An experimental system for ultrasonic attenuation measurements in solids in the upper kilocycle range

AN EXPERIMENTAL SYSTEM FOR ULTRASONIC ATTENUATION MEASUREMENTS IN SOLIDS IN THE UPPER KILOCYCLE RANGE by R. E. BOOKER*

and F. H. SAGAR*

The measurement of acoustic attenuation in small samples of low-loss solids in the range 50 kc/s to 10 MC/S has, up till now, proved difficult. When the pulse method is applied below 10 MC/S it is not easy to perform satisfactory corrections for diffraction and mode-conversion errors. These considerations apart, however, the realization of resolution of successive pulse echoes requires the use of embarrassingly long specimens. On the other hand the resonance method, in the form most frequently used, is suitable for elastic constant and attenuation measurements over the lower part of the kilocycle range, i.e. l-20 kc/s. The frequency-modulation modification devised by Bordoni permits measurements up to 10 MC/S but is restricted to the investigation of elastic constants. In either form, however, a decreasing output signal and deteriorating signal/noise ratio are obtained as the frequency is raised, whether by using a shorter bar or by raising the order of the harmonic employed. This paper is an account of various modifications in the experimental set-up of the resonance method of attenuation measurements, whereby good signal/noise ratios have been realized up to 700 kc/s, attenuation measured from 20 kc/s to 600 kc/s, and output signals generated from 0.5 V down to 10pV.

P rogress

made in growing large single crystal specimens prompted the steps described below to achieve better sensitivities for the conventional electrostatic form of the resonance method of measuring acoustic attenuation in relatively small samples (3-9 cm long) of low-loss solid materials. The experiment was designed to cover as much as possible of the neglected range 20 kc/s-l0 MC/S. Slender rods of crystal of circular cross-section (ratio of rod radius to wavelength less than 0.2) and of maximum length 9 cm are the largest that can be grown. Square crosssections are also acceptable. Because of inability to resolve successive echo pulses, such lengths will be too small to allow the pulse method to be used below 10 MC/S, but in any case for low-loss materials, satisfactory corrections for diffraction, mode conversion and other losses cannot easily be performed below this frequency. Consideration shows that rods of such lengths suit the resonance method for the range 20-700 kc/s provided that they are thin. Such rods are only difficult to provide in organic and inorganic crystals. For single crystal and polycrystalline metals, adequately slender rods can usually be obtained. The use of such rods to yield attenuation data over the kilocycle range quoted demands particular experimental features. These will now be mentioned.

In addition to the increase in the excitation field already mentioned a greatly enhanced detection sensitivity is needed to offset the reduction in signal output V, occasioned by use of a small length, L, and by excitation at a harmonic frequency of high order N. Refer to Equation 2. RESONANCE

METHODS

Fig. 1A shows the conventional electrostatic form of the resonance method of velocity and attenuation measurement. The bar, lightly supported at a central node, is set into longitudinal resonance at each of a number of odd harmonics in turn by variation of the driving oscillator frequency. The attenuation coefficient, GI,is obtained from observation of the Q of the bar (see Appendix A, Equation 6) which in turn is obtained from measurements on a response curve for low Q or on a decay curve for high Q. If a level recorder is used with a decay curve, the formula quoted below is employed, in which w is the resonant angular frequency in radians per second and S is the decay curve slope in decibels per second : Q = 8.68 w/S The Bordoni frequency

Screening

Satisfactory electrostatic and electromagnetic screening of the input from the output electrodes and from the components of the apparatus shown in Fig. IA is necessary. The reduction of the rod length brings these into unavoidable proximity. Two factors are present. For the output signal to remain constant despite the decrease in the length of the bar, the exciting field must be increased by the amount predicted by Equation 2. Hence, with short rods, it is difficult to provide adequate screening. The lower end of the kilocycle range (840 c/s) has recently been investigated by Zemanek and Rudnickl who used a 10 ft length of 0.5in diameter aluminium alloy (24ST). *University

Excitation jieId and detection sensitivity

of Auckland

modulated

method

Both forms of this modification of the resonance method are briefly considered to show that they are unsuitable for measuring attenuation in low-loss solids. In the Bordoni method the exciting electrode has the a.c. driving voltage applied to it and is at the same time part of an h.f. oscillator (frequency about 70 MC/S) which is frequency modulated by the capacitance variations in the electrode-bar capacitor that are produced by the bar vibrations. A low pass filter prevents a short-circuit of the h.f. oscillations, which would occur if they reached the 1.f. oscillator terminals. Similarly the 1.f. oscillations are prevented from reaching the h.f. oscillator terminals by means of a high pass filter. The geometry of the electrodebar capacitor must therefore be constant, which calls for a semi-rigid bar suspension. The method has been used by

ULTRASONICS/~~~~~~~-December 1963

223

jYTI*l-,

1

, SUSPENSION

7

ENCLOSURE

IA TUNABLE BAN)-PASS FILTER

Fig. 1. Excitation and detection circuit for resonant bar A. High Qs are obtained from a decay curve recorded by the component X on either a level recorder or an oscilloscope plus camera. For a high-loss material exhibiting a low Q, the Q is obtained from a resonance curve plotted from the readings of the voltmeter Y. This curve must be calibrated against signals of known frequency and magnitude injected into the main amplifier circuit through the inner circuit ZZZ. The cathode follower is frequency independent. The time constant RC is kept greater than llf B. Rigid needle-point suspension, unsuitable for low-loss materials C. Modified rigid suspension (Bordoni), used for high frequency investigation of metallic elastic constants: E. metal electrodes; F. flange; M. Mica layers; S. Light springs D. and E. Wire loop suspensions of minimal constraint, that in E being superior to that in D: T. polished tungsten wire of diameter 0.01 cm

ID

Bordoni2g3 for elastic-constant temperature dependence at individual frequencies up to 80 kc/s and by Pursey and Pyett,4 who incorporated additional modifications, for similar work to 250 kc/s.

IE

Single electrode method I This was the initial Bordoni method and employs a single

electrode. It possesses high sensitivity and good signal/ noise ratio, but cannot be used with a slack wire suspension to minimize external energy losses because, as was mentioned in the last section, the geometry of the capacitor is fixed. The need for minimal constraint to be introduced by the suspension when working with high Q rods is discussed in the next main section. The slack wires used in the present work, while reducing external damping, permit changes in the electrode-bar spacing of up to 20 y/oas a result of external vibrations that produce twisting or slipping of an initially horizontal bar. In the wire suspension method, any such bar movements which tend to reduce the output voltage V, can be compensated for by electrode adjustments made outside the vacuum chamber (described later) and by alterations to the driving voltage amplifier output and/or the output amplifier. In other words, neither of the two bar-end capacitors is part of a critically tuned circuit. Double electrode method

Work in the range 100 kc/s-5 MC/S was performed by Bordoni5 with a double electrode form of the frequency modulated apparatus which used : (4 A specimen in the form of a thin circular plate, supported by flanges turned from the plate periphery and supported in ring-like clamps ; (6) Separate electrodes (Fig. 1C) for excitation and detection, pressed by weak springs against the plate

224

ULTRASONICS/~~~~~~~-December 1963

faces and insulated from the faces by thin mica sheets. The high dielectric strength and dielectric constant of the mica sheets permit the application of a high driving voltage and give an attractive force on the specimen (and thus vibration amplitudes) at least a hundred times greater than those produced by electrodes of the same spacing in air or vacuum, sparking over difficulties apart. At these frequencies, two electrodes are essential. A large electrode to drive the plate with little excitation of parasitic frequencies, and a small one, on the opposite face, to tune an h.f. oscillator of adequately high frequency. Because of the relatively high external damping of the specimen produced by the rigid flange supports and the

mica/electrode contact, this method is more suitable for elastic constant work than for attenuation measurement. ESSENTIAL FEATURES IN THE RESONANCE METHOD UNDER REVIEW

Sinusoidal driving voltage

To minimize direct pick-up between input and output circuits, it is necessary to use the method in which we apply to the driving electrode a large sinusoidal voltage of frequency half that of the resonating bar rather than the alternative method in which we use a large d.c. polarization voltage together with a small sinusoidal voltage of the same frequency as that of the bar. Earthing of central support

The central suspension point of the rod must be a good electrical earth, as otherwise a fraction of the driving voltage appears on the output electrode via the potentiometer constituted by the input and output capacitors, the conducting bar and the intermediate resistance to earth (Fig. IA). (A non-metallic rod is rendered conducting by a thin evaporated aluminium layer.) A metal bar would be easily earthed were it permissible to use the needle-point suspension shown in Fig. 1B but low-loss metals, in which Q is more than, say, 104, require one of the low-loss wire suspensions illustrated in Fig. 1D or 1E; in these, the wires must be cleaned periodically, because the small pressure between wire and rod will otherwise give relatively bad electrical contact. External damping due to acoustic and viscous losses

In past work the degree of evacuation of the chamber enclosing the vibrating bar did not need to exceed the point at which surface acoustic radiation loss and viscous loss in the air layers between electrodes and end-faces became negligible. In the present work electric fields of up to 60,000 V/cm are sometimes applied between the electrode and end-face so that continued evacuation down to blackout is necessary to quench point discharges. Layers of insulating varnish applied to the electrodes are not considered adequate to withstand such fields. External damping due to energy losses to the support

Some preliminary measurements of the Q (order of magnitude = 3 x 104) performed at 240 kc/s on an aluminium polycrystalline bar showed that the measured Q varied with the type of support employed. Changing the tension of the vertical wires (thin tungsten of diameter 0.1 mm) from minimum to maximum practical values decreased the Q by 10%; a larger decrease occurs with the rigid needle point suspension (Fig. 1B). For the arrangement shown in Fig. 1 E in this investigation, the taut horizontal wire supports the rod and the loose vertical spaced pair merely centres the rod and aidsinits equilibrium. Thisis achieved by a shallow groove around the periphery at the central node. Tungsten is preferred to other metals because of its high tensile strength, which permits the mass of the support and hence the energy loss of the bar to the support to be minimized: the energy losses are due to the “Poisson’s ratio effect” in which lateral surface motion occurs at a longitudinal node.2 It is of interest to consider the practical difficulties of achieving low-loss suspensions for long low-loss bars

investigated at low frequencies: such a one was the 10 ft aluminium bar employed by Zemanek and Rudnickl for which the observed Q range was (15-30) x lo4 over the frequency range l-101 kc/s. Nothing superior to a horizontal bar lying in three or more thin wire loops, the arrangement employed by Zemanek and Rudnick, can easily be visualized, but only for work at a single fixed frequency can one tolerate the tedious procedure of trial and error adjustment of the wires to coincide with longitudinal nodes. Work over a wide frequency range means that in general the wire loops are distant from the nodes and consequently the external damping also exhibits a frequency dependence resulting in a scatter of points in a plot of Q versus f. The scatter in Q reported by these authors is IO-20% which is, incidentally, the variation found in the present work between Qs measured in the 3-wire suspension (Fig. 1 E) with maximum and minimum tension. It would seem, therefore, that when attenuation is the main interest, which it was not for Zemanek and Rudnick, the lowest frequency of investigation possible is determined by the greatest length of a rod that can be supported centrally in a vertical position. The capacitor microphone pick-up

circuit

As it is necessary to retain a constant charge on the capacitor C, the time constant RC (Fig. 1A) must exceed the signal period T for all frequencies of measurement, a condition easily fulfilled at high frequencies and also at low frequencies provided that the bars are of more than 1 in diameter, to maximize the capacitor plate area. The use of rods of slender form, mentioned earlier and to be discussed again in the next section, sets a maximum on the possible plate area of the parallel plate capacitor C, so that the required RC can be achieved only by maximizing R and minimizing the plate spacing, dI. If, for example, we want to investigate aluminium over the range 20-700 kc/s by measurements made on a single rod, we must calculate the data in Table 1. Table 1 DATA FORAN ALUMINIUM SINGLE‘RODAT 20 ph 700 xc/s Rod length to give fundamental frequency of 20 kc/s Wavelength, X, at 700 kc/s ’ : : Rod diameter for d/X CO.4 at 700 kc/s . Maximum practical i : : : C, calculated from CR > l/f when f = 20 Plate spacing, d, calculated from C above

:

:

:

12.5 cm 0.7 cm

:

: . .

: . .

0.25 cm 200 MD O-25pF 0.16 mm

kc/s

.

Spacings from O-1 mm to 0.4 mm are practical and have been used here but frequently the realization of a bar diameter as small as 0.25 cm is not possible, so 20 kc/s is the lower frequency limit of the method if the frequency range quoted is to be covered by means of a single bar only. Increasing the lower limit to 60 kc/s increases the permissible plate spacing to 0.48 mm but there seems little prospect of using the method for frequencies above 700 kc/s owing to the condition on d/h given in Table 1. CONDITIONS NECESSARYFOR SATISFYING SIMPLETHEORY In

order

conversion coefficient, other than extensional

apply Equation 6 in Appendix A for the of the measured Q or d into an attenuation it is necessary to avoid the excitation of modes the first longitudinal mode (also called the first or the Young’s modulus mode). Fig. 2 illus-

to

ULwsoNIcs

/ October-December

1963

225

trates the dispersion present in the phase velocity in the various modes for various values of the ratio bar radius/ wavelength a/x. Curve 1 applies to the first longitudinal mode and shows that for a/A < 0.2, the generation of and confusion with the modes represented by curves 2 and 3 can be avoided. Fig. 2 applies to finite cylindrical bars and is obtained from data secured by Davies6 from numerical solutions to the Pochhammer-Chree frequency equation’?s. Full discussion of the topic is presented in the text by Redwood9 and much experimental verification confirms the correctness of the theory of propagation of the various modes of vibration in bars.lJO Generation of radial modes can occur simultaneously with longitudinal modes when the half wavelength for the latter is approximately equal to the rod diameter d. Energy conversion from the longitudinal to the radial mode causes a reduction in the detected output signal which can easily be confused with an increase in the intrinsic attenuation, since the plot of u versus f superficially resembles a relaxation peak (see later section on Q measurements). The upper frequency limit for any particular bar is hence also determined by the condition that 2d
AFFECTING

SENSITIVITY

End-face displacement

amplitude

The displacement amplitude E of the end-face of a supported bar (Fig. 1A) subjected to a sinusoidally varying attractive force of given magnitude and frequency depends directly on the length of the bar, L, and on its Q, and inversely on Young’s modulus E for the material and the square of the order of the harmonic generated, N. The relevant formula is given in Equation 1. •J’l~.mz LQ . . . . . . . . . . . . (1) E 2?r2 N2d,2 E * where E = 8.85 x lo-l2 F/m. The force of attraction on the end-face depends on the square of the applied field, i.e. on (VI/dI)2 so that this is the most important term in determining 5 (here dI is the spacing between the electrode and the bar-end). m5z

=

Output voltage Clearly the voltage V, appearing at the output electrode depends on the magnitude of the applied d.c. voltage V, and on the ratio of displacement amplitude to mean electrode spacing, i.e. f/d,; the suffixes 1 and 2 apply respectively to the input and output electrodes. Hence we can write 2

LQ

V2

. . . . . . (2) EN2 dz Choice of the material fixes Q-and E while choice of the frequency of measurement fixes L and N. If we adjust V,, V,, dI and d, to give an acceptable value of V,,,,,.+

226

ULTRASONICS

-

I October-December

1963

I.5 I.4 I.3 I.2

V 1

_____

V0

1.1 I.0 0.9 0.8

V,

0.7

v, v r

*____ -----

0.6

Vo

0.5 0.4 0

0.2

0.4

0.6

0.8

I.0

I.2

14

I.8

lb6

20

2.2

:-

Fig. 2. Phase velocity Vof extensional waves of wavelength h in a cylindrical bar of radius a for the Poisson ratio of 0.29 plotted from data obtained bg Davies from numerical solutions to the frequency equation of Pochhammer and Chree 1. First extensional or Young’s modulus mode v, = d/~lp 2, 3. Other extensional modes V, = Velocity of dilational waves VP = Velocity of distortional waves V, = Velocity of Rayleigh surface waves

i.e. a satisfactory signal/noise ratio, we can compensate for a small L and a large N in Equation 2 only up to the practical upper limits of VI and V, and lower limits of dI and d,. Calculation of output voltage for a high order harmonic Use of Equation 2 gives, for the 25th harmonic of the 8.888 cm aluminium bar discussed in the section on measurements, the following data for equal drive and d.c. polarization voltages : Table 2 DATA

FOR 25TH HARMONICOFAN

8.888 CM ALUMINIUM

Order of harmonic, N . , . . Peak driving voltage and d.c. polarization voltage (V, and Ve) . Experimental Q at N = 25 Experimental Young’s modulus, k : : Length of rod, L Electrode spacing,

4 ‘and d,

Output voltage V, -, calculated from Equation 2

: .

,

BAR

25

6oov 2 x 10’ 7 x 10” dyn/cms 8.888 cm 0.2 mm

0.5 mV

Since V, is directly proportional to Q, the signal will be much less for high-loss materials such as plastics (Q = 200) if the effect of the smaller Young’s modulus is neglected. If in Table 2 the figure 200 replaces 2 x lo4 (a reduction factor of lOO), the signal will be reduced to 0.06 mV which, with our equipment, gives a rather poor signal/noise ratio. With high Q metals, however, the large maximum V, shown in Table 2 facilitates work with larger electrode spacings if so desired. The value of 600 V for VI and V, in Table 2 is the value realized in the present apparatus and it permits the measurement of Q even though the average air gaps in the capacitors are increased by becoming non-parallel owing to deterioration of the bar alignment during the initial period of evacuation: the deterioration is caused by vibrations arising from running the backing pump.

It is of interest to consider briefly the factor by which the output signal will be reduced by use of conventional values of the quantities Vi, V,, d, and d,, i.e. about 100 V, 200 V, and 0.5 mm, a reduction in V,, which Equation 2 shows to be about 1700. In the recent (1961) experiment of Zemanek and Rudnickl the maximum L and N employed were 10 ft (305 cm) and 102, together with a material of somewhat ‘higher Q (15 x lo4 at N = 102), but with much the same electrode spacings as used in the work here reported. The larger drive and polarization voltages now employed would give the following: Table 3 IMPROVEMENT

Improvement due to . Increase in Vi . . Increase in V, . Overall increase .

IN

. . . .

OUTPUT

. .

VOLTAGE,

. . . .

.

. . . .

V,,

Amount improved

. . .

. . .

. . .

36 3 108

Changing both electrode spacings from the conventional 0.5 mm to 0.1 mm gives a further improvement of 125 and an overall improvement in V,, of 13,500 over the figure realized in most of the work before 1961.

ESSENTIAL FEATURES OF APPARATUS

UNDER REVIEW

The apparatus required for the investigation (Figs. 1 and 3) can be described under four headings. Oscillators

High stability oscillators to 10-400 kc/s, approximately.

cover

the

overall

range

Amplljiers

These comprise a driving voltage amplifier of the range mentioned above, employing an LC tuned output circuit of high Q (not less than 25 at any point in any one range) which will provide a sinusoidal driving voltage of 500-700 V and a wide range output amplifier. The output amplifier constructed was conventional in nature, in performance it exhibited a frequency range of 2~5-900 kc/s, to the 3 dB down points, and at all useful frequencies resulted in a final signal in excess of 100 V on the c.r.0. plates. The variable measured is d, the logarithmic decrement, obtained from the envelope to a closely packed decay curve photographed from the c.r.0. display; see Appendix A.

Fig. 3. Vacuum chamber and diffusion pump unit Chamber contains bar and suspension, exciting and detecting electrodes and cathode follower unit andis made of 4 in steel plate. Vertical, lateral and axial adjustments are provided for each electrode unit and the bar unit. Final adjustment of pick-up electrode spacing is performed after final evacuation to black-out pressure by means of an externally operated worm and gear. During Q measurements, only the diffusion pump,operates. This evacuates the vacuum chamber into a 2.0 ft* backing tank, a procedure which removes the mechanical vibrations of the backing pump from the system and maintains blackout pressure in the chamber for at least 35 min, during which Q is measured. D. Diffusion pump (1 and 2 connected respectively to the backing pump and the tank) H. Hinge for opening chamber lid L. Braced steel supporting legs resting on platform supported by anti-vibration mounts which are not shown P. Half inch plate glass ports R. Circular grooves containing O-rings W. Worm and gear connected to electrode spacing adjuster

Vacuum system

A robust steel vacuum vessel (Fig. 3) houses both electrode assemblies and the rod held in its wire supports, plus a cathode follower and preferably also a small preamplifier. The vessel needs provision for easy and quick opening, resealing and evacuation, and is made in one piece with a large capacity (3 in) oil diffusion pump. (Construction details are given in Fig. 3.) Other essentials are: (1) Two oppositely situated plate glass portholes (0.5 in thick) for monitoring the electrode spacing; (2) Fine thread electrode-spacing adjustments at least one of which, obviously that operating the driving electrode spacing, being manipulated externally by a worm drive and vacuum cable mechanism; (3) Vertical, lateral and axial adjustments to the electrode holders and bar; (4) A steel backing tank (volume 2 ft3) directly coupled to the vacuum vessel and diffusion pump.

Coupling the backing tank directly to the vacuum vessel and diffusion pump means that black-out pressure can be established and the backing pump dispensed with for some thirty minutes, during which measurements of Q are made. During this interval final geometrical and electrical adjustments and the recording are performed in the absence of mechanical vibrations arising from the operation of the backing pump, so yielding an enhanced signal/noise ratio. The need to evacuate down to black-out pressure arises from the need to suppress the spark discharges across the capacitor gaps, caused by large fields, up to 60,000 V/cm, the necessity of which has already been considered. With smaller fields spark discharges can be suppressed by coating the faces of the electrode and the rod end with insulating varnish (Mylar).

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1963

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Noise rejection

transformers with high Q (> 25) capacitance-tuned secondaries, the width of the tuning range being maximized by employing suitable windings to give minimum self capacitance in the secondaries; 20-strand Litz wire was employed in the latter, with the windings wrapped on ferroxcube, a material of low hysteresis and eddy-current loss and adequately high permeability. Subdivision of the complete frequency range into seven overlapping ranges was obtained by a switching procedure in which the secondaries of the transformers were variously connected either in series or in parallel with other ferroxcube core selfinductors of special design, as indicated below: Ranges 3, 6 and 7: covered by a transformer selected from a group of three. Ranges 1, 2, 4 and 5 : covered by switching in to the secondary an appropriate inductor used either in series or in parallel or by switching in a pair of inductors. The LC circuits (Fig. 4) produced by this scheme possessed Q values never less than 25, varying in individual ranges from extreme to extreme by about 20-300/ and sometimes reaching 100; e.g. in range 7 the extreme Q values were 97, at 150 kc/s, and 60, at 400 kc/s. Tuning to a particular frequency was effected with high voltage (1,000 V) variable air capacitors of range 7(r920 pF values which give, on any one range, an optimum maximum to minimum frequency ratio of 3.6: 1. However, the selfcapacitance effect of the transformer secondary, of the electrode-bar system, and of the leads, reduces this to 2.1 :l. This explains the subdivision into seven ranges to achieve overall coverage of 64 octaves. The circuit of the complete amplifier (Fig. 4) shows that the transformer primary is push-pull fed, with consequent

Disconnection of the backing pump during recording leaves the mechanical vibrations caused by earth tremors and by oil bumping in the diffusion pump. It is conventional to reject these and valve noise by use of a high-pass filter and a tunable variable width electronic band-pass filter; comparable filters were employed here except that an inductor-tuned LC filter was preferred for reasons of lower cost and reduced thermionic noise. Such filters also reduce the residual voltages picked up directly from the nearby input circuit owing to the large exciting field mentioned.

Recording apparatus and other ancillary items

A conventional frequency meter, a trigger-switch for c.r.0. timebase initiation, a low speed timebase c.r.0. (l-20 sweeps/set) and an f = 2.5, 35 mm camera for trace photography were employed, the determination of d rather than Q being dictated by the absence of a pen recorder. Of this equipment further description is given below of two items only: the driving-voltage amplifier and the inductor-tuned filter.

FURTHER DETAILS OF ELECTRONIC APPARATUS

The driving-voltage amplijier

The chief reauirements are: (1) An ability to deliver over the range 4-400 kc/s a large (not less than 600 V) frequency independent undistorted signal, second harmonic distortion in particular being absent. (2) No d.c. leak-through to the excitation electrode, this condition being necessary because simultaneous application of an a.c. voltage of frequency f and a d.c. voltage, yields three terms in the resultant attractive force on the bar; a term inf, one in 2f, and a steady d.c. term, so that, in the presence of leak-through, two modes of vibration can exist simultaneously. The 6+-octave range finally decided on could not be covered with a single transformer. The construction of the amplifier shown in Fig. 4 was hence based on a set of three

Fig. 4. Wide range large voltage-output exciting amplifier employing tuned transformer output and fed by constant-current generator Continuous frequency variation over a limited range is provided by high voltage variable capacitors of maximum value 1000 pF. Hence the overall frequency range desired (4-392 kc/s) is covered by means of seven overlapping ranges achieved by a bank of three transformers, used alone for ranges 3, 6 and 7, and with four additional inductors in series and in parallel with the secondary to cover the remaining ranges, 1, 2, 4 and 5. These seven components were fabricated from multiple Litz wire wound on selected ferroxcube pot cores to achieve adequately low overall secondary self-capacitance and high Q. The pentodes, EL83, are high current, high impedance valves. The transformer primary is hence fed from a push-pull constant-current generator, and a highly sinusoidal voltage of not less than 600 V appears across the output terminals

--..._. -_. 470n

+ 658V

_lTRODE

t

IO.05uF

l---h

-m d---l

82kn

EFBO

I ELK3

228

ULTRASONICS/October-December

1963

EL83

IO.OSuF

f

dOY ! A.C.

_

1

4PF

Y

!

+ 270V

IOOOpF

002pF

I

I

I

E #OF

Fig. 5. Circuit diagram of the Harris variable band-pass filter The contamination in the signal appearing on the output electrode (Fig. lA), driving frequency voltages picked up by electrostatic and electromagnetic coupling, is removed by the low noise tunable band-pass filter shown. This uses an LC circuit with Q maximized by feedback and frequency controlled by a variable ferroxcube core inductor L. A bank of three switched series pairs of capacitors covers three ranges Range 1. 8-55 kc/s Ct = 0.07 pF Range2. 38-253 kc/s Cs = 6300 pF Range 3. 180-900 kc/s Cs = 330 pF The measured ratio of maximum to minimum inductance of the inductor is 49:l (maximum 6.29 mH, minimum 133 PH). Details of its construction are shown in Fig. 6. The frequency range is thus 7:1, against 3:l obtained with a variable capacitance. The 7:l ratio is oreferable in that tunine of the filter is tediol,s and becomes easier, the fewer the ranges tb be traversed

output circuits and the magnitude of the driving voltage field across the input electrode and bar end, we have a much greater directly radiated signal than occurs from long rods of, say, 50-300 cm. Tuning. In the circuit due to Harris,ll a cathode follower is fed from a LC circuit consisting of a fixed inductor and two equal variable ganged series connected capacitors with the centre point joined to the earthed cathode. A simple modification suitable for ultrasonic frequencies and giving a wider tuning range* for any one pair of capacitors is to make the latter fixed and to use a variable ferroxcube core inductor (Figs. 5 and 6). As pointed out, the output frequencies are double the driving frequencies and so require the construction of a filter of range, say, 8-800 kc/s. This overall range can be covered by subdivision into no more than three 7:l sub-ranges by the use of a variable L instead of a variable C. (Appendix B). The three overlapping ranges achieved were: Range 1,8-55 kc/s; Range 2, 38-252 kc/s; Range 3, 180-900 kc/s. Construction of variable inductor. Fig. 5 gives details of the switching of various pairs of fixed capacitors into the circuit, and the frequencies mentioned above indicate that the variable inductor to be fabricated must have L,,,JL,,,,. about 49 : 1. The inductor finally constructed and described in Appendix B employed a fixed stator coil coupled to a

even-harmonic suppression, from a constant-current generator employing pentodes of high internal impedance and large current handling capacity (EL 83 valves). The amplifier performs well, has a low noise level, and gives over the whole range an output in excess of the maximum for which it was designed. Variable band-pass jilter

These are low noise (as few valves as possible employed in the construction), frequency independent stability, and, over the working range, independence of the stability from the Q. The “simplified Q multiplier” described by Harris,ll employing one electronic valve with a simple variable feedback circuit, possesses these qualifications and has been used here. Frequency variation of the electrical Q exists but this is no detriment if the mechanical Q of the rod is large enough to permit use of the decay method of measurement (of the logarithmic decrement) but in materials of low Q such as plastics the response curve method becomes necessary. Now the frequency variation of Q of the LC circuit is a complication to be overcome by the “injection method” of calibrating the response curve, that is, by injecting, either immediately before or after the frequency independent cathode follower shown in Fig. lA, signals of known magnitude and frequency for selected points over the response curve. The noise in the output signal is found to be less than that of many conventional electronic band-pass filters employing two or more valves. The function of the filter is to reject the driving frequency voltage component whose frequency is half that of the resonating rod. Because of the proximity of the input and SpeciJc requirements.

* The tuning of the filter is not simple since it is done simultaneously with that of the driving oscillator and driving voltage am lifier during the search for the resonant frequency. The effort of tuning tPIe filter is smaller, the smaller the number of ranges it possesses. The three ranges of the inductance tuned filter described here are therefore preferable to the 5 to 7 ranges which use of most variable air capacitors would give.

E

u~~~~so~~cs/October-December

I963

229

Fig. 6. Construction details of variable inductor used in band-pass filter shown in Fig. 5 The largest ratio of maximum to minimum inductance and hence the greatest tuning range is realized when the inductances of the rotor and stator coils are equal, in the position illustrated in this figure and when the coupling factor k is near unity, since: L maz

-=_zz?-

2L+2M

l+k

I-k 2L-22M L&l The air-gap was minimized to about 0.003 in to maximize k. The inductance ratio was measured as 49, giving a frequency ratio of 7:1, so making possible coverage of the whole range 8-900 kc/s by means of the three ranges mentioned under Fig. 5. The use of 21 strand Litz wire for the windings gave Q > 40 at any frequency between 8 kc/s and 900 kc/s A. C. E. G.

radius/wavelength, and Q obtained at ,a number of frequencies in the range 12-700 kc/s, the measurements being performed on the fundamental and the odd harmonics. The lowest frequency (12.5 kc/s) was given by the fundamental of a Perspex rod (length and diameter respectively 8.872 cm and O-633 cm), and the highest frequency investigated was the twenty-fifth harmonic of an aluminium rod of length 8.876 cm and diameter 0.312 cm. In general it was found unnecessary to reduce the electrode spacings, 4 and CE,,below 0.2 mm and the Qs were adequate to permit their measurement directly by the decay curve method. However, the plastic rod exhibited (2s of the order of 200 only and here the response curve method had to be employed. Owing to its more tedious nature, Q for Perspex was determined approximately only, the interest in any case focusing more on the magnitude of the output signals than on the actual Qs. Table 4 gives the range of variables investigated and shows that the limit of the method would be reached for high order Perspex rod harmonics, a little of reserve existing for copper (range of measured Qs 850-1060) and plenty for aluminium [measured Qs (1.87-4.13) x 104]. Two aluminium rods have been investigated, nearly identical except for diameter (& in and & in, i.e. 0.625 cm and 0.312 cm). For both it was possible to verify that the measured Qs corresponded to the presence of first longitudinal mode resonances by plotting curves of V/V, (phase velocity/l/E/p) versus (radius/wavelength, i.e. a/A) and securing a plot corresponding to the first part of the curve 1 in Fig. 2. Converting the graph of Q versus f for the thick rod into attenuation coefficient, GCversus f revealed a pronounced and mainly fictitious absorption peak at a frequency corresponding to a first order radial mode (417 kc/s). Fig. 7 shows the relevant graph of Q versus f for this mode. Repetition of the experiment on the & in diameter rod gave little indication of this absorption maximum, which should occur (if the frequency could be achieved) at double the frequency, i.e. approximately 830 kc/s. No work above 700 kc/s has yet been performed by this method, We assume therefore that the dip in the

Layer of paper B. Rotor coil Turning spindle for rotor D. Stationary coil F. Rotor Layers of rubber (3) H. Brass holder Ferroxcube E cores K. Brass bottom plate and supports -30

rotor coil, all coils being wound on ferroxcube cores. To ratio of 49: 1 the large coupling achieve the L,,,/L,,, factor k must be made large (O-96), which in turn means a small air-gap between the cores of the stator and rotor coils; in our experiments the average gap giving this k was OGO3 in. The inductor cores were shaped by careful grinding from pieces of ready moulded E core to provide the composite stator and rotor cores illustrated in Fig. 6. Moulding could well give a smaller gap with better values of k. Were a figure of 0.99 achieved for example, the ratio would be 199, corresponding to a ratio LmazIL1IIIR fL/L‘VI of 14:1, and representing a significant improvement on the 7 : 1 ratio achieved. MEASUREMENTS OF Q ON SELECTED TEST SPECIMENS

To test the performance of the equipment, measurements were performed on rods of Perspex, of polycrystalline Duralumin type alloy, and of single crystal copper, of assorted lengths and diameters to cover a range of values of

230

~~~~~so~~cs/October-December

1963

t : 9 x 0

-20

IS - 2.0

t

i=JC)

?..o-

I.0

t-

-

-‘.O

200

300 FREQUENCY,

400 kc/s __r

w

-5

500

Fig. 7. Experimental attenuation plots for polycrystalline altinium rod 8.888 cm long and 0.625 cm in diameter. Measurements made on first 23 overtones. Pseudo absorption peak at about 415 kc/s is due to transfer of energy from the longitudinal to the first radial mode. Data obtained from a rod of half the diameter do not display this peak. Calculated voltage on output electrode (Equation 2) for the nineteenth harmonic, which is marked on the graph of Q above, is 0.8 mV ~-__-__------

Q against frequency CL against frequency a/f2 against frequency

EXPERIMENTAL RANGE

2. Frequency

APPENDIX

Table 4 OF VARIABLES EXPLORED

of measurement

A:

CALCULATION

OF

.

.

.

Lowest value: Perspex

Highest value: ’ Aluminium

N=l 12.5 kc/s

N= 25 about 700 kc/s

4. Voltage obtained on pick-up electrode (calculated) . .

about 200

FROM

Simple theory for the propagation of a continuous longitudinal plane wave through an extended medium relates the amplitude attenuation coefficient, a, of the wave and the mechanical Q of a longitudinally vibrating rod, by Equation 6 below, as follows: By means of the two equations A = A, e-M and I = I,, em2‘, in which k and CIare respectively the damping coefficient or reduction in amplitude per second and the amplitude attenuation coefficient, while the remaining symbols have their usual significance, we obtain, by putting t = T and x = X, kT

a=3. Q obtained

ATTENUATION

MEASUREMENTS OF Q

about 4x 10’

N=3

N=l

4.2 mV

0.5 v

x

. ........

The symbols T and X are the period and wavelength. The logarithmic decrement d, or reduction in amplitude per cycle of vibration, and k, are related by k=;

Lo west value: Aiuminium N= 5. Voltage obtained on pick-up electrode (calculated) . .

while for large Qs it is true that QA =r

25

O-5 mV

-

Hence

k=

6

. . . . . . . . . . . . . . . . . . (4)

This allows Equation 3 to be written graph of Q (Fig. 7) is due to transfer of part of the generated energy at 417 kc/s from the first longitudinal to the radial mode, which is not detected by the capacitor microphone pick-up employed. Further work to be reported later is in progress on Q in diameter polycrystalline metal rods and on iin diameter single crystal rods, for the first of which the frequency limit of the method will be encountered before the onset of first radial mode vibrations; i.e. work on odd harmonics up to about N = 27 should be possible for thin polycrystalline rods, and the frequency dependence of attenuation should be able to be reliably determined.

a =&

. . . . . . . . . . . . . . . .(5)

Ignoring distinctions between the propagation of a continuous wave in an extended medium and of a stationary wave in a suspended resonant bar of length L (for which, if N is the order of overtone generated, we have h = ‘$

>

,

we can finally write Equation 5 as ?i-N a=mL . . . . . . . . . . . . . . . . . .@I APPENDIX B: INDUCTANCE

RANGE OF VARIABLE FERROXCUBE

ACKNOWLEDGEMENTS

CORE INDUCTOR

The purchase and fabrication of equipment used in this investigation were facilitated by a grant from the University of New Zealand Research Grants Committee to whom the thanks of the authors are gratefully tendered. Thanks are due also to Mr. Frank Blair of the Physics Department Workshop staff who constructed the vacuum-chamber/ diffusion-pump unit and shaped the ferroxcube core’ for the band-pass filter.

Consider a variable inductor whose construction, except for the use of ferroxcube cores, is similar to that of the conventional air-core mutual inductor possessing a fixed inductance coil and an equal coupled rotating coil. If L is the resultant inductance, the ratio L-/L,,,,, is obtained from the formulae L,=L, +L,f2MandM=k1/LlL, where L, and L, are the inductances of the stator and rotor coils and M the mutual inductance between them, k being the coupling coefficient; L, = L, = L which leads to

REFERENCES 1. ZEMANEKand RUDNICK, J. Acoust. Sot. Amer., 33. No. lo,1283 (1961). 2. BORDONI,P. G., Nuovo Cimento, 4, 177 (1947). 3. BORDONI,P. G., J. Acoust. Sot. Amer., 26, No. 4, 495 (1954). 4. PURSEY,and Pvsrr, J. Sci. Instr., 31, 248 (1954). 5. BORDONI,P. G., and Nuovo. M., Acustica, 7, 1 (1957). 6. DAVIES, R. M., Phil. Trans., Roy. Sot. Lond., A240, 375, 457 (1946-48). 7. POCHHAMMER,J., Fiir die reine und angewandte Mathematik, 81, 324 (1876). 8. CHREE,Trans. Camb. Phil. Sot., 14, 250 (1889). 9. REDWOOD,M., “Mechanical waveguides,” Pergamon, Oxford (1960). 10. STANFORD,E. G., Nuovo Cimento., Supp. Vol. VII Series IX, 1 (1950). 11. HARRIS, H. E., “Simplified Q multiplier,” Electronics, 24, 130 (May 1951).

L-/L,,

= E ; ;;

= g

. ...(7)

a formula indicating the need for a large k if a large tuning range is to be achieved. As discussed earlier, the tuning range given by the ferroxcube core inductor fabricated here was 7: 1, corresponding to a ratio of 49: 1 for L_/L,r,. From. Equation 7 k was therefore O-96. A still higher k could no doubt be achieved if smaller air-gaps were employed. The average gap realized in the present experiments with the rotor turned to the fully coupled position (Fig. 5) was O-003 in, and although this probably represents the limit possible when the core pieces are ground to shape, superior performance could be achieved by moulding the rotor and stator cores to a more exactly determined shape.

uLTRAsoNIcs/October-December 1963

231