Fundamentals of Electronic Design: Resistor Bridge Basics (I)

Overview

The Wheatstone bridge was used in the early days of electronics to accurately measure resistance values ​​without the need for an accurate voltage reference or high-resistance meter. In practice, resistance bridges are rarely used for their original purpose, but are instead widely used in sensor detection. This article examines the reasons for the popularity of bridge circuits and discusses some key considerations when measuring bridge output.

Note: This article is divided into two parts. Part I reviews basic bridge architectures and focuses on bridge circuits with low output signals, such as wire or metal foil strain gauges. Part II, “Resistive Bridge Basics (Part II)” introduces high output signal bridges using silicon strain gauges.

Basic bridge configuration

Figure 1 shows a basic Wheatstone bridge, where the bridge output Vo is the differential voltage between Vo+ and Vo-. When using a sensor, the resistance of one or more resistors will change depending on the parameter to be measured. The change in resistance will cause a change in the output voltage. Equation 1 gives the output voltage Vo, which is a function of the excitation voltage and all the bridge resistors.

Figure 1. Basic Wheatstone bridge block diagram.

Formula 1: Vo = Ve (R2/(R1 + R2) - R3/(R3 + R4))

Equation 1 may look complex, but it can be simplified for most bridge applications. When Vo+ and Vo- are equal to 1/2 of Ve, the bridge output is very sensitive to changes in resistance. The above equations can be greatly simplified by using the same nominal value, R, for all four resistors. The change in resistance caused by the measurement is represented by the increment of R, or dR. Resistors with a dR term are called "active" resistors. In the following four cases, all resistors have the same nominal value, R, and one, two, or four resistors are active resistors or resistors with a dR term. When deriving these equations, dR is assumed to be positive. If the actual resistance is reduced, it is represented by -dR. In the following special cases, all active resistors have the same dR value.

Four active components

In the first case, all four bridge resistors are active components. The resistance of R2 and R4 increases as the value to be measured increases, while the resistance of R1 and R3 decreases accordingly. This situation is common in pressure sensing using four strain gauges. When pressure is applied, the physical orientation of the strain gauge determines the increase or decrease in value. Equation 2 shows the relationship between the output voltage (Vo) and the change in resistance (dR) that can be obtained in this configuration, which is a linear relationship. This configuration can provide the maximum output signal. It is worth noting that the output voltage is not only linear with dR, but also with dR/R. This subtle difference is very important because the resistance change of most sensor units is proportional to the volume of the resistor.

Equation 2: Vo = Ve (dR/R) Bridge with four active components

An active component

The second case uses only one active component (Equation 3) and is often used when cost or routing is more important than signal amplitude.

Equation 3: Vo = Ve (dR/(4R+2dR)) Bridge with one active element

As expected, the output signal amplitude of the bridge with one active element is only 1/4 of the output amplitude of the bridge with four active elements. The key to this configuration is the presence of the dR term in the denominator, which results in a nonlinear output. This nonlinearity is small and predictable, and can be calibrated out via software if necessary.

Two active components with opposite response characteristics

The third case, shown in Equation 4, contains two active elements, but with opposite resistance change characteristics (dR and -dR). The two resistors are placed on the same side of the bridge (R1 and R2, or R3 and R4). As expected, the sensitivity is twice that of the single active element bridge and half that of the four active element bridge. In this configuration, the output is a linear function of dR and dR/R, with no dR term in the denominator.

Equation 4: Vo = Ve (dR/(2R)) Two active components with opposite response characteristics

In the second and third cases above, only half of the bridge is in effective operation. The other half simply provides a reference voltage that is half the voltage Ve. Therefore, the four resistors do not actually have to have the same nominal value. What is important is that the two resistors on the left side of the bridge match each other and the two resistors on the right side of the bridge match each other.

Two identical active components

The fourth case also uses two active components, but these two components have the same response characteristics, and their resistance increases or decreases at the same time. To work effectively, these resistors must be located at diagonal positions in the bridge (R1 and R3, or R2 and R4). The obvious advantage of this configuration is that the same type of active component is used in two locations. The disadvantage is that there is a nonlinear output, and the denominator in Equation 5 contains a dR term.

Equation 5: Vo = Ve (dR/(2R+dR) In a voltage driven bridge there are two identical active components.

This nonlinearity is predictable and can be eliminated by software or by driving the bridge with a current source instead of a voltage source. In Equation 6, Ie is the excitation current. It is important to note that Vo in Equation 6 is only a function of dR, not proportional to dR/R as mentioned above.

Equation 6: Vo = Ie (dR/2) In a current-driven bridge there are two identical active components

It is important to understand the structure of the four different sensing element configurations mentioned above. However, many times there may be a bridge with an unknown configuration inside the sensor. In this case, it is not very important to know the specific configuration. The manufacturer will provide relevant information, such as linearity error of sensitivity, common mode voltage, etc. Why is the bridge the preferred solution? This question can be easily answered by the following example.

Load Cell

A common example of a resistor bridge is a load cell with four active elements. Four strain gauges are arranged in a bridge configuration and fixed to a rigid structure that deforms slightly when pressure is applied. When a load is applied, two of the strain gauges increase in value while the other two decrease in value. This change in resistance is small, and the full-scale output of the load cell is 2mV at a 1V excitation voltage. From Equation 2 we can see that this is equivalent to a 0.2% change in full-scale resistance. If the output of the load cell requires 12-bit measurement accuracy, it must be able to accurately detect a 1/2ppm change in resistance. Directly measuring a 1/2ppm change in resistance requires a 21-bit ADC. In addition to requiring a high-precision ADC, the ADC reference must be very stable and must not change by more than 1/2ppm over temperature. These two reasons are the main reasons for driving the use of a bridge structure, but there is an even more important reason for driving the use of a bridge.

The resistance of the load cell responds not only to the applied pressure, but also to the thermal expansion of the device that holds the load cell and the TCR of the gauge material itself. These unpredictable resistance changes may be greater than the resistance change caused by the actual pressure. However, if the unpredictable change is the same for all bridge resistors, their effect can be ignored or eliminated. For example, if the unpredictable change is 200ppm, it is equivalent to 10% of the full scale. In Equation 2, a 200ppm change in resistance R is less than 1 LSB for a 12-bit measurement. In many cases, the change in resistance dR is proportional to the change in R. That is, the ratio dR/R remains unchanged, so a 200ppm change in R will have no effect. The value of R can be doubled, but the output voltage will not be affected because dR will also be doubled.

The above examples show that using a bridge can simplify the measurement of small changes in resistance. The following describes the main considerations for bridge measurement circuits.

Five key factors of bridge circuit

There are many factors to consider when measuring a bridge with low output signals. The five most important factors are:

Excitation voltage

Common mode voltage

Offset voltage

Offset Drift

noise

Excitation voltage

Equation 1 shows that the output of any bridge is directly proportional to its supply voltage. Therefore, the circuit must maintain the bridge supply voltage constant during the measurement (with the same accuracy as the measurement) and must be able to compensate for changes in the supply voltage. The simplest way to compensate for supply voltage changes is to derive the ADC reference voltage from the bridge excitation. In Figure 2, the ADC reference voltage is divided down from the bridge supply. This suppresses supply voltage changes because the ADC voltage resolution changes with the bridge sensitivity.

Figure 2. ADC reference voltage proportional to Ve. Gain error due to Ve variation can be eliminated.

Another approach is to use an additional channel of the ADC to measure the bridge supply voltage and compensate for the bridge voltage variation in software. Equation 7 shows the corrected output voltage (Voc) as a function of the measured output voltage (Vom), the measured excitation voltage (Vem), and the excitation voltage during calibration (Veo).

式7： You = VomVeo/Come

Common mode voltage

One disadvantage of the bridge circuit is that its output is a differential signal and a common-mode voltage equal to half the supply voltage. Usually, the differential signal must be level-shifted to ground reference before entering the ADC. If this step is necessary, it is important to pay attention to the system's common-mode rejection ratio and the effect of changes in Ve on the common-mode voltage. For the above pressure cell example, if an instrumentation amplifier is used to convert the bridge's differential signal to a single-ended signal, the effect of changes in Ve needs to be considered. If the allowable range of Ve change is 2%, the common-mode voltage at the bridge output will change by 1% of Ve. If the common-mode voltage deviation is limited to 1/4 of the accuracy specification, the amplifier's common-mode rejection must be equal to or better than 98.3dB. (20log［0.01Ve/（0.002Ve/（40964））］ = 98.27). Although such a specification is achievable, it is beyond the capabilities of many low-cost or discrete instrumentation amplifiers.

Offset voltage

The offset voltages of the bridge and the device being measured can pull the actual signal higher or lower. Calibration of these drifts is easy as long as the signal remains in the valid measurement range. If the bridge differential signal is converted to a ground referenced signal, the offsets of the bridge and amplifier can easily produce an output below ground potential. When this happens, a dead spot is created. The ADC output remains at zero potential until the bridge output becomes positive enough to cancel the negative offset voltage of the system. To prevent this from happening, a positive bias must be provided internally in the circuit. This bias voltage ensures that the output is in the valid range even when the bridge and device have negative offset voltages. One problem with the offset is reduced dynamic range. If this disadvantage is not acceptable for the system, higher quality components or offset adjustment may be required. Offset adjustment can be achieved with a mechanical potentiometer, a digital potentiometer, or an external resistor connected to the ADC's GPIO.

Offset Drift

Offset drift and noise are important issues that need to be addressed in bridge circuits. In the above load cell, the full-scale output of the bridge is 2mV/V, and the required accuracy is 12 bits. If the supply voltage of the load cell is 5V, the full-scale output is 10mV, and the measurement accuracy must be 2.5&m ic ro;V or better. In short, an offset drift of only 2.5µV will cause 1 LSB error in a 12-bit converter. For traditional op amps, achieving this indicator is very challenging. For example, the OP07 has a maximum offset TC of 1.3µV/°C and a maximum long-term drift of 1.5µV per month. In order to maintain the low offset drift required for the bridge, some effective offset adjustment is required. Adjustment can be achieved through hardware, software, or a combination of both.

Hardware offset adjustment: Chopper-stabilized or auto-zero amplifiers are purely hardware solutions. They are special circuits integrated into the amplifier that continuously sample and adjust the inputs to keep the voltage difference between the input pins at a minimum. Because these adjustments are continuous, drift over time and temperature becomes a function of the calibration circuit and not the actual drift of the amplifier. Typical offset drift for the MAX4238 and MAX4239 is 10nV/°C and 50nV/1000 hours.

Software Offset Adjustment: Zero calibration or tare measurement are examples of software offset calibration. The output of the bridge is measured at one state of the bridge, such as without a load, and then a load is added to the load cell and the value is read again. The difference between the two readings is related to the excitation source, and taking the difference between the two readings not only eliminates the offset of the device, but also eliminates the offset of the bridge. This is a very effective measurement method, but it can only be used if the actual result is based on the change in the output of the bridge. If the absolute value of the bridge output needs to be read, this method will not work.

Hardware/Software Offset Adjustment: Adding a two-pole analog switch to the circuit allows software calibration to be used in the application. In Figure 3, the switch is used to disconnect one side of the bridge from the amplifier and short the amplifier inputs. Leaving the other side of the bridge connected to the amplifier inputs maintains the common-mode input voltage, thereby eliminating errors caused by changes in common-mode voltage. Shorting the amplifier inputs allows the offset of the system to be measured, and the system offset can be subtracted from subsequent readings to eliminate all device offsets. However, this method does not eliminate the offset of the bridge.

Figure 3. Adding a switch to implement software calibration

This auto-zero calibration is widely used in current ADCs and is particularly effective in removing ADC offsets. However, it does not remove the offset of the bridge or any circuitry between the bridge and the ADC.

A slightly more complex form of the offset correction circuit is to add a double-pole, double-throw switch between the bridge and the circuit (Figure 4). Switching the switch from point A to point B reverses the polarity of the bridge connection to the amplifier. If the ADC reading when the switch is at point A is subtracted from the ADC reading when the switch is at point B, the result is 2VoGain, which is the absence of the offset term. This approach not only eliminates the offset of the circuit, but also improves the signal-to-noise ratio by a factor of two.

Figure 4. Adding a double-pole, double-throw switch enhances software calibration.

AC bridge excitation: This method is not often used, but in traditional designs, AC excitation of a resistor bridge is a common and effective method to eliminate DC offset errors in circuits. If the bridge is driven by an AC voltage, the output of the bridge will be an AC signal. This signal is capacitively coupled, amplified, and biased, and the AC amplitude of the final signal is independent of any DC offset of the circuit. The amplitude of the AC signal can be obtained by standard AC measurement techniques. When using AC excitation, the measurement can be completed by reducing the common-mode voltage change of the bridge, which greatly reduces the circuit's requirements for common-mode rejection.

noise

As mentioned above, noise is a big problem when dealing with small signal output bridges. In addition, the low frequency nature of many bridge applications means that "flicker" or 1/F noise must be considered. A detailed discussion of noise is beyond the scope of this article, and there are many articles on this topic. This article will focus on four noise sources that need to be considered in the design.

Keep noise out of the system (good grounding, shielding, and wiring techniques)

Reducing internal system noise (architecture, component selection, and bias levels)

Reducing electrical noise (analog filtering, common-mode rejection)

Software compensation or DSP (using multiple measurements to increase the valid signal and reduce the interference signal)

The high-precision Σ-Δ converters developed in recent years have greatly simplified the work of digitizing bridge signals. The following will introduce the effective measures these converters take to solve the above five problems.

High-precision Sigma-Delta Converter (ADC)

Today, 24-bit and 16-bit Σ-Δ ADCs with low-noise PGAs offer a perfect solution for resistance bridge measurements in low-speed applications, solving the main problem of quantizing the analog output of the bridge (see above discussion, Figure 2 and following).

The Ve buffered reference voltage input simplifies the construction of a ratiometric system. To obtain a reference voltage that follows Ve, only a resistor divider and noise filter capacitor are required (see Figure 2). In a ratiometric system, the output is insensitive to small changes in Ve, and a high-precision voltage reference is not required.

If a ratiometric system is not used, a multi-channel ADC can be selected. One ADC channel is used to measure the bridge output, and another input channel is used to measure the bridge excitation voltage. The change of Ve can be calibrated using Equation 7.

Common mode voltage

If the bridge and ADC are powered from the same supply, the bridge output signal will be a differential signal biased at 1/2VDD. These inputs are ideal for most high-precision Σ-Δ converters. Also, due to their extremely high common-mode rejection (greater than 100dB), there is no need to worry about small common-mode voltage changes.

Offset voltage

When the voltage accuracy is in the sub-microvolt range, the bridge output can be directly connected to the ADC input. Assuming there is no thermal coupling effect, the only source of offset error is the ADC itself. To reduce the offset error, most converters have internal switches that can be used to apply a zero voltage to the input and measure it. Subtracting this zero voltage measurement from the subsequent bridge measurement removes the ADC offset. Many ADCs can automatically perform this zero calibration process, otherwise, the user needs to control the ADC offset calibration. Offset calibration can reduce the offset error to the noise floor of the ADC, less than 1µVP-P.

Offset Drift

对ADC进行连续地或频繁地校准，使校准间隔中温度不会有显著改变，即可有效消除由于温度变化或长期漂移产生的失调变化。需要注意的，失调读数的变化可能等于ADC的噪声峰值。如果目的是检测电桥输出在较短时间内的微小变化，最好关闭自动校准功能，因为这会减少一个噪声源。

noise

处理噪声有三种方法，比较显著的方法是内部数字滤波器。这个滤波器可以消除高频噪声的影响，还可以抑制电源的低频噪声，电源抑制比的典型值可以达到100dB以上。降低噪声的第二种方法依赖于高共模抑制比，典型值高于100dB。高共模抑制比可以减小电桥引线产生的噪声，并降低电桥激励电压的噪声影响。最后，连续的零校准能够降低校准更新频率以下的闪烁噪声或1/F噪声。

Practical tips

Connecting the output of the bridge directly to the input of a high-precision Σ-Δ ADC is not the answer to all problems. In some applications, a matched signal conditioner is needed between the bridge output and the ADC input. The signal conditioner performs three main tasks: amplification, level shifting, and differential-to-single-ended conversion. A good instrumentation amplifier can perform all three functions, but it can be expensive and may lack a measure of offset drift. The following circuit can provide effective signal conditioning at a lower cost than an instrumentation amplifier.

Single Operational Amplifier

If only amplification is required, the simple circuit shown in Figure 5 will suffice. This circuit may not seem like the best choice because it is asymmetrical and adds a load to the bridge. However, this load is not a problem for the bridge (although it is not encouraged). Many bridges have low-impedance outputs, usually 350Ω. Each output resistor is half that, or 150Ω. The 150Ω resistor only slightly reduces the gain by adding resistor R1. Of course, considering the tolerance of the 150Ω resistor and the temperature coefficient of resistance (TCR), the TCR of resistors R1 and R2 cannot be exactly matched. Compensating for this extra resistance is simple, just choose R1 to be much higher than 150Ω. Figure 5 includes a switch for zero calibration.

Figure 5. Example of connecting a low-impedance bridge.

Differential and Instrumentation

For many applications, the instrumentation amplifier can be replaced with a differential amplifier. This not only reduces cost, but also reduces sources of noise and offset drift. For the above amplifiers, the bridge resistance and TRC must be considered.

Dual power supply

The circuit structure of Figure 6 is very simple. The bridge output uses only two operational amplifiers and two resistors to complete amplification, level conversion, and output a ground-referenced signal. In addition, the circuit doubles the bridge power supply voltage, which doubles the output signal. However, the disadvantage of this circuit is that it requires a negative power supply and has a certain degree of nonlinearity when an active bridge is used. If only one side of the bridge uses active components, placing the non-active side of the bridge in the feedback loop can generate -Ve, thereby avoiding linear errors.

Figure 6. Alternative circuit for connecting to a low-impedance bridge.

Summarize

Resistor bridges are very effective at detecting small changes in resistance and rejecting changes in resistance caused by interfering sources. New analog-to-digital converters (ADCs) have greatly simplified bridge measurements. Adding one of these ADCs provides the key features of a bridge detection ADC: differential inputs, built-in amplifiers, automatic zero calibration, high common-mode rejection ratio, and digital noise filters, helping to solve key issues in bridge circuits.