Ultrasonic Transducers and Arrays

! Transducers" Piezoelectric effect" Transducer parameters" Transducer models" Field emitted by a piezoelectric transducer

! Ultrasonic arrays" Beamforming" Steering" Focusing" Spatial resolution

! Imaging using arrays! Selected array applications! Homework

Tadeusz Stepinski, Signals and Systems, Uppsala University

ts@signal.uu.se

Transducers

Typical acoustic transducer Typical ultrasonic transducer

Mech.-acoustical transformer

Electro-mech.transformer

Electro-acoustical transformer

Piezoelectric effect

The inverse effect - production of strains within the material by an applied electric field.

The direct effect - the rearrangement of the charge distribution within a crystal with the application of a stress, resulting in a volume polarization of the material.

Piezoelectric crystal at different states.

Transducer parameters

Bandwidth and pulse length define axial resolution

B – bandwidth

τ – pulse length

δ – axial resolution

τδ

τ

cB

tt

ffB

21

12

121

=

=−=

−=

Transducer parameters

Spatial parameters

Θ – lobe width

∆r – range resolution

Transducer designs

Angle transducersNormal incidence (0°) transducers

RLC model of a piezoelectric transducer

Electrical RLC equivalent circuit

d+ V3 -

I3

A

I3

Co

R1

L1

C1

V3

+

-

PiezoelectricCrystal

RLC CircuitModel

( )( ) 111

211

211

211

11

CCLCCCRjCLjCRZ

ooRLC_in ωωωω

ωω+−−

−+=

( )( ) ( )2

1122

11

112

111

1

1

−+

−−+=

CLCRCLjCR

CCjY oRLC_inωωωωωω

Impedance

Admittance

RLC equivalent circuit is valid for one resonance only!

Magnitude of the input impedance of a loaded transducer

11

1CLr =ω

Resonance frReactance of a series branch goes to zero

Antiresonance faResonance with C0, impedance achieves maximum

RLC model of a piezoelectric transducer

10

1 +=CC

ra ωω

Electro-acoustical KLM model

211

φωa

oKLMin

ZjXCj

Z ++=−

Input impedance

Za – acoustic impedance

Z1,2 – acoustical impedances loading the transducer

KLM model of a piezoelectric transducer

KLM model describes several consecutive resonances

Circles showing equal wave fronts from part of the transducer split up into point sources. Plane transducer (left). Focused transducer (right)

Field of a piezoelectric transducer

Transducer diffraction effects

Field of a piezoelectric transducer

Near and far field

Normalized pressure on z-axis for a 5MHz 1x1 mm element

Near field Far field

0 5 10 15

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dis tance z/ λ

Nor

mal

ized

am

plitu

de

λ44

22 dcfdN =≅

Near field length for a circular element

d – diameterf – frequencyc – sound velocityλ – wavelength

Field of a piezoelectric transducer

Lobe width

d – diameterf – frequencyc – sound velocityλ – wavelength

-50 -40 -30 -20 -10 0 10 20 30 40 50

10-3

10-2

10-1

100

Angle [deg]

Nor

mal

ized

am

plitu

de

Bea m pa tte rn for cp = 1500 [mm], and f = 1 [MHz]

dfdc λγ 5.05.0sin =≅

Beam spread (-6dB) -

γ

Field of a piezoelectric transducer

Lobe width

d – diameterf – frequencyc – sound velocityλ – wavelengthdfd

c λα 22.122.1sin =≅

Beam spread (first zero)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Polar plot of beam pa tte rn for cp = 1500 [mm], and f = 1 [MHz]

α

Field of a piezoelectric transducer

Transducer field in water, transducer 1x1 mm

Fie ld for λ = 0.5 [mm]

Z / λ

X coord. [mm]-2 -1 0 1 2

0

1

2

3

4

5

6

Fie ld for λ = 0.3 [mm]

Z / λ

X coord. [mm]-2 -1 0 1 2

0

1

2

3

4

5

6

fc=λ

f – frequencyc – sound velocityλ– wavelength

cH2O = 1480 m/s

ULTRASONIC ARRAYS

! Array types" Phased arrays

• Beam steering• Beam focusing

" Linear arrays• Beam focusing• Electronic scanning

ARRAY TYPES

! Most of modern arrays are made of 1-3 piezocomposite

ARRAY FUNCTIONS

! Beam focusing

! Beam steering

! Electronic scanning

ARRAY FORMS

Ultrasonic arrays from Imasonic, France

THEORY – Beamforming

Beamforming equation

where:a - a window function, X(t) - all sensor outputs, and r - phase delays of

individual sensors

raXa ⋅⋅⋅=⋅= jketxtty )()()(

[ ][ ]

[ ]TM

jkTM

M

rretxtxtxt

aa

10

10

10

)()()()(

−

⋅−

−

=⋅==

=

K

K

K

rX

ar

THEORY – Beamforming

Uniform (rectangular) window

-100 -80 -60 -40 -20 0 20 40 60 80 1000

5

10

15

20

25

30

35BEAM PATTERN

Incident wave direction

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 100-80

-60

-40

-20

0P OWER

Incident wave direction

Pow

er (d

B)

-100 -80 -60 -40 -20 0 20 40 60 80 1000

5

10

15

20BEAM PATTERN

Incident wa ve direction

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 100-120

-100

-80

-60

-40

-20

0P OWER

Incident wa ve direction

Pow

er (d

B)

Hamming window

Theoretical beam patterns in far field 32-element arrays in water

THEORY – Electronic beam steering

Beam steering principle

0

0

sin0

0)(

0

),(),()()(),(

ψψψωψ

⋅⋅⋅

−⋅

⋅=⋅=⋅⋅=

m

rr

aa

djk

jk

etbbtxetxty

where

ψ0 - desired bearing (steering direction)

[ ]TM 11,0 −= Km

THEORY – Electronic beam steering

Theoretical beam patterns in far field 32-element arrays steered with 60° in water

-100 -80 -60 -40 -20 0 20 40 60 80 1000

10

20

30

40BEAM P ATTERN

Incident wave dire ction

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-80

-60

-40

-20

0P OWER

Incident wave dire ction

Pow

er (d

B)

-100 -80 -60 -40 -20 0 20 40 60 80 1000

5

10

15

20BEAM P ATTERN

Incide nt wa ve dire ction

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 100-120

-100

-80

-60

-40

-20

0P OWER

Incide nt wa ve dire ction

Pow

er (d

B)

Without apodization (left) and with apodization using Bartlett window (right). Frequency 1 MHz, element spacing 0.74 mm, λ = 1.5mm.

THEORY – Focusing

Beam focusing principle

[ ]2)1(,,02

depthfocalwhere),(

),()(),(

−=

−⋅=⋅=

⋅⋅

Mf

df

eftbfbtxfty

m

djk m

Kp

a p

ω

10 15 20 25 30 35 40 45 50-14

-12

-10

-8

-6

-4

-2

0Beam power along z-axis

Distance from a rray in mm

Mag

nitu

de

Pressure on axis for a focused array

THEORY – Focusing

Theoretical radiation pattern of a 32-element array in water

Array without apodization, focused at 30mm.Frequency 1 MHz, element spacing 0.74 mm, λ = 1.5mm.

Radia tion for an a rray focused a t 30 mm

Distance from a rray in mm

X-a

xis

in m

m

15 20 25 30 35 40 45 50

-30

-20

-10

0

10

20

30

THEORY – Steering and focusing

Theoretical radiation pattern

Array steered with 20°, focused at 30mm.Frequency 1 MHz, element spacing 0.74 mm, λ = 1.5mm.

PrincipleRadia tion for a n a rra y focused a t 30 mm

Dis

tanc

e fro

m a

rray

in m

m

X-axis in mm-30 -20 -10 0 10 20 30

15

20

25

30

35

40

45

50

THEORY – Spatial aliasing

Example

Radiation patterns for an array steered with an angle 30° for spacing d =0,75λ

Principle

cdfwhere

Mb

=

⋅⋅⋅=

δ

ψπδψπδψω)sinsin(

)sinsin(),(

Theoretical beampattern of an array in far field

has several peaks at points

These peaks are called grating lobes.

If d = λ/2 then grating lobes will appear at ±90°

If d > λλλλ/2 then grating lobes will appear at ψ < |90°°°° |

L,2,1,0 ±±==⋅ nfornπψπδ

-100 -80 -60 -40 -20 0 20 40 60 80 1000

5

10

15

20

25

30

35BEAM P ATTERN

Incident wave direction

Mag

nitu

de

-30 -20 -10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1BEAM P ATTERN

Incident wave dire ction

Mag

nitu

de

THEORY – Spatial resolution

Normalized radiation patterns for the 16 element array (left) and 32 element array (right). Element spacing 0.5λ , no apodization.

Beam resolution is defined by the width of the main lobe.

The 3dB (half power) beam width defines an angle Θ3dB that is used as a measure of spatial resolution

⋅=⋅= −

MdMdB

λδθ arcsin89.0)arcsin(89.0 13

-30 -20 -10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1BEAM P ATTERN

Incident wave dire ctionM

agni

tude

16 elements 32 elements

LINEAR ARRAY

! Electronic scanning using linear array eliminates one axis of a mechanical scanner" A small active aperture is shifted along the array" Aperture´s beam can be focused or/and steered

CIRCULAR ARRAY

Electronic beam rotation

Array

Rotating ultrasonic beam

IMAGING USING ARRAYS

Ultrasonic imaging

" A-scan (a single echo) " B-scan (a vertical cross section)

IMAGING USING ARRAYS

Ultrasonic imaging

" C-scan (a horizontal cross section)

IMAGING USING ARRAYS

Ultrasonic imaging using phased array

Side drilled holes

B-scanPhased array

IMAGING USING ARRAYS

B-scans of side drilled holes

Unfocus ed array

20 40 60 80 100 120 140 160 180 200

200

400

600

800

1000

1200

1400

1600

1800

200020 40 60 80 100 120 140 160 180 200

200

400

600

800

1000

1200

1400

1600

1800

2000

Side drilled holes

Unfocused array Focused array

TYPICAL APPLICATIONS

Turbine ring inspection (R/D Tech)

Using conventional transducers Using phased array

SPECIFIC APPLICATION - EB weld inspection

SKB copper canisters for spent nuclear fuel

Phased array

SPECIFIC APPLICATION – natural defects

C-scan of the specimen with natural defects

Specimen with natural defects

Homework

1. Show that the relation between resonance frequency fr and antiresonancefrequency fa for a RLC transducer model is

2. Beam spread (-6 dB lobe width) for a circular transducer at 1 MHz in water is 30°. Calculate frequency for which beam spread will be 10°. What will be the near field length for both frequencies?

3. A linear ultrasonic array has 64 elements spaced at 1 mm. Calculate time difference between the consecutive elements required for steering its beam, with 30° respective in water and in steel for frequency 3 MHz (assume sound velocity in water cwater = 1500 m/s and steel csteel = 5900 m/s).

4. Calculate position of the first grating lobe for the array defined at p. 3 in water.5. Calculate the shape of the ultrasonic response at a B-scan obtained from a

small, side drilled hole in steel block scanned with a normal incidence transducer as shown in figure below. Hint: calculate arrival time of an echo as a function of transducer position.

10

1 +=CCff ra

Scan direction xTransducer

Hole