Page 76 - Mechanical Engineers' Handbook (Volume 2)

6 Measurement Systems 65
1 (R /R ) RC
1
s
ƒ
ƒ
1 (R /R ) (R /R ) ƒ ƒ 1
RC ƒ (61)
i
ƒ
i
s
Our filter has its expected characteristics, R C and K 1, only if both
ƒ
ƒ
ƒ
ƒ
and
approach zero, a circumstance that requires R → 0 and R → while R remains
s
i
ƒ
finite. As the instrument resistance (R ) decreases, the static gain and time constant decrease
i
as well, which means that the break frequency of the filter increases from the designed value
because the instrument provides another path for the discharge of the capacitor in the filter.
As the transducer resistance (R ) increases from zero, the static sensitivity (K) again de-
s
creases, but this time the break frequency decreases as well.
This analysis shows us that source loading by a filter and filter loading by a readout
instrument can cause significant changes in both the designed filter gain and break frequency.
Only if and
in Eqs. (60) and (61) remain equal does the filter break frequency remain
unscathed by nonideal source and load impedances. This condition requires that
R R
R s s ƒ

R R
ƒ i
or
2
R RR RR 0 (62)
i
ƒ
s
ƒ
s
Dividing the second of Eqs. (62) by R 2 s and solving it for R /R yield
s
ƒ
1 4
R ƒ 1 R i R i
R 2 1 R R (63)
s s s
since any realistic measurement system has R /R 1, and since the resistances must be
i
s
positive, the negative solution is discarded. This approximation is equivalent to saying that
R should be chosen to be the geometric mean of the estimated or known values of R and
ƒ
i
R ; that is, R RR s presuming only that R R . For this choice of filter resistance, the
ƒ
i
s
s
i
resulting gain is given by
1 1
K 1 for R R s (64)
i
1 (R /R ) 3/2
s
1 R /R R /R i s i
s
i
If the filter fails these conditions, it must either be carefully designed for the task or it should
be an active* filter with a high input impedance and low output impedance.
6.3 Null Instruments
Many instruments are servos, active systems designed to oppose the variable they measure
so as to decrease their demand on the complementary energy or power variable to zero. In
a steady-state sense, these instruments draw no power or energy from the measurement
interface because these energetic requirements are provided by the instrument’s power supply.
These instruments therefore have infinite or zero input impedance in steady state. Examples
abound: Slidewire potentiometers for strain readout and thermocouple readout both measure
voltage by servoing to zero current. Servo accelerometers avoid the problems with temper-
ature inherent in the elasticity variations of metals and crystals by servoing the motion of
*Incorporating amplification, usually an operational amplifier.