11. Bridges

Before precision instruments, bridge measurements were the gold standard for component values. The principle: balance the bridge so the detector reads zero, then compute the unknown from the known elements.

11.1 Wheatstone bridge

Four resistors in a diamond, with a galvanometer or detector across the middle:

            ┌── R1 ──┬── R2 ──┐
            │        │        │
        +V ─┤    [Detector]   ├─ GND
            │        │        │
            └── R3 ──┴── R4 ──┘

The detector measures the voltage between the midpoints. Balance is reached when: $\frac{R_1}{R_2} = \frac{R_3}{R_4}$ At balance, no current flows through the detector, regardless of source voltage. The unknown $R_4 = R_2 \cdot R_3 / R_1$ (or similar, depending on which arm holds the unknown).

Wheatstone is supremely sensitive because at null we are looking for zero, not a small magnitude in a sea of noise. Even a 1 part-per-thousand change unbalances the bridge measurably. Used historically for all resistance measurement; today the Wheatstone configuration lives on in strain gauges, load cells, and pressure sensors where four matched resistors form the bridge with the strain gauges varying with applied force.

11.2 AC bridges

For inductors and capacitors, replace the DC source with an AC source and the galvanometer with a tuned detector. Each bridge configuration is optimized for a particular range of values.

Maxwell-Wien bridge (Maxwell's inductance bridge). Measures L of an unknown inductor with a Q in the medium-low range. Two arms have R, two have L and C in suitable combinations. At balance: $L_x = R_2 R_3 C_4, \quad R_x = R_2 R_3 / R_4$

Hay bridge. For high-Q inductors. Series RC arm balances against the unknown $R_x + jX_{Lx}$.

Schering bridge. For capacitance and the loss factor (dissipation factor) of a capacitor. Common in HV insulation testing; used in cable and transformer testing labs. The bridge uses two ratio arms and one variable C and one variable R; balance gives both $C_x$ and the dissipation factor.

Anderson bridge. A clever modification of Maxwell-Wien for measuring self-inductance with high accuracy, even when Q is moderate. Has extra branches that improve sensitivity.

Wien bridge (frequency). When used to measure frequency: the bridge balances at a single frequency $f = 1/(2\pi RC)$. The same network is used in oscillator design (section 6.1).

11.3 Practical issues

• Detector sensitivity at the bridge frequency must be high; tunable detector amplifiers or lock-in amplifiers are common.
• Stray capacitance to ground from each node must be controlled; Wagner earthing schemes drive a phantom ground to suppress stray currents.
• Excitation purity: if the source has harmonics, balance is hard because each harmonic balances at a different point.

In modern labs, dedicated LCR meters (Keysight E4980, GW Instek LCR-6300, Hioki IM3536) replaced manual bridge balancing. They sweep frequency, measure complex impedance directly, and report L, C, R, Q, D in real time.

11.4 Q-meter

Measures the quality factor $Q$ of an inductor or capacitor at RF. A signal generator drives a series resonant circuit (the unknown inductor in series with a known variable cap); a high-impedance voltmeter measures the voltage across the cap. At resonance, the voltage across the cap is $Q$ times the source voltage (because at resonance, the impedance of the cap and inductor cancel, and only the equivalent series resistance limits the current). So $Q = V_C / V_S$ at resonance.

Used in RF coil design, LC filter design, antenna characterization.