Transducers

 

 

Transducers

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Chapter 4 : Transducers - Generation and Detection of Ultrasound

• Introduction

 

• Transducers are used as both transmitters and receivers, converting electrical energy to acoustical energy and vice versa. In low frequency applications (below 20KHz), microphones and loudspeakers are two well-known examples. For diagnostic ultrasound, higher frequencies are required and piezoelectric materials are most commonly used.

• Piezoelectricity is defined as the generation of an electrical polarization in a substance by the application of a mechanical stress and, conversely, a change in the shape of a substance when an electric field is applied. In other words, a material is strained when an electric field is applied to it.

 

• Commonly used piezoelectric materials include naturally occurring crystals, such as quartz and certain man-made ceramic materials, such as lead zirconate-titanates (PZTs). Crystals such as quartz are inherently piezoelectric, with properties determined by their crystallographic features. In contrast, man-made ceramics are polarized above the Curie temperature (typically around 320-370oC for PZTs) by the application of strong electric fields to induce anisotropy responsible for their strong piezoelectric properties. PZTs are the most commonly used piezoelectric materials for diagnostic ultrasonic imaging.

• In addition to PZTs, PVDFs and composites are two other commonly seen materials in medical ultrasound. PVDF is often used for acoustic field measurements due to its broad bandwidth and sufficiently small thickness. Note that small thickness is necessary in order to minimize the interference of sound field due to the presence of the hydrophone. Composite materials, on the other hand, have gained wide interest due to the potential of improving imaging performance.

 

• Piezoelectric Constitutive Relations

• Consider the following figure, in which the equilibrium spacings between neighboring rows of atoms are L, a1 and a2, and q is the magnitude of the charge of the atoms,

 

 

 

 

the polarization (per unit volume) of the object is

.

• In the presence of strain S, a1 and a2 change to a1+Da1 and a2+Da2, respectively, and the polarization changes to P+DP. Since Da1=a1žS and Da2=a2žS, under the conditions that Da1 and Da2 are small, we obtain

 

,

where e is defined as the piezoelectric stress constant. The total change in electric displacement (or electric flux density) in the presence of an electric field E is

,

where e is the permittivity with zero or constant strain. We find that the electric displacement in a piezoelectric material is dependent on both the electric field and the strain.

• Similar to the above derivation, we can determine the stress in a piezoelectric medium due to an electric field E. Since the forces per unit area on the positive and negative atoms are ±qE/L2, the stresses in the regions of length a1 and a2 are therefore

 

.

The average internal stress in the medium due to the electric field is

.

The total stress () applied to the object is the sum of the externally applied stress T (or ) and the internal stress TE (or ). According to Hooke’s law and defining  as the elastic constant (under a constant electric field condition), we have

 
(or ).

• Equations  and  are known as piezoelectric constitutive relations. The above derivation assumed one-dimensional situations, where a single scalar e is adequate to represent the coupling between the elastic and electric properties.

 

• Wave Propagation in Piezoelectric Materials

 

• Equations governing the acoustic wave propagation in a piezoelectric material are obtained by using Newton’s second law with the constitutive relations. Based on previous derivation of one-dimensional wave propagation in a non-piezoelectric material, we can replace the bulk modulus B by the elastic constant  and

.

Re-arranging the above equation, we obtain

.

• Imagine the piezoelectric material in the form of a plate with metal electrodes on each face. If the electrodes on opposite faces are short circuited, the electric field is reduced to zero and the above equation is reduced to a homogeneous wave equation. On the other hand, if the electrodes are open-circuited or if the medium is infinite long in z, there would be no free charges in the transducer medium (i.e., D is a constant in z, but may be a function of time). Under this condition, the above equation takes the following form by applying the other constitutive relation

 

.

• Note that represents the propagation velocity in a non-piezoelectric medium. From the above equation, we have

 

.
It is then obvious that acoustic waves propagate at a higher velocity when the material is being piezoelectrically stiffened. The elastic constant under a constant electric displacement condition () is related to that under a constant electric field condition () by

.

The quantity , where  , is known as the electromechanical coupling constant. The magnitude of the electromechanical coupling constant is a useful index of the strength of the piezoelectric effect in a particular material. Note that  is the dielectric permittivity under a constant strain condition.

 

• Piezoelectric Generation and Detection of Ultrasound

• Ultrasonic waves are generated by the application of an external electric field to a piezoelectric material. A previously derived inhomogeneous wave equation indicates that a gradient in eE (the product of the piezoelectric stress constant and the electric field) is the source for the generation of mechanical disturbances. Usually the surfaces of a piezoelectric material offer the sharpest discontinuity in both e and E, hence they are the strongest sources of sound. This can be shown by the following simplified figure. Assuming a parallel plate capacitor containing a piezoelectric material of dielectric constant e.

 

 

      

 

• Consider the application of a charge density to the capacitor shown in figure (a). Since there are no free charge between the plates, we have  and hence D is a constant as illustrated in the figure (b). Additionally, the permittivity is typically higher inside the piezoelectric material, thus making the electric field E smaller than E in the outside. Therefore, figure (c) can be used to represent E and it becomes apparent that the only gradient in the electrical field occurs at the surfaces of the material. Consequently, the surfaces of the piezoelectric material are the predominant sources for the generation of ultrasound.

 

• Piezoelectric detection of ultrasonic waves is reciprocal to the process of wave generation. In other words, the conversion of mechanical energy into electrical energy is also a phenomenon dominated by the behavior at the surfaces of the piezoelectric material. As shown below, the voltage measured across a piezoelectric plate is the integral of the electric field over the thickness of the crystal

 

 

 

 

 

Using the constitutive relation relating the electric field to the strain and the electric displacement, the voltage becomes

From Gauss’s law, we have  where is the total charge on area . Therefore, the above equation reduces to

.

where  is the capacitance of the piezoelectric plate. The magnitude of the net displacement can be calculated from the boundary conditions placed at the surfaces of the material. In addition, the above equation shows that under open circuit situations, i.e.,  is not time-varying, the voltage developed across the plate is directly related to the relative displacements of the front and back surfaces of the material. If the thickness of the plate corresponds to an odd integral number of half-wavelengths of the ultrasonic wave impinging on the material (i.e., the two surfaces oscillate 180o out of phase), then the relative displacement of the front and back surfaces, , is the largest. In contrast, if the thickness of the crystal is an even number of half-wavelengths, then the amplitude of oscillation of the two surfaces is in phase, and therefore, .

• Equivalent Circuits

 

• Based on the piezoelectric detection equation that we previously derived, we have .

Taking the partial derivative with respect to time on both sides, we obtain

which can be represented by the following equivalent circuit.
 

 

 

• Defining force as we do voltage in electrical circuits, and particle velocity as we do current in electrical circuits, the non-piezoelectric component can be described using the other constitutive relation (i.e., ) and represented by the following circuit (a transmission line)

 

 

 

 

 

 

where  due to symmetry. Let represent the characteristic impedance of the piezoelectric material, we can obtain the following equations using methods similar to those used by deriving the acoustic wave equations

.

• In order to obtain  and , the following relations need to be used.

 

.

Letting , the following equation yields

.

Therefore
.

and

Furthermore,  (=) can be obtained as the following:

• By using a transformer to couple the electrical components with the acoustic components, we then obtain the following equivalent circuit (a.k.a. Mason Equivalent Circuit)

 

 

 

 

• With the front and back faces of the transducer being loaded by mechanical impedances and , the equivalent circuit can be re-drawn as the following:

 

 

 

• The above circuit can be further transformed to the following circuit identity with Z4=1/Z1.

 

• In the neighborhood of a mechanical resonance, impedance of the branch containing 2Z1 is large and the branch can be neglected. Therefore, the model can be approximated with the following circuit. Such an approximation is adequate to describe the operation of a transducer near resonance driving a wide variety of mechanical loads.

 

 

 

• An alternative equivalent circuit (KLM Equivalent Circuit) has been developed to be better suited for broadband operations. Please refer to the following paper if interested.

 

- D. Leedom, R. Krimholtz and G. Matthael, “Equivalent circuits for transducers having even or odd symmetry piezoelectric excitation”, IEEE Trans. on Sonics and Ultrasonics, vol. SU-18, No. 3, pp. 128-141, July 1971.

• Design Considerations for Broadband Transducers

 

• Broadband transducers are necessary for pulse-echo imaging applications in order to achieve high range resolution, which is inversely proportional to the pulse bandwidth. However, a short pulse is usually achieved by sacrificing sensitivity. Considering the following piezoelectric transducer with no matching or damping layers, it rings and produces an unacceptable long pulse.

 

 

 

By placing a lossy material (highly attenuating) which has a similar acoustic impedance as the PZT, the reflection at the back of the transducer can be reduced and therefore the pulse can be shortened. Apparently, sensitivity is degraded due to attenuation.
 

 

The acoustic impedance of a typical PZT material is around twenty times higher than that in the body, therefore, part of the sensitivity loss can be recovered by adding one or multiple quarter wave matching layers at the front surface of the transducer.
 

 

 

• Two-way insertion loss, defined as the ratio of the available electrical power generated by the device as a receiver to the electrical power dissipated in the device as a transmitter under the conditions in which the acoustic wave produced is reflected from a perfectly reflecting interface and received by the same transducer, is often used as a measure of the electromechanical efficiency of the transducer. The lower the insertion loss is, the higher the sensitivity can be achieved.

 

• Equivalent circuits can be used to match both the mechanical impedance and the electrical impedance. Ideally, real part of the electrical impedance should be 50Ohms in order to match a typical transmitter output impedance. Imaginary part of the impedance, on the other hand, should be to zero in order to obtain optimal efficiency. This is typically done by placing an inductor (as a tuning element) to cancel the capacitance of the transducer (and sometimes the transducer cable). However, the improvement in sensitivity is often gained at the price of bandwidth of the spectrum.

• An acoustic lens is often placed on the front of the transducer in order to provide a fixed geometric focusing. This is particularly important for imaging using one-dimensional arrays, in which case the geometric focusing is provided along the non-scan direction.

 

• A transducer array consists of many piezoelectric elements. The elements are arranged depending on specific scan formats. A typical diagram of a one-dimensional transducer array is shown in the following.

 

 

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Chapter 1

Transducers

Basically transducer is defined as a device, which converts energy or information from one form to another. These are widely used in measurement work because not all quantities that need to be measured can be displayed as easily as others. A better measurement of a quantity can usually be made if it may be converted to another form, which is more conveniently or accurately displayed. For example, the common mercury thermometerconverts variations in temperature into variations in the length of a column of mercury. Since the variation in the length of the mercury column is rather simple to measure, the mercury thermometer becomes a convenient device for measuring temperature. On the other hand, the actual temperature variation is not as easy to display directly. Another example is manometer, which detects pressure and indicates it directly on a scale calibrated in actual units of pressure.
Thus the transducer is a device, which provides a usable output in response to specific input measurand, which may be physical or mechanical quantity, property or condition. The transducer may be mechanical, electrical, magnetic, optical, chemical, acoustic, thermal nuclear, or a combination of any two or more of these.

1. Mechanical transducers:

Are simple and rugged in construction, cheaper in cost, accurate and operate without external power supplies but are not advantageous for many of the modern scientific experiments and process control instrumentation owing to their poor frequency response, requirement of large forces to overcome mechanical friction, in compatibility when remote control or indication is required, and a lot of other limitations. All these drawbacks have been overcome with the introduction of electrical transducers.

1.2 Electrical Transducers:

Mostly quantities to be measured are non-electrical such as temperature, pressure, displacement, humidity, fluid flow, speed etc., but these quantities cannot be measured directly. Hence such quantities are required to be sensed and changed into some other form for easy measurement.
Electrical quantities such as current, voltage, resistance inductance and capacitance etc. can be conveniently measured, transferred and stored, and therefore, for measurement of non-electrical quantities these are to be converted into electrical quantities first and then measured.  The function of converting non-electrical quantity into electrical one is accomplished by a device called the electrical transducer. Basically an electrical transducer is a sensing device by which a physical, mechanical or optical quantity to be measured is transformed directly, with a suitable mechanism, into an electrical signal (current, voltage or frequency). The production of these signals is based upon electrical effects which may be resistive, inductive, capacitive etc in nature. The input versus output energy relationship takes a definite reproducible function. The output to input and the output to time behavior is predictable to a known degree of accuracy, sensitivity and response, within the specified environmental conditions.

 

1.3 Basic Requirements of a Transducer:

The main function of a transducer is to respond only for the measurement under specified limits for which it is designed. It is, therefore, necessary to know the relationship between the input and output quantities and it should be fixed. Transducers should meet the following basic requirements.

1. Ruggedness. It should be capable of withstanding overload and some safety arrangement should be provided for overload protection.
2. Linearity. Its input-output characteristics should be linear and it should produce these characteristics in symmetrical way.
3. Repeatability. It should reproduce same output signal when the same input signal is applied again and again under fixed environmental conditions e.g. temperature, pressure, humidity etc.
4. High Output Signal Quality. The quality of output signal should be good i.e. the ratio of the signal to the noise should be high and the amplitude of the output signal should be enough.
5. High Reliability and Stability. It should give minimum error in measurement for temperature variations, vibrations and other various changes in surroundings.
6. Good Dynamic Response. Its output should be faithful to input when taken as a function of time. The effect is analyzed as the frequency response.
7. No Hysteretic. It should not give any hysteretic during measurement while input signal is varied from its low value to high value and vice-versa.
8. Residual Deformation. It should be no deformation on removal of local after long period of application.

1.4 Selection of Transducers:
In a measurement system the transducer (or a combination of transducers) is the input element with the critical function of transforming some physical quantity to a proportional electrical signal. So selection of an appropriate transducer is most important for having accurate results.
The first step in the selection procedure is to clearly define the nature of quantity under measurement (measurand) and know the range of magnitudes and frequencies that the measurand is expected to exhibit. Next step will be to examine the available transducer principles for measurement of desired quantity.
The type of transducer selected must be compatible with the type and range of the quantity to be measured and the output device.
In case one or more transducer principles are capable of generating a satisfactory signal, decision is to be taken whether to employ a commercially available transducer or build a suitable transducer. If the transducers are available in the market at a suitable price, the choice will probably be to purchase one of them, otherwise own transducer will have to be designed, built and calibrated.
The points to be considered in determining a transducer suitable for a specific measurement are as follows:

1. Range. The range of the transducer should be large enough to encompass all the expected magnitudes of the measurand.
2. Sensitivity. The transducer should give a sufficient output signal per unit of measured input in order to yield meaningful data.
3. Electrical Output Characteristics. The electrical characteristics-the output im­pedance, the frequency response, and the response time of the transducer output signal should be compatible with the recording device and the rest of the measuring system equipment.
4. Physical Environment. The transducer selected should be able to withstand the environmental conditions to which it is likely to be subjected while carrying out measurements and tests.

Such parameters are temperature, acceleration, shock and vibration, moisture, and corrosive chemicals might damage some transducers but not others.

1. Errors. The errors inherent in the operation of the transducer itself, or those errors caused by environmental conditions of the measurement, should be small enough or controllable enough that they allow meaningful data to be taken.

However the total measurement error in a transducer-activated system may be reduced to fall within the required accuracy range by adopting the following techniques.

1. Calibrating the transducer output against some known standards while in use under actual test conditions. This calibration should be performed regularly as the measurement proceeds.
2. Continuous monitoring of variations in the environmental conditions of the transducer and correcting the data accordingly.

Controlling the measurement environment artificially in order to reduce possible transducer errors artificial environmental control includes the enclosing of the transducer in a temperature-controlled housing and isolating the device from external shocks and vibrations.

1.5 Classification of Transducers:

The transducers may be classified in various ways such as on the basis of electrical principles involved, methods of application, methods of energy conversion used, nature of output signal etc as shown in
table (1.1).

1. Primary and Secondary Transducers: Transducers, on the basis of methods of applications, may be classified into primary and secondary transducers. When the input signal is directly sensed by the transducer and physical phenomenon is converted into the electrical form directly then such a transducer is called the primary transducer.

For example a thermistor used for the measurement of temperature fall in this category the thermistor senses the temperature directly and causes the change in resistance with the change in temperature.
When the input signal is sensed first by some detector or sensor and then its output being of some form other than input signals is given as input to a transducer for conversion into electrical form, then such a transducer falls in the category of secondary transducers.
For example, in case of pressure measurement, bourdon tube is a primary sensor which converts pressure first into displacement, and then the displacement is converted into an output voltage by an LVDT. In this case LVDT is secondary transducer.

1. Active and Passive Transducers:Transducers, on the basis of methods of energy conversion used, may be classified into active and passive transducers. Self-generating type transducers i.e. the transducers, which develop their output the form of electrical voltage or current without any auxiliary source, are called the active transducers. Such transducers draw energy from the system under measurement. Normal such transducers give very small output and, therefore, use of amplifier becomes essential.

Transducers, in which electrical parameters i.e. resistance, inductance or capacitance changes with the change in input signal, are called the passive transducers. These transducers require external power source for energy conversion. In such transducer electrical parameters i.e.resistance, inductance or capacitance causes a change in voltages current or frequency of the external power source. These transducers may draw sour energy from the system under measurement. Resistive, inductive and capacitive transducer falls in this category.

1. Analog and Digital Transducers:Transducers, on the basis of nature of output signal, may be classified into analog and digital transducers. Analog transducer converts input signal into output signal, which is a continuous function of time such as thermistor, strain gauge, LVDT, thermo-couple etc. Digital transducer converts input signal into the output signal of the form of pulse e.g. it gives discrete output. These transducers are becoming more and more popular now-a-days because of advantages associated with digital measuring instruments and also due to the effect that digital signals can be transmitted over a long distance without causing much distortion due to amplitude variation and phase shift.

Sometimes an analog transducer combined with an ADC (analog-digital convector) is called a digital transducer.

1. Transducers and Inverse Transducers:Transducer, as already defined, is a device that converts a non-electrical quantity into an electrical quantity. Normally a transducer and associated circuit has a non-electrical input and an electrical output, for example a thermo-couple, photoconductive cell, pressure gauge, strain gauge etc. An inverse transducer is a device that converts an electrical quantity into a non-electrical quantity. It is a precision actuator having an electrical input and a low-power non-electrical output. For examples a piezoelectric crystal and transnational and angular moving-coil elements can be employed as inverse transducers. Many data-indicating and recording devices are basically inverse transducers. An ammeter or voltmeter converts electric current into mechanical movement and the characteristics of such an instrument placed at the output of a measuring system are important. A most useful application of inverse transducers is in feedback measuring systems.

Table (1-1)
Classification of Electrical Transducers

 

 

 

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Transducers

 

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Transducers