THE MODELLING AND DESIGN OF A LINEAR VARIABLE …

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THE MODELLING AND DESIGN OF A LINEAR VARIABLE DIFFERENTIAL

TRANSFORMER

KANG-YUL BAE1 & YOUNG-SOO YANG2

1Department of Mechatronics Engineering, Gyeongsang National University, Jinju, Korea

2Department of Mechanical Engineering, Jeonnam National University, Gwangju, Korea

ABSTRACT

In the design and improvement of an LVDT, theoretical analysis or numerical analysis can facilitate optimal design for

the sensor performance by effectively and quickly predicting the measurement range and the sensitivity with the changes

ofdesign and process variables. In this study, analysis models of the LVDT were proposed through theoretical analysis

and the finite element method (FEM), and the effects of design and process variables on the sensitivity and linear region

of the LVDT according to the core motion were analyzed by the proposed models. The theoretical model for the relation

between the output voltage and the change in core position was developed by deriving the change in the mutual

inductance of the primary and secondary coils. Meanwhile, the core, coil, magnetic shell, electric circuit, and core

movement of the LVDT were constructed as models for the FEM to obtain the voltage output using a commercial

analysis program. The results of the LVDT output characteristics analyzed by the theoretical and the finite element

models were mutually verified. By the verified models, a series of the analyses of the LVDT were performed with

changes in the supply voltage, core size, number of primary and secondary turns, distance between coils, coil length,

initial core position, and permeabilities of core and magnetic shell. The effects of those variables on the sensitivity and

linear region of the LVDT could then be revealed.

KEYWORDS: Linear Variable Differential Transformer (LVDT), Characteristic Variables, FEM Model, Theoretical

Model, Sensitivity & Linear Region

Received: Apr 25, 2021; Accepted: May 14, 2021; Published: May 29, 2021; Paper Id.: IJMPERDJUN202140

1. INTRODUCTION

A Linear Variable Differential Transformer (LVDT) is a magnetic position transducer for linear displacement

measurement, and as it offers high resolution, accuracy, and good repeatability, is widely applied (Masi et al., 2011;

Al-Sharif et al., 2011; Baidwan et al., 2015). An LVDT sensor is composed of three coils that include one primary

coil with a cylindrical shape in the center and two secondary coils wound on each side of the primary one, has a

high permeability core inside the coils, and has a magnetic shell outside (Martino et al., 2010). When a high

frequency of several kHz or less and a low voltage of 3 to 15 V are applied to the primary coil, magnetic flux is then

generated to link to each of the secondary coils located around it, and a voltage is finally induced in the secondary

coil. The magnetic flux generated between the primary coil and each secondary coil changes according to the

position of the core, and accordingly, the magnitude of the voltage induced in the secondary coil changes. For the

measurement of the displacement with an LVDT, two secondary coils are connected differentially, and the voltages

at the free ends of the two coils are measured. When the core is located in the center, the output voltage becomes 0;

and when the position of the core is shifted, the output voltage appears proportional to the displacement of the core.

The characteristic variables of an LVDT can be divided into process variables and design variables.

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International Journal of Mechanical and Production

Engineering Research and Development (IJMPERD)

ISSN (P): 2249–6890; ISSN (E): 2249–8001

Vol. 11, Issue 3, Jun 2021, 513-526

© TJPRC Pvt. Ltd.

514 Kang-Yul Bae & Young-Soo Yang

Impact Factor (JCC): 9.6246 NAAS Rating: 3.11

Process variables are the magnitude and frequency of the supply voltage, whereas design variables include the size of the

core and coil, and the number of turns. Numerous studies on the relation between the characteristic variables and output,

and on the techniques to solve the nonlinearity of the output, have been conducted (Petchmaneelumka et al., 2019;

Santosh et al., 2012; Mandal et al., 2018). However, it takes a lot of time and cost to perform the actual fabrication of the

LVDT to verify the studies.

Because theoretical analysis makes it possible to effectively analyze the effect of the characteristic variables,

theoretical models of LVDT have been proposed to select characteristic variables that can increase the linear range of

output. In particular, in these studies, a method of obtaining the mutual inductance by introducing the magnetic

conductance considering the shape of an LVDT was applied (Baidwan et al., 2015; Souza et al., 2008), and the voltage

induced in each secondary coil was directly derived from the time change of the magnetic flux linking the secondary coil

(KSRI, 1989; KAERI, 2008). However, in such previous studies, the inductance values when the core does not exist in

each secondary coil, or when the core deviates from the boundary of the secondary coil, were not considered; therefore, the

analysis results were considered to be inaccurate, due to the edge effect (Al-Sharif et al., 2011).

In addition to the theoretical analysis, the finite element method (FEM) has been applied to effectively analyze

the effects of the characteristic variables of an LVDT. The analysis based on the method makes it possible to estimate the

influence of local environmental conditions that cannot be grasped by theory or experiment. In particular, the finite element

model enables analysis of the influence of the surrounding interference condition and the influence of the induced current.

The method of analyzing magnetic flux between coils according to characteristic variables and obtaining the mutual

inductance of the secondary coil at a specific position has been studied using finite element analysis (FEM) (Al-Sharif et

al., 2011). The FEM for the characteristics of LVDT was reported to simulate the results of the experiment very well

(Mashi et al., 2011; Al-Sharif et al., 2011; Baidwan et al., 2015). The effect of magnetic interference caused by

peripheral devices on the operation of LVDT was studied through the FEM, and it was shown that a design that could

reduce the magnetic effect through analysis was possible (Mashi et al., 2011; Martino et al., 2010).

In this study, both a theoretical analysis model and a finite element analysis model are proposed to understand

the influence of design and process variables on the performance of an LVDT. In the theoretical model, the magnitude of

mutual inductance occurring in between the primary and each secondary coil is first derived according to the position of

the core in the coils, and is then used to obtain the differential voltage in the secondary coils. Based on the FEM, the core,

coil, magnetic shell, and electric circuit are modeled using a commercial analysis software (Altair, 2020), and the finite

element model of an LVDT is then presented to enable the analysis of the output voltage. By comparing the results of the

theoretical model and the finite element model, the suitability of each proposed model is verified. The ratio of the change

in the output voltage according to the change in the design and process variables is set as the sensitivity, and the limit of the

linear range of the output voltage according to the change in position of the core is set as the linear region. With the

proposed theoretical and finite element models, a series of analyses on the changes in the sensitivity and the linear region

of the LVDT according to the design and process variables are carried out.

2. LVDT ANALYSIS MODEL

2.1. Theoretical Model

Figure 1(a) shows a schematic of the LVDT, which is composed of three coils that include one primary coil in the center,

and two secondary coils on each side of the primary one, a core inside the coils, and a magnetic shell outside. Figure 1(b)

The Modelling and Design of a Linear Variable Differential Transformer 515


shows that an LVDT can be simplified to an electrical model consisting of a power source, a resistor, and a coil, where, 𝑒𝑒𝑥

is the supply voltage with the frequency 𝑓; 𝑅𝑝, 𝐿𝑝, and 𝑖𝑝 are the resistance, the inductance, and the current of primary

coil, respectively; 𝑀1 and 𝑀2 are the mutual inductances between the primary coil and the secondary coil 1, and between

the primary coil and the secondary coil 2, respectively; 𝑒1 and 𝑒2 are the induced voltages in the secondary coil 1 and the

secondary coil 2, respectively; 𝐿𝑠 and 𝑅𝑠 are the total inductance and the total resistance of the secondary coils,

respectively, 𝑅𝑚 is the resistance of a load, 𝑖𝑠 is the current of the secondary coil, and 𝑒𝑜 is the output voltage.

(a) Components (b) Equivalent Electrical Circuit

Figure 1: Schematic of the LVDT.

The ratio of the supply voltage (𝑒𝑒𝑥 ) and the output voltage (𝑒𝑜 ) by arranging the voltage potential equations

expressed with Laplace transform for the primary circuit and the secondary one becomes as follows (Doebelin, 1990;

Souza et al., 2008):

𝑒𝑜

𝑒𝑒𝑥=

𝑅𝑚(𝑀1−𝑀2)𝑠

𝑋𝑠2+𝑌𝑠+(𝑅𝑠+𝑅𝑚)𝑅𝑝 (1)

where, 𝑋 = −(𝑀1 −𝑀2)2 + 𝐿𝑝𝐿𝑠, Y = 𝐿𝑝(𝑅𝑠 + 𝑅𝑚) + 𝑅𝑝𝐿𝑠 . Meanwhile, by dividing the denominator of Eq.

(1) by 𝑅𝑚, and assuming that the values of 𝑅𝑝 and 𝑅𝑠 are negligible and 𝑅𝑚 is infinity, the ratio of the magnitudes of the

supply voltage and the output voltage can be presented as follows:

|𝑒𝑜

𝑒𝑒𝑥| =

𝑀1−𝑀2

𝐿𝑝 (2)

where, when 𝐸 is the root mean square(rms) voltage of 𝑒𝑒𝑥 , the rms output voltage is as follows:

𝑒𝑜 = 𝐸𝑀1−𝑀2

𝐿𝑝= 𝑒1 − 𝑒2 (3)

Equation (3) shows that the output voltage of the LVDT is proportional to the difference in the mutual

inductances, and is inversely proportional to the self-inductance of the primary coil. Accordingly, the output voltage can be

calculated by deriving the mutual inductance between the primary coil and each secondary coil, and the self-inductance of

the primary coil.



(a) Sectional Drawing (b) Penetration of Core towards Secondary Coil 1

Figure 2: Geometrical Schematic of the LVDT.

Figure 2(a) shows the cross section of an LVDT, revealing the geometric shape that consists of a primary coil,

two secondary coils on the left and right, a core, and a magnetic shell. The radius and length of the core are 𝑟𝑐 and 𝐿𝑎 ,

respectively; the outer and inner radii of the coil are 𝑟𝑜 and 𝑟𝑖 , respectively; the lengths of the primary and secondary coils

are b and m, respectively; and the separation between the primary and secondary coils is represented by d. The length of

the magnetic shell is expressed as 𝐿𝑠, and the inner and outer diameters of the magnetic shell are expressed as 𝑟𝑠𝑖 and 𝑟𝑠𝑜 ,

respectively. Meanwhile, 𝐿1 and 𝐿2 represent the lengths of the core inserted into the secondary coils 1 and 2 plus the

distance d, respectively.

If the magnetic flux from the end surfaces of the core is ignored and 𝐵1 and 𝐵2 are the leakage magnetic flux

densities at the core surface across 𝐿1 and 𝐿2, respectively, the magnetic flux density along the surface of the core can be

expressed as follows (KSRI, 1989), assuming that the magnetic potential drop in the core and the magnetic shell with high

relative permeabilities is very small, compared to the potential drop in air.

𝐵1 =(2𝐿2+𝑏)

𝐿𝑎

𝜇𝑜𝑁𝑝𝑖𝑝

2𝑟𝑖ln(𝑟𝑜𝑟𝑖) ; 𝐵2 = −

(2𝐿1+𝑏)

𝐿𝑎

𝜇𝑜𝑁𝑝𝑖𝑝

2𝑟𝑖ln(𝑟𝑜𝑟𝑖) (4)

where, 𝜇𝑜 is the permeability of air and 𝑁𝑝 is the number of turns in the primary coil.

Figure 2(b) shows that at a distance of 𝑥 ′ from one end of the secondary coil 1 with a number of turns of Ns, the

number of turns is 𝑁𝑠𝑑𝑥′/𝑚 within the infinitesimal increment of 𝑑𝑥 ′, and when the outer diameter of the core (𝑟𝑐) is

assumed to be the same as the inner diameter of the coil (𝑟𝑖), the magnetic flux generated at the primary coil becomes

∅𝑥 ′ = 2𝜋𝑟𝑖𝑥′𝐵1(KAERI, 2008). Considering that the core moves a distance of 𝑥1, the total magnetic flux passing through

the the secondary coil 1 (𝑁𝑠∅12) can be expressed as follows:

𝑁𝑠∅12 = ∫ ∅𝑥 ′𝑁𝑠

𝑚𝑑𝑥 ′

𝑥1

0 (5)

Meanwhile, when the rms current flowing through the primary coil is expressed as 𝐼𝑝, the mutual inductance

between the primary coil and the secondary coil 1 can be expressed as follows:

𝑀1 =𝑁𝑠

𝐼𝑝∅12 (6)

In Eq. (5), the magnetic flux changes according to the range of 𝑥1 . When 𝑚 ≥ 𝑥1 > 0 , the mutual

inductance becomes as follows:

𝑀1 =1

𝐼𝑝∫ ∅𝑥 ′

𝑁𝑠

𝑚𝑑𝑥 ′

𝑥1

0 (7)



If the magnetic flux is replaced by the product of the magnetic flux density and the passing area in Eq. (7), the

equation can be expressed as follows:

𝑀1 =2𝜋𝑟𝑖𝐵1𝑁𝑠

𝐼𝑝𝑚∫ 𝑥 ′𝑑𝑥 ′𝑥1

0=

𝜋𝑟𝑖𝐵1𝑁𝑠𝑥12

𝐼𝑝𝑚=

𝜋𝜇𝑜𝑁𝑝𝑁𝑠

2ln(𝑟𝑜𝑟𝑖)∙(2𝐿2+𝑏)𝑥1

2

𝑚𝐿𝑎 (8)

Therefore, the voltage induced in the secondary coil 1 is obtained as follows:

𝑒1 = 𝐸𝑀1

𝐿𝑝+ 𝑒0 = 𝜔𝐼𝑝𝑀1 + 𝑒0 =

𝜋2𝑓𝜇𝑜𝐼𝑝𝑁𝑝𝑁𝑠

ln(𝑟𝑜𝑟𝑖)

∙(2𝐿2+𝑏)𝑥1

2

𝑚𝐿𝑎+ 𝑒0 (9)

where, 𝑒0 is the voltage when 𝑥1 ≤ 0, and can be calculated as 𝑒0 = 𝐸𝑀0

𝐿𝑝, and 𝑀0 is the mutual inductance

between the primary coil with the core and the secondary coil without the core. Meanwhile, assuming that the resistance of

the primary coil is very small, the rms current of the coil has the following relationship with the self-inductance of the coil:

𝐼𝑝 =𝐸

2𝜋𝑓𝐿𝑝 (10)

where, in order to calculate the rms current (𝐼𝑝) in Eq. (10), it is necessary to obtain the inductance (𝐿𝑝) of the

primary coil and core system, and the following relationship between the magnetic flux and the inductance can be used:

𝑑(𝑁𝑝∅)

𝑑𝑡=

𝑑(𝐿𝑝𝑖𝑝)

𝑑𝑡 (11)

When the total number of magnetic fluxes exiting and entering the coil and core system is divided by 2, the total

number of magnetic fluxes exiting the system can be obtained as follows:

𝑁𝑝∅ = 𝑁𝑝(𝐵1𝐿1 + 𝐵2𝐿2)(2𝜋𝑟𝑖)/2 (12)

Therefore, 𝐿𝑝 can be derived from Eqs. (11) and (12) as follows:

𝐿𝑝 =𝐿1𝐿2(4+2𝑏)

2𝐿𝑎

𝜋𝜇𝑜

ln(𝑟𝑜𝑟𝑖)𝑁𝑝

2+ 𝐿𝑝0 (13)

where, 𝐿𝑝0 represents the inductance when 𝐿1 or 𝐿2 is 0.

In Eq. (5), when the range of 𝑥1 is 𝑥1 > 𝑚, the mutual inductance can be expressed as follows:

𝑀1 =2𝜋𝑟𝑖𝐵1𝑁𝑠

𝐼𝑝(𝐿1 − (𝑚+ 𝑑) +

1

𝑚∫ 𝑥 ′𝑑𝑥 ′

𝑚

0

) =𝜋𝑟𝑖𝐵1𝑁𝑠

𝐼𝑝(2𝐿1 −𝑚 − 2𝑑)

=𝜋𝜇𝑜𝑁𝑝𝑁𝑠

2ln(𝑟𝑜𝑟𝑖)∙(2𝐿2+𝑏)(2𝐿1−𝑚−2𝑑)

𝐿𝑎 (14)

At this time, the following voltage is induced in the secondary coil 1.

𝑒1 =𝜋2𝑓𝜇𝑜𝐼𝑝𝑁𝑝𝑁𝑠

ln(𝑟𝑜𝑟𝑖)

∙(2𝐿2+𝑏)(2𝐿1−𝑚−2𝑑)

𝐿𝑎+𝑒0 (15)

When the core moves in the 𝑥2 direction, the voltage induced in the secondary coil 2 can be derived in the same

way. The output voltage of the LVDT can then be obtained as the difference in the voltage induced in each of the secondary

coils, as shown in Eq. (3).



2.2. Finite Element Model

A FEM to simulate an LVDT sensor is proposed using the FLUX software (Altair, 2020), which is suitable for the

electromagnetic field analysis problem. Using the proposed model, we try to verify the theoretical model by comparing the

results analyzed with the models, and also to simulate the characteristics of an LVDT due to various changes in the design

and process variables. As the analysis type of the model is a magnetic field analysis using an AC power source of a

constant frequency, a steady state AC magnetic field 3D analysis is selected.

Figure 3(a) is the solution domain of the FEM for an LVDT, showing the finite element division of the core and

the magnetic shell regions, excluding the air region outside the magnetic shell, while Fig. 3(b) shows the domain including

the primary and secondary coils that are not divided into elements. Outside of the magnetic shell, an infinite box with a

diameter of 120 mm and a length of 240 mm, beyond which range was assumed to have no influence of magnetic flux, was

set as the analysis domain. In the domain, the rest, except for the core, coil, and magnetic shell, was set as the air region.

The lengths of the primary and secondary coils were basically set to 32 mm, the inner and outer radii of the coils were 10.5

and 29.5 mm, respectively, and the numbers of turns of the primary and secondary coils were both set to 1,000. The radius

and length of the core were set to 10 and 70 mm, respectively. The radial separations between the core and the coil, and

between the coil and the magnetic shell, were both set to 0.5 mm, respectively. The magnetic shell is a hollow cylinder

with an inner diameter of 30 mm, a thickness of 2 mm, and a length of 100 mm.

(a) Core and Magnetic Shell Regions (b) Non-Meshed Coils in the Domain

Figure 3: Solution Domain of the FEM for LVDT Analysis.

Tetrahedral elements were applied to the element division for the FEM, and the core and the magnetic shell were

divided into 2 mm intervals in the longitudinal direction, 1 mm intervals in the circumferential direction, and 1 mm

intervals in the radial direction. The air region outside the magnetic shell was automatically segmented by setting elements

with a size of 10 mm in the length direction, and 10 mm in the circumferential direction at the edge of the infinite box. The

solution domain was totally divided into about 100,000 tetrahedral elements. The primary and two secondary coils were set

as non-meshed coils.

When the position of the center of the core was set to 0, the voltage level of each secondary coil was analyzed

while moving the position of the core from -40 to 40 mm at 1 mm intervals. The mesh division process was repeatedly

performed according to the change in the core position. The materials of the core and the magnetic shell were set to ferrite



and ferromagnetic steel, respectively, and the permeabilities of the materials were both assumed to be 10,000. The

electrical conductivities of the core and the magnetic shell were neglected. In the electric circuit model of the LVDT, an AC

power source with an rms voltage of 5 V and a frequency of 2 kHz was basically applied to the primary coil. The primary

coil with a resistance of 0.1 Ω was directly connected to the supply power, and a resistance of 100 kΩ was placed in serial

between the ground and each secondary coil with a resistance of 0.1 Ω.

3. APPLICATION OF THE THEORETICAL MODEL AND THE FEM

By applying the proposed theoretical and finite element models, the output characteristics of the LVDT according to the

design and process variables related to the dimension and operation of the LVDT were analyzed. Table 1 shows the design

and process variables of the LVDT, and presents the basic value and its variation for analysis. The size of the core was kept

constant, and the voltage, frequency, number of coil turns, coil length, separation distance between coils, and

permeabilities of the core and the magnetic shell were set as variables.

Table 1: Design and Process Variables for LVDT.

Variables Basic value Variation for analysis

Supply voltage (rms; V) 5 3 - 12

Frequency (Hz) 2,000 500 - 2,500

Length of all coils (mm) 32 24 - 40

Length of secondary coils (mm) 32 20 - 48

Inner radius of coil (mm) 10.5 6.5 - 14.5

Outer radius of coil (mm) 29.5 23.5 - 35.5

No. of turns of all coils 1,000 300 - 2,000

No. of turns of secondary coils 1,000 300 - 2,000

Separation between coils (mm) 2 1 - 5

Relative permeability of core 10,000 100, 500, 1,000, 2,000, 6,000, 10,000, 14,000

Relative permeability of magnetic shell 10,000 1, 2,000, 6,000, 10,000, 14,000

In the outputs of the theoretical and finite element models, the rms current of the primary coil and the rms

voltage of each secondary coil were used as the results for the current and the voltage, respectively. In order to obtain the

output voltage according to the change of a specific design or process variable, only the variable was changed while the

other variables were kept at their basic values.

4. ANALYSIS RESULTS AND DISCUSSION

By applying the theoretical and the finite element models, changes in the voltages inducing in the secondary coils

according to the displacement of the core were obtained while considering the changes of the design and process variables

of the LVDT.

Figure 4(a) is the output voltages according to the displacement of the core, derived from the theoretical model

for the LVDT with basic process and design variables, and shows the voltage at each secondary coil, and the differential

voltage in the secondary coils. As for the movement of the core, when the longitudinal center of the core was at the center

of the primary coil, it was set as the null position, the direction to the secondary coil 2 was set as positive, and the direction

to the secondary coil 1 was set as negative. As shown in Eqs. (8) and (14), it can be determined that the increase in the

voltage induced in the secondary coil with the displacement of the core is due to an increase in the mutual inductance

between the corresponding secondary coil and the primary coil.



The theoretical equation cannot define the result when the core is completely out of one of the secondary coils,

that is to say, when 𝑥1 < 0, as shown in Eq. (9). Therefore, by setting the change of the core center position to a maximum

of 19 mm, solutions were derived to the extent where the edge of the core does not deviate from one of the secondary coils.

It can be seen that even when the core is positioned to be deflected to one of the secondary coils, the voltage in the other

coil remains constant. This is a result showing that basic electromagnetic induction is also performed in the air-core coil.

(a) Theoretical Model (b) FEM Model

Figure 4: Output Voltages of Secondary Coils with Change of Center Position of Core Obtained from the Analyses.

Figure 4(b) shows the output voltages and differential voltage induced in the secondary coils according to the core

displacement obtained from the FEM for the LVDT with basic process and design variables. In the case of the FEM, it is

possible to analyze even if the position of the core deviates from one of the secondary coils, so the movement range of the

core was further expanded and analyzed by setting up to 40 mm. In the region where the core is out of one of the secondary

coils, it shows a distinct nonlinear characteristic. The results of the theoretical model show that even when the core is

positioned to be deflected to one of the secondary coils, the voltage in the other coil remains constant.

Figure 5: Changes of Current and Inductance According to the Length of the

Primary Coil and the Center Position of the Core.

Figure 5 shows the changes in the inductance of the primary coil system and in the rms current of the primary

coil according to the changes in the position of the core, analyzed by the FEM, when the length of the primary coil is

increased from 24 to 40 mm at 4 mm intervals. As the length of the coil increases, the inductance decreases, and the current



increases. When the position of the core is placed in the center, the inductance becomes high and the current becomes low;

and as the position of the core is biased toward the secondary coil, the inductance decreases, and the current increases.

Accordingly, when the core is deflected, it can be said that an increase in the voltage induced in the secondary coil is due to

not only an increase in the mutual inductance, but also an increase in the current of the primary coil. Meanwhile, when the

length of the primary coil increases, the current in the primary coil increases and the voltage induced in the secondary coil

increases, so that an increase in the sensitivity of LVDT is expected.

Figure 6: Comparison of Output Voltages of Secondary Coils Obtained from the Analyses with the FEM and

Theoretical Models.

Figure 6 shows the result of comparing the LVDT output voltages obtained by the theoretical model and the

finite element model for the core movement from 0 to 19 mm in the positive direction, and the two output voltages can be

seen to agree well. This shows that it is possible to examine the effects of LVDT design and process variables using either

the theoretical model, or the FEM. Meanwhile, the FEM, as shown in Fig. 4(b), is a more effective one, because the

analysis range can be extended to the region where the core is outside of the secondary coil, and the effects of the eddy

current and the changes in the permeabilities of the core and coil can also be investigated.

Meanwhile, in the theoretical model, there were few changes in the output voltages from the result shown in Fig.

4(a) with the changes of frequency, inner and outer diameters of the coils, and number of turns of the coils. This is because

each of these variables is itself removed, as shown in the theoretical model. In the theoretical model, the frequency cancels

out the frequency that induces the current, so it has no effect on the output voltage. The inner and outer diameters of the

coils also cancel out those diameters at the inductance function that induces the current, so that they do not affect the

output voltage. If the number of turns of the primary and secondary coils is the same, the product of the number of turns

appearing in the output voltage of the theoretical model and the square of the number of turns of the primary coil in the

inductance function inducing the current cancel each other. Therefore, the number of turns does not influence the output

voltage. For these reasons, it was judged that there were no changes in the output voltage according to the changes of these

variables. The same phenomena also appeared in the results of the FEM, and the results according to the changes of these

variables showed the same as in Figure 4(b).

The output characteristic of an LVDT can be expressed as the relationship between the core displacement and the

output voltage. The relationship between the voltage output and the core movement initially shows a linear characteristic; a



slope line can be obtained using this, and the slope of this line can be called the sensitivity of the LVDT. As the core

movement increases, the output voltage exhibits nonlinear behavior, so it is necessary to set a linear measurement range for

use as a displacement sensor. Therefore, in this study, in order to show the output characteristic of the LVDT, the slope of

the linear range, that is, the voltage output per unit travel distance of the core, was defined as the sensitivity. In addition,

the moving distance of the core with an error rate of less than 1 % between the voltage output and the slope line of the

linear range (the percentage to the difference between the output value and the linear output value at the maximum travel

distance) was defined as the linear region. Therefore, in order to show the changes of the output voltage with the design

and process variables, and according to the core displacement for the characteristic of LVDT, the sensitivity and the half

range of linear region (in one directional movement of the core) are utilized.

(a) Supply Voltage(b) Number of Turns in Secondary Coils

Figure 7: Sensitivities and Linear Regions According to Variables Obtained from the Analayses with FEM and

Theoretical Models.

Figure 7(a) shows that when the supply voltage is changed from 3 to 12 V, the sensitivity increases linearly from

0.08 to 0.32 V/mm. On the other hand, it is found that the linear region remained constant, with a size of 10.5 mm.

Figure 7(b) shows that when the number of turns of the two coils on the secondary side is changed from 300 to

2,000, the sensitivity increases linearly from 0.04 to 0.27 V/mm, while the linear region is maintained at around 10.5 mm.

Figure 8: Sensitivities and Linear Regions According to the Separation between Coils Obtained from the Analyses

with the FEM and Theoretical Models.



Figure 8 shows that when the separation between each coil is increased from 1 to 5 mm, the sensitivity decreases

from 0.14 to 0.12 V/mm, and the linear region decreases from 11 to 8.6 mm. Both the sensitivity and the linear region

show a slightly larger decrease in the theoretical model than in the FEM.

(a) Length of Secondary Coils (b) Length of Primary and Secondary Coils

Figure 9: Sensitivity and Linear Region According to Lengths of Coils Obtained from the Analysis of FEM Model.

Figure 9(a) shows the sensitivity and linear region of the LVDT obtained from the finite element model when the length of

the primary coil is set to 32 mm and the length of each secondary coil is increased from 20 to 48 mm at 4 mm intervals. At

this time, the lengths of the 70 mm long core initially inserted into the secondary coils 1 and 2 are constant at 17 mm,

respectively, but the ratio of the insertion length to the length of each coil changes from 85 to 35.4 %. As the length of the

secondary coil increases, that is, as the ratio between the core length and the insertion length decreases, the linear region

increases, and the sensitivity decreases. Therefore, it can be seen that when the initial insertion length of the core in the

secondary coil is increased in an LVDT design, the sensitivity increases, and the linear region decreases.

Figure 9(b) shows the sensitivity and linear region obtained with the FEM when the length of both the primary

and secondary coils of the LVDT is increased from 24 to 40 mm at 4 mm intervals. At this time, the length of the 70 mm

long core initially inserted into the secondary coils 1 and 2 is changed from 21 to 13 mm, and the ratio of the insertion

length to the coil length changes from 87.5 to 32.5 %. As the length of the coil increases, that is, as the ratio of the core

length and the insertion length decreases, the linear region increases, and the sensitivity decreases. However, it can be seen

that as the length of the coil decreases, the increase in the sensitivity slows down, and as the length of the coil increases,

the decrease in the sensitivity slows down. This can be judged as the result that is shown in Fig. 5, that as the length of the

primary coil becomes shorter, the inductance increases and the current decreases, resulting in a relative decrease in the

sensitivity; and conversely, as the length of the coil increases, the current increases, resulting in a relative increase in the

sensitivity.



Figure 10: Sensitivity and Linear Region According to the Relative Permeabilities of the Core and Magnetic Shell

obtained from the Analysis of the FEM Model.

Figure 10 shows the analysis results of the FEM for the sensitivity and linear region according to the change of

permeabilities of the core and the magnetic shell. If the relative magnetic permeability is as low as 1,000 or less, an

increase in the current, an increase in the magnetic flux, and an increase in the induced voltage can be expected, due to the

decrease in the inductance of the primary coil. However, the decrease in the permeability results in a decrease in the mutual

inductance with the secondary coil, resulting in a decrease in the magnitude of the induced voltage, and thus a decrease in

the sensitivity. This occurs when the relative permeability of the core, or that of the magnetic shell, are low. However,

when the relative permeability is high enough, at about 1,000 or more, those effects cancel each other, and it can be

determined that the sensitivity and the linear region are not affected by the permeability.

5. CONCLUSIONS

Models that can design an LVDT by applying theoretical analysis and finite element analysis were presented, and the

results of each model were mutually verified. In the prediction of the output of an LVDT with the change of the design and

process variables, the results of the proposed theoretical model were in good agreement with those of the FEM. The

proposed models can be used as the means to design an LVDT, and also to understand its output characteristics.By

applying the theoretical and the finite element models, the effects of the design and process variables on the output voltage

of the LVDT were examined, and the following conclusions were then derived, considering the sensitivity and linear region

of the output voltage:

The increase in the voltage induced in the secondary coil with the displacement of the core is due not only to

an increase in mutual inductance, but also to an increase in the current in the primary coil.

Changes in the frequency, the inner and outer diameters of the core, and the number of turns when the

number of turns of the primary and secondary coils are the same, do not affect the voltage output.

Increments in the magnitude of the supply voltage and in the number of turns of the secondary coil increase

the sensitivity linearly, respectively, but do not affect the linear region.

As the separation between coils increases, the sensitivity and linear region decrease.

As the length of the primary coil decreases, the current in the coil decreases, and the voltage induced in the



secondary coil also decreases, reducing the sensitivity.

When the initial insertion length of the core in the secondary coil is increased, the sensitivity increases and

the linear region decreases.

When the permeabilities of the core and the magnetic shell are 1,000 or more, the magnitude of the

permeability does not affect the sensitivity and linear region.

ACKNOWLEDGMENT

This work was supported by the research invigoration program of 2020 Gyeongnam National University of Science and

Technology.

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