Traveling Wave Cylindrical Induction Heating System

The paper deals with the traveling wave cylindrical heating systems. The analysis presented is analytical and a multi-layer model using cylindrical geometry is used to obtain the theoretical results. To validate the theoretical results, a practical model is constructed, tested and the results are compared with the theoretical ones. Comparison showed that the adopted analytical method is efficient in describing the performance of such induction heating systems.


LIST OF PRINCIPAL SYMBOLS
B magnetic flux density, T E electric field strength, V/m H magnetic field strength, A/m k wave length factor = 2π/λ P w power induced in the charge, W P number of poles I max peak of phase current, A J' amplitude of line current density, A l axial coil length, m m number of phases N eff effective number of series turns per phase Z t terminal impedance, Ω F supply frequency, Hz r,θ,z subscripts for cylindrical coordinates

INTRODUCTION
The primary object of this work is to propose a general mathematical model for the system Fig.
(1) using the actual topology for three-phase excitation with any number of poles in the axial direction.As a second object, the paper employs the multi-layer approach with an appropriate current sheet representation to calculate the flux density components, induced power in the charge, terminal impedance and electromagnetic force in the direction of the traveling field.
The primary coil construction may be explained in a similar manner with the aid of Fig.
(2) which shows the cylindrical windings in its tubular form with multi-polar system distributed axially.The model shown in

2.MATEHMATICAL MODEL
A general multi-region problem is analyzed.The model is taken to be a set of infinitely long concentric cylinders, with a radialy infinitesimally thin and axially infinite current sheet excitation of radius r g .It is further assumed that magnetic saturation is neglected.
Maxwell's equations for any region in the model are

Assumptions
Maxwell's equations are solved using cylindrical coordinates system subject to the following assumptions and boundary conditions.1.The induction heating system is infinitely long in the axial z-direction.

The primary current density
The primary winding considered is of cylindrical geometry and the excitation wave produced is assumed to be a perfect sinusoidal traveling wave.The line current density may be represented as The field produced will link all regions (1) to (N).

The field equations of a general region
As a first step in the analysis, the field components of a general region are derived.
Assuming that all fields vary as ( ) kz Using equation ( 1) and ( 6) and taking only the θ-component from both sides, gives, After rearranging, equation ( 11) may be written in the form The solution is given by and ω is replaced by (sω) for any region with slip s.I 1 and K 1 are the modified Bessel functions of the first order and of general complex argument.A and D are arbitrary constants to be determined from boundary conditions.Using equations ( 2), ( 7) and ( 13), it can be shown that From equation ( 13) and ( 14), it can be shown that

Field calculation at the region boundaries
H can be found by replacing (r n ) in the above expressions by (r n-1 ).Now, for regions where n ≠ 1 or N, Where [T n ] is the transfer matrix [3,4] for region n, and is given by Therefore, from equation ( 15) and ( 16) one may obtain that (A=0) and ( ) Considering the first region (1), then Therefore, from equation ( 15) and ( 16) one may obtain that (D=0) and ( ) ( ) It should be appreciated that equations (23 … 26) describing the field components at the boundaries of regions ( 1) and (N) still contain arbitrary constants.However, the ratios of θ E to z H at these boundaries contain no arbitrary constants, and it is only these ratios that are needed for a complete solution.The next section shows how this may be accomplished.The ratios of θ E to z H have been termed the surface impedance [5].

Surface impedance calculations
The surface impedance looking outwards at a boundary of radius r s is defined as and the surface impedance looking inwards a boundary is defined as Where Z in is the input surface impedance at the current sheet, and Z g+1 and Z g are the surface impedances looking outwards and inwards at the current sheet.Substituting for g Z and 1 + g Z using equations ( 28) and ( 27) respectively, and with rearranging to get Thus, the input impedance at the current sheet (Z in ) has been determined.This means that all the field components can be found by making use of this and equations ( 28), ( 21) and ( 22).

Terminal impedance
The terminal impedance per phase can be derived in terms of in Z [7], as

Power calculations
Having found θ E and z H at all boundaries, it is then a simple matter to calculate the power entering a region through the concept of pointing vector.The time average power passing through a surface is given [7], as Using equations.( 28), ( 31) and ( 33), it can be shown that the total charge power is

Axial force
It follows that the axial acting force on the region [8] is

Model description
Fig.( 4) shows the experimental test rig.The coils (primary winding) were wound circularly on a plastic tube, and held in positions through using circular guides.These guides were fixed on the tube in such away to make ditches between them.These ditches represent the slots of the primary circuit.Since the core has no backing iron then the primary circuit is of the open type.Each slot (which is open) is filled with a coil, and the coils are connected in star to a variable voltage supply.It is worth mentioning that these star connected coils represent the heating part of the system (heater).The heater was then mounted and fixed on a board using a suitable mechanical structure.This enables fine adjustment for a uniform air-gap surrounding the workpiece that represents the load (charge).The type of the charge used is a solid aluminum cylinder.The magnetic circuit for this material provides the required flux paths.The conductivity of the charge was measured using standard DC measurement.Table 1 shows the parameters of the experimental model.
To calculate the magnetic flux density on the charge surface, simple type of independent probe (search coil) was used.The magnetic flux density in the axial direction was measured using a (B -probe), as will be explained in the next section.

B -probe
A search coil is used to investigate the axial component of magnetic flux density at different positions on the surface of the charge.Five identical B-probes, displaced (5 cm) from each other, were used to measure the magnitude of the magnetic flux density.Each probe consists of (200) turns of thin wire of size (SWG 30) wound around the cylindrical charge, of ( 2 w r ⋅ π ) area.The coil ends were twisted together and connected to a digital voltmeter.Using Faraday's law, the induced voltage V C across the search coil is Where w r : radius of the charge, (mm)

Load power measurement
The load power was measured experimentally as follows: a.The input power to the heater was measured by using two wattmeter method.b.Primary phase current was measured using an ammeter.Then, the copper loss for the three-phase primary winding (coils) was calculated, as Where cs P : Copper loss of coil conductors, (w).(Ω) c.The load power in charge was found by subtracting the copper losses from the input power, as follows cs t ch Where ch P : Load power in charge, (w).

Axial force measurement
An experimental measurement of the axial force has been implemented using the experimental set-up shown in Fig. (5).Each point was found, by adjusting the suspended mass ( z M ), to balance the force produced by the system.The force is then calculated using the following equation Where z F : Axial force, (N).

RESULTS
Theoretical and experimental results were executed with different number of poles.Variable three-phase current excitation was used at a test frequency of 50 Hz.6), and (7) show the variation of charge power with exciting current per phase for 2,and 6 poles respectively.
From the results, it is clear that increasing number of poles reduces the power induced in the charge.This of course is to be expected since increasing the number of poles is accompanied by reducing the effective number of turns per pole per phase.
The layer theory approach has been used for the analysis of induction heating system with rotational symmetry with three-phase excitation.The analysis presented is quite general in that it lends itself to the analysis of traveling wave induction heating systems with any number of poles.
The displayed results show clearly that the theoretical results correlate well with the experimental ones.This may be considered as fair justification to the method adopted for the analysis in this work.

Conclusions
The layer theory approach has been used for the analysis of induction heating system with rotational symmetry with three-phase excitation.The analysis presented is quite general in that it lends itself to the analysis of traveling wave induction heating systems with any number of poles.
The displayed results show clearly that the theoretical results correlate well with the experimental ones.This may be considered as fair justification to the method adopted for the analysis in this work Fig.(2-a) has two axial poles, and Fig.(2-b) has six axial poles.

Fig
Fig.(3-a) shows a general region n, where n , E θ

cn
: number of turns of the search coil.

cA
: cross sectional area of the charge around which search coil is wound, (m 2 )

phR:
Resistance of primary coil per phase,

Fig. (
Fig.(6), and (7)  show the variation of charge power with exciting current per phase for 2,and 6 poles respectively.From the results, it is clear that increasing number of poles reduces the power induced in the charge.This of course is to be expected since increasing the number of poles is accompanied by reducing the effective number of turns per pole per phase.The layer theory approach has been used for the analysis of induction heating system with rotational symmetry with three-phase excitation.The analysis presented is quite general in that it lends itself to the analysis of traveling wave induction heating systems with any number of poles.The displayed results show clearly that the theoretical results correlate well with the experimental ones.This may be considered as fair justification to the method adopted for the analysis in this work.