Biomedical Engineering Reference
In-Depth Information
A first-order differential equation describes the electrical portion of the motor
L
di
r
(
)
θ(
)
t
K
e
d
t
+
Ri
r
(
)
=
V
i
(
)
−
t
t
(4.42)
dt
dt
Alternatively, equations (4.41) and (4.42) can be written in terms of the speed of the motor
d
ω(
t
)
(
J
r
+
J
L
)
+
b
ω(
t
)
=
K
m
i
r
(
t
)
(4.43)
dt
L
di
r
(
t
)
+
Ri
r
(
t
)
=
V
i
(
t
)
−
K
e
ω(
t
)
(4.44)
dt
4.5.5 Similarities of the Two Models
SPICE software used to analyze electrical circuits is both accurate and easily avail-
able; therefore, mechanical models are often converted to their electrical counterparts
to make use of this software. For example, the electrical equivalent of the standard mass-
spring-damper block diagram is the parallel capacitor-inductor-resistor circuit shown in
Figure 4-10.
Table 4-3 lists the equations that describe the equations for the linear and torsional
mechanical elements along with their electrical equivalents.
4.5.6 Fluid Flow Elements
Similar models can be derived for fluid flow components. However, in this instance there
are two different formulations: one for incompressible hydraulics; and another for com-
pressible pneumatics, where changes in pressure result in changes in density. This topic
examines only the incompressible hydraulic model.
The relationship between the volume rate of flow,
q
(m
3
/s), and the pressure difference,
p
2
, is determined by the hydraulic resistance,
R
. This is the hydraulic equivalent
to Ohm's law, where the hydraulic resistance is analogous to electrical resistance, the
volume flow rate is equivalent to current, and the pressure difference is equivalent to the
potential difference.
p
=
p
1
−
p
=
p
1
−
p
2
=
qR
(4.45)
FIGURE 4-10
Equivalent 2nd
order models
(a) Mechanical.
(b) Electrical.
