Biomedical Engineering Reference
In-Depth Information
v
1
¼ Av
2
Scaling
ð
Dissipative element
Þ
v
1
¼ A
dv
2
dt
Time differentiation
ð
Inertial elements
Þ
v
1
¼ A
Ð
v
2
dt
Time integration
ð
Capacitive elements
Þ
[Eq. 2.3.5]
where
v
1
and
v
2
are the variables associated with the
analog element. The specific variables depend on the type
of elements. For electrical elements, they are voltage and
current, whereas for mechanical elements they are force
and velocity (
Table 2.3-2
).
One of the advantages of analog models is that al-
though the systems they represent can be quite compli-
cated, the individual analog components behave in
a straightforward manner as noted in Eq. 2.3.5. Analog
models become complicated because of the number of
elements involved and their configuration, but the ele-
ments themselves are simple. Another advantage of
analog analysis is that given the configuration of elements,
a mathematical description of the overall model follows
directly. We will find that by applying the conservation
laws (conservation of charge, energy, and force) to
a specific configuration of elements, a mathematical de-
scription follows in algorithmic fashion. One need merely
follow a set of rules to obtain a mathematically complete
description of the model.
It is possible to introduce nonlinear components into
an analog model, but this complicates the analysis. For
example, the model shown in
Figure 2.3-5
is actually
nonlinear because the capacitance changes its value as
blood pressure changes. Often, a piecewise linear ap-
proach can be used where the model is analyzed over
several different operating regions within which the
nonlinear elements can be taken as linear.
Figure 2.3-7 Analog model of the lateral and medial rectus
muscles and associated mechanical involved in directing
horizontal eye position. The neural signals, N
ANT
and N
AG
, are the
inputs and the angular position,
q
, is the output. The function of
the various analog components is discussed in the text. (Adapted
from Bahill and Stark, 1979.)
muscles, the lateral and medial rectus (
Figure 2.3-7
).
These muscles are the mechanical elements involved in
controlling the horizontal position of the eye. Each of the
two extraocular muscles shown, the lateral and medial
rectus, is represented by a force generator
F
ANT
(or
F
AG
),
a viscous element
B
ANT
(or
B
AG
), a series elastic element,
k
SE
,
and a parallel elastic element,
k
PE
.
The two muscle
representations also include an additional elastic element
k
LT
.
Three other elements represent the mechanical
properties of the eyeball and the orbit: an inertial com-
ponent,
J
, representing the moment of inertia of the
eyeball; a viscous element
Bo,
representing the friction
between the eye and orbit; and a parallel elastic element,
k
P¢,
representing the elastic properties of the eye in the
orbit. The neural signals,
N
ANT
and
N
AG
, are the inputs
and the angular position, q, is the output.
With the aid of a computer, this model, and all
quantitative models, can be tested to see if they predict
reasonable results. This is one of the primary motiva-
tions for construction of any model, the ability to 'try
out' the model to see if it behaves in a manner similar to
the process it represents. Programming a model into
a computer to see how it behaves is known as
simula-
tion.
Simulations of the model in
Figure 2.3-7
have
produced highly accurate predictions of the behavior of
real eye movements and have also provided insight into
the nature of the neural signals that activate the two
muscles.
An appealing aspect of the analog-modeling approach
is the relative simplicity of the mathematical description
of the elements. All linear analog elements can be
represented by scaling, integration, or differentiation
operations between the associated variables:
Example 2.3.2: A constant force of 4 dyne is applied to
a 2-g mass. Find the velocity of the mass after 5 seconds.
Table 2.3-2 Variables associated with analog elements and
related conservation laws
Element
type
Variable
Conservation
law
Element
(type)
Electrical
Voltage, V
(volts)
Charge
(Kirchhoff's
current law)
Resistor
(dissipative)
Current, i
(amps)
Energy
(Kirchhoff's
voltage law)
Inductor
(inertial)
Capacitor
(capacitive)
Mechanical
Force, F
(newtons)
Force
(Newton's law)
Friction
(dissipative)
Velocity, v
(cm/sec)
Mass
(inertial)
Elasticity
(capacitive)
