Loading Errors
Consider again the voltage divider circuit:
nd
Figure 6.16 in 2
rd
and 3 Edition
The equivalent resistance of the circuit is given
by:
R1 R m
=
+
=
+
R eq R 2 R L R 2
R1 + R m
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November 14, 2008
(6.361 , 6.322,3 )
Page 1
The output voltage Eo is related to the input
voltage Ei by:
1
Eo =
(6.371 , 6.352,3 )
E i 1 + ( R 2/ R1 ) ( R1/ R m + 1)
We can define a loading error as the normalized
difference between the output voltage with an
infinite meter resistance and that with a finite
value:
ei
=
Ei
'
Eo - Eo
Ei
Ei
(6.381 , 6.37 2,3 )
e
- + ( - )( /
+ 1)
= R 1 RT 2 RT R 1 R 1 R m
E i RT +( RT / R1 - RT )( R1 / R m + 1)
Here
is the ideal value and
is the
actual value. (Notice the correction of the
equation in the First Edition of the text). The
value of
is constant. Let's plot the error
as a function of R1/RT for various meter
resistances.
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For a meter resistance Rm equal to the total
resistance RT, the error is a significant fraction of
the signal. To get accurate measurements, we
must therefore keep Rm >> RT.
When we consider the effects of loading, we
must distinguish between measurement of effort
variables such as voltage and measurement of
flow variables such as current.
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Effort Variable
Flow Variable
Voltage
Current
Pressure
Flow
Temperature
Heat Flux
Force
Velocity
We can define the input impedance of the
instrument and the output impedance of the
system. For the voltage divider circuit, the
output impedance is RT and the input impedance
of the meter is Rm.
For effort variables, the requirement is that the
input impedance of the instrument should be
much greater than the output impedance of the
system.
nd
Figure 6.17 in 2
rd
and 3 Edition
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November 14, 2008
Page 4
1
e1
= 11 + Z 1/ Z m
E1
(6.39)
For flow variables, the input impedance of the
instrument should be much less than the output
impedance of the system.
nd
Figure 6.18 in 2
rd
and 3 Edition
e1
Zm
= 2
E1 Z 1 + Z 1 Z m
Zm / Z1
=
1+ ( Z m / Z 1 )
I
e1
(6.42)
If our instrument has more than one component,
the same rules are applied to each stage,
treating the output of one stage as the input of
the next.
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In designing an instrument, we want impedance
mismatching. If we wanted to transfer a
maximum amount of power, say to the speakers
of a sound system, we would want impedance
matching.
Let us consider an example which is not
electrical, the measurement of the temperature
of a cup of water using a thermometer.
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November 14, 2008
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An energy balance gives:
C t ( T f - T o ) = C w( T i - T f )
where C is the heat capacity, and t refers to the
thermometer and w to the water.
Here we will define the impedances as:
Z 1 = 1/ C w , Z m = 1/ C t
The error in the measurement e = (Ti - Tf) can be
written:
e
Z
1/ Z m
=
T i - T o 1 + Z 1/ Z m
1
= 11 + Z 1/ Z m
which is the same as Equation (6.39)
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