0195136047.pdf

12.1 BASIC PRINCIPLES OF ELECTROMECHANICAL ENERGY CONVERSION 513

EXAMPLE 12.1.3

A loudspeaker is a common electrochemical transducer in which vibration is caused by changes in
the input current to a coil which, in turn, is coupled to a magnetic structure that can produce time-
varying forces on the loudspeaker diaphragm. Figure E12.1.3(a) shows the schematic diagram of
a loudspeaker, Figure E12.1.3(b) is a simplified model, Figure E12.1.3(c) is a free-body diagram
of the forces acting on the loudspeaker diaphragm, and Figure E12.1.3(d) is the electrical model.
The force exerted on the coil is also exerted on the mass of the loudspeaker diaphragm.

S

N turns

N

N

Mass of

loudspeaker

diaphragm

Sound

output

Electric

input

Permanent

magnet

v

Spring

Moving

coil

−

+

k

d

x

m

x

m

f = m

fk

fe = Bli

fd

du

dt

u = dxdt

u = dxdt

(b) (c)

(a)

−

+

−

+

v e = Blu

L R

(d)

Figure E12.1.3Loudspeaker.

Develop the equation of motion for the electrical and mechanical sides of the device and
determine the frequency responseU(jω)/V(jω)of the loudspeaker using phase analysis, and
neglecting the coil inductance.

Solution

The electrical side is described by

v=L

di

dt

+Ri+e=L

di

dt

+Ri+Blu

whereBis the flux density andlis the length of the coil given by 2πNr, in whichNis the number
of coil turns andrthe coil radius. Note thateis the emf generated by the motion of the coil in the
magnetic field.
The mechanical side is described by