A Wheatstone Bridge

A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. Its operation is similar to the original potentiometer. It was invented by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. One of the Wheatstone bridge’s initial uses was for the purpose of soils analysis and comparison.[1]

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.

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 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. 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 (nonlinear 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 Rmand 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.

Wheatstone Bridge
For measuring accurately any resistance Wheatstone Bridge is widely used. There are two known resistors, one variable resistor and one unknown resistor connected in bridge form as shown below. By adjusting the variable resistor the current through the Galvanometer is made zero. When the current through the galvanometer becomes zero, the ratio of two known resistors is exactly equal to the ratio of adjusted value of variable resistance and the value of unknown resistance. In thi Wheatstone Bridge Theory

The general arrangement of Wheatstone bridge circuit is shown in the figure below. It is a four arms bridge circuit where arm AB, BC, CD and AD are consisting of resistances P, Q, S and R respectively. Among these resistances P and Q are known fixed resistances and these two arms are referred as ratio arms. An accurate and sensitive Galvanometer is connected between the terminals B and D through a switch S2. The voltage source of this Wheatstone bridge is connected to the terminals A and C via a switch S1 as shown. A variable resistor S is connected between point C and D.

The potential at point D can be varied by adjusting the value of variable resistor. Suppose current I1 and current I2 are flowing through the paths ABC and ADC respectively. If we vary the electrical resistancevalue of arm CD the value of current I2 will also be varied as the voltage across A and C is fixed. If we continue to adjust the variable resistance one situation may comes when voltage drop across the resistor S that is I2.S is becomes exactly equal to voltage drop across resistor Q that is I1.Q. Thus the potential at point B becomes equal to the potential at point D hence potential difference between these two points is zero hence current through galvanometer is nil. Then the deflection in the galvanometer is nil when the switch S2 is closed. Now, from Wheatstone bridge circuit

current I1 =| V|
| P + Q|
and
current I2 =| V|
| R + S|
Now potential of point B in respect of point C is nothing but the voltage drop across the resistor Q and this is I1.Q =| V.Q|
——————-(i)|
| P + Q| |
Again potential of point D in respect of point C is nothing but the voltage drop across the resistor S and this is I2.S =| V.S| ——————(ii)|
| R + S| |

Equating, equations (i) and (ii) we get,
V.Q| =| V.S| ⇒| Q| =| S|
P + Q| | R + S| | P + Q| | R + S|

⇒| P + Q| =| R + S| ⇒| P| + 1 =| R| + 1| ⇒| P| =| R| | Q| | S| | Q| | S| | | Q| | S|

⇒ R = SX| P|
| Q|
Here in the above equation, the value of S and P ⁄ Q are known, so value of R can easily be determined. The resistances P and Q of the Wheatstone bridge are made of definite ratio such as 1:1; 10:1 or 100:1 known as ratio arms and S the rheostat arm is made continuously variable from 1 to 1,000 Ω or from 1 to 10,000 Ω The above explanation is most basic Wheatstone Bridge theory.