Table of Contents
- Overview
- Resistance to Voltage Conversion
- Increasing Sensitivity with Second Element
- Excitation Voltage Independence
- DC Offset and Dynamic Range
- Summary
Overview
The Wheatstone bridge is a configuration of components, typically but not always resistors, that can be used to minimize disturbances from environmental conditions. These conditions include power supply variation, noise, and component tolerances. Wheatstone bridges are used in many instrumentation systems including force (strain gauge applications), pressure, temperature, and LCR meters to name a few.
In this article a mathematical analysis is performed showing the performance improvements of a Wheatstone bridge architecture for resistive elements in comparison to a voltage divider. A Wheatstone bridge is used to convert a changing resistance into a differential voltage. This differential voltage can then be measured by an analog to divider converter (ADC) to determine the equivalent resistance value from the known resistors.
Resistance to Voltage Conversion
The Wheatstone bridge can be used to create a differential voltage output with a change in resistance. This differential output voltage can then be used to determine the unknown resistance given the measured voltage and known resistive elements.
A potential difference is simply the difference in potential (voltage) between two nodes. Mathematically, this could be described as \(V_D = V_A - V_B\). Where the choice of A, and B nodes is arbitrary.
A Wheatstone bridge can be thought of as two voltage dividers in which 3 of the resistors are precisely known. As the unknown resistance deviates from the nominal resistance, a voltage difference across the nodes will appear that is proportional to the change in resistance.
\[\Large V_D = (V_A-V_B)\]Expanding the model into the two separate voltage dividers yields the following equation where \(V_{exc}\) is the excitation, or supply voltage of the bridge.
\[\Large V_D = V_{exc} \left( \frac{R_2}{R_1 + R_2} - \frac{R_x}{R_x+R_3} \right)\]Rearranging the differential voltage equation allows determination of the unknown resistor from the existing knowns:
\[\Large R_x = \frac{-V_D \cdot R_1 \cdot R_3 - V_D \cdot R_2 \cdot R_3 + V_{exc} \cdot R_2 \cdot R_3}{V_{exc} \cdot R_1+V_D\cdot R_1 + V_D\cdot R_2}\]Factoring out \(R_3\) and combining terms yields the following more concise equation:
\[\Large R_x = \frac{R_2\cdot V_{exc} - (R_1 + R_2) \cdot V_D}{R_1 \cdot V_{exc}+(R_1 + R_2) \cdot V_D} \cdot R_3\]Increasing Sensitivity with Second Element
With a Wheatstone bridge the sensitivity (\(V/V\)) can be doubled by adding a second sensing element to the opposite side of the bridge. $R_1$ and $R_x$ respectively. This can be particularly advantageous when using sensors that only vary slightly in order to optimize the output for the measurement system.
Recall that a voltage divider has the following formula for division ratio:
\[\Large \frac{R_2}{R_1 + R_2}\]When the resistances in this divider are equal the ratio is \(0.5\) or \(1/2\).
\[\Large R_2 \equiv R_1\] \[\Large \frac{1}{1+1} = \frac{1}{2} = 0.5\]If \(Rx\) is the sensing element, or changing resistance, and \(R_1\), \(R_2\), and \(R_3\) are the same then the equation simplifies to the following:
\[\Large V_D = V_{exc} \cdot \left( \frac{1}{2} - \frac{R_x}{R_3 + R_x} \right)\]When two changing resistive elements are used then the output level will be exactly twice the output of a single element, without the use of an amplifier.
\[\Large V_{exc} \cdot \left( \frac{R_n}{\textcolor{WildStrawberry}{R_x}+R_n} - \frac{\textcolor{WildStrawberry}{R_x}}{\textcolor{WildStrawberry}{R_x}+R_n} \right) \equiv 2 \left( \frac{R_n}{2R_n} - \frac{\textcolor{WildStrawberry}{R_x}}{\textcolor{WildStrawberry}{R_x}+R_n} \right) \cdot V_{exc}\]Four Element (Full Bridge) Configuration
| Element | $1$ | $2$ | $4$ |
|---|---|---|---|
| $R_1$ | $1$ | $0.9$ | $0.9$ |
| $R_2$ | $1$ | $1$ | $1.1$ |
| $R_3$ | $1$ | $1$ | $1.1$ |
| $R_x$ | $0.9$ | $0.9$ | $0.9$ |
| $V_D$ | $26$mV | $53$mV | $100$mV |
Consider the full bridge load cell in which four strain gauges are used. This configuration increases the voltage amplitude by a factor of 4. Attention should be given to the configuration of a full bridge load cell in that two resistive elements (strain gauges) are in tension, and the other in compression. That is, two resistive elements are increasing$\uparrow$ with applied force, and two resistive elements are decreasing$\downarrow$ with applied force. Without this opposition the voltage magnitude would not change.
Excitation Voltage Independence
Wheatstone bridges excel in their ability to reject power supply variations due to the differential nature of the bridge.
Consider a Wheatstone bridge circuit in which 3 of the resistors are equivalent. The unknown resistor varies over some range $\pm \Delta R$ which could be in either direction.
\[\Large V_D = V_{exc} \left( \frac{1}{2} - \frac{R_2 \pm\Delta R}{(R_1 + R_2) \pm\Delta R} \right)\]When all resistors are equivalent then $V_D \equiv 0V$. As the excitation voltage increases the deviation remains minor. In contrast, a voltage divider circuit will vary substantially more. This voltage divider circuit can be modelled as follows:
\[\Large V_o = V_{in} \left( \frac{R_2 \pm \Delta R}{(R_1 + R_2) \pm \Delta R} \right)\]To demonstrate this visually consider the following charts in which a nominal $1000 \Omega$ resistance is varied -$500 \Omega$ to $+1000 \Omega$ and then plotted over three different voltages nominal, and then $\pm 5\%$ of nominal.
Mathematically, the equation for this curve is as follows:
\[\Large V_o = V_{exc} \cdot \left( \frac{R_2 \pm \Delta R}{\left( R_1 + R_2 \right) \pm\Delta R} \right)\]In comparison, the bridge configuration has much less variation, due to the differential nature of the bridge.
The deviation is much less significant over the excitation voltage deviation compared to a voltage divider.
The equation used for this example is as follows:
\[\Large V_D = V_{exc} \left( \frac{1}{2} - \frac{R_2 \pm \Delta R}{\left( R_1 + R_2 \right) \pm \Delta R} \right)\]PSRR and Noise Performance
Another way to quantify the performance of the bridge with power supply variation is power supply rejection ratio. This can be determined using the PSRR formula.
\[\Large PSRR_{[dB]} = 20\cdot \log_{10} \left( \frac{\Delta V_{exc}}{\Delta V_{out}} \right)\]The following table was created using the PSRR formula and the standard equations for a voltage divider as well as Wheatstone bridge. The resistances were normalized where the changing resistance is varied from $0.9$ to $0.1$, and the other resistors are held constant at $1$.
| Normalized R2 | PSRR (dB) | Bridge PSRR (dB) |
|---|---|---|
| 0.9 | -6.490 | -31.596 |
| 0.8 | -7.044 | -25.105 |
| 0.7 | -7.707 | -21.087 |
| 0.6 | -8.519 | -18.062 |
| 0.5 | -9.542 | -15.563 |
| 0.4 | -10.881 | -13.380 |
| 0.3 | -12.736 | -11.398 |
| 0.2 | -15.563 | -9.542 |
| 0.1 | -20.828 | -7.764 |
This suggests that for a small change in resistance, the bridge has much better PSRR than an equivalent voltage divider. This also translates to noise rejection as well. Bridges are typically used with sensors that have a small change in resistance and can be designed to have optimal components such that the ratios are best suited for the sensor used to optimize PSRR and sensitivity.
DC Offset and Dynamic Range
Another property of the Wheatstone bridge is that it can remove the large DC offset that is inherent in a voltage divider. This allows for better dynamic range and minimizes the need for signal conditioning circuitry, potentially simplifying a design.
Summary
In summary, the Wheatstone bridge is an excellent architecture for sensitive circuits such as instrumentation due to the low dc-offsets which translate to higher dynamic range and high noise rejection capability. Additionally, it allows for multiplying the output voltage using additional sensors to enhance sensitivity even further.