Study on differential transformer displacement sensors

. This paper studies magnetic circuits of known and developed differential transformer sensors of large linear motion, which refer to circuits with special structure of magnetic resistance parameters distribution of long ferromagnetic cores and magnetic capacitance between them. It is shown that in the known sensor the distribution of the working magnetic fluxes along the length of the ferromagnetic cores has a nonlinear character, as a result of which the dependence between the output signal in the form of the electromotive force and the input linear motion of the sensor also has a nonlinear character. It was found that by selecting the law of changing the working gap between the middle and inner cores along the length of displacement of the moving measuring winding by making the middle concentric core in the form of a paraboloid of rotation, a linear distribution of the working magnetic flows along the length of the ferromagnetic cores and linearity of the transformation characteristic of the differential transformer sensor of large linear displacements are achieved.


Introduction
Automation of various technological processes, requiring the use of a set of primary transducers -sensors, makes it quite easy to perform labor-intensive processes, save energy resources, reduce the cost of production and improve its quality by ensuring close to optimal modes of operation of the control and management objects, both in normal and in emergency situations [1,2].Meanwhile, one of the main obstacles to this is the lack of sensors for motion parameters (displacement, speed, acceleration, vibration, etc.) that meet modern requirements when working in extreme operating conditions with increased dustiness, humidity, significant temperature fluctuations, e t.c.A comparative analysis of the main characteristics of the existing sensors of motion parameters shows that the most promising direction in solving the actual problem of more effective use of control and management systems is the use of differential transformer sensors (DTS), which by their properties (high reliability and stability of the characteristics in extreme operating conditions, as well as large output power) best satisfy the modern requirements of control and management systems [3,4,5].However, the existing design schemes of DTS for measuring large linear displacements are practically unsuitable due to the instability of sensitivity in the entire measuring range and nonlinearity of the static characteristic.As a result, the accuracy of monitoring and control decreases, and in some cases, the system stability margin decreases.The need for more accurate and stable measurement of technological parameters of motion of the control and management systems objects puts forward the problem of developing new DTS for measuring large linear movements with improved information and technical characteristics, which explains the relevance of the research problem.First, we investigate a magnetic circuit of the known DTS [5], used to measure large linear displacements, in order to determine the pattern of change of magnetic fluxes in the cores and the magnetic tension between them depending on the coordinate of the moving element.The structural diagram of the DTS with the corresponding designations is shown in Fig. 1.The magnetic core of the sensor contains three coaxial cylindrical cores 1, 2, 3, with cores 1 and 3 connected at the ends by flanges 4 and 5 and cores 2 and 3 connected in the central part by ring jumper 6.The excitation winding consists of two sections 7 and 8 connected in series and in opposition.The measuring coil 9 encloses the rods 3 and can be moved along the magnetic coil by means of a non-magnetic rod 10 located inside the hollow rods.When an alternating voltage is supplied to the excitation winding, due to the constancy of the magnetic flux tubes length along the steel in the working range of movement of the moving part, a magnetic field along the length of the magnetic wire is formed in the sensor working gap.

Materials and Method
As a consequence, the output e.d.s.varies linearly as a function of the moving part movement, and the phase is constant over the moving part movement range.The phase of the output e.d.s. is reversed at zero crossing.To eliminate the influence of short-circuited body circuits on the sensor's amplitude-phase characteristics, the sensor is cut in the middle part along the generatrix."Zero" of the sensor is adjusted by turning the covers, which changes the complex magnetic resistance of the left and right relative to the transverse axis of symmetry parts of the magnetic wire.The cylindrical body of the sensor is a shield for the magnetic fields acting in the transverse direction.For a magnetic field acting perpendicular to the plane of the coils of the measuring coil, the presence of the gap significantly increases the resistance.In this case, the external magnetic flux is short-circuited mainly along the case.It should be noted that in contrast to magnetic circuits of sensors with П-shaped magnetic cores, where the length of magnetic field lines along the steel throughout the length of the working gap is unequal, in the considered magnetic circuit the length of magnetic field lines along the steel throughout the length of the working gap is the same.For this reason, the former are called magnetic circuits with the usual structure of resistance and conductivity parameter distribution, and the latter, to which the investigated magnetic circuit belongs, are called magnetic circuits with the special structure of resistance and conductivity parameter distribution.Magnetic circuits with distributed parameters of all existing sensors belong either to the first group or to the second group of magnetic circuits.In theoretical studies of magnetic circuits, we neglect the nonlinearity of steel magnetic resistance characteristics, convex fluxes at the ends of the magnetic circuit, in the area of the moving part, the measuring winding, and its longitudinal size.These assumptions do not introduce perceptible inaccuracies, but greatly simplify the calculation.To determine the current values of magnetic voltage   and magnetic flux   , created along a long magnetic circuit by two sections of the field winding, the coordinate x of the moving part of the considered core sections 2 and 3 is most convenient to count from the central part of the magnetic circuit.Since the magnetic system is symmetrical relative to the vertical axis passing through the central part of the magnetic core, it is enough to calculate the magnetic circuit for one, for example, the left half from the central part of the magnetic core.The number in the indexes of values and parameters means that they refer to the corresponding core.

Results and Discussions
Changes of magnetic flux and magnetic voltage on the elementary section of magnetic circuit dx (Fig. 2), created by the left section of the field winding, are found by making up the following differential equations on the basis of Kirchhoff's laws [6,7,8]: where -values of magnetic resistances of cores 2,3 and magnetic conductivity of working gap δ_r between them, per unit length of magnetic circuit; ,  0 = 4 • 10 −7   ⁄ -relative magnetic permeability of steel and magnetic constant respectively;  2 ,  3 , ℎ 2 , ℎ 3 -radii and thickness of cores 2 and 3 respectively.The following equality is true for the considered magnetic circuit [6]: ) here  м is the total magnetic flux created by one section of the excitation winding (we neglect the scattering fluxes from the surface of core 2 to core 1 through gap δs3 and from the end parts of core 2 to flanges 4,5 through air gaps δs1 due to small values of magnetic conductivities of these gaps).Differentiating (3) and inserting ( 1) and ( 2) into it, we obtain the following homogeneous linear differential equation of the second order: The general solution of the differential equation ( 5) has the following form [9]: where  = √( 2п +  3п ) п is the value of the coefficient of magnetic flux propagation through the magnetic circuit, From (4) we find the magnetic flux  2 :  2 =  м −  3 .
(7) Substituting ( 7) into (3), from there we find the value of magnetic flux  3 : The integration constants  1 and  2 are determined from the following boundary (boundary) conditions: Substituting in (9) the boundary values of  3 by (8), we obtain the following algebraic equations: Solving together equations ( 10) and (11), we obtain the following values of integration constants: Substituting the found values of  1 and  2 into expressions ( 6) and ( 8), we finally have the following: At  2п =  3п (14) and ( 15) takes the following form: The presence of hyperbolic functions in expressions ( 15) and ( 17) already means that the dependence  3 = () has a nonlinear character.
In order to graph a function  3 = () and   = () for ease of analysis, go to relative units: The graphs based on ( 22) and ( 23) show (Fig. 3 ), that the operating magnetic flux  3 is non-linear depending on the coordinate of the moving part of the sensor and the degree of non-linearity of the function  3 Degrees of nonlinearity of magnetic flux distribution  3 * along the length of the cores for the cases  2п ≠  3п and  2п =  3п , respectively, calculated by formula (24) for the above magnetic circuit parameters, for the prototype are equal:  1 = 5,42 % and  2 = 2,73 %.
(26)  In order to make the displacement sensor with long cores, along the length of which the moving element in the form of a concentrated measuring winding is moving, have a linear static characteristic, it is required a linear distribution of operating magnetic flux in the range of movement of this measuring winding [7].
In the developed sensor this is achieved by selecting the law of variation of the working gap between the middle and inner cores on the length of displacement of the moving measuring winding by making the middle concentric core in the form of a paraboloid of rotation.In order to determine the law of variation of the working gap let us make and solve differential equations for the elementary section dx of the magnetic circuit, considering a linear value of magnetic conductivity of the working gap along the length of the cores to be variable, i.e.  п = .
The design diagram of the left half of the magnetic core of the developed DTS is shown in Fig. 5, a, and the substitution diagram of the elementary section dx of the magnetic circuit in Fig. 5, b, which differs from the substitution diagram of the elementary section dx of the magnetic circuit of the prototype (Fig. 2, b) only  п instead of  п .
It should be noted that in the developed sensor to create the same conditions for the passage of magnetic field lines of force across the middle and inner cores the following ratio is required: It should be noted that in the developed sensor, in order to create the same conditions for the passage of magnetic field lines along the middle and inner cores, the following ratio is required: From there: Given the relation (28) we can assume that  2п =  3п =  п .The changes of magnetic flux and magnetic voltage on the elementary section of magnetic circuit dx (Fig. 5), created by the left section of the excitation coil, are found by composing the following differential equations on the basis of Kirchhoff's laws [8]: here w meas. is the number of turns in the movable measuring winding.

Conclusions
Thus, the design of the developed DTS creates strictly the same magnetic resistance in the path of magnetic field lines along the whole length of the working gap due to the implementation of the linear value of magnetic capacitance of the working gap between the middle and inner cores of the magnetic core by (43), as a result of which the working magnetic fluxes are distributed along the length of the cores strictly according to linear law, and the static characteristic of the developed DTS is linear in the linear motion measurement range.In the developed detector, this is achieved by selecting the law of variation of the working gap between the middle and inner cores along the moving length of the movable measuring winding by making the middle concentric core in the form of a paraboloid of rotation.

Fig. 1 .
Fig.1.Design diagram of the magnetic circuit of the known DTS

Fig. 3 .
Fig. 3. Plots of the dependence of the working magnetic flux  3 on the coordinate of the moving part x at different values of the magnetic field attenuation coefficient in the magnetic circuit β

Fig. 4 .
Fig. 4. Design diagram of the magnetic circuit of the developed transformer large linear displacement sensor