Wheatstone bridge calculator: bridge output, strain gauges and load cells

The output voltage of a Wheatstone bridge from its four resistors, balanced or not, and the unknown arm from a measured output; the output of a quarter, half or full bridge for a resistance change or a strain, with the exact value beside the straight-line approximation and the nonlinearity between them; and a load cell rated in mV/V, through the amplifier gain that fills an ADC, down to counts per kilogram. The equations are TI's, from SLOA034 and the SLYP163 bridge measurement seminar, and every result is worked through the same four-resistor equation.

Fig 1 — Full bridge, arms of 1.000 kΩ at rest, ΔR/R = 1.00 % in the principal gauge, excited at 5.00 V: V_O = 50.00 mV between the midpoints. Right: V_O per volt of excitation for the quarter, the two half and the full bridge, the selected one with its straight-line approximation dashed; below it the nonlinearity, (exact − linear)/linear, which is TN-507's −η: −x/(2 + x) for the quarter bridge and the opposite-arm half bridge, −(1 − ν)x/(2 + (1 − ν)x) for the Poisson half bridge and column, zero for the rest.

Bridge

Configuration

Excitation voltage V_E (V)

Arm resistance at rest R (Ω)

Change given as

Change in the principal arm ΔR (Ω)

Gauge factor GF

Lead wire resistance R_L, each (Ω)

Bridge output V_O, full bridge: +50.00 mV
ΔR/R · equivalent strain: 1.000 % · 5000 µε at GF 2
Output per volt of excitation: 10.00 mV/V
Straight-line V_O ≈ V_E·ΔR/R · nonlinearity (TN-507 case 7): +50.00 mV · none: exact
A strain indicator at GF 2 reads 4·V_O/(V_E·GF) · TN-507 correction to the strain under one gauge: 20 000 µε · 5000 µε
Arms R1 · R2 · R3 · R4: 1.010 kΩ · 990.0 Ω · 1.010 kΩ · 990.0 Ω
Left node V_L · right node V_R: 2.475 V · 2.525 V
Same ΔR/R: quarter · half, opposite · half, adjacent · full: +12.44 mV · +24.88 mV · +25.00 mV · +50.00 mV
Source resistance across the output, R1∥R2 + R3∥R4: 999.9 Ω
Excitation current · power in each arm at rest: 5.00 mA · 6.25 mW

How this is calculated

Standard: TI SLOA034, Signal Conditioning Wheatstone Resistive Bridge Sensors (Karki, 1999), §1–§2.3; TI SLYP163, Bridge Measurement Systems (Precision Analog Applications Seminar), slides 5–18; Micro-Measurements TN-507-1, Errors Due to Wheatstone Bridge Nonlinearity, Table 1; Micro-Measurements TN-514, Shunt Calibration of Strain Gage Instrumentation, Eqs. 3–9, 23–38, 44–45 and Table 1; HBM, Hoffmann, Applying the Wheatstone Bridge Circuit (S1569-1.3), Eqs. 1–5 and 16–23

$V_O = V_R - V_L = V_E\left(\frac{R_3}{R_3 + R_4} - \frac{R_2}{R_1 + R_2}\right)$: Two voltage dividers across one excitation, read between their midpoints. The arms are numbered around the diamond from the top left: R1 and R2 form the left divider, R4 and R3 the right one, with R3 at the bottom. HBM's Fig. 1 numbers them the same way, and its Eq. 2, U_A/U_E = R1/(R1 + R2) − R4/(R3 + R4), is this equation rearranged. SLOA034 does not number the arms; this numbering puts its SIG+ on the right. With SLOA034 Figure 3's arms, 1010 Ω, 990 Ω, 1010 Ω and 990 Ω at 5 V, it gives 2.525 V and 2.475 V, which are the SIG+ and SIG− that SLOA034 prints, and 50.0 mV between them.
$V_O = V_E\left[\frac{R_1}{R_1 + R_G} - \frac{1}{2}\right]$: SLYP163 slide 7, as printed (slide 8 repeats it): the single-point bridge, with the gauge R_G in the upper arm of one divider and three fixed resistors R1, and VO+ at the midpoint below R_G. It is the general equation with R4 = R_G, so the output falls as R_G rises. Slide 8 redraws the circuit as a diamond with the + and − terminal labels on the opposite midpoints; with those labels the same equation would carry the other sign. The calculator puts a quarter-bridge gauge in R3 instead, where the same change gives the same size of output with a positive sign. SLYP163 slide 6 gives the lone divider the other way round: R_g = R1(V_E/V_O − 1).
$R_1 R_3 = R_2 R_4 \;\Rightarrow\; V_O = 0, \qquad R_x = R_3 = \frac{R_2 R_4}{R_1}$: The balance condition. SLYP163 slide 5: "if three resistances are known, and the current in the cross branch is zero, the fourth resistance can be calculated." For an unbalanced bridge the calculator finds R3 from a measured V_O instead: V_R = V_L + V_O, then R3 = R4·V_R/(V_E − V_R).
$V_{sig} = V_{exc} \times \frac{\Delta R}{R}, \qquad V_{sig} = V_{exc} \times F \times S$: SLOA034 page 2, "With four active elements", two arms at R + ΔR and two at R − ΔR; S is "the sensitivity, or output voltage, of the sensor in mV per volt of excitation with full scale input", F the fraction of full scale. With ±ΔR in every arm each divider keeps a total of 2R, so the first equation is exact, not an approximation.
$\frac{\Delta R}{R_G} = F_G\,\varepsilon$: The gauge factor: TN-514 Eq. 4, "the definition of the gage factor", and HBM Eq. 1, ΔR/R0 = k·ε, where "The exact value is specified on each strain gage package. In general, the gage factor for metal strain gages is about 2." The calculator takes it as an input, 2 by default.
$\frac{E_o}{E} = \frac{x}{4 + 2x}\;(1), \quad \frac{x}{2}\;(3), \quad \frac{x}{2 + x}\;(4), \quad x\;(7), \qquad x = F\varepsilon$: TN-507-1 Table 1, cases 1, 3, 4 and 7: one active gauge; two in the same divider with equal and opposite strains, "typical of bending-beam arrangement"; two with "equal strains of same sign" in opposite arms; four with pairs at equal and opposite strains. These are the calculator's quarter, adjacent-arm half, opposite-arm half and full bridges, and each is the four-resistor equation on those arms. TN-507 numbers its arms R1 upper left, R2 upper right, R3 lower right, R4 lower left, so its R2 and R4 are this page's R4 and R2; moving a gauge to the mirror-image arm of the same divider does not change V_O. SLYP163 slides 8–10 draw cases 1, 4 and 7 as its single-, two- and four-point bridges.
$\frac{E_o}{E} = \frac{(1+\nu)\,x}{4 + 2(1-\nu)\,x}\;(2), \quad \frac{(1+\nu)\,x}{2 + (1-\nu)\,x}\;(5), \quad \frac{(1+\nu)\,x}{2}\;(6)$: TN-507 Table 1, the cases with "Poisson" gauges mounted across the load, which see −ν times the strain: a half bridge (2), a full bridge on a column (5) and on a beam (6). ν is Poisson's ratio; TN-507 plots its Figure 2 at 0.30 and HBM gives "around 0.3 for metals".
$\frac{E_o}{E} = K\varepsilon\times10^{-3}\,(1 - \eta), \qquad \eta = \frac{F\varepsilon\times10^{-6}}{2 + F\varepsilon\times10^{-6}}\;(1, 4), \quad \frac{F\varepsilon(1-\nu)\times10^{-6}}{2 + F\varepsilon(1-\nu)\times10^{-6}}\;(2, 5), \quad 0\;(3, 6, 7)$: TN-507's nonlinearity column, with ε in microstrain and E_o/E in mV/V; K is F/4, F(1 + ν)/4, F/2, F/2, F(1 + ν)/2, F(1 + ν)/2 and F. The calculator reports (exact − linear)/linear, which is −η: the quarter bridge reads 0.50 % low at ΔR/R = 1 %. TN-507 p. 3: the output is linear only when the resistance changes are such that "the currents through the bridge arms remain constant".
$\varepsilon = \frac{2\varepsilon_i}{2 - F\varepsilon_i\times10^{-6}}\;(1), \qquad \varepsilon_i = \frac{4}{F_I}\,\frac{E_o}{E}$: TN-507's corrections column for case 1, from the strain an instrument indicates, ε_i, to the actual strain under one gauge; the table has one for each case, correcting both the nonlinearity and the number of active gauges. A strain indicator set to gauge factor F_I reads ε_i = C·e_O/E with C = 4/F_I (TN-514 Eqs. 33 and 38). TN-507 p. 4: at F = 2, "Substituting F = 2.0 and εi = 15 000 ... gives ε = 15 228"; the calculator's function gives 15 228.
$k^{*} = k\,\frac{R}{R + R_K}, \qquad U_{EK} = U_E\,\frac{R_B}{R_B + R_{K2} + R_{K3}}$: Lead-wire desensitisation. HBM Eqs. 18 and 20: a lead of R_K in series with a gauge makes its apparent gauge factor k·R/(R + R_K), which is TN-514 p. 5's three-wire quarter bridge, "attenuated by the factor RG/(RG + RL)", and HBM Eq. 23; two wires put both leads in the gauge's arm, R/(R + R_K1 + R_K2) (HBM Eq. 21). In a full bridge the leads carry the excitation instead, and Eq. 16 divides it. (HBM's Eq. 22 is printed without the ε1 that Eq. 23 carries; the calculator uses Eq. 21 and Eq. 23.)
$R_C = \frac{R_G\times10^{6}}{F_G\times N\times\varepsilon_{S(\mu)}} - R_G - 2R_L, \qquad \varepsilon_S = \frac{R_G}{F_G\,N\,(R_G + R_C + 2R_L)}$: Shunt calibration, TN-514 Eq. 9: the resistor that, across one arm, simulates a strain ε_S in microstrain. N is the number of active gauges (Eq. 8): 1, 1 + ν, 2, 2(1 + ν) or 4. With N = 1 and no leads it is Eq. 7, R_C = R_G/(F_G·ε_S) − R_G, whose Table 1 lists 22 values at F_G = 2; the calculator reproduces each of them to the ohm the table truncates to, 87 150 Ω for 2000 µε on 350 Ω among them. The second form is the inverse, Eq. 5 generalised. TN-514 prints Eq. 6 without the leading minus that Eq. 5 requires, then explains that the minus is customarily dropped because the simulated strain is always compressive; Eq. 7 is written that way.
$R_C = \frac{R_G}{F_G\,\varepsilon_{ST1}}\;(23), \qquad R_C = \frac{R_G}{2F_G\,|\varepsilon_S|} - \frac{R_G}{2}\;(28), \qquad R_C = \frac{R_G}{N F_G\,|\varepsilon_S|} - R_G\left[\frac{1 + 3R_L/R_G}{2(1 + 2R_L/R_G)}\right]\;(29)$: TN-514's large-strain relationships, exact at any strain: shunting a quarter bridge's dummy to simulate tension (Eq. 23; Eq. 23a with leads), a bending half bridge (Eq. 28; Eq. 27 with leads) and the linear full bridges (Eq. 29). Across the gauge of a quarter bridge Eq. 7 is already exact (p. 12). The calculator also carries Eqs. 24, 25, 30 and 31 for the Poisson arrangements, and each relationship is tested by building the shunted bridge and the strained bridge from the four-resistor equation and checking they read the same. Eq. 24 is printed with +R_G in its first term, which with the compressive strain's sign carried gives a negative resistor; Eq. 30, its full-bridge counterpart, has −R_G, and that is the form used. Eqs. 27 and 28 print the leading 2 as "s".
$\frac{e_O}{E} = \mp\frac{R_G}{4R_C + 2R_G}, \qquad R_C = \frac{R_G}{F_I\,\varepsilon_I} - \frac{R_G}{2}$: TN-514 Eq. 35, the bridge output while one arm is shunted, and Eq. 38, the resistor that makes an ideal indicator set to F_I register ε_I: instrument verification rather than scaling, independent of the gauge factor of the gauge.
$\frac{U\varepsilon_S}{\varepsilon_S} = \sqrt{\left(\frac{R_1 + 2R_C}{R_1 + R_C}\right)^{2}\left(\frac{UR_1}{R_1}\right)^{2} + \left(\frac{R_C}{R_1 + R_C}\right)^{2}\left(\frac{UR_C}{R_C}\right)^{2} + \left(\frac{UF_G}{F_G}\right)^{2}}$: TN-514 Eq. 45, the probable error in the simulated strain from the tolerances on the installed gauge resistance R1, on R_C and on the gauge factor; Eq. 44 is the worst case, the same terms added. Its p. 18 example, 350 Ω, 87 150 Ω, ±1 %, ±0.01 % and ±1 %, prints ±2.23 %; the calculator gives ±2.23 %.
$G = \frac{V_{FS,ADC}}{S \, V_E}, \qquad R_f = G\left(R_{in} + \frac{R_B}{2}\right), \qquad \text{counts per kg} = \frac{G\,S\,V_E / m_{cap}}{V_{FS,ADC}/(2^N - 1)}$: The gain that takes the full-scale output to the ADC's range; for SLYP163's 2 mV/V at 10 V into 0–5 V that is 250. A single op amp difference amplifier sees the bridge's source resistance, R_B/2 per side, in series with its input resistors (SLOA034 Figure 4): for 1 kΩ inputs and a 1 kΩ bridge, gain 100 needs 150 kΩ. The LSB is the adc-resolution calculator's FSR/(2ⁿ − 1), from TI SLAA013.

Assumptions

Constant-voltage excitation and an ideal voltmeter across the output: nothing loads the midpoints. TN-507 and TN-514 make the same two assumptions (TN-514 p. 1: "the input impedance of any instrument applied across the output terminals of the bridge circuit is effectively infinite"). An amplifier with finite input resistance loads the bridge's source resistance; SLOA034 §2.1 works the effect for a difference amplifier.
In the sensor mode every arm has the same resistance R at rest and the bridge starts balanced, as TN-507 assumes (p. 3); the active arms change by exactly ΔR, a Poisson gauge by −ν·ΔR. Real gauges are matched only so closely: SLYP163 slide 7 assumes "that all the R1s are very closely matched", and the mismatch shows up as an offset. TN-507 p. 4 shows how an initial unbalance enlarges the nonlinearity.
Lead resistance, when entered, is the same in every wire. In the sensor mode it is applied as a sensitivity factor, HBM's apparent gauge factor; in the shunt calibration mode it enters TN-514's relationships directly, and the shunt goes directly across the arm (TN-514 p. 6) unless the remote option is chosen.
A load cell is treated as a straight line through zero and its rated output: V_O = S·V_E·load/capacity. Its own nonlinearity, offset and drift come from its data sheet.
The gauge factor is a single constant, and temperature is not modelled: neither the resistance change with temperature nor the gauge factor's drift.

How a Wheatstone bridge turns a small resistance change into a voltage

A strain gauge changes its resistance very little. TI's SLYP163 seminar: "The resistance change per unit of strain is very small, and requires sensitive circuitry to measure it accurately." SLOA034's example sensor moves by 1 % at full scale. Put such an element in a voltage divider and the output moves by a small fraction of a large standing voltage; SLYP163 again: "for a sensor like a strain gauge, this circuit will tend to produce very small changes in voltage, offset by a large amount." A 1 kΩ gauge that rises by 1 % in a divider with a 1 kΩ resistor at 5 V moves its midpoint from 2.500 V to 2.488 V: 12.4 mV of signal on top of 2.5 V of offset.

The bridge removes the offset by building a second divider beside the first and measuring between the two midpoints. SLYP163: "The large offset can be eliminated by adding a voltage divider and measuring the output differentially." TI's application report SLOA034 describes the resting state: "When an excitation voltage is applied between Vexc and GND and all resistances are equal, the voltage at SIG+ and SIG– is 1/2 Vexc." Both midpoints sit at the same voltage, the difference is zero, and any change in an arm appears as a difference from zero rather than a small change on a large number. The half-excitation at both terminals does not go away; it becomes the common-mode voltage the amplifier after the bridge has to reject, which SLOA034 puts as "a common mode component equal to 1/2 Vexc and a differential component equal to 1/2 Vsig" on each side.

A sensor bridge is designed so that the arms move in the directions that add. SLOA034: "Sensors are designed so that when acted upon, opposite resistors in the bridge change resistance as shown by ±ΔR." With two arms rising and two falling it gives the output in one line: "With four active elements, the output voltage is: Vsig = Vexc × ΔR/R." And because ΔR is proportional to the force or pressure, it "can be rewritten as Vsig = Vexc × F × S, where F is the force or pressure and S is the sensitivity, or output voltage, of the sensor in mV per volt of excitation with full scale input". That second form is where the load cell's mV/V rating comes from.

Wheatstone bridge formula with four resistors, and the unknown arm

The calculator numbers the arms around the diamond from the top left: R1 from the excitation to the left midpoint, R2 from there to ground, R3 from ground up to the right midpoint and R4 from there to the excitation. The left midpoint is V_L = V_E·R2/(R1 + R2), the right one V_R = V_E·R3/(R3 + R4), and the output is their difference, positive when the right midpoint is the higher. Neither TI document numbers the four arms (SLOA034's R1 to R4 are its amplifier's resistors); HBM's "Applying the Wheatstone Bridge Circuit" does, the same way, and its Eq. 2, U_A/U_E = R1/(R1 + R2) − R4/(R3 + R4), is the same equation rearranged. The numbering puts SLOA034's SIG+ on the right. SLOA034's Figure 3 bridge, 1010 Ω, 990 Ω, 1010 Ω and 990 Ω at 5 V, gives 2.525 V and 2.475 V, the SIG+ and SIG− the report prints, and 50.0 mV between them.

The output is zero when the two dividers have the same ratio, which is R1·R3 = R2·R4: the products of opposite arms are equal. That is the balance condition, and it is the original use of the circuit. SLYP163: "The principle of the circuit is that if three resistances are known, and the current in the cross branch is zero, the fourth resistance can be calculated." For Figure 3's other three arms, the R3 that would balance the bridge is 990 × 990 / 1010 = 970.4 Ω.

A sensor bridge is not nulled like that, and SLYP163 is explicit that the sensor is a descendant rather than the same thing: "This circuit is actually not used in bridge sensors, despite appearances", and of the single-gauge bridge, "this circuit is not really a Wheatstone bridge, since the legs are not connected, and voltage is measured instead of current." The unbalanced bridge is read as a voltage, and the unknown arm can still be recovered from it. With three 1 kΩ arms at 5 V and 10 mV measured, the right midpoint sits at 2.5 V + 10 mV, and R3 = R4·V_R/(V_E− V_R) = 1008.03 Ω. The straight-line estimate, ΔR/R ≈ 4·V_O/V_E, gives 1008.00 Ω; the difference is the quarter bridge's nonlinearity, below. SLYP163's slide 6 gives the same inversion for a lone divider, R_g = R1(V_E/V_O − 1).

Quarter, half and full bridge: sensitivity and linearity

SLYP163 draws three sensor bridges. The single-point bridge has one active arm: "a single resistor changes, while the other three are fixed", and "the output voltage does not change linearly with the changing resistance." The two-point bridge has two: "Higher-end load cells are commonly constructed this way, with two strain gauges connected oppositely. This makes the cell twice as sensitive." The four-point bridge has all four, "done in cases where a very linear and sensitive output is needed. Matching the elements can be difficult, so this is seldom done in inexpensive sensors."

Micro-Measurements' Tech Note TN-507-1 tabulates seven arrangements in its Table 1, with the exact output of each, its nonlinearity and the correction from an indicated strain back to the real one. Its cases 1, 4 and 7 are SLYP163's three bridges. Case 3 is the linear half bridge, two gauges in the same divider with "equal and opposite strains — typical of bending-beam arrangement". Cases 2, 5 and 6 add "Poisson" gauges, mounted across the load so that they see −ν times the strain of the main gauge, ν being the material's Poisson's ratio. Every one of TN-507's printed outputs is the four-resistor equation with the gauges in those arms and x = ΔR/R = F·ε, which is how the calculator computes them. TN-507 numbers its arms differently, with R2 upper right and R4 lower left, and draws its single gauge upper left; the calculator keeps SLOA034's orientation and puts the quarter-bridge gauge lower right. Moving a gauge to the mirror-image arm of its own divider changes neither the size nor the sign of V_O.

SLYP163's two-point bridge draws its gauges in opposite arms; with both rising, the output is x/(2 + x), exactly 2 times the quarter bridge's at every x, which is the "twice as sensitive". It is also exactly as nonlinear: TN-507 gives cases 1 and 4 the same nonlinearity, because each active gauge still sits in a divider whose total resistance changes. A half bridge made linear has its two gauges in the same divider, one rising and one falling; the divider's total then stays 2R and the output is x/2 with no error. TN-507 puts the rule in one line: the output is a linear function of strain only when the resistance changes are such that "the currents through the bridge arms remain constant". The full bridge is both, and SLOA034's Vsig = Vexc × ΔR/R is exact for it.

Output per volt of excitation for a strain of 1000 µε under the principal gauge, a gauge factor of 2, which HBM gives as "about 2" for metal gauges, and ν = 0.3, the value TN-507 plots its nonlinearity at: ΔR/R = 0.20 %. The 1000 µε is a round value chosen to illustrate.

Configuration	TN-507 case	V_O/V_E	Exact, mV/V	Straight line, mV/V	At 5 V	Nonlinearity
Quarter bridge	1	x/(4 + 2x)	0.4995	0.5000	2.498 mV	−0.0999 %
Half bridge, gauge and Poisson gauge	2	(1 + ν)x/(4 + 2(1 − ν)x)	0.6495	0.6500	3.248 mV	−0.0700 %
Half bridge, adjacent arms	3	x/2	1.0000	1.0000	5.000 mV	none
Half bridge, opposite arms	4	x/(2 + x)	0.9990	1.0000	4.995 mV	−0.0999 %
Full bridge, Poisson gauges (column)	5	(1 + ν)x/(2 + (1 − ν)x)	1.2991	1.3000	6.495 mV	−0.0700 %
Full bridge, Poisson gauges (beam)	6	(1 + ν)x/2	1.3000	1.3000	6.500 mV	none
Full bridge	7	x	2.0000	2.0000	10.00 mV	none

Three things stand out. The full bridge gives four times the quarter bridge's output for the same strain, which at millivolt signal levels is four times the signal against the same amplifier noise. A Poisson gauge adds ν to the signal and, being the same kind of gauge at the same temperature, compensates it: TN-514 describes the Poisson half bridge as providing "an augmented bridge output, along with excellent temperature compensation". And at this strain the quarter bridge's error is small, 0.0999 % of the reading; it is not at larger changes, as the next table shows.

Quarter bridge nonlinearity, computed

For one active arm, V_O/V_E = x/(4 + 2x) against the straight line x/4, so the reading is low by x/(2 + x) of itself for a rising resistance and high by the same form for a falling one. TN-507 writes the output as K·ε·(1 − η) with K = F/4 and η = Fε/(2 + Fε); η is that same fraction, and the calculator's nonlinearity is −η. The table runs it from 0.1 % to 20 %; the strain column is the same ΔR/R read at a gauge factor of 2, for scale only.

ΔR/R	Strain at GF 2	Exact, mV/V	x/4, mV/V	Error, rising	Error, falling
0.10 %	500 µε	0.2499	0.2500	−0.0500 %	+0.0500 %
0.20 %	1000 µε	0.4995	0.5000	−0.0999 %	+0.100 %
0.50 %	2500 µε	1.247	1.250	−0.249 %	+0.251 %
1.0 %	5000 µε	2.488	2.500	−0.498 %	+0.503 %
2.0 %	10000 µε	4.950	5.000	−0.990 %	+1.01 %
5.0 %	25000 µε	12.20	12.50	−2.44 %	+2.56 %
10 %	50000 µε	23.81	25.00	−4.76 %	+5.26 %
20 %	100000 µε	45.45	50.00	−9.09 %	+11.1 %

The error is very nearly half the fractional change: 0.498 % at 1 %, 4.76 % at 10 %. TN-507 puts it in strain: "the error is about 0.1% at 1000µε, 1% at 10 000µε, and 10% at 100 000µε; or, as a convenient rule of thumb, the error, in percent, is approximately equal to the strain, in percent." At a gauge factor of 2 the exact figures are 0.0999 %, 0.990 % and 9.09 %. It is not symmetrical, so a quarter bridge that sees tension and compression reads them with slightly different slopes: TN-507, "Wheatstone bridge nonlinearity causes indicated tensile strains to be too low, and indicated compressive strains too high." For a metal strain gauge in its elastic range the change is small and the error with it; for a resistive sensor whose resistance moves by tens of percent over its range, it is an error of several percent unless the exact equation is used to linearise it, which is what the calculator's "work back from V_O" option does. SLYP163 treats the same error from the sensor maker's side: load cells "are typically 0.1% - 1% accurate, due to non-linear output curve", and the inherent nonlinearity "can be corrected for to a degree, but commonly is not, as it is (with a good sensor) smaller than most other errors."

Strain gauge formula: from microstrain to millivolts

A strain gauge's resistance change is its strain times its gauge factor: ΔR/R = F·ε. TN-514 calls this "the definition of the gage factor" (its Eq. 4), and HBM's Eq. 1 is the same relation, with the value: "The exact value is specified on each strain gage package. In general, the gage factor for metal strain gages is about 2." The calculator takes it as an input, 2 by default, for you to replace with the figure on the package. Strain is dimensionless and small, so it is written in microstrain, µε, parts per million of length. From there the chain is the bridge equation: ε and F give x, the configuration gives V_O/V_E, and the excitation gives millivolts. At the illustrative 1000 µε and F = 2, a quarter bridge at 5 V gives 2.498 mV, a full bridge 10.00 mV.

Run the other way, a measured output gives the strain: the calculator inverts each configuration's exact equation, x = 4v/(1 − 2v) for the quarter bridge with v = V_O/V_E, then divides by the gauge factor. Using the straight-line inverse instead carries the nonlinearity above straight into the strain reading.

A strain indicator does the straight-line inverse. It is built for one active gauge: TN-514 gives its reading as ε_I = C·e_O/E with C = 4/F_I (Eqs. 33 and 38), F_I being its gauge factor setting. So a half or full bridge reads high by its number of active gauges, and a quarter bridge reads off by its nonlinearity; the 1000 µε full bridge above reads +4000 µε. TN-507's last column undoes both. For the quarter bridge it is ε = 2ε_i/(2 − Fε_i), strains in plain ratios; each case has its own, and the calculator shows the corrected strain beside the indicated one.

Worked example: TN-507's nonlinearity corrections

TN-507 works three quarter-bridge examples at a gauge factor of 2.0. In the first the bridge starts balanced and the indicator reads 15 000 µε in tension, then −15 000 µε in compression. In the other two it starts unbalanced, which TN-507 treats by correcting the total indicated unbalance before and after the load and subtracting: "the initial unbalance (expressed in strain units) must be added algebraically to any subsequent observed strains so that the nonlinearity correction is based on the total (or net) unbalance". The left column is the calculator's arithmetic, the right what TN-507 prints.

1         indicated +15 000 µε: 2εi/(2 − Fεi)          = +15 228 µε TN-507: 15 228
          indicated −15 000 µε                         = −14 778 µε TN-507: −14 778
2         initial unbalance, indicated −4500 µε        = −4480 µε   TN-507: −4480
          total, indicated −12 500 µε                  = −12 346 µε TN-507: −12 346
          applied, the difference                      = −7866 µε   TN-507: −7866
3         before loading, indicated −48 000 µε         = −45 802 µε TN-507: −45 802
          after, indicated −45 000 µε                  = −43 062 µε TN-507: −43 062
          applied, the difference                      = +2739.3 µε TN-507: +2740

Every value but the last agrees to the microstrain. The last line is the one to look at: the indicator saw the load as +3000 µε, the bridge was really strained by +2739 µε, 8.69 % less, because it was working −45 802 µε from its balance point. TN-507's figure, +2740, is the difference of its two rounded results; unrounded it is 2739.3. TN-507: "even with relatively modest working strains the nonlinearity error can be very significant (about 10% in this instance) if the Wheatstone bridge is operated far from its resistive balance point."

Lead wires: how much signal the cable takes

In a quarter or half bridge the gauges are at the end of a cable and the completion resistors are in the instrument, so each lead wire is part of a bridge arm. A lead of R_L in series with a gauge dilutes its resistance change: HBM gives the apparent gauge factor as k* = k·R/(R + R_K) (Eq. 18). TN-514: "In a three-wire quarter-bridge circuit, for instance, the signal will be attenuated by the factor RG/(RG + RL), where RL is the resistance of one leadwire in series with the gage." The three-wire connection puts one lead in the gauge's arm and one in the dummy's, so the leads cancel from the balance; a two-wire connection puts both in the gauge's arm (HBM Eq. 21). In a full bridge the leads carry the excitation instead and divide it, R_B/(R_B+ 2R_L) (HBM Eq. 16).

For 350 Ω gauges and 5 Ω a lead, an illustrative figure: three-wire quarter bridge 1.41 % of the signal lost, two-wire 2.78 %, half bridge 1.41 %, full bridge 2.78 %. HBM's own example: "The sensitivity of a bridge with RB = 120 Ohms will be reduced by, say, 5.8 % when using a special cable with a length of 100 meters". The loss is a gain error, not an offset, so it is what shunt calibration exists to remove: TN-514, "The usual way of correcting for leadwire desensitization is by shunt calibration".

Shunt calibration: the resistor that simulates a strain

A strain gauge is hard to calibrate with a known strain, so the usual check is electrical. TN-514: "decreasing the resistance of a bridge arm by shunting with a larger resistor offers a simple, potentially accurate means of simulating the action of a strain gage." A resistor R_Cacross a gauge of R_G changes it by ΔR/R_G = −R_G/(R_G + R_C) (Eq. 3), which the gauge factor turns into a strain, and solved for the resistor: R_C = R_G/(F_G·ε_S) − R_G (Eq. 7). Shunting always lowers the resistance, so the strain it simulates in the gauge is compressive. Shunting the arm next to it instead, the dummy of a quarter bridge, moves the output the other way and reads as tension; the calculator's gauge is R3 and the adjacent arm R4, which is TN-514's active gauge R1 and dummy R2 renamed.

TN-514's Table 1 lists the resistors for 120, 350 and 1000 Ω gauges at a gauge factor of 2.000. Here it is recomputed from Eq. 7. TN-514 truncates rather than rounds: 1000 Ω at 3000 µε is 165 666.7 Ω, which the table prints as 165 666. Truncated the same way, all 22 values agree with it to the ohm.

Simulated strain	120 Ω gauge	350 Ω gauge	1000 Ω gauge
100 µε	599 880 Ω	—	—
500 µε	119 880 Ω	349 650 Ω	999 000 Ω
1000 µε	59 880 Ω	174 650 Ω	499 000 Ω
2000 µε	29 880 Ω	87 150 Ω	249 000 Ω
3000 µε	19 880 Ω	57 983 Ω	165 666 Ω
4000 µε	14 880 Ω	43 400 Ω	124 000 Ω
5000 µε	11 880 Ω	34 650 Ω	99 000 Ω
10 000 µε	5880 Ω	17 150 Ω	49 000 Ω

With more than one active gauge, the bridge gives N times the output for the same strain and the resistor has to simulate N times as much: Eq. 8 divides by N, "the number of active gages", which is 1 for a quarter bridge, 1 + ν for a gauge and its Poisson gauge, 2 for a bending half bridge, 2(1 + ν) for the Poisson full bridges and 4 for a full bridge in bending or torsion. If the resistor sits at the instrument and reaches the gauge through two extra leads, Eq. 9 subtracts their 2R_L.

Those relationships ignore the bridge's nonlinearity, which TN-514 accepts up to a limit: "Since the nonlinearity error at 2000µε is normally less than 0.5%, that level has been taken arbitrarily as the upper limit of small strain". Above it, its Section V gives exact relationships, derived by making the shunted bridge's output equal to the strained bridge's. Shunting a quarter bridge's gauge is already exact, "because the nonlinearity due to shunting is the same as that caused by compressive strain in the gage". Shunting its dummy is not: for tension the exact resistor is R_G/(F_G·ε) (Eq. 23), and using Eq. 7's instead simulates an error that is, in TN-514's words, "approximately equal to the gage factor times the strain, in percent". For 350 Ω at 2000 µε, Eq. 23 gives 87 500 Ω against Eq. 7's 87 150 Ω, and Eq. 7's resistor across the dummy simulates +2008 µε. The table has each arrangement at the same 2000 µε on 350 Ω, ν = 0.3.

Arrangement	TN-507 case	N	Eq. 8, Ω	Exact, across the gauge, Ω
Quarter bridge	1	1	87 150	87 150 (Eq. 7)
Half bridge, gauge and Poisson gauge	2	1.3	66 958	67 038 (Eq. 24)
Half bridge, adjacent arms	3	2	43 400	43 575 (Eq. 28)
Half bridge, opposite arms	4	2	43 400	none in TN-514
Full bridge, Poisson gauges (column)	5	2.6	33 304	33 432 (Eq. 30)
Full bridge, Poisson gauges (beam)	6	2.6	33 304	33 479 (Eq. 29)
Full bridge	7	4	21 525	21 700 (Eq. 29)

Each exact relationship in the calculator is tested the same way TN-514 derives it: the bridge with the resistor across one arm and the bridge under the strain are both built from the four-resistor equation, leads included, and must read the same. That test is how one of TN-514's typesetting slips shows up. Eq. 24, for compression in the Poisson half bridge, is printed with +R_G in its first term alongside the instruction that "the signs of the simulated strains must always be carried"; carried, a compressive strain gives a negative resistor. Eq. 30, the full-bridge counterpart, has −R_G, and that form passes. Two more are plain on the page: Eqs. 27 and 28 print the leading 2 of 2F_G as "s". And Eq. 6, the step before Eq. 7, lacks the minus that Eq. 5 carries; the text after it explains that the minus is customarily dropped, which is how Eq. 7 is written.

Worked example: TN-514's 350 Ω gauge at 2000 µε

TN-514 p. 18: "assume that a 350-ohm gage with a gage factor of 2.0 is to be shunted to simulate a strain of 2000µε." It then asks how accurate the simulation is, with the calibration resistor at ±0.01 %, the gauge factor at ±1 % and the installed gauge resistance at ±1 %, wider than the gauge's ±0.3 % because "the gage resistance may have been shifted during installation". The left column is the calculator's arithmetic, the right what TN-514 prints.

R_C       Eq. 7: 350 × 10⁶/(2 × 2000) − 350            = 87 150 Ω   TN-514: 87150 ohms
ΔR/R      Eq. 3: −350/(350 + R_C)                      = −0.4000 % 
strain    Eq. 5: (ΔR/R)/F                              = −2000 µε  
V_O/V_E   Eq. 35: −350/(4R_C + 700)                    = −1.00200 mV/V
reads     Eq. 36, ideal indicator at F_I = 2           = −2004 µε  
coeffs    (R1 + 2R_C)/(R1 + R_C), R_C/(R1 + R_C)       = 1.996, 0.996 TN-514: 1.996, 0.996
U         Eq. 45: √(1.996²·1² + 0.996²·0.01² + 1²) %   = ±2.233 %   TN-514: ±2.23%
worst     Eq. 44: the same terms added                 = ±3.01 %

The gauge resistance dominates: its tolerance enters twice over, because it sets both the resistance change and the resistance being changed, and TN-514 notes that "the percent error in simulated strain is about twice that in the gage resistance". The resistor's own tolerance hardly counts. Two more lines are worth reading. An ideal indicator set to F_I= 2 registers −2004 µε for a simulated −2000 µε: the quarter bridge's nonlinearity, which reads compression high. TN-514 separates the two uses of shunt calibration for exactly this reason. Scaling adjusts the instrument until it registers the simulated strain, taking out lead loss and N at the same time; verification checks an instrument against Eq. 38's R_C = R_G/(F_I·ε_I) − R_G/2, which does not involve the gauge's own gauge factor. The same note prints its resistors' tolerance as ±0.02 % on p. 4 and ±0.01 % on p. 18; the worked example uses its own p. 18 figure.

Load cell mV/V: output, gain and counts per kilogram

A load cell's data sheet gives its rated output, and SLYP163 explains the unit: "Sensitivity – measured in mV/V; voltage output when the bridge is excited with 1 volt and the sensor is at full scale; 2mV/V is common." And what follows from it: "A bridge with 2mV/V sensitivity will deliver 2mV if the measurement is at full-scale and the bridge is excited with 1V. This shows that the signals to be measured are very, very small; a 10V excitation can excite a 2mV/V sensor to at most 20mV!" The full-scale output is simply the rating times the excitation:

Rated output	V_E = 1 V	V_E = 5 V	V_E = 10 V
1 mV/V	1.00 mV	5.00 mV	10.0 mV
2 mV/V	2.00 mV	10.0 mV	20.0 mV
3 mV/V	3.00 mV	15.0 mV	30.0 mV
10 mV/V	10.0 mV	50.0 mV	100 mV

SLOA034 puts the usual range as "Normally full scale output voltages are in the 10 mV to 100 mV range, and need to be amplified in a data acquisition system." The rest of the calculation is the gain that stretches that range over the converter's input, and what one code is then worth. SLYP163's most common arrangement for a SAR converter amplifies the signal "from 0-20mV to the converter's input range, often 0-5V". Worked with SLYP163's figures, a 10 kg capacity and a 16-bit converter (both illustrative):

full      V_FS = 2 mV/V × 10 V                         = 20.0 mV    TI: 20 mV
current   I = 10 V / 500 Ω                             = 20.0 mA    TI: 20 mA
power     P = 10 V × 20 mA                             = 200 mW    
gain      G = 5 V / 20 mV                              = 250        TI: 0–20 mV to 0–5 V
per kg    V_FS / 10 kg                                 = 2.00 mV   
LSB       5 V / (2¹⁶ − 1)                              = 76.3 µV   
at input  LSB / G                                      = 305 nV    
counts    per kg                                       = 6553.5    
          per count                                    = 153 mg    
5 kg      code                                         = 32767.5

When the gain fills the converter exactly, the counts per kilogram reduce to (2^N − 1)/capacity, whatever the mV/V rating: a more sensitive cell needs less gain, not fewer bits. What the rating does change is the voltage one count stands for at the bridge, here 305 nV, and whether a count that small is above the amplifier's and converter's noise is a separate question; the ADC noise floor calculator takes it up. The excitation current is not small either. SLYP163: "the bridge resistance determines power consumption, which can be rather high; a 10V excitation consumes 20mA for a 500-ohm bridge." That is 200 mW in the bridge.

Worked example: TI's 10 mV/V sensor and a difference amplifier

SLOA034 §2.1 works a complete case: "Given a sensor having 1 kΩ elements and a sensitivity of 10 mV/V is being used with 5 V of excitation. At full-scale, the resistors will have ΔR=10 Ω and 50 mV will be seen from SIG– to SIG+ if measured with a high impedance voltmeter." Then: "Assuming full scale at the output of the amplifier is 5 V, a gain of 100 is needed. Choosing R1 = R2 = 1 kΩ and R3 = R4 = 100 kΩ seems to be correct, but when tested the output is 30% lower than expected." The left column is the calculator's arithmetic, the right what TI prints.

signal    V_sig = 5 V × 10 mV/V                        = 50.0 mV    TI: 50 mV
ΔR/R      V_sig / V_exc = 50 mV / 5 V                  = 1.0 %      TI: ΔR = 10 Ω
SIG+      5 V × 1010 / (1010 + 990)                    = 2.525 V    TI: 2.525 V
SIG−      5 V × 990 / (1010 + 990)                     = 2.475 V    TI: 2.475 V
gain      5 V / 50 mV                                  = 100        TI: 100
R_TH      1010 Ω ∥ 990 Ω, each side                    = 499.95 Ω   TI: 500 Ω
naive     100 kΩ / (1 kΩ + 500 Ω)                      = 66.7       TI: "30% lower"
fix       R_f = 100 × (1 kΩ + 500 Ω)                   = 150.0 kΩ   TI: 150 kΩ

The cause is the bridge's own resistance. Each output terminal is a Thévenin source of R1∥R2 or R3∥R4, which SLOA034 takes as R/2 per side, 500 Ω here, and a single op amp difference amplifier puts that in series with its 1 kΩ input resistors. The gain becomes 100 kΩ / 1.5 kΩ = 66.7, which is 33.3 % low; SLOA034 rounds it to "30% lower". Its fix: "R3 and R4 need to be 150 kΩ to get the required gain of 100", which is the calculator's R_f = G·(R_in + R_B/2) = 150.0 kΩ, the "50% Higher" in the title of its Figure 4. It adds a caution: "Note that the impedance seen at the negative input (SIG–) is not constant. It varies with the output voltage which causes slight non–linearity."

The instrumentation amplifiers avoid the problem. Of the three op amp in-amp, SLOA034 says it "has high input impedance, and source impedance does not play a role in calculation of gain", and the same of the two op amp version. All three circuits "require resistor matching to achieve good CMRR", because the half-excitation common-mode voltage sits on both inputs. The load cell mode computes the difference amplifier's feedback resistor when that amplifier is selected; the op amp gain calculator covers the resistor choice in general.

Ratiometric measurement: why the excitation need not be accurate

The bridge output is proportional to its excitation, so any drift in the excitation is a gain error in the reading. The standard cure is to make the converter measure a ratio. SLYP163, of its amplifier-and-SAR circuit: "This circuit shows a ratiometric connection. The excitation for the bridge is also used for the ADC's reference." Its summary slide lists "Use ratiometric connections where possible". The ADC's code is its input over its reference; with the reference taken from the excitation, both scale together and V_E cancels out of the code.

In the worked load cell, if the 10 V excitation sags by 1 % to 9.9 V, the code at 5 kg with a fixed reference falls from 32768 to 32440, an error of 328 counts, or 50 g. With a ratiometric reference it does not move. The cancellation is only as good as the match between the voltage the bridge sees and the voltage the reference sees, which is where the cable comes in.

Where the ideal-bridge model stops being valid

Lead resistance. The excitation current flows through the cable, and the drop in the two excitation wires takes voltage from the bridge, U_E·R_B/(R_B + R_K2 + R_K3) in HBM's Eq. 16. With a 500 Ω bridge and 1 Ω in each wire (an illustrative figure) the bridge sees 9.960 V instead of 10 V, a gain error of 0.398 %; at 5 Ω a wire it is 1.96 %. It also drifts, because copper's resistance changes with temperature. SLYP163's six-wire connection returns the voltage at the bridge on separate sense wires to the reference: "The Kelvin connection aims to reduce errors arising from voltage drops in the excitation line. The drops are caused by the relatively high current required to excite the load cell. Since the ADC's reference draws very little current, these drops do not occur in the return line, which is usually called the sense line." And: "Kelvin connections are usually needed only for load cells connected through a long cable."

Mismatch and offset. The equations assume the arms are equal at rest. SLYP163's differential divider "assumes that Rg at rest is roughly equal to R1, and that all the R1s are very closely matched." A mismatch leaves an output at zero load, an offset: "Offset error is the voltage produced when the sensor's measurement parameter is zero. This can easily be calibrated out." Full-scale error, "the deviation in slope from the expected value", is corrected the same way, with a known weight.

Temperature and time. The calculator does not model temperature. SLYP163 names it as the hard part: "Drift is very important: it is the change of the above values with temperature and time. It is much more difficult to compensate for drift – especially drift with time." Calibration removes offset and gain at one temperature; drift is what remains.

Loading. The output is computed for a voltmeter that draws no current. Anything else across the midpoints, a difference amplifier's input resistors or an input filter, divides against the bridge's source resistance, which the calculator reports, R1∥R2 + R3∥R4. SLYP163 on filter resistors: "The resistors should likewise be very small, no more than 100-200 ohms. If they are too large, increased noise and gain errors will result."

Common Wheatstone bridge mistakes

Using the full-bridge formula for a quarter bridge. V_E·ΔR/R is SLOA034's result for four active arms; one active arm gives a quarter of it, 2.498 mV rather than 10.00 mV in the table above.
Assuming any two-gauge bridge is linear. Two gauges that both rise, in opposite arms, double the output and keep the quarter bridge's nonlinearity. Only a rising and a falling gauge in the same divider cancel it.
Designing a difference amplifier's gain without the bridge's source resistance. SLOA034's example comes out 33 % low for exactly this reason.
Getting the sign from a drawing. SLYP163's single-point equation has its gauge in the upper arm, so its output falls as the gauge's resistance rises; the same gauge in the lower arm gives the opposite sign. Its slide 8 also redraws the bridge with the + and − terminals on the midpoints opposite slide 7's, under the same equation. Check which midpoint the amplifier's + input is on.
Referencing the ADC to a fixed voltage while exciting the bridge from a regulator that drifts. A ratiometric connection takes the excitation out of the reading.
A four-wire connection over a long cable. The excitation wires' drop is a gain error that changes with temperature; a six-wire cell senses the bridge voltage directly.
Reading a half or full bridge on a strain indicator without accounting for the number of active gauges. The indicator assumes one; TN-507's corrections column, or a shunt calibration with Eq. 8's N, turns the reading back into the strain under one gauge.
Shunting from the signal lead to an excitation lead instead of across the arm. TN-514: "the calibration resistor must be connected directly across the dummy to produce the desired result. Gage strain cannot be accurately simulated by shunting from S– to P– (or from S– to P+)."
Using Eq. 7's resistor across the dummy to simulate a large tensile strain. It is exact only across the gauge, in compression; for tension the exact resistor is Eq. 23's.
A poor differential filter capacitor. SLYP163: "The differential capacitor is critical, and should be of very high quality. Ceramic X5Rs and worse should never be used. Even X7R-grade capacitors should be avoided. C0G or film capacitors should be used here."