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The circumferential propagation process for the magnetization of tape wound cores

Davis, George W.

Monterey, California: U.S. Naval Postgraduate School

This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.

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ia) LIBRARY Dudley Knox Library / Naval Postgraduate School

411 Dyer Road / 1 University Circle Monterey, California USA 93943


1963 DAVIS, G.







George W. Davis



By George W. Davis U4

Lieutenant, United States Navy

Submitted in partial fulfillment of the requirements for the degree of



United States Naval Postgraduate School Monterey, California

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George W. Davis

This work is accepted as fulfilling the thesis requirements for the degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING from the

United States Naval Postgraduate School


A theory is developed describing the process by which a signal induced at one point on a ferromagnetic toroidal core propagates to other locations about the core. Both qualitative and quantitative arguments are presented to support this description. Am experiment is discussed which tests the proposed theory. A complete analysis of the results of this experiment is made and includes a quantitative comparison of the measured results with those predicted by the theory. In as much as this comparison shows excellent agreement between the experimental results and theoretical predictions, the proposed theory appears to explain quite adequately the circumferential propagation process in magnetic cores.

The author wishes to express his appreciation for the assistance and encouragement given by Professor Charles H. Rothauge, and Mr. Raymond B. Yarbrough of the U. S. Naval Postgraduate School and of

Mr. Bernard M. Loth of the Lawrence Radiation Laboratory.



TABLE OF CONTENTS Title Introduction A Theory on the Circumferential Propagation Process

Experimental Verification of the Circumferential Propagation Process

Analysis of Experimental Results Conclusions Appendix I Appendix IT





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Radiation Pattern for a Single Current Carrying Conductor

Radiation Pattern within a Thin Ring Magnetic Core Derivation of Core Trigonometry

Theoretical Propagation Curve for Impulse Signal Normalized Magnetic Field Attenuation Curve Representation of a Ramp MMF Function

Theoretical Propagation Curve for Ramp MMF Signal

Equipment Arrangement for Measuring the Circumferential Propagation Velocity

Oscilloscope Presentation of Signal Pulse

Comparison of Experimental and Theoretical Propagation Curve for a 225 A-T Pulse

Ramp Function Idealization of MMF Pulse

Comparison of Experimental and Theoretical Propagation Curve for a 117 A-T Pulse

Comparison of Experimental and Theoretical Propagation Curves for Pulses of Various Rise Times

Normalized Predicted and Experimental Attenuation Curves



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1. Introduction

There is a relatively large volume of information in today’s literature explaining the processes believed to occur in the normal magnetization or "switching" in tape wound ferro-magnetic cores. Since most of the interest has been eeneeen around the magnetization in small cores with uniformly distributed windings, little information is avail- able on the process this author chooses to call "the circumferential propagation process"; that is, the process by which a signal induced at one point on a magnetic core migrates to other points about the core circumference. Such a process is not normally of consequence when the core has a uniform distribution of windings since the signal is in- duced at all points about the core circumference. When large diameter cores are to be used in very fast pulse circuits, such a process becomes quite important. To illustrate this point, assume that a core of diameter 25 cm is excited at one point by a single turn primary. If the signal traveled on or in the core at the speed of light, it would arrive at a secondary winding placed diametrically opposite the primary after a 2.5 ns delay. One would guess that the actual velocity of signal propagation, that is the circumferential propagation velocity, would be somewhat less than the speed of light. If the velocity were an order of magnitude less, a delay of 25 ns would result. A 25 ns delay is somewhat longer than rise times encountered in modern high power pulse circuits. Even though these delays are significant in such circuits, this author could find no published reference explaining or even theorizing on the process by

which a signal might propagate from the region of an exciting winding to

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other locations about a magnetic core, It appears that further improve- ment in high power pulse circuits might well depend on an understanding of this process. Therefore, it is the intent of this thesis to develop

a theory, based on presently accepted ferro-magnetic concepts (such as those presented in reference 1) to explain the process of circumferential

propagation in ferro-magnetic torroidal cores.

2. A Theory on the Circumferential Propagation Process

2.1 Development of the Theory

Under the action of an applied field, flux reversal in magnetic tape begins with the formation of domains of reversed magnetization, In the normal polycrystaline magnetic tape, these domains form at crystal imper- fection such as voids, inclusions, etc. on or near the tape surface; then grow inward into the tape. The formation of these domains, the so called nucleation process, requires a certain minimum applied field and hence energy. Once this nucleus of reversed magnetization is formed, it expands by domain wall motion at a rate dependent on the magnitude of the applied field. As these domain walls grow, colliding and consolidating with those from neighboring nucleation sites, the region of reversed flux continues to enlarge. This process will continue until either all of the flux in the tape has been reversed or the applied field is reduced below some minimum coercive force. (1)

While the above constitutes only a very brief description of the magnetization process, two concepts essential to the development of the subject theory are introduced, First, the nucleation process will take place whenever a suitably oriented applied field at a nucleation site ex- ceeds some minimum magnitude. Second, provided the applied field is maintained above some other minimum value*, the domain wall will grow into the tape from the nucleation sites. Hence, there exist only two modes by which a signal induced at one point on the circumference of a magnetic toroid can propagate to some other point on the core. Once applied, the

signal must either travel internally by means of domain wall motion or

*The applied field necessary to move a domain wall is less than chat required to form the wall. See reference (1).


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over the surface in the form of an E-M wave originating flux reversals at surface nucleation sites wherever they occur in the path of travel of the wave.

Domain wall motion constitutes a linear transfer of energy in the direction of this wall movement. Since this energy is supplied by the magnetic forcing field, the direction of transfer must be that of the Poynting vector associated with this field. Therefore domain wall motion is in a direction perpendicular to the applied magnetic field. This domain wall motion is always accompanied by eddy currents which circulate in accordance with Lenz's law in such a manner as to oppose the applied field. These currents increase with increase in wall velocity, thus creat- ing a requirement for higher applied fields to maintain the initial wall motion, A signal, then, could not be propagated in the circumferential direction by this mode if the applied field were always parallel to the core lamenations. Certainly the leading edge of the signal wave has a component of its Poynting vector in the direction of the lamenations (or else no energy would ever transfer in this direction) and hence it is possible for domain growth to occur in such a direction. However, this growth is severely damped by the eddy current effect and its rate of propagation is many orders of magnitude less than that of the E-M signal wave over the surface of the tape.

Finally, the propagation velocity of a domain wall is directly related to the applied field. Hence, if the induced signal were propagated in this manner, the propagation velocity would be dependent upon the magni- tude of this field. (2) That this is not the case will be shown experi- mentally.

From the above, it must be concluded that the signal applied to a


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core is propagated from the point of application to the various points about the core by an electromagnetic wave. The signal radiates from

the exciting winding and becomes evident in the core whenever it reaches a nucleation site with a suitably oriented component of its magnetic vector greater than the nucleation force required by the site. This is the manner in which the signal must make its circumferential progress. Since the E-M propagation velocity is independent cf its amplitude, the signal applied to a toroidal core will likewise be independent of signal magnitude. Those factors which affect the time of transmission of the E-M wave from the exciting winding to outlying points on the core will also affect the time of signal transmission to the various points about the core, These effects have been investigated experimentally and the

results are included in chapter 4,

2.2 The Propagation Process(for an Ideal Impulse Signal) To illustrate the above theory graphically, consider the radiation

from a single current carrying conductor as shown in figure 2-1.

Fig. 2-l(a) Cylindrical Radiation Pattern for Impulse of Current in a single Conductor (b) Decay in Magnetic Field Intensity as a Function of Distance of Wave Travel.




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If a unit impulse of current is applied to the wire, a cylindrical pulse of magnetic field propagates radially outward at the speed of light. (3) The magnitude of this pulse field decreases in amplitude as the inverse of its radial distance of travel, as predicted by Ampere's law. This magni- tude decay is illustrated in Figure 2-l(b). This attenuation can be thought of in terms of a thinning out of the energy radiated in any one segment of the cylindrical field. Once this energy has been committed to a particular direction of travel it remains so directed as long as it is in the same medium.* Hence the energy density becomes less at the more remote points in space.

These ideas can now be applied in the analysis of magnetization in a large diameter magnetic ring as shown schematically in Figure 2-2 below. When an impulse of current flows in the exciting winding, a radiation

pattern similar to that described above results. The characteristic of

Fig. 2-2 Schematic Representation of the Magnetic Field Radiation Pattern Within a Thin Ring Magnetic Core. The Pattern is Produced by an Assumed Impulse of Current Through the Single Turn Exciting Winding.

*The direction of energy transfer will be altered when the E-M wave strikes the core,

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this process that is of most interest in this discussion is the time required for the applied signal to reach points A, B, C, etc. about the core. The relationship between these time increments and the are distance of the points A, B, C, etc. from the exciting winding will yield the ap- parent circumferential propagation velocity. This apparent velocity can be examined in the light of the proposed theory as follows. By trigono- metric manipulations as shown in Figure 2-3, we can obtain the arc lengths OA, OB, OC, etc. as functions of time elapsed between application of

the impulse and the arrival of the impulse signal at any point on the core.

diameter of core

NR Guwcud de st) d


Sin ae = = (2-2) cA. = half angle subtended by an arc with respect to the toroid center or a = Ca ch Cc = velocity of light

By solving the above equations graphically with an assumed core diameter of 40 cm, the plot of arc length against elapsed time shown in Figure 2-4 was obtained. The slope at any particular point on this graph represents the apparent circumferential propagation velocity for the corresponding location on the core. Obviously this slope, that is velocity, is not constant even though the energy associated with the signal travels over the chord lengths at a constant rate. In fact, this apparent point velocity must approach infinity as r approaches the diameter of the core.

This can be shown by substituting sin! ch for o0 in (2-1). Then:

Are = d sim. of

differentiating arc length with respect to time to obtain apparent propaga-

tion velocity:



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The amplitude of the signal (or the energy) arriving at the points A, B, C, etc, around the core varies inversely as the chord length of travel. Using the trigonometric relations introduced earlier, Figure 2-5 was constructed to illustrate this decrease of signal amplitude with increasing arc length.

In the above discussion, second order effects or signal noise resulting from reflections of the H field from the tape surfaces have been neglected, This is justifiable in the first approximation since they cannot effect the arrival time of the radiation at remote points about the core. Also, the magnitude of these reflections should be small since the reflecting surface is a lossy magnetic material. None-

theless, they may be apparent as oscillation in the signal.

2.3 Propagation Process for Signals with Finite Rise Times

In the above discussion a.unit impulse of sufficient amplitude to cause nucleation at the furthest point on the core was assumed, In reality, the signal cannot show the abrupt discontinuity of the unit impulse but must be accompanied by some finite rise time. This alters the results of the above in a small but important way. As was pointed

out earlier, the signal amplitude (or energy level) must exceed 2


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certain minimum level before nucleation can take place. Hence, when

the finite rise time of a realistic pulse is encountered, a certain delay will result while this pulse increases from zero to the minimum nucleation requirement for each point on the core. This delay is ampli- fied by the decrease in pulse magnitude with increase in radial travel. To illustrate these two points, consider Figure 2-6. A ramp of current* is assumed to be applied to the exciting winding to produce the magnetic

intensity pulse shown in the figure. This pulse can be expressed as a

; to


Fig. 2-6 The Ramp of Magnetic Field Intensity Resulting from a Ramp Current Function Showing H Critical and Hx as a Function of Time.

function of time in the following manner:

H = a = (2-4) Where K is the slope of the applied

ramp of current in the units of amp-

turn per second,

*As was the case with an impulse, a true ramp of current cannot be realized. However, the ramp is closer to reality than the impulse and serves well to illustrate the pertinent points.


If H crit is defined as the minimum magnetic intensity level which must exist at some point X on the core in order to produce nucleation, then the magnitude of this critical field at the exciting winding must have been, (from Equation 2-4) H crit = i: Where r_ is the chord distance from 2nr x the exciting winding to point X on

the core.

Solving fort:

k= a aaa (2-5) Thus, the time Ee must expire after the application of the ramp current function before a field of sufficient initial magnitude to cause nuclea- tion at some point X on the core is radiated from the exciting winding. An additional time, defined as At, must elapse while this H field travels the distance r to the nucleation sites at point X on the core, Again referring to the trigonometric relations of Figure 2-3, a relation-

ship can be developed between core arc length and the total elapsed time

from application of the ramp to nucleation at points on the core.

arc =qd and smde oe

vy =Kty and Tretar= tz + At By assuming a value of K' and using r as the independent parameter, a graph of arc length as a function of total elapsed time can be construct- ed. The slope of this plot will represent the apparent circumferential propagation velocity for the ratio of signal slope to coercive force, K'.* For the sake of concreteness, a value of K' = 3.33 = is assumed, For artitrary value of Es and Z\t, and hence T total, arc length can be calculated. A plot of values so obtained is shown in Figure 2-7.

*The derivation of numerical values for K' will be discussed in Chapter IV.


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The relatively long delay in the arrival of the signal at points about the core is obvious from this graph. Comparison with Figure 2-4 shows the total time required for an impulse signal to reach the point on the core diametrically opposite the exciting winding is only 1.3 ns while in the case of the ramp pulse just considered, the time required was over 13 ns. This large difference in time is clearly the result of the slope on the ramp pulse.

At this point, it is appropriate to bring together the various ideas presented and to sketch a single picture of the circumferential propaga- tion process. The proposed theory suggests that all signal transmissions from the exciting winding to other points on the core takes place by means of an E-M wave propagating across the inner core diameter at the velocity of light. However, the intensity of this wave attenuates at a rate inversely proportional to the distance of travel from the source. If consideration is restricted to signal pulses having finite rise times,* then it can be shown that this attenuation becomes a major determinant of the apparent propagation velocity. Recall that domains of reversed magnet- ization are formed by the nucleation process providing that the magnetic field intensity at the nucleation site exceeds some minimum or threshold magnitude, Therefore, the signal level at the exciting winding must exceed this minimum nucleation field by the appropriate attenuation factor if it is to cause flux reversal at some remote point on the core. Since the signals being considered have a finite rise time, a delay will result while the signal reaches this “remote nucleation level" (that is, an intensity level which includes both the minimum nucleation field and the

attenuation factor). This delay will increase with the distance from the

*Any physically realizable MMF pulse will have a finite rise time.


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exciting winding and will thus give an apparent retardation to the propagation velocity. This is readily seen by comparing Figure 2-7 with Figure 2-4, Figure 2-4 illustrates the time required for a pulse with zero rise time to reach various points about the core. Figure 2-7 il- lustrates the time required for pulse with a certain finite rise time to deliver the minimum nucleating field to the various points about the core. In the former illustration, a time of 1.3 ns is required for the signal to reach a point diametrically opposite the exciting winding while in the latter case, 13 ns are required. The difference, 11.7 ns, is identically the time required for the ramp excitation to reach the "remote nucleation level.'' As far as the core is concerned, it took 11.7 ns longer for the ramp pulse to reach the remote point in question than it did for the pulse with a zero rise time. Therefore the apparent signal velocity of the ramp pulse is less than that of the impulse.

A general verification of the proposed theory by experiment will be considered next. An analysis of these experimental results (Chapter IV) will further illustrate the effect of finite rise time on apparent

propagation velocity.


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3. Experimental Verification of the Circumferential Propagation Process

3.1 Purpose

This experiment was designed to measure the circumferential propa- gation velocity defined in Chapter I and to observe the effect of varia- tions in pulse amplitude and rise time on this velocity. A comparison of these measurements with those predicted by Chapter II will be made in

Chapter IV. 3.2 Procedure

The general procedure used to measure the signal velocity was to induce a fast rising step of current in an exciting winding located at one point on a magnetic core and then measure the time required for

this signal to become apparent at other points about the core.

3.3 Equipment Setup

To carry out this procedure, a thin ring of 1 mil Ni-Fe tape with a ring cross section of one-half square centimeter and a mean magnetic path of 127 cm was wound with a three turn primary and six (6) single turn secondaries (or pickups) spaced about the core as shown in Figure 3-1. A reference winding, designated G, was placed inside the primary. A 2 megawatt pulser capable of delivering a 20 KV square pulse with a L5 ns rise time was used to deliver the signal to the primary. A non induc-

tive resistance R was placed in series with the pulser to give it 4 current


Shunt for Measuring Pulse Current



V Choke 6

NOh— | ———__—_—_—.—


Bias Current

Fig. 3-1 Equipment Arrangement for Measuring the Circumferential Propagation Velocity


drive capability. The magnitude of the applied current was changed both by varying the applied voltage and by changing R while holding the

applied voltage at 20 KV.

3.4 Velocity Measurement Techniques

Various methods for measuring the signal velocity were considered.

The two methods used were selected because of their simplicity, reliability and reasonable accuracy.

The first method was a photo technique - consisting simply of present- ing the output signals from the six pick up windings on a tektronix's 517 oscilloscope using a common time base for the horizontal sweep. When displayed on a multiply exposed photograph, the time difference between signals could be measured and compared with the known relative positions of the pick ups to obtain the velocity.

A second method using a tektonix's 576 oscilloscope with dual beam, sampling, plug in units presented an even more attractive method of obtaining the necessary data. Here, the signal from the reference winding* was presented on the "A'' trace with the pickup signals being presented one at a time on the "B" trace, The two signals were simultaneously presented on the scope face as shown in Figure 3-2, The time difference between corresponding points on the reference trace and the pickup trace was record- ed for each pickup. The points chosen were typically the time from 507% of the peak voltage on the "A" trace to 50% of the peak voltage on the

"B" trace. A Tektronix's RS 1 digital readout unit was used to automatically

*It was found necessary to use the signal from the reference winding as the common time base for all time measurements. This eliminates the problems of accounting for time delays which might result from differences in the "A" and "B" channel probe leads. Also, since pulse comparison is the method used to determine the time delay, it is necessary that the pulses be of similar shape.


AL ShEace. or Reference Signal

is Sage gl

"B' Trace or Pickup Signal

Lt = Time delay between 50% Level of "A" to 50% level of yoy

Fig. 3-2 Oscilloscope Presentation of Signal Pulses read the time difference between the selected points. A plot of the re- lative positions of the pickups against the time differences so measured

readily yields velocity.

3.5 Variation of Pulse Rise Time

In order to observe the effect variations in pulse rise time have on the apparent propagation velocity, the circuit of Figure 3-1 was altered to produce pulses with rise times of 45 ns, 55 ns, and 75 ns. To ace complish this, small valved (that is, 1 to 5 ohm) wire wound resistors were placed in series with the non inductive resistor, R, of the original circuit. This technique satisfactorily produced the desired variations

without significantly altering the final pulse amplitude.

3.6 Attenuation Measurement

An effort was made to determine the attenuation of the signal at the various points on the core. These measurements consisted of record- ing the peak magnitude of the first voltage spike appearing on the pickup signal. Under most circumstances, the spike was quite evident and little

room for doubt remained as to what relative point was being measured,

3.7 Summary of Experimental Results

The above procedures were carried out by successively pulsing the


core with MMF steps of 225, 168, 117, and 50 ampere turns. The rise time was maintained constant at 30 ns. Time delays between the “A' trace reference signal and the "B" trace pickup signal were recorded for the

20%, 50%, and 80% levels of the peak voltages. The position of the

pickups on the core were plotted as functions of these time delays. Graphs 1 through 7 of. Appendix I show these plots. The rise time for the 225 a-t pulse was altered successively to rise times of 45 ns, 55 ns, and 75 ns. Graph #8 is a plot of the apparent propagation velocity for these various rise times, Finally, pulse amplitudes at the various pickups were measur- ed for each of the MMF pulses listed above. These are plotted in Graph #9.

Graph #10 is a normalization of the data from Graph #9.



4. Analysis of Experimental Results

In an analysis of the results of the experiment described in Chapter III, the proposed theory will be examined in two respects. First, the results will be used to support the theoretical argument