteristics In this section the characteristics of conventional LVDT and FOLVDT are studied by considering a commercial general purpose AC LVDT (MHR010) supplied by Measurement Specialties TM , and its specifications are given in Table 1. Table 1. Specifications of LVDT MHR 010 Parameter Stroke range (mm) Length of primary (h1) in mm Length of secondary (h2) in mm Inner radius of the coil (d) in mm Outer radius of the coil (D) in mm Core length (ha) in mm Primary impedance in ohm Secondary impedance in ohm

Value 0.25 4.50 4.50 1.59 4.76 6.00 165 300

4.1. Conventional LVDT For α=β=1 the equation (6) represents the output voltage of conventional LVDT and is given by

Vo 

L L s

Vp Rm (m)s

p

 (m)2  s 2   Lp  Rs  Rm   Ls Rp  s  Rp  Rs  Rm 

(9)

Stroke range, sensitivity and non-linearity are the important characteristics of an LVDT and hence these characteristics are studied in detail. The output voltage of the LVDT with respect to the input displacement is shown in the Fig. 3(a). The input-output characteristics of the LVDT shows that the variation in the output with respect to the displacement is linear only for small range of displacement x, for the LVDT model MHR010 the linear operating range is mm, which is only 30% of the total possible stroke. The total possible stroke is mm. The sensitivity remains constant in the linear operating range and it diverges for larger displacement as shown in Fig. 3(c).

Fig. 3. Characteristics of conventional LVDT (a) Magnitude (b) Phase and (c) sensitivity

4.2. FOLVDT In analysing FOLVDT, three different cases are considered: In case 1, self-inductance alone is considered to exhibit fractional behaviour, in case 2 mutual inductance alone is considered to exhibit fractional behaviour and in case 3 both self and mutual inductance are considered to exhibit fractional behaviour. 4.2.1 Case 1: The input-output characteristics of FOLVDT for different α with β=1 is shown in Fig. 4; It is observed that the sensitivity increases with decrease in α. The results in Fig. 5 shows the variation in sensitivity and nonlinearity with α. It is evident from the results that the sensitivity of FOLVDT is considerably higher as compared to the sensitivity of conventional LVDT although this is desirable, the nonlinearity is found to increase. From these results it is concluded that the presence of fractional order self-inductance in the system increases the sensitivity at the cost of reduced linearity.

Fig. 4. Input-output characteristics of FOLVDT with fractional self-inductance

Fig. 5. Variation in sensitivity and nonlinearity of FOLVDT with α

4.2.2 Case 2: The input-output characteristics of the FOLVDT with fractional mutual inductance of different order β with α=1is shown in Fig. 6. It is observed from the results in Fig.6 that the output of FOLVDT is linear even for displacements larger than that of the given stroke range. This suggests that stroke range of LVDT can be enhanced with fractional mutual-inductance. The variation in sensitivity and nonlinearity with respect to β is shown in Fig. 7. It is evident from the results that the nonlinearity of FOLVDT with fractional mutual inductance is less compared to the nonlinearity of the conventional LVDT with a decrease in the sensitivity.

Fig. 6. Input-Output characteristics of FOLVDT with fractional mutual-inductance

Fig. 7. Variation in sensitivity and nonlinearity of FOLVDT with β

4.3.3 Case 3: The input-output response of FOLVDT with fractional self and mutual inductance of same order (α=β) is shown in Fig. 8. It is observed that the variation in the input-output characteristics with different fractional order is not following any trend as observed in case 1 or case 2. The results in Fig. 9 show the variation in sensitivity and nonlinearity for 100 % stroke (0.25mm) as compared with the conventional LVDT. The following conclusions are drawn (i) For (α=β) > 0.8, sensitivity is found to be higher with reduced nonlinearity as compared to the conventional LVDT. (ii)For (α=β) < 0.8, the nonlinearity of FOLVDT is less compared with conventional system but with decreased sensitivity.

Fig. 8. Input-Output characteristics of FOLVDT with fractional self and mutual-inductance

Fig. 9. Variation in sensitivity and nonlinearity of FOLVDT with fractional order

In case 1, the fractional order system has sensitivity up to 12 times the sensitivity of the conventional system. In case 2 the fractional order system has reduced sensitivity, in this case the sensitivity of the fractional order system reduces below 0.0625 (1/16) times the sensitivity of the conventional system, which is clearly seen in zoomed portion of Fig. 7. Case 3 is a combination of the above two cases, hence it accounts for both increase in sensitivity and decrease in sensitivity as seen in case 1 and 2 separately. So the overall change in the sensitivity of the fractional order system with fractional self and fractional mutual inductance has a maximum sensitivity of 1.6 times the sensitivity of the conventional system. Further analysis on FOLVDT is carried out to explore the possibility of increasing the stroke range. In this analysis 100% (0.25 mm), 150% (0.375 mm), 200% (0.5 mm) and 300% (0.75 mm) stroke ranges are considered for different fractional order values of α=β. Output voltage at stroke end, sensitivity and nonlinearity obtained from this analysis is presented in Table 2. It is observed from the results in Table 2 that for certain values of fractional order the FOLVDT has higher sensitivity and reduced nonlinearity as compared to conventional LVDT even for increased stroke range. The variation in the sensitivity of FOLVDT with respect to the fractional order for stroke ranges

and

is shown in Fig. 10. It can be concluded from the results that by choosing the

suitable fractional order self and mutual inductance the FOLVDT can be used for larger stroke range, with no loss in sensitivity and linearity as compared to the conventional LVDT, and these combinations are highlighted in Table 2. Moreover if FOLVDT is operated in the specified stroke range (0.25 mm) enhanced sensitivity and reduced nonlinearity is obtained as compared to the conventional LVDT.

α=β 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

Table 2. Variation in sensitivity and nonlinearity of FOLVDT for different stroke range Output at stroke end (mV) Sensitivity (mV/V/mm) Non-linearity % of FR 100% 150% 200% 300% 100% 150% 200% 300% 100% 150% 200% 300% 60.4 88.8 115.2 160.3 240.4 236.1 230.0 213.3 0.603 1.361 2.428 5.507 83.7 124.2 163.0 233.6 333.5 330.3 325.4 311.0 0.333 0.761 1.379 3.261 95.8 143.3 190.4 282.2 381.6 381.2 380.1 375.8 0.070 0.165 0.312 0.827 89.1 133.9 179.0 270.4 354.8 356.0 357.2 360.1 0.062 0.137 0.240 0.510 69.2 104.0 139.3 211.3 275.5 276.7 278.0 281.4 0.086 0.193 0.345 0.785 47.3 71.2 95.2 144.3 188.5 189.3 190.1 192.2 0.077 0.174 0.312 0.716 30.1 45.2 60.5 91.6 119.8 120.2 120.7 121.9 0.069 0.156 0.280 0.643 18.4 27.6 36.9 55.9 73.1 73.4 73.7 74.4 0.065 0.148 0.265 0.610 11.0 16.5 22.0 33.3 43.7 43.8 44.0 44.4 0.064 0.145 0.260 0.599 6.5 9.7 13.0 19.6 25.7 25.8 25.9 26.2 0.064 0.145 0.259 0.595 3.8 5.7 7.6 11.5 15.0 15.1 15.1 15.3 0.064 0.144 0.259 0.595

Fig. 10. Sensitivity of FOLVDT with fractional order for different stroke range

Sensitivity and nonlinearity of FOLVDT in the given stroke range of the LVDT MHR010 is determined for different combinations of α and β and the results are presented in Table 3 and

Table 4. It is concluded that it is possible to achieve higher sensitivity and reduced nonlinearity by appropriately choosing the fractional order values, such combinations are highlighted in Table 3 and Table 4. Table 3. Variation in sensitivity of FOLVDT for different combinations of α and β Sensitivity mV/V/mm at 0.25mm stroke β α 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

1 240.4 570.9 1149.7 1981.1 2951.3 3840.9 4505.3 4951.1 5241.6 5430.5 5552.9

0.95 139.9 333.5 661.5 1087.1 1510.8 1844.1 2071.8 2220.6 2317.7 2381.0 2422.0

0.9 80.9 193.2 381.6 618.0 841.3 1008.9 1120.5 1192.9 1240.1 1271.0 1291.0

0.85 46.6 111.5 220.1 354.8 479.8 572.2 633.1 672.5 698.1 714.9 725.8

0.8 26.8 64.2 126.8 204.2 275.5 327.9 362.4 384.6 399.1 408.6 414.7

0.75 15.4 37.0 73.0 117.5 158.5 188.5 208.2 221.0 229.2 234.7 238.2

0.7 8.9 21.3 42.0 67.6 91.2 108.5 119.8 127.1 131.9 135.0 137.0

0.65 5.1 12.2 24.2 38.9 52.5 62.4 68.9 73.1 75.9 77.7 78.8

0.6 2.9 7.1 13.9 22.4 30.2 35.9 39.7 42.1 43.7 44.7 45.4

0.55 1.7 4.1 8.0 12.9 17.4 20.7 22.8 24.2 25.1 25.7 26.1

0.5 1.0 2.3 4.6 7.4 10.0 11.9 13.1 13.9 14.5 14.8 15.0

Table 4. Variation in nonlinearity of FOLVDT for different combinations of α and β Non-linearity % of FR at 0.25mm stroke β α 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

1 0.603 0.916 0.222 2.300 5.960 9.245 11.409 12.652 13.356 13.771 14.024

0.95 0.177 0.333 0.167 0.615 1.755 2.745 3.394 3.779 4.006 4.142 4.225

0.9 0.021 0.091 0.070 0.157 0.518 0.842 1.062 1.195 1.275 1.323 1.353

0.85 0.035 0.006 0.000 0.062 0.174 0.281 0.356 0.403 0.432 0.449 0.459

0.8 0.054 0.043 0.037 0.052 0.086 0.121 0.146 0.162 0.172 0.178 0.182

0.75 0.061 0.056 0.053 0.056 0.066 0.077 0.086 0.091 0.094 0.097 0.098

0.7 0.063 0.061 0.060 0.060 0.063 0.066 0.069 0.071 0.072 0.073 0.073

0.65 0.064 0.063 0.062 0.062 0.063 0.064 0.065 0.065 0.066 0.066 0.066

0.6 0.064 0.064 0.063 0.063 0.063 0.064 0.064 0.064 0.064 0.064 0.064

0.55 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064

0.5 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064

5. FOLVDT vs. conventional LVDT Sensitivity and nonlinearity of FOLVDT depends on the choice of fractional orders α and β. To compare the performance of the proposed FOLVDT with conventional LVDT, four different FOLVDTs having fractional orders α and β as given in table 5 are considered. The input-output characteristics and the sensitivity of FOLVDTs in Table 5 are shown in Fig. 11 and 12. From the results, it is concluded that FOLVDT provides higher sensitivity as compared to conventional LVDT and also the sensitivity remains almost constant in FOLVDT. The results are summarized in Table 6 for different stroke range.

Table 5. Fractional orders of systems System LVDT FOLVDT-1 FOLVDT-2 FOLVDT-2 FOLVDT-4

Fractional order α β 1 1 0.85 0.9 0.85 0.85 0.8 0.85 0.7 0.8

Fig. 11. Input–output characteristics of FOLVDT and Conventional LVDT

Fig. 12. Sensitivity of FOLVDT and conventional LVDT Table 6. Performance of FOLVDT for different stroke range 100% 150% (0.25 mm) (0.375 mm) S NL S NL System α β LVDT 1 1 240.43 0.6032 236.11 1.3610 FOLVDT-1 0.85 0.9 618.04 0.1571 621.84 0.3337 FOLVDT-2 0.85 0.85 354.82 0.0617 356.00 0.1371 FOLVDT-3 0.8 0.85 479.84 0.1739 483.21 0.3895 FOLVDT-4 0.7 0.8 362.38 0.1460 364.60 0.3297 S and NL indicate sensitivity and nonlinearity respectively. Stroke

200% (0.50 mm) S NL 229.98 2.4280 625.98 0.5429 357.20 0.2395 487.35 0.6879 367.32 0.5893

300% (0.75 mm) S NL 213.35 5.5066 633.90 0.8865 360.08 0.5105 498.58 1.5150 374.89 1.3457

Further to illustrate the advantage of FOLVDT over conventional LVDT the characteristics are compared with their respective linear characteristics obtained through linear approximation and the results are shown in Fig. 13. It is observed that characteristics of FOLVDT are in close agreement with their liner approximation as compared to the conventional LVDT.

Fig. 13. Comparison of FOLVDT and LVDT characteristics

5.1. Electrical equivalent of FOLVDT FOLVDT design presented in section 3 is emulated by its equivalent electrical circuit shown in Fig. 14, where the mutually coupled circuits are replaced by its ‘T’ equivalent. The equivalent circuit is simulated using TINA Design Suite. Equivalent rational approximation of the fractional element is derived using Oustaloup approximation method [45]. In this method the order of approximated transfer function is fixed depending on the acceptable tolerance in the response. Here the fractional term is approximated to 10 th order transfer function, so that the error in the approximation is less than 2% in the frequency range of 1 to 10 kHz.

Lp  m

Ls  m

Rp

Rs Rm

Vp m

Fig. 14. Electrical equivalent of FOLVDT

Lp

The system FOLVDT-3 in Table 5 with fractional order α=β=0.85 is constructed and simulations are carried out to determine the output voltage for the input range of -0.25 mm to 0.25 mm. The simulation results are presented in Table 7 along with the analytical results obtained from the mathematical model of the FOLVDT given in eq. (6). The results show that the voltage obtained from electrical equivalent circuit is in close agreement with the analytical results and hence it is concluded that the proposed FOLVDT design is valid. Table 7. Results of electrical equivalent of FLVDT-3 S. no 1 2 3 4 5 6 7 8 9 10 11

x (mm) -0.25 -0.2 -0.15 -0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

Analytical magnitude (mV) Phase (degree) 95.794 9.980 76.685 9.769 57.543 9.605 38.376 9.488 19.192 9.418 0.000 0.000 19.192 -170.582 38.376 -170.512 57.543 -170.395 76.685 -170.231 95.794 -170.020

Electrical equivalent Magnitude (mV) Phase (degree) 94.64 12.56 75.02 12.08 57.23 11.92 38.65 11.57 18.51 11.14 0.000 0.000 18.95 -168.21 38.57 -168.07 56.98 -167.84 76.38 -167.21 95.42 -166.91

6. Conclusion This paper attempts to present the design of LVDT using fractional self and mutual inductance. In addition to the conventional structural parameters, fractional order system provides extra degree of freedom in terms of fractional orders of the self and mutual inductance (α and β) for designing. The characteristics of LVDT designed with fractional elements provide higher sensitivity and reduced nonlinearity with increased stroke range. The performance of the proposed FOLVDT design is demonstrated by considering the commercially available LVDT through analytical and electrical equivalent simulation results. The use of fractional calculus for designing LVDT is first of its kind and it provides way to improve the transducer specification and performance.

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