Question 6.6.1: An Electromagnetic Speaker Develop a model of the electromag...

Figure 6.6.3a shows a simplified model of the mechanical subsystem, along with its free body diagram. The mass m represents the combined mass of the diaphragm and the coil. The spring constant k and damping constant c depend on the material properties of the diaphragm. The force f is the magnetic force, which is related to the coil current i by (6.4.1), f = nBLi, where
n is the number of turns in the coil. Let K_{f} = nBL. From Newton’s law

f = BLi (6.4.1)
m \frac{d^{2}x}{dt^{2}}= −c \frac{dx}{dt}  −  kx + K_{f} i (1)
Figure 6.6.3b shows the electrical subsystem. The coil’s inductance and resistance are L and R. The coil experiences a back emf because it is a current conductor moving in a magnetic field.
This back emf is given by K_{b} \dot{x}. The voltage v is the signal from the amplifier. From Kirchhoff’s voltage law,
v = L \frac{di}{dt} + R_{i} + K_{b} \frac{dx}{dt}        (2)
The speaker model consists of equations (1) and (2).
Transforming equation (1) and solving for X(s) gives
X(s) = \frac{K_{f}}{ms^{2}  +  cs  +  k} I(s)
Transforming equation (2) and solving for I(s) gives
I(s) = \frac{1}{Ls  +  R} [V(s)  −  K_{b}s X(s)]
Eliminating I(s) from the previous two equations, we obtain the desired transfer function.
\frac{X(s)}{V(s)} = \frac{K_{f}}{mLs^{3}  +  (cL  +  m R)s^{2}  +  (kL  +  cR  +  K_{f} K_{b})s  +  k R}       (3)