Bridge circuits
No text on electrical metering could be called complete without a
section on bridge circuits. These ingenious circuits make use of a
null-balance meter to compare two voltages, just like the laboratory
balance scale compares two weights and indicates when they're equal.
Unlike the "potentiometer" circuit used to simply measure an unknown
voltage, bridge circuits can be used to measure all kinds of
electrical values, not the least of which being resistance.
The standard bridge circuit, often called a Wheatstone bridge,
looks something like this:
When the voltage between point 1 and the negative side of the
battery is equal to the voltage between point 2 and the negative
side of the battery, the null detector will indicate zero and the
bridge is said to be "balanced." The bridge's state of balance is
solely dependent on the ratios of Ra/Rb and R1/R2,
and is quite independent of the supply voltage (battery). To measure
resistance with a Wheatstone bridge, an unknown resistance is
connected in the place of Ra or Rb, while the
other three resistors are precision devices of known value. Either
of the other three resistors can be replaced or adjusted until the
bridge is balanced, and when balance has been reached the unknown
resistor value can be determined from the ratios of the known
resistances.
A requirement for this to be a measurement system is to have a
set of variable resistors available whose resistances are precisely
known, to serve as reference standards. For example, if we connect a
bridge circuit to measure an unknown resistance Rx, we
will have to know the exact values of the other three
resistors at balance to determine the value of Rx:
Each of the four resistances in a bridge circuit are referred to
as arms. The resistor in series with the unknown resistance Rx
(this would be Ra in the above schematic) is commonly
called the rheostat of the bridge, while the other two
resistors are called the ratio arms of the bridge.
Accurate and stable resistance standards, thankfully, are not
that difficult to construct. In fact, they were some of the first
electrical "standard" devices made for scientific purposes. Here is
a photograph of an antique resistance standard unit:
This resistance standard shown here is variable in discrete
steps: the amount of resistance between the connection terminals
could be varied with the number and pattern of removable copper
plugs inserted into sockets.
Wheatstone bridges are considered a superior means of resistance
measurement to the series battery-movement-resistor meter circuit
discussed in the last section. Unlike that circuit, with all its
nonlinearities (logarithmic scale) and associated inaccuracies, the
bridge circuit is linear (the mathematics describing its operation
are based on simple ratios and proportions) and quite accurate.
Given standard resistances of sufficient precision and a null
detector device of sufficient sensitivity, resistance measurement
accuracies of at least +/- 0.05% are attainable with a Wheatstone
bridge. It is the preferred method of resistance measurement in
calibration laboratories due to its high accuracy.
There are many variations of the basic Wheatstone bridge circuit.
Most DC bridges are used to measure resistance, while bridges
powered by alternating current (AC) may be used to measure different
electrical quantities like inductance, capacitance, and frequency.
An interesting variation of the Wheatstone bridge is the
Kelvin Double bridge, used for measuring very low resistances
(typically less than 1/10 of an ohm). Its schematic diagram is as
such:
The low-value resistors are represented by thick-line symbols,
and the wires connecting them to the voltage source (carrying high
current) are likewise drawn thickly in the schematic. This
oddly-configured bridge is perhaps best understood by beginning with
a standard Wheatstone bridge set up for measuring low resistance,
and evolving it step-by-step into its final form in an effort to
overcome certain problems encountered in the standard Wheatstone
configuration.
If we were to use a standard Wheatstone bridge to measure low
resistance, it would look something like this:
When the null detector indicates zero voltage, we know that the
bridge is balanced and that the ratios Ra/Rx
and RM/RN are mathematically equal to each
other. Knowing the values of Ra, RM, and RN
therefore provides us with the necessary data to solve for Rx
. . . almost.
We have a problem, in that the connections and connecting wires
between Ra and Rx possess resistance as well,
and this stray resistance may be substantial compared to the low
resistances of Ra and Rx. These stray
resistances will drop substantial voltage, given the high current
through them, and thus will affect the null detector's indication
and thus the balance of the bridge:
Since we don't want to measure these stray wire and connection
resistances, but only measure Rx, we must find some way
to connect the null detector so that it won't be influenced by
voltage dropped across them. If we connect the null detector and RM/RN
ratio arms directly across the ends of Ra and Rx,
this gets us closer to a practical solution:
Now the top two Ewire voltage drops are of no effect
to the null detector, and do not influence the accuracy of Rx's
resistance measurement. However, the two remaining Ewire
voltage drops will cause problems, as the wire connecting the lower
end of Ra with the top end of Rx is now
shunting across those two voltage drops, and will conduct
substantial current, introducing stray voltage drops along its own
length as well.
Knowing that the left side of the null detector must connect to
the two near ends of Ra and Rx in order to
avoid introducing those Ewire voltage drops into the null
detector's loop, and that any direct wire connecting those ends of Ra
and Rx will itself carry substantial current and create
more stray voltage drops, the only way out of this predicament is to
make the connecting path between the lower end of Ra and
the upper end of Rx substantially resistive:
We can manage the stray voltage drops between Ra and Rx
by sizing the two new resistors so that their ratio from upper to
lower is the same ratio as the two ratio arms on the other side of
the null detector. This is why these resistors were labeled Rm
and Rn in the original Kelvin Double bridge schematic: to
signify their proportionality with RM and RN:
With ratio Rm/Rn set equal to ratio RM/RN,
rheostat arm resistor Ra is adjusted until the null
detector indicates balance, and then we can say that Ra/Rx
is equal to RM/RN, or simply find Rx
by the following equation:
The actual balance equation of the Kelvin Double bridge is as
follows (Rwire is the resistance of the thick, connecting
wire between the low-resistance standard Ra and the test
resistance Rx):
So long as the ratio between RM and RN is
equal to the ratio between Rm and Rn, the
balance equation is no more complex than that of a regular
Wheatstone bridge, with Rx/Ra equal to RN/RM,
because the last term in the equation will be zero, canceling the
effects of all resistances except Rx, Ra, RM,
and RN.
In many Kelvin Double bridge circuits, RM=Rm
and RN=Rn. However, the lower the resistances
of Rm and Rn, the more sensitive the null
detector will be, because there is less resistance in series with
it. Increased detector sensitivity is good, because it allows
smaller imbalances to be detected, and thus a finer degree of bridge
balance to be attained. Therefore, some high-precision Kelvin Double
bridges use Rm and Rn values as low as 1/100
of their ratio arm counterparts (RM and RN,
respectively). Unfortunately, though, the lower the values of Rm
and Rn, the more current they will carry, which will
increase the effect of any junction resistances present where Rm
and Rn connect to the ends of Ra and Rx.
As you can see, high instrument accuracy demands that all
error-producing factors be taken into account, and often the best
that can be achieved is a compromise minimizing two or more
different kinds of errors.
- REVIEW:
- Bridge circuits rely on sensitive null-voltage meters to
compare two voltages for equality.
- A Wheatstone bridge can be used to measure resistance
by comparing unknown resistor against precision resistors of known
value, much like a laboratory scale measures an unknown weight by
comparing it against known standard weights.
- A Kelvin Double bridge is a variant of the Wheatstone
bridge used for measuring very low resistances. Its additional
complexity over the basic Wheatstone design is necessary for
avoiding errors otherwise incurred by stray resistances along the
current path between the low-resistance standard and the
resistance being measured.
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