Maxwell bridge balance equation and suitability

The Maxwell inductance-capacitance bridge measures an unknown inductance by comparing it with a standard capacitor.

Circuit

          A
        /   \
  L1,R1(Z1)  R3 (Z4)
      /         \
     B---[D]----D
      \         /
    R2 (Z2)   R4 || C4 (Z3)
        \     /
          C

Arm AB: unknown coil (L1, R1). BC: R2. CD: R4 in parallel with a standard capacitor C4. DA: R3.

Balance equation

Using Z1 Z3 = Z2 Z4, with Z3 = R4/(1 + jωC4R4):

(R1 + jωL1) · R4/(1 + jωC4R4) = R2R3

(R1 + jωL1) R4 = R2R3 + jωC4R2R3R4

Real parts: R1 = R2 R3 / R4

Imaginary parts: L1 = R2 R3 C4

Quality factor: Q = ωL1/R1 = ωC4R4

Both results are independent of frequency. If R4 and C4 are the variable elements, the two balance conditions are independent (R4 sets R1 and C4 sets L1).

Why it is suitable for Q between 1 and 10

• Q = ωC4R4. For Q up to about 10, R4 and C4 have reasonable values, so a standard decade resistor and a standard capacitor can be used.
• For Q > 10 (high-Q coils), R4 would have to be very large; such resistors are expensive and inaccurate and suffer from stray effects, so the Hay bridge is used instead.
• For Q < 1 (very low Q), the bridge has balance-convergence problems. In practice C4 is fixed (a variable standard capacitor is costly), so R3 (or R2) and R4 are adjusted; R3 appears in both balance equations, so the adjustments interact. With low Q this causes a "sliding balance", and many alternate adjustments are needed to reach the null.
• Only a standard capacitor is needed, which is cheaper, smaller and more accurate than a standard inductor, and the balance is independent of frequency.
• Disadvantage: the range of Q covered is limited; the bridge is not suitable for high-Q or very low-Q coils.