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loading, displacement, or any physical variable. When we apply a step function input to a
measurement system, we obtain information about how quickly a system will respond to a change
in input signal. To illustrate this, let us apply a step function as an input to the general first-order
system. Setting F(t) ¼ AU(t) in Equation 3.4 gives
t_ y þ y ¼ KAU t
ð Þ ¼ KF t
ð Þ
with an arbitrary initial condition denoted by, y(0) ¼ y 0 . Solving for t ! 0
þ yields
y t
ð Þ
|ffl{zffl}
Time response
¼ KA
|ffl ffl{zffl ffl}
Steady response
þ y 0 À KA
ð
Þ e
Àt=t
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Transient response
ð3:5Þ
The solution of the differential equation, y(t), is the time response (or simply the response) of the
system. Equation 3.5 describes the behavior of the system to a step change in input. This means that
y(t) is in fact the output indicated by the display stage of the system. It should represent the time
variation of the output display of the measurement system if an actual step change were applied to
the system. We have simply used mathematics to simulate this response.
Equation 3.5 consists of two parts. The first term is known as the steady response because, as
t ! 1, the response of y(t) approaches this steady value. The steady response is that portion of the
output signal that remains after the transient response has decayed to zero. The second term on
the right side of Equation 3.5 is known as the transient response of y(t) because, as t ! 1, the
magnitude of this term eventually reduces to zero.
For illustrative purposes, let y 0 < A so that the time response becomes as shown in Figure 3.6.
Over time, the indicated output value rises from its initial value, at the instant the change in input is
applied, to an eventual constant value, y 1 ¼ KA, at steady response. As an example, compare this
general time response to the recognized behavior of the bulb thermometer when measuring body
temperature as discussed earlier. We see a qualitative similarity. In fact, in using a bulb thermometer
to measure body temperature, this is a real step function input to the thermometer, itself a first-order
measuring system.
Suppose we rewrite the response Equation 3.5 in the form
G t
ð Þ ¼
y t
ð Þ À y 1
y 0 À y 1
¼ e
Àt=t
ð3:6Þ
The term G t
ð Þ is called the error fraction of the output signal. Equation 3.6 is plotted in Figure 3.7,
where the time axis is nondimensionalized by the time constant. We see that the error fraction
– 1
0
1
2
0
1
2
U(t)
Time, t
Figure 3.5 The unit step function, U(t).
86 Chapter 3 Measurement System Behavior
15:24:52 Page 86
loading, displacement, or any physical variable. When we apply a step function input to a
measurement system, we obtain information about how quickly a system will respond to a change
in input signal. To illustrate this, let us apply a step function as an input to the general first-order
system. Setting F(t) ¼ AU(t) in Equation 3.4 gives
t_ y þ y ¼ KAU t
ð Þ ¼ KF t
ð Þ
with an arbitrary initial condition denoted by, y(0) ¼ y 0 . Solving for t ! 0
þ yields
y t
ð Þ
|ffl{zffl}
Time response
¼ KA
|ffl ffl{zffl ffl}
Steady response
þ y 0 À KA
ð
Þ e
Àt=t
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Transient response
ð3:5Þ
The solution of the differential equation, y(t), is the time response (or simply the response) of the
system. Equation 3.5 describes the behavior of the system to a step change in input. This means that
y(t) is in fact the output indicated by the display stage of the system. It should represent the time
variation of the output display of the measurement system if an actual step change were applied to
the system. We have simply used mathematics to simulate this response.
Equation 3.5 consists of two parts. The first term is known as the steady response because, as
t ! 1, the response of y(t) approaches this steady value. The steady response is that portion of the
output signal that remains after the transient response has decayed to zero. The second term on
the right side of Equation 3.5 is known as the transient response of y(t) because, as t ! 1, the
magnitude of this term eventually reduces to zero.
For illustrative purposes, let y 0 < A so that the time response becomes as shown in Figure 3.6.
Over time, the indicated output value rises from its initial value, at the instant the change in input is
applied, to an eventual constant value, y 1 ¼ KA, at steady response. As an example, compare this
general time response to the recognized behavior of the bulb thermometer when measuring body
temperature as discussed earlier. We see a qualitative similarity. In fact, in using a bulb thermometer
to measure body temperature, this is a real step function input to the thermometer, itself a first-order
measuring system.
Suppose we rewrite the response Equation 3.5 in the form
G t
ð Þ ¼
y t
ð Þ À y 1
y 0 À y 1
¼ e
Àt=t
ð3:6Þ
The term G t
ð Þ is called the error fraction of the output signal. Equation 3.6 is plotted in Figure 3.7,
where the time axis is nondimensionalized by the time constant. We see that the error fraction
– 1
0
1
2
0
1
2
U(t)
Time, t
Figure 3.5 The unit step function, U(t).
86 Chapter 3 Measurement System Behavior