Page 148 - Organic Electronics in Sensors and Biotechnology

Integrated Pyr oelectric Sensors 125

relative voltage at the layers, i.e., the percentage V /V with respect to
1,s
V /V of the applied voltage, exponentially approaches V /V with
2,s 1,s,0
respect to V /V corresponding to the ratio between C and layer
2,s,0 tot
capacitance (see Fig. 4.5).
1 1 1
= + (4.7)
C C C
tot 1 2
R = R + R + R ) (4.8)
(
tot 1 2 c
However, simultaneously but with a much larger time constant τ ,
l
the relative layer voltages V /V and V /V exponentially approach
1,l 2,l
V /V and V /V which is the ratio between layer resistance and
1,l,0 2,l,0
total resistance R (see Table 4.1).
tot
In contrast to τ , the time constant τ is the product of parallel layer
s l
resistances and the parallel layer capacitances. Since the layer resist-
ances are much larger than the contact resistance, τ is much larger
l
than τ . In the ideal case of infinite layer resistances, τ is infinite and
s l
the layer voltage will stay at its first reached level V .
s
A very good approximation for the evolution of the ith layer volt-
age is given by the following expressions:
Vt() = V ( − e − t/τ s )
1
,
is i s,0
,
(4.9)
Vt() = V ( − e −t/τ l )
t
1
,
il , il,0
Vt() = V () −τ l + V t()
t/
t e
,
i i s il , (4.10)
The Capacitive Three-Layer Structure
The behavior of the capacitive three-layer structure is largely analogous
to that of the double-layer structure. However, the analytical solution of
the underlying differential equation is more sophisticated. The equiva-
lent circuit of the capacitive three-layer structure is shown in Fig. 4.6.
The system of differential equations for the evolution of the
voltages at the three layers is analogous to that of the double-layer
structure:
Vt () VV t ()− − V t () dV t ()
1 + 1 2 + C 1 = 0 (4.11)
R R 1 dt
1 c
Vt () VV t ()− − V t () dV t ()
2 + 1 2 + C 2 = 0 (4.12)
R R 2 dt
2 c
−
Vt () VV t () − V t () dV t ()
3 + 1 2 + C 3 = 0 (4.13)
R R 3 dt
3 c