handbook of modern sensors

114 3 Physical Principles of Sensing
A zero-order response is a static or time independent characteristic
S(t)= Gs(t), (3.148)
where G is a constant transfer function. This relationship may take any form—for
instance, described by Eqs. (2.1—2.4). The important point is that G is not a function
of time; that is, a zero-order response to a step function is a step function.
A first-order response is characterized by a first-order differential equation
a 1
dS(t)
dt
+ a 0 S(t)= s(t), (3.149)
where a 1 and a 0 are constants. This equation characterizes a sensor that can store
energy before dissipating it. An example of such a sensor is a temperature sensor
which has a thermal capacity and is coupled to the environment through a thermal
resistance. A first-order response to a step function is exponential:
S(t)= S 0 (1 − e −t/τ ), (3.150)
where S 0 is a sensor’s static response and τ is a time constant which is a measure of
inertia. A typical first-order response is shown in Fig. 2.9B of Chapter 2.
A second-order response is characterized by a second-order differential equation
a 2
d 2 S(t)
dt 2
+ a 1
dS(t)
dt
+ a 0 S(t)= s(t). (3.151)
This response is specific for a sensor or a system that contains two components which
may store energy—for instance, an inductor and a capacitor, or a temperature sensor
and a capacitor. A second-order response contains oscillating components and may
lead to instability of the system. A typical shape of the response is shown in Fig. 2.11E
of Chapter 2.Adynamic error of the second-order response depends on several factors,
including its natural frequency ω 0 and damping coefficient b. A relationship between
these values and the independent coefficients of Eq. (3.151) are the following:
√
a0
ω 0 = , (3.152)
a 2
b = a 1
2 √ a 0 a 2
. (3.153)
A critically damped response (see Fig. 2.10 of Chapter 2) is characterized by b = 1.
The overdamped response has b>1 and the underdamped has b