handbook of modern sensors

250 6 Occupancy and Motion Detectors
happen when a person walks into a warm room from the cold outside. The surface
of his clothing will be cooler than the sensor and the flux will be negative. In the
following discussion, we will consider that the object is warmer than the sensor and
the flux is positive.
A maximum operating distance for given conditions can be determined by the
noise level of the detector. For reliable discrimination, the worst-case noise power
must be at least three to five times smaller than that of the signal.
The pyroelectric sensor is a converter of thermal energy flow into electric charge.
The energy flow essentially demands the presence of a thermal gradient across the
sensing element. In the detector, the element of thickness h has the front side exposed
to the lens, and the opposite side faces the detector’s interior housing, which normally
is at ambient temperature T a . The front side of the sensor element is covered with
a heat-absorbing coating to increase its emissivity ε s to the highest possible level,
preferably close to unity. When thermal flux s is absorbed by the element’s front
side, the temperature increases and heat starts propagating through the sensor toward
its rear side. Because of the pyroelectric properties, electric charge is developing on
the element surfaces in response to the heat flow.
Upon influx of the infrared radiation, the temperature of the sensor element increases
(or decreases) with the rate, which can be derived from the absorbed thermal
power s and thermal capacity C of the element:
dT
≈ s
dt C , (6.15)
where t is time. This equation is valid during a relatively short interval (immediately
after the sensor is exposed to the thermal flux) and can be used to evaluate the signal
magnitude. The electric current generated by the sensor can be found from the
fundamental formula
i = dQ
dt , (6.16)
where Q is the electric charge developed by the pyroelectric sensor. This charge
depends on the sensor’s pyroelectric coefficient P , the sensor’s area s, and the temperature
change dT:
dQ= PsdT. (6.17)
Thermal capacity C can be derived through a specific heat c of the material, area s,
and thickness of the element h:
C = csh. (6.18)
By substituting Eqs. (6.15), (6.17), and (6.18) into Eq. (6.16), we can evaluate the
peak current which is generated by the sensor in response to the incident thermal flux:
i = PsdT = Ps s
dt csh = P hc s. (6.19)
To establish relationship between the current and the moving object, the flux from
Eq. (6.14) has to be substituted into Eq. (6.19):
where T = (T b − T a ).
i = 2Paσγ
πhc
bT 3 a
T
L 2 , (6.20)