handbook of modern sensors

16 Temperature Sensors 459
from which we may find the sensor’s temperature as
T S = T B − (T B − T 0 ) r 1
r 2
= T B − T r 1
r 2
, (16.2)
where T is a thermal gradient between the object and the surroundings. Let us take
a closer look at Eq. (16.2). We can draw several conclusions from it. The first is
that the sensor temperature T S is always different from that of the object. The only
exception is when the environment has the same temperature as the object (a special
case when T = T B − T 0 = 0). The second and the most important conclusion is that
T S will approach T B at any temperature gradient T when the ratio r 1 /r 2 approaches
zero. This means that for minimizing the measurement error, one must improve a
thermal coupling between the object and the sensor and decouple the sensor from the
surroundings as much as practical. Often, it is not easy to do.
In the above, we evaluated a static condition; now let us consider a dynamic case
(i.e., when temperatures change with time). This occurs when either the object or the
surrounding temperatures change or the sensor has just been attached to the object
and its temperatures is not yet stabilized. When a temperature-sensing element comes
in contact with the object, the incremental amount of transferred heat is proportional
to a temperature gradient between that sensing element temperature T S and that of
the object T B :
dQ= α 1 (T B − T S )dt, (16.3)
where α 1 = 1/r 1 is the thermal conductivity of the sensor–object interface. If the
sensor has specific heat c and mass m, the absorbed heat is
dQ= mc dT . (16.4)
If we ignore heat lost from the sensor to the environment through the connecting
and supporting structure (assuming that r 2 =∞), Eqs. (16.3) and (16.4) yield the
first-order differential equation
We define the thermal time constant τ T as
then, the differential equation takes the form
This equation has the solution
α 1 (T 1 − T)dt= mc dT . (16.5)
τ T = mc
α 1
= mcr 1 ; (16.6)
dT
T 1 − T = dt
τ T
. (16.7)
T S = T B − T e −t/τ T
, (16.8)
where, initially, the sensor is assumed to be at temperature T B . The time transient of
the sensor’s temperature, which corresponds to Eq. (16.8), is shown in Fig. 16.2A.