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Figure 1651 A more realistic representation of the transducer of Figure 1650
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The device shown in Figure 1652 is called a seismic transducer and can be used to measure the displacement, velocity, or acceleration of a body The permanent magnet of mass m is supported on the case by a spring, k, and there is some viscous damping, d, between the magnet and the case; the coil is xed to the case You may assume that the coil has length l and resistance and inductance Rcoil and Lcoil , respectively; the magnet exerts a magnetic eld B Find the transfer function between the output voltage, vout , and the acceleration of the body, a(t) Note that x(t) is not equal to zero when the system is at rest We shall ignore this offset displacement
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;; ;;
x(t) a(t)
Figure 1652 An electromagnetomechanical seismic transducer
Solution:
First we apply KVL around the electrical circuit to write the differential equation describing the electrical system: di dx + (Rcoil + Rout )i + Bl =0 dt dt Also note that vout = Rout i Next, we write the differential equation describing the mechanical system The magnet experiences an inertial force due to the acceleration of the supporting body, a(t), and to its own relative acceleration, d 2 x/dt 2 ; thus, we can sketch a free-body diagram and apply Newton s second law to the permanent magnet, as shown in the sketch L M a+ d 2x dt 2 +d dx + kx = Bli dt
N S N Case k d Accelerating body
+ vout Rout i
Part III
Electromechanics
Finally, using the Laplace transform, we determine the transfer function from A(s) to Vout (s) Let R = Rcoil + Rout Then (Ls + R)I (s) + BlsX(s) = 0 BlI (s) (Ms 2 + Ds + K)X(s) = MA(s) Since we need the transfer function from A to Vout , we use the expression Vout (s) = Rout I (s) and, after some algebra, nd that I (s) = or Vout (s) MBsRout = A(s) (Ls + R)(Ms 2 + Ds + K) + B 2 l 2 s MBlsA(s) (Ls + R)(Ms 2 + Ds + K) + B 2 l 2 s
EXAMPLE 1613 A Loudspeaker
Problem
A loudspeaker, shown in Figure 1653, uses a permanent magnet and a moving coil to produce the vibrational motion that generates the pressure waves we perceive as sound Vibration of the loudspeaker is caused by changes in the input current to a coil; the coil is, in turn, coupled to a magnetic structure that can produce time-varying forces on the speaker diaphragm A simpli ed model for the mechanics of the speaker is also shown in Figure 1653 The force exerted on the coil is also exerted on the mass of the speaker diaphragm, as shown in Figure 1654, which depicts a free-body diagram of the forces acting on the loudspeaker diaphragm
Electrical input + v N
N turns
S N
Figure 1653 Loudspeaker
;; ;; ;;;; ;;
k m Sound output f d x N Spring u= dx dt fk fi Mass N S N fd m x Coil N
Figure 1654 Forces acting on loudspeaker diaphragm
16
Principles of Electromechanics
L v + _
+ _ e
The force exerted on the mass, fi , is the magnetic force due to current ow in the coil The electrical circuit that describes the coil is shown in Figure 1655, where L represents the inductance of the coil, R represents the resistance of the windlings, and e is the emf induced by the coil moving through the magnetic eld Determine the frequency response, U (j )/V (j ) of the speaker
Figure 1655 Model of transducer electrical side
Solution
Known Quantities: Circuit and mechanical parameters; magnetic ux density; number of coil turns; coil radius Find: Frequency response of loudspeaker, U (j )/V (j )
Schematics, Diagrams, Circuits, and Given Data: Coil radius = 005 m; L = 10 mH;
; m = 0001 kg; d = 2275 N s2 /m; k = 5 105 N/m; N = 47; B = 1 T
Analysis: To determine the frequency response of the loudspeaker, we write the
differential equations that describe the electrical and mechanical subsystems We apply KVL to the electrical circuit, using the circuit model of Figure 1655, in which we have represented the Blu term (motional voltage) in the form of a back electromotive force, e: v L or L di + Ri + Blu = v dt di Ri e = 0 dt
Next, we apply Newton s second law to the mechanical system, consisting of: a lumped mass representing the mass of the moving diaphragm, m; an elastic (spring) term, which represents the elasticity of the diaphragm, k; and a damping coef cient, d, representing the frictional losses and aerodynamic damping affecting the moving diaphragm m du = fi fd fk = fi du kx dt
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