E1C03 09/14/2010
15:24:52 Page 85
provides the zero-order response equation between output displacement and input pressure
and gives
y ¼ A=k
ð
Þ p À p atm
ð
Þ
The term p À p atm
ð
Þrepresents the pressure relative to atmospheric pressure. It is the pressure
indicated by this gauge. Direct comparison with Equation 3.2 shows that the input pressure
magnitude equates to the piston displacement through the static sensitivity, K ¼ A/k. Drawing
from the concept of Figure 3.2, the system operates on pressure so as to bring about the relative
displacement of the piston, the magnitude of which is used to indicate the pressure. The equivalent of
spring stiffness and piston area affect the magnitude of this displacement—factors considered in its
design. The exact static input–output relationship is found through calibration of the gauge. Because
elements such as piston inertia and frictional dissipation were not considered, this model would not be
appropriate for studying the dynamic response of the gauge.
First-Order Systems
Measurement systems that contain storage elements do not respond instantaneously to changes in
input. The bulb thermometer discussed in Section 3.2 is a good example. The bulb exchanges energy
with its environment until the two are at the same temperature, storing energy during the exchange.
The temperature of the bulb sensor changes with time until this equilibrium is reached, which
accounts physically for its lag in response. The rate at which temperature changes with time can be
modeled with a first-order derivative and the thermometer behavior modeled as a first-order
equation. In general, systems with a storage or dissipative capability but negligible inertial forces
may be modeled using a first-order differential equation of the form
a 1 _
y þ a 0 y ¼ F t
ð Þ
ð3:3Þ
with _
y ¼ dy=dt. Dividing through by a 0 gives
ty þ y ¼ KF t
ð Þ
ð3:4Þ
where t ¼ a 1 =a 0 . The parameter t is called the time constant of the system. Regardless of the
physical dimensions of a 0 and a 1 , their ratio will always have the dimensions of time. The time
constant provides a measure of the speed of system response, and as such is an important
specification in measuring dynamic input signals. To explore this concept more fully, consider
the response of the general first-order system to the following two forms of an input signal: the step
function and the simple periodic function.
Step Function Input
The step function, AU(t), is defined as
AU t
ð Þ ¼ 0 t 0
À
AU t
ð Þ ¼ A t ! 0
þ
where A is the amplitude of the step function and U(t) is defined as the unit step function as depicted
in Figure 3.5. Physically, this function describes a sudden change in the input signal from a constant
value of one magnitude to a constant value of some other magnitude, such as a sudden change in
3.3 Special Cases of the General System Model 85
15:24:52 Page 85
provides the zero-order response equation between output displacement and input pressure
and gives
y ¼ A=k
ð
Þ p À p atm
ð
Þ
The term p À p atm
ð
Þrepresents the pressure relative to atmospheric pressure. It is the pressure
indicated by this gauge. Direct comparison with Equation 3.2 shows that the input pressure
magnitude equates to the piston displacement through the static sensitivity, K ¼ A/k. Drawing
from the concept of Figure 3.2, the system operates on pressure so as to bring about the relative
displacement of the piston, the magnitude of which is used to indicate the pressure. The equivalent of
spring stiffness and piston area affect the magnitude of this displacement—factors considered in its
design. The exact static input–output relationship is found through calibration of the gauge. Because
elements such as piston inertia and frictional dissipation were not considered, this model would not be
appropriate for studying the dynamic response of the gauge.
First-Order Systems
Measurement systems that contain storage elements do not respond instantaneously to changes in
input. The bulb thermometer discussed in Section 3.2 is a good example. The bulb exchanges energy
with its environment until the two are at the same temperature, storing energy during the exchange.
The temperature of the bulb sensor changes with time until this equilibrium is reached, which
accounts physically for its lag in response. The rate at which temperature changes with time can be
modeled with a first-order derivative and the thermometer behavior modeled as a first-order
equation. In general, systems with a storage or dissipative capability but negligible inertial forces
may be modeled using a first-order differential equation of the form
a 1 _
y þ a 0 y ¼ F t
ð Þ
ð3:3Þ
with _
y ¼ dy=dt. Dividing through by a 0 gives
ty þ y ¼ KF t
ð Þ
ð3:4Þ
where t ¼ a 1 =a 0 . The parameter t is called the time constant of the system. Regardless of the
physical dimensions of a 0 and a 1 , their ratio will always have the dimensions of time. The time
constant provides a measure of the speed of system response, and as such is an important
specification in measuring dynamic input signals. To explore this concept more fully, consider
the response of the general first-order system to the following two forms of an input signal: the step
function and the simple periodic function.
Step Function Input
The step function, AU(t), is defined as
AU t
ð Þ ¼ 0 t 0
À
AU t
ð Þ ¼ A t ! 0
þ
where A is the amplitude of the step function and U(t) is defined as the unit step function as depicted
in Figure 3.5. Physically, this function describes a sudden change in the input signal from a constant
value of one magnitude to a constant value of some other magnitude, such as a sudden change in
3.3 Special Cases of the General System Model 85