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Part III
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Electromechanics
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and L= N2 5002 = 0157 H = R 159 107
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Finally, we can calculate the stored magnetic energy as follows: 1 2 1 Li = (0157 H) (01 A)2 = 0785 10 3 J 2 2 Part 2 To calculate the induced voltage due to a time-varying magnetic ux, we use equation 1616: Wm = e= d dB d =N = NA = NAB0 cos( t) dt dt dt
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= 500 00001 06 377 cos(377t) = 1131 cos(377t) V
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Comments: The voltage induced across a coil in an electromagnetic transducer is a very
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important quantity called back electromotive force, or back emf We shall make use of this quantity in Sec 165
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Magnetic Reluctance Position Sensor
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A simple magnetic structure, very similar to those examined in the previous examples, nds very common application in the so-called variable-reluctance position sensor, which, in turn, nds widespread application in a variety of con gurations for the measurement of linear and angular velocity Figure 1623 depicts one particular con guration that is used in many applications In this structure, a permanent magnet with a coil of wire wound around it forms the sensor; a steel disk (typically connected to a rotating shaft) has a number of tabs that pass between the pole pieces of the sensor The area of the tab is assumed equal to the area of the cross section of the pole pieces and is equal to a 2 The reason for the name variable-reluctance sensor is that the reluctance of the magnetic structure is variable, depending on whether or not a ferromagnetic tab lies between the pole pieces of the magnet
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+ eS Magnet
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FOCUS ON MEASUREMENTS
Figure 1623 Variable-reluctance position sensor
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lg lg a a Tab Steel disk
16
Principles of Electromechanics
The principle of operation of the sensor is that an electromotive force, eS , is induced across the coil by the change in magnetic ux caused by the passage of the tab between the pole pieces when the disk is in motion As the tab enters the volume between the pole pieces, the ux will increase, because of the lower reluctance of the con guration, until it reaches a maximum when the tab is centered between the poles of the magnet Figure 1624 depicts the approximate shape of the resulting voltage, which, according to Faraday s law, is given by eS = d dt
The rate of change of ux is dictated by the geometry of the tab and of the pole pieces, and by the speed of rotation of the disk It is important to note that, since the ux is changing only if the disk is rotating, this sensor cannot detect the static position of the disk
Magnet Tab eS (V)
Maximum flux
Figure 1624 Variable-reluctance position sensor waveform
One common application of this concept is in the measurement of the speed of rotation of rotating machines, including electric motors and internal combustion engines In these applications, use is made of a 60-tooth wheel, which permits the conversion of the speed rotation directly to units of revolutions per minute The output of a variable-reluctance position sensor magnetically coupled to a rotating disk equipped with 60 tabs (teeth) is processed through a comparator or Schmitt trigger circuit (see 15) The voltage waveform generated by the sensor is nearly sinusoidal when the teeth are closely spaced, and it is characterized by one sinusoidal cycle for each tooth on the disk If a negative zero-crossing detector (see 15) is employed, the trigger circuit will generate a pulse corresponding to the passage of each tooth, as shown in Figure 1625 If the time between any two pulses is measured by means of a high-frequency clock, the speed of the engine can be directly determined in units of rev/min by means of a digital counter (see 14)
Part III
Electromechanics
Variable reluctance sensor voltage eS
Schmitt trigger + R1 Vout
R2 Vref
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T Vclock N
Vout
Figure 1625 Signal processing for a 60-tooth-wheel RPM sensor
Voltage Calculation in Magnetic Reluctance Position Sensor
This example illustrates the calculation of the voltage induced in a magnetic reluctance sensor by a rotating toothed wheel In particular, we will nd an approximate expression for the reluctance and the induced voltage for the position sensor shown in Figure 1626, and show that the induced voltage is speed-dependent It will be assumed that the reluctance of the core and fringing at the air gaps are both negligible
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