FerroxcubeBulletin_550

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Linear Ferrite' MAGNETIC

DESIGN MANUAL

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a o

:The. " . -'

. FERRDXCUaE FERRITE LINE

is the most complete and the most diversified range of ferrite-based

devices ever offered to the electronics industry

Q o

Toroidal Cores fOf linear and saturating magnetic devices. Pot Cores for precision linear magnetic devices ..

Special-Purpose Machined Ferrites.

Ferrite Shielding-Beads and Wide-Band Chokes.

Bulletin 550

Prlnted In U.S.A.

\

Square Cores for optimum packing densities.

E, U, and I Cores for linear and saturating devices.

COPljlught FVtJt.Oxc.ube. 1971

ITB 2. 5m 11/77

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FERROXCUBE MAGNETIC DESIGN MANUAL

TABLE OF

CONTENTS

Page

Introduction .......... "". . . . . . . . . . . . . . . . . .. 2

SECTION I - LINEAR, LOWLEVEL TRANSFORMER DESIGN THEORY

1.0 Linear Low-Level Transformers Defined ..... 3 1.1 Basic Transformer Theory ............. 3 1.2 Simplifying Observations on Practical

Transform ers III III 4

1.2.1 Calculations of F lux Density and Magnetizing Current ................ 4

1.2.2 Calculation of Primary Current ........... 4 1.2.3 Calculation of Required Turns Ratio ....... 4 1.3 Additional Design Considerations ......... 4 1.4 Resonating the Transformer ............. 5 1.5 Pulse and Broadband Requirements ........ 5 1.6 Calculating Leakage Reactance and

Self-Resonant Capacitance ............ 6 1.7 Low-Frequency Performance of Pulse and

Broadband Transformers ............. 7 1.8 High-Frequency Narrow-Band Transformers .. 8

DESIGN SUMMARY ................. 8

SECTION 11.- POWER TRANSFORMER DESIGN THEORY ............ " 9

2.0 Power Transformers Defined . . . . . . . . . . .. 9 2.1 Equivalent Circuits for Low and High

Frequencies ... . . . . . . . . . . . . . . . . .. 9 2.2 Inductance Calculation -

2.3 2.4 2.5 2.6 2.7 2.8

AC Excitation Only. . . . . . . . . . . . . . . .. 9 Transformer Design Procedure - Phase I .... 9 Low-Frequency Design Procedure - Phase II .. 10

Special Power Transformers ............. 11 Inverter/Converter Transformer Requirements. 11 Selecting a Core for an I nverter Transformer .. , 1 Completing the Design of an

I nverter Transformer. . . . . . . . . . . . . . . . 12 DESIGN SUMMARY .................. 12

SECTION III - LINEAR, LOWLEVEL INDUCTOR

3.0 3.1 3.2

3.4 3.5 3.6 3.7 3.8 "

3.9 3.10

3.11 3.11.1 3.11.2

DESIGN THEORY .............. 13

Low-Energy Inductors Defined .......... 13 A Note on Excitation Limits ............ 13 Frequency Ranges ................... 13 Pot-Core Construction ................. 13 Effective Permeability ................. 13 Inductance Calculation ................ 14 Importance of the Air Gap .............. 14 Checking the Wire-Winding "Fit" .......... 14 The Q Curve ....................... 14 Temperature Coefficient of Effective

Permeability (I nductance) .. ' ........ 15 Design Procedures ................... 16 Class I - L/V Dominant ............... 16 Class II - Land Q Dominant ... '.' ....... 16

3.11.3 Class III - 0 and T. C. Dominant ......... 16 3.11.4 Class IV - L/V and T. C. Dominant ........ 16 3.11.5 Class V - 0, L/V, and T. C. All Dominant ... 16

DESIGN SUMMARY ................. 18

SECTION IV - POWER INDUCTOR DESIGN THEORY ................... 19

4.0 Power Inductors Defined ............... 19 4.1 General Remarks on Design Procedure ...... 19 4.2 Effective Permeability vs Flux Density ...... 19 4.3 Effective Permeability vs AC Flux Density

4.4 4.5 4.6

4.7

4.8 4.9 4.10 4.11

with D. C. Excitation ............... 19 Variation in Permeability with Temperature .. 19 Core Loss vs Flux Density and Frequency .... 19 Core Losses vs Flux Density and

Temperature .................... 20 Inductance Calculation -

AC Excitation Only ................ 20 Design Procedure - Phase I ............. 21 Completing The Design - Phase II .. '.' ..... 21 "Trimming" the Design ................ 21 Inductance Calculation - AC and DC ....... 22

4.13 Estimating the Required Turns ........... 23

DESIGN SUMMARY ................. 24

Appendix ............................... 25 Wire-Winding Table ......................... 25 Design Factors k2 & Po for Ferroxcube Ferrite Cores .. 26 Glossary ................................ 27 Ferroxcube Ferrite Representatives and

Stocking List .. " ........................ 28

INTRODUCTION

The design of wirewound magnetic components is an exact science. All of the fundamental relationships have been dis-covered and investigated; their equations have been worked out, and reduced to practicable terms and practical quan-tities for real materials. The design procedures are rather tedious and protracted, but they are feasible. Standard en-gineering texts containing all of the details are available, and if the indicated routines are followed, the results will be as predicted -more or less. The design may be an exact science, but the implementation has some slack in it. The selections of standard transformer steels, lamination sizes, and shapes (not to mention the manufacturing tolerances in their mag-netic, electrical and mechanical properties) being what they are, practical approaches to the theoretical possibilities have been subject to various inhibitions and compromises.

The advent of magnetic ferrites, and their subsequent development and expansion, have made radical inroads into the designer's dilemma. A liberal selection of magnetic media, in standard core types and sizes, are now available, each offering a unique combination of magnetic, electrical, and mechanical properties that can be exploited dependably to satisfy specific design requirements. They represent, at times, the only means by which a set of design objectives may be met. The statements that follow, while not inflexible rules by any means, will be found to apply over a remark-ably broad range of energy levels, sizes, and circuit applications.

Inductors: For applications demanding the optimum com-bination ofhighQ,high stability of inductance, and minimum volume, there is no beating the ferrite approach, for fre-quencies up to 20 MHz. In filters and tuned circuits, the superiority of ferrite cores is not to be challenged.

Transformers: For small signal transformers, pulse and broad-band transformers of almost any kind, and power trans-formers over the mid-audio to radio-frequency range, ferrite cores are the first choice to be considered, particularly when size and cost are important factors.

Self-Shielding: There is no economical way of improving upon the inherent self-shielding properties of standard ferrite pot and toroidal core configurations. (This self-shielding per-mits absolute freedom in the "packaging" of the circuit around the magnetic components.)

Mechanical Convenience: The nature of the ferrite manu-facturing process permits us to design the core and its associated "hardware" to independently optimize both the magnetic properties and the mechanical configurations. Thus, ferrite components are the easiest to wind, assemble,mount, and wire to the circuit.

Adjustability: Available in most "preadjusted" Ferroxcube pot cores, this feature, in combination with the inherently high uniformity of the cores themselves, reduces or elimin-ates completely the problem of "strays" and tolerances in other circuit components.

2

Range: Ferrite cores are avai1able in an enormous range of standard sizes, shapes, and magnetiC characteristics (includ-ing, of course, the internationally-standardized line of pot cores), at mass production prices, for immediate delivery. If a custom size or shape of core is required, the cost of tool-ing for its manufacture is relatively low. For a suitable ferrite core, compared to the alternatives, the lead-time is shorter.

Easy Design. It is easy to learn to design with ferrite cores, and once the process has become familiar, there is no faster way to design magnetic components than by the use of standard ferrite cores.

In other words, for the very broad range of applications for which ferrites are suitable, they offer the designer many advantages over any alternative design approach. Further-more, as the subsequent sections of this Design Manual demonstrate, the reliability, uniformity, and repeatability of their characteristics, and the precision and stability of their dimensions, make it possible for the first time to offer practical, sensible abridgements of the classic design pro-cedures, that can save you untold amou~ts in man-hours and materials, at the same time affording a marked improve-ment in accuracy of results.

Short-form procedures for the design of low-level, linear inductors and transformers, as well as power inductors and transformers, are presented separately in the subsequent four sections of this manual. These sectiQns describe and re-late the fundamental elements that enter into the design of practical, efficient magnetic components using ferrite cores; to facilitate recognition of these fundamentals, we have inter-referenced the sections, where related procedures are employed. On the other hand, we have i~tentionally repeat-ed information in certain cases, to preserve continuity of the exposition. The design procedures are not empirical; the derivation and development of the equations used are given. so that it is unnecessary to accept the recommended techniques on faith.

At the end of each Section a summary of the design pro-cedure is presented, with appropriate references to specific paragraphs within the Section. With practice, the designer will eventually be able to complete many of his designs by reference to these summaries alone, referring to the theoreti-cal treatment only in critical or unorthodox situations.

This Design Manual is intended to be used in conjunction with the Ferroxcube Linear Ferrite Catalog (ava'Hable free of charge from your local F erroxcube Field Engineer, Repre-sentative or Stocking Distributor). The Catalog provides complete technical data on all of the standard ferrite core types and sizes manufactured by the Company, as well as on the characteristics of the eight standard ferrite materials in which they'are produced. Except for the wire table, ferrite chart, and k2 -factor chart given in the Appendix of this manual, all reference data required in design - mechanical and magnetic dimensions, Inductance Indices (AL), Q-Curves, winding areas, etc., - are provided in the Catalog,

SECTION 1-LINEAR, LOW-LEVE'L TRANSFORMER DESIGN THEORY

1.0 Linear, Low-Level Transformers Defined. In a linear transformer. the excitation of the core is low enough so that the slope of the B vs H curve (See Figure 1-1) may be considered to be essentially constant. In this region, at or below flux density Bo ,

B (1-1) H = Po and a linear relationship

exists between the excitation current and the resultant flux. In other words, the permeability of the core is essentially constant at the value Po.

In a low-level transformer. neither the core losses nor the losses in the windings cause sufficient temperature rise to affect significantly the behavior of the core, or limit the performance of the transformer in any other way.

Linear, low-level transformers fall into two broad categories: 1. Narrow-band or low-frequency types. 2. Broad-band or pulse types

(A special case of type 1 is the tuned transformer, one or both of the windings of which is resonated at either high or low frequency.)

1.1 . Basic Transformer Theory. Figure 1-2 shows a simple two-winding transformer schemati-cally. From the magnetic circuit parameters given, as as-suming that the linear operating region is employed, the inductance, Lm" of a primary winding having Np turns may be calculated as follows:

2 Ae (cm ) [ 2J Lm = 0.4 7rN P J.lo e (cm) X 10-8 Henries (1-2) For many standard core configurations and core materials, the above calculation has been further simplified by listing a "compound parameter," AL, the so-called ~'Inductance Index," which reduces equation (1-2) to:

Lm = N~ AL x 10.9 Henries (1-3)

(The dimensions of AL are "millihenries per thousand turns", incidentally.)

If it is desired to calculate the flux density, B, so as to verify operation in the linear region, one may employ the relation-ship:

Ex 108 B = GauD 4.441Np A e

where E is in volts RMS (or volts peak x 0.707) lis in Hz

and Ae is in cm 2

( 1-4)

As long as B ~ Bo , linear operation is assured. If B ~ BL, then linear, low-level operation is assured.

The current that must flow through Lm in order to establish B, at an excitation voltage E and a frequency f is:

I -' E Amperes nz - 2 1T I Lnz (l5)

3

Ep

. ----(, CORE (SHAPE INDETERMINATE) HAVING "EFFECTIVE" PARAMETERS AS FOLLOWS:

C .... &--..cti .... 1 A, ... Ae Mo ... tic"""", L_,tIo ,.

Figure 11

Figure 1-2

LEAKAGE REACTANCE (ASSUMED MEGLIGlILE 1M THIS APPROXIMA TtOM).

I. ~ (.,2R1)\ (( ___ ----4Iloo SI,,) )-J'V v'-C_-.-....-.(VV'"6~r:':': ..

Figure 1-3

Figure 1-3 shows one of the simplest practical approximate equivalent circuits one may draw for Figure 1-2. The term "approximate"must be applied to this circuit because of the following assumptions, upon which its derivation is based:

Capacitance effects of all kinds are negligible (or may be accounted for as part of the external circuitry) for the frequency Tange of interest.

Leakage reactance effects for the frequency range of in-terest are negligible; that is, the coupling between primary and secondary is perfect - all flux established by ex-citation of either winding links both windings, equally and fully.

Winding resistances are so proportioned that the turns ratio, n, may be considered to be equal to the primary! secondary terminal-voltage ratio; that is:

Np Ep' n = - = - (1-6)

Ns - Es

A simple, "lumped" lqss path (Rc) may be considered to represent all core losses.

All significant winding losses may be represented as series elements (Rp and n2 Rs).

SECTION 1 . ..:---LIN-EAR,. L.OW-LEVEL .TRANSFORMER DESIGN THEO.RY

I. '. .,' ... '. : .'. _ . ..J '. .." '. ~. .... _'. ': 4 .... ~ :. ..

For the equivalent circuit of Figure 1-3, one may write a set of familiar equations relating the parameters of the equivalent electrical circuit. From these equations, and a consideration of the magnetic circuit parameters, an admirably exact analysis may be made, predicting the performance charac-teristics of a given transformer. Analysis, however, is not our objective. We seek a design approach - perferably a simple, explicit one.

1.2 Simplifying Observations on Practical Transformers. In almost all practical designs, the equivalent circuit of Figure 1-3 may be further simplified in several ways, for the purposes of creating a "first-round" design, ~fter which, the significant effects .- if any - of these approximations may be removed by a set of second-round, or "refining" design procedures_ In studying the simplifications outlined below, the reader should always bear in mind two precautions: they can be unwaiTan(ed, in certain special instances; and some of them apply only to the simplification of the linear, low-level narrow-band, low-frequency, equivalent circuit of Figure 1-3.

1.2.1 Cal culation of Flux Density and Magnetizing Current. In practical transformers, the voltage drop across Rp may be neglected in calculating B and 1m, at least for the first-round design. Th~n Em = Ep = E in equations (I -4) and (1-5).

1.2.2 Calculation of Primary Current. In practical low-level transformers, the core-loss component (Ie) of the primary current may be ignored. Then the primary current consists of only two components, in quadrature:

In many instances, the magnetizing current is so small that it, too, may be ignored, particularly since it is in quadrature with the load component of current (ls/n)-

1.2.3 Calculation of Required Turns Ratio. It will be recalled - see equation (1-6) - that the turns ratio does not exactly relate the terminal voltages, due to the resistances of the primary and secondary windings. The following expressions. exactly relate primary and secondary voltages, and permit calculation of an "adjusted" turns ratio that achieves the desired (full load) ratio of primary to secondary voltage_

This assumes only that core-loss and 1m may be neglected in process of correcting the value of n.

( /8)

This expression becomes far more manageable when one as-sumes that Rp = n2 Rs; that is, that primary copper loss equals secondary copper loss.

4

(AI PRIMARY lUNING

(Bl SECONOARY TUNING

Figure' 1-4

wL." 7L!c; 0"--2-2- C,.). L..,f" R+~ ...,.

P R ...

o CALCULATION PRIMARY RESONANCE

Figure 1-5

Then equation (1-8) becomes:

n = :: lRL ~L 2R,] (19) In the "first round", however, it is only necessary to assume that equation (1-6) applies.

1.3 Additional Design Considerations. In order to design a narrow-band, low-frequency, linear, low-level transformer, the reader need only add two more simple calculations to his design technique: calculation of the required wire size, and checking of the winding "fit".

In low-level transformers, the selection of wire size is not usually warranted by current density (heating) but by the need to achieve reasonable regulation - values of Rp and Rs that are low enough to satisfy the requirements of the application. It must be emphasized that in many instances a design with quite high regulation will, indeed, perform acceptably, since neither efficiency nor heating considera-

tions govern.

For each core configuration, the Ferroxcube Linear Ferrite Catalog furnishes, either directly or indirectly. the exact dimensions andlor area of the available winding space. By apportioning this available space to the bobbin, the primary winding, and the (one or more) secondary windings, togeth-er with allowances for insulation, imperfect utilization of the winding space, and wire insulation, the designer is able to assign a particular "available winding space"(Ap) to the primary windings alone.

~:::)I::' '-' I 1'-' I "I 1-

LINEAR, . LOW-.LEVE~ ,TRANSFq~MER DESIGN 'THEORY. ".. . . '. ..... . '." .! .. ,.' "" '-:

Then knowing the number of turns he desires to wind on the primary (as dictated by the necessity to achieve a particular

~ value for Lm), he can calculate the number of turns per square inch. J

N Turns/Square Inch = 2

Ap (1-10)

Using the wire table in the Appendix, the largest wire that may be employed to fit that number of turns into the core can be determined.

The primary resistance (Rp) may then be calculated by esti-mating (from the geometry) the mean length of turn, and from that, the total length of the primary winding. The wire tables will give the resistance per unit length for that wire gauge. (The MLT for bobbins is given in the catalog.)

1.4 Resonating the Transformer. Figure 1-4 shows the equivalent circuits for the two configu-rations in which the linear, low-level transformer may be resonated at low frequencies. It is immediately apparent that the Q (quality factor) of the two circuits is quite differ-ent, although the resonant frequency may be made exactly the same, merely by using a secondary capacitor that is n2

times the primary capacitor. Figure 1-5 shows the method used for calculating the Q with primary resonance, and the derivation of the formula. Similarly, Figure 1-6 deals with the secondary resonance case.

In a tuned transformer, the bandwidth is related to the Q of a resonant circuit by the relationship:

II. fo BW = 2 ~f = Q (1-11)

where ~ f is the difference between the center frequency and the low-frequency cut-off point on the resonance curve.

Using the series-parallel equivalency expressions in Figure 1-5 and Figure 1-6, the Q of a double-tuned transformer may also be derived, as may be the expressions for the resonant frequency, attenuation, phase shift, and bandwidth of circuits having significant external capacitive loading.

(NOTE: The reader will, of course, have observed that leak-age reactance is still ignored in these low-frequency examples. When high-frequency or broadband transformers are dis-cussed,later in this section, this restriction will be remove~.)

1.5 Pulse and Broadband Transformer Requirements.* Figure 1-7 describes the significant characteristics of a pulse, and Figure 1-8 shows the equivalent bandwidth required of a linear transformer to reproduce a pulse with the fidelity implicit in the requirements of the application. As a good first order approximation, it is assumed that the transform-er behaves in much the same manner as an uncompensated, Re-coupIed, single-stage amplifier. The validity of this as-sumption will be examined later. The ability to reproduce a pulse with minimum "droop" of the top (or bottom) of the

u)ZLmZ

Rc

Lm

WLm Q=------------------~-----

WZLm2Z n2 Rp+ Rc +n Rs+ w2c$2RL

W= ~o~_ VLmCs

APPROXIMATE EQUIVALENT CIRCUIT FOR "a" CALCULATION (VALID FOR Q~IO)

Pd= 100B . A =2001Ttpft

t - Pd ,- 2001ftp

FigUre 1-6

Figure 1-7

O------r-----~-=--------------------------------------------~

-3db,--~------------------~

fl . f2

Figure 1-8

waveform, is directly related to the value of the "low-frequency cutoff' - the frequency at which th~ sinusoidal amplitude vs frequency characteristic of Figure 1-8 has fallen from the mid-frequency level by 3 dB (0.707 in relative voltage response). This low-frequency cutoff, fl' is related to the percentage droop (P d) by the expression:

Pd (1-12) h = 200 1f t

P

where tp is the duration of the longest flat portion of the waveform to be reproduced (either the bottom or the top of the pulse) with no more droop than Pd.

'" The short-form design procedure is provided here. Designers using conventional long-form procedures will refer to the pulse-magnetization data charts given in Section J of the Ferrite Catalog. 5

,SECTION ,1-' . ..., .... ..' LINEAR, 'LOW-LEVEL TRANSFORMER DESIGN THEO~Y

In a similar manner, the required rise-time characteristic is related to f2 , the "high-frequency cutoff' point on the sinus-oidal characttfristic of Figure 1-8. The expression relating the longest allowable time for the pulse to rise from 10% of its final amplitude to 90% of its final amplitUde (tc) is given by

Rg

Ein

h = 0.35 'r

Rp Lt

Figure 1-9

(1-13)

(nEs)

Figure 1-9 shows the high-frequency equivalent circuit of a transformer. including internal resistance of the signal generator, Rg. Both Lt. the leakage inductance, and Ct, the equivalent shunt self-resonating capacitance, are functions of the interwinding geometry and materials selected. We shall, in the next paragraph, consider how to estimate these parameters. but let us first turn to an examina'tion of their

effect on the high-frequency transmission properties of the broadband or pulse transformer. It can be shown that the fastest possible rise-time that may be obtained in the trans-former having the configuration shown in Figure 1-9, with-out overshoot or post-pulse ringing, is given by:

tr (critically damped) = 3.35 JCi.LtCI (1-14)

Note that this rise-time is designated as the fastest obtainable under "critically damped" conditions, which is another way of stating that there is no overshoot or ringing. (In this condition, the transformer behaves very much like the RC-coupled amplifier mentioned earlier.) Faster rise-times may be obtained by allowing some overshoot, and this condition may be achieved by increasing RL, or Rs; or decreasing Rg or Rp.

The factor "Ci." in equation (l-14) is known as the "attenu-ation constant" for the resonant circuit. and expresses the

mid-frequency reduction in amplitude from the generator (Ein) to the voltage established across Ct. This constant is given by the expression:

n2 (Rs + RL) a = ---------

Rg + Rp + n2 (Rs + RL) (l-15)

6

The designer will find Ci. convenient for use in calculating the actual total attenuation from the generator to the output terminals (EsfEin) which we shall call Ci.t. and which is given by:

at - (RL \ Ci. RL +Rs') n

(1-16)

The criterion for criticai damping is:

(1-17)

The next step in determining the rise-time available from a given design is to estimate Lt and Ct.

1.6 Calculating Leakage Reactance and Self-Resonant Capacitance_

(NOTE: This derivation is appro>;imate, and, while it is use-ful for most conventional structures, may be in serious error for special geometries. In general, the results it gives are con-servative, in that they predict a longer rise-time for the trans-former than it will actually exhibit.)

Space does not permit derivation of. the equations to be

introduced herein, but reference to Figure 1'-10 will indicate the reasoning employed. As shown irr Figure 1-10, the primary and secondary windings are both assumed to be well separated from the core (by the bobbin or other insu-lating structure) and from each other by a layer of insu-lation having a dielectric constant, K, and a thickness, d inc~es. Windings have, respectively, Np and Ns turns, with a

ratio of Np/Ns = n: The mean diameter of the insulating layer is D inches. The width of both windings is winches.

The reader will recognize that the configuration shown in Figure 1-10 (although restricted to a very regular geometry) is really quite general, and essentially independent of the specific core geometry employed, provided only that both windings are substantially spaced from the core, at least by a distance several times d.

Lt can be shown to be approximated by:

2rrN2 d D LI == : x UY8 Henries (1-18)

and Ct (which is not the interwinding capacitance, but 'rather the equivalent shunt capacitance represented by the inter-winding capacitance) is approximated by:

Ct == K2 ~n~ x 10- 12 Farads (1-19)

One interesting fact comes to light at once when we mUltiply equations (l-I8) and (l-19) together, and take the square

root:

JL;C; = N p D .JK1T x 10- 10 seconds n

( 1-20)

This expression, which is called the "natural period" of Lt and Ct, does not contain either the factor d or the factor w. This important result indicates that, so long as d is substan-

. :::,t=,\...J I IUI"l 1-

LINEAR, LOW-LEVEL TRANSFORMER .. PESIGN .THEORY

tially smaller than the spacing of either winding to any part of the core, the exact interwinding spacing is not significant; nor is the width of the winding! In fact, the natural high-frequency period of the transformer depends only on the number of turns, the turns ratio, the mean diameter, and the dielectric constant of the interwinding insulation. Sub-stituting equation (1-20) into equation (1-14) yields:

6 NpD r;;-;:, tr = -- V K a x 10- 10 seconds

n (1-21 )

which relates the fastest obtainable rise-time to the para-meters just enumerated, with the single addition of the da~ping constant, a. It is obvious, therefore, that in order to achieve shorter rise-time in a pulse transformer (higher upper cutoff frequency in a broadband transformer), one should attempt to:

Reduce the number of primary turns - this is directly opposite, as we shall soon see, to the improvement of the low-frequency behavior of the transformer_

Reduce the mean diameter - this, too, will render more difficult the "fit" of sufficient primary turns for the establishment of good low-frequency characteristics.

Design the transformer for the largest possible value of n - this means, in the limit, a large "step-down", result-ing in loss of "amplification n. In other words, the greater' the "gain" of the transformer, the smaller will be its maximum bandwidth. This is, once again, very consistent with the behavior of an RC-coupled amplifier, in which the gain-bandwidth product is constant.

Use the lowest possible dielectric constant for the inter-winding insulation.

Reduce the attenuation constant - this is possible only over a limited range, since the condi'tions imposed by equation (1-17) for critical damping will essentially fix the attenuation constant~ for most problems.

I NOTE: If some overshoot may be tolerated, the transform-er may be designed so as to cause the right-hand side of equation (1-17) to be smaller than the left. As an example, if 20% overshoot may be tolerated, the right-hand side of equation (1-17) may be multiplied by 0.5, yielding a rise-time of approximately 60% of the critically-damped value, with only a small amount of ringing after the initial over-shoot.]

The expressions contained within this subsection are em-pirical, and are most accurate when applied to a single-layer winding, in which w is at least 10 times d and in which D is of the same order of magnitude as w. As the number of lay-ers increases, Ct increases slightly, as does Lt, and the minimum rise-time becomes progressively longer.

External capacitance, if appreciable, will increase the mini-mum achievable rise-time obtainable with any given trans-former. As a good first approximation, provided that Rs is very much lower than RL, it is valid to simply add external,

7

... ...... 1---- W ----lIP Np

NS=-n--

Np Turns

\ II

CORE

dielectric constant = K

Figure 1-10

( nEs)

Figure 1-11a

R-

Lm E=Es

Figure loll b

secondary-connected capacitance to the value of Ct in equation (1-14). Primary-connected capacitance, on the other hand, has less effect, in general, since it is "driven" from Rg and its effect is more accurately represented by first calcula ting the rise-time of the RC circuit formed by this external primary-connected capacitance with Rg, and then combining that rise-time with the rise-time of the trans-former, calculated as described above, by taking their geo-merric mean.

1.7 Low-Frequency Performance of Pulse and Broadband Transformers.

Figure 1-11 a is a low-frequency equivalent circuit of a pulse transformer, including the internal resistance, Rg, of the sig-nal generator. This diagram may be reduced to the simple

. ,

iECTION, ,1-' .. , , " . ' . " JNEAR, LOW-LEVEL'TRANSFORMER DESIG~ T~EORY,

form of I-lIb, as far as the low-frequency determining ?perties of the transformer are concerned. In this simpli-.:d form, the low-frequency performance is readily calcu-

lated. If a flat-topped pulse having a duration tp must be reproduced with a maximum allowable "droop" of Pd %, then the values of Rand Lm are given by:

where R

100 R tp Pd == ----'-

Lm

for small values of Pd,

(1-22)

(1-23)

It is obvious that, in order to extend the low-frequency cut-off characteristic of the transformer, so that it can repro-duce long-flat-topped pulses with rela'tively low distortion (tilt), it is necessary to make Lm as large as possible, for a given value of R. R must be minimized to extend the low-frequency response.

Maximizing Lm is accomplished, as we have seen in the foregoing paragraph, only at the expense of the high-fre-quency performance of the transformer. Thus, a particular transformer will have an inherent bandwidth limitation, for \ particular set of input and output impedance levels. On

_fIe other hand, reducing the input and output impedances aids both the low and the high-frequency response charac-teristics of the :transformer, and the designer should strive to select the external circuitry employed so as to reduce both Rg and RL.

It is interesting to note that the value of Lm is directly pro-portional to the effective permeability of the core, while the value of Lt, the leakage reactance, is not affected by this

permeability, provided that it is high compared to that of air_ Thus, for maximum bandwidth, the designer should select the highest permeability core material.

Similarly, the core loss, which determines R, should be minimized to improve the low-frequency performance of the transformer; but Rc does not, in general, playa signifi-cant role in determining the high-frequency cutoff charac-teristic, unless of course, the core loss becomes excessive in the high-frequency region of interest.

Therefore, the designer should strive to select a core rna terial having high permeability at the low-frequency end of the bandwidth of the frequency range of interest, and low losses in the low end, maintained reasonably low through-out the entire frequency range. This requirement is almost a description of those Ferroxcube Ferrite core materials recommended for broadband applications. (See Appendix table, and Section I of the Linear Ferrite Catalog).

1.8 High-Frequency Narrow-Band Transformers.

It is obvious from the material developed in Section 1.6 that the same factors that limit the rise-time of a trans-former also limit its upper cutoff frequency. Although various circuit "gimmicks" may be employed to "equalize" or otherwise compensate the pulse perf?rmance of a trans-former, nothing can be done to avoid the inevitable attenu-ation above the high-cutoff frequency given by: .

1 12= (1 -24)

Below f2 , however, the transformer may be tuned effective-ly without significant attenuation beyond that given by equations (1-15) and (1-16). A good rule is to limi t such applications to a frequency of approximately one-third of f2 or lower, so that the effects of Lt will be minimized.

DESIGN SUMMARY FOR LINEAR, LOWLEVEL TRANSFORMERS

I. Lm = N; A LxI ()"9 (Primary Inductance - Par. 1.1)

Ex 108 2. B = (Flux Density - Par. 1.1, Fig. 1-1) 4.441Np A e

3. I = E (Magnetizing Current - Par. 1.1) m 21f1 Lm

4. Ip = j I~ 2 + I,~ (Primary Current _ Par. 1.2.2) 5. 11

(Pulse Requirements - Par. 1.5, Fig. 1-7, 1-8)

6. a =

7.

Rg+ Rp + n2 (Rs + RL)

(Attenuation - Par. 1.5)

2 21f N dD

---"P--=-g- (Leakage inductance - Par. 1.6) wx 10

KDw x J(r 12 Ct = (Self-resona ting capacitance - Par. 1,6)

2dn2

8

100Rlp 8. Pd =

(Low-frequency performance of Pulse, Broadband Transformers - Par. 1.7)

9. h = __ 1 ___ (High cur-offfreqllellcy - Par. 1.S) 2rr V aLter

'. . SECT.ION 11-POWE~ TRANSFORMER D,ESIGN T'HEORY

2.0 Power Transformers Defined

In a power transformer, the excitation is high enough to "cause the effective permeability of the core to vary signi-ficantly from the small-signal value We) used in linear, low-level transformer design calculations. At the same time, core losses become appreciable, the self-heating effect* may be-come a factor, and the quality factor (Q) is significantly lower. Finally, the design limits are usually dictated by ex-cessive heating, rather than by lowered Q or Stability.

2.1 Equivalent Circuits for Low and High Frequencies.

Referring back to Section I, figures 1-2, 1-3, and 1-9 will be found to apply as well to power transformer design as to low-level transformer design. For most low-frequency trans-formers (up to perhaps 50kHz, or even 100kHz) the simple circuit of figure 1-3 applies. Above that range, the effects of leakage reactance and distributed capacitance become more and more significant, while the shunting effects of the mag-netized inductance become less and less important, and figure 1-9 is a more accurate representation of the circuit. In speCial designs involving many turns of relatively fine wire, resonances may develop that invalidate the procedures outlined here, but such instances are rare, and are usually predictable.

2.2 Inductance Calculation - AC Excitation Only.

Let us begin by repeating two fundamental equations stated in Section 1, paragraph 1.1, for inductance and flux density.

Lm = 0.4 1f N~ Il[ ~] x 10-8 Henries (12)

B = E x 108

Gauss 4.44fNp A e

(14)

From equation (1-2), we see that, for a given core geometry, the relationship between inductance and the number of turns is known if one can determine the permeability. Since the permeability depends upon B, the flux density, one must turn to equation (14) to calculate B. Since B also depends upon the number of turns, it is necessary combine equations (1-2) and (1-4) which produces a new equation:

(21)

or B = kl ..fIT (2-2)

where kl - 2500 E - f..;r;;;v;- (23) and Ve is defined by:

(2-4)

"'Self-heating is defined as tlie temperature rise in the will ding and core due to excitation alone as distinguished from rhe temperature rise caused by heat exchange with the ambient ellvironment.

9

Note that aU the factors in kl are known:

Ae is the effective core area in cm2 (listed in core group charts in the Linear Ferrite Catalog);

Qe is the effective magnetic length of the core in cm (also listed in core-group charts);

Ve is the effective core volume in cm3 , given by equation (2-4) as the product of Ae and Qe (also listed in core-group charts);

L m is the minimum magnetizing inductance in Henries, obtained as indicated in Par. 2.3 below.

E is the excitation voltage (in volts RMS), part of the problem statement;

and f is the frequency of excitation (in Hz), also stated initially.

It is convenient to restate equation (2-2) in the form:

(22a)

A very useful parameter in power transformer design is~t~b .... e _____ _ ratio of its Voltampere rating to the (lowest) frequency at which it will be used.

The Voltampere rating is given by:

EI = E: Voltamperes (25) 2rcf Lm

Therefore the ratio of Voltamperes to frequency is,

El EI - Voltamperes/Hz

- 21ff2 Lm

For convenience, we shall define a new factor, k2 , as:

k '- 21fEI -~ 2 --f- - Lmfl

(2-6)

(27)

It can be shown* that the maximum value of k2 for any particular core shape, size, and material is given by:

(27a)

~ = 15.6 x 10-8 r Ve (Bmaxpl Voltamperes/Hz 211 21f [Pe J

Ve and Ile have 'already been defined, and Bmax is the maximum recommended flux density for the core material, as listed in the Ferrite table in the Appendix.

2.3 Transformer Design Procedure - Phase I. In an efficient transfonner, the excitation current (drawn by Lm) should be small compared to the load current. Since they are in quadrature, the excitation current need not be made very much smaller to have little effect; for example, a ratio of 3: 1 (load current to magnetizing current) will result in a total current only 5.7% higher than the load current! We shall

*Equation (2-7a) is derived as follows: (1) Solve (J4) for E: (2) Substitute that expression inlO (2-7): (3) into this new form of (2-7), substitUll! equation (1.2) for Lm; (4) clear and simplify, using (24). The result should agree with (27a).

SECTION II~" POWER TRANSFORMER DESIGN THEO'RY

,define the ratio of the load component of primary current ~o the magnetizing current as:

Is M=--

nIm (28)

The minimum magnetizing inductance Lm may be calculated by using a form of equation (I -5):

p Lm = 2rrfIm

( ISa)

For best utilization of a given core, when size, weight, and cost must be minimized, M should be between 10 and 20. For best regulation (lowest IpRp drop) M should be lower ... perhaps between 2 and 4. At higher frequencies, M need not be higher than 5 for good utilization of the core.

2.3.1 Select a likely core of the desired shape, being guided by the values of k2 * listed in the Appendix.

2 k ---

2 - LmP (27)

2.3.2 Solve equation (2-3) for kl , using Ve from the Linear Ferrite Catalog for the core selected.

k I = 2S00 E f-V Lm Ve

(23)

\l.3.3 For the core selected, a particular core material is J specified and for this material there is a set of curves of

Jl vs B in Section I of the Catalog. Using the value of kl cal-culated in the preceding step, find a point on the appropriate Jl vs B curvet for which:

B2 J1= -

k 2 1

2.3.4 Substitute the value of B so obtained into:

B = Ex J08

4.44fNpAe

and solve for N p, the primary turns.

(22a)

(1.4a)

2.3.5 Use the wire table in the Appendix to estimate the wire size from the primary current. Check that Np turns will fit into approximately 40% of the available winding area, as given on the core-group charts for the selected core and bobbin, in the Linear Ferrite Catalog. (A good rule of

thumb is to allow a current density of 1000 amperes/sq.in.)

If not, select a core having at least as high a k2 rating, but

with a larger window, and repeat the above procedure.

For minimumsize designs below 10k/h, at ambient temperatures below 7SoC, a core hailing about SO% of the calculated k2 may be practical, since the core losses will be very low.

t Use the high-temperature curve for this firstround design.

ttFor example, if 10% regulation is specified, (1 - Ct.) = 0.1, or Ct. = 0,9, (This nomenclature is consistent with the use of Ct. as an attenuation constant, ill Section 1.)

10

2.4 Low-Frequency Design Procedure - Phase II.

Ha ving successfully selected a core t,hat will accommoda te Np into 40% of the available winding area in Phase I, it is now possible to "trim" the design, to check and accommodate the regulation and/or efficiency specifications. The recommend-ed procedure is as follows:

2.4.1 From the wire table, determine Rp, the resistance of Np, at 20C. Assume that half of the regulation of the trans-former is in the primary, and check that:

Rp = % n2 RL (1 - CX) (2-9)

where (I -a) is the regulation of the transformer, expressed as a numeric t t, and n2 Rt is the equivalent load resistance, referred to the primary side. To determine n, use equation (I -9). If Rp is too high, Phase I of the design must be re-worked, using a core with a larger window.

Correct Np to allow for the voltage drop in Rp:

Np = Np [I - (I; "IJ 2.4.2 Set Ns, the adjusted secondary turns, equal to:

I _ N P [1 (1 -Ct.) ] Ns - n + --2-

(2~1 0)

(2-Il)

and, from the wire table in the Appendix and the secondary (load) current,.determine the secondary wire size, and check that Ns will fit into about 35-40% of the available winding area.

2.4.3 Compute the t9tal copper losses, and add them to the core losses, as obtained from the core-loss curves for the materials of the selected core, using the known values of B and f. The sum is then the total power (Pt) dissipated in the transformer. Having calculated this power, one may deter-mine the temperature rise from Po, the temperature rise fac-tor given for the core in the Appendix:

50Pt Trise =--p;;- (2J 2)

2.4.4 Having calculated T rise, and knowing the highest ambient temperature in which the transformer must func-tion, one may determine the actual maximum operating temperature, T max, which is their sum. This temperature may be used as a "final trim," to adjust the turns ratio for the inevitable rise in Rp and Rs, by adjusting Ns to a new

value, Ns', such that R'= R I J + r (Tmax - 20) I (21 J)

where r is the temperature coefficient of resistivity of copper, and is equal to 0.0040, approximately.

The fit of Ns and Np, plus adequate insulation, should be checked once morc.

2.4.5 As a further check on the design, the value of J1 at T max should be compared to the value obtained fron. the 100C curve in step 2.3.3, of Phase I. Unless the new value

SECTION 11-POWER' TRANSFORMER' DESIGN "THEORY

differs radically from the old - say by 50% or more - the consequent effect on Lm, Np, and B should not be signifi-cant. If it is, a small increase in the number of turns in both windings should correct for it.

2.4.6 As a final check on the design, compute (or estimate) the leakage inductance, or determine its effect on regulation for the highest operating frequency.

One method of calculating the leakage inductance, Lt, is given in Section 1.6 - note particularly equation (1-18) and figure 1-10. The effect of leakage reactance on regulation may be estimated by comparing Xt (= 2 1TfLt) with the equivalent (primary-related) resistance of both windings, (Rp + n1 Rs). If Xt is significant (remember that it is to be added in quadrature), then repeat equation (2-9) in step 2.4.1, using:

J (2 Rp)1 + (XcJ 2 ~n2 RL (1 - a) (2-14) to determine the allowable maximum Rp. (If Xt is by itself too large to permit achieving the desired regulation, a new design will be necessary. . . one in which the leakage induc-tance is markedly reduced by altering the winding geometry).

2.5 Special Power Transformers. The reader now has at his command a set of procedures that will be found to yield, if not optimum designs, certainly designs that are practical and close to the optimum, needing only the slight modifications that may be suggested by test data on the laboratory per-formance of a prototype.

2.6 Inverter/Converter Transformer Requirements. The procedures described in the preceding paragraphs of this section have all assumed conventional sinusoidal, linear operation, despite the relatively high power levels involved. In the succeeding paragraphs, which are concerned with power transforme.rs designed to pass square waves, a design approach is described that lends itself to the development of many s~e~ial transformers.

Be~ause of their high permeability and inherent low losses at relatively high power and audio frequencies, ferrite cores are particularly well suited to use in square-wave applica-tions, in which the core is rapidly switched from full mag-netization in one direction to full magnetization in the opposite direction. Characteristic of such applications is the large class of inverters and converters in which the power transformer is not operated in a saturated mode, but is driven up to very near saturation by a transistor switching circuit having an independent drive-signal source.

2.7 Selecting a Core for an Inverter Transformer. We will find it useful to apply the nomenclature associated with figure 2-1, which establishes a value for maximum "linear" flux density, Bmax, for a given core material, and an equiva-lent average permeability, J.1av. The relationship between

11

EUIIII'Ie uf POWC\" Tl'lInd'ormc( Dcsicn

A ~nl:lll (lOwer supply is In be lIIountcU direclly 011 a pluj;in prinlC

;eCTIO.N 11-' .. lOWER TRANS.FORryU~:R .DESIGN 'THEORY

Bmax and the peak value of the primary voltage, Epmax, ~l1ay be shown to be:

Ep max = 4fNpAeBmax x /0-8 Volts (2-15)

in which Np is the number of primary turns across which Epmax is impressed, Ae is the effective core area in cm2 , and f is the frequency of the applied square-wave, in Hz.

The peak input current to the transformer, Ipmax. may be estimated from the load power. in the first-round design phase, by assuming a conservative 80% efficiency, so that

I Pload (Watts) Amperes pmax = Epmax xO.BO

(2-16)

If the primary-winding current density is established, as be-fore, at 1000 amperes/square inch, the cross-sectional area of the wire, Ax, is given approximately by:

Ax = Ipmax. ____ PI_oa_d __

square inches 1000 BOO E pmax

(2-17)

Pload or Ax = cm2

124 Epmax

and the number of turns may then be related to the avail-able winding area of the core, Ac (allowing for a winding space factor of 70%, and assigning 45% of the winding area to the primar:y). For a split primary we can state:

2N _ Ac(0.70) (0.45) _0.315Acx124Epmax

p - - Tums (2-1 B) Ax Pload

39AcEpmax 2Np = Tums

Pload

Substituting this expression for Np into equation (2-15), we may relate the minimum product of Ae and Ac to the known parameters:

1.3 Pload x 106

AcAe = [ Bmax

(2-19)

To select a core of a particular style (as dictated by the usual considerations of geometry and production methods) one need only look up the Bmax for the material of which it is made, solve equation (2-19), and then select a core size that provides at least that minimum product of Ae and Ac.

2.8 Completing the Design of an Inverter Transformer. One important requirement of the inverter transformer is that it have adequate magnetizing inductance, Lm, to prevent excessive droop [See Section I, paragraphs 1.5 and 1.7, particularly equations (1-12), (l-22), and (1-23).1 Having translated the droop requirements of the specification into an inductance requirement, Lm, check the design so far established by using the expression:

2 (J..lav) 0.4,1T Np -. Ae L = 2 x /0.8 Hellries

Qe ( 2-20)

which is a form of equation (1-2), obtained by assuming

Pal' J..l = -2- at B = Bmax

12

Assuming that the value of L found in equation (2-18) is at least as large as the minimum allowable Lm set by droop considerations, the trimming of the design may begin.

The trimming process is almost identical to that used for conventional power transformers:

Check Rp and Rs to verify that regulation will be satisfactory (See 2.4.1 and 2.4.2.)

Determine temperature-rise by calculating copper and core losses. (See 2.4.3.)

Trim regulation adjustment tor actual operating tem-perature (See 2.4.4.)

Check 11 variation with temperature, assuming that Pav varies in proportion to 11. (See 2.4.5.)

Finally, check that Lt and Ct will not cause sluggishswitch-ing or excessive "ringing" during switching. This can be done by assuming that the rise-time requirements of the square-wave, if not given in the specification, are such

that [2 = 20 f. [See paragraph 1.5, especially equations (I-13), (I-14), (I-IS), (1-16), and (I-17). Also see para-graph 1-6, especially equations (I -18), (1-19), and (1-20).]

DESIGN SUMMARY FOR POWER TRANSFORMERS

Is Ep 1. M = --: Lm = --- (Magnetizing .

n 1m 21TJlm Inductance - Par. 2.3)

E2 2. k2 = --2 (Core Selection Factor - Par. 2.:3.1)

Lmf

2500E ---- (Permeability/Flux [ .J Lm Ve Density [actor - Par. 2.3.2)

2 4. J..l =L (Practical Permeability value - Par. 2.3.3)

k 12

5. B Ex /Oa

--'------ (Preliminary design Check - Par. 2.3.4) 4.44 fNpAe

6. I = _E_

SECTION 111-" LINEAR, LOW-LEVEL INDUCTOR DESIGN THEORY

3.0 Low-Energy Inductors Defined.

1n this class of components are included all devices in which a specific inductance" value must be achieved, often at a specific maximum value, usually at a specific minimum Q. usually to a specific tolerance, over specific frequel1(.:v and temperature ranges and over a specific excitation (voltage) range They are called low-energy inductors, because the maximum excitation specified never drives the core into a non-linear region~ hence the effective permeability (Me) is essentially constant, and we may assume that the inductance is essentially independent of excitation . .. in other words, that the inductor is a linear magnetic device.

3.1 A Note On Excitation Limits

The excitation applied to a linear inductor will not cause the core on which it is wound to operate beyond the linear B vs H region of its magnetization curve. For AC excitation, Bmax may be calcuiated from the expression:

Erms x lOs Bmax (A C) = 4.44 fN Ae gauss (3-1)

where Ae is the "equivalent" area of the magnetic path in cm 2 Erms is the effecth1e value of the applied alternating vollage. N is the number of turns, alld fis the frequency of the excitation. in Hertz.

For excitation containing superimposed DC, as well as AC, we must modify the expression for Bmax as follows:

Bmax (total) == Bmax (AC) + Bmax (DC)

trms x JOB N IDe AL +----

4.44fNA e JOAe (3-2) Bmax (lolal)

A safe limit for Bmax, which permits application of the de-sign procedures described below with full confidence, is 250 gauss.

Higher flux densities will cause non-linear behavior, which will introduce errors in the computation of inductance and Q. Consult the Ferroxcube Engineering Department for specific details.

An approximate value for Ae may be obtained, for the twelve standard pot-core sizes, from this chart:

Ae Ae Core Group

cm2 in2 Core Group

cm 2 in:!

743 .0433 .0067 2616 .948 .147

905 .101 .0156 3019 1.38 .214

1107 .167 .026 3622 2.02 .313

1408 .251 .039 4229 2.66 .413

1811 .433 .067 45 2.91 .451

2213 .635 .0985 6656 7.15 1.11

13

3.2 Frequency Ranges

Standard Ferroxcube pot cores for low-energy inductors are manufactured from four types of ferrite material. Each type exhibits desirable characteristics (high effective permeability, low losses, high time stability, controlled temperature co-efficient) over the specific frequency ranges for which it is recommended.

3.3 The following table summarizes the properties of these four types of material. Complete technical information on all types is provided in Section 1 of the Ferroxcube Linear Ferrite Catalog.

Ferroxcube Material

367

369

303

4C4

Recommended Frequency

Range

power frequencies to 300KHz

power frequencies to 300KHz

200KHz to 2.5MHz

lMHz to 20MHz

3.4 Pot-Core Construction

Summary Of Other Properties

Essentially zero T.C. Higher Q than 369, permitting larger AL values.

Linear, controlled T.C. over a wide temperature range (_30 to +70C) to compensate polystyrene capacitors.

Linear,controlled T.C. over a wide temperature range (_30 to +70C) to compensate polystyrene capacitors.

Negative, "linear, controlled T_C. to compensate silver-mica capacitors.

A pot core forms a nearly closed ferromagnetic Circuit hav-ing a very small inherent air gap between the two halves and, in the cores used herein, a. recessed center post to increase the air gap to some larger value. The permeability can be regulated in a very effective manner by varying the length of the air gap in the center post. .

3.5 Effective Permeability

Figure 3-1 is an approximate but useful representation of a pot-core inductor.

Figure 3-1

3ECTION 111.-..... _iNEAR, LOW-LEVEL INDUCTOR DESIGN THEORY

. . . . ... ' .. .. ~

'11 such an inductor, the inductance is given by:

L = OAn N2 x J(r 8 (3-3)

~+!L J.lo Am Ag

The denominator is usually written as L~A' the reluctance J.l..e

of the magnetic circuit. It is convenient to think of the pot core as providing a closed. homogeneous magnetic path, having an "effective" permeability, /-Ie such that

(3-4)

For small air gaps, Po == Pe, the effective permeability of the pot core assembly, and

L = Jle Lo (3-5)

For a given ferrite material and a given set of pot core dimensions (including air gap) then, one can relate induct-ance to turns by simply stating the effective permeability, /-Ie. for that pot core.

j.:tually, for convenience in caluc1ation, a further simplifi-cation in the statement of Pe and / A is made in Ferroxcube pot core literature. An "inductance index" (AL) is stated for each pot core, taking into account the dimensions, the ferrite material of which the core is manufactured, and the particular gap adjustment made at the factory. This index is expressed directly in "millihenries per lOGO-turn coil." Algebraically,

(3-6)

3.6 Inductance Calculation

Once the AL of the core is known, the specific number of turns (N) required to achleve a specific inductance, L, may be calculated from equation (3-7), which takes into account the fact that L varies directly as N2 (See equation (3.3).)

where L is in millihenries

(3-7)

Curves are also provided in the core-group charts in the 'atalog showing the typical decrease of AL as a function of

.:he relative winding height on a partially-filled bobbin, and showing the range of inductance adjustment achieved with standard adjustors available for use with each core size and material.

14

3.7 Importance Of the Air Gap

We have already seen how the size of the air gap determines the effective permeability, Pe, and therefore AL, the induct-ance index of a particular core. The air gap also exerts two other powerful influences on the behavior of any low-energy inductor wound on the core:

The larger the air gap (lower AL), the higher will be the stability of the inductance with time and temperature.

The larger the air gap (lower AL), the lower will be the maximum Q obtainable at any given inductance on that core.

Withln these two relationships lies a potential design conflict, for many applications require both high stability of induct-ance and high Q. When we add to this conflict the need to minimize the size of the core,. the full range of design limitations is imposed, and core selection becomes a critical step in the design procedure.

3.8 Checking The Wire-Winding "Fit"

The complete speCifications for all standard Ferroxcube pot cores are given in the fon., of charts and Qcurves in Section 2 of the Ferroxcube Linear Ferrite Catalog_ In these "core-group charts" as can be seen from the typical examples reproduced in Figures 3-2, 3-3, and 3-4, thOe Winding Area (equivalent to "window" cross-section) is given for each core for each type of bob~in available for ~se with that core.

After determining the turns required for a given inductance this number of turns may be divided by the Winding Area, as given by the core-group chart. The resulting quotient is the required number of turns per square inch, which deter-mines the largest wire that may be employed to fit that number of turns into the core - assuming 100% utilization of the winding area, which is only possible with exact layer winding. The Wire Table provided in the Appendix of this Bulletin allows for insulation thickness and will, therefore, determine the largest practical size of wire for exact layer winding_ A practical commercial-winding figure probably lies between 90% and 95% of the value given.

An example of the use of the winding table and core-group chart to determine "fit" is given at the end of the Design Example in paragraph 3.11 .1 .

3.9 The Q-Curves

For each standard core-group listed in Section 2 of the Ferroxcube Linear Ferrite Catalog a set of curves is given, relating Q to frequency for each standard air-gap (Ad value, assuming that the winding space is completely filled. One such curve is given here, for reference as Figure 3-5.

,;;;;,,:;;;;. '-' I 1'-' I "I. ~ 11-

LINEAR, LOW-LEVEL INDUCTORDESIGN THEqRY

800

700

600

0500

,00

GAPPED POT COR ES . .. NON- ADJUSTABLE-

ADJUSTABLE GAP

GAPPED POT CORE CORE

POT CORE ASSEMBLY MATERIAL

PART NO. PART NO.

l408P-A 100-387 l408C-A 100387 Vellow Adjustor

1408P-A160J87 l408C-A 160-387 387 While Adjustor (To 300 KHz)

1408pA250387 1408C-A250387 Brown Adjustor

l408P-A60-389 1408C-A60-389 Red Adjustor

~9.A. l408CA 100-389 Vellow AdiuJtoc .-.... 389

Figure 3-.:!

INDUCTANCE AND WINDING INFORMATION

TYPICAL FULL-80BBIN

WINDINGS

TURNS WIRE

6' 20 10' 19/46 16' 19/46 25' Xl/46

iN-C

N.1 OC 0 / ~~

'Ke

CORE INDUCTANCE FOR STANDARD 387,389. 303 and 4C4

AL 25 AL 40

0.92 J.lH 1.46pH 2.5 pH 3.7 pH

6.3pH 10J.lH 16pH 25 pH

2S.6J.1H 42 pH 64pH

Figllr(' 3-3

I"'", It-

N.~O p 1.1 1)

II Ii'IJI\

t-- . N.1 if rr'"iI ~

1\ ~.25~ TZ \

t!, ~ ~~j~ ~ k '\ r'\\~1

" \F::i ,0Ke FREOUENCY

Figure 3-5

POT CORES

AL ro AL 100 AL 160

38.3 pH 62.6 pH 101 pH

~~p~ 2JO~~~~

.\ 1\ 1\ \ \ \-

~ 1\ ~ \ \ 1\ \ \ 1\ \ 1\ b\\ \ \ f\ f\

100Ke

Note that the Qcurves are given for windings using both solid wire (for lower frequencies), and Litz wire (for higher frequencies) with the number of turns selected in geometrical

Apf>ROX, TEMPERATURE

GAP COEFFICIENT PPMrC

ALt P. LENGTH VALUE {REF.! lin.!

MIN-MAX TEMP. RANGE

100 63 .012 -3810 +38 1%

160 100 .0065 -EO to +60 +20 1.5% to

250 156 .0039 -94to~4 +70C

2%

60 37 .024 +51 to +81 1%

100 63 .012 +88 to +139 1%

~ 100 .0065 +140 to +220 -30"

\~ .:9Q..39 _~218tO+~~ --- ---- --

STANDARD BOBBIN DIMENSIONS All dimensions in inches eJ(

SECTION 111-LINEAR~ LOW-L.EVEL""iNDUCTO"R DESIGN THEORY"

There is also a small but observable variation in Q with tem-iperature. Each of these loss factors: Copper resistance losses

(includitzg "skl:n effect "), eddy-c.1J.rrent losses in the windings, dielectric losses ill the windillgs, alld three core-loss com-ponents: hysteresis, eddy-current, and residuals - will affect the Q of the coil. These loss factors are somewhat influen-ced by temperature variations~ thus the Q will also be affect-ed by temperature change. For critical applications, consult the Ferroxcube Engineering Department for guidance in estimating the effect of temperature on Q.

It should be kept in mind that the values given for T.C. are themselves not truly constant, nor are they without a manufacturing tolerance. When the designer wishes to take advantage of the inherent stability of the Ferroxcube ferrite, by using it to neutralize the T.C. of a capacitor, he is seeking, not minimum T.C., but an accurately-known, controlled, constant temperature coefficient of effective permeability (inductance). For this reason, the range of variation in the T.C. is given on the core-group charts.

3.11 Design Procedures , Here are descriptions of the five most common classes of inductor-design problems, leading to five (increasingly com-plex) design procedures. We shall consider each class separate-ly, recommending a procedure for optimizing each design,

cand giving an example of the calculations involved.

3.11.1 Class I - L/V Dominant In this class of applications, the required Q and stability are so low that almost any design that achieves the required inductance will satisfy. In this class, we desire to minimize the volume, Ve, reqUired to achieve the specified inductance. The procedure is extremely simple:

First, select a suitable core material - one that is recom-mended for the frequency range of interest.

Second, find the smallest-volume group of pot cores made from the selected material, and choose the one with the highest AL value (smallest gap).

Third, determine the turns required (N) using equation (3-7).

Fourth, check that the required riumber of turns, of a wire size that is not too fine to wind economically, will actually fit the core. (Use the Wire Table in the Appendix and the procedure of section 3.8.) If the required number of turns fits, a rough check of Q may be made from the curves (see 3.9) and the temperature coefficient may also be verified as low enough (see 3.10) after which the design is complete.

If the required turns of a practical wire size will not fit, choose a larger core of the same material, and repeat the third and fourth steps above. Continue the process until the smallest satisfactory core is found. An example of the above design procedure is given below.

16

EXAMPLE: Class I Design

Design the smallest possible 400 mH inductor for 150 kHz use, using no finer than # 46 single-Formvar wire. Low drift of inductance with temperature is desired.

1. Referring to Table 1 in par. 3.3, we would select 387 material (over 389) because it has low T.C. and is recommended for 150 kHz use.

2. In the charts in the introduction to Section 2 of the catalog, the smallest core-group for 3B7 ferrite is the 905 group, for which the highest AL value is 160.

3. Using equation (3-7), with L = 400 mH and AL = 160, we solve for N:

N = 103 j 400 = 1580 tIlrns 160 .

4. Using a single-section bobbin, the winding area of which, from the core-group charts, is 0.0053 sq. inches, we find that the turns/sq. inch are

0158~ = 30,000. The maximum .size of wire .00 3

possible (from the Wire Table in the Appendix) is . #46 single Formvar, which satisfies the original restriction. The design is complete.

3.11.2 Class II Land Q Do minant

In this class, the required stability is low, and the volume occupied by the inductor is of secondary importance.

First, select a suitable core material- one that will have the highest Q for the frequency range of interest (see Q-curves and 3.9).

Second, find the smallest core-group manufactured of the material) that exhibits, for its highest AL value (smallest gap), at least the minimum required Q over the frequency range of interest.

Third, using that AL value, find the number of turns required to. achieve the specified inductance, using equation (3-7).

FOllrth, using the Q-curves as a guide (see 3.9), verify that the required number of turns of a wire of sufficient size to maintain the Q at the minimum required will fit the core. (Use the winding table, as described in 3.8).

If the wilH.ling fits, a rough check of the temperature coefficient (as in 3.10) completes the design.

\

SECTION 111-LINEAR, LOW-LEVELINDUCTORDESIGNTHEORY

If the winding will not fit, select a larger core-group, repeat-ing the second, third, and fourth steps. Continue the pro-cess until the smallest satisfactory core is found.

NOTE: To maximize Q, select the largest core-group that may be tolerated, dimensionally and economically.

3.11.3 Class III - Q and T.C. Dominant

In this class, almost always a tuned-circuit application, the specific value of L is not important, but very high Q and stability (low or controlled temperature coefficient) are required. At this point, please refer back to section 3.10, in which the distinction is made between minimum temperature coefficient and controlled temperature coefficient. This procedure examines both kinds of T.C. requirement, be-ginning with minimum T .C.

First, select a suitable core material - one that will have the highest Q for the frequency range of interest (see Q curves and 3.9.)

Second, select the largest acceptable core-group that is dimenSionally and economically acceptable, and manufac-tured of that material.

Third, refer to the Q-curves for that group, forlowest value of AL that will, yield, over the frequency range of interest, at least the minimum acceptable Q.

Fourth, refer to the charts for that group, and verify that the T .C. is low 'enough for the core with that value of At.

If the T.C. is found to be far lower than required, a core with a smaller gap (higher AL) will probably yield a better compromise, since it will have higher Q, and still have ade-quate stability. If both the Q and the T.C. are found to be much better than needed, a smaller core-group should be tried, repeating the third and fourth steps until the smallest core-group yielding satisfactorily-high Q and stability are found.

In applications in which a particular controlled T.C. is de-sired, so as to compliment the (opposite) T.C. of the ieso-nating capacitor, the third and fourth steps should be reversed - viz:

Third (alternate), refer to the charts for the group selected. Choose the core having a gap (Ad value that most closely approximates the T .C. desired.

Fourth (alternate), check that the AL value of the selected core will maintain satisfactorily high Q over the frequency range of interest, using the Q-curves for that group.

17

Here again, it may develop that the largest acceptable core-group will not yield the best compromise. It would be ad-visable to check the smaller groups for a closer T.C. match, particularly if the Q determined in the fourth (alternate) step is much higher than required.

Another reason for investigating smaller cores is that they offer choices of T .C. that are not available in larger sizes, provided that the lower Q they exhibit is tolerable.

3.11.4 Class IV - LtV and T .C. Dominant

In this class, primarily encountered in instrumentationnet-work and lower-frequency filter applications for extreme environments, the Q required is moderate, but the required inductance must be obtained in a minimum volume, and the temperature stability required is high.

First, select a suitable core material - one that is recom-mended for the frequency region of interest.

Second, find the smallest core-group manufactured of the selected material that exhibits (in the group-chart) an accept-ably-low T .C. over the temperature range of interest. This will correspond to a particular gap (AL value).

Third, using equation (3-7), determine the turns required, using that AL value, to achieve the inductance required.

Fourth, verify that a practical wire size may be used to fit this value of N into the core selected. (See 3.8).

Assuming that the winding fits, the Q-value may be checked roughly (see 3.9) and the design is complete.

If the winding does not fit, the next larger core-group exhi-biting an acceptably-low T.C. must be tried, repeating the third and fourth steps. Continue selecting larger cores until an acceptable design is found.

3.11.5 Class V - 0, LN, and T.e. All Dominant In this relatively small, but nonetheless important, class of applications, the designer (and the ferrite pot core) are often pushed to the limit before a satisfactory design is found. The procedure recommended is as follows:

First, select a material having the highest Q over the free quency range of interest.

Second, find all core-groups manufactured from the selected material that exhibit, in any of their air-gap sizes, a suffici-ently-low T .C. We shall call these "acceptably-stable cores",

Third. Working with the acceptably-stable cores only, segre-gate out only those whose Q-curves indicate an acceptable Q or higher over the frequency range of interest. We shall call these "acceptable Q - T.C. cores".

SECTION 111- LINEAR, LOW-LEVELINDUCTOR DESIGNTHEO,RY

Fourth. start with the smallest "acceptable Q - T.C." core, and use its AL value in equation (3-7) to determine the num-ber of turns.

Assuming that the winding fits, and that the Q is accept-able, the design is successful. If not, proceed to the next largest of the cores in the "acceptable Q - T.C." group of cores, repeating the fourth, fifth, and sixth steps. Continue selecting larger cores until a successful design is achieved.

Fifth. Check the winding fit (see 3.8) and verify that the inducta~ce can be achieved with a practical wire size.

If the largest core available still does not yield an acceptable Q at an acceptable T .C., the available recourses are:

Sixth. If the wire size differs greatly from that used to ob-tain the Q-curve data (as noted on the curve), check the Q, using the approximation given in (3.9).

a special ferrite material

a special core, gapped for this application

a special winding structure

two or more cores, with windings connected in series.

DESIGN SUMMARY FOR LINEAR, LOW-LEVEL INDUCTORS

STEP

STEP

CLASS I - LN Dominant CLASS" - L and a dominant

OPERATION REFERENCE STEP OPERATION REFERENCE

S~1!C1 cor~ material.

Select smallest cor~group, and core with highest A L .

Find turns requir~ (tl)

ChecI< fit by dividing N by Mooing arQ (bobbin aru from cor~group chartsl. Then ch~k N/Aw (turm!SQ. in.l agaimt .... inding tabt~ for wire !oize.

If fit is not satisfactOf"Y. ~ a larger eo

SECTION IV~ POWER INDUCT.OR DESIGN THEORY

4.0 Power Inductors Defined.

~n power inductors, the excitation is high enough to cause the effective permeability to vary significantly from the small-signal value(l1e) used in Low-Energy Inductor calcula-tions. Coincidentally. core losses become appreciable, the self-heating effect* may become a factor, and the quality-factor (Q) is significantly lower. Finally, the design limits are usually dictated by excessive heating. rather than by lowered Q or Stability.

4.1 General Remarks on Design Procedure.

The procedure recommended here for designing power in-ductors is almost exactly the same as that used for power transformers, since the same set of design limitations** are imposed, with only.two relatively minor differences:

Q and/or DC resistance in the inductor design take the place of regulation in the transformer design.

Only one winding must fit into the available core window

of the inductor design as against two in the transformer design.

4.2 Effective Permeability vs Flux Density.

The curves of Figure 4-1 relate Jl to Bmax, the peak flux density corresponding to the peak of the applied excitation, for various AL values of the core. This family of curves is typical, having been drawn for a particular size and shape of core, manufactured of a particular Ferroxcube ferrite ma-terial, with the core material at a particular temperature. Each curve in the family shown corresponds to a particular air gap, which is described, as usual, in terms of the inductance index, AL, that the core would exhibit with that air gap at low (nearly zero) excitation. As might be expected, the larger the gap, the more constant is the permeability with, and the closer does it conform to, the value given at

Bmax = O. As might also be expected, as the core saturates very fully, the effective permeability drops to a relatively low value.

4.3 Effective Permeability vs AC Flux Density with DC Excitation.

The curves of Figure 4-2 are drawn for a particular core material, and a particular core structure and air gap. They relate the permeability, Jl, to Bmax, for different values of Ho. the DC magnetization of the core upon which the AC excitation is superimposed. It is obvious from these curves that the effect of DC magnetization is to reduce the effective permeability of the core, and to cause the core to saturate at a lower value of peak flux density. (It should be emphasized that the flux density referred to throughout this paragraph is that due to the AC excita tion ollly_)

Scl[-heating is defined as the temperature rise ill the will ding and cure due 10 excitatiun alolle as disrillg"uis/zed from rhe tC'mperature rise caused by heat exchange with the ambient elll'ironmel11.

Primarily, ~'ariatioll of permeability with excitation, and temperature-rise due to core and cupper losses: secondarily, Q and/or DC resistan ceo

4.4 Variation in Permeability with Temperature.

The curves of Figure 4-3 are drawn fm a variety of core materials,andare independent of the core structure or air gap.

They relate the core-material permeability. 11. to the tern perature of the core, for a single value of flux density. It is obvious from these curves that the effect of temperature of' permeability, while significant, is relatively small, and may usually be accounted for in the "second-round" design phase.

4.5 Core Loss vs Flux Density and Frequency.

The curves of Figure 44 are drawn for a particular core material, but are independent of core structure or air gap. They relate the core loss to flux density, for various values of freq uency. Similar curves can be dra wn for core loss vs frequency at various flux densities. Note that core loss is expressed in terms of milliwatts/cm3 of core volume.

19

Figure 4-1

Figure 4-2

Figure 4-3

~~t 400 t 300

200

\ASCENOING AL I VALUES

Smal (PEAK FLUX DENSITY) AC EXCITATION ONLY

Bma. (PEAK AC FLUX DENSITY) WITH DC EXCITATION (QUE TO Ha)

100 -----+,,~=-"7.._:...j

o -.=~~==-:= o 1000 2000 3000 GAUSS

Figure 4-4 Bmax(PEAK FLUX DENSITY)

SECTION IV~-POWER INDUCTORDESIGN THEORY

4.6

11 lO~.----"""------.r-----.---""7'"":"""-'

10 100 1000 10.000 GAUSS

Bmax (PEAK FLUX DENSllY)

Figure 4-5

Core Losses vs Flux Density and Temperature.

The curves of Figure 4-5 are drawn for a particular core material, but are independent of the core structure or air gap. They relate the core loss to flux density for several core temperatures. As was true in the case of permeability, it can be seen that temperature, while significant, has a second-order effect on core loss, so that the variation in loss factor with temperature may be entirely neglected, or applied only as a second round design correction.

4.7 lndu(:tance Calculation - AC Excitation Only.

Let us begin by repeating two fundamental equations stated in Section 1, paragraph 1.1, for inductance and flux density.

L = O.41CN'1l f ~ J. 10-& Hem:ies {1-2} x 108

B = Gauss 4.44fNAe

(1-4)

From equation (1-2), we see that, for a given core geometry, the relationship between inductance and the number of turns is known if one can determine the permeability. Since the permeability depends upon B, the flux density, one must turn to equation (14) to calculate B. Since B also depends upon the number of turns, it is necessary to combine equa-tions (1-2) and (14) which produces a new equation:

B =[ 2500E 1 F f~J

(4-1)

or 8 = kl F- (4-2) 2500E

fF (4-3) where kl = and Ve is defined by:

Ve 2eAe (4-4)

20

Note that all the factors in kl are known:

Ae is the effective core area in cm2 (listed in core-group charts in the Ferroxcube Linear Ferrite Catalog, and tabulated in paragraph 3.1)~

Q e is the effective magnetic length of the core in cm (also listed in core-group charts)~

Ve is the effective core volume in cm3 , given by equation (4-4) as the product of Ae and Qe~

L is the required inductance in Henries, which is part of the problem statement;

E is the excitation voltage (in volts RMS), also part of the problem statement;

and f is the frequency of excitation (in Hz), also stated initially.

It is convenient to restate equation (4-2) in the form:

8 2 Jl=-

k 12

(4-2a)

A very useful parameter in power inductor design is the ratio of its Voltampere rating to the (lowest) frequency at which it will be used.

The Voltampere rating of an inductor is given by:

2 EI = -- Voltamperes

2rrfL

Therefore the ratio Qf Voltamperes to frequency is,

(4-5)

EI 2 - = --- Voltamperes / Hz (4-6)

f 2rrf2 L

For convenience, we shall define a new factor. k2 , as:

(4-7)

It can be shown* that the maximum value of k2 for any par~icular core shape, size, and material is given by:

k2 _ 15.6 x 10-8 [Ve (Bmax) 2] 21i - 2rr L J.Le Voltamperes/Hz (4-7a)

Ve and Jle have already been defined, and Bmax is the maxi mum recommended flux density for the core material, as listed in the specifications for that material in Section 1 of the Linear Ferrite Catalog, and in the Appendix at the rear of this Manual.

"'Equation (4 7a) is derived as follows: (1) Solve (1-4) for E: (2) Sub-stitute that expression into (4.7): (3) illlo this new form of (4.7). sub!.'titute equation (1-2) for L: (4) clear and simplify. using (4.4). The result should agree with (4 7a).

~ ..... '-' .. ""'., ... -POWER INDUCTOR DESIGN THEORY

4.8 Design Procedure - Phase I. In the Table on page 26, we list k2 for every core suitable for \ower inductor (or power transformer) use. Thus, one Jegins the design by evaluating equation (4-7), and then finds the smallest core with a k2 value equal to or greater than that required. The Catalog charts list Ve for each core, which may then be used, in equation (4-3), to find k 1

Having calculateu kl' one need only find a point on the Permeability vs Flux Density curves (Section 1 of the Catalog) that satisfies equation (4-2a) - that is, a point at which J.I. is equal to 8 2 /kl 2 . Then after adjusting this value of J.I. for the core configuration, one returns to equation (1-2) and, with the new value of J.I. determined, finds the number of turns, N, required to achieve the required inductance. As a check, one may find the flux density, B, for the value of N, from equation (1-4). The example shown at right takes the design through Phase 1.

4.9 Completing The Design - Phase II.

Having determined the number of turns that will yield the desired inductance,it is only necessary to check the "fit" of the winding with an appropriate wire size for the effective current using a form of equation (1-5):

I = _E __ Amperes 2rrfL

( 1-5a)

,A good rule of thumb is to allow a current density of 1000 )amperes/sq. in. The core-group charts provide information on the available winding area. This winding area cannot, of course, be completely filled with copper. One must allow for the fact that the wire is insulated; that it is (generally) round, not square; for the fact that the winding efficiency may not be much higher than 80%; for the fact that some of the win-dow area is taken up by the bobbin (if used) and by inter-layer insulation (if used). The procedure is similar to that of paragraph 1.3, Section I, except that, in determining hAp", no allowance need be made for a secondary winding. Using a form of equation (I-IO),one may determine the largest wire that will fit.t

Turns/Square Inch = ~ AC

(l-lOa)

If this size is equal to or larger than the wire size selected, the first-round design is complete. If not, a core providing a larger window (and at least as high a k2) must be selected, and the design repeated.

4.10 "Trimming" The Design.

Having selected the wire size, it will be possible to calculate the resistance of the winding, and estimate the total wind-ing copper loss. Having both the winding loss and the core loss (from the curves for the material selected) one can compute the total power (Pt> dissipated in the inductor.

tTJIC wire table actually lists the turns per square inch (theoretical) /0 speed this calculation.

Having calculated this power, one may determine the tem-perature rise from:

21

50 Pt Trise = ---p;;- (4-8)

in which the factor Po for each core is listed in the Table on page 26. (Po is the power-

SECTION IV--POWER 'INDUCTOR DESIGN THEORY'

Having determined the temperature rise of the core, it is ';lOW possible to check and "adjust" the design for the effect of this temperature rise on inductance (permeability) and on the Q. Since these adjustments are second-order effects, they do not usually cause a design to change by more than a few per cent. _ . a small change in the number of turns is usually sufficient, unless the first-round design was very marginal.

4.11 Inductance Calculation - AC and DC Excitation_

In many power inductors,a large DC component of excitation is present. In practice, it is quite frequently true that the DC component of the excitation represents many times as much magnetomotive force as that reSUlting from the peak AC excitation. Under these circumstances, the introduction of an air gap into the magnetic circuit permits the use of a smaller core than would be possible with a closed magnetic circuit, all other things being equal.

The introduction of the air gap, however, complicates the relationShips amongst the various parameters involved in the inductance. There is now an equivalent or "hybrid" flux path, part of which is through a relatively long, high-perme-abilitY,ferrite circuit,and part of which is through a relative-ly short but low-permeability air gap. Since the air gap permeability may be taken as unity. the expression for the effective length of the magnetic path, 2e, is (see figure 4-6):

(4-9j

where J.l. is the permeability of the magnetic material, 2m is

the length of the magnetic path (formerly considered to be Qe for zero gap, and so listed in the core group charts), and 2g is the length of the gap, all lengths being expressed in cni in the equations that follow.

Although there is sometimes significant fringing of flux in the air gap, there is no need, at least in the first phase of design, to attempt to compensate for it, so that we may assume that:-

Ae = Am = Ag (41 OJ

where Am is the value formerly considered to be Ae for zero gap, and so listed in the core group charts, and all areas are expressed in cm:

4.12 The Concept of Average Permeability.

Figure 4-7 shows a typical hysteresis curve for core materials employed in Ferroxcube ferrite cores. Saturation becomes evident at a value of flux density, Bmax, corresponding to a magnetizing force, Hmax. If, as a first approximation, we

22

Figure 46 J-l

Bmax

Figure 47

Bma.x. 7"--_-L. Slope:--:~v

Hma.x.

Hmax

ignore the fact that the hysteresis curve between the 0, point and the Brnax, Hmax, point, departs significantly from a straight line, we may define an average permeability, J.l.av, as the slope of such a straight line. This value is liste~ in Section 1 of the Catalog, and in the AppendiX of this Manual, for each of the core -materials.

As a first approximation, then, we may rewrite equation (4-9) in the form:

(41lj

and we may also rewrite the familiar expressions for maximum flux density, Bmax, and inductance, L, in terms of J.l.av:

0.4 1T N [max J.lav Bmax == Gauss

2e (412)

where lmax is the peak excitation current CAC plus DC);

L _ x 10.8 Henries (4-13)

It should also be noted that these two equations may be combined in such a way as to eliminate the assumed, approxi-mation, since both Pav and 2e drop out, yielding:

N= L [max ---x AeBmax

(4-14)

. I I ,

!::SI::,L..; I H. .. ~I"'. I V - .

POWER INDUCTOR DESIGN THEORY'

Ioc+vz lAC = lmox. L EAC

lAC: 2rrfL

Figure 4-8

4.13 Estimating the Required Turns.

The problem statement will always contain values for Ioc, EAC, f, and L - from which we may compute lmax.

From Figure 4-8, which assumes a sinusoidal current wave-form for the AC excitation, we obtain:

(4-15)

assuming that the inductor resistance may be neglected. From this,

lmax = IDC + j2 lAC (4-16) The value of lmax will permit us to select a provisional wire size, having a cross-sectional area, Ax. The total winding area, Ac, required, allowing 80% for the winding efficiency, is then:

Ac = NAx

0.8 (4-17)

Then we may substitute equation (4-14) into equation(4-17), yielding:

A - AxLlmax x 108 cm 2 c - 0.8 Ae Bmax

which may be written in the form:

AcAe Ax L [max

0.8 Bmax

(4-18)

(4-19)

Note that all the factors in the right-hand side of the equation are known, once a core material has been selected, yielding Bmax. Therefore, we know the value of the product of the bobbin area and effective magnetic cross-sectional area for a core that will satisfy. Having this AcAe product, we may move directly to tr..~ core-group charts for the type of core that best suits the geometry. and select a provisional core size to satisfy our requirements.

The following example will serve to illustrate the method.

23

Example of Inductor Design for AC plus DC Excitation

Required: A fiter chci:e (wound on a pot core) for a 3-phase.,

60 Hz" ful-wavo-bridge rectific:r. Ambient temperature is 25 - lOC.

L '"' 10 mH loe .,. 0.10 Ampc:rcs EAC 1.6 Volts. RMS f c 360 Hz

1. From (4-15) and (4-16) we calculate I max :

lmax = 0.10 + v'i~ 1.6 ] == 0.2Amperes L x 360