#### 3.1. Generation Mechanism and Theorical Calculation of the Blind Zone in a UDM System

As mentioned earlier, three piezoelectric ceramic transducers are adopted in the tool. During the excitation stage, an electric pulse with a certain frequency is applied to the transducer, which generates a mechanical vibration that causes the transducer to emit ultrasonic waves. When the excitation pulse is removed, the transducer continues to vibrate due to mechanical inertia. Such a vibration is called the residual vibration or “tailing” of the transducer. During the tailing stage, the echo signal is superimposed on the residual vibration of the transducer, which cannot be distinguished from the residual vibration until the vibration stops or is sufficiently small. Then, half of the acoustic travel distance is called the blind zone length of the UDM system.

If tailing time is

t_{b}, then blind zone size

l_{b} can be calculated using:

where

ν is the acoustic velocity of the medium.

The residual vibration of transducers is a type of underdamping vibration, and the blind zone can be reduced if the vibration is rapidly attenuated. The ultrasonic transducer has an equivalent circuit, as shown in

Figure 4a. The entire circuit can be regarded as an

RLC underdamping oscillation model, where

R_{1} is the dynamic resistance, also known as the mechanical loss resistance, and

C_{0},

C_{1}, and

L_{1} are the static capacitance, dynamic capacitance, and dynamic inductance, respectively. If the transducer is excited by an electric pulse with amplitude

U_{0} and frequency

${f}_{s}=1/2\pi \sqrt{{L}_{1}{C}_{1}}$, then series-resonance occurs in the branch

L_{1}C_{1}R_{1}. This branch has minimum impedance and functions as a pure resistance, where

f_{s} is the series-resonant frequency of the transducer. When the excitation pulse is removed,

C_{0} and

C_{1} are connected in series. Let

${C=C}_{0}{C}_{1}{/(C}_{0}{+C}_{1})$, then the equivalent circuit can be transformed into the form shown in

Figure 4b. Capacitor

C is used as the energy storage element and is discharged through

R_{1} and

L_{1}. Voltage

u_{c} on capacitor

C is a function of time, and its initial value is

U_{0}, which means

${u}_{c}\left(0\right){=U}_{0}$.

In the circuit shown in

Figure 4b, the voltage at

C is equal to the sum of the voltages at

R_{1} and

L_{1}; that is:

There are relationships

${u}_{l}{=\u2122L}_{1}{di/dt=\u2122L}_{1}{Cd}^{2}{u}_{c}{/dt}^{2}$ and

${u}_{r}{=\u2122R}_{1}{i=\u2122R}_{1}{\mathit{Cdu}}_{c}{/d}_{t}$, then Equation (5) can be rewritten as:

Voltage at

C at a certain time can be evaluated by solving the second-order differential equation:

In Equation (7),

${\alpha}_{d}={R}_{1}/2{L}_{1}$,

${\omega}_{s}=2\pi {f}_{s}$,

${\omega}_{d}=2\pi {f}_{d}=\sqrt{{{\omega}_{s}}^{2}{{-\alpha}_{d}}^{2}}$, and

$\phi ={\mathit{tan}}^{-1}({\omega}_{d}/{\alpha}_{d})$. As shown in Equation (7),

u_{c} decays exponentially along with time, and simultaneously, voltage oscillates periodically with angular frequency

ω_{d}.

Table 1 displays the main impedance–admittance parameters of a given transducer, where

Q and

K_{eff} are the quality factor and effective electromechanical coupling coefficient, respectively. As shown in

Figure 5, the residual vibration waveform of the transducer can be drawn by using the aforementioned parameters.

Let variable

τ represent the time when

u_{c} decays from its maximum amplitude

$\frac{{U}_{0}}{{\omega}_{d}\sqrt{{L}_{1}C}}$ to a value that 10

^{−5} times of the amplitude. Then:

Variable

τ can be obtained by solving Equation (8), as follows:

where

$Q={\omega}_{d}{L}_{1}/{R}_{1}$. When the parameters in

Table 1 are inputted to Equation (9), we can obtain

$\tau \text{}\approx \text{}98\text{}\mathsf{\mu}s$. Then, the blind zone length

l_{b} can be calculated by combining this result with that of Equation (4). If the acoustic velocity of the medium is 1600 m/s, then

${l}_{b}=0{.5\text{}\nu}_{m}\tau \text{}\approx \text{}78.5\text{}\mathrm{mm}$.

A large blind zone is harmful to measuring borehole diameter, and residual vibration must be reduced as quickly as possible. To solve this problem, an acoustic emission circuit is provided.

#### 3.2. Circuit Design for Reducing Blind Zone

In general, the underdamping oscillation frequency of a transducer is similar to its series-resonant frequency. In accordance with the parameters in

Table 1, the calculated frequency

f_{d} is 214 kHz, which is extremely close to

f_{s} (221 kHz). Hence, the residual vibration can hardly be distinguished from echo signal via filtering. Equations (7) and (9) show that the methods for increasing attenuation coefficient

α_{d} and reducing excitation pulse voltage

U_{0} can reduce the residual vibration. However,

α_{d} is determined by the impedance characteristics of the transducer, which cannot be changed by the external circuit. The reduction of emission power will reduce the detection range of the UDM system. Therefore, these two methods are undesirable.

The energy stored in the static capacitor

C_{0} must be released quickly to reduce residual vibration. As shown in

Figure 6, the solution adopted for this circuit design is as follows: connecting the transducer in parallel to a switch

K and a resistance

R_{d} in series. During the excitation stage, switch

K is off and

R_{d} cannot access the loop. When the excitation pulse is removed, switch

K is closed,

R_{d} and

C_{0} form an

RC discharge circuit, and energy stored in

C_{0} is consumed by

R_{d}. The time constant of the

RC discharge circuit is

τ_{d}, and

${\tau}_{d}={R}_{d}{C}_{0}$. Moreover, when

τ_{d} is smaller, the discharge time is shorter. Evidently, to eliminate residual vibration as soon as possible, the smallest

R_{d} should be selected while ensuring that the current flowing through

K is lower than the maximum pulse current that can be tolerated by the switch.

The role of switch

K can be replaced by a high-speed N-channel MOSFET (NMOS). As shown in

Figure 7, NMOSs

Q_{1},

Q_{2},

Q_{3}, and

Q_{4} form a full bridge excitation circuit. Freewheeling diodes

D_{2},

D_{3},

D_{5}, and

D_{6} are connected to the source and drain of NMOSs, which will protect the NMOSs from the reverse current produced by their sudden shutdown. According to the command of the complex programmable logic device (CPLD), bootstrap gate drivers

BGD_{1} and

BGD_{2} control the on-off function of the NMOS by adjusting

V_{gs} (NMOS gate–source voltage) [

8,

9].

The control sequence of the CPLD is shown in

Figure 8. During the acoustic emission stage

t_{f}, the CPLD controls the bootstrap drivers

BGD_{1} and

BGD_{2} to turn on

Q_{1} and

Q_{4}, and turn off

Q_{2} and

Q_{3}. The voltage of pin

V_{s} of the bootstrap driver is pulled to

HV and

HVGND levels, respectively; and the transducer is excited by a rectangular pulse with amplitude

HV and frequency

f_{s}. A simplified diagram of the excitation circuit at this stage is provided in

Figure 9a. After excitation,

Q_{1} and

Q_{3} are turned off, whereas

Q_{2} and

Q_{4} are turned on. The circuit is simplified into the form shown in

Figure 9b, and the transducer is discharged through resistors

R_{3} and

R_{7}. The residual vibration of the transducer has been eliminated outside

t_{b}. During this stage, all NMOSs are turned off to prepare for echo detection, and the simplified circuit is shown in

Figure 9c.

#### 3.3. Circuit Test

To test the function of the emission circuit and the actual blind zone length, a reasonable CPLD control sequence is designed; thus, the circuit can produce a rectangular single pulse (15 V, 2.25 μs) to excite the transducer.

Figure 10 shows the waveforms after the excitation pulse being removed. Observed by a digital comparator, when

t_{d} is continuously changed, residual vibration is eliminated at

${t}_{d}\ge 10\mathsf{\mu}\mathrm{s}$, and it will not affect the echo detection anymore.

In mud medium with acoustic velocity ${\nu}_{m}=1600\text{}\mathrm{m}/\mathrm{s}$, and blind zone time ${t}_{b}={t}_{f}+{t}_{d}=12.25\text{}\mathsf{\mu}\mathrm{s}$, on the basis of Equation (4), the blind zone length can be calculated: ${l}_{b}=0.5\times 1600\text{}\mathrm{m}/\mathrm{s}\times 12.25\text{}\mathsf{\mu}\mathrm{s}=9.8\text{}\mathrm{mm}$, which shows that the blind zone of the UDM system has been significantly reduced.

During drilling and well logging, the influence of the blind zone on diameter measurement is inevitable when the outer surface of the tool is close to the borehole wall. As mentioned in

Section 2 and

Figure 3a, the transducers are indented into the tool housing to cope with this situation. “Zero distance” measurement can be realized as long as the indentation length

${l}_{\mathit{in}}>9.8$ mm. When the mechanical structure of indented-in transducers is adopted, the equivalent length of the blind zone can be considered zero.