handbook of modern sensors

118 3 Physical Principles of Sensing
and, in the result, we obtain the differential equation
dT h
dt
+ T h
RC = Q 1
C + T a
RC . (3.161)
This is a first-order differential equation which is typical for thermal systems. A
thermal element, if not a part of a control system with a feedback loop, is inherently
stable.Aresponse of a simple thermal element may be characterized by a thermal time
constant, which is a product of thermal capacity and thermal resistance: τ T = CR. The
time constant is measured in units of time (s) and, for a passively cooling element, is
equal to the time which it takes to reach about 37% of the initial temperature gradient.
3.14.3 Electrical Elements
There are three basic electrical elements: the capacitor, the inductor, and the resistor.
Again, the governing equation describing the idealized elements are given in Table
3.4. For the idealized elements, the equations describing the sensor’s behavior may be
obtained from Kirchhoff’s laws, which directly follow from the law of conservation
of energy:
Kirchhoff’s first law: The total current flowing toward a junction is equal to the total
current flowing from that junction (i.e., the algebraic sum of the currents flowing
toward a junction is zero).
Kirchhoff’s second law: In a closed circuit, the algebraic sum of the voltages across
each part of the circuit is equal to the applied e.m.f.
Let us assume that we have a sensor whose elements may be represented by a circuit
shown in Fig. 3.50B. To find the circuit equation, we will use the Kirchhoff’s first
law, which is sometimes called Kirchhoff’s current law. For the node,
i 1 − i 2 − i 3 = 0, (3.162)
and for each current,
i 1 = e − V 3
R 1
= 1 L
∫
(V 3 − V 1 )dt,
i 2 = V 1 − V 2
R 3
= C dV 2
dt ,
i 3 = V 1
R 2
. (3.163)
When these expressions are substituted into Eq. (3.162), the resulting equation becomes
V 3
+ V 1 − V 2
+ 2 V 1
+ C dV 2
− 1 ∫
(V 3 − V 1 )dt= e . (3.164)
R 1 R 3 R 2 dt L
R 1
In Equation (3.164), e/R 1 is the forcing input, and the measurable outputs are V 1 ,
V 2 , and V 3 . To produce Equation (3.164), three variables i 1 , i 2 , and i 3 have to be