# Carey Foster bridge

In electronics, the Carey Foster bridge is a bridge circuit used to measure low resistances, or to measure small differences between two large resistances. It was invented by Carey Foster as a variant on the Wheatstone bridge. He first described it in his 1872 paper "On a Modified Form of Wheatstone's Bridge, and Methods of Measuring Small Resistances" (Telegraph Engineer's Journal, 1872–1873, 1, 196).

## Use

The Carey Foster bridge. The thick-edged areas are busbars of almost zero resistance.

In the diagram to the right, X and Y are resistances to be compared. P and Q are nearly equal resistances, forming the other half of the bridge. The bridge wire EF has a jockey contact D placed along it and is slid until the galvanometer G measures zero. The thick-bordered areas are thick copper busbars of almost zero resistance.

1. Place a known resistance in position Y.
2. Place the unknown resistance in position X.
3. Adjust the contact D along the bridge wire EF so as to null the galvanometer. This position (as a percentage of distance from E to F) is l1.
4. Swap X and Y. Adjust D to the new null point. This position is l2.
5. If the resistance of the wire per percentage is σ, then the resistance difference is the resistance of the length of bridge wire between l1 and l2:
${X-Y=\sigma(l_2-l_1)}\,$

To measure a low unknown resistance X, replace Y with a copper busbar that can be assumed to be of zero resistance.

In practical use, when the bridge is unbalanced, the galvanometer is shunted with a low resistance to avoid burning it out. It is only used at full sensitivity when the anticipated measurement is close to the null point.

### To measure σ

To measure the unit resistance of the bridge wire EF, put a known resistance (e.g., a standard 1 ohm resistance) that is less than that of the wire as X, and a copper busbar of assumed zero resistance as Y.

## Theory

Two resistances to be compared, X and Y, are connected in series with the bridge wire. Thus, considered as a Wheatstone bridge, the two resistances are X plus a length of bridge wire, and Y plus the remaining bridge wire. The two remaining arms are the nearly equal resistances P and Q, connected in the inner gaps of the bridge.

A standard Wheatstone bridge for comparison. Points A, B, C and D in both circuit diagrams correspond. X and Y correspond to R1 and R2, P and Q correspond to R3 and RX. Note that with the Carey Foster bridge, we are measuring R1 rather than RX.

Let l1 be the null point D on the bridge wire EF in percent. α is the unknown left-side extra resistance EX and β is the unknown right-side extra resistance FY, and σ is the resistance per percent length of the bridge wire:

${P \over Q} = {{X + \sigma (l_1 + \alpha )} \over {Y + \sigma (100 - l_1 + \beta)}}$

and add 1 to each side:

${{P \over Q} + 1} = {{X + Y + \sigma (100 + \alpha + \beta)} \over {Y + \sigma (100 - l_1 + \beta)}}$       (equation 1)

Now swap X and Y. l2 is the new null point reading in percent:

${P \over Q} = {{Y + \sigma (l_2 + \alpha )} \over {X + \sigma (100 - l_2 + \beta)}}$

and add 1 to each side:

${{P \over Q} + 1} = {{X + Y + \sigma (100 + \alpha + \beta)} \over {X + \sigma (100 - l_2 + \beta)}}$       (equation 2)

Equations 1 and 2 have the same left-hand side and the same numerator on the right-hand side, meaning the denominator on the right-hand side must also be equal:

\begin{align} &Y + \sigma(100 - l_1 + \beta) = X + \sigma (100 - l_2 + \beta) \\ \Rightarrow {} &X - Y = \sigma(l_2 - l_1) \end{align}

Thus: the difference between X and Y is the resistance of the bridge wire between l1 and l2.

The bridge is most sensitive when P, Q, X and Y are all of comparable magnitude.

## References

• Carey Foster Bridge (PDF) (St.Stephen's College, Delhi University)
• A. H. Fison (1919). "Obituary notices: .... George Carey Foster, 1835–1919; .....". J. Chem. Soc., Trans. 115: 412–427. doi:10.1039/CT9191500408.