Analysis of ratio characteristics related to the device output, the ratio to be measured, and other voltage or current ratios

The ratio characteristic mentioned in this article refers to the ratio between the output of the device and the voltage or current to be measured.
sensor and resistive detection element
The output of many sensors is proportional to their power supply voltage. This is usually because the sensing element that produces the output is a ratiometric device. A common ratio element is a resistor, whose resistance changes with the changes being measured. Resistance temperature detectors (RTD) and strain gauges are typical resistive sensitive components.
The ratio of resistive elements is due to the fact that their impedance cannot be directly measured. Its value is determined by the ratio of the voltage across the resistor to the current through the resistor.
R = V/I Formula 1 (Ohm's theorem)
Sensors that use resistive elements usually pass a current through a resistor and measure its voltage. Before outputting the sensor, the voltage can be amplified or level shifted, but its magnitude is still related to the current flowing through the resistor. If the current comes from the power supply voltage, then the output of the sensor is proportional to the power supply voltage. Equation 2 describes the output of this type of proportional sensor (Figure 1), where Vs is the output signal, Ve is the excitation voltage, S is the sensitivity of the sensor, P is the value of the measured parameter, and C is the offset of the sensor.
Vs = Ve (P x S + C) Formula 2

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Figure 1. Proportional sensor
Honeywell? [1] The MLxxx-C series pressure sensor is a representative device among many automotive proportional sensors. When working at a nominal power supply voltage of 5V, the offset voltage is 0.5V and the full-scale output is 4.5V. If you change the excitation voltage, the offset voltage and full-scale output will change proportionally.
Need to know the excitation voltage to use the output signal, which is very inconvenient in many applications. To solve this problem, the manufacturer added a voltage reference to the circuit. This kind of device can provide very high voltage, and has nothing to do with temperature and power supply voltage. If the current flowing through the sense resistor comes from the reference voltage, then Ve in Equation 2 can be replaced with a constant. This results in Equation 3, where the new constants are contained in S2 and C2.
Vs = P x S2 + C2 Formula 3
Because the output signal is only a function of the measured parameter, formula 3 is not a proportional relationship. Honeywell's MLxxx-R5 series pressure sensors are non-proportional sensors. When working at any supply voltage between 7V and 35V, the offset is 1V and the full-scale output is 6V.
Analog-to-digital converter (ADC) and resistive devices
The ADC used to digitize the sensor signal is also a proportional device. Regardless of its internal architecture, all ADCs work by comparing an unknown input voltage with a known reference voltage. The digitized output of the converter is the ratio of the input voltage to the reference voltage multiplied by the ADC's full-scale reading. Taking into account the diversity of internal amplification and design, a scale factor K is also required. Regardless of the value of K, as long as the ADC configuration has not changed, the value of K remains fixed. Equation 4 describes the relationship between the digital reading (D) and input signal (Vs), reference voltage (Vref), full-scale reading (FS), and scale factor (K) of an ADC (Figure 2) in a general sense.
D = (Vs/Vref) FS x K formula 4

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Figure 2. Analog-to-digital converter in a general sense
The reference voltage is related to the specific design of the ADC. In some ADCs, the reference voltage is the power supply voltage, while in other ADCs, the reference voltage comes from an internal reference source. In other designs, the user must connect the reference voltage to the Vref input of the ADC. If an internal or external voltage reference is used to make the reference voltage a constant value, then Equation 4 can be simplified to Equation 5, where K2 is a new constant whose value is FS x K/Vref.
D = Vs x K2 Formula 5
Sensor measurement
The output of a small system consisting of a non-proportional sensor and an ADC with a fixed reference voltage can be obtained by substituting Vs (input of the ADC) in Equation 3 (the output of the sensor) into Equation 5. As shown in Equation 6.
D = P x S2K2 + C2K2 formula 6
Equation 6 gives the exact relationship required. The magnitude of the digital value (D) is proportional to the change in P and is only affected by the change in P. D is not affected by changes in temperature and power supply voltage.
omit voltage reference
The use of voltage references to stabilize sensors and ADCs is an effective and necessary technology. However, it is not always a technology.
The rest of this article will discuss how to creatively use the ADC's reference voltage input, thereby eliminating the need for voltage references and current sources in many sensor circuits. This design saves component cost, board space, and voltage "headroom." Because the voltage reference is omitted, the error associated with the non-ideal reference no longer exists, so the accuracy is also improved. This technology has been used in the automotive industry for many years. Once the ratio between the sensor and ADC and the power supply voltage is determined, no voltage reference is needed.
The similar technology that uses current-driven sensors and single-element resistive sensors (such as RTD) is no longer commonly used. The sensitivity of the ADC in these circuits varies with temperature or power supply voltage. Nevertheless, the combination of ADC and sensor input is quite stable.
Sensor proportional to power supply voltage
Substituting the input signal (Vs) in formula 2 into formula 4, the ADC output when measuring the proportional sensor can be obtained. Formula 7 is obtained, which says: D is a function of P, Ve and Vref.
D = P (S x FS x K x Ve/Vref) + C (FS x K x Ve/Vref) Formula 7
At first glance, the method in Equation 7 does not seem to be ideal, because the output (D) is a function of three variables, not just a function of P. However, a closer look will reveal that the ratio of Ve/Vref is very important, and the value alone does not have much meaning. If the Ve and Vref voltages come from the same power source, it is easy to get a constant Ve/Vref ratio. Once this is the case, D will be proportional to the change in P and only related to the change in P. Assuming that the ratio of Ve/Vref is a constant, Equation 7 can be simplified to a form similar to Equation 6. Therefore, this means that the same performance can be achieved without a voltage reference.
From the perspective of practical applications, Ve and Vref must be large enough to avoid noise interference; at the same time, Ve and Vref must also be within the range specified by the ADC and the sensor. Using positive power supply voltage as the voltage source of Ve and Vref can usually meet the above requirements, and allows a large number of parallel sensors to be powered, as shown in Figure 3 [2].

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The front end of the MAX1238 in Figure 3 has a 12-input multiplexer and a built-in voltage reference. In this case, although there is no additional cost associated with the ADC benchmark, adding a benchmark to each of the 10 sensors will significantly increase the cost. MAX1238 also allows AN11 input as a reference voltage. Using AN11 as a reference input and connecting it to a 5V power supply, the ADC's full-scale input can be set to 5V, which is convenient for use with a proportional sensor. In Figure 3, the internal reference voltage of the MAX1238 is not idle. The internal voltage reference can be controlled by software and used for diagnosis, such as measuring the power supply voltage. This can be achieved through a voltage divider connected to the input AN10.
Figure 3. MAX1238 ADC allows AN11 input as a reference voltage, so ADC can be used in conjunction with a proportional sensor.
The topology of Figure 3 is very suitable for automotive applications and those applications that are powered by a single power supply and have a small voltage drop on the power supply line. It is not suitable for sensors that have to use long wires in their work or applications where ADC and sensor are powered by different power sources.
current-driven bridge
In a low-noise environment or system, if the pressure sensor is placed close to the ADC, it may not be necessary to use a sensor with signal amplification. In these applications, low-cost bridge output sensors are more suitable. In order to reduce the cost of the sensor, while providing good performance in the entire temperature range, many such pressure sensors, such as Nova Sensor's NPI-19 series [3] are powered by a current source rather than a voltage source. (For a more detailed discussion, please refer to Appendix 1). Equation 8 gives the output of this current-driven sensor, where Ie is the excitation current.

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  Vs= Ie (S x P+C) Formula 8
Figure 4 shows a current source commonly used in bridge output sensors. The current source consists of a low temperature coefficient resistor, an operational amplifier and a voltage reference. If the ADC and the pressure sensor are integrated in one component, the voltage reference of the current source can also provide a reference voltage for the ADC. In the circuit of Figure 4, the voltage reference is used to stabilize the sensor and ADC at the same time, so that they are not affected by changing temperature and power supply voltage.
Figure 4. The current source of the current-driven sensor in this design consists of a resistor, an operational amplifier and a voltage reference.
Another method similar to Fig. 4 is the circuit shown in Fig. 5 without a current source or voltage reference. It should be noted that although the combination of the sensor and the ADC is stable over the entire temperature range, both the ADC and the sensor have a large temperature drift. If measured separately, the sensitivity of the sensor will decrease with increasing temperature, while the sensitivity of the ADC will increase. Since the ADC output is not stable over the entire temperature range, special care must be taken when using this method in circuits with multiple inputs to the ADC.

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Figure 5. Another design method of sensor and ADC combination, without a separate current source or voltage reference.
Formula 9 can be derived from Figure 5:
Vref = Ie x R1 Formula 9
Substituting Vref in Equation 9 and Vs in Equation 8 into Equation 4 of the above ADC, Equation 10 is obtained.
D = [Ie (S x P+C)/(Ie x R1)] (FS x K) Formula 10
Because the excitation current (Ie) is contained in the numerator and denominator, it can be eliminated. Thus, formula 11 can be obtained, which indicates that the output has nothing to do with the excitation current. If the constant terms in formula 11 are combined, the equivalent formula of formula 6 will be obtained again: a system with a voltage reference.
D = P (S x FS x K/R1) + C (FS x K/R1) Formula 11
If R1 is used as a constant, it must have a lower temperature coefficient. Compared with Figure 4, Figure 5 requires R1 to have good temperature stability, which is not its disadvantage, because the resistor in Figure 4 must also have good temperature stability.
There is no R2 in formula 11, and R2 is not needed in the circuit. However, the analysis of R2 is to show that it does not affect ADC readings. R2 can be replaced by another current-driven pressure sensor, RTD, or resistance of a solid-state switch without affecting ADC readings.
Theoretically, a multi-channel input ADC and several current-type sensors driven in series can be used. However, the series connection of sensors will make the excitation current (Ie), sensor signal (Vs) and reference voltage (Vref) lower. When the sensors are connected in series, special attention should be paid to the ADC Vref requirements and system noise.
RTD
RTD is another sensor that is usually used in conjunction with a current source. The commonly used material for RTDs is platinum, which usually has a positive temperature coefficient of about 3,800 ppm/°C. The traditional method of measuring RTD is to use it as a terminal of a resistance bridge. However, in practical applications, resistance bridges are rarely used. The existence of low-cost and high-resolution ADCs makes it more economical to drive a current through the RTD and directly measure the voltage across the RTD. This method avoids the non-linear problem of the unbalanced bridge and eliminates the three precision resistors that make up the resistance bridge.

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The circuit in Figure 6 also does not require a bridge or a stable current source to measure RTD (Rt). This circuit only needs a stable reference resistor (R1) and a low-grade current-limiting resistor.
Figure 6. Circuit without resistance bridge or stable current source to measure Rt
The following formula can be derived from Figure 6:
Vs = (V+) x Rt/(R1+R2+Rt) Formula 12
Vref = (V+) x R1/(R1+R2+Rt) Formula 13
Substituting Vs in Equation 12 and Vref in Equation 13 into Equation 4, the output of the ADC in Figure 6 is obtained. After simplification, formula 14 can be obtained. Equation 14 shows: If R1 is a constant value, D will be proportional to and only change with the change of Rt, which is exactly the desired result.
D = FS x K x (Rt/R1) Formula 14
It can be seen from Equation 14, R2 does not affect the reading; R2 reduces the power consumed by Rt. If there is no R2, Rt's own heat will cause a large error in the temperature indication. R2 also reduces the common-mode input voltage of the ADC. This is very necessary for some ADCs whose common-mode input voltage range is less than the supply voltage.
The ADC similar to the MAX1403 contains a current source for driving the RTD. However, they are not precision current sources, and some calibration is required. Calibration is usually achieved by using an additional ADC input to measure a reference resistance driven by the same current source. Then, the software is used to determine the measured value of the unknown resistance in proportion to the measured value of the known resistance. Although this technique works well, it is simpler to use R1 as a reference resistor and no additional ADC input is required. The current source on the board can still be used to excite the RTD and the reference resistor. Replacing R2 in Figure 6 with a current source will not affect Equation 14.
Some ADCs can provide two matched current sources for measuring remote RTDs. In these applications, the resistance of the long wires will increase the impedance of the RTD, resulting in errors, which must be removed. The cost solution is to use a three-wire RTD. As shown in Figure 7, the current source 1 can be used to generate a voltage drop across the RTD. This current source also creates an additional voltage drop on the upper wire leading to the RTD. In order to compensate for this excess voltage drop, a current source 2 is used to generate a voltage drop on the middle wire. The two current sources flow to ground through the wires at the bottom of the RTD. The length and material of the three wires on the RTD are the same, so that the resistance between each other can be very close. The matched resistance delivers matched current to produce a matched voltage drop. Therefore, the voltage drops of the upper two wires cancel each other out, and the differential input voltage on the ADC is the same as the voltage across the RTD.

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Figure 7. The MAX1403 ADC has two matched current sources. In this circuit, current source 1 is used to generate a voltage drop across the RTD.

Source 2 is used to generate the voltage drop of the intermediate conductor.
temperature and pressure
Figure 8 combines the design concepts in Figure 5 and Figure 6, using a very simple circuit to measure temperature and pressure simultaneously with a single resistor as a reference. The amplitudes of Vs1 and Vs2 are very different. This difference can be adjusted by changing the gain of the built-in programmable gain amplifier (PGA) of the ADC (such as MAX1415). These converters allow the PGA to set different gains for each channel. The change of gain can change the value of K in Equation 4, thus allowing a single reference voltage to adapt to a wider range of input voltages.

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Figure 8. A simple circuit that uses a single resistor as a reference to measure temperature and pressure
Wheatstone Bridge
The 　　 Wheatstone bridge was invented by Sir Charles Wheatstone (1802 to 1875) in the early stages of electronics development. The Wheatstone bridge measures resistance by comparing three known resistance values with an unknown resistance value. When the bridge just reaches equilibrium, the resistance measurement has nothing to do with the excitation voltage, meter accuracy, or meter load in the circuit. In an era when voltage standards and high-quality meters are not yet available, this condition is very important. However, bridge circuits are still very popular today, because when all bridge resistors have the same temperature coefficient, they will not produce a large offset and can suppress the temperature effect.

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Figure 9 is a Wheatstone bridge composed of two voltage dividers powered by the same voltage source. It is customary to draw the bridge as a rhombus, because this shape emphasizes the importance of the same voltage source powering each voltage divider. The output of the bridge (Vo) is the difference between the output voltages of the two voltage dividers (Equation 15). When Vo is zero, the bridge is said to reach equilibrium. Under this condition, because Ve is multiplied by a zero term, the value of the excitation voltage (Ve) is not important. Equation 16 can calculate the resistance value of the unknown resistance (Ru) in the balanced bridge. In practical applications, Ra = Rb is usually used, so that Equation 16 can be simplified to Ru = Rc.

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Figure 9. Schematic diagram of Wheatstone bridge composed of voltage divider powered by the same voltage source
Vo = Vb(Rc/(Rc+Ru)-Rb/(Ra+Rb)) Formula 15
If Vo = 0, then Ru = Rc x Ra/Rb Formula 16
At present, the balanced bridge circuit is rarely used to measure resistance, but it is quite common to use an unbalanced bridge in the sensor. During factory calibration, the bridge is usually balanced at a preferred operating point; the deviation from this point is measured by measuring the imbalance in the bridge. The following examples illustrate the advantages of using the bridge in this way.
Assume that a silicon stress gauge is bonded to a thin film to form a pressure sensor with the desired pressure resolution (0.1%). Under the condition of 0psi and 25°C, the resistance value of the resistor is 5000Ω. Under the conditions of 100psi (full scale pressure) and 25°C, the resistance value increases by 2% to 5100Ω. In addition to being sensitive to stress, resistance is also sensitive to temperature, with a temperature coefficient of resistance (TCR) of 2000ppm/°C.
Because the resistance changes by 100Ω in the entire pressure range, it must be able to distinguish the resistance of 0.1-#937; in order to obtain a pressure resolution of 0.1psi (0.1%). Measuring 0.1Ω in 5000Ω is equivalent to 1/50,000 or 15.6 bits of resolution. A more serious problem than resolution is the effect of temperature changes. Because the resistance has a high TCR, every 1°C change in temperature is equivalent to the influence of a pressure change of 10 psi on the resistance. The influence of temperature change per degree Celsius on resistance is equivalent to 10% of full scale.
Now consider the situation when the same resistance is used in the bridge circuit and the excitation voltage is 2V. The other three resistors are all 5000Ω and have the same TCR as the sense resistor. The installation conditions of these resistors can ensure that they are isothermal. This approach has two significant advantages.
The advantage of the bridge in this application is that it can suppress temperature-induced changes. Analyzing formula 15 finds that TCR is no longer a problem. The output remains unchanged even if the bridge resistance is doubled. As long as all resistances change in the same proportion, the output will not change!
The second advantage of the 　　 electric bridge is that it reduces the resolution requirement. When the pressure is 0psi, the bridge output is 0mV, and when the pressure is 100psi, the bridge output is 10mV. To measure a pressure of 0.1psi, you need to distinguish 10 from 10mV? V. In contrast to the 15.6-bit resolution required for direct resistance measurement, only 10-bit resolution is required.
From the point of view of practical application, can 10-bit ADC not directly measure 10? V’s signal. The signal must be amplified. The cost of signal amplification may make high-resolution ADCs that do not require external amplifiers more attractive. The advantage of the low-resolution scheme lies in its requirements for benchmarks. It is usually impractical to design a voltage reference, current source, or reference resistor that can stably achieve 16-bit resolution over the entire time and temperature range.
The value selection in this example is not used to deliberately highlight the importance of the electric bridge. These values are very typical for many piezoresistive pressure sensors (see Appendix 2).
Linearization of Wheatstone Bridge
The disadvantage of using an unbalanced Wheatstone bridge is its non-linearity. The Ru term in the denominator of Equation 15 indicates that the output of the bridge is not a linear function relationship with Ru. When the resistance change is very small, the linear error is also very small, and the linear error becomes larger when the bridge is unbalanced. Fortunately, if the ADC reference voltage comes from the bridge, this error can be eliminated.
Figure 10 shows a simple temperature sensor with digital display. The temperature sensing element (Rt) is a platinum RTD. Platinum was chosen because its resistance changes linearly with temperature. The bridge circuit removes the excess signal at 0°, so that the ADC reading can be equal to the temperature. Equation 17 gives the bridge signal (Vs) in Figure 10. Equation 18 is the reference voltage of the ADC. Both signals are nonlinear functions of Rt, but the result of their combined action is linear.
How to use the ratio characteristics of the sensor and ADC to improve the accuracy of the electronic system
Figure 10. In a simple temperature sensor with a digital display, the bridge circuit removes the excess signal at 0°, making the ADC reading equal to the temperature.
Vs = (Vb) (R3/(R2+R3)-(R1/(R1+Rt)) Formula 17
Vfer = (Vb)(R1/(R1+Rt) Formula 18
The ADC output (Equation 19) is obtained by substituting Vs and Vref in Equations 17 and 18 into Equation 4. Equation 19 shows that when this reference voltage is used, the ADC output becomes a linear function of Rt, and the expected offset term is subtracted.
D = Rt(R3/(R1(R2+R3))-R2/(R2+R3) Formula 19
In Figure 10, R3b and R1b adjust the offset and sensitivity, respectively. When adjusting, the display will directly display the temperature in °C or °F. One of the obvious errors comes from the nonlinearity of the RTD itself. The error is only a few tenths of a degree Celsius in the range of 0°C to 100°C.
Through the serial interface of MAX1492 ADC, it is also possible to digitally correct the offset error and sensitivity error of the circuit in Figure 10. This calibration method not only eliminates the need for R1a and R3a, but also provides an opportunity to correct linear errors in the RTD. If you need a higher measurement resolution, you can replace the MAX1492 with the MAX1494, which can increase the resolution by one bit.
According to Equation 19, the value of R4 will not affect the reading. Increasing R4 in the circuit can reduce the RTD's own heat. At the same time, the signal from the bridge is weakened and the reference voltage is lowered. Although the MAX1492 has no internal PGA, it allows the use of a smaller reference voltage. Using a smaller reference voltage can save additional amplifying circuits.
concluding remarks
In many sensor applications, a simple circuit is used to maintain an appropriate relationship between the sensor output and the ADC reference input, which can save the voltage reference and current source. In addition to reducing costs and saving space, these circuits can also eliminate errors introduced by non-ideal references and improve performance.