A. Pokryvailo, M. Kanter, and N. Shaked

Discharge Efficiency of Cylindrical Storage Coils

Copyright © 1998 IEEE. Reprinted from IEEE Trans. on Magnetics, Vol. 32, No. 2, pp. 497-504, March 1996.

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491 IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 2, MARCH 1996

Discharge Efficiency of Cylindrical Storage Coils A. Pokryvailo, M. Kanter, and N. Shaked

Abstract-The influence of eddy currents on the energy trans- fer from a storage coil to a resistive load in systems with a long charge was studied. The magnetic diffusion equation for a cur- rent exponential fall in an infinite slab, as a model problem, was solved analytically. The dependence of the energy loss on the slab characteristics and time constant of the current decay was determined. More complicated cases of cylindrical coils were treated numerically. Both electromagnetic and stress analyses were performed. A jellyroll coil was found to be su- perior to a pancake coil in terms of discharge efficiency. A jel- lyroll coil of the Brooks type is able to supply to the load ap- proximately 25% more energy than a pancake coil of identical size, providing both carry the same current. However, the en- ergy density of a pancake coil is approximately 20% higher than that of a jellyroll coil, if the coils are charged up to the limit of their mechanical strength. To calculate the discharge efficiency of a pancake coil, a time-consuming transient analysis could be replaced by a dc or low-frequency ac analysis. Discharge effi- ciency increases for a faster charge, whereas the total trans- ferred energy decreases, owing to a slight field diffusion into conductors. The experiments with a pancake coil yielded re- sults in close agreement with the theoretical analysis.

I. INTRODUCTION NERGY density, defined as the ratio of the stored en- E ergy to the weight or volume, is one of the most im-

portant parameters for an energy storage system. The ul- timate limit of the stored energy is set by the mechanical stress in the storage element [ 11. For a storage coil, a lim- iting parameter is a magnetic field that results in mechan- ical stresses in the coil. For this reason, it is important to a designer to have information on the magnetic field dis- tribution. A body of publications is available on the cal- culation of the magnetic field using analytical and numer- ical methods (see, eg, a recent book by Binns et al. [2] and its bibliography). Moon [3] merges the magnetic field with stress analysis, thus enabling calculation of the at- tainable stored energy for a given geometry. However, some of the stored energy is lost during its transfer to the load. Therefore, the discharge efficiency, defined as the ratio of the energy transferred to the load, to the stored energy, is of no less importance than the stored energy density. Considerable losses can be caused by a long opening time of the switch, commutating the coil current to the load. However, even with a very fast switch not all the stored energy is transferred to the load, due to the eddy currents arising in the coil when its current falls rap-

Manuscript received January 8, 1995; revised July 3, 1995. The authors are with the Propulsion Physics Laboratory, Soreq NRC,

Publisher Item Identifier S 0018-9464(96)00662-0. Yavne 81800, Israel.

idly. Mathematically, the problem is described by the magnetic diffusion equation which has been treated by several authors, eg, by Knoepfel [4], and with an as- sumption of a small skin depth compared to the conductor size, by Schneerson [SI. An extensive collection of ana- lytical solutions of the diffusion equation for thermal problems for different geometries that can be used suc- cessfully for magnetic problems as well is given by Carslow and Jaeger [6]. These works, however, do not provide us with sufficient information for a straightfor- ward analysis of the coil discharge efficiency. The aim of the present work is to fill this gap, especially when the skin depth is commensurable with the coil conductor size. The analysis is limited to the case of cylindrical coils, namely, of the pancake and jellyroll types.

11. GENERAL CONSIDERATIONS A circuit diagram of a typical inductive energy storage

system with a long charge time is presented in Fig. 1. The battery charges the coil when the switch is closed, and upon the switch opening the coil current is commutated to the load. The transfer efficiency is defined here as the ratio of the energy EL transferred to the load, to the stored energy Eso at the moment preceding the coil discharge:

E L

E . 7 = -.

For a nonferromagnetic medium, at any moment, the coil energy content can be calculated as a volume integral

Es(t) Es,(t) + Esa(t)

where the first integral represents the energy E,, stored in volume V, of the metal, the second integral is the energy E,, stored in volume Vair of the air, H ( t ) is the magnetic field, and po is the permeability of free space. In other terms, the stored energy can be calculated as

(3)

where i is the coil current and L is the coil inductance that according to (2) depends on the field distribution and thus is a function of time. The energy transferred to a resistive load RL is

T, 2

EL = Io 2 dt (4)

0018-9464/96$05.00 0 1996 IEEE

498 IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 2 , MARCH 1996

VB

Fig. 1. Circuit diagram o f a typical inductive storage system.

where the integration begins at the start of the current commutation into the load, T,, is the transfer time, and vL is the voltage across the load. The coil inductance and resistance are not constant but vary during both the charge and the discharge periods due to eddy currents. There- fore, neither charge nor discharge is strictly exponential even if RL does not change. However, for the sake of sim- plicity, we assume in the subsequent analysis that the cur- rent changes exponentially both at the charge and the dis- charge, and that RL = const. Accordingly, the term time constant is employed throughout the text. At the dis- charge, the current falls as

iL(t) = Ioe-r’7. (5) Then (4) can be rewritten as

T I T

EL = RL iozr iidt = R& io dt. (6)

It is reasonable to choose T,, L 27, because after that time EL does not change more than by factor e-4, or by 1.8 % if T,, -, 03.

There are two major mechanisms of energy loss during the discharge stage. The first is the ohmic heating of the coil by eddy currents, resulting from the skin and prox- imity effects, while the second part of the stored energy

is lost in the switch during commutation time r,, . It is useful in the following discussion to outline qual-

itatively the energy flow balance in the coil during the charge and discharge in the circuit of Fig. 1. The balance is represented in Fig. 2 in accordance with (2). It can be seen that while E,, is transferred to the load, E,, , trapped in the metal, decreases more slowly than the current. Thus, after 27, practically no energy flows to the load, but a considerable part of the stored energy E, remains in the metal and is dissipated in the form of heat by eddy currents.

In the case of a fast current rise which is characteristic of a coil charge from a capacitor bank, the magnetic field does not diffuse into the conductors and E,, can be ne- glected. In this case all the stored energy is concentrated in the nonconducting space. For a slow charge, where the skin depth is much greater than the conductor thickness, the magnetic field diffuses into the conductors com- pletely. This means that at the end of the charge the field

Fig. 2. Schematic representation of the energy flow balance in a coil.

distribution is close to that of the dc case. This mode of charge is shown in Fig. 2.

Assuming zero switch losses, we can write the energy balance in the form

E, Esm + E,, = E L + Elost = EL + Ej + E, (8)

where E,,,, is the sum of losses in the coil during the dis- charge, consisting of the joule losses

E, = s: 1 v u 2 d u d t (9)

and of the remaining energy E,. at time T,, . Hence, (1) transforms to

enabling the calculation of the coil discharge efficiency qd using (2).

ID. MAGNETIC FIELD AND STRESS ANALYSIS A. General Equations

ducting medium is written as The magnetic diffusion equation for an isotropic con-

where k,, = l /up and 0, p are, respectively, the conduc- tivity and magnetic permeability of the conductors ; they are considered to be constants. The current density can be calculated from the equation

or j = V x H (12)

The above equations, together with appropriate boundary and initial conditions, constitute the electromagnetic problem to be solved. Once the solution of (11)-(13) has been obtained, the magnetic body force density can be calculated by

f = j x B (14)

and used as a driving force for stress analysis. All mate- rials are considered to be isotropic and the deformations to be elastic.

POKRYVAILO et al. : DISCHARGE EFFICIENCY OF CYLINDRICAL STORA( 499 3E COILS

The electromagnetic analytical solutions that can be found for a limited number of simple cases are rather clumsy, especially for hollow cylinders [6]. An analytical solution is not available for a multiturn coil and an arbi- trary combination of the boundary and initial conditions. Therefore, numerical analysis seems to be the most ap- propriate approach for this work. PC-Opera software de- veloped by Vector Fields Ltd. [2], [7] that uses the finite element method has been employed in this work. The software contains static, ac, transient, stress, and other solvers for two-dimensional geometries that yield prompt solutions for a wide scope of electromagnetic problems. Some simple cases are treated analytically in order to achieve a better insight.

B. Current Fall in an Infinite Slab Consider an infinite conducting slab positioned as

shown in Fig. 3. The slab carries a uniformly distributed current with densityjYo in the direction of the y-axis, cre- ating a magnetic field H, (x) . The current for a unit height of the slab is Io = j y o l , where 1 is the slab thickness. At the time moment t = 0, the current begins to fall expo- nentially with the time constant 7, causing the change of the magnetic field. For a one-dimensional problem, (1 1) reduces to

(15) I aH,(x, t ) a2Hz(X, t )

ko at ax

and for the static case ( t < 0) it is further simplified to

Integration of (16) yields an initial field distribution in the slab

H,(x, 0) = 2Ho(; - k) where Ho is the initial field value on the outer surfaces of the slab.

Applying the Ampere's law in integral form

$I Hdl = I

and considering the symmetry, we obtain the boundary conditions

I 1 2 2

-H,(O, t ) = Hz(l , t ) = - = - I0e-"' = Hoe-"' (18)

where Ho = Z0/2. Finally, the problem is stated as a diffusion equation

(15) with an initial condition (17) and boundary condi- tions (18). Koshlyakov et al . [8] give a solution to a more general problem with initial and boundary conditions as arbitrary functions of the time and the coordinate x :

H,(x, 0) = 4 4 (19)

(20) H,(O, f) = +I(% Hz(L 0 = + 2 w

Fig. 3 . Infinite slab conductor.

The solution is sought as a series

03 nax H,(x, t ) = c T,(t) sin -

n = l 1

nax

and X is a dummy variable of integration. Making use of (17)-(20) and integrating, we obtain

m

nax H,(x, t ) = n = 2 C ~ , ( t ) sin -, 1

. (e((n%/l)2k0 - 1/T)f - 111, n = 2 , 4, 6, * - .

The current density can be calculated using (12) in the form

CO

aH, lr nax j = --= _- c nT,(t) cos -, ax 1 n = 2 1

(24) n = 2 , 4, 6, - - .

Let us introduce nondimensional variables

and a parameter cp = kodZ2 = ( s + , / Z ) ~ , characterizing the ratio of skin depth s+, = k07 (defined in such a form by Knoepfel [4, p. 551 for the case of an exponential bound- ary condition) to the slab thickness. For the sake of sim- plicity, let Ho = 1. Equations (23) and (24) then take the

500

form

IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 2, MARCH 1996

n = 2, 4, 6, * *

W

j y ( t , 6) = T C nTn(0) COS ( n n ~ ) . (27) n = 2

These expressions are greatly simplified when T + 0 or 6 -+ 0, ie, when the current stops suddenly:

a

j y ( t , 0) = 4 C cos n n t . (29) n = 2

Equations (28) and (29) depend only on the conductor's properties and the slab dimensions. They are plotted in Fig. 4 for several values of the nondimensional time 0 (the curves o f j are normalized with respect to the initial value j y o , lettingjyo = 1). While the net current is zero, the field does not drop abruptly to zero but decays gradually. The eddy currents flow in the slab, in the central region in one direction and at the outer surfaces in the opposite one, so that the net current is zero. Haines [9] named a similar phenomenon inverse skin effect, emphasizing that there is a current reversal, contrary to that of the usual skin effect.

Note that while series defining H converges fairly rap- idly, the corresponding series defining j does not con- verge satisfactorily for small time-values. Our interest, however, is in the energy aspects of the problem. With this in mind, let us introduce a function

showing the ratio of the trapped energy E, that remained in the conductor after the nondimensional time B = 28 had elapsed, to the energy Esmo stored initially inside it. Equation 26 is to be substituted into (30) in these calcu- lations. The function KE(8) is plotted in Fig. 5 , illustrat- ing that, depending on the rate of the current decay, a considerable part of the energy may remain in the storage element after the transfer to the load has been completed. This energy is eventually dissipated by eddy currents. Note that the term EJ is not accounted for in the plot of Fig. 5 .

The introduction of the function

where (28 ) is substituted for the integration, somewhat simplifies the representation of the energy flow in a slab

2

0

-2 2 e ..-) -4

-6

-8 0 0.2 0.4 E 0.6 0.8 1

Fig. 4. (a) Magnetic field and (b) current density distribution in an infinite slab for a current cutoff.

I

- N i l i

6 Fig. 5.Normalized trapped energy at 2tP after the beginning of the current

fall.

e Fig. 6 . Normalized trapped energy as function of time.

for a current cut-off. This function is shown in Fig. 6. Both functions KE(0) and &e) may be used for the as- sessment of the energy that is lost in conductors during coil discharge. For example, let us calculate K i ( 0 ) for two aluminum slabs (a = 3.4 lo7 Q-' m-') having a thickness of 0.15 m and 0.015 m, respectively, for 2 ms after the current cutoff. Making use of (25) and Fig. 6, we find that K i = 0.64 and K i = 0, respectively.

C. LOSSES IN CYLINDRICAL COILS Two-coil designs are examined: a pancake coil (PCC)

and a jellyroll coil (JRC) (see Fig. 7). Both are close to a

POKRYVAILO et al. : DISCHARGE EFFICIENCY OF CYLINDRICAL STORAGE COILS

~

501

Conductor (AI-1 100) Insulator (G10)

No. of turns: N=20 a=150, b=298, e 8 . 2 , d=2.6, h=213. Dimensions are in mm

(b) Fig. 7 . (a) Pancake and (b) jellyroll coil configuration.

Brooks-type coil that possesses a maximum energy den- sity [lo], albeit with an accompanying shortcoming of a high fringe field. A numerical analysis is adopted in this section, considering the intricacy of the problem. The coil was excited by a current waveform comprised of a charge and a discharge stage. The first one was an exponentially rising current i(t) = Io( 1 - exp( -t/TCh)). After t = 37,,, , the discharge stage was simulated by an exponentially falling current i = lo exp ( - t / ~ ) , where Io = i(37ch). Static and ac solutions were analyzed as well. A mesh of ap- proximately 8000 linear elements was used that yielded less than 1 % error for static and ac solutions and less than 1.5% and 4% in the charge and discharge stage, respec- tively, for its transient solution.

Field lines for the PCC are shown in Fig. 8 at the end of the charge (t = 3 ~ , ~ = 225 ms).and for 27 after the beginning of the discharge (t = 227 ms). Only half of the coil cross section is shown in view of mirror-image sym- metry of the solution in the median plane. In contrast to the end of the charge and to an ac field distribution, where the field is concentrated close to the coil’s inner surface, the field of a decreasing current exhibits the opposite be- havior, moving outward. Correspondingly, the current density reverses its direction close to the inner surface (see Fig. 9) and remains quite large even after 47 of the discharge, when the coil current has reached almost zero. The current density at the end of the charge is approxi-

Z. m 0.2

0.16

0.12

0 08

0.04

00 0 1 0 2 0.3 0 4 radius, m

(a)

0 16

0 12

0 08

0 04

0 0 0.1 0.2 0.3 0.4 radius, m

‘(b)

Fig. 8. Magnetic field lines (a) at the end of the charge and (b) 27 after the beginning of the discharge.

I I 1 I I 0.15 0.18 0.21 0.24 0.27 0.3

Radius, m

Fig. 9. Current density in the median plane of the pancake coil at the end of the charge (-) and 27 after the beginning of the discharge (---).

mately inversely proportional to the radius. Such a distri- bution is the result of a uniform voltage drop across the conductors prescribed by the program designers for the ac and transient solvers. Eddy currents have almost no influ- ence on it.

The current density distribution in the JRC is shown in Fig. 10. At the end of the charge the distribution is almost uniform, contrary to that of the PCC. Owing to this, the latter has a lower inductance of 0.125 mH (see Fig. 11) compared to that of 0.137 mH of the JRC, or in other words, a lower energy storage capacity. Note that the sec- ond value practically coincides with the inductance cal- culated using the data of Grover [lo], which are based on the assumption of a uniform current distribution. Hence, the conclusion may be drawn that a JRC is able to store 10% more energy than a PCC of the same size, provided they carry the same current.

Assuming the same filling ratio, the two coils have the same dc resistance. As a result, the time constant of the JRC is approximately 10% more than that of the PCC. Therefore, a less powerful primary source is required for charging the JRC to the same energy.

502 IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 2, MARCH 1996

4.5 , 1

Radius, m

Fig. 10. Same as in Fig. 9, for jellyroll coil.

I I I ! P6 0.14

- m- 0.13 E $ 0.12

5 0.11

.E 0.1

E - 0.09

1.4 5

1.2 f U,

1 ;

0.8 2 rn

1-

0.6 0 0.05 0.1 0.15 0.2 0.25

charge time, seconds

Fig. 11. Inductance and resistance of 20-tum pancake coil at the charge.

1

0.1

0

," 0.01 B E 0.001

1

'=

0.0001 0 225 0226 0227 0228 0229

E B

discharge time, seconds

Fig. 12. Inductance and resistance of 20-tum pancake and jellyroll coils at the discharge. ---: JRC inductance. ---: PCC inductance. -: JRC resistance. -: PCC resistance.

During the discharge, both the coil inductance L and resistance R increase markedly, by more than two orders of magnitude, after 37 (see Fig. 12). This is understand- able, because the energy stored at the end of the discharge remains quite large, whereas the current is very small. This effect is opposite to that of a decrease in inductance at high frequency [ 101. On the other hand, the nonuniform current distribution causes the coil resistance to increase. This is similar to an increase of the conductor's resistance at high frequencies. Note that neither L nor R, nor their ratio, remains constant during the discharge.

The discharge efficiency q of the JRC, calculated by (10) for T,, = 47, was found to be 87 % . Compared with the efficiency of 76% of the PCC, the increase is quite remarkable. Since a JRC is able to store 10% more en- ergy, we obtain a 26% increase of the load energy com- pared to a PCC carrying the same current.

At the end of the charge, the ratio of the energy stored outside the conductors E,, to the total stored energy E,, is K,, = 0.73 for the JRC and K,, = 0.75 for the PCC.

The latter figure coincides with the value of r] for the PCC, within the accuracy of the calculations. This means that all the energy trapped in the metal is dissipated in it, as could be foreseen from Fig. 5. Indeed, neglecting the coil curvature, we obtain 6 = 0.001; this supports the above conclusion. Therefore, the calculation of the discharge ef- ficiency for a PCC may be reduced to a static or a low- frequency field analysis, instead of a time-consuming transient analysis, provided the charge is long enough for the field to reach a steady state.

The JRC turns' thickness in the direction of the field penetration is approximately 20 times less than that of the PCC. This results in a parameter 6 = 0.35 that according to Fig. 5 should yield a high efficiency, approaching unity. Unfortunately, this is not the case, because of the prox- imity effect well known in multiturn coils at high fre- quencies. The resulting efficiency lies somewhere in be- tween the ratio K,, and unity.

Heretofore, a slow coil charge had been considered. With a fast charge, the discharge efficiency defined by (1) increases for the PCC because most of the energy is stored in the nonconducting space. For instance, with 7,h = 1 ms and keeping the other parameters unchanged, we ob- tain r] = 93%, whereas Kam = 0.9. However, the coil inductance at the end of the charge is only 0.1 mH. On the whole, the coil supplies to the load 10% less energy than with 7& = 75 ms. This example illustrates the ne- cessity of a sufficiently long charge time if the maximum energy density is to be reached.

D. Stress Analysis For the sake of simplicity, we model the coil as a thick

hollow homogeneous cylinder having the general dimen- sions of the coils shown in Fig. 7 . Thus the role of the magnetic field distribution is important, whereas the de- tailed structure of the coil is not. We use the following values of the Young modulus E and Poisson ratio E , char- acteristic for hard aluminum: E = 6.9 * 10" Pa, E = 0.33. With a mesh of approximately 2000 elements, a magnetic solution was obtained for a dc case that corre- sponds to a uniform current distribution in the JRC at the end of the charge, and for a low-frequency ac case that imitates closely the current distribution in the PCC, also at the end of the charge. The net current through the coil cross section is 1.846 MA, and that through one turn of a 20-tum coil is 92.3 kA. The Lorenz force of (14) was used as the input for the stress solver, with no constraints applied to the coil. All stress components were obtained. It follows that the hoop stress is dominant. Its maximum oemax is in the median plane on the inner surface, being 1.35 times greater for the JRC than for the PCC, as shown in Fig. 13. Note that the stresses computed by the sem- ianalyfical method of Gray and Ballou [ 111 for a uniform current distribution agree within 10% with the present calculations for the JRC.

Assuming both coils are capable of withstanding the same mechanical stress, it follows that the PCC is able to

POKRYVAILO et al. : DISCHARGE EFFICIENCY OF CYLINDRICAL STORAGE COILS

PCC

~

503

Es 1 EL I Kam v 9 5 I EL

Same current same m8-

1 1 1 075 076 0001 1 1

I I I 1 0.15 0.18 0.21 0.24 0.27 0.3

Radius, m

Fig. 13. Hoop stress in the median plane of the pancake ( - ) and jel- lyroll (. . . .) coils.

carry f i = 1.16 times more current than the JRC. Taking into account the difference between the induc- tances (125 pH and 137 pH, respectively), we deduce that the energy density of the PCC can be 1.23 times higher than that of the JRC. Calculations show that this value holds, with small deviations, for similar coils close to a Brooks' shape. It should be emphasized that the above conclusion is valid for unsupported, unprestressed coils, with no technological aspects taken into account.

It follows that for a given charge energy, with the same mechanical stress, a PCC is able to deliver to the load approximately 10% more energy than a JRC, at the ex- pense of higher losses and higher power drawn from the primary source. Therefore, a conclusion can be drawn that with a powerful primary source a PCC may be preferable, whereas a JRC is superior to its counterpart when the pri- mary source is not able to charge the coil up to the limit of the mechanical strength.

For the sake of convenience, the results of the calcu- lations obtained for PCC and JRC are summarized in the Table I, where the energy and the stress are given in rel- ative units.

IV . EXPERIMENTAL We use an experimental setup comprising a battery

bank, a closing and opening switch (OS), a 20-turn PCC described in detail elsewhere [12] and shown in Fig. 7, a resistive load, and measurement means. The battery bank consists of 360 car batteries where 72 strings, each com- prising five units, connected in series, are connected in parallel. It has an open circuit voltage of 70 V and is ca- pable of a short circuit current of over 100 kA. Two types of OS were employed: a GTO OS for low-current exper- iments and a two-stage OS for high-current experiments. It consisted of a commercial vacuum circuit breaker as a

first stage and an exploding wire as a second one, similar to that described by Braunsberger et al. [13]. The circuit breaker was employed as a closing switch as well. Car- borundum noninductive resistors in series with a high cur- rent diode formed the load branch. The routinely mea- sured parameters were (see Fig. l): OS current is,, voltage across the OS v,, , and load voltage vL . In the low-current experiments, coil current i was recorded as well. Ro- gowski coils and high-voltage compensated dividers were employed for the current and voltage monitoring. The data have been evaluated as follows.

The energy stored in the coil at the moment preceding the current break, Es0, was calculated using (3), with the value of the coil inductance taken from Fig. 11. Such a method allows one to account for the change of the coil parameters, which are hard to measure directly in the time domain. The energy transferred to the load, EL, was com- puted using of (4), neglecting the voltage drop across the cutoff diode D and assuming RL = const, which was, typ- ically, 87 ma. During the discharge, the load resistors' temperature rise was no more than 30°C. Thus, with the specified resistance temperature coefficient of 0 to -O.I%/"C, the error of E L calculation does not exceed 3%.

Further, the energy transfer efficiency 7 was calculated by use of (1). In order to find the coil discharge efficiency qd we must account for the energy E,,, drawn from the coil to the OS-battery branch during the switching:

E,,, = ir vLisw dt. (32)

With an ideal switch, this energy would be supplied to the load. Thus, the discharge efficiency can be calculated as

(33)

Note that E,, is only a part of the O S losses. The rest is the dissipation of the energy stored during the charge in the parasitic inductance of the buswork.

For numerical field analysis, the coil current wave- forms of the low-current experiments were used as a cur- rent driving function. The coil was charged to l .6 kA dur- ing 40 ms, or about five time constants, whereas the discharge time constant was approximately 0.5 ms. The OS losses were neglected in view of a fast GTO switch- ing, namely, less than 10 ps. The calculated value of dis- charge efficiency, vd = 0.8, practically coincides with that obtained by the processing of the experimental data and with the value of Kam .

At higher currents of the order of 30 kA, Es0 = 60 kJ and RL = 87 ma, the typical discharge efficiency was con- siderably lower, on the order of 65%. This decrease is caused by a much longer charge of 150 ms that allows the field to diffuse into the conductor (Kam = 0.75) and by considerable OS losses. The latter were accounted for using (32) and found to constitute 5-8% of the stored en- ergy. Thus with an ideal switch, according to (33), an

504 IEEE TRANSACTIONS ON MAGNETICS, VOL. 32, NO. 2 , MARCH 1996

efficiency of 70-73 % would be obtained that agrees fairly well with the above analysis. The discrepancies may be attributed to measurement and calculation errors. Other energy loss mechanisms, such as shock losses, do not seem probable at the present current level. Indeed, esti- mates based on the data of Singer and Hunter [14] show that these losses do not exceed several joules and there- fore can be ignored. Thus, in an inductive storage system with a long charge, the coil appears to be a major source of losses, due to eddy currents.

V. CONCLUSIONS A storage coil design must include an optimization not

only of the stored energy density but also an optimization with respect to the discharge efficiency. Of the two coil types examined in this paper, a JRC is superior to a PCC in terms of the discharge efficiency. A Brooks’ type JRC is able to supply the load with approximately 25% more energy than a PCC of the identical size, provided both coils carry the same current. However, the energy density of a PCC is approximately 20% higher than that of a JRC, if the coils are charged up to the limit of their mechanical strength. With this provision, a PCC is able to deliver to the load approximately 10 % more energy than a JRC. This is valid for unsupported, unprestressed coils, neglecting technological manufacturing aspects. It was found that for calculation of the discharge efficiency of a PCC, in the case of a long charge, a time-consuming transient analysis can be replaced by a dc or low-frequency ac analysis. For 6 < 0.001, the discharge efficiency can be calculated as q = KO, = EO/Es0. The discharge efficiency increases for a faster charge, while the total transferred energy de-

creases, owing to a smaller field diffusion into conduc- tors.

ACKNOWLEDGMENT The authors thank Dr. N. Spector for his valuable com-

ments.

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