# A Wheatstone Bridge

**Topics:**Resistor, Wheatstone bridge, Electrical resistance

**Pages:**7 (2270 words)

**Published:**February 16, 2013

Operation

In the figure, is the unknown resistance to be measured; , and are resistors of known resistance and the resistance of is adjustable. If the ratio of the two resistances in the known leg is equal to the ratio of the two in the unknown leg , then the voltage between the two midpoints (B and D) will be zero and no current will flow through the galvanometer . If the bridge is unbalanced, the direction of the current indicates whether is too high or too low. is varied until there is no current through the galvanometer, which then reads zero. Detecting zero current with a galvanometer can be done to extremely high accuracy. Therefore, if , and are known to high precision, then can be measured to high precision. Very small changes in disrupt the balance and are readily detected. At the point of balance, the ratio of

Alternatively, if , , and are known, but is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of , using Kirchhoff's circuit laws (also known as Kirchhoff's rules). This setup is frequently used in strain gauge and resistance thermometer measurements, as it is usually faster to read a voltage level off a meter than to adjust a resistance to zero the voltage. -------------------------------------------------

[edit]Derivation

First, Kirchhoff's first rule is used to find the currents in junctions B and D:

Then, Kirchhoff's second rule is used for finding the voltage in the loops ABD and BCD:

The bridge is balanced and , so the second set of equations can be rewritten as:

Then, the equations are divided and rearranged, giving:

From the first rule, and . The desired value of is now known to be given as:

If all four resistor values and the supply voltage () are known, and the resistance of the galvanometer is high enough that is negligible, the voltage across the bridge () can be found by working out the voltage from each potential divider and subtracting one from the other. The equation for this is:

where is the voltage of node B relative to node D.

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...

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