Bridges formulas

Products of opposite arms are equal. The condition does not involve $E$, $G$ or the source resistance, and it is unchanged if the supply and the galvanometer are interchanged (only the sensitivity changes).

Limiting error is a guaranteed (worst-case) bound: for a product or quotient, the relative limiting errors of the factors add, whether a factor multiplies or divides. For example, arms each known to $\pm 0.05\%$ give $R$ to within $\pm 0.15\%$. Any lack of detector sensitivity (the smallest unbalance that can be seen) adds to this.

Open-circuit voltage of B with respect to D, and the resistance seen from B–D. This simple $R_{\text{th}}$ holds for a supply of negligible internal resistance; with a source resistance, find $R_{\text{th}}$ by a delta–star conversion instead.

Current in a galvanometer of resistance $G$ connected across an unbalanced bridge. $S_i$ is the galvanometer's current sensitivity (e.g. mm/μA).

Change in the open-circuit output when the unknown moves from its balance value $R$ to $R+\Delta R$, with $\Delta R\ll R$ for the approximate forms. For a given $R$, $S/(R+S{)}^2$ is largest at $S=R$, so equal arms give the largest open-circuit output per ohm of unbalance. The middle form is exact and shows the output is linear only for small $\Delta R$.

Open-circuit output with one, two or four active arms (e.g. strain gauges), each changing by $\pm \Delta R$. The active arms must be placed so their effects add: two adjacent arms must change in opposite senses, two opposite arms in the same sense. The full bridge, and the half bridge with two adjacent arms changing in opposite senses, are exactly linear. The quarter bridge, and a half bridge with two opposite arms changing in the same sense (exact output $E\Delta R/(2R+\Delta R)$), are linear only for small $\Delta R$.

$P,Q$ = outer ratio arms; $p,q$ = inner ratio arms ($P$ and $p$ on the side of the unknown $R$, $Q$ and $q$ on the side of the standard $S$); $r$ = resistance of the link between $R$ and $S$. The second term is the error caused by the link, and it has the sign of $(P/Q-p/q)$.

In practice the two ratio pairs are ganged so they stay equal, the link is made short and thick so any residual mismatch multiplies a small $r$, and every ratio arm is made much larger than the lead and contact resistances in series with it. Used for resistances below about 1 Ω, such as shunts and busbar sections.

A capacitor $C$ charged to $V_0$ discharges through the unknown $R$; $V$ is the voltage after time $t$, read on a voltmeter that draws no current (an electrostatic voltmeter). It suits very high resistances, where the decay is slow enough to time accurately; with a small $R$ the voltage collapses too quickly.

$R_1$ is the leakage resistance of the capacitor itself (together with that of the voltmeter), found by a second test with $R$ disconnected. It is always in parallel with $R$, so the first test gives $R^{\prime }<R$.

Products of the impedances of opposite arms are equal. The admittance form is convenient when arms are parallel combinations.

Both must hold at once. $\theta$ is the impedance angle: positive for an inductive arm, negative for a capacitive arm, zero for a pure resistance.

Subscript $s$ = series equivalent circuit, $p$ = parallel equivalent circuit; in the parallel model the coil's $Q=R_p/(\omega L_p)$. $\delta$ is the loss angle: $90^{\circ }$ minus the angle by which the capacitor's current leads its voltage. For small $\delta$ the capacitor's power factor $\cos \varphi =\sin \delta \approx \tan \delta =D$. For a coil, $D=1/Q$. From here on $Q$ denotes a quality factor, not a ratio arm of the DC bridges.

Converts between the series model (natural for an arm like $R+j\omega L$) and the parallel model. For $Q\ge 10$, $L_p$ and $L_s$ agree within 1%; for small $D$, $C_p\approx C_s$.

Arm 2 is a variable standard inductor $L_2$ with total series resistance $R_2$ (its own resistance plus any added resistor); arms 3 and 4 are resistors, $R_4$ opposite the unknown and $R_3$ opposite the standard. From $(R_1+j\omega L_1)R_4=(R_2+j\omega L_2)R_3$. Frequency-independent; its drawback is the need for a variable standard inductor.

Arm 4 = $R_4$ in parallel with a standard capacitor $C_4$. From $Z_1=R_2{R}_3{Y}_4=R_2{R}_3(1/R_4+j\omega C_4)$. The balance equations are free of frequency, and if $R_4$ and $C_4$ are the adjusted elements the two conditions are independent. Suited to medium-Q coils (roughly $1<Q<10$).

Arm 4 = $R_4$ in series with $C_4$. From $Z_1=R_2{R}_3/(R_4-j/(\omega C_4))$. Both $L_1$ and $R_1$ depend on $\omega$, so the frequency must be known. Suited to high-Q coils ($Q>10$).

Because ${\omega }^2{C}_4^2{R}_4^2=1/Q^2$. The approximation overstates $L_1$ by the factor $1+1/Q^2$: 1% at $Q=10$, 0.25% at $Q=20$. Use the exact form when a question gives $\omega$ and all the element values.

Topology as described above: the series $r$–$C$ branch is in parallel with $R_4$, and the detector taps its midpoint. $R_1$ is the total resistance of the coil's arm: if a variable resistor $r_1$ is added in series with a coil of resistance $R_c$ to obtain balance, $R_c+r_1=R_2{R}_3/R_4$. The balance is frequency-independent, and $r$ does not appear in the resistance condition.

From $Z_1=Z_2{Z}_3{Y}_4=\frac{R_3}{j\omega C_2}\left(\frac{1}{R_4}+j\omega C_4\right)$. Both equations are free of frequency. $R_4$, the resistor in the arm opposite the unknown, is in the numerator for $C_1$.

Series-model dissipation factor of the unknown; unlike $C_1$ and $r_1$, it contains $\omega$. With $R_4$ fixed, the $C_4$ dial can be calibrated directly in $D$ at one frequency, and $R_3$ is varied for the $C_1$ balance.

From the imaginary part of $Z_1{Y}_2=R_3/R_4$: $\omega C_2{R}_1=1/(\omega C_1{R}_2)$.

Real part of the same equation. $R_3$ is the resistor opposite the parallel-RC arm, $R_4$ the one opposite the series-RC arm.

With ganged $R_1=R_2$ and equal fixed capacitors, one dial reads frequency directly. At $f_0$ the series-RC/parallel-RC divider passes exactly one third of the voltage with zero phase shift, which is why a Wien-bridge oscillator's amplifier needs a gain of 3.

At resonance the loop's reactances cancel, so the current is $E/R$ ($R$ = coil resistance) and the capacitor voltage is $Q$ times the injected voltage. Assumes $R_{\text{sh}}$ and the capacitor's losses are negligible compared with $R$, the voltmeter does not load the circuit, and the self-capacitance is neglected.

$C_d$ = the coil's self (distributed) capacitance, which adds to the tuning capacitance as far as the resonant frequency is concerned. Neglect it only when $C\gg C_d$.

The shunt is part of the resonant loop, so the measured $Q_m$ is slightly low. The correction is negligible when $R_{\text{sh}}\ll R$ and matters for coils of very low resistance.

$C_1$ and $C_2$ are the tuning-capacitor settings for resonance at $f_1$ and at the higher frequency $f_2$. From ${\omega }^{2}L(C+C_d)=1$ at both frequencies: $C_1+C_d=n^2(C_2+C_d)$.

$Q_{\text{ind}}$ is the reading taken at tuning setting $C$. The self-capacitance acts across the coil's terminals, so the indicated Q falls short of the coil's true $\omega L/R$ by the factor $C/(C+C_d)$.

$l_s$ and $l_x$ are the balance lengths for the standard emf $V_s$ and the unknown $V_x$ at the same working current $I_w$, on a uniform wire of resistance ${\rho }_l$ per unit length; $k$ is the potential gradient (volts per unit length).

A precision resistive divider presents $V/K$ to the potentiometer ($K$ = ratio, e.g. 10 or 100) and $V_p$ is the potentiometer reading. Unlike the potentiometer itself, the divider draws current from the circuit under test.

The potentiometer reads the drop $V_p$ across a four-terminal standard resistor $R_s$ carrying the current. This is how ammeters are calibrated.

$R_x$ and a standard $R_s$ carry the same current in series, and the potentiometer reads the two drops in quick succession. Suitable for low resistances, because only the potential terminals are involved.

$V_1$ = in-phase component, $V_2$ = quadrature component, both measured against the potentiometer's reference phase; the signs of $V_1$ and $V_2$ fix the quadrant of $\theta$. The polar type gives $V$ and $\theta$ directly.