Many laboratory sensors—including strain gauges, RTDs, and thermistors—produce their output in the form of a change in electrical resistance. To measure these often very small changes accurately, engineers commonly use the Wheatstone bridge, a configuration of four resistors arranged so that small changes in one leg create a measurable differential voltage. The Wheatstone bridge offers much higher sensitivity, superior noise immunity, and better drift rejection than a simple voltage divider.
This section introduces the Wheatstone bridge for resistive measurement, explains why it is preferred over a simple divider, and develops the essential concepts used in practical measurement systems. The presentation follows Doebelin §10.1.
It is tempting to measure a sensor's resistance by placing it in a voltage divider:
\[ v_{\text{out}} = V_{\text{ex}} \frac{R_s}{R_f + R_s}. \]
Although this works in principle, it has several practical drawbacks:
Strong nonlinearity
The function \(R_s / (R_f + R_s)\) is nonlinear, which complicates calibration.
Poor sensitivity for small fractional changes
Typical sensors change by only a small percentage (e.g., 0.1–1% for strain gauges).
A single divider produces only a small voltage change in response.
Strong susceptibility to temperature drift
Any change in the fixed resistor or lead resistance looks exactly like a sensor change.
No common-mode rejection
Noise and excitation-voltage drift directly affect the output.
The Wheatstone bridge addresses each of these issues by using two dividers and measuring their difference.
A Wheatstone bridge consists of four resistors arranged in a diamond configuration (Figure 10.1). One diagonal is excited by a voltage source \(V_{\text{ex}}\), and the output voltage \(v_{AC}\) is measured across the other diagonal.
Figure Placeholder 1 — Basic Wheatstone Bridge Schematic (corresponds to Doebelin Fig. 10.1)
The bridge is balanced when
\[ \frac{R_1}{R_2} = \frac{R_3}{R_4}, \]
in which case
\[ v_{AC} = 0. \]
A change in any of the resistances unbalances the bridge and produces a nonzero differential voltage.
Because the bridge output is the difference between two voltage dividers, it is more sensitive to small resistance changes and less influenced by uniform drifts affecting all legs.
One common configuration uses only a single active sensing element placed in one leg of the bridge, typically \(R_1\). The other three resistors serve as fixed bridge-completion resistors, often located in the instrumentation device.
Assuming the measuring instrument has very high input impedance (Doebelin explicitly considers the case \(i_m \approx 0\)), node voltages are simply:
\[ v_A = V_{\text{ex}} \frac{R_2}{R_1 + R_2}, \qquad v_C = V_{\text{ex}} \frac{R_4}{R_3 + R_4}. \]
Thus the bridge output is
\[ v_{AC} = v_A - v_C = V_{\text{ex}} \left( \frac{R_2}{R_1 + R_2} - \frac{R_4}{R_3 + R_4} \right). \]
A small change in the sensor resistance \(R_1 \to R_1 + \Delta R_1\) results in a small but nearly linear change in \(v_{AC}\), provided the bridge is initially balanced and the changes are small (Doebelin notes this explicitly for practical sensors such as strain gauges).
In the deflection method, all resistors are fixed except the sensor. The output voltage \(v_{AC}\) is measured directly using a high-impedance voltmeter, oscilloscope, or instrumentation amplifier. Advantages:
Drawback:
In the null method, one resistor (e.g., \(R_2\)) is variable. When the sensor resistance changes, the operator adjusts the variable resistor until the output returns to zero. The adjustment directly corresponds to the sensor's resistance change.
Advantages (Doebelin):
Disadvantages:
Doebelin shows that the open-circuit output is
\[ v_{AC} = V_{\text{ex}} \left( \frac{R_2}{R_1+R_2} - \frac{R_4}{R_3+R_4} \right). \]
In general the relationship between \(v_{AC}\) and the resistor values is nonlinear. However:
Special cases of perfect linearity occur when equal and opposite resistance changes occur in opposing legs, such as:
\[ +\Delta R_1,\ -\Delta R_2,\ +\Delta R_3,\ -\Delta R_4. \]
Consider a quarter-bridge with one active gauge in leg \(R_1\), and the other three resistors fixed and equal to the nominal gauge resistance \(R\). At the balanced condition, \[ R_1 = R_2 = R_3 = R_4 = R, \] and if the gauge experiences a small strain \(\varepsilon\), its resistance becomes \[ R_1 = R + \Delta R, \qquad \frac{\Delta R}{R} = GF\,\varepsilon. \]
Substituting these into the bridge expression and linearizing for small \(\Delta R / R\) gives the well-known approximation \[ v_{AC} \approx -\,\frac{V_{\text{ex}}}{4}\,\frac{\Delta R}{R} = -\,\frac{V_{\text{ex}}}{4}\,GF\,\varepsilon. \]
The sign depends on whether the gauge is in tension or compression and on which leg it occupies; the important result is that the magnitude of the bridge output is proportional to \(V_{\text{ex}}\), the gauge factor, and the applied strain: \[ \lvert v_{AC} \rvert \approx \frac{V_{\text{ex}}}{4}\,GF\,\varepsilon. \]
This shows explicitly how a strain gauge with a given gauge factor translates mechanical strain into an electrical signal in a quarter-bridge configuration.
If two strain gauges are used in a half-bridge configuration (for example, one in tension and one in compression so that their resistance changes are equal and opposite), the bridge sensitivity increases. A standard analysis with \[ R_1 = R + \Delta R,\quad R_2 = R - \Delta R,\quad R_3 = R_4 = R \] yields the approximate result \[ \lvert v_{AC} \rvert \approx \frac{V_{\text{ex}}}{2}\,GF\,\varepsilon. \]
In a properly wired full-bridge configuration with two gauges in tension and two in compression, \[ \lvert v_{AC} \rvert \approx V_{\text{ex}}\,GF\,\varepsilon. \]
Thus, for the same excitation and strain, a full bridge provides roughly four times the output of a quarter bridge, and a half bridge provides roughly twice the output of a quarter bridge. This is one of the main reasons why multi-gauge bridge configurations are preferred in high-sensitivity strain measurements.
Doebelin evaluates loading using Thévenin's theorem (Figure 10.2). When the meter has finite input resistance \(R_m\), the measured output voltage is reduced:
\[ v_{AC}^{\text{(loaded)}} = v_{AC}^{\text{(open)}} \frac{R_m}{R_m + R_{\text{th}}}, \]
where \(R_{\text{th}}\) is the Thévenin resistance of the bridge (Doebelin Eq. 10.17).
If \(R_m \gg R_{\text{th}}\), loading effects are negligible; otherwise, sensitivity is reduced.
Figure Placeholder 2 — Thévenin Equivalent Representation of Wheatstone Bridge (corresponds to Doebelin Fig. 10.2)
Modern instrumentation amplifiers typically have input resistances of 1 MΩ or more, making loading insignificant for most strain-gauge or RTD bridges.
Doebelin describes a practical and accurate method for calibrating the sensitivity of the bridge using a precision shunt resistor temporarily switched across one leg (Figure 10.3). When the switch is closed, the sensor resistance changes by a known amount:
\[ \Delta R = R_{\text{eff}} - R, \]
where \(R_{\text{eff}}\) is the parallel combination of the sensor and the shunt resistor.
The corresponding bridge output \(v_{AC}\) allows direct computation of the overall system sensitivity:
\[ S = \frac{v_{AC}}{\Delta R}. \]
This method takes into account:
Figure Placeholder 3 — Shunt Calibration Arrangement (corresponds to Doebelin Fig. 10.3)
Doebelin notes several features commonly included in commercial bridge instrumentation (Figure 10.4):
Figure Placeholder 4 — Bridge with Sensitivity, Balance, and Calibration Controls (corresponds to Doebelin Fig. 10.4)
Additional real-world considerations include:
Although the quarter-bridge configuration is common, other arrangements offer better performance:
Two resistors change, often in opposite directions.
Improves sensitivity and compensates for uniform temperature changes.
All four legs contain active sensors (e.g., four strain gauges on a load cell).
Maximizes sensitivity and yields perfect linearity under matched conditions (consistent with Doebelin's symmetry example).
Provides compensation for long leads connected to a remote sensor.
Used in some NASA applications (Doebelin notes the Anderson Loop).
These are not strictly Wheatstone bridges but address similar measurement requirements.
The Wheatstone bridge is the preferred method for converting small resistance changes into measurable voltages. Compared with a simple voltage divider, it provides:
Although our focus here has been the resistive Wheatstone bridge, the same principles extend to capacitance and inductance bridges, which are widely used for capacitive and inductive sensors in displacement, humidity, pressure, and proximity measurements. As Doebelin notes, capacitance and inductance bridges are important variants even though resistive bridges are the most common in practice.