**1. Introduction**

Acoustical measurement is ubiquitous in industrial applications, scientific research, and daily life, e.g., mobile and internet communication [1,2], exploration of underground mineral resources (oil, gas, coal, metal ores, etc.) [3], measurement of the in situ stresses of underground rock formation [4], and the inspection of mechanical properties of concrete [5,6], as well as intravascular ultrasound [7], medical imaging [8], biometric recognition [9], implantable microdevices [10], rangefinders [11], nondestructive detection [12–14], experimental verification of acoustic lateral displacement [15], inspection of a specific polarization state of a wave propagating in layered isotropic/anisotropic media [16,17], wave energy devices [18], and more. One of the key factors toward achieving a high-quality acoustic measurement is a good understanding of the properties of the acoustic transducers, e.g., the type of the transducers, the material property, and the geometric structure of the transducers.

A unique characteristic of piezoelectric materials is their electric–mechanical transduction ability, converting mechanical energy to electrical energy and vice versa. This property has been exploited extensively in the construction of acoustic transducers for industrial applications [19], e.g., in electrical engineering, biomedical engineering, and geophysical engineering, among others. Along with technological progress, the quality of piezoelectric transducers has also been improving dramatically, e.g., smaller geometric dimensions, reduced noise level [20], lowered power consumption [21], etc. The typical geometric structures of acoustic transducers in practical applications are cylindrical, schistose, spherical, among others. Thin cylindrical transducers are widely used in industry, e.g., in petroleum logging tools. The physical properties of cylindrical piezoelectric transducers are also widely studied, including radiation, electric–acoustic and acoustic–electric conversions, and more. The radiation of a cylinder transducer with harmonic vibration was reported by Bordoni et al. [22] and Williams et al [23]. Fenlon [24] reported calculations for the acoustic radiation field at the surface of a finite cylinder using the method of weighted residuals. Wu [25] described an application of a variational principle for acoustic radiation from a vibrating finite cylinder. The e ffect of the length and the radius of a cylinder on its radiation e fficiency was discussed and reported by Wang and coworkers [26,27].

We are concerned here with the thin cylindrical piezoelectric transducers used in petroleum logging tools, either as an acoustic source or as a receiver. Conventionally, in studies of acoustic-logging, the simplified analytical-models have been used for measuring acoustic signal waveforms and in processing the measured acoustic signal. Many times, the Tsang wavelet has been used as an acoustic source in acoustic logging [28,29]. Oversimplified acoustic-source functions have been used in forwarding-model research of acoustic logging and/or in inversion analysis and processing of the measured acoustic-logging signal, e.g., Ricker wavelet [30], Gaussian impulse wavelet [31], etc. However, the mathematical expressions of the simplified models have not been able to provide the practical relationship between a driving-voltage signal, geometrical and physical properties of the transducer, and radiated acoustic signal-wavelet.

The transient response of a transducer driven by a sinusoidal electrical signal was reported by Piqtuette [32,33]. However, the reported results were not adequate for practical considerations for either acoustic logging or other acoustic measurements due to the driving-voltage signal of an exciting transducer containing multiple frequency components with di fferent amplitudes and phases. In the process of providing a radiating acoustic signal, the transducer is also counteracted by the acoustic field radiated by itself and creates a radiation resistance and radiation mass, which are functions of the vibration frequency on the transducer's surface.

The complex e ffect of radiation resistance and radiation mass on the radiated acoustic-signal wavelet was reported by Fa and co-workers, where the deriving-voltage signal contained multi-frequency components. These researchers reported the case of thin spherical-shell transducers polarized in the radial direction, which radiated acoustic waves omnidirectionally [34–37]. Even though these reports are meaningful for modeling, the adopted transducers in the acoustic-logging tools and many other practical applications are mostly cylindrical, with radiation directivities quite di fferent from that of the thin spherical-shell transducers.

It is understood that correct analysis and inversion interpretation of the measured acoustic-logging signal wavelet rely on accurate acoustic measurement instruments, which must be established on a solid physical foundation with a strict engineering mechanism. Based on our current knowledge of the thin cylindrical piezoelectric transducers used in petroleum logging tools, further development is highly desirable to achieve an enhanced understanding to develop applicable measurement instrumentations.

In this paper, we report a study of thin cylindrical transducers widely used in petroleum acoustic-logging tools. By adopting the method describing a thin spherical-shell transducer's transient response [34] for the excited driving-voltage signal wavelet with multi-frequency components, we established the parallel-connected equivalent circuits for the thin cylindrical-shell transducers polarized in the radial direction. By solving the corresponding equations of motion, we analyzed the physical properties related to the transient response. Technically, we employed a measurement system with a high-resolution, high-sampling-rate digitizer to perform an experimental measurement, e.g., with a resolution range from 16-bit to 24-bit and a sampling rate from 500 KS/s to 15,000 KS/s. From this system, we were able to achieve good measurements in a large frequency range, even for weak acoustic signals. Based a proper analysis using the experimentally acquired data, we obtained the relationships between the various physical quantities, e.g., the driving electric-signal wavelet, the electric–acoustic/acoustic–electric conversion factors, the propagation media, and the measured

acoustic signal. This, in turn, provided us with the ability to inspect the properties of the transducers, obtain the physical parameters of the measured fluid and solid material, check the quality of the measured objects, and perform a verification of existing and/or on-going scientific research. We report that the calculated results of the physical properties and transient response of the thin cylindrical-shell transducer were in good agreemen<sup>t</sup> with those of experimental observations.

#### **2. Theory and Modeling**

Let us consider a piezoelectric, thin, cylindrical transducer, with an average radius ρ0 and a wall thickness *lt*, polarized in the radial direction. The electrodes were connected to the inner and outer surfaces, as shown in Figure 1. Because the radius of the thin cylinder is much larger than its thickness (ρ0 *lt*), we have the following approximations: ρ0 ≈ ρ*a* ≈ ρ*b* and ρ0 = (ρ*a* + ρ*b*)/2 From the axis symmetry of the particle displacement, tangential and axial stresses are equal zero, i.e., *<sup>T</sup>*ρ*z* = *<sup>T</sup>*ρφ = *Tz*φ = 0.

**Figure 1.** A piezoelectric thin spherical-shell transducer.

If the inner and outer surfaces are free from any other forces, i.e., the normal stress in the radial direction is *T*ρ = 0, then the equation of motion for the particle vibrations of the transducer can be simplified to:

$$
\rho\_n \frac{\partial^2 u\_\rho}{\partial t^2} = -\frac{T\_{\varphi}}{\rho\_0},
\tag{1}
$$

$$
\rho\_n \frac{\partial^2 \mu\_z}{\partial t^2} = -\frac{\partial T\_z}{\partial z},
\tag{2}
$$

where *T*φ and *Tz* are the normal stress in tangential and axial directions, respectively; *<sup>u</sup>*ρ and *uz* are the particle displacement in the radial and axial directions, respectively; and ρ*n* is the density of the transducer material.

Employing subscripts {1, 2, 3} in place of subscripts {ϕ, *z*, ρ}, the piezoelectric equations with respect to radial polarization can be expressed as:

$$S\_1 = s\_{11}^E T\_1 + d\_{31} E\_{3\prime} \tag{3}$$

$$D\mathfrak{z} = d\mathfrak{z}\_1 T\mathfrak{z}\_1 + \varepsilon\_{33}^T E\mathfrak{z}\_1 \tag{4}$$

where *S*1 and *T*1 are the strain and stress in ϕ-direction, respectively; *D*3 and *E*3 are the radial components of the electric displacement and electric field vectors, respectively; and *<sup>s</sup>E*11, <sup>ε</sup>*T*33, and *d*31 are the compliance, piezoelectric, and dielectric constants of the piezoelectric material, respectively. Because the height ( *H*) of the transducer is much larger than its thickness, the coupling between the axial and radial vibrations can be neglected and the axial stress *T*2 can be ignored. The vibration of the thin cylindrical transducer can be simplified as being one-dimensional in the radial-direction.

Substituting Equation (3) into Equation (1) yields:

$$m\frac{d^2u\_\rho}{dt^2} = -\frac{2\pi Hl\_t}{s\_{11}^E}S\_1 + \frac{2\pi Hl\_t d\_{31}}{s\_{11}^E}E\_{3\prime} \tag{5}$$

where *m* = <sup>2</sup>π*Hlt*ρ0ρ*n* is the mass of the transducer.

We set the transducer to be in coupling fluid, where only its outer surface was in contact with the coupling fluid. The transducer vibrates in the radial direction, where the vibration of the thin cylindrical transducer's outer surface causes the surrounding medium to expand and contract alternately in the radial direction. Consequently, the acoustic waves are radiated outward. Meanwhile, the thin cylindrical transducer is in the acoustic field radiated by itself. Then, it is acted on by a counterforce caused by the acoustic field. On the outer surface, by using a similar method described in References [34,35], we can obtain this counter-force as follows:

$$\begin{cases} F\_r = -R \Big( \frac{k^2 \rho\_0^2}{1 + k^2 \rho\_0^2} + j \frac{k \rho\_0}{1 + k^2 \rho\_0^2} \Big) \frac{d\mu\_\rho}{dt} = -\Big( R\_\rho + iX\_\rho \Big) \frac{d\mu\_\rho}{dt} = -Z\_\rho \frac{d\mu\_\rho}{dt} \\\ R\_\rho = R \frac{k^2 \rho\_0^2}{1 + k^2 \rho\_0^2}, \; X\_\rho = R \frac{k \rho\_0}{1 + k^2 \rho\_0^2} \end{cases} \tag{6}$$

where *R* = 2πρ0*H*ρ*mvm*; ρ*m* and *vm* are density and acoustic velocity of coupling fluid around the transducer, respectively; *i* is the unit imaginary number; and the radiation resistance and the radiation reactance are *<sup>R</sup>*ρ.

If the thin cylindrical transducer vibrates harmonically with various frequencies, its radiation resistance and radiation reactance would also be di fferent, where *k* = <sup>ω</sup>/*vc*.

Due to the viscosity of the coupling liquid, the vibration of the transducer creates a frictional resistance force, which can be expressed as:

$$F\_f = = -R\_{\text{ll}} \frac{du\_\rho}{dt},\tag{7}$$

where *Rm* is the frictional resistance on the surface of the transducer, which is proportional to the viscosity coe fficient of the coupling fluid and the outer side wall area of the transducer. The transducer's radiation surface is approximately *A* ≈ 2πρ*bH*, and the total outer force acted on the transducer is:

$$F = F\_r + F\_f = -(R\_\rho + R\_m + iX\_\rho)\frac{du\_\rho}{dt}.\tag{8}$$

The axial symmetry of thin-cylindrical transducer leads to:

$$S\_1 = \frac{u\_\rho}{\rho\_0}.\tag{9}$$

For the harmonic vibration, substituting Equations (8) and (9) into Equation (5) yields:

$$u\_{l} = \frac{2\pi Hl\_{l}d\_{31}/s\_{11}^{k}}{-\omega^{2}\left(m+m\_{\rho}\right) + j\omega\left(R\_{\rho}+R\_{m}\right) + 1/C\_{m}}E\_{3\prime} \tag{10}$$

where *Cm* = <sup>ρ</sup>0*sE*11/(<sup>2</sup>π*Hlt*). Also, from Equations (3) and (4), we have:

$$D\_3 = \frac{d\_{31}}{s\_{11}^E \rho\_0} u\_\rho + \varepsilon\_{33}^T (1 - K\_{31}^2) E\_3 \tag{11}$$

where *K*31 = *<sup>d</sup>*31/ -*sE*11ε*T*33.

Because the thickness of the cylindrical transducer is small enough, the edge effect of the upper and lower cross-section can be neglected. From the Gauss theorem, the total charge on each electrode (either inner or outer surface) of the thin cylindrical transducer is *Q* = 2πρ0*HD*3. The instantaneous current into the electrodes is the time derivative of *Q*. For a harmonic driving electric signal, we have:

$$I = \frac{dQ}{dt} = i\omega \mathbf{C}\_0 V + N^2 \frac{V}{R\_\rho + R\_m + i\omega (m + m\_\rho) + 1/i\omega \mathbf{C}\_m},\tag{12}$$

where *C*0 = <sup>2</sup>πρ0*H*ε*T*33(<sup>1</sup> − *<sup>K</sup>*2*p*)/*lt*; *N* = <sup>2</sup>π*Hd*31/*sE*11, which is the electric–mechanical turn coefficient of the thin cylindrical transducer; and *V* = *ltEr*, which is the voltage across the two electrodes of the thin cylindrical transducer.

Suppose that the driving circuit outputs a sinusoidal signal *<sup>U</sup>*1(*t*) with an angular frequency ω and output resistance *Ro*. Based on Equation (12), the electric–acoustic equivalent circuit of the transducer for harmonic vibration is shown in Figure 2a. Its corresponding *s*-domain network is shown in Figure 2b, where *<sup>v</sup>*ρ(*t*) and *<sup>v</sup>*ρ(*s*) are defined as particle vibration speeds at the transducer's surfaces, and *<sup>m</sup>*ρ is defined as the transducer's radiation mass.

**Figure 2.** Equivalent circuits of a thin cylindrical transducer for electric–acoustic conversion: (**a**) equivalent circuit in the time domain and (**b**) equivalent circuit in the *s*-domain, where it is excited by a harmonic sinusoidal electric signal. *<sup>U</sup>*1(*t*) is the driving voltage source and *R*o is its output resistance; *V*(*t*) is the voltage signal at the electric terminals of the source; *<sup>m</sup>*ρ, *<sup>R</sup>*ρ, *Cm*, *m*, *Co*, *N*, and *Rm* are the radiation mass, radiation resistance, elastic stiffness, mass, clamped capacitance, mechanical–electric conversion coefficient, and fraction force resistance of transducer, respectively; and *<sup>v</sup>*ρ(*t*) is the vibration speed at the transducer surface.

At the electric terminals of the *s*-domain in the network, as shown in Figure 2b, we have:

$$\mathcal{L}I\_1(\mathbf{s}) = V(\mathbf{s}) + I(\mathbf{s})R\_0 \tag{13}$$

and

$$I(s) = s\mathbb{C}\_0 V(s) + I\_1(s) = s\mathbb{C}\_0 V(s) + Nv\_r(s). \tag{14}$$

*Micromachines* **2019**, *10*, 804

The transient response process of the thin cylindrical transducer can be held as a zero-state response. In the *s*-domain, we define the electric–acoustic conversion system function as a ratio of the vibration speed of the transducer's outer surface to the sinusoidal driving voltage signal. In terms of Figure 2b and Equations (13) and (14), this electric–acoustic conversion function can be expressed as:

$$H\_1(s) = \frac{v\_r(s)}{lI(s)} = \frac{ds}{s^3 + as^2 + bs + c},\tag{15}$$

where:

 $a = \frac{R\_m + R\_\rho}{m + m\_\rho} + \frac{1}{R\_0 \mathbb{C}\_0}$   $b = \frac{R\_m + R\_\rho}{(m + m\_\rho)R\_0 \mathbb{C}\_0} + \frac{1}{(m + m\_\rho)\mathbb{C}\_m} + \frac{N^2}{(m + m\_\rho)\mathbb{C}\_0}$   $c = \frac{1}{(m + m\_\rho)\mathbb{C}\_0 \mathbb{C}\_0}$   $d = \frac{N}{(m + m\_\rho)\mathbb{C}\_0 \mathbb{R}\_0}$ .

By applying the residue theorem to Equation (15), we obtain the electric–acoustic impulse response of the thin cylindrical transducer:

$$h\_1(t) = \sum\_{i=1}^{L} \text{Res}[H\_1(s\_i)e^{\varsigma\_i t}],\tag{16}$$

where *L* is the pole number of Equation (15). The denominator of this equation is a cubic polynomial with one unknown variable with the three roots:

$$s\_1 = \mathfrak{x} + \mathfrak{y} - a/\mathfrak{z},\tag{17}$$

$$s\_{2,3} = -(\mathbf{x} + \mathbf{y})/2 - a/3 \pm j\sqrt{3} (\mathbf{x} - \mathbf{y})/2,\tag{18}$$

where:

$$\begin{array}{c} \text{x} = \sqrt[3]{-q/2 + \sqrt{D}}, \; y = \sqrt[3]{-q/2 - \sqrt{D}},\\ p = b - a^2/3, \; q = c + 2a^3/27 - ab/3, \; D = \left(p/3\right)^3 + \left(q/2\right)^2. \end{array}$$

In theory, there are three cases with different D parameters—*D* > 0, *D* = 0, and *D* < 0—which correspond to the three motion modes: over-damping, critical-damping, and under-damping (oscillatory), respectively. Practically, the physical properties of a piezoelectric material, i.e. its physical and piezoelectric parameters guarantee that the parameter *D* is greater than zero, means that the transducer works only in the oscillatory mode and its electric–acoustic impulse response can be written as:

$$h\_1(t) = A\_3 \exp(-\alpha\_1 t) + B\_3 \exp(-\beta\_1 t) \cos(\omega\_1 t + q\_1) \tag{19}$$

where:

$$\begin{aligned} A &= (\mathbf{x} + \mathbf{y})/2, \ B = (\mathbf{x} - \mathbf{y})/2, \ \boldsymbol{\beta}\_1 = A + a/3, \ \boldsymbol{\alpha}\_1 = a/3 - 2A, \ \boldsymbol{\sigma}\_1 = \boldsymbol{\beta}\_1 - \boldsymbol{\alpha}\_1, \\\ A\_3 &= \frac{-da\_1}{\sigma\_1^2 + 3B^2}, \ B\_3 = -\frac{d(\boldsymbol{\sigma}\_1 - \boldsymbol{\beta}\_1)}{\sigma\_1^2 + 3B^2}, \ \boldsymbol{\omega}\_1 = \sqrt[4]{3}B, \ \boldsymbol{\varrho}\_1 = \arctan\frac{\boldsymbol{\beta}\boldsymbol{\sigma}\_1 + 3B^2}{\sqrt{3}B(\boldsymbol{\sigma}\_1 - \boldsymbol{\beta}\_1)}. \end{aligned}$$

The acoustic–electric conversion of the transducer is the inverse process of the electric–acoustic conversion. By repeating the discussed process in reverse order, we obtain the acoustic–electric impulse response of the transducer as follows:

$$h\_3(t) = \overline{A}\_3 \exp[-\alpha\_3 t] + \overline{D}\_3 \exp[-\beta\_3 t] \cos(\alpha\_3 t + q\_3),\tag{20}$$

where:

*a* = (*m*+*mr*)+*CoRo*(*Rr*+*Rm*) *CoRo*(*m*+*mr*) , *b* = *CoRo*+*Cm*(*Rr*+*Rm*)+*CmRoN*<sup>2</sup> *CmCoRo*(*m*+*mr*) , *c* = 1 *CmCoRo*(*m*+*mr*), *d* = ρ0*vcN Co*(*m*+*mr*), *x* = 3 - −*q*/2 + . *D*, *y* = 3 - −*q*/2 − . *D*, *p* = *b* − *a* 2/3, *q* = *c* + 2*a* 3/27 − *ab*/3, *D* = (*p*/3) 3 + (*q*/2) 2 , *A* = (*x* + *y*)/2, *B* = (*x* − *y*)/2, β3 = *A* + *a*/3, α3 = *a*/3 − 2*A*, σ3 = β3 − α3, *A*3 = −*d*3α<sup>3</sup> σ2 3+3*<sup>B</sup>* 2 , *B*3 = −*d*3(<sup>σ</sup>3−β3) σ2 3+3*<sup>B</sup>* 2 , *D*3 = - *B* 2 3 + *C* 2 3, *C*3 = −*d*3(β3σ3+3*<sup>B</sup>* 2 ) √ <sup>3</sup>*<sup>B</sup>*(σ<sup>2</sup> 3+3*<sup>B</sup>* 2 ) , ω3 = √ 3*B*, ϕ3 = arctan β3σ3+3*B* 2 √ <sup>3</sup>*<sup>B</sup>*(<sup>σ</sup>3<sup>−</sup>β3) .

For a harmonic vibration, in the equivalent circuits in Figure 2, the two mechanical components, i.e., radiation resistance and radiation mass, are functions of vibration frequency. Also, for most cases, either the electrical signal of an exciting source transducer or the acoustic signal arriving at the receiver transducer is a signal wavelet with multi-frequency components. Excited by an electric/acoustic signal-wavelet with multi-frequency components, the vibration of the transducer's surface also consists of multiple sinusoidal frequency components.

The Fourier transform of the electric/acoustic signal of an excited transducer can be expressed as a linear superposition of sine-wave components with di fferent frequencies, amplitudes, and phases. The electric–acoustic/acoustic–electric excitation can be processed by the parallel-connected network as shown by parts I and III in Figure 3. Each of these equivalent circuits in the network has its own unique electric–acoustic/acoustic–electric impulse response resulting from its individual radiation resistance and radiation mass.

A continuous driving electric signal *<sup>U</sup>*1(*t*) with amplitude spectrum *<sup>S</sup>*(ω) and phase spectrum φ(ω) can be decomposed into *N* frequency components using an *N*-point discrete Fourier transform. Each frequency component can be written as:

$$dL\_{1\rangle}(t) = \left| S(w\_{\rangle}) \right| \cos[w\_{\rangle} t + \phi(w\_{\rangle}) \Big|\tag{21}$$

where *j* = 1, 2, 3, ... , *N*; and *<sup>S</sup>*(<sup>ω</sup>*j*) and φ(<sup>ω</sup>*j*) are the amplitude and the phase of the *j*th sinusoidal frequency component. Therefore, a normalized driving electric signal can be expressed as:

$$\mathcal{U}I\_1(t) = \sum\_{j=1}^{N} \mathcal{U}\_{1j}(t) / \max \left[ \sum\_{j=1}^{N} \mathcal{U}\_{1j}(t) \right]. \tag{22}$$

The output from the *j*th circuit in part I of Figure 3 is a convolution of the *j*th sinusoidal frequency component of the driving electric signal, with the *j*th electric–acoustic impulse response function being:

$$\left.v\_{r1j}(t)\right|\_{\omega\_j} = \left[\!\!\!\!\!I\_{1j}(t) \* h\_{1j}(t)\right]\!\!\!\!\!\!\/\_{\omega\_j}.\tag{23}$$

Then, the normalized vibration speed of the surface of the thin cylindrical transducer, i.e., the radiated acoustic signal, is defined as:

$$v\_{r1}(t) = \sum\_{j=1}^{N} v\_{r1j}(t)\big|\_{\omega\_j} / \max[\sum\_{j=1}^{N} v\_{r1j}(t)\big|\_{\omega\_j}].\tag{24}$$

For the *j*th frequency component of the radiated acoustic signal, if the propagation medium produces an acoustic impulse response *<sup>h</sup>*2*j*(*t*) <sup>ω</sup>*j* , it would yield the *j*th frequency component arriving at the receiver transducer as:

$$\left.v\_{r3j}(t,\omega\_{\dot{j}}) = \left[v\_{r1j}(t) \* h\_{2j}(t)\right]\right|\_{\omega\_{\dot{j}},t\_{1j}}.\tag{25}$$

where *t*1*j* is the propagation time of the *j*th sinusoidal frequency component from the source transducer to the receiver transducer.

The acoustic–electric conversion of a transducer is the inverse of the electric–acoustic conversion. The *j*th frequency component of an acoustic signal arriving at the receiver transducer passing the *j*th circuit (part III of Figure 3) is converted to an electric signal according to:

$$dI\_{\mathfrak{H}}(t) = \left[\upsilon\_{r\mathfrak{H}j}(t) \* h\_{\mathfrak{H}j}(t)\right]\Big|\_{a\circ\_{j}t\mathfrak{H}j}.\tag{26}$$

Finally, the measured acoustic signal, i.e., the electric signal at the electric terminals of the receiver transducer, is a collection of the outputs from all circuits in part III of the network (Figure 3), which is normalized as:

$$\mathcal{U}\_3(t) = \sum\_{j=1}^{N} \mathcal{U}\_{3j}(t)|\_{\omega\_j} / \max \left[ \sum\_{j=1}^{N} \mathcal{U}\_{3j}(t)|\_{\omega\_j} \right]. \tag{27}$$

The above discussion shows that an acoustic-measurement process can be achieved through a parallel-connected transmission network, as shown in Figure 3.

**Figure 3.** Schematic representation of a transmission network, which shows an acoustic-measurement process. *N* is the total number of components in the frequency spectrum of a driving electric signal. *<sup>U</sup>*1*j* is the *j*th frequency component in the driving electric signal. *vr*1*j* is the *j*th sinusoidsal frequency component of the vibration speed on the surface of the transducer. *vr*3*j* is the *j*th sinusoidal frequency component in the acoustic signal arriving at the receiver transducer. *<sup>U</sup>*3*j* is the *j*th frequency component of measured acoustic signal (i.e. electric signal at the electric terminals of the receiver transducers) created by the acoustic–electric conversion of the receiver transducer, where *j* = 1, 2, ... , *N*.
