Lakeshore 330 User Manual page 102

Lake Shore Model 330 Autotuning Temperature Controller User's Manual
The measured offset voltages shown in Figs. 4 and 6 can be
understood by using the well-known result from p-n junction theory:
I = I [exp(eV / nkT) - 1]
s
where I = the forward current through the junction, I
saturation current, e = the electron charge, V = the voltage across the
junction, k = Boltzmann's constant, and T = the absolute temperature.
n is a parameter depending on the location of the generation and
recombination of the electrons and holes and typically has a value
between 1 and 2. This expression for the IV characteristics of a p-n
junction is valid from approximately 40 K to above 300 K for the silicon
diodes discussed here. Below 40 K, a new conduction mechanism
becomes dominant, suggesting the influence of impurity conduction,
carrier freezeout, increased ohmic behavior of the bulk material, and
1-6
p-i-n diode type behavior.
The only adjustable parameter in Eq. 1 which is necessary for the
present analysis is the parameter n . This parameter can be
determined quite easily from the IV characteristics of the silicon diode
temperature sensor. The parameter I
IV curve to an arbitrarily chosen point on the curve. The value of n =
1.8 was found to give a relatively good fit to the IV data for both 305
and 77 K and has been assumed in the present discussion.7 Equation
(1) can now be solved for V(I):
V(I) = (nkT / e)ln(I / I + 1)
s
Substituting a dc current with an ac modulation, I
average voltage read by the voltmeter in the dc voltage mode can be
calculated from:
z
1 T
ω
= +
V V I ( I cos t dt
dc ac
T
0
where T = the period of integration of the voltmeter or approximately
2 π / ω . Implied in this derivation is the assumption that ω is sufficiently
small so that effects from diode capacitance (on the order of
picofarads) can be ignored.
On carrying out the integration of Eq. (3) and subtracting V(I
offset voltage is:
nkT
∆V = − =
V V I ( ) ln
dc
e
≤ I
where I + I . If a small signal (linear) model is used, the rms
ac dc s
voltage across the diode can be easily related to I
F I
H G K J
I dV
= ac =
V
rms
dI
2
=
I I dc
Evaluation of Eq. (5) and substitution back into (4) yields:
L
F
M M
G G
nkT 1
∆V = + −
ln 1 1 2
H
e N 2
2
≤ 1 for a physical solution. Equation (6) predicts an
where 2(eV / nkT)
rms
offset voltage which is independent of both frequency and dc operating
current and is shown plotted in Fig. 4 by the solid line. The agreement
with the experimental measurements is quite good, verifying the overall
picture as to the effect of induced currents on diode temperature
sensors. The results recorded at 305 K are described equally well by Eq.
(6).
B-16
(1)
= the reverse
s
is eliminated by normalizing the
s
(2)
+ I
dc
)
(3)
L O
F I
F I
M M P P
2
G G J J
H G K J
1 eV
+ − rms
1 1 2
H K
N 2 nkT Q
:
ac
F I
F I
H G K J
1 nkT I
H G K J
ac
+
e I I
2
dc s
O
I
F I
P P
2
J J
H G K J
eV
rms
K
nkT Q
ω
cos t , the FIGURE 4. DC offset voltage as a function of rms
ac
ac voltage across a silicon diode temperature
sensor operating at 77 K. The symbols represent
data recorded at three different dc operating
currents with a 60 Hz signal superimposed. The
solid curve gives small signal model results while
the dashed curve represents the extended
calculations. Equivalent temperature errors are
indicated along the right edge.
), the dc
dc
(4)
(5)
(6)
FIGURE 5. DC offset voltage as a function of rms ac
voltage across a silicon diode temperature sensor
operating at 4.2 K. The symbols represent data
recorded at three different dc operating currents with
a 60 Hz signal superimposed. Equivalent
temperature errors are indicated along the right
edge.
Application Notes