SENSOR TRANSDUCER DETECTOR: nonlinearity

Sunday, November 14, 2010

Typically a sensor cannot be directly connected to the instruments that record, monitor, or process its signal, because the signal may be incompatible or may be too weak and/or noisy. The signal must be conditioned—i.e., cleaned up, ampliﬁed, and put into a compatible format.

Resistive elements are some of the most common sensors. They are inexpensive to manufacture and relatively easy to interface with signal conditioning circuits. Resistive elements can be made sensitive to temperature, strain (by pressure or by ﬂex), and light. Using these basic elements, many complex physical phenomena can be measured, such as ﬂuid or mass ﬂow (by sensing the temperature difference between two calibrated resistances) and dew-point humidity (by measuring two different temperature points), etc. Bridge circuits are often incorporated into force, pressure and acceleration sensors.

Sensor elements’ resistances can range from less than 100 Ω to several hundred kΩ, depending on the sensor design and the physical environment to be measured. For example, RTDs (resistance temperature devices) are typically 100 Ω or 1000 Ω. Thermistors are typically 3500 Ω or higher.

Bridge Circuits

Resistive sensors such as RTDs and strain gages produce small percentage changes in resistance in response to a change in a physical variable such as temperature or force. Platinum RTDs have a temperature coefﬁcient of about 0.385%/°C. Thus, in order to accurately resolve temperature to 1°C, the measurement accuracy must be much better than 0.385 Ω, for a 100 Ω RTD.

Strain gages present a signiﬁcant measurement challenge because the typical change in resistance over the entire operating range of a strain gage may be less than 1% of the nominal resistance value. Accurately measuring small resistance changes is therefore critical when applying resistive sensors.

Bridges offer an attractive alternative for measuring small resistance changes accurately. The basic Wheatstone bridge (actually developed by S. H. Christie in 1833) is shown in Figure 1. It consists of four resistors connected to form a quadrilateral, a source of excitation (voltage or current) connected across one of the diagonals, and a voltage detector connected across the other diagonal. The detector measures the difference between the outputs of two voltage dividers connected across the excitation.

Fig 1. The Wheatstone bridge

A bridge measures resistance indirectly by comparison with a similar resistance. The two principal ways of operating a bridge are as a null detector or as a device that reads a difference directly as voltage.

When R1/R4 = R2/R3, the resistance bridge is at a null, regardless of the mode of excitation (current or voltage, AC or DC), the magnitude of excitation, the mode of readout (current or voltage), or the impedance of the detector. Therefore, if the ratio of R2/R3 is ﬁxed at K, a null is achieved when R1 = K·R4. If R1 is unknown and R4 is an accurately determined variable resistance, the magnitude of R1 can be found by adjusting R4 until null is achieved. Conversely, in sensor-type measurements, R4 may be a ﬁxed reference, and a null occurs when the magnitude of the external variable (strain, temperature, etc.) is such that R1 = K·R4.

Null measurements are principally used in feedback systems involving electrome-chanical and/or human elements. Such systems seek to force the active element (strain gage, RTD, thermistor, etc.) to balance the bridge by inﬂuencing the parameter being measured. For the majority of sensor applications employing bridges, however, the deviation of one or more resistors in a bridge from an initial value is measured as an indication of the magnitude (or a change) in the measured variable. In this case, the output voltage change is an indication of the resistance change. Because very small resistance changes are common, the output voltage change may be as small as tens of millivolts, even with VB = 10 V (a typical excitation voltage for a load cell application).

Constant Voltage Excitation Bridge

In many bridge applications, there may be two, or even four, elements that vary. Figure 2 shows four commonly used bridges suitable for sensor applications and the corresponding equations which relate the bridge output voltage to the excitation voltage and the bridge resistance values. In this case, we assume a constant voltage drive, VB. Note that since the bridge output is directly proportional to VB, the measurement accuracy can be no better than that of the accuracy of the excitation. .

Fig 2. Output voltage and linearity error
voltage for constant voltage drive bridge conﬁgurations

In each case, the value of the ﬁxed bridge resistor, R, is chosen to be equal to the nominal value of the variable resistor(s). The deviation of the variable resistor(s) about the nominal value is proportional to the quantity being measured, such as strain (in the case of a strain gage) or temperature (in the case of an RTD).

The sensitivity of a bridge is the ratio of the maximum expected change in the output voltage to the excitation voltage. For instance, if VB = 10 V, and the full-scale bridge output is 10 mV, then the sensitivity is 1 mV/V.

The single-element varying bridge is most suited for temperature sensing using RTDs or thermistors. This conﬁguration is also used with a single resistive strain gage. All the resistances are nominally equal, but one of them (the sensor) is variable by an amount ∆R. As the equation indicates, the relationship between the bridge output and ∆R is not linear.

For example, if R = 100 Ω, and ∆R = 0.152, (0.1% change in resistance), the output of the bridge is 2.49875 mV for VB = 10 V. The error is 2.50000 mV – 2.49875 mV, or 0.00125 mV. Converting this to a percent of full scale by dividing by 2.5 mV yields an end-point linearity error in percent of approximately 0.05%. (Bridge end-point linearity error is calculated as the worst error in % FS from a straight line which connects the origin and the end point at FS, i.e. the FS gain error is not included).

If ∆R = 1 Ω (1% change in resistance), the output of the bridge is 24.8756 mV, representing an end-point linearity error of approximately 0.5%. The end-point linearity error of the single-element bridge can be expressed in equation form:

Single-Element Varying Bridge End-Point Linearity Error ≈ % Change in Resistance ÷ 2

It should be noted that the above nonlinearity refers to the nonlinearity of the bridge itself and not the sensor. In practice, most sensors exhibit a certain amount of their own nonlinearity which must be accounted for in the ﬁnal measurement.

In some applications, the bridge nonlinearity may be acceptable, but there are various methods available to linearize bridges. Since there is a ﬁxed relationship between the bridge resistance change and its output (shown in the equations), software can be used to remove the linearity error in digital systems. Circuit techniques can also be used to linearize the bridge output directly, and these will be discussed shortly.

There are two possibilities to consider in the case of the two-element varying bridge. In the ﬁrst, Case (1), both elements change in the same direction, such as two identical strain gages mounted adjacent to each other with their axes in parallel. The nonlinearity is the same as that of the single-element varying bridge, however the gain is twice that of the single-element varying bridge. The two-element varying bridge is commonly found in pressure sensors and ﬂow meter systems.

A second conﬁguration of the two-element varying bridge, Case (2), requires two identical elements that vary in opposite directions. This could correspond to two identical strain gages: one mounted on top of a ﬂexing surface, and one on the bottom. Note that this conﬁguration is linear, and like two-element Case (1), has twice the gain of the single-element conﬁguration.

Another way to view this conﬁguration is to consider the terms R + ∆R and R – ∆R as comprising the two sections of a centertapped potentiometer. The all-element varying bridge produces the most signal for a given resistance change and is inherently linear. It is an industry-standard conﬁguration for load cells which are constructed from four identical strain gages.

Constant Current Excitation Bridge

Bridges may also be driven from constant current sources as shown in Figure 3. Current drive, although not as popular as voltage drive, has an advantage when the bridge is located remotely from the source of excitation because the wiring resistance does not introduce errors in the measurement. Note also that with constant current excitation, all conﬁgurations are linear with the exception of the single-element varying case.

Figure 3. Output voltage and linearity error

for constant current drive bridge conﬁgurations

In summary, there are many design issues relating to bridge circuits. After selecting the basic conﬁguration, the excitation method must be determined. The value of the excitation voltage or current must ﬁrst be determined. Recall that the full scale bridge output is directly proportional to the excitation voltage (or current).

Typical bridge sensitivities are 1 mV/V to 10 mV/V. Although large excitation voltages yield proportionally larger full scale output voltages, they also result in higher power dissipation and the possibility of sensor resistor self-heating errors. On the other hand, low values of excitation voltage require more gain in the conditioning circuits and increase the sensitivity to noise.

Regardless of its value, the stability of the excitation voltage or current directly affects the overall accuracy of the bridge output. Stable references and/or ratiometric techniques are required to maintain desired accuracy.

Bridge considerations

Selecting Configuration (1,2,4 - Element Varying)
Selection of Voltage or Current Excitation
Stability of Excitation Voltage or Current
Bridge Sensitivity: FS Output / Excitation Voltage, 1mV / V to 10mV / V Typical
Fulscale Bridge Outputs: 10mV - 100mV Typical
Precission Low Noise Amplification / Conditioning Techniques Required
Linearization Techniques May Be Required
Remote Sensors Present Challenges.

Handbook Of Sensor, Transducer, Actuator,MEMS, Nano Technology, etc. : Sensor & Transducer Store

Saturday, January 9, 2010

Sensor Examples:

7. Hysteresis

A hysteresis error is a deviation of the sensor’s output at a specified point of the input signal when it is approached from the opposite directions (Fig. 4). For example,a displacement sensor when the object moves from left to right at a certain point produces a voltage which differs by 20 mV from that when the object moves from right to left. If the sensitivity of the sensor is 10 mV/mm, the hysteresis error in terms of displacement units is 2 mm. Typical causes for hysteresis are friction and structural changes in the materials.

8. Nonlinearity

Nonlinearity error is specified for sensors whose transfer function may be approximated by a straight line [Eq. (1)]. A nonlinearity is a maximum deviation (L) of a real transfer function from the approximation straight line. The term “linearity” actually means “nonlinearity.” When more than one calibration run is made, the worst linearity seen during any one calibration cycle should be stated.

Usually, it is specified either in percent of span or in terms of measured value (e.g, in kPa or °C). “Linearity,” when not accompanied by a statement explaining what sort of straight line it is referring to, is meaningless. There are several ways to specify a nonlinearity, depending how the line is superimposed on the transfer function. One way is to use terminal points (Fig.5A); that is, to determine output values at the smallest and highest stimulus values and to draw a straight line through these two points (line 1). Here, near the terminal points, the nonlinearity error is the smallest and it is higher somewhere in between.

Another way to define the approximation line is to use a method of least squares (line 2 in Fig. 5A). This can be done in the following manner. Measure several (n) output values S at input values s over a substantially broad range, preferably over an entire full scale. Use the following formulas for linear regression to determine intercept a and slope b of the best-fit straight line:

a = (ΣSΣs² - ΣsΣsS) / (nΣs² - (Σs)² , b = (nsS-ΣsΣS)/(nΣs² - (Σs)² (16)
where Σ is the summation of n numbers.

In some applications, a higher accuracy may be desirable in a particular narrower section of the input range. For instance, a medical thermometer should have the best accuracy in a fever definition region which is between 37°C and 38°C. It may have a somewhat lower accuracy beyond these limits.

Usually, such a sensor is calibrated in the region where the highest accuracy is desirable. Then, the approximation line may be drawn through the calibration point c (line 3 in Fig. 2.5A). As a result, nonlinearity has the smallest value near the calibration point and it increases toward the ends of the span. In this method, the line is often determined as tangent to the transfer function in point c. If the actual transfer function is known, the slope of the line can be found from Eq. (5).

Independent linearity is referred to as the so-called “best straight line” (Fig. 5B), which is a line midway between two parallel straight lines closest together and enveloping all output values on a real transfer function. Depending on the specification method, approximation lines may have different intercepts and slopes. Therefore, nonlinearity measures may differ quite substantially from one another.Auser should be aware that manufacturers often publish the smallest possible number to specify nonlinearity, without defining what method was used.

9. Saturation

Every sensor has its operating limits. Even if it is considered linear, at some levels of the input stimuli, its output signal no longer will be responsive. A further increase in stimulus does not produce a desirable output. It is said that the sensor exhibits a span-end nonlinearity or saturation (Fig. 6).

10. Repeatability

A repeatability ( reproducibility) error is caused by the inability of a sensor to represent the same value under identical conditions. It is expressed as the maximum difference between output readings as determined by two calibrating cycles (Fig. 2.7A), unless otherwise specified. It is usually represented as % of FS:

δr = Δ / FS × 100% (17)

Possible sources of the repeatability error may be thermal noise, buildup charge, material plasticity, and so forth.

11. Dead Band

The dead band is the insensitivity of a sensor in a specific range of input signals (Fig. 7B). In that range, the output may remain near a certain value (often zero) over an entire dead-band zone.

12. Band Resolution

Resolution describes the smallest increments of stimulus which can be sensed. When a stimulus continuously varies over the range, the output signals of some sensors will not be perfectly smooth, even under the no-noise conditions. The output may change in small steps. This is typical for potentiometric transducers, occupancy infrared detectors with grid masks, and other sensors where the output signal change is enabled only upon a certain degree of stimulus variation. In addition, any signal converted into a digital format is broken into small steps, where a number is assigned to each step.

The magnitude of the input variation which results in the output smallest step is specified as resolution under specified conditions (if any). For instance, for the occupancy detector, the resolution may be specified as follows: “resolution—minimum equidistant displacement of the object for 20 cm at 5 m distance.” For wire-wound potentiometric angular sensors, resolution may be specified as “a minimum angle of 0.5°.” Sometimes, it may be specified as percent of full scale (FS). For instance, for the angular sensor having 270° FS, the 0.5° resolution may be specified as 0.181% of FS. It should be noted that the step size may vary over the range, hence, the resolution may be specified as typical, average, or “worst.” The resolution of digital output format sensors is given by the number of bits in the data word. For instance, the resolution may be specified as “8-bit resolution.” To make sense, this statement must be accomplished with either the FS value or the value of LSB (least significant bit). When there are no measurable steps in the output signal, it is said that the sensor has continuous or infinitesimal resolution (sometimes erroneously referred to as “infinite resolution”).

13. Special Properties

Special input properties may be needed to specify for some sensors. For instance, light detectors are sensitive within a limited optical bandwidth. Therefore, it is appropriate to specify a spectral response for them.

14. Output Impedance

The output impedance Zout is important to know to better interface a sensor with the electronic circuit. This impedance is connected either in parallel with the input impedance Zin of the circuit (voltage connection) or in series (current connection).

Figure 8 shows these two connections. The output and input impedances generally should be represented in a complex form, as they may include active and reactive components. To minimize the output signal distortions, a current generating sensor (B) should have an output impedance as high as possible and the circuit’s input impedance should be low.

For the voltage connection (A), a sensor is preferable with lower Zout and the circuit should have Zin as high as practical.

15. Excitation

Excitation is the electrical signal needed for the active sensor operation. Excitation is specified as a range of voltage and/or current. For some sensors, the frequency of the excitation signal and its stability must also be specified. Variations in the excitation may alter the sensor transfer function and cause output errors.

An example of excitation signal specification is as follows:
Maximum current through a thermistor in still air 50 µA in water 200 µA