blog.hirnschall.net
Enter the four bridge components at balance. The unknown inductance \( L_x \) and its series resistance \( R_x \) are calculated from the balance conditions. Providing the test frequency also yields the quality factor \( Q \).
The Maxwell bridge (more precisely the Maxwell-Wien bridge) is an AC bridge circuit used to measure the inductance and series resistance of an unknown inductor. It places the unknown inductor \( L_x \) with its winding resistance \( R_x \) in one arm, balances it against a known capacitor in an adjacent arm, and reads off \( L_x \) and \( R_x \) directly from the values of the calibrated components at null.
The bridge has four arms. The unknown arm contains \( L_x \) in series with \( R_x \). The opposite arm (the standard arm) contains a known resistor \( R_4 \) in parallel with a known capacitor \( C_4 \). The two remaining arms are plain resistors \( R_2 \) and \( R_3 \). An AC source drives one diagonal and a null detector (galvanometer or oscilloscope) monitors the other.
At balance, no current flows through the detector. Setting the complex impedance products of opposite arm pairs equal and separating real and imaginary parts gives two independent equations: \[ L_x = R_2 \cdot R_3 \cdot C_4 \] \[ R_x = \frac{R_2 \cdot R_3}{R_4} \] These two conditions are independent of frequency, which is one of the key practical advantages of the Maxwell bridge: the source frequency does not need to be known precisely. The ratio arms \( R_2 \) and \( R_3 \) always appear as a product, so only their product \( R_2 R_3 \) matters — in practice one is often kept fixed while the other is adjusted in decade steps to set the range.
The quality factor of the measured inductor at the test frequency \( f \) is: \[ Q = \frac{\omega L_x}{R_x} = \omega C_4 R_4 \] where \( \omega = 2\pi f \). \( Q \) can be read directly from the standard arm components once the bridge is balanced.
The Maxwell bridge works well for medium-Q inductors, typically in the range \( 1 \leq Q \leq 10 \). Outside this range, balance becomes difficult to achieve because the real and imaginary balance conditions become nearly dependent, making the null hard to locate. For high-Q inductors (\( Q > 10 \)) the Hay bridge is the better choice: it uses a capacitor in series rather than parallel with \( R_4 \), which suits high-Q measurements naturally. The calculator outputs a warning when the computed Q falls outside the useful range.
In a physical measurement, the bridge is brought to null by adjusting the variable components (typically \( R_4 \) for the resistive balance and \( R_3 \) or \( C_4 \) for the reactive balance). Once the detector reads zero, the component values at that point are entered here to recover \( L_x \) and \( R_x \). No frequency measurement is needed for \( L_x \) and \( R_x \); frequency is only required if Q is also wanted.
The formulas can also be rearranged for component selection when designing a bridge for a target inductance range. Fixing \( C_4 \) and choosing \( R_2 R_3 = L_x / C_4 \) sets the inductance scale, and choosing \( R_4 = R_2 R_3 / R_x \) sets the resistance scale.
More calculators: blog.hirnschall.net/tools/.
