56
2 Fundamental Properties of Mem-Elements
By exploiting the POP it is easily seen that there exists a unique EP ¯
T = T 0P
which attracts all solutions. This means that, for any initial condition T (0), the
corresponding solution tends to T 0P as t → +∞, i.e.
G(T (t)) → G(T 0P ) =
1
R 0P
as t → +∞.
The power-off PTC thermistor is then equivalent to a linear passive resistor
with resistance equal to R 0P at ambient temperature 300 K. Such a memristive
system is said to be volatile because the initial condition T (0) (which accounts
for the past history of T (t)) has no effect on the value of the small-signal resistance
(δv(t)/δi(t)) measured from this device by applying an infinitesimal voltage δv(t)
across the device and measuring the corresponding current response δi(t). Said
another way, the effect of the past input signals that this memristive device has been
subjected to is forgotten once a sufficient time has elapsed.
Similar considerations hold for the Negative Temperature Coefficient (NTC)
thermistor defined by
i = W (T )v
and
˙
T =
δ N
H CN
(T 0N − T ) +
W (T )v 2
H CN
where
W (T ) =
R 0N exp
β N
1
T
−
1
T 0N
−1
and δ N , H CN , T 0N , R 0N , β N are positive parameters.
Example 2.17 (Discharge Tube) A discharge tube can be described as a currentcontrolled memristive device via the state-dependent Ohm’s law
v =
F
n
i
dn
dt
= −βn + α
F
n
i
2
where α, β, and F are device constants depending on the dimensions of the tube and
gas filling, and n is the internal scalar state variable. The POP gives the unique EP
¯
n = 0 that attracts all solutions, thus the discharge tube operates as an open circuit
at the steady state.