ANALYSIS AND APPLICATION OF MAGNETOSTRIC!FION DELAY LINES

M e t B. Thompson Oklahoma Agricultural and Mechanical College,

St i l lwater , Oklahoma and

John A. M. Lyon Northwestern University

Evanston, I l l i n o i s

Summary -- Uniformly variable sordc delay lines are discussed, with par t icular a t tent ion to the factors affect ing the formation of the mechanical signal of the transmitting transchcer and the e lec t r ica l s igna l of the receiving transducer.

' Examples of pulse shapes for different conditions are calculated and sketched, For comparison with these, some experimental results are described ancl i l l u s - trated. Consideration is given to va r i a t ion of delay, use of multiple trans- ducers and the control o f the polarity of the response. Several devices i n which magnetostriction delay Nnes may be used are briefly described.

INTRODUCTION

The extensive use of intentional delay o f s igna ls has developed largely since 1940, beginning i n radar and other mili tary equipment and found now i n maw f o r delays longw t h a n a feu microseconds because of the long paths required a t the high velocity of e l ec t r i c wave propogation. Ultrasonic systems, requiring less space for the lower ve loc i t ies of mechanical uave propogation and corre- spondingly shorter paths, are firmly established where the longer delays are required. The wide extent of the literature o f ultrasomc delay lines has been shown i n the 1954 bibliography compiled by M. D. Fagan l i s t i n g seventy-eight t i t 1 e s . l The at tent ion has been given almost exclusively t o systems other than magnetostriction types. T h i s has been due mainly 'to the frequency limitation imposed by the use of the lat ter.

b additional applications. Purely e l e c t r i c a l means of delay become impractical

Three groups of researchers have published papers dealing with magneto- stric%ion delay lines. The ear l ies t , by E. M. Bradburd, appeared i n 1951 describing lines used as components o f aerial navigation equipment.* R. J4illersh$p, R. C. Robbins and A. E. DeBarr have described applications t o digital computers made i n England.3-5 The laboratorg note of H. EDstein and 0. Stram was published i n ,1953.6 It i s evident that, i n some instances, factors such as greater reliability an3 ruggedness, ease of delay variatLon, saving i n cost, space anl weight, as well as other advantages will dic ta te the use of magnetostriction delay Urns i n spite of t h e i r shortcomings i n other respects.

The magnetostriction deliry l ine retards an e lec t r ic s igna l by converting it in to a mechanical wave i n an elastic medium. After i t s re la t ive ly slow passage through this material it i s reconverted into an e1ectrI.c signal. Such a delay l i n e is shoun,in Flg. 1. The transmitting and'receiving transducers are solenoidal coils placed on the delay Une a c h m a y be a wire or tube of nickel or some other magnetostrictive material. The ends of the l ine me clamped between damping pads t o reduce the unwanted reflections. The magnets represent the magnetic Masing f i e l d i n the receiving transducer. The e lec t r i c signal

a

died t o the transnittim c o i l wl .ll change the magnetic f ie ld i n the nickel core. Because it is magnetostrictive the core corrtracts. o r expands, thereby launching mechanical waves along the delay l ine . When such a disturbance passes

MAGNETS I TRANSMITTER

I l r I

DAMPING IN OUT OUT DAMPING PA D RECEIVERS PAD

LTUBE OR WIRE

Fig. 1 - Magnetostriction delay l ine .

through the recsiving coil the change i n stress i n the nickel will a l t e r the biasing field, thus generating an e lec t r i c signal i n the receiving transducer. Since the v e l o d w . o f the mechanical wave i s much less than that of an e lec t r ic s i g n a l a delay of the signal i s obtaimd.

THE MECHANICAL SIGNAL FOR STEP INPUT

The configuration resulting when the magdtude o f t h e s t r a i n i n the medium i s plotted against distance along the longitudinal h s o f the line i s designated here as the shape o f the mechanical pulse. It is known tha t f o r a given magneto- s t r i c t ive medium the shape will be determined by three principal factors: the effective length of the transducer Coil, 11; the time, T, required f o r the flux i n , t h e medium to reach a stable value; and the strength of the magnetic fields * applied .S

T h i s can be demonstrated i n a simple way by considering incremental. charges of the f l u x density occurring a t short time in te rva ls , nt, from 0 t o T. A t each o f the instants , WAt up through time T, an incremental deformation 11 i n length will be launched along the l i ne a t velocity @. The shape result ing *om the summation of these incremental strain pulses i s shown i n M g . 2. The width o f this disturbance (outlined with the heavy l i n e ) i s seen t o be the sum o f the transducer length plus the product Tc.

For low flux densities the deformation will be p ropor t iona l t o the square o f the flu densityO7 Thus

where & i s the s t ra in .

Since B i s a function of time,

From Eq. (1) it can be seen tha t a unit step of B w i l l . produce a strain pulse of in tens i ty and 11 i n length. T h i s respozlse, the analog of the i n d i c i a admit- tance of an e lec t r ica l netrjork, can be expressed as a pulse,

9

Tc I_

DSPLACEMENT

b

~~

Fig. 2 - The strain pulse formeti i n the medium by the transmitting transducer i s represented by the heavy outline. The sha2ed rectangles repre- sent the strain increments which are summed to ge t t he pulse.

oc [S(t-t0) - S(t-to- L:] 1

C

where S(t-to) represents a unit s t e p f a c t t o n a t to.

If a mathematical expression of B as a function of time is not readily aka i lab le the approhat ion methods of time series can be used.899 I n aqy case, the approximation i s fast and conveeent. The following example i l l u s t r a t e s the method. If the f lux density i s considered t o change linearly with time, Eq. (2)

& ( t ) = 6 k2t2

Time in te rva ls of 1/2 psec, represented by A t are used with the time equivalent of the transducer length equal to 2 A t and 6 k 3 equal to unity. The s t r a i n response to a unit step of flux applied t o the medium m y be written, .

L 1 8 = 1 + x + x 2 (5 1 I n this notation the coefficients of the terms x*, XI, x2, ..-. x" are e-

10

respect ively to the value of 6 a t t-s 0 , v, 2v, , , m, The superpos f o r T equal to 3 A t is performed as a polynomial summation:

1 + x + x 2

9x2 + 9x3 + 9&

It will be noted tha t one i s not l imited to easily expressed relations between B and t. Empirical data or oscillographic traces showing this re la t ion are easily used t o determine & ( t ) . Equation (6) represents a function with ordinates 1, 5, 14, 29, etc., a t the times 0, v, 2v, 3v, etc., as plotted i n Flg. 3. This function i n time corresponds t o an inverted space function, since the first e lemnt i n time, located a t the origin, corresponds t o the element o f

l'

32 !-

Fig, 3 - Configuration representing the response of t ,ransmitter to a step input, with stra.in proportional t o the square of the f l u x density.

the mechamcal pulse located fart'nest from the origin i n space representation. The pulse shape i n space then may be described as an inversion of Eq. (6),

6. (1) 0 16x t 2Sx2 + 29x3 + Ik& + 5x5 + x6 (7)

The accuracy of the approemation i s improved by reducing the t h e interval , to I n the limit as t approaches zero the solution i s exact.

THE OUTPUT OF THE REcEIVIExj COIL

For a given delay medium, the four principal factors involved i n the shape of the electric output signal are: the effective length of the receiver coil, the magnetization levels i n t h e medium, the shape of the mechanical signal received and t h e m b e r of turns i n the coi l . A magnetic bias f ie ld for the medium enclosed by the receiving coil must be provided since the V i l l a r i effect occurs only i n t h e presence of a magnetic f i e ld .

As the strain pulse passes through the receiver coil the change i n t h e flux present will cause poten t ia l s to be generated i n the receiver coil . Such a c o i l and a trapezoidal pulse somewhat shorter are shown i n =g. h (a) . The pulse enters the coi l and i t s leading edge generates a voltage as it progresses. Since, under strain, the permeability of the magnetostrictive medium undergoes a change, the passage of the pulse into the coi l produces a time rate of change f o r the reluctance of the magnetic path l i n k i n g t he co i l windings. This will produce the change of flux which generates the voltage i n t h e c o i l windings. The magnitude of the voltage depends on the number of t u r n s i n t h e c o i l and the time rate of the flux change. The lat ter will be affected by the shape of the pulse, the reluctance of the flux-path 3inking t he ' co i l , and the level of magnetization establlshed i n t h e c o i l by t he bias f i e ld . This voltage appears as the posit ive potential pulse i n Fig. 4 (b) . As the t ra i l ing edge o f the pulse enters the coil there i s no further change i n reluctance and flux, there- fore no voltage appears a t t h e c o i l terminals while the pulse is ent i re ly within the coi l . As the leading edge of . the pulse leaves the co i l the potent ia l of opposite polarity generated by the same act ion i n reverse will appear, extending i n time until the pulse has completely passed the coil. T h i s produces the second

- . - . pulse shown as negative i n Fig . 4 (b) Thus, fo r s t r a in pulses 'of aqy shape,

COIL

A

12

Fig. 4 - Response of a receiver. The strain pulse shown with the coil i n (a), produces the output shown i n (b) .

, . - - .- , . . -

.. r . r .. . . . -. _ - , :, _ _ . . .

t h rere W ~ U appear t ;m. voltage pulses, simlar i n shape to the inver'se of the s t r a i n pulse,- one posit ive. and one negative, separated by an in t e rva l of time equivalent t o the coil length. The width of t h i s response i n space is 11+ Tc +l2 and i ts time duration T + (11 , + 12) /C.. For cases,. discussed below, where the posit ive and negative pulses overlap, these are reduced in proportion, both i n amplitude and duration.

I n t h e time series notation the electrical output f o r a receiver of length equivalent to 1.5~ sec i s found by polynomial additi.on with a displacement of six time in te rva ls (3 psec) between the two pulses of opposite sign:

e ( t ) = l + Sx + Ux2+2*3 + 25& + 16x5 - x6 (8)

- 5x7 - k8- 29x9- 25x1' - 16$- l

This appea.rs in Fig. 5; as tke ccmbirtztion of poei.t.T.m p l s s A , and negative pulse B. A longer receiving coil would produce a hider spacing between the positive

TIME, MICROSECONDS

Fig. 5 - Responses of delay l ines to s tep inputs . A plus B represents the response o f a l i n e having a longer receiver t h a n the one producing A plus C.

and mgxtive pulses. I n Fig. 5; the combination of pu3.se -4 ,and pulse C represents the respcnse o f a receiving coil of lergth equivalmt t o 1.2 Hsec.

If the receiver coil i s shortened t o 1/2 psec equivalent length the output becornes,

1 + sx + 1 ~ x * + 29x3 t 2 5 4 + 166Xs ( 9 )

- x - 5x2 - 4 x 3 - 2& - 25x5- 16x6

e ( t ) P 1 + 4~ + pX2 + 152 - LUE)' - d- ' 1 6 ~ ~

13

b

This is Shawn as response A i n Fig. 6. Here, B represents the response for a transducer length equivalent t o l ~ s e c . These two responses i l l u s t r a t e t h e par t ia l cancel la t ion of voltage that occurs when the equivalent length of the

-3 Ok l I 2 I I I I 3 4 5 TIME, MICROSECONDS

Fig. 6 - The e f f ec t of overlapping o f the posi t ive and negative portions of the output signal. This occurs when the time equivalent t o the length of the receiver is less than T, result ing i n the reduction of voltage.

receiver coil becomes l e s s than the time T. It is this ef fec t which has reduced the.peak voltage although all factors, except the receiver coil length, are the same f o r A and B.

It should be noted that the response of the 3 i n e described above, having the 1.5 vsec transmitter and 12 wsec receiver as shown i n Fig. 5, i s not altered i f the roles of the coils are interchanged, with all other conditions the same. The response of the 1.5 psec receiver t o the signal output of the 1 2 Hsec transmitter is found to be,

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T h i s response is i den t i ca l t o that of the original combination of coils appearing as A and c in f i g . 5. Thus, with other factors remaining the same, the inter- change of the lengths of the transmitting and receiving- transducers does not affect the output o f the delay line.

RESPONSE TO A PULSE IhTUT

When a step voltage i s applied to the input terminals o f the delay l ine a pulse of compression, f o r example, is formed fn the delay medium. With the assumption that B is linear with time, the compression increases with the square o f the time and decreases abruptly as shoun i n Fig. 3. When the voltage i s removed a pulse of the opposite kind, tension i n this case, will be formed, with tension r i s i n g abruptly and f a l l i n g a t a rate inversely proportional t o t he time. Using the same numerical example, the pulse formed a t the removal of the voltage i s expressed, . i

1

1

-16 - 25x - -l4x3 - S& - x S V

The complete strain pulse formed uhen a pulse of voltage is applled is, there- fore, the sum of these two different pulses separated by a time interval equal t o the electric pulse duration. For a 3 psec electr ic pulse the delay line output is ,

1 + x + x 2 (12 1

h+ h 2 + 4x3

9 x 2 + 9x3 + Sd.L 16x3+ 16x4 c 1 6 2

-16x6 - 16x7 - 16x 0

15

For a 1.5 vsec e lec t r ic pu lse appl ied the s t ra in s igna l muLd be,

l + x + x

4.x t 4x2 + 4x3

9x2 + px3 t 9x4

- 9x3 - & - 9x5

-lupk-Q-jJ&

b

15r

16

TIME, MICROSECONDS

Fig. 8 - Response of a delay l ine to pulse input when the receiver is shortened t o cause partial overlapping.

FACTORS AFFEc!lTNG R E S P O N S

It has been shown that the shape o f the output of the magnetostriction deley l i n e f o r a given medium deperds principally on the effect ive length of the trans- ducers, t he &er of turns on the coils, the saturat ion time and t h e levels of magnetization used. Other factors, most of uhich contribute to these principal ones, will be mentioned, Although external ele,ctric drcuits.connected t o the transducers have not been considered i n this paper, it uill be understood tha t the line! must be matched t o these far.best results.

Blos -- It has been poirrted out t ha t a magnetic b ias f i e l d i s required f o r the operation o f the receiving transducer. The magnitude of this f ie ld deter- mines the region of operation on the saturation Curve of the delay.medim. The amplitude of the 'output can be modified by adjustment of the bias. Reversal of i t s direct ion will reverse the polarity of the output. The shape o f the response can be changed by the appl icat ion of bias Fields not uniformly d i s t r ibu ted i n the medium. The t ransmit t ing coi l does not require a bias f ie ld ; however, one may be used t o alter the magnitude and shape of mechamcal pulse, since the s t r a i n produced i s not a linear function of the magnetization.

-

Magnetic c i r c u i t -0 The reluctance of the magcetic c i r cu i t o f the trans- mitter will affect the amplitude of the mechanical signal. By enclosing the coil

with a magnetic covering, one m a y increase the levels of magnetization produced i n t h e delay medium f o r a given current i n the transmitter. The result i n the receiver i s to increase the effect of the pulse on the coi l flux and thus increase the magnitude of the output, The dimensions of the air gaps a t the coil aperture will af fec t the flux dens i ty in the medium, as well as the fringing of flux, and thereby, the effective length of the transducer.

Medium -- The time f o r f lux saturat ion can be appreciably reduced by the use of the delay medium i n forms such as ribbons,* tubing6 o r f i ne wires. There i s usually some mechanical advantage i n t h e use of the tube since it i s easier t o get a close fit t o a round coi l aper ture and the tube i s more l i k e l y t o be self-supporting . Reduction i n volume of the medium enclosed i n t h e transducer will reduce hysteresis and eddy current losses. The physical experience of the medium, such as heat treatment, w i l l af fec t i t s magnetostrictive characteristics, as well as the velocity of propogation of the wave i n t h e medium.lo This l a t t e r f a c t will, of course, determine the time of delay per unit length. To a lesser degree the response shape i s affected by the treatment of the medium i n t h a t the dispersion of the s t ra in pulse and velocity of propogation are f a c t o r s i n i t s detezpination. Applying tension or tors ion to the medium will a l t e r i t s magnetostrictive characteristics, i n some cases permanently.1°

Transducer Coils -- Integrat ion of the delay line w i t h the complete system w i l l require sui table e lectr ic Character is t ics for i t s coils.7 The relat ions existing among quant i t ies such as number of turns, coil length and diameter,

will make some of t h e m dependent upon the values selected for the others. This limits the choice of values for shaping the response o f the delay l ine so that , as i s usual i n such cases, a compromise i s required. These coil character- i s t i c s a f f ec t t he pulse shape by modifying the pr incipal factors mentioned a t the beginning o f this section.

b wire s ize and insulation, aperture diameter, coil impedance and current capacity

EXPERIMENTAL RESULTS

Factors i n shaping the output -- A number of delay l ines were assembled t o determine the effects of parameter changes on the responses to.the step and pulse voltage inputs. Traces on the oscilloscope showing some of these a re i l lus t ra ted i n Figs. 9 , 10 and 11. They may be compared with the calcuLated responses shown i n Figs. 3 , 5 and 6. It should be kept i n mind that the flux-time relationship which was assumed- there i s now different. I n addition, factors such as the lack of resolution i n t h e pulse launched and i t s f'urther dispersion, not considered i n the calculations, have an e f f ec t on these responses. Furthennore, the nomni- formity of the bias f i e l d s adds distor t ion t o some of these outputs.

The t races ..shown i n Fig. 9 illustrate the par t played by the length of the receiving transducer i n the shape and magnitude of the lins output. The same transmitter, equivalent to 0 ,33-~sec . in ac tua l l ength , was used fo r A and B. The measured length of the receiver used i n making A, 0,33 ysec, i s one-eighth as long :as the coil producing B, The spacing between the mEudmum and. mLnimum for each output i s .proportional t o t he length -of the 'receiver. used. ,. The .effect '"., of the short rne&anicaYpulse on the ' flux of the sho'kt ' receiver , ' k S greater, than ,' i t s ef fec t on the flux of the longer receiver. This caused the peak voltage t o be lower f o r B. I n this graph the parts played by the .effect ive length of the transmitter ard the time t o saturate, T, may a l s o be seen. . .

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Fig. 9 - Oscillograph traces of the responses of two delay lines, ident ica l except that the B receiver i s longer than tha t of A . Overlapang does not occur.

t v Fig. 10 - Oscillograph traces

of the responses of two delay lines, ident ica l except that the B receiver i s shorter than t h a t of A causing overlapping i n t h e output which reduces the peak values of B

V

Figure 10 i l l u s t r a t e s , with actual responses, the reduction i n voltage which occurs uhen the reduced length of the receiver coil causes the posit ive and negative pulses t o overlap. The only difference i n the.conditions for'maklry the two traces was the use of a receiver f o r B on ly one-fourth as long as fo r A, causing partial cancellation of voltages, resulting i n reduction i n the peaks of the output.

figure 11 (a) shows the effect of the shape of the l ine cross-section and s ize on the output of a line using the same transducers and annealed mckel throughout . The lowest peak was produced u i th wire having a diameter of 0.005 inch; the next l o w s t peak was" produ.ced u i th a 0.010 inch wire. The highest voltage peak, a t 5.4 cm deflection which is above the range of the figure, came ui th the use 'of a tube having 0.020 inch outside diameter and 0.010 inch inside diameter . N e x t louer is tb fourth trace uhich was produced with a 0.020 inch &re. The closeness of fit of the la rger l ines to the co i l s resu l ted in h igher flux-density levels. The reduced cross-section of the tube forced the flu density even higher, producing the highest output of the lines tested. Required time t o saturate, for the same l eve l of flux density, i s l e s s f o r t h e tube than f o r the wire. I n Fig. ll (b) the e f fec t of current change from 120 t o 200 ma r e s u l t s i n a proportional. r i s e of the response voltage, depending on a number of fac tors such as condition of saturation.

Other results -- The number of transducers which can be applied to a delay line appears t o be limited only by their physical s ize and space along t h e l im. No effect on the output by the addition of a number of transducers d t h high

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l Fig . 11 - (a) Oscillograph traces

of the responses of four delay l i nes using as delay medium: A, .005 in. wire; B, .010 i n . wire; C, .020 i n . wire; D, -020 in . th in wall tubing. (b) Oscil- lograph traces of the responses of a delay lines with current producing A sixty per cent of t ha t produclng B.

b impedance loading could be observed. The length of delay afforded by any delay medium i s determiEd by the speed of sound i n it. T h i s i s calculated a s the square root of the quotient Young*s modulus divided by the density. For nickel the value is near 1.6 by l& fee t per pecond. T h i s is 0.192 inches per micro- second, Wng the delay nearly 5.3 sec per inch. During the experiments, lengths varying from a few inches t o 71 f ee t were used. The delay along the lat ter was 5100 p sec.

The space required for the l ine may be reduced by coiling the delay medium. Care must be taken to avoid small-radius bends i n the Une since they cause ref lect ions of the signal. Moreover, damping and ref lect ions due t o contact with supports are to be minind.zed. One 27 foot l ine was coi led into a space less than 7 inches i n diameter with no serious detrimental effect on the signal. The damping effect required a t the ends i s easily supplied by neoprene pads clamped to the delay medium. The length of delay can be varied smoth ly by moving the receiver coi l along the l ine . To change the delays for several receivers simul- taneonsly the transmitter position m a y be changed. If the s igmficant s i g n a l i s a reflection, the delay m a y be varied by moving the reflecting discontinuity with respect t o the receiver. Variation.by steps can be obtained by switching to separate coils located along the line.

UPIJ CATI ONS . . . -

Limited frequency response and high inser t ion loss have restr ic ted the use of the magnetostziction system i n applications requiring delay alone. . However, where these handicaps can be tolerated -- when, f o r exanple, the signal can be shaped aftgr delay -- the relative advantages i n cost, temperature, stability,2 r e l i ab i l i t y , space and weight have made the use of magnetostriction practical..

S .

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In addition, some of i ts unique charac te r i s t ics make it particularly useful i n other si tuations. Onti of these features i s t h e l a t i t u d e i n the choice of location and number o f transducers used on the l i n e . The use of the single pai r of transducers g i n s 'a deley l i n e which can be made continuously variable over some specified range by 'changing the,distance separating the two along the delay medium. A scheme for dividing the pulse rate. provides for the signdl received aFter delay to t r igge r a gate uhich admits the next pulse simultaneously into the transrtdtter and the output .S O n e transmitter and a number o f receivers properly spaced along the Una can be used t o change the series dist r ibut ion of a t r a i n of pulses t o a parallel distribution.6 This process may be reversed i f the roles of the transducers are exchanged. A similar arrangement can be used for multiplying the frequency of a pulse sequence by combimng tk output of the transducers i n cascade.

The second characterist ic largely responsible for the applicat- Lon o f the magnetostriction delay l i n e i s the revers ibi l i ty of the polarity of the output. If the direct ion of the bias field i s reversed the polarity of the output i s changed. Information i n binary code can be easily stored and read i n the polar i ty of the magnetic bias field of a number of receivers. The readiry: is done without destruction of t h e stored information by the sonic pulse sent down the delay line, reading each b i t o f information through the output of i t s receiver.5,6 Changes i n the stored information i s made by a change of bias field which i s remanent magnetism i n t h e l i n e . This m a y be done by a pulse through ,the receiver. "he same system with fixed bias f i e l d s may be sed as word generators which supply, f o r example, constants used repeatedly.? By using reshaping equipment a circulating memory can be assembled with the information, supplied as positive or negative pulses, circulating down the l i n e and through the restoring system arad back down the l ine un t i l requi red i n the computation.5

REFERFNCFS

1. Fagan, M. D. : "Bibliography on ultrasonic delay l ines ." Trans. E, FGUE-2, pp. 3-8 (NOV. 1954).

3. Millership, R., Robbins, R. C., ancl DeBarr, A. E. : lWagneto&rfction storage systems for a high speed digital computer .l1 Brit ish J0u.1-d of Applied Physics, vol. 2, p. 304 (Oct. 1951). -

4. DeBarr, A. E.: " D i g i t a l storage using ferromagnetic materials." E l l i o t t J o ~ n a l , vol. 1, pp. 116-120 ( M a y 1953)

5. Robbins, R. C., and Millership, R.: lfApplications o f Magnetostriction Delay Lims." Automatic D i g i t a l Computation, proceedings of a symposium held at the National Pwsical Laboratory (London), 1953.

6. Epstein, H., ard Stram, 0. r "Magnetostrictive scmc delay line." --- Rev. ' S d . Instr . , vol. 24, pp. 231-2 ( M a r . 1953) .

7. Mason, W. P. : TE2ectromechdcal Transducers and .Wave Filters.11 New York, N. Y o , D. Van Nostrand Company, Inc., Chapt. 6, 1948.

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8. "ustin, A.: "A method of analyzing the bshaviour o f linear systems i n terms of time ser ies ." J. Inst. Elec. Engrs. (London) vol. 94, par t 11-A, pp 13O-fi2 (MW 1947)

- - - 9 . Bubb, F. W.! "A New Line= Operational Calculus." Report No. 6581,

Wright Air Development Center, Dayton, Ohio (1951) .

b

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