A Note on Linear Variable Differential Transformers
R.J. Vaccaro
There are four essential elements of an LVDT: (1) a case made of a material with high magnetic
permeability µ; (2) a moveable high-µ core; (3) a low-µ plunger attached to the core; and (4) one
primary and two secondary windings. These elements are shown in Fig. 1 below, which is a crosssectional diagram of an LVDT:
Figure 1: A cross sectional view of an LVDT. Each secondary has NS turns of wire uniformly distributed
over a length T . The core is shown at a displacement of zero.
The secondary windings are wired in such a way that the difference of secondary voltages is measured. The amplitude of this sinusoidal signal is proportional to the displacement of the core. An
LVDT is called a differential transformer because the output is the difference of the two secondary
voltages. An LVDT is a variable transformer because the core is moveable, which changes the number
of flux-linking turns of wire in one or both of the secondaries. A schematic diagram of an LVDT is
shown in Fig. 2.
Figure 2: A schematic diagram of an LVDT. The core is shown at a displacement of x.
The voltage across secondary S1 is called V21 . The voltage across secondary S2 is called V43 .
Consider a closed path starting from the positive terminal of VO and proceeding through the terminals
3, 4, 2, 1, the negative terminal of VO , and back to the positive terminal of VO . While traversing this
path we add the voltage rises and subtract the voltage drops as follows:
V43 − V21 + (V21 − V43 ) = 0.
(1)
The sum around this closed path is zero, thus the output voltage is indeed the difference of V21 and
V43 .
Both secondaries have the same number of turns, NS . Consider core displacements in the range
0 < x < T , as shown in Fig. 2. In this region, the flux path passes through all NS turns of secondary
S2 , thus
dφ
V43 = NS .
(2)
dt
However, for secondary S1 , the flux path passes through only a fraction of the NS turns. The effective
number of turns of secondary S1 when the core is at a displacement x is
T −x
Ns .
T
(3)
We can check that when x = 0, the number of turns is NS and when x = T , the number of turns is
zero. Thus
dφ
T −x
NS .
V21 =
(4)
T
dt
The output voltage VO is obtained by taking (4)-(2):
VO = V21 − V43 =
T −x
T
− 1 NS
dφ
NS dφ
= −x
.
dt
T dt
(5)
The amplitude of the output voltage is proportional to the displacement x.
For core displacements in the range −T < x < 0 it can be shown that we get the same formula for
VO . Thus, over the range −T < x < T , the output voltage is
VO = −x
NS dφ
.
T dt
(6)
Suppose that T is one inch and dφ/dt = A sin ωt. The following graphs show the output voltage
waveform for three different values of core displacement. When x = −0.75in the amplitude of VO is
0.75ANS . When x = −0.1in the amplitude of VO is 0.1ANS . Finally, when x = 0.3 the amplitude
of VO is −0.3ANS . The negative sign on this last amplitude implies a change of sign (or 180◦ phase
change) in VO , as shown in the third graph.
Figure 3: Plots of VO for three different values of core position.