LINEARIZATION OF WHEATSTONE-BRIDGE
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On: 26 Jun, 2019
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Wheatstone bridge circuits have been in the field for a very long time and still are among the first choices for front-end sensors. Whether the bridges are symmetric or asymmetric, balanced or unbalanced, you can accuratelymeasure an unknown imped...
Wheatstone bridge circuits have been in the field for a very long time and still are among the first choices for front-end sensors. Whether the bridges are symmetric or asymmetric, balanced or unbalanced, you can accuratelymeasure an unknown impedance using the bridge. In fact, the simplicity and effectiveness of a bridge circuit makes itvery useful for monitoring temperature, mass, pressure, humidity, light, and other analog properties in industrial andmedical applications.
The Wheatstone bridge has a single impedance-variable element that is inherently nonlinear away from the balancepoint. Bridge circuits are commonly used to detect the temperature of a boiler, chamber, or a process situatedhundreds of feet away from the actual circuit. Usually a sensor element, typically a resistance temperature detector(RTD), thermistor, or thermocouple, is situated in the hot/cold environment to provide information about resistancechange to temperature.
In the following discussion, we will consider this resistance-variable element in a Wheatstone bridge. We willexamine its behavior and explain how to linearize the bridge circuit to optimize performance. Note finally, that whenwe speak generally about "bridges," this article is focused on circuit design for a Wheatstone bridge.
Single Variable-Resistance Wheatstone Bridge
Resistance-variable Wheatstone bridge circuits perform most of the front-end tasks in a design. They useinexpensive, accurate discrete parts. By incorporating an RTD element, the bridge's inherent resistance variationsare kept within the accepted linearity and tolerance limits, depending on the manufacturer of the RTD.
RTD devices have a very detailed data sheet characterizing their behavior with look-up tables and even transferfunction equations down to four or more orders of error compensating terms. To ensure a high-precision system,designers must consider both the inherent nonlinearity of the RTD element and the Wheatstone bridge, then painfullycalibrate the front-end, and linearize the front-end at the microcontroller side. Increasing the order of the equation inthe microcontroller is going to improve the linearity. A typical bridge circuit (Figure 1) detects milliohms of changesin resistance (ΔR).
Assuming that R1 = R2 = R3 = R4 = R in Figure 1, the bridge is balanced with nodes A and B at a constant V/2 (volts) and with adifferential voltage of 0V across VAB. When there is a change in resistance (ΔR) from R3, then the output differential voltage created is:
when R1 = R2 = R3 = R4 = R, the bridge is balanced.For a single variable-resistance element, for R3 = R + ΔR and R1 = R2 = R4 = R
Equation 2 suggests that increasing the constant supply voltage, V, to the bridge will increase the output voltage, i.e., the swing rangeacross the bridge. This also suggests that having a dual supply across the four-legged resistance arrangement could be helpful not onlyto increase the range, but also to help maintain a 0V common-mode voltage across the AB nodes.
The voltage V is usually amplified by using a subsequent amplifying stage typically done with a differential amplifier. There is acaveat, however. Changing the common-mode voltage across V adds more error and complexity in the amplifying second stage,which is usually realized as a instrumentation quality differential amplifier. Therefore, a common-mode voltage centered around 0V ispreferable and certainly easier to manage.Figure 2 illustrates the natural tendency of the bridge's single variable element: inherent nonlinearity in its transfer function
Look closer at the trend line in the figure. The curve's absolute deviation from an ideal straight line or, in fact, the linearity error is about0.62%. This is obtained by comparing the curved trend line with the line of best fit, i.e., the straight line relative to the curve. In this way,we actually quantify the worst-case linearity error for the above curved line. In some cases 0.6% is certainly not acceptable, and thisarticle illustrates a way to achieve better than 0.1% accuracy.
Besides the inherent nonlinearity of the bridge, the designer must also manage the nonlinearity of the temperature sensor element,RTD, or even thermistor as discussed in the prior section. When sensing the differential voltage across the nodes A and B, theinstrumentation amplifier (Figure 3) has a common-mode voltage of V/2. The amplifier is usually a differential amplifier with fourresistors or a three-op-amp instrumentation amplifier integrated in a single package.
When a differential amplifier is used, the nodes A and B are connected to the amplifier's input gain-setting resistors, as shown in Figure3. The choice of the op amp and the input resistors is important as this path directs current away from the bridge, hence affecting theaccuracy.Also the choice of resistors affects the bridge performance, as even 0.1%-tolerant resistors used with the amplifier provide only 60dB ofcommon-mode rejection.
Linearize the Bridge Output Without the Instrumentation Amplifier
From the previous discussion it seems logical to have dual supplies across the resistor bridge to increase the dynamic range, and tohave the sensing nodes centered around the 0V common mode. The advantage of this design is that the transfer function from the nodeB is going to be linear with a change in resistance. The range of output swing from the bridge is doubled versus the output from thecircuit in Figure 1.
The circuit implementation in Figure 4 uses two op amps to replace the more complex instrumentation amplifier. Now the linearized bridge output avoids the unnecessary current paths created by the differential amplifier. This circuit eases the design process comparedto the circuit in Figure 3. The only issue here is having positive and negative supplies to the amplifiers which are providing twice theswing range. The added benefit is improved common-mode rejection performance, as the second amplifier operates comfortably around0V.
From Figure4, node A sees GND as it is the summing node of amplifier 1. Thus, a constant current of is forced through theR1|R3 branch, producing an equal and opposite voltage on the other side of the bridge with -V. When the single variable-resistance R3changes (from R3 to R ±ΔR), then Ix (the change in current due to change in the resistance) flowing through this resistance produces avoltage V ±ΔV. A factor of this ΔV is manifested across node B by the balancing of the resistance bridge (for a balanced bridge, ofcourse), as the current forced through resistor branch R2|R4 is equal to (V+ - (V- + ΔV))/(R3 + R4). Since node B is centered at 0Vcommon mode, the voltage produced across node B is going to be gained by a noninverting amplifier. Furthermore, filtering can be doneon this gain stage to optimize the bandwidth making the noise level acceptable for the application.
At balance, when R1 = R2 = R3 = R4 = R, the voltage at nodes A and B is:
This suggests that the output from the second op amp is inverting in nature.Figure 5 shows the transfer function and its nonlinearity from the Figure 4 implementation.
The absolutedeviation from an ideal straight line, i.e., the linearity error, in Figure 5 is less than 0.02%. Improvement in absolutenonlinearity means that the full-scale error or the relative error is also going to improve.
There are no interacting resistive branches, so precision matching of resistors is not required. The variation of the Rx and Rg will onlyprovide a gain error, which can be calibrated at the same as the RTD device.
The above data suggest that this approach can be a viable implementation for 12-, 14-, 16-, even 18-bit applications. The design issimple and very little calibration is needed by the microcontroller. This circuit has, in fact, been widely used in the field for many years.
To implement the Figure 4, circuit you need a dual-supply voltage for the front-end. This negative supply also needs additional board
space and components, a requirement that often may not be a viable option if this is the only place in the entire system where thenegative supply is needed. Low- offset voltage, low-offset drift, and low-noise performance are additional requirements for a high-precision bridge sensor.
Implementing the Bridge Design with a Dual Op Amp
What if the amplifier used in Figure 4 needs only one power supply? The MAX44267 operates from a single supply and is capable ofoutputting bipolar voltages. Unlike other single-supply amplifiers which need headroom above ground, the MAX44267 provides a true-zero output, making it a great fit for bridge sensors (Figure 6). The MAX44267 integrates charge-pump circuitry that generates thenegative voltage rail in conjunction with external capacitors. This allows the amplifier to operate from a single +4.5V to +15V powersupply, but it is as effective as a normal dual-rail ±4.5V to ±15V amplifier.
The MAX44267can be implemented in the Figure 6 circuit with just one supply voltage (positive supply, V ). The integrated negativeV generator or charge pump generates a negative supply voltage. This architecture provides a good advantage to the designerbecause it eliminates the need for negative supply regulators and reduces board layout space and cost.Figure 7 includesthe MAX6070_A25 voltage referenceto generate a 2.5V reference. A dual op amp (again, the MAX44267) is usedwith the resistance bridge where R1 = R3 = 1kΩ and R2 = R4 = 10kΩ. An additional 1.8kΩ is used in series to reduce the amount ofcurrent flowingthrough the bridge and to reduce the power dissipation. The V(+) node becomes one-third the voltage reference's outputat a balanced condition. This is followed with second-stage amplification with a gain of 11 at the OUTA node.
A Fluke RTD calibrator was used as the temperature-dependent resistance element (as PT1000) in place of R3; a temperaturechange from -50°C to +155°C is evaluated. For the given temperature change using a PT1000, the change in resistance (±R) is about800Ω and an equivalent range of 325mV is effected (see Equation 4). Because amplifier 2 has an internal negative supply, it canaccommodate this swing (-242mV to -83mV) at its input below ground, and it provides an output gain of 11.Figure 7 utilizes a Sallen-Key filter in the second stage to filter the input signal to the required bandwidth (50Hz used in this case). Full-scale error accuracy within ±0.05% is obtained from the bridge output at node B without any calibration or trim. In this way, the transferfunction of the bridge circuit is made linear; improved performance of the front-end circuit is realized using MAX44267 in the subsequentsection.
Test Measurements 1.
Figure 8 shows the absolute bridge voltage output versus change in resistance (a linearity curve output), under 0.02%.
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