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ABSTRACT The longstanding IEEE definition of induction heating‘the heating of a nominally conducting material in a varying electromagnetic field due to its internal losses’t@)- is interpreted here as including processes in which these losses are associated with conducted, as well as induced, currents. During thepastthreedecades, induction heating,utilizing induced currents, has grown into amultimillion dollarindustry - heating for melting,forging, hardening, brazing, welding, and many more applications. Equipment ratings are from a few hundred watts to over 60000 KW, at frequencies ranging from 50Hz to several MHz. More recently, a number of processes known generally as High Frequency Resistance Heating have received worldwide acceptance, especially in the tube and pipe welding fields. They employ thesameelectromagnetic phenomena to cause internal losses in nominally conducting material, but these losses are associated with current which is introducedthrough contacts.Thispaper describestheseelectromagnetic phenomena, and emphasizes the similarity of these commonly-called different processes.


resistivjty, ohm-cmp= relative permeability; f = frequency, hz; c = reference depth, c,m; a, x, N , L, and I, are defined inFig 1.

Solutions of (1)have been used since the 1930’s for designinginductionheating coils (2*3s4,596y7). They all contain Bessel functions of the dimensionless parameter a/c. The derivations are in the references and will not he repeatedhere. Fig 2 illustrates agroup of solutionsin the form of the vector distribution of magnetizing force H and current density S for various bar radii at various depths as noted. Radii and depths areexpressedas multiples of c: a/c and x/c respectively.Eachdiagram is for the same airgap H, taken as 1 + j0. Vectors42 and 12 are the total induced current respectively:

workpiece flux and


=JpH 2’ii’xdx ; 12 = L Sdx 0 -la (31 E2 is the voltageinduced in each coil turn by(&, and leads& by 90’. 12 opposes an identical component of ampereturnsinthecoil. The power density (not shown in Fig 2) at depth x/c is S 2 r , and the total heating power is : 2 P2 = L S2f dx = 12 R2 = -E212 cos& watts(4)

BASICELECTROMAGNETICPHENOMENA A typical classical inductionheatingconfiguration is shown in Fig 1. Actually, this is onlyone of many possiblevariations of thegeneralprocess. A coil(single o r multiturn) sets up an alternating magneticfield along the surface of a solid round conducting bar (not necessarily steel).


The impedance 22 reflected into the coil



22 = R2


X2 =

2 L 12

= N


2 7 (3 (KR + j Kx) L 2c


% a r e plotted againsta/cin Fig3a.They both contain terms which areBessel functions of a/c. Fig 3b shows the locus of the vector KR + K;I for various values of a/c.

KR and +L-----I

Fig 1 Coil and Solid Round Workpiece Imagine the bar to be composed of many thin coaxial sleeves. Themagnetizing force H at theoutersleeve equalstheairgap magnetizing force Hg. The entire flux through the bar is surrounded by the outer layer, and induces a voltage inthatlayer.Thiscausescurrentto flow aroundthe outer layer in a direction opposite to that of theprimary(coil)current.Thiscurrentproduces ?R heating in the outer layer and affects H at the next lower layer, reducing its magnitude and retarding its time phase. The voltage induced in the second layer therefore lags that in the outer layer by a small phase angle, and its magnitude is lowerbecause it surrounds less The process continues, layer after layer, until H and S (current density)disappearentirely,or until the center is reached, where S has no layer in which to flow. This is described by the differential equation:

The impedance

20 looking into the coil terminals


R1 and X1 are the resistance and reactance of the usually thin layer on the inside of the coil where 11 flows; Xg is the reactance arising from the voltage induced in the coil by the flux along the airgap between the work and thecoil. Usually, inpractical,reasonably efficient applications, Xg> Xz>Xi and RGR1.


The factor 2 G a r / L z c in (5) would equal the DC or ohmic resistance around the thin tube in Fig 4 if a were taken as the average, rather than the outside, radius


The electrical efficiency%, defined as powerdeveloped in theworkpiece/coilinput,canbe seen from (6) to R2 (7 ) 694 Authorized licensed use limited to: National Taiwan Univ of Science and Technology. Downloaded on May 17, 2009 at 08:46 from IEEE Xplore. Restrictions apply.

tions, such as the need for shallow surface heating. Fig 3b illustrates another penalty for operating below a/c = 3, namely, that % + j X2 becomes highly reactive, therefore requiring large values of reactive power.






Fig 4 Tube with Wall = v c



All equations derivedfrom (1) contain a/c.Similar parameters appear with other shapes, such as t/2 c for depth c is an slabs, where t = slab thickness. Reference ever-presentfactor. Various terms have beenapplied to c (also t o ' 6 c ) by different authors, such as "skin depth" o r "reference dimension". Electricalsize,a/c, is frequently called "index ratio'!. Dependent upon resistivity,permeability, and frequency, reference depth, c, is anything but constant during a typical heating cycle. f increases ten-fold in steel fromroomtemperatureto forging temperature. Other metalsexperience somewhat milderincreasesin The temperature distribution depends upon power distribution, heating rate,specificheat,thermal conductivity, and surfacelosses. In melted charges it also depends upon the stirring action from nagnetic forces. In surfacehardening, only theouterlayersareheated. The permeability of paramagnetic loads depends upon magnetic saturation and thetemperaturerelativetoCurietemperature. At the start of most steel heating cycles, the surface layer is magnetically saturated, while for large values of a/c (Fig 2) H is vanishingly low at thecenter.Reference depth almost loses its identity in this heterogeneous, rapidly changing situation. It shouldrightfully be treated as a variable, dependent on anumber of other interdependent variables. Useful approximate methods fordeal(*). Today, ing with this have been in use for some time aided by computers, more sophisticated research is underway (').


Fig 2 hternal Vector Diagrams

In spite of these limitations the practical concept of c as reference depthcontinues to be useful. It is usually treated as a factor which increases during the heating cycle.Thisshrinkstheelectricalsize of the workpiece, a/c,reducing bothaand the power-factor (Fig3). This effect is mostdrastic with paramagneticloads. For example, in steel at 1 0 KHz, c is a few thousandths of a cm cold, but more than .5 cm above Curietemperature (as inhardening, forging, and melting).The rate of change of c is highest when passing through Curie temperature. Should this rendera/c<3(Fig3),furthertemperature increase costs unduly large expenditure of power, and may even be impossible. Except for materialswhere f decreases with temperature and,&= 1 (e.g., carbon), the choice of minimum frequency depends upon final conditions.

Fig 3 KR and Kx vs a/c The KR curve in Fig 3a can therefore be taken as a measure of It shows that when the bar radius is less than3 reference depths (a/c<3),7&plunges rapidlytoward zero. Above a/c = 3, rises at a slow rate. Thedecision to use more costly higher frequency sources in order to operate far into this range is based on other considera-



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Long cycles for heating steel for forging o r melting i f power is to be held require external circuit adjustments constant. Staticinverters can be programmed to adjust frequency to maintain a/c and power. Motor-alternators having constantfrequency and voltage require adding capacitor steps and changing transformer ttips during such cycles.

rates are in hundreds of feet 'per minute, dependent on wall thickness and material.

RESTRICTEDHEATEDAREAS Many applications require heating only a small portion of a workpiece. In Fig5a theheated band is in parallel with nearby circumferential paths which are unaffected. It follows by Kirchhoff's voltage law thatthere can be no net voltage drop along the heated area, the voltage to drive the heating (b) current being exactly counterbalanced by the voltageinduced by the coil (i.e., transformer action).



(a) Fig 6 Two Contact Heating Applications








A Section B-B

Section A-A

Fig 7 CurrentPenetration


Fig 5 Restricted Heated Areas

In Fig 5b a tlpancaketl coil heats a well-defined closed path on a flat conducting surface. Equation (1) applies. Themagnetizing force H and thecurrent density S decrease with depth from their maximum values next to thecoil, becoming more laggingin phase,in amanner similar to Fig 2 . However, i f the electrical thickness t / C is small, H and S are not zero at the back surface. AS in Fig s a , there is no measurable voltage drop along the heated path. CONTACT APPLICATIONS In Figs 1, 5a, and 5b,and indeed, in any heating applicationusing induced currents, the heated area must In many applications this is bea closedelectricalpath. not thecase. A solution forthese is to introduce the current throughcontacts and control its flowby means of a return ("proximity)conductor closely spaced to the desired path as suggested in Fig 6a. An early a lication was hardening the edges of chain saw guide bars ( 6 Another is the continuous high speed welding of tube and pipe (Fig6b),wherecurrent flows from one sliding contact along one edge of the movingopen tube to the apex where theedges meet and back along the other edge to a secondsliding contact. Heating occurs only where it is useful - along the edges as theyapproach a cluster of weld pressurerolls.Each edge actsasthereturn conductorfortheother, Muchof the world's welded tube and pipe is made by this process, from brass radiator tubing 1.25 cm 0.1). by .02 cm wall to 2 cmthicklarge diameter, high strengthline pipe.The power ranges from 60 KW to 600 KW, at about 400 KHz. Welding


Fig 7 shows a more recently introducedcontact process which statically heats the edges of two flat steel pieces to welding temperature while they are forced togetherunderpressure. It is usedâ&#x201A;Źor applications such as welding automotive side bar blanks and wheel rims. A weld 20 cm long by .25 cm thick requires about 1.0 second with an input of 100 KW, 10 KHz. Thename Current Penetration Welding comes from the choice of electrical size. In hot steel at 1 0 KHz c = . 6 cm, so t/c is about .4, and this low value oauses the current t o peretrate through thethickness. Theefficiency, F[e, is high, because the heated path is restrained by the proximity conductor, which is surrounded by a suitable core, to a narrowpath along theseam. Another core beneath the work provides a low reluctance path â&#x201A;Źor the flux which penetratesthe work. Thesystem is in effect a one-toone transformer, with the primary (proximity conductor) in series with thesecondary (heated seam). A s inFigs 5 and 6a, there is no measurable voltage along the heated path. Thisstatement obviously agrees with Kirchhoff's voltage law, and is not arevocation of Ohm's law. The resistive voltage required is precisely induced by transformer action in the proximity conductor and appears as a component of the input voltage. A s in inductionheating using induced currents, this lack of workpiece voltage drop is an obvious advantage in terms of mechanicalfixturing. CONCLUSION The contact heatingapplications described here obey the same fundamental relationships that apply to induction heating with induced currents. They are truly "the heat696

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ing of a nominally conducting material in a varying eledromagneticfield due to its internal losses". REFERENCES


Proposed Standard, Test Code, and Recommended Practice for Induction and Dielectric Heating Equipment. AIEE Standard No. 54 Oct. 1952.


H. B. Dwight, M.M. Bagai, CalculationsforCoreless Induction Furnaces. AIEE Transactions, vol. 54, March 1935 pp 312-15.


J. T. Vaughan, J. W. Williamson, Design of Induction Heating Coils for Cylindrical Nonmagnetic Loads. AIEE Transactions, vol 64, Aug. 1945, pp 587-92.

4. G. H. Brown, C. N. Hoyler, R.A. Bierworth, Radio FrequencyHeating, Van Nostrand 1947. 5. R. M. Baker, Design and Calculation of Induction Heating Coils, AIEE Transactions, vol 76, pt. 11, March 1957, pp 31-40. 6. H.H. Storm,Surface Heating by Induction, AIEE Transactions, vol 63, Oct. 1944, pp 749-54.



C A. Tudbury , Basics of Induction Heating, Rider, 1960.


J. T. Vaughan, J. W. Williamson, Design of Induction Heating Coils for Cylindrical Magnetic Loads, AIEE TransactionsPaper 46-124, 1946.

9. J. D. Lavers, P. P. Biringer, R. S. Segsworth, Current Distribution, Forces and Circulations in the Coreless Furnace - IEEE Conference Record of TAS Seventh Annual Meeting 1972, pp 343-48. 10. Conduction Improves Heat-Treating, Metal-Working April 1955.

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electromagnetics in induction heating 1974  

repeated here. Fig 2 illustrates a group of solutions in the form of the vector distribution of magnetizing force H and current density S fo...

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