Phased Array Leseprobe

2.2.1 Delay laws
In what follows, formulas for beam steering are given for a one-dimensional linear array of
identical and regularly spaced rectangular elements. This is the simplest case mathematically, and
will be sufficient to illustrate basic phased array principles. There will be different formulae that
determine beam steering characteristics for more complex phased array probe geometries, such as
irregularly spaced linear elements, annular arrays and two-dimensional arrays, but these are
beyond the scope of this Handbook.
Delay laws determine the focal characteristics and steering behavior of the ultrasonic beam. Angle
beam steering is accomplished by applying a linear time delay function to the individual elements.
For unfocused beams, the angle of sound in the material is directly proportional to the applied time
delay; as the applied time delay increases, so does the angle. The formula governing this behavior
is shown in Equation 2-1, where θ = requested angle of incidence, n = 0, 1, 2, ... – number of
array elements with respect to the central array element, с = sound velocity in material, t 0 = a
constant value for avoiding negative delay values, and d = distance between the centre of two side
by side elements.
t
d
n sin
Equation 2-1
c
n t o
When using this equation, the central array element time delay is calculated by using n = 0 in the
equation, which in this case is t n = t 0 . The time delay of the element directly on one side is the
central element is then calculated by setting n = 1, and the element directly on the other side by
n = –1. By varying the value of n over all of the elements in the virtual aperture, the time delays of
all of the elements are derived.
Focusing is accomplished by applying a delay function; various functions such as parabolic
function can be used for this purpose. The focal depth is inversely proportional to the applied time
delay; therefore, as the maximum applied time delay decreases, the focal depth increases. The
formula governing this behavior is shown in Equation 2-2 for a focussed ultrasonic beam
propagating at = 0 into material, where F = requested focal depth, and the other variables are
defined above.
t
n
2
F nd 2
1
1
1/
to
Equation 2-2
c F
Equation 2-3 is the equation governing the creation a focused angle beam by combining angle
beam steering and focusing simultaneously, and reduces to the first two equations for simpler
cases. If Equation 2-3 is considered in the limit F (i.e. in the limit of an unfocussed ultrasonic
sound beam), then the result is Equation 2-1. If Equation 2-3 is considered in the limit of
(i.e. normal incident sound), then the result is Equation 2-2. Lastly, if Equation 2-3 is considered
in the limit of n = 0 only (i.e. only one element in the phased array probe), then t n = t 0 , or the time
delay is arbitrary, as is expected with conventional ultrasonic testing with single crystal probes.
Note that Equations 2-1, 2-2 and 2-3 apply only when the phased array probe is directly coupled to
the test material with normal incidence, and do not take into consideration a wedge attached to the
probe. Equation 2-3 and Figure 2-4 are the sum of equations 2.1 and 2.2.
2
F nd nd 2
t 1/
n 1 1 2 sin
t o
Equation 2-3
c F F
5