NASA Contractor Report 3332 

Coupled Wave Model 

for Large Magnet Coils 

G. J. Gabriel 

GRANT NSG-3182 

SEPTEMBER 1980 ’ 
NASA Contractor Report 3332 

Coupled Wave Model 

for Large Magnet Coils 

G. J. Gabriel 

University of Notre Dame 

Notre Dame, In&ma 

Prepared for 

Lewis Research Center 

under Grant NSG- 3 18 2 

NASA 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

TECH LIBRARY KAFB, NM 

llulll~lllllllllylllR

00639511 

1980 

TABLE OF CONTENTS 

SUMMARY 

I. INTRODUCTION 

A. Background 

B. Statement of Problem 

C. Salient Results 

II. THEORY OF COUPLED WAVE MODEL 

A. Evolution of Model 

B. Formal Solution 

C. Two-Turn Coil 

III. TESTS ON A TWO-TURN COIL 23 

A, Frequency Response 23 

B. Decay of Initial Current 26 

C. An Experimental Test 35 

IV. CONCLUSION AND RECOMMENDATIONS 

APPENDIX 

REFERENCES 

iii 

9 

9 

15 

20 

44 

47 

49 

- - 
SUMMARY 

An important problem in the design of large superconducting 

magnets is that of protecting the coil against high voltage breakdown due to electromagnetic transients which might be initiated 

by any of a number of causes such as faults, quenches, or switching. 

An understanding of such transients as the key to coil protection 

provided the motivation for this investigation and its objective 

of evolving a suitable analytical model that would help in the design 

of protective measures. Inadequacy of existing circuit theoretic 

models to account for effects observed in an earlier work necessitated 

a detailed field theoretic analysis which has resulted in the 

coupled wave model (CWM) based on the theory of multiconductor transmission line theory. Significantly, this model shows that, unlike 

the predictions of circuit theory, the temporal response of a coil 

at its terminals unfolds in discrete steps having durations equal 

to the travel time of a wave front around a single turn of winding. 

It is during one or more of these steps that high voltages are likely 

to occur. Thus the size of a coil becomes an important consideration 

in design. 

Successful results of detailed computational and experimental 

analysis of a two turn coil indicate that the model is of sufficient 

merit to warrant further refinement, if need be, and to justify 

detailed analysis of the general case of a multiconductor coil. 

Notably,‘ both the high frequency and low frequency behavior are 
incorporated in the model where, it is found that the turn-to-turn 

coupling coefficients play a significant role. While these coefficients are related to such factors as wire size, shape, and packing 

factor, design specifications cannot be firmly established at this 

stage of the development. 

2 
I. INTRODUCTION 

A. Background 

Superconductivity is finding an important application in large, 

energy efficient d.c. magnets such as those intended for use in magnetohydrodynamic energy conversion, thermonuclear fusion, and pulsed high 

power generation. In the current state of the art, such magnets are 

typically in the form of solenoid or saddle coils of wire composed of 

superconducting metal imbedded in copper substrate and cryogenically 

cooled to liquid Helium temperature. Because of their large scale in 

every respect--current magnitude, energy, physical size--superconducting 

magnets, as envisaged for the above mentioned applications, pose new 

problems which heretofore had not been encountered in smaller conventional coils. 

Protection of the coil against possible damage, especially that due 

to high voltage breakdown within the coil, is one such problem which is 

of major concern to superconducting magnet programs. To guard against 

damage due to excessive ohmic heating resulting from a sudden quench, 

typically the coil is switched from the power supply to a set of damping 

resistors intended to limit the terminal voltage during the current 

decay, as well as to dissipate the stored magnetic energy. Despite this 

conventional method or protection, evidence of interlayer voltage 

breakdown has been observed at NASA-Lewis Research Center upon disassembling a damaged coil [l]. That the voltages anywhere within the 

coil should not exceed the prescribed terminal voltage, however, is a 

conclusion based on the circuit theory of lumped inductors and, as such, 

3 
ignores the spatial distribution of voltages and currents. Properly, 

the problem falls in the realm of electrodynamic field theory. 

In support of the field theoretic viewpoint, evidence of multiple 

reflection of waves as well as a multiplicity of resonances due to standing 

waves on a prototype coil was reported in an earlier work [2]. It was 

also proposed, then, that multiply reflected wave fronts could provide 

a mechanism for voltage buildup at appropriate points within the coil. 

In view of these earlier findings, the need for a more realistic 

analytical model for predicting the coil behavior became apparent. 

The research reported herein was undertaken as a first step toward 

a better understanding of the electrodynamics of coils which hold the 

key to the objective of evolving a suitable analytical model that would 

help in the design of protective methods. To this end, the coupled 

wave model resulting from this investigation, as discussed in Sec. II, -- 

represents a significant refinement over earlier models. It is, however, 

one step in what is seen to be a continuing evolutionary process. 

B. Statement of the Problem 

Analytical modeling of the transient behavior of large coils is 

not new but has a long history in the power industry in connection with 

large transformers. A search of the literature, however, reveals that 

existing models are hypothetical, founded exclusively on the principles 

of circuit theory [3]-[5]. Typically, the coil is divided into an 

arbitrary number of meshes composed of inductors and capacitors. 

Moreover, most of the data appear to be restricted to a time scale 

measured in tens of microseconds which is too coarse for the events 

4 
we have observed. The significance of this point in the context of 

large coils having diameters of, say, one ft or more is appreciated when 

one considers that an electromagnetic disturbance travels 30 cm (1 ft) 

in one nanosecond. 

A purely network theoretic approach suffers from several disadvantages 

as follows [6]: 

1. There is no principle by which one determines beforehand the 

number of meshes into which a coil must be subdivided; 

2. There is no principle by which one determines the necessity 

of a particular topography of interconnections among the 

assumed circuit elements; 

3. The nodes and branches of the assumed network model cannot 

be uniquely identified with actual spatial locations on the 

physical coil; and 

4. Geometric factors pertaining to the coil are not readily 

manifested in the assumed circuit elements. 

Consequently, the thrust of our effort was shifted to a search for 

a model founded on field theory, even if approximate. A detailed field 

theoretic solution of the problem, though necessary and desirable in 

the long run, is replete with difficulties owing to the unorthodox 

reference frames that even the simplest coil geometry requires. Nevertheless, evidence from a number of experiments on a variety of helical 

coils, leads to the conclusion that electromagnetic waves propagate 

along the coil wire itself, and that to a fair degree of approximation 

they behave as transverse electromagnetic (TM) waves. Moreover, the 

waves on one turn of the coil are coupled to those on the others. 

5 
On the other hand, analysis of the field equati ons in a helical coordinate 

system appropriate to solenoids has shown that , strictly speaking, the 

electromagnetic field is not separable into the form demanded by the 

conditions for the existence of TEM waves. A perturbed version of 

TEM waves, however, is theoretically possible, and these would exhibit 

certain periodicities owing to the inherent periodicity of helical 

coordinates. Based on these considerations, the proposed coupled wave 

model was evolved as a foundation which is amenable to further refinement 

if need be. This model, together with corroborative computational tests 

on a two-turn coil, is discussed in the next sections, which are then 

followed by our conclusions and recommendations. 

C. Salient Results 

The coupled wave model (CWM) has many interesting features as 

analytical model for coils in general. However, it is appropriate 

this point to present the three most significant findings as perta 

to the objective of this investigation: 

an 

at 

ining 

1. All temporal events on the coil, such as decay of initial 

current through a resistor, evolve in discrete steps having a characteristic 

duration T, which is the travel time of a wave front around a single turn. 

In agreement with observations in time domain reflectometery (TDR) 

experiments, the theoretical calculations show that over long time 

scales these steps are smoothed so that response curves approach those 

expected from circuit theory. This phenomenon is interpreted as being 

a manifestation of coupled multiple reflections due to proximity and 

periodicity inherent in the physical structure of any coil. 

6 

&- ._ ._ 

-2-m --.- -. 
2. The discrete nature of the coil respon.se is a very significant 

finding, because high voltages are likely to occur during one of these 

discrete steps. Thus, if the probability of, dielectric breakdown 

increases with the duration of a high voltage pulse, a large coil is 

more likely to experience breakdown than a smaller one. Theoretical 

confirmation is provided by analysis of the decay of initial current in 

a two-turn coil connected to a resistor. With appropriate combination 

of parameters, the terminal voltage is found to exceed that predicted 

by circuit theory by a factor of 1.125. Although this excess of 12.5 

percent might not be sufficient to cause breakdown, it serves to 

illustrate the point that high voltages can occur, given the right 

conditions. 

3. In the earlier work [2] on a prototype four-layer coil, the 

multiplicity of periodically spaced resonances (at frequencies upward 

of 600 kHz) was roughly explained in terms of standing waves on a 

single straight transmission line. This, however, did not account 

for the low frequency resonance (of the order of 15 kHz) which bears 

no discernible relation to wire length as expected of standing waves. 

Existence of this resonance has been known for many decades and it has 

generally been interpreted in a circuit theoretic context as being a 

consequence of distributed shunt capacitance. Such circuit models, 

on the other hand, do not fully describe the high frequency (or short 

time) behavior of the coil. The coupled wave model bridges the gap. 

The low frequency resonance, in the sense of a minimum in the terminal 

admittance, is found to be a direct manifestation of coupling between 

turns. At frequencies which are small compared to l/~, the terminal 

7 
admittance calculated from the CWM approaches that of a parallel combination 

of a lumped inductor and capacitor. Although much attention has been 

given to the low frequency resonance in the literature on transformers, 

it does not appear to be a serious contender as a cause for high voltage 

breakdown, as shown in the sequel. It is, however, still important to 

the extent that it provides one more datum in support of the wave model, 

and may also prove useful in experimental estimation of coil parameters. 
II. THEORY OF COUPLED WAVE MODEL 

A. Evolution of the Model- _- 

Consider M straight wires, each having a length 2L, arranged in a 

uniform parallel array lying on a plane as shown in Fig. 1. Such an 

array is known -to support dynamic electromagnetic fields in the TEM 

mode. The geometry of the array is such that it permits exact solution 

of Maxwell's equations whereby the electric and magnetic fields, E and 

tJ, are separated into the product of functions, i.e. 

E(x,y,c,t) = v’(r;,t) e(xsY) 

t$x,y,c,t) = i’kd) +Y) . 

(1) 

(2) 

The amplitude functions v' and i' satisfy the one dimensional wave 

equation in the St manifold, and, with appropriate geometric normalization, they can be interpreted as voltage and current insofar as their 

product yields the total power. The vector functions e and h, on the 

other hand, are strictly dependent on the cross sectional geometry of 

the array, since they are derived from a scalar potential which satisfies 

the two dimensional Laplace quation subject to the boundary constraint 

imposed by the conductors. 

As a consequence of some properties of the Laplace equation, it is 

possible to associate with each conductor in the array, say the m th 

one, 

a "voltage" v,(G,t) and a current im(<,t). Each of these voltages and 

currents is porportional, respectively, the the field amplitude functions 

v' and i' , while the geometry of the array is manifested in coupling 

coefficients ymn relating the m th voltage to the n th current, vice versa. 

9 
r I 

. 

i,(-L St) 

. 

. 

. 

im(L,t) 

t 1 

i,,+-Lt) 

b 4 

. 

. 

. 

. 

. 

1 

2 

m 

m+l 

M 

-L 0 L 

Fig. 1: Straight wire array. 

10 
It is most convenient to employ the formalism of matrices and introduce 

the column vectors 

v(u) = 

s m 

v,(r .t) 

. 

. 

. 

v& St) 

. 

, ~bzst) = 

. 

ilk4 

i&z ,t) 

m 

and the coupling coefficients matrices 

Lzpy-1 , pcy 

where E and u are the permitivity and permeability of the medium surrounding 

the wires. Throughout this report matrices are denoted by a bar over the 

symbols, unless otherwise stated. 

With these definitions, the electromagnetic behavior of the array is 

governed by the system of equations, in matrix form, 

(3) 

(4) 

These equations are well known as the basis of the theory of coupled TEM 

transmission lines. For convenience of reference, we call rthe inductive 

coupling matrix and E the capacitive coupling matrix. They must always 

satisfy the condition 

11 
rc=$r (5) 

where c is the speed of wave propagation and T is the identity matrix. 

Consequently, when Equations (3) and (4) are diagonalized, the result 

is the one dimensional wave equation satisfied by each component of 1 

and 1, 

a2 

a52 

v 

:I 

i 

-pg 

at 

- 

1 

i: 

i - 1 1 = 

. . 

0 

0 

I . 1 . (‘3 

In other words, all component voltages and currents travel in unison. 

To apply the theory of coupled lines to a coil, let us bend the 

plane in which the wires lie into a cylindrical surface, joining the 

right end of the m th wire to the left end of the (m+l)th wire. The 

result is a single layer solenoidal coil whose terminals correspond to 

the left end of the first wire and right end of the Mth one, as illustrated 

in Fig. 2. 

With each turn of the solenoid now viewed as one line in the Mconductor system, the geometry of the solenoid amounts to imposition of 

the periodic boundary constraints 

im+l(-L,t) - im(L,t) = 0, 1 (m 5 M-l (7) 

ipA = iM(L,t) = i,(t) 

vm+l(-L,t) - vm(L,t) = 0, 1 (m 5 M-l 03) 

q-LA - vM(L,t) = v,(t) 

12 
ma1 

. 

. 

. 

M 

Fig 

. 

2: Helical Coil geometry* 

13 
where vt and it are the terminal voltage and current. These may be 

readily put in matrix form by introducing the interconnection matrix 

7 whose elements in the case of a solenoid are 

T mn = ‘r-i m+l 

, 

1 zrn 2 M-l 

(9) 

T =$,I Mn , 

where the Kronecker delta has the usual meaning. The boundary constraints 

then take the compact form 

Tl(-L,t) - y(L,t) = Vta+j , (10) 

r I(-LJ) - i(LJ) = 0 (11) 

where a+ is the normal column vector having zeros in every row except M, 

and 1 in row M. 

It should be noted that every interconnection, including taps, 

corresponds to a matrix 7 , whose columns and rows contain only one 

non-zero element, which is usually 1. Further, the matrix is always 

orthogonal having eigenvalues which are the complex roots of 1 on the 

unit circle in the complex plane. These eigenvalues are significant 

since they are directly related to the res0nance.s of the coil in frequency 

domain. Thus, a coil with M turns, having an M x M matrix 7, can 

exhibit at least M resonances, if not more. 

The telegraphist equations (3) and (4) which are valid for TEM waves 

only, together with the interconnection matrix 7 and boundary constraints, 

forms the gist of the coupled wave model. As mentioned earlier, however, 

-- 

it represents a first order approximation to the field equations in helical 

14 
coordinates. If the coupling coefficients are to be computed analytically 

from the geometry of the coil, rather than empirically, then a detailed 

solution of the field equations is necessary. 

It would be a mistake to think of the elements of the coupling 

matrices, L mn and 'rnn) as distributed inductances and capacitances, 

because the concept of distributed circuit elements does not always have 

justification in the electromagnetic field [6]. The important point to 

keep in mind is that one matrix is porportional to the inverse of the 

other. Thus when the geometric structure is such that capacitance 

coefficients can be calculated in the electrostatic sense, e.g., the 

straight array in Fig. 1 without interconnections, then it is helpful 

to think of c as a capacitance matrix. However, r would have no more 

significance than that of being an inverse. On the other hand, in a 

coil the r matrix is related to the low frequency inductance, as 

illustrated for a two turn coil in the next section. In this case, the 

concept of distributed capacitance would be inappropriate, and c should 

be regarded as the inverse of the inductive matrix. In fact, the relation 

between the inductive coupling matrix of the wave theory and ordinary 

inductance observed at low frequencies may provide a way for estimating 

the coupling coefficients. 

B. Formal Solution 

The system of equations (3) and (4) may be solved formally only to 

the extent of relating the terminal voltage and current, v,(t) and i,(t), 

to the M voltages and currents on each turn. A complete description of 

the coil behavior, however, cannot be given without prior specification 

15 
of its connections to external components such as generators and damping 

resistors. As to method o-f solution, the Laplace transform (over time) 

offers the most direct route since it also allows inclusion of initial 

currents. Transforms here are denoted by capital symbols, e.g., v(~,s) 

is the transform of v(<,t). 

As to be expected, the solution is dominated by properties of the 

interconnection matrix 7 and the coil geometry as manifested in a 

characteristic wave admittance matrix defined as 

Y(s) = ‘1, 3s) (12) 

where II, = (E/P) 

?- 

' is the wave admittance of the medium surrounding the 

coil wires. After transformation of the matrix wave equation (6), the 

general solution is found to be 

l(<,S> = A(s) ewsc" + B(s) esg/' 

I(C.,S) = Y@(s) emsc" - g(s) esg'c] 

where A and B are as yet undetermined column vector coefficients. By 

imposition of the periodic boundary constraints stated in (7) and (8), 

one determines these vector coefficients in terms of the terminal 

voltage V,(s) to be 

A(s) = m SW w a+ v,(s) 

e(s) =/p-o- Q-s) P&s) G V,(s) 

where 

B(s) = emST 

T = 2L/c 

16 

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 
F-l(s) = T - B(s) T (19) 

s-l(s) = 27 - C(s) (20) 

G(s) = 7 - l+(s) Y-l(s) K(s) V(s) (21) 

x-l(s) = R(-s) = f3T - (l-B*) P(s) . (22) 

It is noticed that the important matrices in this formal solution 

are the characteristic admittance matrix and the delay matrix p, defined 

in (19) in terms of the interconnection matrix 7. Further, the term 

pfl-r;r in the expressions for the vector coefficients A and B is the column 

vector composed of the Mth column of p. In the particular case of a 

uniform single layer helix, r has the elements defined in (9), so that 

7 takes the form 

-p(s) = 1- 

1 - BM 

#1-l @M-2 

. . . B 1 

1 Jl-1 gM-2 B2 B 

B 1 SM-1 B2 

. 

1 

(23) 

The pattern is such that each row is obtained from the preceeding one 

above it by shifting and folding the columns one step to the right. 

Significantly, this matrix is strictly a function of the delay 

parameter B = ems'. Unlike ordinary circuit elements, then, any transfer 

functions or admittance functions which might characterize the coil as a 

system would be functions of 6 rather than the frequency image parameter s. 

17 
Translated to time domain, this means that the temporal events within 

the coil evolve in discrete steps each having a duration T = *L/c, 

which is the time of travel of an electromagnetic disturbance around a 

single turn. As stated earlier, it is on this time scale that high 

voltages are likely to occur. 

The Laplace transform expressed in terms of the variable eBS' 

instead of the usual s is sometimes called the Z-transform [7], which 

is found useful in manipulating discrete time sequences. However, so as 

not to confuse the issue, whereas the Z-transform is applicalbe to time 

functions that exist only at discrete points and are zero in the intervals 

between, here the time functions exist during the entire interval making 

abrupt jumps from one interval to the next. Physically, this behavior 

is a manifestation of multiple reflections and turn-to-turn coupling. 

The phenomenon becomes more meaningful in the sample computations on a 

two-turn coil given in the next section. 

We now turn our attention to the terminal admittance which is an 

important parameter, playing a role in the way the coil interfaces with 

external components. It is defined as 

It(s) 

Y,(s) = ‘v,(57 * (24) 

Since the terminal current, It, is the boundary current, II(-L,s), of 

the first turn, the terminal admittance is simply the first row element 

of the vector solution L(-L,s) in Eq. (14). After substitution of (15) 

and (16) into (14) and using the form of P, one finds 

Yt = * j& ml1 'lj [5jm(S)Bm-1 + Sjm(-S)B"-m+ll * 

= 

(25) 

18 

- I
It is evident that the problem of calculating Yt centers around finding 

the matrix element Sjm , which is far from being an easy task. The 

difficulty arises largely from the fact that two unrelated matrices, l? 

and v, enter the definition of 3-l given in (20) and (21). Consequently, 

some of the well known techniques of analytical inversion, such as 

expansion in eigenvectors or idempotents, are not readily applicable to 

any significant advantage. That is, using the eigenvectors of one 

matrix only complicates the other. A possible technique that might 

prove useful consists of expanding 5 in an operator power series, 

suggested by the form of 3-l given in (20). Thus, assuming that such 

a series converges, one writes 

(26) 

Retaining only the first term, %T, we have examined the simplified 

approximate solution for M turns and found the results to be in accord 

with the conclusions based on exact analysis of the two-turn coil. It 

is interesting to note that the first term of s yields an exact solution, 

even in the general case of M turns, under the very special condition when 

the product RP is commutative. For, in that case, as evident from (21), 

F vanishes identically. 

At this stage, it appears that a numerical technique utilizing 

computers offers the only practical course of action. However, because 

of the magnitude of the task, lying well beyond the scope of this investigation, evolution of computer programs for the general case was not 

undertaken. Instead, owing to its mathematical tractability, the twoturn coil has been analyzed largely for the purpose of appraising the 

merit of the model. 

19 
C. Two-Turn Coil 

Highlights of the reduction of the general solution to the particular 

case M = 2 are given here to theextentneeded for the analysis presented in 

Sec. III. In this case, the characteristic admittance matrix has only 

two distinct elements 

while the interconnection matrix takes the simple form 

0 1 

y= [ 1 . 

1 0 

(26) 

(27) 

Under these conditions, the product i?v is commutative so that the matrix 

s reduces to ST, thus simplifying Eqs. (15) and (16) substantially. The 

result is 

A(s) = q(s) -@L 1 

l-B2 [I B 

B(s) t = -q(s) + B 1-B c3 1 

Using these coefficients in Eqs. (13) and (14), one can calculate the 

voltage and current at any point on the two turns. Our interest, 

however, centers mainly on the terminal admittance function, because 

this provides a direct way of testing validity of the model. Since 

s. =ti. Jm Jm’ Eq. (25) reduces to 

20 

(28) 

(29) 
Y,(s) = 

% 

*(l-8*) 

[v,(l+B2) - *Y*Bl * (30) 

Alternately, the admittance may be put in the form 

Y,(s) = 3 [y, coth (ST) - Y2csch (s-c)] (31) 

which is found more convenient in analysis of steady-state frequency 

response. 

It is evident that even for the two turn-coil, the wave theory 

admittance Yt is far from the expression l/sL, obtained for the lumped 

inductance L, found in circuit theory. However, the expression in (31) 

contains the circuit theoretic form as a limit. This may be shown by 

expanding the hyperbolic functions in a power series, the first two terms 

of which are 

rlo Yl -Y2 Yl v,(s) -f 2 yy- [l + t2S2T2 ___ Y1-Y*l ’ 

This asymptotic form would be valid in the low end of the frequency 

response or, equivalently, in the long term limit of the time domain 

response. The quantity multiplying the term s2 in (32) is properly 

interpreted as a resonance frequency, 

2 w = 

2(Y1-Y*) 

. 0 Yp* 

(33) 

It is also recognized that the asymptotic form of Yt corresponds exactly 

to the admittance of a hypothetical circuit comprised of an inductor and 

capacitor in parallel having values 

21 
LeTj?$$ 

vo= .c, = - 4 

(34) 

(35) 

While L, can be related to the quasistatic inductance of a coil as 

given by the Neumann formula [8], one would be hard pressed to find field 

theoretic justification for a coil capacitance having the above value 

'es This point emphasizes the limitations of the circuit theoretic 

approach. That the inductance couplihg matrix elements are related to 

the quasistatic inductance is a consequence of our assumption that the 

waves on a coil behave as transverse electromagnetic (TEflwaves [9]. 

22 
I 

7 i! i 

III. TESTS ON A TWO-TURN COIL 

A. Frequency Response 

The frequency response of a coil, as seen at its terminals, is 

essentially a manifestation of its terminal admittance function. In 

a typical experiment, one observes the terminal current as a function 

of frequency. It is best, therefore, to examine this current when 

the coil is connected to an oscillator whose voltage and output resistance 

are V,(jw) and R , 

9 

respectively, where w is the angular frequency and 

j = J--T. The terminal current phasor is then 

I&j4 = Vg(ju) 

Y,(jw) 

l+RY g t (5-J . 

To convert Y,(s) from (31) to phasor form, one simply makes the substitution s = jw to obtain 

It(jw) = R-l g V,b) 

Yl COS(WT> - Y* 

y1 COS(WT) - y2 + j sin(wT) 

where 

yi = %TI y.R 

0 1 g’ 

i = 1,2 

(36) . 

(37) 

(38) 

are the normalized admittance matrix elements. 

Inspection of (37) reveals that, for values yl f 0 and y2 # 0, the 

magnitude of the current reaches a maximum when the imaginary term in 

the denominator vanishes, and a minimum when the numerator vanishes. That 

is, for integer vi:lues of m and n, the maxima occur at uniformly spaced 

frequencies, 

23 
"m = rnn/T , 

and the minima at non-uniformly spaced frequencies, 

(39) 

wn 

= $ [cos-~(Y~/Y$ + 2nd . (40) 

It is evident that the turn-to-turn coupling plays a significant 

role in determining the minima of the frequency response. Typical graphs 

of normalized current vs. we are shown in Fig. 3 for two values of the 

ratio y2/y1. In the absence of coupling, y2 = 0, the minima become 

uniformly spaced at odd multiples of IT/~. This is the result one would 

obtain if the entire length of both turns of the coil were treated as a 

single uncoupled transmission line, as was done in the earlier work [2]. 

However, in the presence of coupling, the maxima are not affected, but 

the minima are shifted symmetrically toward the even numbered maxima. 

The lowest minimum, lying near n = 0, is interpreted as the low frequency 

resonance commonly observed in all coils. As it is apparent from (40), 

when y2 = yI implying perfect coupling, the lowest resonance frequency 

approaches zero. Interestingly, this resonance is actually repeated 

periodically at higher frequencies. Thus, even the low frequency 

resonance, which heretofore has been interpreted in terms of distributed 

capacitance, is seen to be a manifestation of coupled waves. Finally, 

recalling that T = *L/c where 2L is the length of one turn, one observes 

that the relationship between wire length and resonance frequency in 

(40) is not simple enough that it could be readily inferred from experimental data. Certainly, in a coil with a larger number of turns the 

relationship is expected to be more involved. This explains why in our 

earlier work [2], no simple relationship between wire length and the low 

resonance frequency was discernible in the data. 

24 
.8 

+J 

5 

:: .6 

s 

.2 

1 I\ 

;I l-l 

I \ 

II I\ I \ 

\ 

\ 

\ 

\ 

. 

I 

0 

--w 

y2 

= 0

Normalized frequency, o'c 

Fig. 3: Typical frequency responses, yl = 0.1. 
B. Decay of Initial Current 

One of the questions which is of central importance to this 

investigation concerns response of the coil when its terminals are 

suddenly switched from a power source to a damping resistor R . We 9 

consider this now for a two-turn coil. Prior to switching, the 

current within the coil is assumed to have reached a steady state 

value of IO which is uniformly distributed. In that case, the voltages 

anywhere on the coil would necessarily be zero. Conditions immediately 

after switching, however, are not so obvious. 

In this context, two points are worth noting. First, if the coil 

were viewed as the lumped inductor of circuit theory, the terminal 

voltage immediately after switching would jump abruptly to a value 

-I R 

0 54' 

and at no time thereafter would the voltage anywhere within the 

coil exceed this value. In the field theoretic viewpoint, however, 

placement of a resistor has the significance of a boundary constraint 

relating only the terminal voltage to the terminal current, and the 

latter need not necessarily be IO at the instant of switching. 

The second point concerns the meaning of Rg as the apparent resistance 

experienced by the coil at its terminals. Thus, if the physical resistor 

is coupled to the coil via any length of wire or cable, then the pertinent 

value Rg is that obtained by impedance transformation appropriate to 

transmission lines. Indeed, the effective impedance referred to the 

coil terminals might not even be purely resistive unless the cable 

impedance is matched to the resistor. We assume this to be the case. 

When an initial current I, is included, the system of equations (3) 

and (4) leads to the general solution 

. 

26 
v(w) = /l(s) e-cs’c + B(s) e-‘c (41) 

IJ S,S) = T [A(s) e-cs’c - k(s) e-‘c] t iI0 (42) 

where the coefficients &and k have the same expressions (15) and (16) 

as the initially uncharged case. Since the terminal current It(s) is 

the boundary current II(-L,s), the following conditions must prevail 

V,(s) ,= -I1(-LsNg (43) 

11(-L,s) = V,(s) Y,(s) + iI0 * (44) 

Equation (43) is merely the condition imposed by the resistor, while 

(41) follows from (42) and the definition of Y,(s) given in (24). 

Combining these two conditions yields the desired terminal current 

It(S) = $ I 1 

o 1 + Rg Y,(s) ' (45) 

From this expression, which is valid for any number of turns, one concludes 

that the response of the coil under the above conditions is equivalent to 

excitation of an initially uncharged coil by a step voltage -IoRg through 

a series resistor R 

g' 

This is also true of any transmission line. 

It is convenient to deal with a normalized terminal current 1; defined 

as It/IO. After substituting (30) into (45) one obtains for the two-turn 

coil, 

Ii(s) = $ 1 - D2 (45) 

WY+ - 2Y2B - Cl-Yl)B2 

27 
where yI and y2 are defined in (38). Inasmuch as the denominator is a 

polynomial in 6 = ems' instead of s, the inverse of (46) consists of 

discrete steps in intervals of duration T. This is manifested effectively 

by expansion of (46) in a power series in 6, as outlined in the Appendix. 

The procedure also facilitates numerical evaluation of the inverse, 

ii(t), by means of a digital computer. The result is the sequence of 

steps 

ii(t) = F KA u(t - nT) (47) 

n=o 

where u(t) is the unit step function and KI; are determined from the roots 

of the quadratic in the denominator of (46). 

Note that the sum in (47) is cumulative, so that the value of ii at 

the Nth transition is the sum of the first N coefficients. That is, 

ii(t) = SN = ! K;, NT < t < (N+l)T . 

n=o 

(48) 

Since the time is indexed in discrete intervals, the sequence SN which is 

a function of the integer N, is also essentially a function of time. Thus, 

whether ii decreases or increases'monotonically, or whether it oscillates 

in time depends on the behavior of the sequence SN as a function of N. 

This in turn is determined by the quantity 

b = [l _ y12 + Y2 1 2 s . (49) 

When b2 > 0, the sequence decreases monotonically after the first few 

steps following t = 0. When b2 < 0, the sequence breaks into oscillation 

whose period is determined by the values of yI and y2 as well as their 

ratio y,/y,. This condition is reminsecent of the underdamped and 

28 
~~.- .- - _._. ..-.. 

overdamped RLC circuit. Indeed, if one were to observe the current decay 

on instruments having insufficient bandwidth, the sequence would appear 

as a smoothed curve much as the response of an ordinary circuit. 

The above features are illustrated in a selection of graphs 

shown Figs. 4 through 8 , which were obtained by the use of a digital 

computer. Two of these figures are worth singling out. In Fig.6 

during the second interval the current jumps to 1.125 times the 

value of the initial current, unlike the circuit theoretic behavior. 

Similarly, the voltage is 1.125 times I R 

0 g' 

This excess occurs when 

y2/y1 approaches unity (perfect coupling) and yI is 0.33, the latter 

being a condition that depends on the choice of R 

g' 

Effects of 

compression of time scale are shown in Fig.8 for the underdamped case, 

b2 < 0. In particular, the period of oscillation here, as one 

expect, is related to the low frequency resonance discussed in 

previous section. If we define Np-c to be this period, then it 

be shown that the frequency is 

2Ti (yf - yg - 1)4 

i$T = + tan-I [ 

y2 

1 

might 

the 

can 

which is related to the low resonance frequency via (40) with n = 0. 

It is important to note that this oscillation can be easily suppressed 

by the simple expediency of proper choice of R 

g' 

However, this would 

not eliminate the possibility of high voltage buildup because this 

occurs in the overdamped case as seen in Fig. 6 . This theoretical 

conclusion which is based on the simple configuration of two-turns 

agrees with the earlier experiments on a multiturn coil [2] where the 

29 

(50) 
Yl = 2.0 

c, CO--’ z 1 

Y21Y1 = 0.9 

k b* = 0.240 

z + 

; 40-- 

N *F 7 - 

2 

2 

= 20- 

80- 

0 

1 

4 

! I 

8 12 16 20 24 

Time index, NT 

Fig. 4: Decay of initial current in short time scale, overdamped. I, = 100. 

30 
y2’y, = 0.90 

$ = -0.99 

4 8 12 16 20 24 

Time index, NT 

Fig. 5 Decay of initial current in short time scale, 

underdamped case. I = 100, N = 4. 

0 P 

31 

P 
110 

7c 

yl 

= 0.32 

y2Iy1 

= 0.99 

b* = 0.998 

I 1 I 

, 

0 4 8 12 16 20 

Time index, NT 

Fig. 6: First few steps of current decay under 

conditions when 3t exceeds IO = 100. 

32 
100 

20 

0 

y1 = 0.32 

y2/y1 = 0.99 

b2 = 0.998 

200 400 600 800 1000 1200 

Time index, NT 

Fig. 7: Decay current of Fig. 6 on compressed time scale. 

33 

,,;--TyF-. 
.----AL_. _-.- -Lee--: -.. - . ------ -_:--. - --___ __ 

2( 

l! 

l( 

I 

c, - 

z 

k 

z 

z ( 

N -I- F 

E 

0’ 

= I 

-. 

-l( 

-l! 

-2f 

- 

- 

- 

y1 = 275 

0 100 200 300 400 500 

Time index, Nr 

Fig. 8: Decay of initial current in underdamped 

case on compressed time scale. IO = 100; 

N 

P 

= 201. 

34 

.-___I_.- _ ----._l-- ---.- 

. . :~ . : --.-- 
low frequency ringing was successfully suppressed by choice of resistor, 

which had otherwise little effect on multiple step reflections. 

Returning to the overdamped case, one observes by inspection of 

(46) that because of the numerator the current is actually the difference 

between a sequence and its delayed replica, displaced by the interval 

2T. The denominator shows that this sequence increases monotonically, 

reaching the limiting value 1/2(yI-y2) as obtained by setting s = 0. As 

evident, this limiting value grows without bounds as y2 approaches y1 

with increasing coupling. However, because one sequence is delayed by . 

an exact multiple of the uniform duration of each step, the subtraction 

is complete resulting in a decreasing net response. It is conceivab1.e 

that when the durations of the steps in the sequences are not the same, 

a high voltage pulse might occur due to incomplete cancellation. This 

situation .is possible in a multilayer solenoid, for example, where travel 

times per turn differ from layer to.layer. By this line of reasoning, 

one tentatively concludes that dielectric breakdown between adjacent 

layers is more likely than between turns of the same layer. 

C. An Experimental Test 

Time domain reflectometry (TDR) offers a simple and direct means for 

an experimental appraisal of the model. InLa pulsed TDR experiment, one 

measures not'only reflection coefficients but also pulse travel'times. 

Typically, a cable with known characteristics is connected to a pulse 

generator on one end and to the test component on the other, as shown in 

Fig. 9 . The signal reflected from the termination, which is displayed 

on an oscilloscope coupled to the generator end, is determined by the 

35. 
Coil (Y,) 

Cable To 

Fig. 9: Schematic diagram for pulse TDR experiments. 

36 

.._- .._ .-__. _--_. : ~____.~ .. .._ _..._______. -- .--- --.__ 

...... ... 

... 

._ , - 
reflection coefficient defined in-terms of the cable admittance Y, 

and the terminal admittance Y,(s) as 

PtW = 

y. - Yt 

y. + Yt (51) 

It is well known in the theory of transmission lines that if 

the generator is matched to the cable and its voltage is V,(s), then 

the observed reflected voltage V,(s) at the generator terminal is 

given by 

V,(s) = V,(s) pt(s) eBS'o (52) 

where ho is the fixed delay due to the cable length, which is not of 

consequence. Except for this fixed delay, the reflected signal v,(t) in 

time is essentially the convolution of v,(t) and the inverse of pt(s). 

After substitution of Y,(s) given in (30) into (51), the inversion may 

be effected most conveniently by expanding the result in powers of 

-ST B = e * The reflected signal once again is found to be an infinite 

sequence of successively delayed terms of the form 

v,(t) = T P, vg(t - n-d (53) 

n=O 

where the time origin has been shifted to suppress the fixed delay 

T 

0. 

Thus, if the duration of the generator pulse form v,(t) is shorter 

than the coil travel time -r, each term in the sequence is distinctly 

resolved. However, owing to the damping arising from the finite 

resistivity of the coil wire at room temperature, only the first few 

terms are observable experimentally. 

37 
According to.the theory, the first term, p,, is due..to the . 

discontinuity at the coil terminals and depends only on the diagonal 

element yl. The second term, p1 arriving 'c units later, is proportional 

to the mutual coupling element y2; while the third term, p3, arriving 

2~ units later is dependent on yI and y2, and so on. The arrival 

times are very significant. For, when y2 vanishes in the absence of 

mutual coupling, the theory shows that the successively reflected terms 

would be separated by intervals of 2~ units corresponding to travel 

around the entire length of the two turns. The fact that, theoretically, 

the reflected pulses are separated by intervals corresponding to travel 

around one turn and the fact that the second term is proportional to 

the mutual coupling a.re two important data amenable to experimental 

verification. 

Typical experimental reflection patterns are shown in Figs. 10 

and 11 for loose coupling and tight coupling. The coil under test 

consists of two turns of No. 22 enameled copper wire wound on a wooden 

form having a diameter of 2.0 ft. In the tightly coupled coil, the 

wires are separated essentially by the enamel thickness, while i n the 

loosely coupled case they are separated by l/8 inch nominally. The 

travel time, T, should be roughly 8.2 nsec as determined from measurement 

of the propagation speed on a pair of straight sections of the same wire 

laid out as a transmission line. This measurement yielded 1.3 nsec per 

ft. 

As evident from Figs. 10 and 11, the timing of the pulses and the 

variation of the second pulse amplitude with coupling are in qualitative 

38 
Fig. 10: Pulse reflection pattern from a loosely coupled 

two-turn coil. Scale: 5 nsec/cm 

39 
Fig. 11: Pulse reflection pattern from a tightly coupled 

two-turn coil. Scale: 5 nsec/cm 
agreement with the theory. In the absence of an analytical relationship 

between the coupling coefficients and coil geometry, however, quantitative 

comparisons cannot be made at this stage of development. It is noticed, 

also, that in the tightly coupled case the travel time increases slightly. 

This effect is attributed to the fi,nite resistivity of the wire which is 

not included in the model equations for simplicity. However, when it is 

introduced as an additive diagonal matrix, the product of the inductive 

and capacitive matrices would no longer be l/c*, but it would depend on 

the relative magnitudes of resistivity and coupling coefficients. 

The physical mechanism underlying the observed patterns may be 

visualized with the aid of Fig. 12. To do so, one must keep in mind 

that the observable voltages are ultimately representations of the 

charge distribution on the wires. When such voltages are transmitted 

along a uniform cable they are associated with a positive charge distribution on one conductor and a negative (image) charge distribution on the 

other, both traveling in unison. Thus, an incident positive voltage 

pulse arriving at the coil means that terminal 1 receives a positive 

charge pulse, while terminal 2 receives an equal negative charge pulse. 

Due to the discontinuity, however, a fraction of these pulses is returned 

to the generator end, appearing as the first reflected term. The mismatch 

between the cable and coil terminals determines the polarity of the 

first reflected term relative to that of the incident one. We consider 

here the positive case. 

Now suppose that there is no coupling between turns. Then the 

pulses entering the terminals travel independently, but in opposite 

direction, down the entire length of the coil, the positive pulse arriving 

at terminal 2 and the negative pulse at terminal 1. As evident, the 

41 
fractions of these pulses entering the cable and returning to the 

generator end have polarities opposite to those of the incident pulses 

and their arrival time is 2~ units later than the first reflected'pulse. 

All successively reflected pulses would be separated by the.interval 2-c. 

On the other hand, if there is coupling, each of the incident 

pulses, a and b in Fig. 12, induces a pulse of opposite polarity in the 

neighboring turn. These secondary pulses, c and d, travel in unison 

with their respective primary pulses, but they arrive at the terminals 

after traversing only one turn corresponding to a delay of T units 

relative to the first reflected pulse. Ploreover, they enter the cable 

and return to the generator end in the same polarity as the incident 

pulse. Thus, turn-to-turn coupling causes a periodicity in the coil 

response, having intervals related to one turn. Another periodic interval 

results from the overall length of the coil. 

Extending this line of reasoning to a multilayer solenoid, one 

concludes that a coil should manifest periodicities related to the wire 

length between any neighboring points that are tightly coupled. A 

layer-wound solenoid having M turns per layer may be viewed as a single 

coil that 1s folded on itself at intervals of M turns. The first turn 

in the first layer is then coupled to the 2Mth turn which lies in the 

next layer. Consequently, the coil response should also manifest 

periodicities at intervals ~MT, i.e., intervals determined by the spiral 

length of a layer. Such layer related periodicity has been observed [2]. 

42 

2ziis-” i ._.-. . _-... . . 
Fig. 12: Schematic illustration of pulse 

travel around two-turn coil. 

43 

r 
IV. CONCLUSION AND RECOMMENDATIONS 

We have evolved basically the skeletal structure of an analytical 

model for coils which, unlike earlier models, is founded on principles 

of electromagnetic field theory. Despite the fact that it incorporates 

only the essential features of coupled wave phenomena, the model is 

remarkable in its agreement with several facts observed experimentally. 

The single most important finding demonstrated on a two turn coil is 

the discrete nature of time domain responses which unfold in steps 

having the characteristic duration related to a single turn of winding. 

This feature alone is sufficient to indicate the inadequacy of circuit 

theoretic modeling. Where high voltages are concerned, the analysis of 

a two-turn coil has illustrated that through the mechanism of wave 

reflection the terminal voltage can exceed that predicted by circuit 

theory. In this highly simplified configuration, having only two 

degrees of freedom, the coil terminals are the only location where an 

excessive voltage materializes. Given the larger number of degrees in 

multiturn coils, it is conceivable that high voltages could develop at 

several locations within the coil. As discussed in Sec. III.B, any 

inhomogeneity in travel times per turn , owing.to variations in the radii 

of layers or turns, appears to be a likely predisposing condition for 

high voltages to occur. 

The pertinent parameters have been identified to be the wave coupling 

coefficients as manifested in the admittance matrix v and the characteristic travel time of a wave front around a single turn. At this stage, 

however, any firm conclusions on design criteria would be premature and 

ill-founded. Such criteria cannot be conclusively defined without 

44 
further exhaustive analysis which might proceed in two phases. The 

first would consist of a detailed exploration of the general matrix 

solutions given in Sec. II.B, based on a hypothetical selection of 

parameters to determine their causative relation to the overall electrodynamic behavior of multiturn coils. The second would be directed at 

determining the relationship between physical features of the coil and 

those parameters which have been found theoretically to be most pertinent. 

Only after such conclusive analysis can any meaningful specifications be 

cited for such physical features as wire size, wire cross sectional 

shape, and packing factor, all of -which have direct bearing on the 

magnitudes of coupling coefficients. It is not difficult to estimate 

intuitively and from experience the way the coupling coefficients might 

vary with the above features. However, the significant but inconclusive 

aspect is the link between these coefficients and the coil behavior when 

large numbers of turns are involved. 

Success of the computational and experimental tests on the two-turn 

coil indicates that the coupled wave model has sufficient merit, to 

warrant further evolution and refinement. Among the, desirable features 

which should be included in the future are effects of: line-of-sight 

radiative coupling, finite resistivity,of windings, and inhomogeneities 

in travel times. Glhile the resistivity and travel time are of primary 

concern, the radiative coupling is not, save for the fact thatit might 

account for the fine structure parasitic oscillations generally observed 

. 

on TDR traces for all coils examined in this investigation. 

Finally, no investigation of high speed electromagnetic transients 

on superconducting coils would be realistic without serious attention 

45 

---.- ._-----.. .,, 

: ‘. ._, ,’ -.,. /.. 
directed to questions concerning the current distribution over the wire 

cross section. Given the typical configuration of copper clad superconductor fibers, it is quite certain from the Maxwell field theory 

that the current associated with any externally initiated disturbance 

would reside at the surface of the copper substrate. The more difficult 

question, however, concerns the way a large current initially confined 

within the superconducting fibers might distribute itself when it is 

suddenly subjected to a disturbance in the neighborhood of the associated 

high magnetic field. Whatever the origin of superconductivity on the 

atomic scale, its macroscopic manifestation is an enormously large 

conductivity, of the order of 10 25 mho/m. Even the smallest practical 

fiber is large enough that the Maxwell theory would be very much in 

command of events. The theory of skin effect indicates that, given 

the high conductivity, a disturbance occuring at a rate as slow as one 

cycle per second is sufficient to force the current to the surface of 

the superconductor. Such disturbances could result, for example, from 

slow movements of the coil winding under magnetic stresses. Although 

the distribution of dynamic currents in a superconductor is not fully 

understood, the skin effect in normal conductors suggests that the 

fiber currents could leak into the surrounding copper substrate, thus 

introducing local heating and possibly a quench. Considerations of 

thermal stability notwithstanding, it would appear that spreading 

the superconductor as a thin film on the outer periphery of the copper 

wire would be preferable from the standpoint of electrodynamic stability. 

This is merely a hypothetical remedy, but clearly questions pertaining 

to the current distribution deserve careful research as an inseparable 

part of the overall problem of coil protection. 

46 

---- .--- - ~._. ____ 

.:* -- - --.- ..---.. - - ._-.___ -.____ 

. 
APPENDIX 

INVERSION OF TRANSFORMS 

Laplace transforms of functions encountered in analysis of 

multiply reflected waves typically have the form 

F(s) = ,$$# , D = emST (A.11 

where Q(B) is a polynomial in 6. Such transforms may be'inverted by 

the standard contour integration in the complex s-plane resulting 

invariably in an infinite sum of residues on account of the periodicity 

of e-". A more useful sum for our purposes, however, is obtained by 

first expanding l/Q(B) in partial fractions followed by expansion in a 

power series in j3. This method is convenient for digital comoutations 

and it elucidates the delay feature inherent in multiple reflections. 

Thus, in the particular case when Q(B) is a quadratic having roots 

I$ and D2, one obtains 

F(s) =# [& - & . 

1- 2 - I 

(A.2) 

In regions of the s-plane where 131 < ID11 and 1D21, each of the fractions 

may be expanded as a geometric series, 

~(~1 = ti i (p-1 _ q-1) ,-nst . 

n=O 2 (A.3) 

One sees readily that, by the shifting property of the Laplace transformation, 

the inverse f(t) is the sum of a sequence of delayed and weighted values 

of h(t) 

f(t) = ; K, h(t-nT), (A-4) 

n=O 

47 

1 
where Kn represents the coefficients in (A-3). Although this sum 

is similar to Z-transforms in appearance, it differs in one respect, 

namely, h is defined throughout the entire interval nT to (n+l)-c. 

In the particular case of the decay of initial current, Sec.III.B, 

H(s) has the form 

H(s) = 4 (LB2). 

Consequently, the inverse is the difference of two sequences 

ii(t) = y Kn[u(t-n-c) - u(t-r-T-&)] . (A.5) 

n=O 

When the index in the second term is shifted, one obtains a single sum 

it(t) = -L 

l+Yl u(t) + F P$,+l - K,,J u(t-nd (A.6) 

n=l 

where for the two turn coil, 

Kn = (l+~~)-~ [(y2+b)” - b2-b)“l 

and b is defined in (49). In the underdamped case, when b2 < 0, one 

can conveniently group the terms as trigonometric functions thus manifesting 

oscillation. Omitting details here in the interest of brevity, we indicate 

that with b = ja, the typical n 

th term qn in the sum(A.6)takes the form 

qn 

= c, sin[ne - $1 

where 

tan9 = a/y2; tan@ = y1 tan8. 

Since the index n is essentially the time, it is seen that 8 basically 

defines the frequency of oscillation as given in (50). 

48 

. . ‘. :_ _ ._- -_ 

-. , 

--- 

.- _ ,- 
REFERENCES 

1. Maddock, B J. and James, G. B.,Proc. IEEE, vol. 115, p. 543, 1968. 

2. Gabriel, G. J. and Burkhart, J. A., Proc. 7th Symp Engineering 

Problems in Fusion Research, vol. I, p. 741, 1977 iIEEE Pub. no. 

77CH1267-4-NPS) Also Technical Memorandum, NASA TM;73808. 

3. Fergestad, P. I. and Henricksen, T., IEEE Trans. PAS, vol. 93. 

p. 510, 1973. 

4. McNutt, W. J ., Blalock, T. J., Hinton, R. A., IEEE Trans. PAS, 

vol. 93, p. 457, 1973. 

5. McWhirter, J. H., Fahrnkopf, C. D., and Steele, J. H., AIEE Trans. 

PAS, vol. 75, p. 1267, 1956. 

6. Gabriel, G. J., Proc. IEEE, vol. 68, March 1980. 

7. McGillem, C. D. and Cooper, G. R., Continuous and Discrete Signal 

and System Analysis, New York, Holt Rienhart and Wilson, 1974. 

8. Silvester, P., Modern Electromagnetic Fields, Englewood Cliffs, 

N.J., Prentice-Hall, p. 161, 1968. 

9. Gabriel, G. J., Proc. IEEE, (to be published). 

49 
1. Report No. 

NASA CR-3332 

2. Government Accession No. 

I 

3. Recipient’s Catalog No. 

4. Title and Subtitle 

~~UPLEDWAVEMODELFORURGE MAGNETCOILS 

6. Performing Organization Code 

7. Author(s) 

G. J. Gabriel 

9. Performing Organization Name and Address 

University of Notre Dame 

Notre Dame, Indiana 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D. C. 20546 

9. Performing Organization Report No. 

10. Work Unit No. 

/ 11. ;;xc;8;rant No. 

13. Type of Report and Period Covered 

I Contractor Report 

14. Sponsoring Agency Code 

15. Supplementary Notes 

Final report. Project Manager, James A. Burkhart, Wind and Stationary Power Division, NASP 

Lewis Research Center, Cleveland, Ohio 44135. 

16. Abstract 

A wave coupled model based on field theory is evolved for analysis of fast electromagnetic transients on superconducting coils. It is expected to play a useful role in the design of protection 

methods against damage due to high voltages or any adverse effects that might arise from unintentional transients. The significant parameters of the coil are identified to be the turn-to-tur 

wave coupling coefficients and the travel time of an electromagnetic disturbance around a single 

turn. Unlike the circuit theoretic inductor, the coil response evolves in discrete steps having 

durations equal to this travel time. It is during such intervals that high voltages are likely to 

occur. The model also bridges the gap between the low and high ends of the frequency spectrum, 

7. Key Words (Suggested by Author(s) 1 

Superconducting magnet coil; Electromagnetic 

transients; Coil protection; High voltage 

breakdown; Coupled wave theory 

- I 

19. Distribution Statement 

Unclassified - unlimited 

STAR Category 31 

I I 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 

Unclassified , Unclassified 1 “’ No’go2f Pages ( T&W -~ __ 

* For sale by the National Technical Information Service, Sprrnefreld. Vrrgrnia 22161 

NASA-Langley, 1980 

B -. .---.- -.-_-- -~ II