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Model 2002 Vacuum Gauge Page 19 of 37
5.2. Pirani Sensor
Figure 5.3a shows a thin film Pt resistive element on a one micron thick Si
3
N
4
continuous membrane
surrounded by a thin film Pt reference resistor on a Si substrate. The membrane is heated to a constant 8
0
C
above ambient temperature that is monitored by the substrate resistor. The membrane resistor is
approximately 60 Ω and a constant substrate to membrane resistance ratio is maintained at 3.86. Figure 5.3b
shows the Pirani die in cross section. A parallel Si lid is eutectically bonded to the Au pads and sits 5 microns
above the membrane. As shown, this dimension gives a Knudsen number of greater than 0.01 up to
atmospheric pressure, which insures a molecular flow component. At 10 Torr the region above the
membrane is totally in the molecular flow regime and thus provides a relatively linear output verses pressure
overlapping the linear output versus pressure of the piezo.
The measurement technique is to produce an output signal that is proportional to the power supplied to the
heated resistor by using the product of the current and voltage. This rejects errors introduced by resistance
changes since the sensor resistance is no longer part of the power equation.
A signal proportional to the power is obtained by multiplying the voltage across the heated sensor and the
voltage impressed by the direct current across a constant series resistance. The power supplied to the sensor
resistor must equal the heat dissipated (E
t
). The three main heat loss routes from the heated sensor are
thermal conduction through the silicon nitride membrane to the silicon substrate (E
s
) radiation losses (E
r
)
and thermal conduction through the gas to the silicon substrate (E
g
); thus, as shown in Figure 5.3c,
E
t
= E
s
+ E
r
+ Eg
The first term, Es, is dependent on the thermal conductivity of the silicon nitride (K), the temperature
difference (∆Τ) between the heater and silicon substrate and geometric factors (AM & L). ES is given by
E
s
= (Κ ∆Τ Α
m
)/L
Am is the membrane cross sectional area through which the heat transfer occurs. This is, approximately, the
outer circumference of the membrane multiplied by the membrane thickness. L is the distance from the edge
of (Rs) the heated sensor resistor to the silicon substrate.
For any particular sensor, all of the factors, except DT, are constants dependent on its construction. The DT
is held constant by the control circuit. The thermal loss through the silicon nitride will be a constant value
independent of the thermal conductivity and pressure of the gas.
Radiation is another source of thermal losses. It can be determined from
E
r
= σε(T
h
4
-T
a
4
)A
s
Where
σ = Stefan-Boltzmann radiation constant
ε = thermal emissivity of the silicon nitride membrane
A
S
= surface area of the heated portion of the membrane
T
h
= temperature of R
s
T
a
= ambient temperature
This radiation loss is also independent of the thermal conductivity of the gas. It is somewhat dependent
upon the absolute temperature of R
s
and the ambient temperature, but since DT is kept to less than 20°C,
this loss is only approximately 10% of E
s
. If ambient changes are small compared to the absolute values of
the temperature this loss can approximated as a constant with temperature.