Lake Shore Model 321 Autotuning Temperature Controller User’s Manual
Application Notes
D-3
To illustrate the effect of the sensor, in more detail, consider
the idealized curve (Figure 4) for a Lake Shore silicon diode
which has a nominal sensitivity of -50 mV/K below 30 kelvin
and -2.5 mV/K above 30 kelvin. Figure 3 illustrates the effect
of converting the voltage error signal (horizontal axis) to its
equivalent temperature error for the two sensitivity regions of
the silicon diode sensor. These curves introduce the concept
of loop gain dP/dT (watts/kelvin), which includes the gain of
the sensor as well as that of the deviation amplifier and
power output stage. As the transition in temperature from
above 30 kelvin to below 30 kelvin is made, the loop gain is
increased by a factor of 20 because of the increased
sensitivity of the silicon diode thermometer. Because of
noise and thermal phase lag, the deviation amplifier gain will
normally have to be reduced by the same factor so that the
loop gain remains relatively constant.
In order to maintain any desired temperature above that of
the cryogen in a cryogenic system, of course some level of
heater power must be supplied by the controller. We have
seen in Figures 2 and 3 that a non-zero temperature error signal is
necessary to produce an output, and that the magnitude of the
error—or temperature
offset
— is a function of the power output
level and the loop gain. Let us demonstrate the nature of the offset,
also called droop, with an example.
Assume that a system sample block (the mass whose temperature
is to be controlled) has a finite heat capacity, but that its thermal
conductivity is infinite, as is the thermal conductance between the
block and the sensor and heater. The result will be that the
temperature within the block will be isothermal, no matter at what
rate the block is heated or cooled. For the following discussion,
ignore any noise associated with the system and assume that to
control at 20 kelvin, the heating power required is 0.2 watts.
Assume also that 50 watts of heater power is available, reducible in
five steps of one decade each. Figure 5 shows the control offset for
an amplifier gain of 100 and three output power settings which will
deliver enough power to the system to balance the cooling power.
The temperature offsets for a power level of 0.2 watts at 20 kelvin
are easily calculated from Figures 2 and 4 for the three maximum
power settings: 0.1 K for a 50 watt setting, 0.32 for a 5 watt setting, and 1.0 for the 0.5 watt setting. As expected, the
temperature offsets become smaller as the loop gain increases. However, there are limits to this approach as we move
from the idealized example to a real system.
The Real World
Unfortunately, the thermal conductivity within a system is not infinite, and both it and the heat capacity may vary by
several orders of magnitude between 1 K and 300 K. Also, the controller, the sensor, the sensor leads, and the block may
all have electrical noise. This noise is amplified by the controller; for a high enough amplifier gain setting, the output of the
controller will become unstable and oscillate. In addition, the placement of the sensor with respect to the heater and the
sensor construction and mounting itself introduce thermal lags. This is due to the finite thermal conductivity of the block
and the thermal resistances between the heater, sensor and the block. These thermal lags introduce a phase shift
between the controller output and the sensor, which will reduce even further the gain at which the system will be stable.
Therefore, the thermal block design is extremely important in the proper performance of any cryogenic system. No
controller can make up for poor thermal design of the system, nor can good design overcome the inherent limiting
properties of the materials and sensor packages which are currently available.
FIGURE 4.
Idealized curve for Lake Shore Cryotronics, Inc. DT-
500 Series silicon diode temperature sensors.
FIGURE 5.
Effect of output power setting on offset for a
proportional controller
only
.
