Page 150 - Organic Electronics in Sensors and Biotechnology

Integrated Pyr oelectric Sensors 127

However, the final time constant τ is not given by the product of
l
the parallel capacitances and resistances. In addition, an intermediate
time constant with an associated voltage level for each layer arises.

1 = 1 + 1 + 1
C C C C
tot 1 2 3 (4.14)

R = R + R + R +( R )
tot 1 2 3 c
C R R + R + C R R + R + C R R + R )
(
(
(
)
)
A = 1 1 2 3 2 2 1 3 3 3 1 2 (4.15)
2
B = C C C C R R R R (4.16)
12 3 tot 1 2 3 tot
− C R + RR τ + C R [− CR + RR + C RR + R τ ] + CR C R R + R τ )
)
(
(
(
)
fa b c) = a b c a l b b c b c c a a c a b l c c a b a c l
,
(,
+
τ
CR + R R τ + CR [ 2 + R + R ) ]τ + C R [ 2 − C RR + ( R + R ) ]
− C R R
2
(
(
)
− CRR
a b c a l c c a b a b a l b b c c a a a c a c a l
(4.17)
In case one of the three layer resistances is much larger than the
other two, say, R >> R , one can find a simplified approximate
1 2,3
expression for τ :
l
τ = CR + R + C R + C R (4.18)
)
(
l 1 2 3 2 2 3 3
If the time constants of the respective regimes are too close to
each other, the first voltage levels of the respective time domain
(regime) will not clearly show because the layer already charges up to
the voltage level of the second time domain.
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 is,0
,
V () = V ( − e −t/τ m ) (4.19)
t
t
1
,
im im,0
,
Vt () = V ( −1 e − t/τ l )
il , il , ,0
Vt() = [ V t e t − /τ m + V ()] t − /τ l + V t() (4.20)
()
t e
,
,
i s
,
il
im
i

In case R is not much smaller than the layer resistances, a better
v
approximation for the short time regime is obtained by adding a
resistive prefactor:
⎛ ⎡ R ⎞ ⎤
⎢ ⎢
()
Vt() = 1 − v ⎟ Vt e − t/τ m + V ()⎥ e −t/τ l +V t() (4.21)
t
⎜
,
i R ⎠ is , im il ,
⎣ ⎝ ⎢ tot ⎦ ⎥