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Step Function Input The step function, AU(t), is defined as AU(t) = 0 t ? 0? AU(t) = A t ? 0? 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 86 Chapter 3 Measurement System Behavior 2 U(t) 1 0 0 1 2 Time, t Figure 3.5 The unit step function, U(t). 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) = AU(t) in Equation 3.4 gives $\tau\frac{dy}{dt} + y = KAU(t) = KF(t)$ with an arbitrary initial condition denoted by, y(0) = y?. Solving for t ? 0? yields y(t) = \frac{KA}{\tau} + (y? - \frac{KA}{\tau})e^{-t/\tau} Time response Steady response Transient response

          Step Function Input
The step function, AU(t), is defined as
AU(t) = 0  t ? 0?
AU(t) = A  t ? 0?
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
86 Chapter 3 Measurement System Behavior
2
U(t) 1
0
0
1
2
Time, t
Figure 3.5 The unit step function, U(t).
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) = AU(t) in Equation 3.4 gives
$\tau\frac{dy}{dt} + y = KAU(t) = KF(t)$
with an arbitrary initial condition denoted by, y(0) = y?. Solving for t ? 0? yields
y(t) = \frac{KA}{\tau} + (y? - \frac{KA}{\tau})e^{-t/\tau}
Time response     Steady response      Transient response

Step Function Input
The step function, AU(t), is defined as
AU(t) = 0 t ? 0?
AU(t) = A t ? 0?
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
86 Chapter 3 Measurement System Behavior
2
U(t) 1
0
0
1
2
Time, t
Figure 3.5 The unit step function, U(t).
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) = AU(t) in Equation 3.4 gives
τ(dy)/(dt) + y = KAU(t) = KF(t)
with an arbitrary initial condition denoted by, y(0) = y?. Solving for t ? 0? yields
y(t) = (KA)/(τ) + (y? - (KA)/(τ))e^-t/τ
Time response Steady response Transient response

Added by Jennifer L.

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