When an electrically conductive body is placed in the region of a time varying magnetic field, electric currents are induced in the body causing thermal power generation in the body. This effect, known as induction heating, is widely used in industries ranging from the production of optical glass fiber to the heating of 25 tonne steel slabs [examples are given in BNCE (1984)]. The magnetic field is produced by a suitable arrangement of conductors, the induction coil, connected to a source which can provide the required time varying current in the coil. Electrical power supplied to the coil is thus converted to thermal power in the workpiece through the electromagnetic field, without physical electrical connection to the workpiece. Almost invariably, power sources used for induction heating provide an alternating current to the induction coil, the choice of frequency being critical to the particular heating application.

The induced 'eddy' current intensity is greatest at the surface of the workpiece and decreases towards its center as a function of the ratio: thickness/skin depth. As the ratio increases, a greater proportion of the total power is dissipated near the surface, this phenomenon is known as the *skin effect*. The skin depth, δ, is defined as δ = , where ρ is the electrical resistivity (Ωm) and ω = 2πf (rad/s) is the angular frequency of the coil current. The absolute magnetic permeability μ is μ_{r}μ_{0}, where μ_{0} = 4π·10^{−7} (H/m); the relative permeability, μ_{0}, is a function of the applied magnetic field strength for magnetic materials and has the value 1 for nonmagnetic materials such as copper and aluminum.

Power generated in a workpiece and the induction heating efficiency can be derived for regular shapes, such as cylindrical rods or tubes and wide rectangular slabs, from analytical solutions to the diffusion equation of the induced current, supplemented by empirical factors. These derivations are given in Davies & Simpson (1979), Orfeuil (1987) and Davies (1990). The analytical solutions assume constant material properties throughout the workpiece, whereas resistivity and specific heat vary with temperature and the permeability of magnetic materials is a function of field strength and temperature, reducing to μ_{0} above the Curie temperature (≈750 °C for steel). Computer-based numerical solutions are now commonly used to take account of these variations, an early example being Gibson (1973).

For a solid circular billet of diameter d and length L, heated in an enclosing circular coil of diameter D, length L_{c} and having N turns with a current of I amp/turn, the induced power P_{w} is approximately given by:

where Q_{rod} is given in Figure 1 as a function of d/δ, and K_{c} is dependent on the ratios d/D, d/δ and L/L_{c}. Orfeuil (1987) gives empirical values for K_{c}, which tend to unity as d/D and L/L_{c} approach unity. The power induced in hollow cylinders of wall thickness t is calculated with Q_{rod} in the above expression replaced by an equivalent flux factor Q_{cyl}, which is a function of t/d, d/δ and μ_{r} independently of δ. Davies (1990) shows graphs of Q_{cyl} for a range of these parameters.

Similarly, for a rectangular slab of length L, having width W, much greater than its thickness t, the induced power is:

Substitution of the Q factors by P_{rod} or P_{slab} from Figure 1 gives the reactive power (VAR) in the workpiece, which is needed for the evaluation of the power factor of the coil.

The efficiency of conversion of the electrical power supplied to the coil into thermal power in the workpiece, known as the coil or electrical efficiency, η_{c}, is given by:

where Q is the relevant flux factor, K_{A} is the space factor of the coil system and S_{C}/S_{W} is the ratio of the coil perimeter to that of the workpiece in the same plane. Harvey (1976) shows that coil efficiency can be significantly increased by the use of multilayer windings instead of the more conventional single-layer coil. These high-efficiency coils are now commonly used for heating nonferrous billets at mains frequency.

The overall efficiency of induction heating is η_{supply}·η_{thermal}·η_{c} ·η_{supply} is typically 0.8-0.9 (per unit) and accounts for losses in cables, power factor correction capacitors and frequency conversion equipment; the thermal efficiency, η_{thermal}, represents thermal losses from the workpiece and is critically dependent on operating temperature, thermal insulation and method of operation of the heater. Typical values are in the range 0.7-0.9 (per unit).

Transverse flux induction heating is employed for heating continuous metal strips. In this mode, the magnetic field is directed at the broad face of the material rather than through its narrow cross-section, with the induced current flowing across the width of the strip. The advantages of the method include a higher efficiency, particularly for nonferrous strips, at much lower operating frequencies than are possible with conventional axial flux induction heaters. Ireson (1989) gives a useful overall account of the technique and its commercial realization.

Apart from mains frequency installations, power supplies for modern, induction heaters are derived from solid state frequency converters. Unit sizes up to 7 MW have been installed for metal melting at 1-3 kHz and I MW units are now available for frequencies up to 500 kHz, previously the domain of power vacuum tube triodes.

#### REFERENCES

Davies, E. J. (1990) *Conduction and Induction Heating*. Peter Peregrinus Ltd. London.

Davies, E. J. and Simpson, P. G. (1979) *Induction Heating Handbook*.

McGraw-Hill Book Company (UK) Limited. Maidenhead.

Gibson, R. C. (1973) SLEDDY, a computer programme for calculating the induction and other heating of metal slabs and long cylindrical billets.

Report ECRC/MM16. *EA Technology*. Capenhurst, Chester.

Guide to induction heating equipment. (1984) *British National Committee for Electroheat (BNCE)*. 30 Millbank, London.

Harvey, I. G. (1976) The theory of multilayered windings for induction heating and their application to a 1 MW, 50 Hz, longitudinal flux billet heater. Paper H(a)4. *8th UIE Congress*. Liege.

Ireson, R. C. J. (1989) Induction heating with transverse flux in strip-metal process lines. *IEE Power Engineering Journal.* 3: (2). London.

Orfeuil, M. (1987) *Electric Process Heating*. Battelle Press. Columbus, Richmond, Ohio