Keywords: Bridge Circuits, Bridges, Wheatstone Bridge, Sigma Delta, Sigma-Delta Converters
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TUTORIALS 3545

Abstract: Bridge circuits are a very useful way to make accurate measurements of resistance and other analog values. This article, the continuation from Part One, covers how to interface bridge circuits implemented with higher signal-output silicon strain gauges to analog-to-digital converters (ADCs). Featured are sigma-delta ADCs, which provide a low-cost way to implement a pressure sensor when utilizing silicon strain gauges.

The most common example of a MEMS device is the silicon pressure sensor, which first became popular in the 1970s. These pressure sensors are fabricated using standard semiconductor processing techniques, plus a special etching step. The special etch selectively removes silicon from the back side of the wafer to create hundreds of thin square diaphragms with strong silicon frames surrounding them. On the front side of the wafer, one strain-sensitive resistor is implanted on each edge of each diaphragm. Metal traces connect the four resistors around an individual diaphragm to create a fully active Wheatstone bridge. A diamond saw is then used to free the individual sensors from the wafer. At this point the sensors are fully functional, but must have pressure ports attached and wires connected before they are useful. These small sensors are inexpensive and relatively robust. There is a negative, however. These sensors suffer from large temperature effects and have a wide tolerance on initial offset and sensitivity.

Where V

V _{OUT}= V_{B}× (P × S_{0}× (1 + S_{1}× (T - T_{0})) + U_{0}+ U_{1}× (T - T_{0}))(Eq. 1)

Equation 1 uses first-order polynomials to model the sensor. For many applications it may be necessary to use higher-order polynomials, piecewise linear techniques, or even piecewise second-order approximations with a lookup table for the coefficients. Regardless of which model is used, digital calibration requires the ability to digitize V

P = (V _{OUT}/V_{B}- U_{0}- U_{1}× (T-T0))/(S0 × (1 + S_{1}× (T-T_{0}))(Eq. 2)

Key considerations in designing such a circuit are the dynamic range and ADC resolution. The minimum requirements will depend on the application and the exact specifications of the sensor and RTD used. For illustrative purposes, the following specifications are used.

System specifications

- Full-scale pressure: 100psi
- Pressure resolution: 0.05psi
- Temperature range: -40°C to +85°C
- Power supply: 4.75 to 5.25V

- S
_{0}(sensitivity): 150 to 300µV/V/psi - S
_{1}(temperature coefficient of sensitivity): -2500ppm/°C, max - U
_{0}(offset): -3 to +3mV/V - U
_{1}(offset temperature coefficient): -15 to +15µV/V/°C - R
_{B}(input resistance): 4.5k - TCR (temperature coefficient of resistance): 1200ppm/°C
- RTD: PT100
- Alpha: 3850ppm/°C (ΔR/°C = 0.385Ω nominal)
- Value at -40°C: 84.27Ω
- Value at 0°C: 100Ω
- Value at 85°C: 132.80Ω
- For more details on the PT100, see Maxim application note 3450, "Positive analog feedback compensates PT100 transducer."

Using Equation 1 and the appropriate assumptions from above:

ΔVTherefore:_{OUT}min = 4.75V (0.05psi/count 150µV/V/psi × (1+ (-2500ppm/°C) × (85°C -25°C)) ≈ 30.3µV/count

Again, using Equation 1 and the appropriate assumptions from above:

VTherefore:_{OUT}max = 5.25V × (100psi · 300µV/V/psi × (1+ (-2500ppm/°C) ×(-40°C - 25°C)) + 3mV/V + (-0.015mV/V/°C) × (-40°C - 25°C)) - 204mV

V_{OUT}min = 5.25 × (-3mV/V + (0.015mV/V/°C × (-40°C - 25°C))) - -21mV

Typical sigma-delta converters operating from a 5V supply will use a 2.5V reference and have an input range of ±2.5V. To meet the resolution requirements of our pressure-sensor application, such an ADC would need a dynamic range of (2.5V -

An alternate approach to an 18-bit ADC uses a lower resolution converter with an internal amplifier, such as the 16-bit MAX1416. Selecting an internal gain of 8 has the effect of shifting the ADC reading 3 bits toward the MSB, thereby using all the converter's bits and reducing the converter requirement to 15 bits. When choosing between a high-resolution converter without gain, and a lower resolution converter with gain, be sure to consider the noise specifications at the applicable gain and conversion rate. The useful resolution of a sigma-delta converter is frequently limited by its noise.

Selecting a maximum voltage for Vt that is close to the maximum pressure signal ensures that the same ADC and internal gain can be used for temperature and pressure measurement. In this example the maximum input voltage is +204mV. To allow for resistor tolerances, the maximum temperature voltage can be conservatively selected as +180mV. Limiting the voltage across Rt to +180mV also eliminates any problems with self-heating of Rt. Once the maximum voltage has been selected, the value of R1 is calculated to provide

Where T

R1 = Rt × (V _{B}/Vtmax - 1)(Eq. 3) R1 = 132.8Ω × (5.25V/0.18V - 1) ≈ 3.7kΩ T _{RES}= V_{RES}× (R1 + Rt)²/(V_{B}× R1 × ΔRt/°C)(Eq. 4)

TA 0.07°C temperature resolution will be adequate for most applications. If, however, higher resolution in needed, several options are available: use a higher resolution ADC; replace the RTD with a thermistor; or utilize the RTD in a bridge circuit so that a higher gain can be used inside the ADC._{RES}= 30µV/count × (3700Ω + 132.8Ω)²/(4.75V Ω 3700Ω × 0.38Ω/°C) ≈ 0.07°C/count

Note that to achieve a useful temperature reading, the software must compensate for any changes in the supply voltage. An alternate approach connects R1 to V

ADCs also have a ratiometric property; their output is directly proportional to the ratio of the input voltage and the reference voltage. Equation 6 describes a generic ADC's digital reading (D) in terms of the input signal (Vs), the reference voltage (V

V _{OUT}= V_{B}× ƒ(p,t)(Eq. 5)

The performance of the ADC can be seen by replacing Vs in Equation 6 with the equivalent of V

D = (Vs/V _{REF})FS × K(Eq. 6)

In Equation 7 the

D = (V _{B}/V_{REF}) × ƒ(p,t) × FS × K(Eq. 7)

Where R

R1 = (R _{B}× V_{RES})/(V_{DD}× TCR × T_{RES}- 2.5 × V_{RES})(Eq. 8)

Continuing with the previous example and assuming a desired temperature resolution of 0.05°C, R1 = (4.5kΩ × 30µV/count)/(((5V × 1200ppm/°C × 0.05°C/count) - 2.5) × 30µV/count) = 0.6kΩ. This result is valid since R1 is less than half of R

As temperature increases, the resistance of the bridge rises causing more voltage to be dropped across it. This change in V

An equation for the ADC's output in the circuit in Figure 4 can be obtained by starting with Equation 7 and replacing V

The circuit shown in Figure 5 can provide the same performance as the circuit in Figure 4, but without using a current source or voltage reference. This can be shown by comparing the output of the two circuits. The output of the ADC in Figure 5 is found by starting with Equation 7 and substituting the appropriate equations for V

D = (I _{B}× R_{B}/V_{REF}) × ƒ(p,t) × FS × K(Eq. 9)

Repeat of Equation 7: D = (V

In the circuit in Figure 5, V

And V

Substituting these into Equation 7 yields Equation 10.

If R1 is selected to equal V

D = (R _{B}/R1) × ƒ(p,t) × FS × K(Eq. 10)

When using the circuit in Figure 5, it is important to remember that the ADC's reference voltage changes with temperature. This makes the ADC unsuitable for monitoring other system voltages. In fact, if a temperature-sensitive measurement is needed for additional compensation, it can be obtained by using an additional ADC channel to measure the supply voltage. Also, when using the circuit in Figure 5, care should also be taken to ensure that V