Q4Electronic Measurement & Instrumentation

A standard Wheatstone bridge is unsuitable for measuring very low resistances (like copper shunts) because the resistance of the connecting leads and contact points introduces massive errors. Kelvin's Double Bridge incorporates a second set of ratio arms to cancel out these errors.

The circuit consists of the unknown resistance $R$, a standard low resistance $S$, a heavy copper link connecting them of resistance $r$, primary ratio arms $P$ and $Q$, and secondary ratio arms $p$ and $q$. The galvanometer is connected to the junction of $p$ and $q$, which span across the link $r$.

Using delta-star transformation or Kirchhoff's laws at balance (galvanometer current is zero), the equation derived is: $R=S\frac{P}{Q}+\frac{qr}{p+q+r}\left(\frac{P}{Q}-\frac{p}{q}\right)$

To obtain the greatest possible precision, the error term caused by the link resistance $r$ must be driven to absolutely zero. This requires two specific design arrangements:

1. Ratio Matching: The secondary ratio arms must be constructed to exactly match the primary ratio arms at all times: $\frac{P}{Q}=\frac{p}{q}$. When this happens, the term in the bracket becomes zero, and the balance equation perfectly reduces to $R=S\frac{P}{Q}$.

2. Minimizing Link Resistance: Even with perfect ratio matching, slight mechanical variations exist. Therefore, the physical copper link connecting $R$ and $S$ (resistance $r$) must be made as thick and as short as physically possible so that $r\approx 0$, further minimizing any residual error.