USE OF SUBMERSIBLE PRESSURE TRANSDUCERS IN WATER-RESOURCES INVESTIGATIONS

The terms in this glossary were compiled from numerous sources. Some definitions have been modified in accordance with usage of submersible pressure transducers for ground-water-level monitoring and may not be the only valid definitions for these terms.

Absolute Pressure Transducer: A pressure transducer that has an internal reference chamber sealed at or close to 0 psi absolute (full vacumn).

Accuracy (absolute): A measure of the closeness of agreement of an instrument reading compared to that of a primary standard having absolute traceability to a standard sanctioned by a recognized standards organization. This definition of accuracy is rarely used in manufacturer's specifications.

Accuracy (relative): The ratio of the error of an instrument reading to the full-scale output of the instrument or the ratio of the error to a specified output expressed as a percent. Normally this statement of accuracy will include the effects of non-linearity, hysteresis, and repeatability. This definition of accuracy is most commonly used by manufacturers.

Analog output: Transducer output that is a continuous function of the measurand (value of the physical property) as distinguished from digital (discrete) output.

Best-fit straight line: A line chosen to represent the sensitivity of a transducer; from it non-linearity errors may be calculated. The best-fit straight line may be determined from a least-squares linear regression fit of the measurand input and transducer output.

Bridge circuit: An instrument for measuring electrical values by comparing the unknown value with known values. When all values are equal the bridge is said to be 'balanced' or 'nulled'.

Burst pressure: The specified maximum pressure that may be applied to the sensing element of the transducer without rupture of either the sensing element or the transducer case.

Compensation: Provision of a supplemental device, circuit, or special material to counteract known sources of error such as temperature, shock, and vibration.

Creep: A change in output occurring over a specific time period while the measurand and all environmental conditions are held constant.

Data logger: An electronic digital data recorder used by engineers, scientists, and technicians.

Dead volume: The total volume of the pressure port cavity of a transducer under room temperature and barometric pressure.

Diaphragm: The force-sensing membrane that is deformed when pressure is applied.

Digital output: Transducer output that represents the magnitude of the measurand in the form of a series of discrete quantities, coded in a system of notation, as distinguished from analog output.

Drift: A change in output over a period of time that is not a function of the measurand (measured property). Drift is normally specified as a change in zero (zero drift) over time and a change in sensitivity (sensitivity drift) over time.

Error: The algebraic difference between the indicated value and the true value of the measurand.

Error band: The band of maximum deviation of output values from a specified calibration curve or reference line.

Excitation: The external electrical voltage or current applied to a transducer for its operation, usually expressed as a range of voltage or current values.

Force-sensing device: The diaphragm and associated mechanical components that move and cause a change in position that can be sensed by the electrical components in a transducer.

Full-scale output: The algebraic difference between the outputs at the specified upper and lower limits of the measurand inputs.

Gage transducer: A transducer that measures pressure relative to the ambient atmospheric pressure.

Hysteresis: The maximum difference in output, at any measurand value within the specified range, when the value is approached first with an increasing and then with a decreasing measurand. Hysteresis is expressed in percent of full-scale output.

Inductive transducer: A transducer in which the mechanical movement is converted to an electrical signal through electrical induction.

Linearity: The closeness of a calibration curve to a specified straight line. Linearity is expressed as a percent of the full-scale output using the maximum deviation of any calibration point from a specified straight line.

Mean time between failures (MTBF): The total time that a number of sensors operate divided by the number of sensors that fail during the operational period. A value of MTBF usually characterizes a single type of component or a production batch from one manufacturer.

Precision: The ability to reproduce output readings when the same measurand is applied, consecutively, under the same conditions, and in the same direction. Precision is a measure of the repeatability of a measurement. Precision does not imply accuracy. If a measurement system has good repeatability, then consecutive readings will be densely grouped together when the readings are plotted.

Pressure-sensing system: As used in this report, a pressure-sensing system is composed of a data recorder, pressure transducer, electrical connections, power supply, atmospheric pressure sensor or vent tube, and the transducer suspension apparatus.

Pressure transducer: A type of measurement device that converts pressure-induced mechanical changes into an electrical signal.

Proof pressure: The maximum pressure that may be applied to the sensing element of a transducer without changing its performance beyond the specified tolerances.

Range: The upper and lower limit of the measurand within which a transducer is intended to measure.

Reference pressure: The pressure to which a differential pressure transducer measurement is compared.

Reliability: The probability that an item will perform its intended function for a specified interval under stated conditions.

Resolution: The smallest change in the measurand that can be measured or detected in the output reading.

Response time: The length of time required for the output of a transducer to rise to a specified percentage of its final value as a result of a step change in the measurand.

Shunt resistor: A precision resistor with a known value that is used to calibrate a pressure transducer.

Sensitivity: The ratio of the change in transducer output to a change in the measurand. Sensitivity may vary over the operational range of the transducer.

Specific Gravity: The ratio of the mass of a body to the mass of an equal volume of water at 4 °C or other specified temperature.

Stability: The ability of a transducer to retain its performance characteristics for a period of time. Stability is normally expressed in percent of full-scale output over a set period of time.

System effectiveness (reliability): The probability that a system can successfully meet an overall operational demand within a given time when operated under specified conditions.

Test envelope: A line or lines bounding the measured values on a graph of true values versus measured values of a single property (for example, pressure).

Time constant: The length of time required for the output of a transducer to rise to 63 percent of its final value as a result of a step change in the measurand.

CONVERSION FACTORS, VERTICAL DATUM, AND ABBREVIATIONS

Multiply	By	To obtain
inch (in.)	25.4	millimeter
square inch (in.²)	6.452	square centimeter
foot (ft)	0.3048	meter
pounds (lbs)	0.4536	kilogram

Temperature: Water and air temperature are given in degrees Celsius (°C), which can be converted to degrees Fahrenheit (°F) by use of the following equation:
°F = 1.8 (°C) + 32

Absolute temperature in degrees Kelvin (°K) can be converted to degrees Celsius (°C) by use of the following equation:
(°C) = (°K) − 273.15

Sea level: In this report "sea level" refers to the National Geodetic Vertical Datum of 1929 (NGVD of 1929)—a geodetic datum derived from a general adjustment of the first-order level nets of both the United States and Canada, formerly called Sea Level Datum of 1929.

Abbreviations

AC - alternating current
DBLS - depth below land surface
DC - direct current
EDL - electronic data logger
EMF - electromotive force
FPS - foot-pound-second measurement system
Hg - mercury
H₂O - water
lbs/in.² - pounds per square inch
MKS - meter-kilogram-second measurement system
MTBF - mean time between failures
mV - millivolt
µV - microvolt
N - Newtons
Pa - Pascals
psi - pounds per square inch
PVC - polyvinyl chloride
USGS - U.S. Geological Survey
V - volt

ABSTRACT

Submersible pressure transducers, developed in the early 1960s, have made the collection of water-level and pressure data much more convenient than former methods. Submersible pressure transducers, when combined with electronic data recorders have made it possible to collect continuous or nearly continuous water-level or pressure data from wells, piezometers, soil-moisture tensiometers, and surface water gages. These more frequent measurements have led to an improved understanding of the hydraulic processes in streams, soils, and aquifers.

INTRODUCTION

With the improvement of instruments used to collect and record ground-water level data, provision of consistent guidelines for the use of this equipment is important to those who collect the data. Submersible pressure transducers (pressure-sensing devices) and electronic data loggers (digital data recorders) have become an integral part of U.S. Geological Survey (USGS) hydrologic investigations that collect time-series data on water-level fluctuations in wells. Because of the perceived ease of installation and operation of submersible pressure transducers and data loggers, data provided by these systems commonly are not adequately supported by quality-assurance procedures and documentation. Submersible pressure transducers occasionally are misused, either because of a lack of understanding of the capability of a particular transducer, or because of a lack of attention to its calibration and maintenance. This manual describes the operational theory fundamental to all transducers and provides detailed information about the use of submersible pressure transducers in hydrologic investigations conducted by the USGS.

Background

Electrical pressure transducers designed to be immersed in water (submersible pressure transducers) have been used by ground-water scientists since the early 1960s (Shuter and Johnson, 1961; Garber and Koopman, 1968). These pressure-sensing devices, typically installed at a fixed depth in a well, sense the change in pressure against a membrane. Pressure changes occur in response to changes in the height, and thus in the weight of the water column in the well above the transducer. Substantial improvements in design, operation, and accuracy of electronic submersible pressure transducers and data recording systems have led to a significant increase in their use, replacing older systems (described in the box) in some applications.

Submersible pressure transducers also are used to monitor surface water elevations (Meyers, 1994) and variations in soil moisture tension in the unsaturated (vadose) zone (Healy and others, 1986). Several types of water-level sensors can provide time-series data to an electronic data logger, but for many applications only submersible pressure transducers are suitable (Freeman, 1996). Because many types of submersible pressure transducers are available, it can be difficult to determine which type is most appropriate for a particular hydrologic investigation or application.

Earlier Systems

Early systems that recorded continuous water-level changes in wells consisted of a float connected to a steel tape or beaded cable and a counterweight connected to a wheel. The rotating wheel turned a drum, upon which a pen, connected to a clock, recorded water levels on a paper chart. Another system, in use since the 1960s, uses an electrical device to sense the water surface in a well, either with a series of electrodes or with a membrane deflected by the water surface. These probes control an electrically activated drum located at the top of the well. The drum rotates either clockwise or counterclockwise until the sensor is situated at the water surface. The data are recorded on a continuous paper chart or by a digital recorder. A third type of sensor that has been in use since the early 1980's is an acoustic velocity device such as the Polaroid acoustic distance-measuring device. It measures the changes in depth to the water surface by measuring the variations in elapsed time between sending and receiving sound waves.

Purpose and Scope

This manual is intended to be a guide to the proper selection, installation, and operation of submersible pressure transducers and data loggers for the collection of hydrologic data, primarily for the collection of water-level data from wells. The manual describes the basic principles, measurement needs, and considerations for operating submersible pressure transducers, including a discussion of the systematic errors inherent in their use. Standard operational procedures for data collection and data processing, as well as applications of transducers for specific types of hydrologic investigations are included. Basic concepts regarding the physics of pressure and the mechanics of measuring pressure are presented, along with information on the electronics used to make and record these measurements. Guidelines for transducer calibration, proper use, and quality assurance of data also are presented. Field applications of pressure transducer systems are discussed for ground water, surface water, and the unsaturated zone; as are common problems that may corrupt data, and suggestions for field repairs.

BASIC PRINCIPLES OF PRESSURE AND PRESSURE MEASUREMENT

The physics of pressure and types of pressure measurements described in this section provide the background understanding for the application of pressure transducers to hydrologic investigations. An overview of the basic principles of direct current (DC) circuits is provided, since an understanding of the operational theory behind sensing and recording systems requires a working knowledge of DC circuitry. The application of these principles and operations to measure pressure with pressure transducers is addressed in later sections of this manual.

Physics of Pressure

Pressure (P) is defined as the force (F) exerted per unit of surface area (A) over which the force is applied. In classical mechanics, when two solid objects are in contact with each other, the pressure exerted by one object on another object is defined as:

In fluid mechanics, pressure is similarly defined. For purposes of this discussion, fluid refers to both liquids and gases. Unlike solids, however, fluids possess negligible shear strength and will completely or partially occupy the volume of the container in which they are placed. A liquid, if it does not completely fill the container, will present a free liquid surface. A gas, on the other hand, will always fill the volume of its container.

When a gas is confined in a container, molecules of the gas strike the walls of the container. The collision of the molecules against the container walls results in a force exerted against the surface area of the container. Pressure in the container is this force divided by the surface area of the container. Pressure within the container is the same everywhere. For a given mass of gas, pressure inside a container will vary in response to changes in the volume of the container and in the temperature of the gas in accordance with the relations expressed in the combined (Charles's and Boyle's) gas law:

The pressure at a point in a gas that is not confined to the volume of a container, and is at rest, such as atmospheric air, is dependent upon the density and height of the column of gas above the point of measurement. Because gas is compressible, its density will vary in response to position or elevation. Temperature differences, in an otherwise static column of air, also will affect gas densities. For example, atmospheric pressure decreases with increasing altitude; the relationship is not linear, however, because air density and temperature also decrease with increasing altitude. Gas pressure exerted at a point is identical in all directions around the point of measurement.

For a liquid at rest that is not confined, the pressure exerted by the liquid at any point depends upon the density and the height of the column of liquid above the point of measurement. Because water is only slightly compressible, its density can be assumed to be constant, provided temperature and salinity do not vary significantly above the point of measurement. Pressure at any point in a liquid acts perpendicularly against the surfaces it contacts. Within the liquid, the pressure at a point is identical in all directions from the point of measurement. This principle is illustrated in the use of open-ended manometers to measure pressure differentials as shown in figure 1.

In figure 1, pressure P₁ is greater than P₂ by an amount equal to the height (h) of fluid in the manometer times the specific gravity of the fluid. In the absence of capillary forces, the diameters of the reservoir (D) and manometer (d) have no influence on the differential height of fluid between the two liquid surfaces.

Whenever an external pressure is applied to a confined fluid at rest, either liquid or gas, the pressure increases at every point in the fluid by the amount of the external pressure applied. This principle, known as Pascal's law, is the basis for the design of the hydraulic lift, closed-end manometers, and many other types of pressure-sensing or transduction devices. This principle is illustrated in figure 2.

In figure 2, a downward force (F) of 100 pounds (lbs) acting across an area of 1 square inch (in²) produces a pressure (P) of 100 pounds per square inch (lbs/in² or psi). Because the fluid is confined, pressure is the same throughout the fluid, regardless of position. The downward applied force of 100 lbs produces an equivalent force of 5,000 lbs when the pressure is transmitted across an area of 50 in². Unlike the example of the unconfined fluid system shown in figure 1, fluid displacement is dependent on the size of the reservoirs and tubes. Mechanical advantage is gained by difference in the area of the pistons.

Pressure may be expressed in any one of the three major systems of units. The standard unit of pressure in the English, or foot-pounds-second (FPS), system is pounds per square inch, abbreviated as psi. In the meter-kilogram-second (MKS) system of measurement, pressure is expressed as kilograms per square meter. In standard international (SI) units, pressure is expressed in Newtons (N) per square meter, or Pascals (Pa).

In ground-water hydrology, the length unit of measurement commonly is used to represent pressure because it is easily referenced to the total energy at a point in the ground-water flow system. The total energy at a point in a body of water equals the sum of its potential energy and kinetic components. For any incompressible fluid, either at rest or in motion, this relation is expressed as:

Total energy potential or head is simply the summation of position (z) above or below some reference or elevation datum, height of the column of water above the point of measurement, and a kinetic energy term expressed in units of length. In most ground-water hydrology applications, velocity (v) is very nearly zero and the kinetic energy term is therefore assumed to be zero. Because pressure in a static water body is determined by the height of the column of water above the point of measurement, it is convenient to express pressure in units of length. At the water surface, pressure head is zero and total head (or potential) is simply the height of the water surface above or below the reference datum. In ground-water hydrology, potential commonly is referred to as "head." In the FPS system, it is expressed in terms of feet or inches of water; in the MKS system, as meters or centimeters of water.

Pressure-unit conversion factors commonly used in hydrology are listed in table 1. Take care when applying these conversion factors. Pressure expressed in terms of mercury (Hg) or water (H₂O), for example, depends on the specific gravity or density of the fluid, which is temperature dependent. In the case of water, the effects of total dissolved solids and suspended solids (turbidity) on density also must be considered when quoting pressure head in terms of length.

Types of Pressure Measurements

Absolute pressure is measured in reference to a vacuum or zero pressure. Pressure at sea level is 1 atmosphere, 101.3 kPa, or 14.70 psi. Pressures measured by an absolute pressure transducer are always positive because these devices are referenced to a perfect vacuum in which absolute pressure is zero. Zero referencing is accomplished by completely evacuating and sealing the interior or reference side of the transducer.

Gage pressure is measured in reference to atmospheric pressure at mean sea level. Pressures measured by a gage-pressure transducer are positive for pressures greater than sea-level pressure and negative for pressure less than sea-level pressure. Thus, atmospheric pressure measurements above sea-level datum are negative because atmospheric pressure decreases with altitude. Sea-level referencing is accomplished by sealing the interior or reference side of the transducer to atmospheric pressure at sea level. The term "gage pressure" is sometimes used to describe pressure measurements referenced to ambient atmospheric pressure other than to sea level. The relation between absolute pressure and gage pressure is illustrated in figure 3.

Differential pressure is measured with respect to a varying pressure reference such as ambient atmospheric pressure or some other pressure source that is allowed to vary independently of the primary measurement. Pressure transducers constructed in this manner actually sense the difference between two independent pressure sources simultaneously. The output of the differential pressure transducer is proportional to the pressure difference between the two independent sources. Pressure measurements made with open-ended manometers or vented submersible pressure transducers are examples of differential measurements.

Sealed-reference pressure is measured with respect to a sealed reference pressure other than atmospheric pressure at sea level. A sealed-reference pressure is created by evacuating (pressuring down) or pressuring up the interior or reference side of the transducer to a prescribed absolute pressure. Because the sealed-reference side contains a constant volume of gas, the transducer must be maintained at a constant temperature to avoid changes in the reference pressure.

Overview of Direct Current Circuits

The most common type of pressure transducer is the strain-gage pressure transducer. Understanding the operational theory behind these transducers requires a working knowledge of basic DC circuitry. This knowledge applies as well to the operation of the electronic components used to excite and read the device. Many problems encountered in the field can be analyzed and diagnosed by applying some simple techniques that require a working knowledge of DC circuitry. In the following illustrations, V is voltage, I is current, and R is resistance.

Overview of Direct Current Circuit Analysis

Multiply	FPS lb/in²	MKS kg/m² (1)	SI N/m² (pascal)	Temperature Specification
lb/in.² (psi)	1	1.422 x 101	6.895 x 10 3
ft of H₂O	4.336 x 10-1	3.048 x 10 2	2.989 x 10 3	4°C
in of Hg	4.912 x 10 -1	3.453 x 10 2	3.377 x 10 3	0°C
mm of Hg	1.934 x 10-2	1.359 x 10 -3	1.333 x 10 2	0°C
atmosphere (atm)	1.470 x 101	1.033 x 10 4	1.013 x 10 5
N/m² (Pascal)	1.450 x 10 - 4	1.020 x 101	1
bar	1.450 x 101	1.020 x 10 4	1.000 x 10 5
kg/m²(1)	1.422 x 101	1	9.807
(1) One kilogram mass under standard gravitational acceleration.

The overview of DC circuit analysis illustrates how a DC circuit can be reduced to an equivalent single voltage source and a single resistor. For example, in the top part of figure 4, a circuit consists of a battery and two resistors, R₁ and R₂, in series. The battery supplies a voltage (V) causing a current (I) to flow in the circuit. Each resistor produces a voltage drop (V₁ and V₂). Current flow is continuous through the circuit, so I = I₁ = I₂. The total resistance of the circuit is the sum of the individual resistances (R = R₁ + R₂); therefore the circuit in the lower part of figure 4 is functionally the same as the circuit in the top part.

Resistors in Series:

V = IR

(Ohm's law)

R = R₁+ R₂...+ R_n

(resistance is additive)

I = I₁ = I₂ = I_n

(current is constant across each resistor)

V = V₁ + V₂ ... + V_n

(voltage drops are additive)

V = I (R₁+ R₂...+ R_n)

(voltage divides in proportion to resistance)

For resistors in parallel, the voltage drop across each resistor is the same (V = V₁ = V₂). From Ohm's Law, V₁ = I₁R₁ and V₂ = I₂R₂, so I₁R₁ = I₂R₂ and the total current flow in the circuit is the sum of the current in each branch (I = I₁ + I₂). Because I₁ = V/R₁ and I₂ = V/R₂, I=V/R₁+V/R₂ and I=V/R as shown in the lower part of figure 5.

Resistors in Parallel:

V = IR

(Ohm's law)

1/R = 1/R₁ + 1/R₂...+ 1/R_n

(the reciprocal of resistance is the sum of the reciprocals of each resistance)

R = R₁R₂

R₁+ R₂

(special case for two resistors in parallel)

I = I₁ + I₂ ... + I_n

(current is proportional to resistance)

V = V₁ = V₂= V_n

(voltage is constant across each resistor)

For multiple parallel and series components the procedure is virtually the same. First, convert independent or isolated parallel components to an equivalent resistor, then convert any resulting series components to an equivalent resistor. Repeat this process until all resistors are reduced to one equivalent resistor as shown in the box and in figure 6.

Example Mixed CircuiAnalysis

Convert the mixed circuit in figure 6 to a single

voltage source and resistor and find I, I₁, I₂, I_a, I_b,

V₁,V₂,V_a, V_b, R_c, and R_eqv

Given:

V = 10 volts

R₁ = 91 ohms R₂ = 100 ohms

R_a = 90 ohms R_b = 10 ohms

Convert parallel-bR_anch component to an

equivalent resistor, R_c

1/R_c = 1/R_b + 1/R_a

R_c = (R_bR_a)/(R_b + R_a) = 900/100 = 9 ohms

Convert series resistors to an equivalent resistor

R_eqv = R₁ + R₂ + R_c = 91 + 100 + 9 = 200 ohms

1. Compute I, I₁, I₂, I_c

I = V/R_eqv = 10 volts/200 ohms = 0.05 amps

I = I₁ = I₂ = Ic = 0.05 amps

2. Compute V₁, V₂, V_c

V₁ = IR₁ = (0.05 amps x 91 ohms) = 4.55 volts

V₂ = IR₂ = (0.05 amps x 100 ohms) = 5.00 volts

V_c = IR_c = (0.05 amps x 9 ohms) = 0.45 volts

V = V₁ + V₂ + V_c = 10.0 (volts check)

3. Compute V_a, V_b, I_a, I_b

I = I₁ = I₂ = I_a + I_b

V_c = V_a = V_b = 0.45 volts

I_a = V_a /R_a = 0.45/90 = 0.005 amps

I_b = V_b /R_b = 0.45/10 = 0.045 amps

I = I_a + I_b = 0.05 amps (check)

Complex resistance circuits cannot be broken down into a simple equivalent resistor. More complex DC circuits require the application of Kirchhoff's laws to analyze the separate components of the circuit. Kirchhoff's laws state:

An example of a circuit requiring the application of Kirchhoff's laws is the Wheatstone bridge, variations of which are used in many strain-gage pressure transducers (for example, in the common bonded-foil strain-gage transducers). The Wheatstone bridge circuit (fig. 7) consists of four resistors (R₁ through R4) supplied with a fixed voltage (V_in). When all resistances are equal, the current through each resistor is equal (I₁= I₂= I₃= I₄) and no voltage difference is measured at the voltmeter (V). The bridge is then said to be 'balanced.'

Reduction of the Wheatstone Bridge to a Thevenin Equivalent Circuit

Step 1: With the input voltage source shorted, reduce the circuit to an equivalent resistance (R_S) as seen across the open circuit (or output voltage leads) shown in figure 8.

Given:

V_in = 10 volts
R₁ = 2000 ohms
R₂ = 2000 ohms
R₃ = 1900 ohms
R₄ = 2000 ohms

With the input voltage shorted, the circuit reduces to two parallel branches that are in series, as seen across the output voltage leads (V_S). Circuit simplification reduces each parallel component to an equivalent resistor followed by adding in series:

R_S =	R₁R₂	+	R₃R₄
	R₁+ R₂		R₃+ R₄

R_S = 1974.36 ohms

Step 2: Determine the open-circuit voltage (V_S) appearing across the output voltage leads using the input voltage (V_in). Determine voltage potential (V_S) across the output leads, where:

V_S = I₂R₂ - I₃R₃ = I₁R₁ - I₄R₄
I_in = I₂ + I₃
(Kirchhoff's first law)

I₂R₂ + I₁R₁ = I₃R₃ + I₄R₄
V_in = I₃R₃+ I₄R₄
(Kirchhoff's second law)

I₁ = I₂, I₃ = I₄
(current is constant across resistors in series)

Compute I₃ and I₄

10V = 1500 I₃ + 2000 I₄
10V = 3900 I₄
I₄ = 2.5641 milliamps = I₃

Compute I₁ and I₂

10V = 2000 I₁ + 2000 I₂
10V = 4000 I₂
I₂ = 2.5 milliamps = I₁
V_S = I₂R₂ - I₃R₃ = 128.2 millivolts

Compute short-circuit current

I_S = V_S/R_S = 0.065 milliamps

The Thevenin equivalent circuit for the example Wheatstone bridge is shown in figure 9.

Understanding DC circuits can be an invaluable aid in error analysis, selecting measurement instruments, optimizing pressure transducer performance, and troubleshooting system malfunctions. Circuit analysis can be applied to test for electrical short circuits, to locate damaged bridge elements, or to find insulation leakage in the circuit.

The Wheatstone Bridge, Thevenin Equivalent Circuit

A Thevenin equivalent circuit is a hypothetical circuit designed to simplify the analysis of complex, two-terminal linear networks, such as Wheatstone bridges. Thevenin's theorem states that any potential source that has only two output terminals and is composed of resistors and a voltage or battery source can be represented by a series combination of a resistor (R_S) and voltage source (V_S) as shown in figure 9. V_S is the open-circuit potential and Rs is the resistance between the output terminals when the battery or voltage source is shorted. The equivalent circuit is constructed by applying Ohm's law to parallel and series components in the network, reducing the circuit to a voltage source (V_S) connected in series with a Thevenin equivalent resistance (R_S). The procedure for constructing a Thevenin equivalent circuit for the common Wheatstone bridge is described on the preceding page.

Constructing the Thevenin equivalent circuit is the first step in evaluating the adequacy of the measurement system to provide the necessary resolution and sensitivity to resolve the measurement of interest. Resolution and sensitivity are discussed in greater detail later in this manual. Once the expected signal levels (open circuit voltage and short circuit current) are defined, probable error sources that may affect the measurement can be identified. The application and value of this error analysis is illustrated by constructing what is referred to as a test envelope for the type of measurement (voltage, current, or resistance) being made. For purposes of illustration, the Wheatstone bridge in the example above will be used to construct the test envelope.

Construction of the test envelope is accomplished with reference to figure 10 and table 2. Determine the required measurement precision, which is 0.1% (1,000 parts per million) in this example; and determine the associated number of decades (table 2). The number of decades in this example is three. A decade is equivalent to one order of magnitude difference in the open-circuit voltage or short-circuit current. In figure 10, plot the computed open-circuit voltage (V_S = 128 millivolts) and short-circuit current (I_S = 0.065 milliamps). Extend a line the length of the number of decades – scaled to either the horizontal or vertical axis – vertically downward and extend a line of the same length horizontally to the left. Construct an arc the length of the total number of decades connecting the horizontal and vertical axes. The resulting envelope, shown in figure 11, defines the minimum values of resistance that can be placed in parallel with the Thevenin equivalent resistance (about 5 MΩ) and the maximum resistance that can be placed in series (about 5 Ω) without affecting the desired measurement precision. All resistance within the envelope, whether in parallel or in series, would seriously degrade the measurement. A second envelope is shown in figure 11 to illustrate the effect of increasing the measurement precision requirement from 0.1% to 0.01% (100 parts per million). In the second case, the minimum resistance in parallel is 50 MΩ, and the maximum resistance in series is 500 mΩ (0.5 Ω). Potential sources of errors due to parallel and series resistance in the measurement circuit are described in the section "Sources of Error When Making Direct Current Measurements."

Percent	Parts per million	Decades
10	100,000	1
1	10,000	2
0.1	1,000	3
0.01	100	4
0.001	10	5

PRESSURE TRANSDUCER CHARACTERIZATON

Pressure transducers are characterized by their mechanical and electrical transduction elements, the performance specifications of the transducer, and the interaction of the transducer with the other components of the measuring system (such as the power supply and data logger).

Basic Types of Transducers for Measuring Pressure

A transducer is a device that converts energy from one form to another. Electrical pressure transducers, which measure changes in pressure, consist of a mechanical-transduction element or force-summing device coupled to an electrical-transduction element, which is connected to a display or recording device, or both. There are two types of electrical transduction elements—active and passive. Electrical-transduction elements that convert pressure-induced mechanical changes directly to an electrical signal are referred to as active transducers. Passive transducers require an external excitation that causes the transducers to respond to pressure-induced mechanical changes. The electrical-transduction element converts mechanical energy into electrical energy and the force-summing device or mechanical-transduction element converts gas or liquid energy into mechanical energy.

Many types of pressure transducers consist only of mechanical-transduction elements. Open-ended and closed-ended manometers, barometers that record changes in the height of a column of liquid in response to some external pressure change, and spring-loaded pressure-sensing devices are examples of mechanical transducers.

Electrical pressure transducers are classified primarily on the electrical principle or method of electrical transduction involved in their operation. Different electrical transduction elements can be coupled to a variety of force-summing devices. Some combinations work better than others, depending on the application and measurement needs. Commonly used types of force-summing devices are illustrated in figure 12. A piezoelectric pressure transducer incorporating the diaphragm in the housing is illustrated in figure 13.

Electrical pressure transducers using force-summing devices are described below. The most common of the many types of pressure transducer is the strain-gage pressure transducer.

Piezoelectric Transducer

The piezoelectric transducer is an example of a self-generating or active pressure transducer. The design of this type of transducer is based on the ability of certain crystals (quartz, tourmaline, Rochelle salt, or ammonium dihydrogen phosphate) and ceramic materials (barium-titanate, or lead-zirconate-titanite) to generate an electrical charge or voltage when mechanically stressed. The crystal geometry of these materials is oriented to provide maximum piezoelectric response in one direction and minimal response in other directions. The transducer develops a voltage proportional to the change in pressure. These transducers cannot be calibrated using normal static-pressure calibration techniques. This type of transducer is used to measure rapidly fluctuating pressures.

Capacitive Transducer

Inductive and Reluctive Transducers

In an inductive transducer, pressure-induced displacements of a diaphragm cause a change in the self-inductance of a single coil. In a reluctive transducer, displacements occur in the magnetic coupling between a pair of coils. An inductive transducer is active and operates on the principle that the relative motion between a conductor and a magnetic field induces a voltage in the conductor (fig. 15). Because the pressure-induced electrical output signal requires relative motion, the inductive design is limited to dynamic measurements. In a reluctive transducer, displacements occur in the magnetic coupling between a pair of coils. A reluctive transducer is passive and requires external AC excitation of a pair of coils. It operates on the principle that the magnetic coupling between the two coils is affected by the displacement of a pressure-driven conductor located in the magnetic field between the two coils. The conductor is either connected to a force-summing device or is itself a force-summing device. Two basic designs have evolved (figs. 15 and 16).

Potentiometric Transducer

The potentiometric pressure transducer consists of a movable contact driven by an active force-summing device (fig. 17). The movable contact, or wiper, travels across a resistive element that may be a wire-wound coil, a carbon ribbon, or a deposited conductive film. The motion of the wiper across the resistive element causes a change in the resistance selected by the wiper. The change in resistance produces an electric signal (either a change in voltage or current) that is proportional to the mechanical displacement of the wiper. This type of transducer may be excited using either AC or DC.

Vibrating-Wire and Vibrating-Cylinder Transducers

Vibrating-wire (fig. 18) and vibrating-cylinder (fig. 19) transducers use a vibrating element, either a fine wire or a cylinder that forms a portion of one leg of a Wheatstone bridge circuit. The vibrating element is located in a magnetic field with one end of the element attached to a diaphragm or other type of force-summing device. Current flowing through the vibrating element causes the element to move in the magnetic field, which in turn induces a current in the element. The resulting voltage, amplified and fed back to the vibrating element, sustains the oscillations at the element's resonating frequency. The resonating frequency of the vibrating element is controlled by the tension exerted on the wire or cylinder by a diaphragm or other force-summing device. Vibrating-wire transducers can be installed in small-diameter (0.5-in.) wells, and because they produce AC signals, they can be used on long wires with little signal degradation.

Strain-Gage Transducer

The strain-gage transducer, sometimes referred to as a resistive transducer, is by far the most widely used type of pressure transducer. Its electrical transduction elements operate on the principle that the electrical resistance of a wire is proportional to its strain-induced length.

The strain-gage transducer uses the gage-factor property of the strain element to convert a mechanical displacement into a change in the electrical resistance of a circuit. Gage factor, defined as the unit change in resistance (R) per unit change in length (L), is expressed as:

Product specification sheets rarely provide gage factors. Instead, they commonly express pressure-transducer sensitivity as the voltage signal output ratio per unit of pressure change:

There are basically two classes of strain-gage transducers, unbonded and bonded. The unbonded strain gage uses a strain-sensitive wire (or wires) with one end fixed and the other end attached to a movable element. Strain, induced on the wire by the displacement of the movable element, produces a change in resistance proportional to the displacement of the movable element. The basic design of this type of transducer is illustrated in figure 20.

Bonded strain-gage transducers (fig. 21) can be grouped into those that require an adhesive to fix the gage to the pressure-sensing element (metal foil and strain-sensitive wires) and those attached to the strain-sensing element by techniques that effectively make the strain gage an integral part of the strain-sensing element (thin film and semiconductor). Thin film and semiconductor strain gages typically are mounted directly on the pressure-sensing element. Metal foil and strain-sensitive wires commonly are mounted on a secondary sensing element, which acts as the deforming member to produce the strain sensed by the strain gage.

Metal foil—Strain gages consist of wire or foil ribbon coated with a thin layer of insulation and cemented to the strain-sensing element. Distortion of the strain-sensing element is communicated by the bonding material directly to the wire or foil filaments. Increasing the length of the gage reduces the cross-sectional area of the conductor and increases the conductor's resistance, causing a change in voltage, proportional to the pressure change, across the output leads.

Thin film—Strain gages employ a metal substrate on which are deposited thin films as an insulation layer and a resistor layer, using either a vacuum-deposition or sputtering process. The strain gage is either masked onto or etched into the thin film resistor layer, making the gage an integral component of the strain-sensing element. The strain gage can be deposited directly onto sensing elements of any configuration, such as diaphragms, beams, or tubes.

Semiconductor—Strain gages are similar to the thin-film strain gages in that the strain-gage circuit is an integral part of the strain member. In integrated silicon strain-gage pressure transducers, the strain elements are diffused directly into the pressure-sensing element, becoming "atomically" bonded to the sensing member. Because silicon is virtually 100 percent elastic to the breaking point, this type of transducer exhibits very little hysteresis. Because gage factors in these types are in some cases more than 50 times greater than those of wire gages, signal output is high, which commonly eliminates the need for signal amplification.

The Wheatstone bridge, introduced earlier in the discussion of the Thevenin equivalent circuit, is one of the most common bridge configurations for strain-gage pressure transducers. In its simplest form the Basic Wheatstone bridge consists of four resistors arrayed to form a closed loop, a pair of sensing leads, and a pair of excitation leads. The bridge is affixed to a pressure-sensitive diaphragm or substrate. Pressure changes distort the substrate or diaphragm and cause the resistance of the bridge to change in response to strain induced on its resistors. In some designs, all bridge elements may be active, while in other designs only one element may be active.

Variations on the basic configuration of the Wheatstone bridge are referred to as Compensated Wheatstone bridges. These variations include additional resistor circuits, diodes, and circuit components designed to provide various types of compensation functions or signal enhancement capabilities, such as zeroing, shunt calibration, temperature compensation, and sensitivity adjustments (fig. 22).

The strain-gage bridge may be excited by either a constant voltage or a constant current, depending on the application and the excitation method used for calibration. There are advantages to each method.

Voltage—Most manufacturers provide calibrations and transducer specifications using voltage as the mode of excitation. The length of the lead wire (and hence its resistance) needs to be considered when selecting a transducer for a remote application. Short leads usually do not create significant measurement problems because the voltage loss on the excitation lead is small as a percent of the total excitation signal. The resistance of the lead increases as its length increases. The resistance of 20-gage annealed copper wire is approximately 0.01 ohm per foot. A transducer operated using a long wire lead should be calibrated with the lead attached. With a long lead, the voltage drop that develops across the sensing circuit is reduced in proportion to the resistance of the lead; the output signal will be reduced accordingly.

Current—Some transducers can be calibrated and operated using a constant current to supply excitation to the measurement bridge. The advantage of a constant-current excitation is that the effects of lead-wire resistance can be eliminated and the necessity of calibrating each unit with the lead wires attached can be avoided. Delivering a constant current to the device is achieved by allowing the voltage across the output leads to seek the necessary potential (within prescribed limits) required to make the current equal to the calibrated current setting. A current-generating source with the capability to regulate the voltage drop across the input leads is required to provide a constant current. Because current is controlled at the input end of the circuit, the same current will be present across the sensing circuit, provided there are no current leaks or shorts in the measuring circuit.

Understanding Pressure-Transducer Specifications

Selection of a pressure transducer requires careful review of the literature from prospective vendors. Comparing instrument specifications is a difficult and time-consuming process. Vendors commonly specify different sets of parameters and, typically, it is not clear which definitions are being applied to properly interpret a stated specification. Confidence levels are rarely specified and reporting specifications vary from one manufacturer to another. When in doubt, the vendor or manufacturer should be consulted for clarification and additional information. When selecting a pressure transducer, carefully consider the specifications of the instruments that will be used to excite and measure the output of the pressure transducer. These components of the measuring system may be the limiting factor in meeting overall performance objectives. An analysis of the circuits, using the principles in "An Overview of Direct Current Analysis," may be helpful in selecting these instruments. The input and output characteristics of the transducer must be compatible with the excitation and recording system used.

Terms frequently used to describe the performance characteristics of pressure transducers appear in the glossary. The definitions of many of these terms are based on terminology adopted by the Instrument Society of America (1970). Terms such as drift, error, error band, resolution, hysteresis, and time constant require additional explanation as illustrated in the following figures.

Drift, an undesired change in output over a period of time that is not a function of the measurand, is normally specified as a change in zero (zero drift) over time and a change in sensitivity (sensitivity drift) over time. Zero drift is illustrated in figure 23 as the vertical shift in the intersection of the best fit straight line between the initial calibration and the later calibration. The sensitivity drift is shown by the difference in slope of the best fit straight line between the initial calibration and the later calibration, where the initial calibration curve, adjusted for zero drift, is shown as the dashed line.

Error, the algebraic difference between the indicated value and the true value of the measurand, can be zero, span, or nonlinear error. Zero error is constant throughout the range of true values of the measurand as shown in figure 24. Span error is an error that changes linearly with the value of the measurand, as illustrated in figure 25. Nonlinear errors change as a nonlinear function of the value of the measurand, as illustrated in figure 26.

The error band is the band of maximum deviation of output values from a specified calibration curve or reference line. The sum of all the defined errors cause the measured values to differ from the true values. The measured values will fall within the error band, as shown in figure 27.

Hysteresis is the maximum difference in output, at any measurand value within the specified range, when the value is approached first with an increasing and then with a decreasing measurand. Expressed in percent of full-scale output, it is usually described in terms of maximum hysteresis in the output, as illustrated in figure 28.

The resolution of a measurement system is the smallest change in the measurand that can be measured or detected in the output reading. The ability to detect significant differences in output can be due to the construction of the transducer, noise in the system, or the numerical resolution of a digital data logger. The definition of resolution is illustrated in figure 29.

The sensitivity of transducer or system is the ratio of the change in transducer output (Δq_o) to a change in the measurand (Δq_i). Sensitivity may either be constant, in which case Δq_o/Δq_i will vary linearly over the operational range of the transducer (fig. 30a) or can vary nonlinearly (fig. 30b), in which case the sensitivity is commonly defined as a linear approximation for short segments of the curve.

The time constant is the length of time required for the output of a transducer to rise to 63 percent of its final value as a result of a step change in the measurand, as shown in figure 31. The time constant of a transducer is analogous to the time constant of an electrical circuit, in which the time constant is the time required for a capacitor charging through a resistor to reach 63 percent of the applied voltage.

PLANNING CONSIDERATIONS FOR SENSOR SYSTEMS

The type and number of sensors and data recorders needed for automated collection of water-level data depend on the objectives of the study. Determine these objectives prior to selecting system components. Options are numerous, but once the study objectives and needs are clearly determined, the selection of appropriate system components will be simplified. Some considerations for planning the installation of a water-level collection system are presented below. For many installations, submersible pressure transducers may not be needed, nor may they be the most suitable water-level sensors. In the following sections, however, submersible pressure transducers are assumed to be the preferred water-level sensors.

Study Duration and System Reliability

Nearly all submersible pressure transducers are capable of providing accurate results for short-term studies (such as aquifer tests or slug tests) but as the study duration increases, the chance of sensor failure and the amount of zero drift increases. Purchasing more expensive sensors, engineered to withstand the added demands of long-term deployment, may be necessary. Sensor maintenance and recalibration also becomes a consideration when designing a long-term data-collection effort.

For long-term investigations, the data logger and power-supply systems need more attention and protection. It may be necessary to recondition and recalibrate the data logger occasionally or to house it in a dry environment to prevent failure of components due to long-term exposure to moisture. Sensor cables may need to be protected with tubing or pipe to prevent long-term damage from ultraviolet radiation, physical weathering, exposure to ozone, or vandalism.

System reliability is among the most important considerations when designing a water-level monitoring system to be operated over a long duration. Redundancy, designed into the system so a partial failure will not result in complete loss of data, can range from multiple sensors in the same borehole connected to one data logger, to two or three completely separate systems logging water-level fluctuations in the same well. If a high degree of reliability is important, the study should be budgeted to provide early warning of system problems and fast access to replacement components to minimize down time.

Many manufacturers use terms such as mean time between failure and reliability to present durability information on their products. Mean time between failure is most commonly defined as the total time that a number of sensors operate, divided by the number of sensors that fail during the operational period. Reliability is the probability that an item will perform its intended function for a specified interval under stated conditions. Specified interval refers to the length of the study or test. Stated conditions refer to the operational environment—weather, humidity, temperature, and electromagnetic interference. Most of the time, the specified interval or the stated conditions supplied by the manufacturer are not the same as those of the hydrologic investigation. Also, reliability specifications usually refer to a single component of what commonly is a multiple-component system. For example, a pressure transducer may have one stated reliability and a data logger may have a different stated reliability, and the reliability of the combination of the two components (the system reliability) will be different from either of the reliabilities of the individual components. Most of the time, the overall system reliability will approximate the reliability of the least-reliable component.

Required Accuracy

Systems of pressure transducers, data loggers, cables and other supporting equipment used for sensing and recording water levels in wells should be sufficiently accurate to meet the needs of most ground-water projects of the USGS. The following have been suggested as standards: a water-level sensing and recording system should be capable of performing within a measurement error of + or – 0.01 ft. for most water-level measurement applications. For the case of large changes in water level (for example, during aquifer tests), this measurement error may not be achievable, and an accuracy of 0.1 percent of the expected range in water-level fluctuation is acceptable. Where the depth to water is greater than 100 ft, an accuracy of 0.01 percent of the estimated depth to water is generally acceptable. In summary, the measurement error and accuracy standard for most situations are 0.01 ft, 0.1 percent of range in water-level fluctuation, or 0.01 percent of depth to water above or below a measuring point, whichever is least restrictive.

Many hydrologic investigations in the USGS require the accuracy of the preceding suggested standard. While most sensor manufacturers produce devices that achieve this accuracy, the added complexities of the wiring, data logger, power source, and environmental variability may unacceptably degrade the overall system accuracy. Investigators may want to test for themselves the overall accuracy by conducting a pilot project before investing in a system that may not meet data objectives. In some cases, the desired accuracy may not be achievable with current technology or within budgetary constraints. For example, it is difficult to achieve a high level of accuracy with long leads, when depths to water are large, or when water-level fluctuations are large. Stringent accuracy constraints require frequent check measurements in the field.

Installation Location and Site Accessibility

If the study site is near the office, then frequent visits to the site to download data, perform site maintenance, replace failed components, or make accuracy check measurements may be reasonable. If, however, the site is remote or difficult to access, then the system needs to be designed to be operated remotely, and contain greater redundancy to better ensure uninterrupted collection of data. Remote sites may need an enhanced power supply, more robust shelters, extra data-storage capacity, equipment to allow communication with the site and transmission of data from the site, two or more transducers in a well, and automated checks for sensor drift.

System Components and Compatibility

When designing a data-collection system, determine which components are necessary and ensure that all of the components can communicate properly. Because power can be supplied by some data loggers, a short-term study might require only a pressure transducer connected to a data logger. For a long-term study, however, additional components including a power supply (batteries, solar panels, voltage regulator), additional data storage devices, a shelter or shelters, and a data-transmission system may be needed. Ensuring compatibility between components becomes more difficult as the number of components increases. For example, some data loggers can not interpret a digital signal from a transducer that makes an analog-to-digital conversion at the sensor. Similarly, the type of analog signal needs to be compatible; if the sensor sends an amperage signal, the data logger needs to be able to receive an amperage signal. Some pressure transducers require a separate measurement of temperature in order to correct the transducer output for changes in temperature in the well. If a data logger is not capable of receiving the temperature signal, the overall system accuracy is reduced. The data logger must also be able to supply the excitation voltage or current required by the sensor. When designing the installation, the number of sensors the data logger can simultaneously record needs to be considered.

Water Quality

Water quality must be considered when planning an installation. If the well will be used for water-quality sampling, the transducer and cable should be easy to clean before installing. Do not use lead or plastic-coated lead weights to apply tension to the cable. If the well is at a contaminated site, consider the possible effects of contaminants in the water that may corrode or otherwise degrade transducer components. Select components that are corrosion resistant, and easily decontaminated. Some manufacturers make chemical-resistant transducers of stainless steel or titanium, and polyfluroethelene-coated cables.

Number of Wells

For several wells in close proximity—for example, a nest of piezometers or multiple wells for a pumping test—one data logger that can receive signals from several pressure transducers usually is much less expensive than dedicating a data logger to each sensor. Data retrieval from one data logger also is much simpler. Instrumentation of many wells requires many pressure transducers, which can become cost-prohibitive for some studies. In some situations it may be possible to prioritize the need for continuous water-level data, and record water levels in key wells with pressure transducers and data loggers while manually measuring levels in other wells. If the study design calls for single, isolated wells to be instrumented, many manufacturers offer water-level sensing systems that allow the pressure transducer, data logger, power supply and cabling to be installed inside the well bore, thus protecting the entire system from the weather, vandalism, or theft.

Well Location, Diameter, and Depth

The setting in which the transducer is installed needs to be evaluated prior to installation. Wells installed in areas subject to strong electromagnetic fields, such as near generators, motors, pumps, power supplies, or similar devices may not be suitable candidates for some types of pressure transducers and may require additional protection and signal conditioning. Natural occurrences such as storms, precipitation, and lightning likewise can affect the transducer and data logger.

Wells installed in remote locations commonly require provision for additional data storage. If site visits are infrequent, the data-collection system should be robust, may need to contain redundancies, and may need to have a data-transmission capability.

If well diameters are small (less than 2 in.), the choice of pressure transducers is limited. For example, many wells installed in peat deposits are of small diameter to reduce the time lag for pressure equilibrium due to the typically low hydraulic conductivity of peat. Although some transducers are as small as 0.39 in. in diameter, most strain-gage pressure transducers will not fit down a well smaller than 1.25 in. in diameter. The investigator may need to choose a different type of transducer, such as a vibrating-wire pressure transducer, remembering that smaller transducers are usually more fragile.

Wells with a large depth to water present special problems for instrumentation. Unusually deep wells, such as those at the Nevada Test Site where water levels range from tens of feet to several hundreds of feet deep, pose many unsolved anomalous data problems—spikes, drift, inexplicable rises and recessions, lost data—as well as correlation problems between manual and continuous data (O'Brien, 1993). Pressure transducers hung from long cables can be affected by cable stretch. The cable also can expand and contract with temperature changes, thus raising and lowering the transducer in the well and introducing errors into the data.

Sensors are susceptible to impact-shock damage from hitting the sides of the well or being rapidly submerged during installation or calibration. Deep wells present more opportunities for this kind of mechanical damage.

Signals can degrade in long cables. The voltage attenuation and interference in long cables can cause erroneous data to be stored in the data logger. In addition to the effects on surface equipment, great differences in temperature and humidity between the surface and the water level, coupled with dissimilar metals, may lead to galvanic effects inducing voltages and transient currents that may distort the signal. The signal wires and their axially wound shields can act as inductive and capacitive circuits, which then may lead to ferromagnetic resonant effects, inducing transient currents into the signal-bearing channels. One solution is to convert the analog signal to a digital signal at the sensor and transmit the digital signal up the sensor cable to the data recorder. Another solution is to transmit the signal using AC current from 4-20 ma current-mode transducers. Current-mode signals are less susceptible to degradation than the more common DC voltage-mode transmissions.

Vent tubes on long cables have an increased chance of becoming clogged simply due to their greater lengths. Because of the great depths in some wells, environmental conditions may cause a variety of problems with both water-level measurements and data recording. The varying temperature and humidity and the atmospheric pressure gradients between the water level and land surface may cause vent tubes to become congested and ultimately allow moisture to be transported down into the sensitive electronic and mechanical portions of a pressure transducer. Preventing moisture entry is discussed in "Desiccation Systems."

Accurate check and calibration measurements are much more difficult when the depth to water is great. Manual wireline or tape measurement is more difficult because of the line stretch caused by weight-induced tension or temperature-induced expansion. These problems may call into question the accuracy of the data record when it is compared against the manual measurements.

Data-Collection Frequency

For most water-level investigations, the frequency of data collection is limited by the data logger and memory-storage system. Frequent observations require more memory in the data logger or storage devices. Commonly, water-level data in wells are collected no more frequently than hourly. For aquifer-test applications, however, the pressure transducer may limit data-collection frequency. In some situations, where recovery from an aquifer test is very rapid, observations should be made on the order of every 0.5 second. Some pressure transducers may take more time than that for the output to stabilize following excitation of the sensor. Some data loggers store the average of measurements taken over a period of several seconds, so recorded measurements of rapidly changing water levels may lag behind the true water levels.

Data Transfer

Data commonly are stored in a data logger or attached storage device, or both. In order to transmit the data from the logger to a computer in the office, a direct datalogger download can be made during a site visit or the data can be accessed remotely. Remote access can include automated transmission by satellite (Jones and others, 1991), phone line, cell phone, or radio signal; or the data can be stored on an onsite computer. The frequency of this transmission would depend on the timeliness of the data, ranging from the need for immediate transmissions to long transmission intervals designed to prevent exceeding the data storage capacity of the data logger or on-site storage device. Consult the manufacturers' manuals for data transmission techniques.

Cost

Most study objectives are compromised to some extent by budgetary constraints. Developing a priority of goals, and determining the cost of these goals, will provide the greatest value for the funding available. For example, if accuracy is the main goal, but maintaining an uninterrupted data record is a lesser priority, a study design could include very high quality pressure transducers but no system redundancies. Conversely, if maintaining a continuous data record is the highest priority, the study may be designed to have more than one sensor in each well, multiple data- storage systems, and backup power supplies, but use less expensive pressure transducers. The availability of personnel to service the data-collection sites commonly is the single most important decision that needs to be made when designing a study. If the study can afford frequent site visits, the accuracy of data and continuity of record nearly always are increased.

ASSEMBLY, CALIBRATION, AND TESTING

The user must be familiar with the behavior of the transducer, its installation, calibration and sources of error in the calibration to ensure that the data collected are reliable and reproducible. Depending on the accuracy requirements of the study as discussed previously, the user may choose to perform simple field checks of the transducer characteristics supplied by the manufacturer, or perform more detailed tests on individual transducers in the office. Studies requiring greater accuracy will require more extensive testing of the transducers. User-assembled transducer systems will require the most extensive testing.

Waterproof Housing

A rugged and inexpensive housing for these transducers can be assembled with a PVC-pipe coupling and two bushings (figs. 32, 33). The bottom bushing is center-drilled with a letter "D" bit (0.246-in diameter) for a pressed fit that allows the pressure port to stick out of the bottom of the housing. Fit the top bushing with a strain relief fitting. Insert the cable through the strain relief and top bushing and attach it to the transducer. The usual wiring convention is red-excitation; black-analog ground; white-signal high; green-signal low. The pressure port is pressed into the bottom bushing, while optionally sealing the contact between the PVC and pressure port with cyanoacrylic glue. Glue the bottom bushing into the coupling with PVC glue, fill the cavity with potting compound, and press the top plug into place. Purchase extra potting material and practice potting a few times on dummy housings without transducers to become familiar with the procedure.

Familiarization With Transducer Performance

Users should take time in the office to become familiar with new pressure transducers and data loggers, as well as with recalibrated transducers before taking them to the field. Failure to do so can result in much wasted time in the field. Before submerging the transducer, a series of performance tests should be done in the office to determine:

To test, connect the transducer to the data logger and set the output interval to about 5 to 30 seconds, depending on transducer response time, with output to a computer so that numbers, a graph, or both appear on the screen. With the sensor upside down, place a few drops of silicone oil in the transducer port. Compare transducer outputs with the transducer in normal vertical position (as it would be in a well) upside down, and horizontal. A fluid mass such as silicone oil below the transducer plane gives a highest reading with the transducer upside down, an intermediate reading sideways, and the lowest reading in normal vertical position. Warm and cool the transducer, noting changes in output. Vary the excitation voltage over its specified range, noting changes in transducer output. Leave the transducer to record for a few days at a constant temperature, then leave it to record while the temperature fluctuates over a temperature range that includes the range of temperatures expected in the field. For gage or differential transducers, the output during these tests should not change.

The constant-temperature test is a test of drift, exclusive of temperature effect. The variable-temperature test is a test of drift including temperature effect. For the variable-temperature test, return the transducer temperature to the starting temperature and subtract the prorated drift to get residual temperature effect. For an absolute transducer, subtract the output of an absolute-pressure standard from the measurements to get drift and temperature effect. Sudden temperature changes applied to a transducer (such as by holding the transducer in your hand, or lowering a transducer into a well) may cause unusual time-varying changes in output. These changes commonly are due to temperature gradients across the transducer element or among circuit board components.

Inexpensive Silicon Strain-Gage Pressure Transducers

Inexpensive pressure transducers can be soldered to inexpensive cable, potted in waterproof housings, and submerged in water to measure water-level changes in wells, piezometers, and stream gages; pore-pressure changes in saturated sediments; and soil-moisture tension. Measure submergence by using one transducer under water and another transducer as a barometer. Submergence is the calibrated output of the underwater transducer minus the calibrated output of the barometer. If multiple absolute transducers are used for water-level measurements, three transducers should be used for the barometer to ensure redundancy. Differential-pressure transducers require a vent tube from the reference port to the atmosphere but do not require a barometer for adjustment to submergence. An ancillary transducer used as a barometer, however, determines the barometric effect on water levels in wells and piezometers. Specifications for the transducers, cable, fittings and potting compound are given in Carpenter, (1994).

Pressure tests to measure hysteresis are relatively easy to perform. Apply a vacuum to the transducer and release, noting output; then apply positive pressure and release, noting output when the output stabilizes. The difference between the stabilized outputs is hysteresis. Temperature tests for hysteresis can be done in a bath of ice and fresh water by warming the transducer and putting it in the bath, noting the output; then cooling the transducer in a bath of ice and salt water and returning it to the bath of ice and fresh water, noting the output. Temperature tests for hysteresis are difficult to perform on absolute transducers because air pressure may have changed by the time the transducer has come to thermal equilibrium at the three different temperatures. In field installations where the water level or temperature fluctuates rapidly, attaining the desired accuracy may require correcting the transducer readings for hysteresis.

An interesting test to perform on differential pressure transducers is to apply the full-scale pressure to both ports. If the transducer were truly differential, no change in output would occur. In fact, the sensing element has undergone a volumetric strain from the pressure change, and a change in output will occur. In silicon strain-gage transducers, the change in output from application of full-scale pressure to both ports can be as much as one percent of the specified full-scale output. This shift is of little consequence in applications using high-range transducers. When using low-range differential transducers to obtain high resolution water-level fluctuations, however, the shift caused by barometric fluctuations, which can be as much as 0.6 ft of water, can produce errors of 1 percent times the ratio of the barometric fluctuation to the full-scale output of the transducer.

The next set of performance tests is done in the office during and after submerging the transducer. If the transducer has an obvious port, fill it with water. A tiny piece of fine screen with openings smaller than 0.0025 inches glued over the port will keep the water in and sand out. Inject water through the screen using a disposable insulin syringe, which has a very fine needle. The first wet test should be the effect of hydration on zero shift. For this test, note the pressure in a fixed orientation immediately after filling the port with water. The next day note the pressure (after assuring that the port is filled to the same level) in the same orientation. Hydration may be noticeable in some transducers that contain silicone oil or gel in the sensing element or in the port. Hydration does not cause continuing drift, but it is necessary to ensure that the transducer is hydrated before it is installed to avoid having to subtract this small effect from the field data.

Perform the right-side-up, upside-down, and horizontal tests again, comparing the wet outputs with the dry outputs. The wet output minus the dry output, in pressure units of feet of water, gives the length of the port which is used to determine the transducer plane. If the port points down, the difference should be positive with the transducer upside down and negative with the transducer right side up. Immersing the transducer in water should give the result of zero output when the transducer plane is even with the water surface. Another approach to determining the negative-standpipe effect is to connect a small-diameter copper or stainless-steel tube to the port, fill the tube and port with water, and bend the tube around until the tip is even with the transducer plane. Determining the transducer plane allows the user to become familiar with the positive and negative standpipe effects of the port. In field operations, if the port is not filled with water, it will capture an air bubble. The pressure inside the bubble does not increase and decrease linearly with water-level changes, because the bubble's air-water interface moves up and down. In addition, the bubble will gradually dissolve, producing drift. Pulling the transducer out of the water to check zero drift will give a negative reading equal to the column of water held in capillary tension below the transducer plane. Without a screen or small tube, some or all of the water may be shaken out, giving irreproducible results for a zero-drift check and different subsequent drifts for bubble dissolution. Not all transducers exhibit these standpipe effects, and the effects may be negative or positive. Experimentation and testing, which will determine the effects for a particular transducer, should be done in the office before going to the field.

Linear Transducer Calibration

A common calibration procedure for pressure transducers uses a standpipe to obtain different values of submergence and a linear regression or straight-line fit. The equation is:

in which V is transducer output, usually in millivolts; P is pressure, in feet of water; a is the y-axis intercept and b is the slope of the best-fit line. Equation 6 is solved for P giving:

In the calibration procedure, the variables P and V are commonly switched, giving:

Linear calibration of submersible pressure transducers can be done in the office using a vertical standpipe capped on the bottom and kept filled to overflowing with water, or in the field using a well. Be sure that no air is captured in the pressure port because, as explained earlier, the pressure at the diaphragm does not increase linearly with an increase in submergence when a bubble is present. Ensure that water temperature is constant throughout the water column, because density stratification in the standpipe will affect the pressure readings.

Calibration in a standpipe is done by inserting and withdrawing the transducer while maintaining the water level at the top of the pipe. The cable is marked and measured from the transducer plane at a minimum of five increments over the desired range of submergence. The transducer is submerged in the standpipe to a mark on the cable, and the distance and transducer output are noted. Repeat the procedure for all the marks. The linear regression coefficients are determined using a hand calculator or a spreadsheet program. Enter the coefficients into the data logger, and repeat the submergence procedure as a final calibration check. For the insertion and withdrawal phases transducer output may not be identical for a given submergence depth. This difference arises from hysteresis and lack of repeatability in positioning the transducer at precisely the same depth. Some experimentation is needed to determine positioning errors. In a standpipe calibration, it may be difficult to maintain or vary the temperature of the transducer in a controlled way. The transducer may also be calibrated in an observation well, as described in "Transducer Field Calibration."

Sources of Error in Linear Calibrations

The problem with linear regression for calibration of silicon strain-gage pressure transducers is that the procedure leaves significant residual errors while giving the false appearance of a good fit when the calibration points are plotted on the same graph as the straight-line fit. In fact, the r² for linear regressions can exceed 0.99999, while errors of as much as ±0.3 percent exist between the calibration points and the individual linear-regression equations (Carpenter, 1994). Major sources of error in linear calibration are thermal effects on the transducer and non-linear response of the transducer. Different manufacturers' transducers exhibit different curvature and temperature effects in their calibrations. The critical test in the calibration procedure involves plotting residuals of the calibration equation minus the calibration points. When the range of that plot is less than the accuracy required by the user, the calibration is adequate.

Thermal effects in a transducer can arise when water temperatures change at the transducer. In wells and piezometers, geologic features such as faults, fractures, and joints can provide conduits of high hydraulic conductivity. If the head varies between these features with time, water can flow vertically in the well, producing temperature changes. Well screens or gravel packs spanning aquifers with intervening confining units can allow flow within the borehole in response to nearby pumping owing to differences in hydraulic conductivity and specific storage between the aquifers. For example, a temperature change of 3.4°C in 15 minutes was observed at a depth of 51 m in an alluvial aquifer in response to turning on a pump in a nearby well (Carpenter and others, 1987).

Inadequate or nonexistent temperature compensation produces thermal effects in pressure transducers. Bridge networks in transducers commonly are compensated to:

The compensating networks commonly consist of series and parallel resistors, some of which exhibit temperature effects opposite to those of the transducer. Because a compensated design is a solution for the average of a transducer model, and because individual transducers vary, a particular design will reduce some errors and, in the process, introduce other, smaller errors. A compensated design may make it difficult to use a transducer below a manufacturer's error specifications and may make temperature correction by the user much more difficult than uncompensated designs. Some manufacturers apply a simple linear shift or offset for temperature compensation. Such compensation does not usually interfere with user-determined temperature correction of transducers to achieve higher accuracy.

Sources of Error When Making Direct-Current Measurements

The act of measuring the response of the pressure transducer can in itself introduce errors due to the effect of the meter on the circuit being measured. These errors are caused by current loading, shunt-resistance loading, magnetic coupling, and ground loops.

Voltage Measurements

Voltage measurement errors, other than those attributable to instrument calibration, can arise from the physical limitations of the voltmeter and from extraneous voltages generated in other parts of the measurement circuit. An ideal voltmeter would have an infinite input resistance and would draw zero current from the circuit being tested. Because the voltmeter resistance is not infinite and is connected between two points of unequal potential (in parallel with the circuit), a current will occur in the voltage measurement. This current draw may change the current magnitudes in the circuit enough to cause a considerable change in potential being measured. These limitations usually can be minimized with proper instrument selection. Some of the most important sources of error associated with voltage measurements are discussed below.

Input-current loading error, caused by the small amount of input current (I_in) that is drawn by the voltmeter during measurement (fig. 34) is proportional to the source resistance (R_s) of the measurement circuit. Because of this resistance, the source voltage (V_s) read by the voltmeter (V_m) is reduced by the loading current error (IinRs), so that V_m = V_s - I_inR_s. To reduce this error, a voltmeter with a small input current should be used.

Shunt-resistance loading error is caused by any resistance (R_shunt) that is in parallel with the test signal (fig. 35). Shunt-resistance sources include the resistance of the voltmeter and insulation leakage across the test leads and connectors. Because of shunt-resistance loading, the source voltage (V_s) read by the voltmeter (V_m) is reduced by the shunt resistance (Rshunt); so that V_m = V_s[R_shunt/(Rs + R_shunt)]. Shunt-resistance loading errors can be minimized by:

Thermal electromotive force (EMF) errors result from small voltages generated at the junction of two dissimilar metals by the thermocouple effect. The size of these thermocouple-induced EMFs depends on the type of metals used and the temperature differences between junctions. These thermocouple effects will either add to or subtract from the desired signal, producing an unpredictable offset because the EMFs generated by the junctions vary in response to temperature. Thermal EMF errors can be minimized by:

Magnetic coupling errors occur when a magnetic field of varying strength passes through a loop in the measuring circuit. Voltage induced by the magnetic field is proportional to the strength of the magnetic field, the rate of change of the magnetic field, and the loop area of the measuring circuit. Magnetic field effects can be minimized by:

Ground loops may produce both noise and error voltages. Ground loops arise from grounding the measurement instruments and output source at two or more different points on a common ground bus. The separation between ground points can result in a voltage between the output source and measurement instruments, causing a current to flow around the loop. Resistance in the loop produces an unwanted voltage in series with the output voltage of the transducer. To avoid ground loops, all test and measurement components should be grounded at one single point. Some pressure transducers may include a ground wire attached to the outer housing. When submerged, the pressure transducer itself may become a ground point in the circuit. To avoid creating a ground loop, the ground wire should not be connected to the measurement instrument if the measurement instrument itself also is grounded elsewhere. Alternatively, it may be better not to ground the measurement instruments but to use the pressure transducer ground as the common ground point for the entire measurement circuit.

Current Measurements

Current-measurement errors, other than those attributable to instrument calibration, can arise from the physical limitations of the current meter and from unknown or unwanted currents generated in other parts of the measurement circuit. An ideal current meter would have zero input resistance and no voltage drop across the input terminals. As in the case of the voltmeter, actual instruments do have some small resistance that can cause measurement inaccuracies. Since a current meter has some internal resistance, its insertion into the circuit (in series) may decrease the current in the measured branch. Current meter errors can be minimized with proper instrument selection. Some of the most important error sources associated with current measurements are discussed below.

Voltage burden errors, caused by the input resistance (R_in) of the current meter, result in a small voltage drop across the input terminals (fig. 36). The source current (I_s) that is measured by the current meter (I_m) is reduced by the input resistance (R_in) so that I_in = V_s/(R_s + R_in) or I_sR_s/(R_s+ R_in). To reduce this effect, Rin should be as small as possible. Current meters measure current either by measuring the voltage across a shunt resistor, or by using an active circuit such as a current to voltage (I to V) converter. Shunt-resistor current meters tend to have very high voltage burdens (in some cases up to several hundred millivolts), whereas the voltage burden of current meters with I to V converters are typically less than 100 microvolts.

Generated currents or stray currents can add to the desired signal current and introduce errors into the measurement. The effects of these can be minimized with careful attention to the measurement environment. Stray currents may be introduced into the measurement circuit from a number of different sources. These include vibration, insulator deformation, and contamination.

Vibration generates static charge and stray currents. Stray currents are generated by friction between the insulation and the electrical cable, a phenomenon known as the triboelectric effect. To minimize triboelectric effects, isolate the test circuit from sources of vibration, such as motors, pumps, and wind, and securely mount or tie down wires and cables.

Insulator deformation generates piezoelectric currents. Stray currents can be minimized by not overstressing terminal connections and using insulating materials that have low piezoelectric-generating properties.

Contamination caused by body oil and solder flux combined with humidity or moisture may create weak battery effects that result in stray currents. To minimize this source of error, keep all critical connections clean and free of contamination.

Resistance Measurements

Resistance measurement requires the use of either a current meter or a voltmeter and is subject to the same instrument limitations, error currents, and error voltages as are generated in other parts of the measurement circuit. Resistance is not measured directly; rather it is calculated from a known input (current or voltage) and a measured output (voltage or current) using Ohm's law. Resistance measurements can be made using either two wires or four wires.

In two-wire configurations, the resultant voltage or current across the circuit is measured across the same pair of wires used to introduce the input current or voltage (fig. 37). Make two-wire measurements either by supplying a constant current and measuring voltage or by supplying a constant voltage and measuring current. A special variant of the transducer is the transmitter. The output current of the transmitter is measured over the same two wires used for voltage excitation.

In four-wire configurations, one pair of wires is used to supply a current or a voltage and the second pair of wires is used to measure the resultant voltage (fig. 38). Most strain-gage pressure transducers are operated in the four-wire configuration. The main advantage of the four-wire configuration over the two-wire configuration is that there is effectively no voltage drop in the sense leads because there is generally very little current in the sense circuit. The voltage measured in this way is essentially the voltage developed across the test resistance.

A voltage source is normally employed when operating strain-gage pressure transducers. The voltage source generates an input current, and the resultant output voltage reflects changes in the bridge resistance caused by changes in pressure. Long lead wires can introduce errors into the voltage delivered to the transducer because of the resistance of the electrical cable. Although the resistance of the excitation leads generally is not large, the input current usually is significant and can result in a large voltage drop. Input voltages usually are much higher than the output voltage, and hence voltage losses in the input signal can be a source of error in the circuit. To compensate for voltage losses over long lead wires, calibrate the transducer with the lead wires attached.

Using a current source to measure voltage provides a constant current to the measurement circuit. Voltage at the source will vary in order to maintain a constant current to the transducer. A current-supply source with a voltage-regulating capability is needed for this type of operation; unregulated battery supply sources will not maintain a constant input current. Problems associated with voltage drops using long lead wires can be avoided by exciting the transducer with a constant current. The transducer must be calibrated using a current source, and the current input needs to be adjusted to match the excitation voltage recommended by the manufacturer.

Temperature-Corrected Transducer Calibration

A procedure that is more complicated than linear calibration, but is entirely manageable, can reduce residual error to ±0.03 percent of full-scale output or less. The pressure transducers are calibrated at five pressures at each of three temperatures in a water bath in a Dewar flask, insulated water jug, or wide-mouth thermos bottle (fig. 39). A calibrated field standard (precision transducer) with specified accuracy of ±0.01 percent of the full-scale reading is used with a hand pump. These components are connected either by plastic tubing and barbed nylon fittings or by 0.25 in. diameter drip-irrigation tubing and fittings that also distribute pressure to several transducers. A data logger collects and stores data from several pressure transducers (through differential analog inputs), from the field standard, and from several temperature sensors in the water bath (through single-ended analog inputs), transmitting the data to a computer screen. Immerse a precision thermistor connected to a display with a specified accuracy (±0.02°F) in the bath for calibrating the temperature sensors. Immerse the transducers in the water bath, allow them to equilibrate for about one-half hour with an occasional stir to maintain even temperature distribution, and make a set of nearly simultaneous measurements of all the sensors for each of five pressures over the pressure range of interest. Repeat the measurements at two more temperatures, allowing one-half hour equilibration at each temperature.

Because the field pressure standard is an absolute transducer, the calibration is straightforward. It has a tare switch for subtracting starting pressure; thus it can be used to calibrate differential transducers if the series of five pressure measurements is made reasonably quickly and the tare value is checked at the end of the five measurements. Reset the tare at each new temperature before each set of five pressure measurements. As a practical matter, atmospheric pressure does not change rapidly enough to be a problem in calibrating differential transducers, although sudden opening of doors (which cause changes in barometric pressure) should be avoided during a set of nearly simultaneous measurements. Note that the standard can be permanently damaged by overpressurization. Although some standards have a pressure-warning alarm that can be set, it is wise to choose a pump that cannot overpressure the standard. A bath filled with finely crushed ice and just enough de-ionized water to make a slush can be within ±0.04° of 0°F. Because of its repeatability and usefulness in calibrating transducers for temperature, the ice-water bath is a good choice for one of the calibration temperatures. Record the output time for each pressure manually, along with the readings from the pressure standard and precision thermometer, and use them to edit the data after they are transferred to a spreadsheet program. This calibration procedure produces values of voltage for ranges of pressure and temperature.

Multiple regression is used to determine the calibration equation for each pressure transducer. An equation that can reduce residual error with respect to the standard to ±0.03 percent over the calibration range of water-level fluctuation is:

in which V is transducer output, in millivolts; P is pressure, in feet of water; T is temperature, in degrees Fahrenheit; and a, b, c, d, e, and f are regression coefficients. Solving equation 9 for pressure gives:

The value of r² using equations 9 and 10 can exceed 0.9999999. In the spreadsheet, columns of P2, T2, and TP are generated from the data columns of T and P. Multiple regression then is used to generate the coefficients a, b, c, d, e, and f. The coefficients c, e, and f of P², T², and TP are all small, but the removal of any of these terms leaves a much larger residual than that from equation 10. If a temperature sensor is installed with the pressure transducer or built into its housing, equation 10 can be programmed into a data logger or calculated in a spreadsheet to give temperature-corrected output. Before removing a group of pressure transducers from the calibration equipment, check the entire calibration procedure by entering the coefficients and checking the temperature-corrected output against the output from the pressure standard for representative temperatures and pressures.

The pressure standard and precision thermometer are expensive. However, if many transducers are to be calibrated or if accuracy of ±0.1 percent of full scale or better is required, the convenience and accuracy of a pressure standard are well worth the cost. The accuracy obtained from inexpensive, user-assembled transducers (±0.03 percent of full scale) can approach the specified accuracy of the standard (±0.01 percent of full scale). A hand pump also is a major convenience. Because temperature is an ancillary measurement, its accuracy is not as critical as that of pressure, and the precision thermometer may be excluded, although an accuracy of ±0.1°C or better in calibrated field temperature sensors can be valuable for hydrologic interpretation (Carpenter and others, 1995). A bicycle pump and Schrader valve can be used for the hand pump. A simple cost comparison of total components and materials for inexpensive transducers, wire, potting, and calibration equipment plus calibration time versus the number of commercial transducers needed gives a breakeven point of three to ten transducers. The benefits of user-assembled transducers include the ability to:

INSTALLATION

For the successful collection of data, the submersible transducer and data-recording system must be correctly installed. This section provides general guidelines and suggestions, including the proper care and handling of transducers, considerations for designing shelters for equipment, discussion of suspension systems and desiccation systems, and field procedures for calibrating the transducer and optimizing the transducer's position in the well.

Care and Handling

Proper care and handling of a transducer is essential to its reliable operation. Most submersible pressure transducers come with a factory warranty, with specific instructions on necessary maintenance, and with a warning about mishandling. Failure to follow the instructions and warnings could cause irreparable damage to the transducer, void any factory warranty, and cause the corruption or loss of valuable data. The most common causes of damage are sudden impacts, freezing or overheating of the transducer, and sudden, extreme pressure changes (water hammer). Improper electrical connections also are a common source of transducer damage or failure.

Sharp impacts can break the body of the transducer and dislodge or break delicate electronic components. Keep the transducer packaged in its original shipping container until it is ready to be installed. After removing it from the packaging, take precautions to prevent the transducer from being dropped. Avoid sharp contact with the sides of the well or piezometer casing during installation and calibration. Do not over-torque a transducer when fitting it to an auxiliary threaded device such as a counterweight or packer housing.

Exposure to temperature extremes beyond the manufacturer's design specifications also can damage a transducer. Freezing or overheating can damage the sensor's electronics and, in extreme cases, damage its waterproofing by causing warping or cracking, disintegration or melting of the waterproofing material. The severity of damage to the waterproofing material of the transducer depends on the methods and materials used. Exposure to temperature extremes prior to installation can be avoided by storing the transducer in a cool, temperature-controlled environment. Most transducers come from the manufacturer with specifications as to the extremes of the operating temperature. The transducers should not be operated under, or exposed to, conditions that exceed the design limitations. Submersible transducers used to monitor changes in water level commonly are designed to function in the range of 0 to 120°F, though transducers operating through a gr

h =	P/γ + z	+	v² / 2g	(4)
	potential energy		kinetic energy

GF =	ΔR/R	(5)
	ΔL/L

P =	–(fT + b) + √(fT + b)2 –4c (a + dT + eT2 – V)	(10)
	2c

P =	V - a	(7)
	b