Instrumentation and Measurements Dr. Mohammad Kilani

Instrumentation and Measurements Dr. Mohammad Kilani
Analog Electrical Devices and Voltage Dividing Circuits

Electric Signal in Measuring Instruments
Electric signal measurement and conditioning. Common components in a large variety of Instruments Variable Conversion Element Signal Processor Sensor Measured Variable Signal Transmission Presentation / Recording Unit Output Use of Measurement at Remote Location

Advantages of Electrical signal forms over Non-Electrical Forms
Many detecting elements provide an output in the form of varying resistance, capacitance or inductance, voltage or current. Compared to mechanical, pneumatic or hydraulic forms, electric signal forms provide the following advantages:: Fast transient response. Inertia effect are negligible compared with hydraulic and pneumatic systems. Light weight and low power consumption in the measuring instrument Possibility of obtaining large amplification for the signal, with good linearity. Lower hysteresis effect. Hysteresis is present in mechanical devices due to friction, backlash, and windup effects. Availability a multitude of proven analog and digital signal conditioning circuits for filtering, compensation, error-rejection, recording and other signal processing operations.

Galvanometer Instruments

Galvanometer Instruments
A galvanometer is an analog device that produces a deflection proportional to the current in passing in its coil. It utilizes the principle that a current-carrying conductor placed in a magnetic field is acted upon by a force that is proportional to the current passing through the conductor This force can be used as a measure of the flow of current in a conductor by moving a pointer on a display.

Galvanometer Instruments: Force on a straight current conductor placed on magnetic field
Consider a straight length l of a conductor through which a current I flows. The force on this conductor due to a magnetic field strength B is: The vector k is a unit vector along the direction of the current flow, and the force F and the magnetic field B are also vector quantities.

Galvanometer Instruments: Force on a Current Loop in a Magnetic Field
A current loop in a magnetic field experiences a torque if the loop is not aligned with the magnetic field. The torque is given by where N Number of turns in the loop I: current A: cross-sectional area defined by the perimeter of the current loop B: Magnitude of magnetic field strength α angle between the normal cross-sectional area of the current loop and the magnetic field

Galvanometer Instruments: Analog DC Current Measurement
One approach to utilizing the previous equation in current measurement is in the D’Arsonval arrangement. In this arrangement, the uniform radial magnetic field and torsional spring result in a steady angular deflection of the coil that corresponds to the coil current. The coil and fixed permanent magnet are arranged in the normal direction to the current loop. The problem with this arrangement is that the deflection is nonlinear with the current. (torque proportional to sin α) The torque becomes zero when α = 0 irrespective of the amount of current in the coil.

To produce a deflection proportional to the current, the coil is usually wrapped around cylindrical iron core with the two poles form circular arcs around the core. The presence of the iron core causes the magnetic field lines to direct themselves radially toward the center of the core. As the field lines act in the radial direction, they always act normal to the inwards and outwards current paths irrespective over the range of motion. The angle α is always 90⁰ and where r radius of the cylindrical iron core l: length of the cylindrical core

In a typical ammeter the deflection of the pointer indicates the magnitude of the current flow. The range of current that can be measured is determined by selection of the combination of shunt resistor and the internal resistance of the meter movement. The shunt resistor provides a bypass for current flow, reducing the current that flows through the movement. A break switch prevents current overload through the meter coil.

A galvanometer is a highly sensitive D’Arsonval movement device calibrated about zero current; the indicator can deflect to the plus or minus direction. The pointer can be used to adjust a circuit to a zero current state. This mode of operation is called the null mode. The highest sensitivity of commercial galvanometers is about 0.1 µA/division.

Galvanometer Instruments: Analog DC Voltage and Resistance Measurement
DC voltage can be measured through a D’Arsonval configuration placed in series with a resistor. The arrangement is basically a current measuring device, but can be calibrated in terms of voltage by using an appropriate known fixed resistor and through Ohm’s law relating it to the measured current. The same device can be used to measure resistance if instead if an appropriate fixed known DC voltage is applied at the input. Pointer movement can, in this case be calibrated to reflect resistance.

Practical analog ohmmeters use circuits employing a D’Arsonval mechanism with shunt resistors for measuring a wide range of resistance while limiting the flow of current through the meter movement. The lower limit on the measured resistance is determined by the upper limit of current flow through it. A practical limit to the maximum current flow through a resistance is imposed by the ability of the resistance element to dissipate the power generated by the flow of current (Joule heating). At a too high current, the sensor resistance may melt.

This basic circuit is employed in the construction of analog voltage dials and volt-ohmmeters (VOMs), which were commonly used for the measurement of current, voltage, and resistance.

Galvanometer Instruments: Analog AC Current and Voltage Measurement
Deflection meters can be employed for measuring AC current by the uses of diodes to form a rectifier that converts the AC current into a DC current. This current then can be measured with a calibrated D’Arsonval movement meter. An electrodynamometer is basically a D’Arsonval movement modified for use with AC current by replacing the permanent magnet with an electromagnet in series with the current coil. These AC meters have upper limits on the frequency of the alternating current that they can effectively measure; most common instruments are calibrated for use with standard line frequency.

Galvanometer Instruments: Analog AC Current and Voltage Measurement
The waveform output of the rectifier must be considered if a steady meter reading is to be obtained. An AC meter indicates a true rms value for a simple periodic signal only, but a true rms AC voltmeter performs the signal integration required to accurately determine the rms value in a signal-conditioning stage and indicates true signal rms regardless of waveform

Galvanometer Instruments: Errors in D’Arsonval Movement
Errors in the D’Arsonval movement include hysteresis and repeatability errors due to mechanical friction in the pointer-bearing movement, and linearity errors in the spring that provides the restoring force for equilibrium. Also, in developing a torque, the D’Arsonval movement must extract energy from the current flowing through it. This draining of energy from the signal being measured changes the measured signal. Such an effect is called a loading error. This is a consequence of all instruments that operate in deflection mode.

Dynamic Response of a Galvanometer
A galvanometer consists of N turns of a conductor wound about a core of length l and radius r that is situated perpendicular to a magnetic field of uniform flux density B. A DC current passes through the conductor due to an applied potential, Ei(t). The output of the device is the rotation of the core and pointer, u. Develop a lumped parameter model relating pointer rotation and current.

Applying Newton’s 2nd law on the iron core we have The rotation of the current carrying coil in the magnetic field generates an opposing electromotive force Em, given by: Applying Kirchhoff’s law to the resulting circuit gives,

The equations are coupled. The current due to potential E produces torque T that moves the pointer. This motion develops an opposing potential Em.

Using Laplace

HW 2: Analyze the dynamic response of a galvanometer to step input and harmonic input. Discuss the effect of system parameters on the stability, settling time, overshoot and other pertinent dynamic characteristics. Use system analysis tools including: 1. Routh’s stability criterion 2. Root-locus plots 3. Bode diagrams 4. Polar plots 5. Log-Magnitude-vs.-Phase plots 6. Nyquist stability analysis Verify the results of your analysis with computer simulations (e.g. simulink), or video recorded experiments.

The Oscilloscope

The Oscilloscope The oscilloscope is a practical graphical display device which provides an analog representation of a measured signal. It is used to measure and to visually display voltage magnitude versus time for dynamic signals over a wide range of frequencies with a signal frequency extending into the megahertz or gigahertz range. The oscilloscope provides a visual output of signal magnitude, frequency, distortion, and a delineation of the DC and AC components. The visual image provides a direct means to detect the superposition of noise and interference on a measured signal, something non visual metering devices cannot do.

The Oscilloscope In addition to signal versus time, a typical unit can also display two or more signals X(t) and Y(t), perform addition and subtraction of signals, and display amplitude versus amplitude (XY) plots and other features. Some digital oscilloscopes have significant internal storage so as to mimic a data-logging signal recorder. Others have internal fast Fourier transform (FFT) circuitry to provide for direct spectral analysis of a signal.

The Oscilloscope In the cathode ray oscilloscope, the cathode tube operates as a time-based voltage transducer. A beam of electrons emitted by a heated cathode are guided vertically and horizontally by pairs of plates to control the location of the impact of the electron beam on the screen The output of the oscilloscope is a signal trace created by the impact of the electrons on a phosphorescent coating on the screen.

The Oscilloscope Input voltages result in vertical deflections of the beam, and produce a trace of the voltage variations versus time (horizontal or x axis) on the screen. The vertical plates are excited by the electrical field to be measured. The horizontal sweep frequency can be varied over a wide range, and the operation of the oscilloscope is such that high-frequency waveforms can be resolved.

Resistance Measurement Circuits
Vs Rm G kRt 0≤k ≤1

The Simple Current-Sensitive Circuit
Vs Rm G kRt 0≤k ≤1 io Many sensing element produce their output in the form of varying resistance. The element may be a sliding contact, a thermistor, a piezoresistive element or other, where the resistance varies between zero and a maximum value, Rt. The simple current-sensitive circuit may be used to determine the resistance of a varying resistance element through measurement of current magnitude. Rm represents the resistance of the remaining parts of the circuit, including the meter resistance and the internal resistance of the voltage source. The current is related to kRt through Ohm’s law.

The Simple Current-Sensitive Circuit
Vs Rm G kRt 0≤k ≤1 The maximum current flow, imax = Vs/Rm takes place when the resistance of the sensing element is zero. The current ratio, io / imax may be written as: The output (current ratio) is nonlinear with the input (resistance change). The sensitivity is small for small values of Rt /Rm . Greater sensitivity may be obtained for larger values of Rt /Rm but it drops rapidly as the resistance approaches its upper limit (k approaches 1). Careful control of the driving voltage is also necessary if calibration relationship between current and resistance is to be maintained. Rt / Rm = 0.5 1 2 4 10 k io/imax

Voltage Dividing Circuits
Vs R1 R2 Vo

The Resistive Voltage Divider
A resistive voltage divider is obtained from a set of resistors connected in series to divide a voltage source. The voltage across each resistor may be calculated by multiplying the value of the current flowing through the circuit by the resistance of the resistor considered. Vs R1 R2 Vo

The Ballast Circuit Vs Rb Rt Vo Voltage Indicator The ballast circuit is simply a voltage divider with one of the resistors representing the resistive sensing element. The voltage across that resistor changes as its resistance varies with the input to be measured. A voltage sensitive device is placed across that resistance which can be calibrated to indicated the input of interest. Assuming a high-impedance meter, the relationship between the output ratio (Vo/Vs) to the input ratio (R/Rt) can be found from Ohm’s law

The Ballast Circuit Vs Rb Rt Vo Voltage Indicator The relationship between the output ratio (Vo/Vs) and the input ratio (R/Rt) is nonlinear and a percentage variation in the supply voltage results in a greater variation in the output than does a similar percentage change in k. Voltage regulation must be strictly controlled for successful operation. Rt/Rb = 4 2 Vo/Vs 1 0.5 k

The Sliding Contact Resistive Voltage Divider
A sliding contact voltage divider is a resistive voltage divider whereby the resistance ratio is adjusted with a movable contact. A voltage Vs is applied across the two ends A and B of a resistance element , RAB, and the output voltage Vo is measured between the point of contact C of the sliding element and the end A of the resistance element. The output voltage from the divider is controlled by the position of the slider, taking the range 0 ≤ Vo ≤ Vs. as the resistance ratio is adjusted from zero to 1 with a movable contact. A Vs x L C B Vo

The Null Type Resistive Potentiometer
The null type potentiometer uses the sliding contact voltage divider to measure DC voltage. It balances an unknown input voltage against a known internal voltage until both sides are equal. The voltage to be measured, Vm, is applied at the divider’s output, and a current-sensing device, such as a galvanometer, is used to detect current flow on the output branch. The output voltage of the divider VAC is adjusted by moving the sliding contact A; any current flow through the galvanometer, G, would be a result of an imbalance between the measured voltage, Vm, and VAC. The value of Vm is determined from knowledge of VAC. A Vs x L C B Vm G

The Null Type Resistive Potentiometer
A null balance, corresponding to zero current flow through G, occurs only when Vm = VAC. With a known and constant supply voltage Vs , VAC is related to the position of the slider, which can be calibrated to indicate Vm. Potentiometers have been supplanted by digital voltmeters, which are deflection mode devices but have very high input impedances so as to keep loading errors small, even at low-level voltages. A Vs x L C B Vm G

The Deflection Type Resistive Potentiometer
The resistive potentiometer is commonly used as a displacement-measuring device. Assuming high galvanometer impendence, a linear relationship exists between the output voltage Vo and the distance x, for uniform resistivity for RAB Displacement (x) Resistance change (ΔR) Voltage change (Δv) A Vs x L C B Vo G

The Resistive Potentiometer: Loading Error
Loading error occurs when the resistance RL is finite. It can be estimated by calculating the voltage across RL for finite values of RL/RAB . Noting that RAC = kRAB and RBC = (1-k)RAB The total resistance seen by the source Vs is: The current through RBC is thus The voltage across RL is A Vs x L C B RL G

Comparing the values of Vo for finite and infinite values of RL/RAB , the error in Vo for a finite RL/RAB is: The percentage error is thus: A Vs x L C B RL G RL/RAB = 1 2 5 10

The error ratio may alternatively be defined as the ratio of the error to the full scale output, Vs, in this case, A Vs x L C B RL G RL/RAB = 1 2 5 10

Direct Voltage Dividing Circuits for Measurement of Variable Resistance
A voltage divider may be used to measure the small changes in resistance from variable resistance detectors such as RTDs, strain gauges and photoresistive elements. These detectors show only a small change in their resistance in response to the detected input. For example, the resistance of metal strain gauges may vary only be 1 x 10-4 in use. The smallness of resistance change has important consequences on the performance of the voltage dividing circuit. Vs R0 R1 Vo

Direct Voltage Dividing Circuits for Measurement of Variable Resistance : Sensitivity
Vs R1 R0 → R0 + ΔR Suppose that a voltage divider is formed from a detector of an initial resistance R0 and a second resistance with R = R0. The initial output is: Assuming no change in the supply voltage during measurement, dVs = 0, Vo → Vo + Δ Vo

Vs R1 R0 → R0 + ΔR The sensitivity of the potentiometer increases with Vs , so = r/(1+r2) and with 1/Ro. Practical considerations place an upper limit on Vs . Ro can not be made too small as δRo is related to the value of Ro . The ratio r = R1 / Ro, can be chosen to maximize so = r/(1+r2) Vo → Vo + Δ Vo

Vs R1 R0 → R0 + ΔR The sensitivity coefficient so is maximum when r = R1 / Ro = 1. The maximum value of so occurs when r = R1 / Ro = 1, and its value is 1/4. Thus R0 is usually chosen to be equal to R1, and the sensitivity is Vo → Vo + Δ Vo

Example: Strain Gauge in Ballast Circuit Configuration
Vs R0 R0 → R0 + ΔR A strain gauge with a nominal resistance R0 = 120 Ω is used in a ballast circuit designed to produce maximum sensitivity. The maximum expected change in the resistance of the gauge when maximum load is applied is ΔR = 240 x 10-6 Ω . What is the percentage change in the measured voltage when maximum load is applied. What is the percentage change in the measured voltage due to a 0.1 % drift in the supply voltage Vs Vo → Vo + Δ Vo

Solution: percentage change in the measured voltage when maximum load is applied. For a ballast circuit with maximum sensitivity, we have Vs R0 R0 → R0 + ΔR Vo → Vo + Δ Vo R0 = 120 Ω ΔR = 240 x 10-6 Ω Maximum sensitivity configuration Δ Vo / Vo ? A meter with a resolution of better than one part in a millionth is needed in order to see any change in Vo . This excludes most common voltmeters, which may resolve to only 0.01%, (10-4)

Solution: percentage change in the measured voltage due to a 0.1 % drift in the supply voltage Vs The percentage change in the measured voltage due to a 0.1 % the drift in Vs is 0.1 % which is a 1000 times greater than the strain induced change in voltage, % Vs R0 R0 → R0 + ΔR Vo → Vo + Δ Vo R0 = 120 Ω ΔR = 240 x 10-6 Ω Maximum sensitivity configuration Δ Vo / Vo ?

Voltage Dividing Circuits for Measurements of Variable Resistance Detectors
A solution to the voltage divider problems may be obtained by a circuit having an output voltage proportional to ΔVo itself, without the large offset voltage, Vo . This can be done by introducing another voltage divider with fixed resistors Ro, which has a midpoint voltage Vo . We now measure the difference between the midpoint voltages of the two dividers as the output voltage of the circuit This eliminates the problem of caused by the offset voltage Vo . The arrangement of two voltage dividers is in fact identical to the Wheatstone bridge circuit to be discussed next. B Vs R0 Vout A

Resistance Bridges B VS R2 R3 R4 R1 A Vo Bridges offer an attractive alternative for measuring small resistance changes accurately. The basic Wheatstone bridge 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. At balance, Vo = 0 Vs R3 R4 R1 R2 Vo + - VB VA

Resistance Bridges: Null Type Measurement
A bridge measures resistance indirectly by comparison with a similar resistance. The two principle 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, irrespective 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. B VS R2 R3 R4 R1 A Vo At balance, Vo = 0 Vs R3 R4 R1 R2 Vo + - VB VA

Resistance Bridges: Null Type Measurement
VS R2 R3 R4 R1 A Vo Similarly, 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 electromechanical 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. At balance, Vo = 0 Vs R3 R4 R1 R2 Vo + - VB VA

Temperature Controlled Room Set temperature through R4
Application: Null Type Measurement in Feedback Systems Example 1: Room Temperature Controller At balance, Vo = 0 The resistor R4 in this system represents the reference input which may be adjusted by the user. Zero output voltage occurs when the magnitude of the external variable (strain, temperature, etc.) is such that R1 = (R2/R3)·R4. The relationship between the bridge output voltage and ΔR is not linear. The system is good only in ‘heating’ mode, and may become unstable if the room temperature becomes above the set temperature. User adjustable resistance Vs R3 R4 RTD R2 Vo + - Heater Temperature Controlled Room R1 Set temperature through R4 Room temperature through RTD Bridge Output Voltage Heater Room RTD

Application: Null Type Measurement in Feedback Systems Example 2: Position Servo
At balance, Vo = 0 The resistor R4 in this system represents the reference input which may be adjusted by the user. Zero output voltage occurs when the magnitude of the external variable (strain, temperature, etc.) is such that R1 = (R2/R3)·R4. The relationship between the bridge output voltage and ΔR is not linear. The system is good only in ‘heating’ mode, and may become unstable if the room temperature becomes above the set temperature. VB User adjustable resistance R3 R4 - + Vo R2 Set Position through R4 Current Position through R1 Bridge Output Voltage Solenoid Magnet R1 R1 Solenoid Coil Magnet

Resistance Bridges: Deflection Type Measurement
VB R2 R3 R4 R1 A Vo For the majority of sensor applications employing bridges, 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 = 10V (a typical excitation voltage for a load cell application). At balance, Vo = 0 VB R3 R4 R1 R2 Vo + -

Deflection Type Resistance Bridges: Variable Resistance Configurations
VB R3 R4 R1 R2 Vo + - General Bridge In many bridge applications, there may be two, or even four elements which vary. 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 voltage. VB R R + ΔR Vo + - Single Varying Element Two Co-varying Elements VB R+ΔR R R + ΔR Vo + - Two Anti-varying Elements VB R R + ΔR Vo + - R - ΔR Four Varying Elements VB R + ΔR Vo + - R - ΔR R + ΔR R - ΔR

Deflection Type Resistance Bridges: Variable Resistance Configurations
VB R3 R4 R1 R2 Vo + - General Bridge 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). VB R R + ΔR Vo + - Single Varying Element VB R+ΔR R R + ΔR Vo + - Two Co-varying Elements VB R R + ΔR Vo + - R - ΔR Two Anti-varying Elements VB R + ΔR Vo + - Four Varying Elements R - ΔR

Single Varying Element
Deflection Type Resistance Bridges: Single Varying Element Configurations The single-element varying bridge may be used 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 ΔR. The relationship between the bridge output and ΔR is not linear. Since there is a fixed relationship between the bridge resistance change and its output, software can be used to remove the linearity error in digital systems. Alternative bridge configurations can also be used to linearize the bridge output directly. VB R R + ΔR Vo + - Single Varying Element

Two Co-varying Elements
Deflection Type Resistance Bridges: Two Co-varying Elements Configurations VB R+ΔR R R + ΔR Vo + - Two Co-varying Elements In this configuration, both elements change in the same direction. 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.

Two Anti-varying Elements
Deflection Type Resistance Bridges: Two Anti-varying Elements Configurations VB R R + ΔR Vo + - R - ΔR Two Anti-varying Elements This configuration 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. Such a configuration could be used for measuring force, pressure, stress, strain, etc. It produces a linear is output, and it has twice the gain of the single-element configuration. F R + ΔR R - ΔR

Deflection Type Resistance Bridges: Four Varying Elements Configurations
VB R + ΔR Vo + - Four Varying Elements R - ΔR The four varying elements 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.

Case Study II: Diaphragms Pressure Sensors using Strain Gauge
The diaphragm, is an elastic element pressure transducer used for low pressure measurement (up to 2000 bar absolute pressure). A diaphragm can also be used to measure differential pressure (up to 2.5 bar) by applying the two pressures to the two sides of the diaphragm. A diaphragm pressure sensors consists of a thin membrane (the diaphragm) attached to the pressure measurement chamber. Applied pressure causes displacement of the diaphragm and this movement is measured by a displacement transducer.

The diaphragm can be either plastic, metal alloy, stainless steel or ceramic. Plastic diaphragms are cheapest, but metal diaphragms give better accuracy. Stainless steel is normally used in high temperature or corrosive environments. Ceramic diaphragms are resistant even to strong acids and alkalis, and are used when the operating environment is particularly harsh. The typical magnitude of diaphragm displacement is 0.1 mm, which is well suited to a strain-gauge type of displacement-measuring transducer, although other forms of displacement measurement are also used in some kinds of diaphragm-based sensors.

If the displacement is measured with strain gauges, it is normal to use four strain gauges arranged in a bridge circuit configuration. The output voltage from the bridge is a function of the resistance change due to the strain in the diaphragm. This arrangement automatically provides compensation for environmental temperature changes.

Older diaphragm pressure transducers used metallic strain gauges bonded to a diaphragm typically made of stainless steel. Apart from manufacturing difficulties arising from bonding the gauges, metallic strain gauges have a low gauge factor, which means that the low output from the strain gauge bridge has to be amplified by an expensive d.c. amplifier. The development of semiconductor (piezoresistive) strain gauges provided a solution to the low-output problem, as they have gauge factors up to one hundred times greater than metallic gauges. However, the difficulty of bonding gauges to the diaphragm remained and a new problem emerged regarding the highly non-linear characteristic of the strain–output relationship.

The problem of strain-gauge bonding was solved with the emergence of monolithic piezoresistive pressure transducers. These have a typical measurement uncertainty of ±0.5% and are now the most commonly used type of diaphragm pressure transducer. The monolithic cell consists of a diaphragm made of a silicon sheet into which resistors are diffused during the manufacturing process. Such pressure transducers can be made to be very small and are often known as micro-sensors.

Silicon measuring cells have the advantage of being very cheap to manufacture in large quantities. Non-linear characteristic is normally overcome by processing the output signal with an active linearization circuit or incorporating the cell into a microprocessor based intelligent measuring transducer. Such instruments can also offer automatic temperature compensation, built-in diagnostics and simple calibration procedures. These features allow measurement inaccuracy to be reduced to a figure as low as ±0.1% of full-scale reading. By varying the diameter and thickness of the silicon diaphragms, silicon diaphragam sensors in the range of 0 to 2000 bar have been made.

Example: Errors Produced by Wiring Resistance in Remote Resistive Bridge Sensor
A 350Ω nominal resistance strain gage is used as a sensing element in a single varying element Wheatstone bridge. The gauge is connected to the rest of the bridge circuit by 100 feet of 30 gage twisted pair copper wire with a resistance of Ω /ft, at 25°C and a temperature coefficient of 0.385%/°C. The full scale variation of the strain gage resistance (with flex) above its nominal 350 Ω value is +1%. Calculate the offset error in the bridge output due to the lead resistance.

Errors Produced by Wiring Resistance in Remote Resistive Bridge Sensor
Offset error due to lead resistance The resistance of each copper cable is 10.5Ω for 100ft, and the total lead resistance in series with the 350Ω strain gage is therefore 21Ω at 25°C. The bridge output voltage is simply the difference between the output of two voltage dividers, each driven from a +10V source. When the load on the gauge is zero, the bridge would have an output offset voltage of mV for a nominal strain gage resistance of 350Ω.

If a 21Ω compensating resistance RCOMP is used as shown to compensate for the lead resistance, Calculate the full scale bridge output at 25°C and the percentage zero error (offset error) and the sensitivity error (gain error) due to a +10°C temperature rise in the cable.

Full scale bridge output at 25°C The full scale bridge output voltage takes place when the gauge resistance is of 350 Ω + (350 Ω × 1%) = Ω The full scale output of the bridge at 25°C is mV.

Example: Errors Produced by Wiring Resistance in Remote Resistive Bridge Sensor
Zero error (zero offset) due to 10°C With +10°C temperature rise, ΔRLEAD = 10 × (0.385/100) × 10.5 = Ω, RLEAD = 10.5 Ω Ω = Ω. Total lead resistance in series with the gauge = 2 × = With the compensating resistance, an offset error of mV is present at 10°C temperature rise which is (5.343/23.45) = 23.2% of full scale output.

Gain error due to 10°C temperature rise The full scale bridge output at 35°C is calculated based on a gauge resistance of 350Ω + (350Ω × 1%) = 353.5Ω. The lead wire resistance, however, is (2 ×10.904Ω = Ω The full scale output due to 10°C temp rise is mV giving a deflection from zero load of ( = mV) with an error of ( – = mV) or (-0.054/23.45) = -0.23% of full scale output

The effects of wiring resistance on the bridge output can be minimized by the 3-wire connection. The sense lead measures the voltage output of a divider: the top half is the bridge resistor plus the lead resistance, and the bottom half is strain gage resistance plus the lead resistance.

The nominal sense voltage is independent of the lead resistance. When the strain gage resistance increases to fullscale (353.5W), the bridge output increases to mV.

Gain error due to 10°C temperature rise Increasing the temperature to +35ºC increases the lead resistance by W in each half of the divider. The fullscale bridge output voltage decreases to mV because of the small loss in sensitivity, but there is no offset error. The gain error due to the temperature increase of +10ºC is therefore only –0.02mV, or –0.08% of fullscale.

Instrumentation and Measurements Dr. Mohammad Kilani

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