Exponential Decay, Percent Of Maximum Error Vs. Time In Units Of Τ; Input Voltage Rise And Transient Decay - Campbell Measurement and Control Module CR10 Operator's Manual

SECTION 13. CR10 MEASUREMENTS
discussed for minimizing input settling error
13.3.1 THE INPUT SETTLING TIME CONSTANT
The rate at which an input voltage rises to its
full value or that a transient decays to the
correct input level are both determined by the
input settling time constant. In both cases the
waveform is an exponential. Figure 13.3-1
shows both a rising and decaying waveform
settling to the signal level, Vso. The rising input
voltage is described by Equation 13.3-1 and the
decaying input voltage by Equation 13.3-2.
(1-e -t/R C
o
V = V
s so
V = V + (V -V
s so eo so
where V is the input voltage, V
s
signal voltage, V
eo
is time in seconds, R
ohms, and C is the total capacitance between
T
the signal lead and ground (or some other fixed
reference value) in farads.
The settling time constant, τ in seconds, and
the capacitance relationships are given in
Equations 13.3-3 through 13.3-5,
τ = R
C
o T
C = C + C L
T f w
C = 3.3 nfd
f
where C is the fixed CR10 input capacitance in
f
farads, C is the wire capacitance in
w
farads/foot, and L is the wire length in feet.
Equations 13.3-1 and 13.3-2 can be used to
estimate the input settling error, V
For the rising case, V
13-4
FIGURE 13.3-1. Input Voltage Rise and Transient Decay
T
), rise [13.3-1]
) e -t/R C
o T
, decay [13.3-2]
the true
so
the peak transient voltage, t
the source resistance in
o
[13.3-3]
[13.3-4]
[13.3-5]
, directly.
e
= V -V , whereas for
s so e
when long leads are mandatory.
the decaying transient, V
Substituting these relationships for V
Equations 13.3-1 and 13.3-2, respectively,
yields expressions in V
e -t/R C
o T
V = V , rise
e so
e -t/R C
o T
V = V , decay
e e'o
Where V = V -V , the difference between
e'o eo so
the peak transient voltage and the true signal
voltage.
NOTE: Since the peak transient, V
causes significant error only if it is several
times larger than the signal, V
calculations made in this section
approximate V by V
e'o
V .
so
If the input settling time constant, τ , is known, a
quick estimation of the settling error as a
percentage of the maximum error (V
rising, V for decaying) is obtained by knowing
e'o
how many time constants (t/τ) are contained in
the 450 µs CR10 input settling interval (t). The
familiar exponential decay relationship is given
in Table 13.3-1 for reference.
TABLE 13.3-1. Exponential Decay, Percent
of Maximum Error vs. Time in Units of τ
Time %
Constants Max. Error Constants
0 100.0
1 36.8
3 5.0
= V +V .
s so e
in
s
, the input settling error:
e
[13.3-6]
[13.3-7]
,
eo
, error
so
; i.e., V = V -
eo eo eo
for
so
Time %
Max. Error
5 0.7
7 0.1
10 0.004