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Linear Variable Differential Transformer (LVDT)
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The linear variable differential transformer (LVDT) is a displacement transducer based on the mutual inductance concept just discussed Figure 167 shows a simpli ed representation of an LVDT, which consists of a primary coil, subject to AC excitation (vex ), and of a pair of identical secondary coils, which are connected so as to result in the output voltage vout = v1 v2 The ferromagnetic core between the primary and secondary coils can be displaced in proportion to some external motion, x, and determines the magnetic coupling between primary and secondary coils Intuitively, as the core is displaced upward, greater coupling will occur between the primary coil and the top secondary coil, thus inducing a greater voltage in the top secondary coil Hence, vout > 0 for positive displacements The converse is true for negative displacements More formally, if the primary coil has resistance Rp and self-inductance Lp , we can write di = vex dt and the voltages induced in the secondary coils are given by iRp + Lp v 1 = M1 v 2 = M2 so that vout = (M1 M2 ) di dt
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Iron core i
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di dt di dt
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Figure 167 Linear variable differential transformer
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Electromechanics
where M1 and M2 are the mutual inductances between the primary and the respective secondary coils It should be apparent that each of the mutual inductances is dependent on the position of the iron core For example, with the core at the null position, M1 = M2 and vout = 0 The LVDT is typically designed so that M1 M2 is linearly related to the displacement of the core, x Because the excitation is by necessity an AC signal (why ), the output voltage is actually given by the difference of two sinusoidal voltages at the same frequency, and is therefore itself a sinusoid, whose amplitude and phase depend on the displacement, x Thus, vout is an amplitude-modulated (AM) signal, similar to the one discussed in Focus on Measurements: Capacitive Displacement Transducer in 4 To recover a signal proportional to the actual displacement, it is therefore necessary to use a demodulator circuit, such as the one discussed in Focus on Measurements: Peak Detector for Capacitive Displacement Transducer in 8
In practical electromagnetic circuits, the self-inductance of a circuit is not necessarily constant; in particular, the inductance parameter, L, is not constant, in general, but depends on the strength of the magnetic eld intensity, so that it will not be possible to use such a simple relationship as v = L di/dt, with L constant If we revisit the de nition of the transformer voltage, e=N d dt (1613)
we see that in an inductor coil, the inductance is given by L= N = i i (1614)
This expression implies that the relationship between current and ux in a magnetic structure is linear (the inductance being the slope of the line) In fact, the properties of ferromagnetic materials are such that the ux-current relationship is nonlinear, as we shall see in Section 163, so that the simple linear inductance parameter used in electric circuit analysis is not adequate to represent the behavior of the magnetic circuits of the present chapter In any practical situation, the relationship between the ux linkage, , and the current is nonlinear, and might be described by a curve similar to that shown in Figure 168 Whenever the i- curve is not a straight line, it is more convenient to analyze the magnetic system in terms of energy calculations, since the corresponding circuit equation would be nonlinear In a magnetic system, the energy stored in the magnetic eld is equal to the integral of the instantaneous power, which is the product of voltage and current, just as in a conventional electrical circuit: Wm = ei dt (1615)
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