understand that a piezo-electric crystal changes shape when a p.d. is applied across it and that the crystal generates an e.m.f. when its shape changes

Production and Use of Ultrasound

Learning Objective

Students should be able to explain how a piezo‑electric crystal changes shape when a potential difference is applied across it, and how the same crystal generates an e.m.f. when its shape is altered by an external force.

1. The Piezo‑electric Effect

The piezo‑electric effect is a reversible phenomenon observed in certain crystals (e.g., quartz, lead zirconate titanate – PZT). It can be described in two complementary ways:

• Direct effect: Mechanical stress produces an electric charge on the crystal faces.
• Converse effect: An applied electric field causes the crystal to deform (expand or contract).

1.1. Direct Piezo‑electric Effect

When a force \$F\$ is applied, the crystal experiences a strain \$S\$ that produces a charge \$Q\$ on its electrodes. The relationship is

\$ Q = d \, F \$

where \$d\$ is the piezo‑electric coefficient (units C·N⁻¹).

1.2. Converse Piezo‑electric Effect

Applying a voltage \$V\$ across the crystal creates an electric field \$E = V/t\$ (with \$t\$ the crystal thickness). The resulting strain \$S\$ is

\$ S = d \, E = d \frac{V}{t} \$

The change in length \$\Delta L\$ of a crystal of original length \$L\$ is

\$ \Delta L = S L = d \frac{V}{t} L \$

2. Producing Ultrasound with a Piezo‑electric Transducer

Ultrasound refers to sound waves with frequencies above the upper limit of human hearing (≈20 kHz). In medical and industrial applications, frequencies from 0.5 MHz to several tens of MHz are common.

2.1. How a Transducer Works

1. A sinusoidal voltage \$V(t)=V_0\sin(2\pi f t)\$ is applied across the crystal.
2. Because of the converse piezo‑electric effect, the crystal oscillates at the same frequency \$f\$, producing longitudinal mechanical vibrations.
3. The vibrating crystal launches pressure waves into the surrounding medium (usually water or tissue). These are the ultrasound waves.

2.2. Resonance

Maximum efficiency occurs when the crystal thickness \$t\$ is an integer multiple of half the wavelength \$\lambda\$ of the sound in the crystal:

\$ t = n\frac{\lambda}{2}, \qquad n = 1,2,3,\dots \$

Since \$\lambda = v/f\$ (with \$v\$ the speed of sound in the crystal), the resonant frequencies are

\$ f_n = \frac{n v}{2t} \$

3. Receiving Ultrasound – The Reverse Process

When an incoming ultrasound wave strikes the crystal, it compresses and expands the crystal, invoking the direct piezo‑electric effect. The mechanical strain generates an alternating voltage \$V_{\text{out}}(t)\$ proportional to the acoustic pressure \$p(t)\$:

\$ V_{\text{out}}(t) = g \, p(t) \$

where \$g\$ is the electromechanical coupling factor (units V·Pa⁻¹).

4. Applications of Ultrasound

• Medical imaging (sonography): High‑frequency transducers produce detailed images of internal organs.
• Therapeutic ultrasound: Focused beams heat tissue for physiotherapy.
• Non‑destructive testing (NDT): Detect cracks or voids in metals and composites.
• Industrial cleaning: Cavitation generated by high‑intensity ultrasound removes contaminants.

5. Typical Piezo‑electric Materials

6. Example Calculation

Problem: A PZT crystal of thickness \$t = 1\,\$mm is driven by a sinusoidal voltage of amplitude \$V0 = 100\,\$V at its fundamental resonant frequency. The speed of sound in PZT is \$v = 4.0\times10^3\,\$m s⁻¹. Find the resonant frequency and the maximum change in length \$\Delta L{\max}\$.

1. Resonant frequency (fundamental, \$n=1\$):

   \$ f_1 = \frac{v}{2t} = \frac{4.0\times10^3}{2\times1\times10^{-3}} = 2.0\times10^6\ \text{Hz} = 2\ \text{MHz}. \$

2. Using \$d = 500\,\$pC·N⁻¹ = \$5.0\times10^{-10}\,\$C·N⁻¹ and \$E = V_0/t = 100/(1\times10^{-3}) = 1.0\times10^5\,\$V m⁻¹,

   \$ S = dE = 5.0\times10^{-10}\times1.0\times10^5 = 5.0\times10^{-5}. \$

   For a crystal length \$L = 10\,\$mm,

   \$ \Delta L_{\max} = S L = 5.0\times10^{-5}\times10\times10^{-3} = 5.0\times10^{-7}\ \text{m} = 0.5\ \mu\text{m}. \$

7. Summary of Key Points

• The piezo‑electric crystal deforms under an applied voltage (converse effect) and produces a voltage when mechanically stressed (direct effect).
• Ultrasound is generated by driving a crystal at its resonant frequency, causing it to vibrate and launch pressure waves.
• Reception of ultrasound is the reverse process: incoming acoustic pressure induces an alternating e.m.f. in the crystal.
• Choice of crystal material determines the efficiency, frequency range, and suitability for a given application.

8. Suggested Questions for Revision

1. Explain why a crystal must be operated at or near its resonant frequency to produce efficient ultrasound.
2. Derive the expression for the resonant frequency of a thickness‑mode piezo‑electric transducer.
3. A medical ultrasound probe operates at 5 MHz. If the speed of sound in the crystal is \$4.2\times10^3\,\$m s⁻¹, what should be the crystal thickness for the fundamental mode?
4. Discuss the advantages and disadvantages of using quartz versus PZT in high‑precision versus high‑intensity applications.