In the chapter 1 we have been discussing the natural limitations of the production possibilities of the field technologies. In this connection the basic electrical physics and mechanical processes under magnetic pulse metals working were considered in a detail. The conducted analysis has allowed to do some important conclusions.

Briefly, we would like to remind a physical essence and reasons of these limitations. Also we will point out the before suggested ways of the problems solving which are appearing in time of the magnetic fields interaction with thin-walled (if to be more exact - transparent!) conductors.

In accordance with relationship (1.3) the magnetic pressure on conductor is directly proportional to a difference of the intensities squares on its boundary surfaces. If the field penetration takes place the present difference falls down. It leads to lowering of the forces acting on conductor to be worked and appearance of so named "magnetic pillow". Practically this phenomenon shows itself in pushing away of the workpiece from matrix. It can be explained by reversal of the pressure forces direction at the expense of the diffusion processes.

Now we will remind the possible ways of the appearing problems solving.

A removal of the negative consequences of the field penetration through workpieces and forming of thin-walled conductors are possible with help of determined constructions of the inductor systems. The general property of technical solutions like these consists in "artificial creation" of such boundary conditions what will provide a zero field on the workpiece surface faced matrix for the magnetic intensity.

After discussing of advantages and demerits of this idea practical realization variants we have dwelled on the more simple construction of the inductor system (we mean its technical execution!) where the physical phenomenon of the plane electromagnetic waves penetration through thin conducting screen into a free space could be modeled. The construction schemes were presented on Fig.1.6. They realize the present phenomenon at the expense of a dielectric matrix usage.

The second suggestion consists in the several sources usage. An interference of their magnetic fields allows getting of the zero intensity in the space between workpiece and matrix practically. It should be marked this idea is more intricate for a practical realization!

The inductor system scheme with two field sources is shown on Fig.1.5.

The pointed out ideas for removal of the diffusion phenomena negative consequences require an experimental justification. Why?

The problem consists in the following:

- how will it be difficult to realize "the artificial creation" of the boundary conditions for the vector magnetic intensity on the workpiece surfaces from the practical view point ?

- if will the force action be effective (and how much ?) on the metal workpiece with thickness what is much less than magnetic field penetration depth ?

It should be marked that the only idea of the boundary conditions "artificial creation" can not provide the magnetic pulse technology practical realization. As a rule it is necessary solving of the following problems:

- getting of the magnetic fields with maximum uniformity in the working zone;

- getting of the high efficiency of the source energy usage in the time of the force action on workpiece what is being deformed;

- suppressing of the heat emission processes in the workpiece since they are becoming very essential by the metal little thickness and the quite big curl currents excited under the external magnetic field action.

Within the scope of the given monograph we will dwell upon the constructions with dielectric matrix only. These constructions are quite simple for the practical execution and effective rather if to speak about the force action on thin-walled metal workpieces.

Conducting analysis we will distinguish the inductor systems with flat multi-turn coils and also systems containing so named paired current-carrying conductors in area of action on workpiece to be deformed. It should be marked that the last ones are near to single-turn systems as to constructive execution.

In advance it is obviously that the inductor systems with the multi-turn coils will be characterized by a high value of the self-inductance unlike constructions with the paired current-carrying conductors which have the rather little inductance.

This main difference determines the practical possibilities and the application area of the distinguished inductor systems for the magnetic pulse thin-walled metals working.

We will tell it over some later when we have the solution results of the according electrical dynamics problems and data of the experimental investigations.
 

In this connection the thin metals were used as workpieces. The round through holes were stamped with help of the two types of inductor systems: the first one consists of the flat single-turn solenoid and dielectric matrix, the second system is the complex construction consisting of two parallel single-turn solenoids between which two conducting plates were placed (in real conditions one of them can accord to a workpiece and another ó to a matrix).

Investigations were being conducted by a magnetic pulse installation with stored energy till 24 kJ in a voltage range of capacitor bank (6ó11 kV). A measurement of current pulse parameters were being executed by an induction coil (so named the Rogovsky coil!) what was connected to an oscillograph output through an integrating circuit.

In practice the ordered boundary conditions for the intensity vector can be realized rather simply with a flat inductor and dielectric matrix. We keep in mind the boundary conditions in the metal plate to be deformed.

We have shown (before in the chapter 1) that the inductor construction like that allows to realize the electrical dynamics phenomenon of the magnetic field penetration into a free space. We have found this problem mathematical solution for rather thin metal sheet. It connected the intensities of external and penetrated magnetic fields.

From the dependence (1.26) what was found in the chapter 1 it follows that

where

A scheme of the experimental inductor system with dielectric matrix is represented on Fig.3.1 (system 1: (1) is the inductor; (2) is the plate to be deformed; (3) is the matrix).

The flat single-turn solenoid was made from an electrical engineering steel with conductivity = 0.157 x 10⁷ (Ohm · m)^-1. The turnís thickness was equal to d₁= 10^-3 m.

A copper foil was used as workpiece. Its thickness was equal to d = 5 x 10^-5 m and the conductivity = 5.96 x 10 ⁷ (Ohm · m)^-1.

conductivity = 5.96 x 10 ⁷ (Ohm · m)^-1.

In the time of the capacitor bank discharge a power current pulse is flowing in the inductor winding circuit. Under action of magnetic pressure forces the holes appear in the plate-workpiece according to the holes in the dielectric matrix.

The specimen of plate from experiment and oscillogramm of the inductor winding current are shown on Fig. 3.2 (this specimen is marked by #1) and Fig. 3.3. In Fig. 3.3, the horizontal scale reads 1.5 x 10^-6 sec/scale graduation, the vertical scale reads 130 kA/scale graduation.

The executed measurements showed that the current pulse with amplitude I_m = 156 kA and with frequency f = 25.64 kHz had flown in the inductor winding. This current value accords to the magnetic field intensity magnitude H_m = 0.78 x 10⁷ A/m.

For the experiment conditions the plate-workpiece is transparent for acting fields because  = 0.03 << 1 (the relationship of thin-walled conductor from (1.9)). The appearing pondermotor forces are directly proportional to square of the magnetic field intensity practically what is being excited by the current in the inductor winding. The pressure magnitude is P_m = 3.7·x 10¹¹ N/m².

The experimental results showed the effective force action has been realized on thin-walled metal workpiece in investigated inductor system in spite of the foil little thickness.

The experimental specimen is shown on Fig. 3.2. It is marked #2. This specimen was got in the experiment with the same flat inductor but the massive copper matrix was used instead of dielectric one. Several recesses were made in the metal matrix (by analogy to the holes in the dielectric matrix). The recesses' depth was 2 x 10^-3 m, their diameter was 10^-2 m.

In this experiments the problem of the magnetic field penetration through thin metal sheet in a cavity what was bounded by "ideal" conductor was being realized. Theoretically, this problem has been considered in the first chapter.

The relationship for harmonic process was previously defined (1.22). This dependence connected the intensities amplitudes of external and penetrated fields ().

We will need this formula later for calculations. Let us rewrite it.

where

;

h - is the cavity depth;

- are the thickness and conductivity of the foil metal;

- is the vacuum magnetic permeability.

The given decrease can be explained obviously by increase of the equivalent resistance (the inductance) of inductor system when the massive metal matrix was being used.

We will conduct some evaluations of the appearing electrical dynamics forces with help of the measurement results and the formula (3.2).

We have from experiments:  = 1.04, H_1m= 0.65 x 10⁷A/m, H_2m = 0.39 x 10⁷A/m and P = 1.6 x 10¹¹ N/m².

As it follows from calculations, the magnetic pressure forces decrease more than two times under the massive conductive matrix usage in comparison with the case of dielectric matrix application.

So far, we can rather evidently see the results of pushing away from metal matrix of some foil parts over recesses on the experimental specimen #2 (Fig. 3.2).

We can do the conclusion in the whole: the thin copper foil deforming occurs if to use a massive conductive matrix, but the efficiency of this process is quite a little.

The some inductor systems possibilities were investigated in the described experiments. The operation principle of these constructions was based on conclusions of the theoretical problems solutions about the field penetration through a thin conducting sheet in a free space and a bounded cavity (dielectric and metal matrix).

There is another technical solution of "the artificial creation" problem of ordered boundary conditions for the magnetic intensity vector. This solution can be practically realized with help of two sources-inductors. As it was pointed out before the summation of their magnetic fields permits to neutralize attenuating action of the diffusion phenomena.

Constructively, this system was made from two parallel, flat single-turn solenoids. As it was pointed out, two metal plates were being placed between them. We remind that in practice one of them can be workpiece to be deformed, another can be matrix accordingly. But it is possible and other variant. They can play other roles: both they can be workpieces which are being deformed according to profile of dielectric matrix what is placed in the inner space between them. In this case, productivity of the working process will be two times as much.

So, this last situation was being modeled in time of conducted experiment.

The experimental scheme is presented on Fig. 3.1 (System 2).

The inductors were made from electrical engineering steel. The turnís width was 2x10^-2 m. Their thickness' were the same and equal to 10^ó2 m. The dielectric flat matrix with through round holes was placed between them. The matrix thickness was equal to 10^-2 m. The holes were situated in front of the inductors working surfaces.

The metal plates were executed from copper foil with thickness  (the analogous specimens were used in time of the previous experiment).

To get the zero field in the inner space between plates the currents in the solenoids windings must be the same but their directions have to be opposite. The simplicity of the present relationship between the fields of the sources is being explained by geometrical and electrical dynamics symmetry of the inductor system of the taken construction.

Not dwelling on details we would like to mark the positive results of conducted experiment!

The force action on the workpiece was quite effective. The holes in the plates have the clean sharp outlines.

So as before, the capacitor bank voltage was U_m = 6.4 kV, the inductor current amplitude was equal to I_m = 125 kA, the frequency f = 18.52 kHz.

In whole, the conducted experimental investigations as to thin-walled metals deforming by the magnetic pulse method have verified the practical possibility of the idea realization about "the artificial creation" of the ordered boundary conditions for the magnetic field intensity vector on the surfaces of the workpiece to be deformed. With help of suggested inductor system constructions the effective magnetic pulse stamping has been realized of the very thin metal specimens. Their thickness has been less by an order of the according medium skin-depth.

Finally, it is the last thing what should be mentioned.

The used inductor systems (we mean their constructive execution) have been the models only. They must have demonstrated the practical possibilities of the magnetic pulse method for the transparent metal workpieces deforming. The carrying out of the real technological operations requires solving of many problems that will discussed some later.

Some quantitative connections (which accord to the present condition of the electromagnetic processes going) between the inductor systems characteristic parameters can be defined for the example of construction with a flat multi-turn coil.

A choice of this object for investigation is not happen. The first, the inductor systems like that are widely spread in practice of the magnetic pulse working, the second, the found results can be generalized for case of any other construction.

From a mathematical view point these connections have to fix the limitations in application of the plane electromagnetic waves approach for the processes calculations in the inductor systems with dielectric matrix and thin-walled sheet workpiece.

A calculation model of such system with the multi-turn coil is shown on Fig. 3.4.

Under solving of the formulated problem we suppose, that

1. an axial symmetry takes place relatively to OZ - axis, and in the according cylindrical coordinate system ;

1. a distance from the coil to the workpiece and also the workpiece thickness are much less than the characteristic dimensions of system, so that

 

 
 
 
 
 
 
 



2. - in the homogenous winding of the coil with current density j(t) flowing; its spectrum has components with angular frequencies  which satisfy the inequalities  (this is the condition of thin-walled conductor) and 

The Maxwell equations system (1.1) for electromagnetic field intensities after the Laplace transform has a view :

From this differential system we can get an equation for the azimuth component of the electrical field intensity :

where

 

where
 

is the Bessel function of the first order.

In accordance with accepted integral transform the equation (3.3) can be conducted to a view:

where
 

is the Fourier-Bessel transform of the function ;

are the wave numbers in metal and vacuum;

In the workpiece metal (the area where ) the tangent components of the electrical and magnetic fields can be found from equation (3.5) and the first relationship of the system (3.3):

where

is the twofold transformed (Laplace and Fourier-Bessel) dependence for the tangent r-component of the magnetic field intensity on the boundary surfaces of the sheet workpiece, where z = h and z = (h + d).  accord to some arbitrary constants of integration in a total solution of the equation (3.5).

In the matrix material the tangent components of the electromagnetic fields intensities which satisfy to the equations (3.3), (3.5), to the conditions of limitation for  and of continuation for z = (h + d) can be presented by the following relationships:

The continuation requirement of the tangent components of the electrical fields intensities on the separation boundaries of the distinguished areas (where z = (h + d)) gives a connection between the tangent components of the magnetic fields intensities on the sheet workpiece surfaces what is analogous to the relationship (1.24) describing the electromagnetic waves packet penetration process:

where
 

is the coefficient of the magnetic field screening;

As it follows from the physical point of view, the rigorously plane electromagnetic waves exciting is impossible in the real inductor systems with axial symmetry. However, if some fixed conditions are satisfied, the electromagnetic processes in the considerable inductor system will almost be identical to the processes in systems where the rigorously plane waves can be excited.

Let us find these conditions and estimate the possible level of identity.

The main property of the phenomenon being modeled is the practically full screening of the magnetic field intensity tangent component under penetration through thin-walled metal. In a quantitative relation it means that a value of the screening coefficient in the formula (3.8) has to be much less than one, this is .

To define the value  we will need an upper estimate of the -value what is the parameter in the integral Fourier-Bessel transform. Physically, this parameter accords to the wave number what characterizes the electromagnetic waves propagation in a radial direction.

Let us suppose that a distance between coil and workpiece is quite a little, this is . In this case the tangent component of the magnetic field intensity in the area where  will be equal to the exciting current density this is

, where

The twofold transformation of the acting field intensity gives a result:

where
 

,

is the Bessel function of the first order.  

From this formula it follows the Fourier-Bessel transformed intensity contains the function . And this function can be calculated for some interval of x-variable values. The results of calculations represent on Fig. 3.5 in a graphical view.

As it is seen from calculations, . It allows estimating of the integrals which contain the present function and which accomplish the reverse Fourier-Bessel transform. Thus, we can suppose the under-integral expressions will not equal to zero only if , where .

Including the executed estimate for  and the main states in the initial formulation of the problem being solved some following relationships can be introduced:

where

is the effective depth of the field penetration, 

The first inequality accords to the condition of the quite strong skin-effect (we keep in mind the comparison of the depth penetration with the characteristic dimensions of the inductor system!). The second relationship is the quasi-stationary condition of the electromagnetic processes in fact.

The condition of the thin-walled sheet workpiece  should be added to the relations from (3.9).

Executing the according limit transform in the formula for the screening coefficient  we are getting following expression

As it is seen from the relationship (3.10),  if . The physical interpretation of the last inequality can be given with help of the following inequalities:

Thus, when  the intensity tangent component of the magnetic field what has penetrated through the sheet workpiece into a free space was the infinite small value of an order .

This result is a coordination condition of the present calculations and the calculations in the plane waves approach. An error of this coordination is the same value of the order . The found result fixes an accordance degree of the electromagnetic processes in the inductor systems with axial symmetry and the processes in constructions where the rigorously plane waves are being excited.

Finishing the present consideration, we will transform the relationships (3.9) and (3.10) to a view that will be comfortable for future usage.

For it we will keep the evident dependence on the acting field frequency and the workpiece parameters, but instead of the coil external radius R₂ we will introduce some generalized characteristic dimension fixing a maximum cross extent of inductor system with any geometry, D = 2R₂. We will unite the got expression with the thin-walled conductor condition.

Ultimately, we will come to a double inequality:

The relationship (3.12) (connecting the inductor system parameters of any geometry and the workpiece characteristics) fixes the usage limitation of the plane waves approach for calculations of the electromagnetic processes in time of the pulse fields interaction with thin-walled conductors. Within the limits of present approach, the approximate boundary condition for the tangent components of the electromagnetic field vectors on the workpiece surface (what was got before in chapter 1) is quite applicable.

The mentioned boundary condition allows essential simplifying of the according calculations and also designing of the real inductor systems where the phenomenon of the plane waves penetration into a free space is being modeled.

3.3. The Magnetic Field and Pressure on the Workpieces in the Flat Inductor Systems with the Multi-Turn Coils.

The present part will be dedicated to the next calculations of the magnetic fields in the inductor systems with the multi-turn coils for the pulse stamping of thin-walled sheet metals. A direction of this calculations is fixed by the following questions:

• a creating of the uniform fields in the working zone (we keep in mind the area of action on the workpiece)

• a providing of the energy usage high efficiency for exciting of the power pressure forces on the workpiece being deformed.

- a system with a dielectric bandage over coil;

- a system with a metal bandage;

- a construction with a dielectric bandage on a metal base.

The simplest variant is the construction with the dielectric bandage what is installed over coil. A demanded stiffness of the system is being provided by a cross griping of packet from the parallel arranged elements: the flat bandage and coil (including an insulating insert between it and the workpiece), the sheet workpiece and the dielectric matrix.

The calculation model of the given inductor system is represented on Fig. 3.6.

The calculation model of the inductor system with the multi-turn coiland dielectric bandage for thin-walled metal stamping,(1) is the multi-turn coil; (2) is the sheet thin-walled workpiece; (3) is the dielectric matrix; (4) is the dielectric bandage for the system strengthening.Under solving we accept the assumptions which was formulated in the problem statement of the previous part of this chapter with the exception of some additions: a current I(t) is flowing in the coil winding with a turns number w, its spectrum contains the components with frequencies , which satisfy to the conditions: (3.13) It should be marked that the double inequality (3.13) is the condition of the plane electromagnetic fields exciting (3.12) for the present inductor system. Besides, we suppose the inductor coil thickness is quite a little. And its influence upon processes being considered is negligible in the time of calculation.The Maxwell equations (1.1) for the electromagnetic fields intensities components after the Laplace transform in view of the zero initial conditions in a space over the sheet workpiece (where z 3 0) will accept a form:

where  is the current density in the coil winding ;

and - are the Heavyside step-function and the Dirac pulse-function accordingly;

The equation for the azimuth component of the electrical field intensity  can be received from the differential system (3.14) ó (3.16):; (3.17)
The integral Fourier-Bessel transform satisfies to the limitation condition of the -radial distribution for r = 0 and r = °:  (3.18)

where  is the Bessel function of the first order.

In accordance with the expression (3.18) the equation is being conducted to a routine differential equation of the second order with non-zero a right part.

is the Fourier-Bessel transform of the current density,

We remind the function F (x) was tabulated before. Its graph is shown in Fig. 3.5.

We will rewrite the equation for  (3.19) in a form what is more suitable for integrating.

where

A total integral of equation (3.20) can be written in the form:

(3.21)

where

are the arbitrary constants of integration .

As it is not difficult to check by an immediate substitution, the function in the expression (3.21) is the solution of the equation (3.20) really. We remind that in the time of this operation it is necessary to use the known formulas connecting the Heavyside step-function, the Dirac pulse-function, their derivatives and the according integrals.

So far, the Fourier-Bessel transform of the magnetic field intensity tangent component can be found with help of the formulas (3.16) and (3.21):

The connection between the constants of integration in the expressions (3.21) and (3.22) can be fixed from the approximate boundary condition for the electromagnetic field vectors on the thin-walled conducting sheet surface (1.32).

For the problem what we are solving this condition is being written:

Substituting (3.21), (3.22) in the correlation (3.23), we receive, that

Now the expressions for the tangent components of the electrical and magnetic fields intensities receive the form (we keep in mind the formulas (3.21) and (3.22) in view of (3.24)):

The coil is not falling by the dielectric bandage in the inductor system being considered. It means a free space is over the coil (where r and z ® °).

The radial limitation of the solutions was satisfied by the usage of the according integral Fourier-Bessel transform. The limitation condition along the

coordinateóz of the functions (3.25) and (3.26) can be used for defining of the unknown constant of integration  .

Executing of the limitís transforming in the relationships (3.25) and (3.26) we are finding, that

After substitution of the (3.27) in formulas (3.25), (3.26) and execution of some identity transforms we are coming to the final relationship for the tangent component of the electrical field intensity:

The dependence for the tangent component of the magnetic field intensity can be defined by the same way:

Let us find the magnetic field intensity on the sheet workpiece surface from the expression (3.29). We are getting for z = 0,

Executing the reverse Fourier-Bessel and Laplace transforms in the formula (3.30) we are finding the original function:

The pressure force on thin-walled sheet workpiece is directly proportional to the square of the magnetic field intensity tangent component on the workpiece surface as it was shown before.

With help of the (3.31) we can define the magnetic pressure explicit dependence on the radial and time variables for the inductor system with the multi-turn coil and dielectric bandage in a view of the formula:

where it was accepted, that ,  is the dependence on time of the current density in the coil winding.

Further, we will pass to a calculation of the electromagnetic fields and pressure in the inductor system with the massive metal bandage.

The calculation model considered before accorded to the real construction where a hard fixation of the coil was being brought off with the dielectric bandage what was placed above the coil (z > h). The bandage retains hardly it against the workpiece surface (naturally, through an insulating insert). The workpiece lies on the dielectric matrix.

In the construction like this some increase of zone where the field will be uniform is possible at the expense of the relation  growing up only what means the increase of the coil cross dimensions practically.

From the physical consideration it is clear this effect can be achieved by other way. The increase of the magnetic field uniformity level (in zone of acting on the workpiece) without of the cross dimensions increase becomes possible at the expense of the bandage usage from the massive well conducting metal.

In this case the electromagnetic energy of the source will be concentrated between the conducting surfaces of the bandage and workpiece (but not between the coil and workpiece only). And the ordered effect can be achieved if a correct choice of a distance between them was made. A calculation model of the inductor system (where the massive metal bandage separated from the coil by the insulating layer with thickness L is used) is shown on Fig. 3.7.

All assumptions what were formulated before remain valid in the field components calculating. All mathematics operations as to the equations integrating and getting of the total solutions for the electrical and magnetic fields intensities are true too. Naturally the approximate boundary condition for the fields vectors on the sheet thin-walled workpiece holds true. But the second condition for defining of the integration constant  in the formulas (3.25) and (3.26) for  and  accordingly should be formulated by other way.

Before we had the system of infinite extent as to the space coordinate z. The requirement of the solutions limitation reduces to their equality to zero at . Now the system being considered is limited for the space coordinate z. The bandage is made from the ideal conducting metal. The according boundary condition reduces to the requirement of equality to zero of the electrical field intensity tangent component on the bandage inner surface this is  = 0. This condition and expression (3.25) usage allows calculating of the constant of integration  unknown in problem being considered.

Let us rewrite the formulas (3.25) and (3.26) in a form what will be more suitable for the future mathematics operations.

Putting the expression (3.33) equal to zero at z = (h + L) we are finding that

After substituting of the (3.35) in the formulas (3.33) and (3.34) we will get the expressions for the electromagnetic field intensities tangent components in the inductor system with the multi-turn coil and massive metal bandage:
 
 

where
 

.

From the formula (3.37) we will find the magnetic field intensity on the sheet workpiece surface.

For z = 0 we are getting that

At  what accords to the metal bandage removal the expression (3.38) goes over into the dependence got before for the magnetic field intensity on the sheet workpiece surface in the inductor system with dielectric bandage (3.30).

We will pass to the limit and check this statement.

Comparison of the got result with the expression (3.30) shows their full identity.

We will execute the reverse Fourier-Bessel and Laplace transforms in the formula (3.38).

The original function for the magnetic field intensity tangent component on the sheet workpiece surface can be written in the form:

(3.39)

where it was accepted that  is the time dependence of the current density in the coil.

A main purpose of the metal bandage in the inductor system (its scheme is represented on Fig.3.7)) consists in a "leveling" of the magnetic field in zone of action on the thin-walled sheet workpiece.

In the first chapter it has been shown the screening action of the massive ideal conductor is physically caused by the electromagnetic energy absorption. The screening efficiency of the magnetic field (the magnetic field only!) is being fixed by essential reflection from the boundary "metal ó dielectric" but not absorption.

Thus, the "leveling" of the magnetic field acting on the sheet workpiece in time of stamping has to take place in the case when the massive metal bandage will be changed by the dielectric one (as in the system on Fig. 3.6) but with a thin metal base.

A calculation model of the inductor system with the multi-turn coil and dielectric bandage (a semi-space where z > (h + L +d)) on the thin metal base (the thickness is equal to d, the electrical conductivity is equal to ?) is shown on Fig. 3.8. An insulating insert (its thickness is equal to L) is placed between the coil and metal base.

As before all assumptions and expressions for the electrical (3.25), (3.35) and magnetic (3.26), (3.36) intensities satisfying to the approximate boundary condition (3.23) on the thin metal workpiece surface remain valid under calculating of the fields components in the inductor system being considered.

The second boundary condition for defining of the integration constant  what is unknown for the given problem conditions has to formulated by other manner.

The metal base represents the thin conducting sheet. Its one side faces a free space. That is why the tangent components of the electromagnetic field vectors on the sheet inner surface must be connected by the relationship of direct proportionality (1.32).

We have in the terms of the problem being solved that

The sign minus in the right side of the relationship (3.41) is caused by the relative arrangement of the vectors  and  on the surface z = (h + L). We remind this arrangement is being fixed by direction of according Umov-Pointing vector.

Substituting (3.25) and (3.26) in the expression (3.41) we are finding that

The relationship (3.42) should be substituted in the formulas (3.25) and (3.26). We will get the expressions for tangent component of the electromagnetic field intensities in the inductor system with the dielectric bandage on the thin metal base:

where
 

.

We will find the magnetic field intensity on the sheet workpiece surface from the formula (3.44). At z = 0 we are getting that

At  what accords to the metal base removal the expression (3.45) goes over into the dependence got before for the magnetic field intensity on the sheet workpiece surface in the inductor system with dielectric bandage (3.30).

Let us execute the limit process in the formula (3.45) and check the high mentioned remark as before.

Comparing the limit passage result in the expression (3.45) with the relationship (3.30) we is being convinced of their full identity.

Now we will take the reverse Fourier-Bessel and Laplace transform in the formula (3.45) and get that

where

As before we will find the pressure force on the thin-walled workpiece in the inductor system with the multi-turn coil and dielectric bandage on the thin metal base. With help of (3.46) we are writing that

where it was accepted that

is the time dependence of the current density in the coil.

Finishing calculations of the electromagnetic processes in the different inductor systems constructions with the multi-turn coils intended for the thin-walled metals working it should be pointed out the further direction of investigations.

Further we will analyze the efficiency of each suggested constructions of the inductor systems with help of the got expressions for the magnetic field intensities and pressure forces being excited.

3.4. The Efficiency of Action on the Workpieces in the Different Constructions of the Inductor System with the Multi-Turn Coils.

Let us rewrite the formulas for the magnetic fields intensities in the considered constructions of the inductor systems and conduct their comparative analysis.

An analysis goal is the answers on questions what were formulated in introduction of this chapter:

• getting of the magnetic fields with a maximum degree of the fields uniformity in the working zone;
• getting of high efficiency of the source energy usage in the force action on the workpiece being deformed, this is achievement of the magnetic pressure maximum possible amplitudes.

1. in the system with the dielectric bandage (the formula (3.30)),

 

 
 


(3.48)

2. in the system with the massive metal bandage (the formula (3.38)),

 

 
 


. (3.49)

3. in the system with dielectric bandage on the metal base (the formula (3.45)),

(3.50)

All formulas contain the function  representing the Fourier-Bessel transform of the current density radial distribution in the coil. This distribution was accepted by "rectangular" function in the considered problems, this is .

Naturally, it should be expected the uniformity highest degree of the magnetic field (what is being excited by the present current) will have place when the field radial distribution is being described by the function  only in the cross-coordinate interval . It accords to the "rectangular" radial distribution in a space of the original functions (we keep in mind the original functions as to the Fourier-Bessel transform).

Further, the integral transform parameters  come in the formulas (3.48) ó (3.50) so that the according function dependencies on these variables can not be divided. It speaks the connection of the time and space processes of transition to a steady state in the inductor systems about.

However, this connection can be torn in some idealized situations. Then the space and time distributions become independent, and some additional possibilities appear for controlling of dynamics of the electromagnetic processes. In their turn these possibilities allow increasing of the force action efficiency on the workpieces to be deformed.

Unconditionally, some practical realization of these idealized cases is limited essentially by the real technical possibilities of the equipment for the magnetic pulse metals working. But nevertheless some defined possibilities exist, and we will try to find them.

The function dependencies on  can be divided in the expression (3.48) if  << 1. Then we have that .

We have met the limitations  << 1 in the part 3.2 during analysis of the magnetic fields diffusion processes through the thin-walled sheet workpieces in the inductor systems with the multi-turn coils. It was shown this inequality gives a condition of the plane waves excitation. And the ratio in its left side represents a qualitative estimate for the magnetic field intensity what penetrated through the workpiece.

There is no surprising in what this parameter fixes the field non-uniformity degree in the working zone of the inductor system being considered. It is obvious the processes of the field space-time distribution are closely connected with the diffusion phenomenon, and these processes must have some common characteristics.

As before we will formulate the limitation being accepted in the terms characterizing the main geometrical, electrical physical and time indices of the processes investigated.

Using the upper evaluation got before for the parameter  we are writing that

The value of  in the inequality (3.51) fixes quantitatively as an intensity of the diffusion processes so the magnetic field non-uniformity degree in the working zone of the inductor system.

Let us evaluate this index significance for a pointed out electrical dynamics processes characteristic.

We will write an expression for the magnetic pressure on the thin-walled sheet workpiece as a function of the parameter  with help of the formulas (1.3), (3.8), (3.11) and (3.51):

Identically, the formulas (3.48) and (3.51) allow evaluating of the pressure radial non-uniformity in the terms of the parameter :

where

 
.

The expressions - (3.52) and (3.53) contain the quadratic and linear dependencies on the parameter  accordingly. It means the pressure forces radial distribution depend upon the distinguished parameter value more essentially than amplitudes of these forces being fixed by the diffusion phenomenon. It is not difficult to check this statement if to take, for example, . Then we are getting from the formulas (3.52) and (3.53) that

We will illustrate the practical content of the inequality (3.51) by an example from the experiment as to justification of the magnetic pulse method possibilities for the thin walled metal plates deforming (the part 3.1).

Substituting the numerical data in (3.51) we will find that the external radius of the coil has to be no less than m for getting of the magnetic field with non-uniformity below 5% (what accords to the pressure forces non-uniformity below 10%) in the adopted inductor system.

We will add the conducted qualitative analysis by some rigorous numerical calculations of the pressure forces on the sheet workpiece in the experimental inductor system (we would like to remind some parameters of this system:

.

Let us transform the expression (3.32) to a form what will be more suitable for the future calculations.

We will assume the time dependence of the current density in the coil by sinusoidal function, this is . The function  reaches its maximum for . We will fix this moment and define a function convolution:

Further, we will introduce an integration new variable .

The radial dependence of the relative pressure on the sheet workpiece for the maximum of the exciting current can be defined after according substitutions and some mathematical transforms in the expression (3.32):

where

The results of calculations got with help of the formula (3.55) are presented on Fig.3.9. The found calculated dependence of the relative pressure on a relative cross-coordinate illustrates visually the main statements of the qualitative analysis of the investigated inductor system with the multi-turn coil and dielectric bandage possibilities.

We will need an average relative pressure for comparison in future. The magnitude of this characteristic accords to the conditional ideally uniform radial distribution of the magnetic field intensity in the calculated system. In the given case it will be equal to = 0.44.

In the whole the following conclusion can be done: the coil inductor system is not perspective variant of a tool for the thin-walled metals working because it does not provide some acceptable level of space non-uniformity of the magnetic pressure forces in the working zone.

Now we will go over to analysis of expression for the magnetic field intensity tangent component on the sheet workpiece surface in the inductor system with massive metal bandage (the formula (3.49)).

At  the hyperbolic functions can be changed by the first summands of their series approximately.

It should be marked, if to speak a practical interpretation about, the present inequality limits the total value of the insulating gaps - . So, we can get that  with attracting of the limit estimate for

.

The expression (3.49) is being transformed to the following form with allowance for the adopted limitation:

As it follows from the formula (3.56), the space and time dependencies were divided. The dependence on a radial coordinate for the Fourier-Bessel transform of the magnetic field intensity is the function  only what accords in the original function space to the rectangular radial distribution. Practically it means that the magnetic field on the workpiece surface in the considered inductor system has to be uniform for the values of the space variable - .

Now the intensity amplitude should be evaluate. We will find the original function according to the Laplace transform of the time dependence in the formula (3.56).

As before we will assume the current density time dependence by sinusoidal, . We will fix the time moment  and find the magnetic field intensity amplitude on the workpiece surface with help of the expression (3.57).

Thus,

With help of the formula (3.58) we will get that  for the conditions of the before mentioned experiment (the part 3.1) as to investigation of the magnetic pulse method possibilities for the thin-walled metals working and for the value 

The got value of the acting field relative amplitude is a little unconditionally. However it should be expected the introduction of the bandage will conduct to falling down of the intensity.

It is clear from a physical point of view that essential part of the excited field energy has to be absorbed in the massive conductor what is situated over the coil. Besides, as the qualitative analysis of the pressure forces space distribution in the inductor system without metal bandage shows the "leveling" of the distribution to a rectangular form decreases, naturally, the acting field magnitudes to values. These figures accord to the average relative pressure .

As it follows from the expression (3.58) increasing of the - magnitude is possible at the expense of - parameter rising. Practically it means increasing of the frequency or insulating gap ó  (or both of them simultaneously!).

But it should be remembered any variation the pointed out parameters is possible in the rigorously determined range. So, the acting field spectrum frequencies are limited by the thin-walled metal condition. The insulating gap magnitude  can be increased to some determined value too. A further  -growing up will conducts to appearance of connection between the space and time dependencies in the relationships for the field being excited. From the practical point of view it means a decreasing of the area where the magnetic pressure forces will be distributed uniformly.

But some optimization is possible even for the ready experimental inductor system (described in the part 3.1) if to introduce a massive metal bandage instead of the dielectric one. It can be pointed out roughly that the upper limit of the magnetic intensity what can be reached by this way is the value . We remind this is the field amplitude according to the conditional uniform distribution in this inductor system but without the metal bandage.

In the whole the conducted analysis allows to suppose the inductor system with the massive bandage is suitable variant (for practice!) of the tool construction for the thin-walled metals stamping by the magnetic pulse method.

Now we will analyze the expression for the magnetic field intensity on the workpiece surface in the inductor system with the dielectric bandage on the thin metal base (the formula (3.50)).

Identically to previous consideration we will assume that . In this case the relationship can be transformed to the dependence:

In contrast to the result for the magnetic field intensity in the inductor system with massive metal bandage got before the space and time dependencies in the formula (3.59) stayed mutually connected. Their dividing is possible if to introduce an additional condition: << 1.

In terms according to the physical interpretation the given condition can be rewritten in a form (identically to the limitation ó (3.51)):

We will mark the relationship (3.60) does not impose any new principle limitations on the connections between the inductor system parameters in comparison with conditions got before. Moreover, the square in the left part of the expression (3.60) does this condition some "weaker" than identical inequality (3.51).

The numerical evaluations for the experimental data (we keep in mind the possibilities investigation of the magnetic pulse method for the thin-walled metals working ó the part 3.1) show the inequality (3.60) is being executed with allowable error -  << 1.

We will mark the error decreasing (what accords practically to growing up of the limitation - (3.60) adequacy to the experimental conditions!) can be achieved rather easy at the expense of the coil external radius non-essential increasing in the inductor system.

Thus, when the inequality (3.60) is executed the formula for the Fourier-Bessel and Laplace transforms of the magnetic field intensity will receive the form:

The space and time dependencies are divided in the expression (3.61). Because  then the magnetic field is distributed uniformly on the workpiece surface for .

Let us evaluate the intensity magnitudes.

The more simple result can be got from the (3.61) if ,

Physically, given result is being explained by the considered system symmetry relatively of the coil.

As it follows from the (3.62) the intensity amplitude on the workpiece surface is equal to . It is almost 30% as much than the identical evaluation for the inductor system with massive metal bandage.

This result gives food for thought: how can the usage efficiency of the coil electromagnetic energy be increased in the inductor system being considered?

The answer is rather simple: a second workpiece and, accordingly, a second dielectric matrix should be installed instead of the bandage. In this case the coil field will work in two opposite directions simultaneously. The magnetic field with intensity equaled to  will act on the workpieces surfaces in each direction.

The energy usage efficiency being defined by the relative pressure magnitude will be equal to  in the system like this. Given magnitude of  is near to the identical magnitude for the conditional uniform field in the inductor system with the dielectric bandage but without any metal base.

Now let us evaluate the magnetic field intensity amplitude if .

For this we will find the original function according to the twofold integral transform in the expression (3.61).

We will get that

where
 

- is the time dependence of the current density in the coil.

As before we will assume the function  by sinusoidal, , fix the time moment, , and find the magnetic field intensity on the workpiece surface with help of the expression (3.63).

where

With help of the formula (3.64) we will get that  for the high mentioned experimental conditions (the part 3.1) and for  (this distance is being matched to the adopted limitation on the insulating gaps values).

According magnitude of the relative pressure will be equal to  Ultimately, this figure defines the electromagnetic energy usage efficiency in the considered inductor system.

Obviously, that the magnitudes of can be increased. But in this case the same problems can appear as in the inductor system with massive metal bandage.

In the whole, the conducted analysis allows to suppose the inductor system with two thin-walled sheet workpieces placed symmetrically with reference to the coil plane is the most perspective construction of all considered inductor systems. In the system like this the maximum degree of acting field uniformity and the most value of the source electromagnetic energy usage efficiency can be achieved.

The constructive execution example of such inductor system will be given in the conclusive part of present chapter. The suggested system is intended for stamping of articles from the cooper foil thin sheets.
 
 

The inductor systems constructions with the paired current-carrying conductors are not traditional for the magnetic pulse metals working unlike the considered coil variants. However they are interesting according to many reasons. Not enumerating all them we will try to distinguish the main ones.

The first, the system with the paired current-carrying conductors allows exciting of the rigorously plane electromagnetic waves (under suppression of any processes in the cross directions!). It increases essentially the magnetic pressure forces on the thin-walled metals if the dielectric matrix is being used.

The second, the systems like these have a low value of the self-inductance. This fact allows increasing of the field frequency and successful solving of the production problems determined class.

Strictly the inductor system with the paired current-carrying conductors is shown on Fig.3.10.

It consists of two rectangular form turns. They are placed on the same plane (practically, it is the plane of the bandage for the system strengthening in the whole). The longitudinal current-carrying side of one turn is parallel to the longitudinal side of other turn. The both turns is connected to the capacitor bank scheme in series. The currents flow along the paired current-carrying conductors in the same direction. Just this area is a working zone of the inductor system

.

As in the case of the coil inductor systems the calculations direction will be determined by the questions of the uniform magnetic fields designing and of the field source energy effective usage providing.

Unlike the previous two parts of given chapter we will limit ourselves by consideration of the constructions with the dielectric bandage only or with the dielectric bandage on the metal base. As it seems to us the variant with the massive metal bandage is not interesting specifically.

So, the first construction is the inductor system with the paired current-carrying conductors and the dielectric bandage.

The calculation model got by a mental cross section of the system in the working zone is given on Fig.3.11. An influence of the current-carrying conductors closing each of the turns towards the electromagnetic processes in the working zone of the inductor system is being supposed negligible.

The total width of the current-carrying conductors in the working zone is equal to 2b, the distance between them is equal to 2a. The dielectric bandage is situated over them (a semi-space, where z > h). The thin metal layer (the workpiece) with thickness  is placed below the conductors. It rests on the dielectric matrix surface, where z < (-d).
 
 

For the problem solving we are supposing

• the electromagnetic processes are quasi-stationary;
• there are a symmetry with reference to the OZ-axis and a quite big extent along the OY-axis, so that 
• the distance from the workpiece surface till the inductor plane and the workpiece thickness are much less than all system characteristic dimensions, this is 
• the same currents  are flowing along the paired conductors in the same direction;
• the spectrum frequencies are limited by the twofold inequality:

(3.65)

where

is the field diffusion time into conducting layer with the electrical conductivity  and the thickness ,

- is the magnetic permeability of vacuum.
 
 

The Maxwell equations have the view for the electromagnetic field the non-zero components (after the Laplace transform!) in the space above the thin-walled sheet workpiece (where ):

where
 

- is the current density in the current-carrying conductors,

,

- are the step and pulse functions accordingly;

.

The equation for  can be got from the differential system (3.66) ó (3.68):

We will apply the integral Fourier cosine-transform for solving of the equation (3.69). This mathematical approach is permissible because the geometrical and electrical symmetry relative to OZ-axis takes place.

Thus, we have

where
 

;

.

Including (3.70) and (3.71) the equation (3.69) can be transformed to a form:

where

Naturally, a total solution of the equation (3.72) will have a form what coincides with the expression (3.21):

where

- are the integration constants.

The expressions (3.73) and (3.74) have to satisfy the boundary condition on the thin-walled surface, precisely , and the condition of limitation for . It should be marked these conditions are identical to the boundary conditions in the coil inductor problem.

Accordingly, the expressions for the electrical and magnetic fields what satisfy the present conditions will coincide with the identical dependencies ó (3.28) and (3.29).

The magnetic field intensity on the sheet workpiece surface is being found from the formula (3.76) if z = 0:

Having executed in the (3.77) the reverse Fourier-cosine and Laplace transforms we will get an according original function:

Now we will write the formula for the magnetic pressure force on the thin-walled workpiece in the inductor system being considered:

where it was accepted that

is the time dependence of the current density in the paired conductors.

The electromagnetic processes analysis for the coil inductor systems has shown the dielectric bandage with thin metal base usage allows essential increasing of the uniformity degree and the amplitude of the magnetic field what acts on the sheet workpiece.

Let us consider an identical construction of the inductor system with the paired current-carrying conductors and the dielectric bandage on a thin metal base.
A calculation model of this system is shown on Fig.3.12.
 
 

Before it was shown the twofold integral transform (not dependently on its view: the Laplace and Fourier-Bessel or the Laplace and Fourier-cosine!) gave the same equations for the field vectors representations (look at (3.20), (3.72)!) also the same total and special solutions what satisfied the identical boundary conditions. We remind the present different combinations of the integral transforms were being applied for calculations of the coil inductor systems and the flat constructions with the paired current-carrying conductors accordingly.

Including the high mentioned statements the previous results (the formulas ó (3.43) and (3.44)) can be used.

After introducing of the symbols according to the problem being solved we will write the expressions for the electrical and magnetic fields intensities tangent components representations.

where

.

We will mark the formulas - (3.80) and (3.81) contain the function- . Its view is being fixed by the characteristic properties of the inductor system considered geometry (look at the dependencies for  in the (3.19) and (3.71) for comparison).

Further, we will get the relationship for the magnetic field intensity on the sheet workpiece surface from the (3.81) when z = 0.

Having done the twofold reverse transform in the formula (3.82) we will find the original function:

where

.

where it was accepted thatis the time dependence of the current density in the conductors.
 

By the same way as it has been done before in the part 3.4 we will analyze the electromagnetic processes in the inductor system with the paired current-carrying conductors.

Let us rewrite the according formulas for the magnetic fields intensities on the thin-walled sheet workpiece in the different considered constructions:

• the system with the dielectric bandage (the formula (3.77)),

, (3.85)

• the system with the dielectric bandage on the thin metal base (the formula (3.82)),

(3.86)

The function  is present in the expressions - (3.85) and (3.86).

It represents the Fourier cosine-transform of a current density cross distribution in the current-carrying conductors. The present distribution was assumed uniform, this is  for y > 0.

Accordingly, the magnetic field on the workpiece surface will be uniform if its distribution for Y-coordinate is being described by the same function  in a space of the transforms.

By the same way as before we will find some conditions what allow dividing of the functional dependencies on the parameters of the Laplace and cosine-Fourier integral transforms in the formulas (3.85) and (3.86).

A distinct of evaluations in the present analysis from the identical operations in the part 3.4 is caused by the following circumstances:

• the inductor system geometry being considered allows to suppose the integral Fourier cosine-transform  can accept the values from an interval ;
• the maximum value  is determined by view of the functions , which come in the expression for  and for the reverse cosine-Fourier transforms; this parameter permits an approximate upper evaluation -since;
• the high mentioned statements allow to suppose (we remind the question is some approximate evaluations!) that for the function - 

Thus, the evaluations being conducted will differ from the identical values what were found in the part 3.4 by magnitude of  only what is being accepted for the electromagnetic analysis execution.

Identically as in the formula (3.48) the functional dependencies on the integral transforms parameters as to time and space can be divided in the expression (3.85) for  << 1. Practically, it means with allowance for the parameter maximum value that

where

is the angular frequency of acting field;

is the time of diffusion into the conducting layer (the workpiece!), .

. (3.88)

If  it follows from the limitation (3.88) that  or  approximately. A practical realization of the similar inductor system is a rather complicated problem.

However, it should be mentioned a cross dimensions decrease of the current-carrying conductors in the working zone with saving of the field uniformity demanded degree is possible at the expense of a frequency increase. So, we have for  (what accords to kHz practically). This result is quite acceptable for designing of the inductor systems really acting constructions with the paired current-carrying conductors.

Let us pass to analysis of the expression for the magnetic field intensity tangent component on the sheet workpiece surface in the inductor system with the dielectric bandage on the thin metal base (the formula (3.86)).

For , if to be more exact (look at identical evaluations in the (3.56)), the expression (3.86) can be conducted to a form:

The assumed limitation of the insulating gaps total value in the inductor system being considered with the paired current-carrying conductors can be formulated in practical terms.

Including the evaluation for  we are getting that

So far, the execution of the condition  is necessary for dividing of the functional dependencies on parameters  in the formula - (3.86). This inequality can be rewritten in the practice terms:

With allowance for the limitation (3.91) the expression (3.89) is being transformed to a view:

The time and space dependencies are divided in the formula (3.92) such that function  is the Fourier-representation of the distribution as to the cross coordinate . Therefore for  the magnetic field acting on the workpiece will be uniform almost.

Let us evaluate some intensity amplitudes in the inductor systems being considered with the paired current-carrying conductors.

By the same way as in the coil variant the most simple result can be got from the expression - (3.92) if . In this case the system is symmetrical relative the plane of the current-carrying conductors.

Thus, we are writing that

From the energetic point of view such inductor system construction is the most optimum if to place a second sheet workpiece with according dielectric matrix instead of bandage. In this case the field of the current-carrying conductors will work in the two opposite directions. In each of them ~ 0.25 energy of the excited field is being realized (as there is a quadratic dependence on the intensity!). Ultimately, the efficiency coefficient will be equal to ~ 0.5 what accords to some magnetic field intensity conditional amplitude equaled to ~ 0.7 of the maximum possible magnitude.

Finally, for constructions with  the main conclusion can be done in accordance with the analysis results got before for the identical coil system (the part 3.4). The quite high efficiency of the magnetic field energy usage is possible really in the flat inductor system with the paired current-carrying conductors under according correct choice of the space-time characteristics (the question is the system geometry plus the frequencies of acting fields!).

Some heat processes caused by the known Joule-Lens phenomenon become rather essential in the time of thin-walled metals working with help of the magnetic pulse fields energy.

As the experiments have shown an intensive heating conducted to melting of the metal plates-workpieces at the expense of the curl currents flowing in the magnetic field with amplitude what was essentially less than it was necessary for the ordered operation execution.

In the present part we will analyze some different methods of the heat emission intensity decrease. These methods will allow rising of the magnetic field intensity amplitude to some determined value without high mentioned overheating of the thin-walled workpiece being deformed.

Let a metal flat plate (with some assigned geometry and physical parameters) is placed in a variable magnetic field (look at Fig.3.13).

The heat quantity being demanded for the plate temperature change by the value - is being described by the well known formula of thermodynamics:

where
 

M ó is the plate mass, ,

- is the plate material density,

S - is the surface area, ,

d₁- is the plate thickness,

c - is the material specific heat capacity.

The heat quantity what is necessary for heating of the plate area unit can be found from the formula (3.94):

From other hand the heat quantity what is given off during time in flowing of the current  is fixed by the known Joule-Lens correlation:

where

R ó is the active resistance of the plate, ,

- is the specific electrical conductivity of the plate metal.

 

As  we have from the formula (3.97):

Integrating the (3.98) we are finding that

where
 

is the function describing the time dependence of the magnetic field intensity, this is .

Equating the right part of the (3.95) to the right part of the (3.99) we will find the intensity amplitude of the magnetic field (with the assigned time function) the action of which conducts to the plate temperature increase by the value during the time interval :

In the simplest case we may accept that  is the angular frequency, .

Calculating the integral in the expression (3.100) we are finding the magnetic field intensity amplitude the action of which during a half-period of the sinusoidal time function changes the plate temperature by the value :

The interesting result follows from the dependence (3.101).

If the permissible heating temperature is fixed then the acting field frequency increase permits rising of the intensity amplitude (in the limits of acceptable heating!) proportionally to the square root from the frequency . For the pressure it means the possibility to increase the ponder-motor forces proportionally to the acting field frequency.

The form of the time function where is the attenuation decrement, is more real for the magnetic pulse metals working.

It should be marked the time functions for the inductor current and the curl current induced in the plate are assumed the same in the present analysis. It is not always so. But such coincidence of the time dependencies takes place for reality practically.

Thus, integral in the formula (3.100) for the exponentially attenuating sinusoid will be equal to

where

There is a sense calculating of the expression (3.102) for some different time intervals limited by values: It means considering of the heat process development in the time equaled to all duration of the pulse, to the half-period and quarter-period of the sinusoid.

For  we are finding that

For we have

For  we are calculating that

The expression for the magnetic field intensity (3.101) contains the acting field frequency (under the attenuation decrement equaled to zero!).

We will introduce so named equivalent frequency  in formulas (3.103) - (3.105). This parameter can be represent by a directly proportional dependence on the real frequency.

Now we can write  value as function of the equivalent frequency. It permits evaluating of the magnetic field intensity in dependence on the real frequency and the attenuation relative decrement.

Thus, we have for  that

For ,

For ,

With allowance for the introduced equivalent frequency the expression (3.101) accepts a form:

It should be marked the formula - (3.109) - was checked in the experiments as to justification of the magnetic pulse method possibilities for the printed circuit boards production in the electrical engineering.

Analysis of the expression (3.109) including the formulas- (3.106)÷(3.108) shows the  value is growing up if is increasing. As consequence the acceptable amplitude -increases in the fixed temperature regime  too. This fact can be explained by the following: the field real amplitude what is equal to the amplitude for the first maximum (Dt = 0.51) of the exponential attenuating sinus falls down with the attenuation decrement increasing, this is the amplitude decreases  times!

Further, the expressions - (3.105) & (3.109) permit to evaluate an acting fields amplitudes possible increase at the expense of a pulse duration shortening with a fixed temperature of the plate heating.

So, we have for the real value  (from the magnetic pulse metals working practice) that

where
 

is the duration of the pulse current exciting the magnetic field.

The attenuation decrement decreasing (what is equivalently to the field frequency increasing with unchanged active losses in the magnetic pulse installation discharging circuit) allows to increase the magnetic field acceptable intensity else more (and consequently, the pressure forces!) without a temperature growth up of the thin-walled workpiece to be deformed.

So, with ,
 

The main result of the conducted consideration is the conclusion about the heating decrease of the thin-walled plate being worked and the possible increase of the magnetic pressure forces at the expense of the acting field pulse duration shortening.

The simplest technical solution of this problem can be realized with help of a breaking down conductor (a fuse link). The similar schemes have been applied in the plasma physics experiments for some pulse generating with rather abrupt front or droop.

An essence of this suggestion consists in a series connection of the breaking down conductorófuse link with the inductor system winding.

This conductor is being broken down because of the heat energy intensive emission in time of the current flowing. A discharging circuit interruption is causing of the current pulse determined form generating. Unconditionally, an ideal shortening of the pulse duration can not be in consequence of the known inertial electrical physics effects. But to reach of desired result is quite possible with some approximation.

The present statement has been confirmed by the conducted experiments.

A cooper foil strip (of thickness  and width ) was being connected to the winding circuit of the inductor system with the paired current-carrying conductors in a series.

We will need a special formula for the experimental data working. Let us get it from the expression (3.109).

where

is the current amplitude.

The formula (3.110) connects the geometrical and electrical physics parameters of the fuse link with the heating temperature also with the amplitude-time characteristics of the flowing current. This formula permits to choose the fuse link geometry what provides the demanded shortening of the field action duration.

In the conducted experiments the copper foil strip with width  and with thickness  was used as the fuse link. A voltage of the capacitor bank was , the working frequency was  the attenuation relative decrement 

The oscillogramms of the current pulses in the inductor winding circuit are shown on Fig.3.14. Their comparison and sensing with the formulas the present part allows to do the following conclusions:

• as it is seen from the oscillogramms the used cooper fuse link provides the electrical circuit break in the moment 

1. the discharge in the circuit without the fuse link;

the average width of the cooper fuse link calculated with help of the formulas (3.110), (3.107), (3.102) will be equal to  what differs from the experimental value () less than 10%

• the fuse link allows the current pulse real shortening, as consequence decreasing of the heating process development time and it gives the possibility for the magnetic field intensity increasing,

However, to increase the intensity in the fixed temperature regime is possible by other way.

Let us return to the heating processes investigation results what have been got for the magnetic field concentrators in the part 2.3.

As it was shown, the heat-removal application allows essential reducing of the heating intensity caused by the Joule-Lens phenomenon.

In the case of the thin-walled conducting plates the heat-removal can be realized with help of a metal insert what is placed between the workpiece and the dielectric matrix. The electrical conductivity and the insert thickness do not have to distort the demanded space-time distribution of the magnetic field along the workpiece thickness (we keep in mind the suppression of the diffusion processes negative consequences!). From other hand the insert parameters have to provide the determined temperature regime for the plate being worked.

The numerical estimates have shown (the part 2.3!) the heat-removing metal layer introduction allowed to increase times the acceptable heating temperature, where  are the electrical conductivity and the thickness of the bimetal layers " sheet workpiece-insert".

A normal operation of the heat-removal was confirmed experimentally. Disagreements of the calculations and measurements results have not exceeded 5% (the chapter 5, the magnetic pulse of the printed circuit boards for electrical schemes).

If the cooper foil being worked has a thickness ~ 5x10^{- 5} m then existence of the steel insert till 10 ^{ó 3} m allows increasing of the melting temperature ~ 2.25 times in an equivalent. It accords to the acting fields amplitudes growth up ~ 1.5 times.

As some estimates show the considered methods combination (we have in mind the pulse shortening plus the heat-removal) can allow to increase the magnetic field intensity more than 2.5 times in the heating acceptable limits!
 
 

The investigation results give a possibility to close in the inductor system constructions what are optimized for parameters characterizing the magnetic field uniformity level in the working zone and the efficiency of the electromagnetic energy source usage in a first approximation.

In contrast to the inductor system with the magnetic field concentrator for a massive metals compression the suggested constructions for the sheet stamping are not optimized as to the temperature and strength indices.

Why?

The simple physical considerations speak about a small significance of these factors in this case.

Really, the magnetic pressure forces level for the thin metals deforming has to be lower essentially than under the massive workpieces working. It means a lower current load of the inductor windings, essentially less the heating processes intensity in scheme elements (but not in the workpiece!) and significantly less amplitudes of the mechanical stresses. Besides the inner stretching stresses which fixed the mechanical strength of the cylindrical concentrators for the massive metals compression are not being excited in the flat constructions. Though if it is necessary to provide some demanded level of these indices in the suggested inductor system for the flat stamping is not complicated problem what can be decided at the expense of some additional constructive elements.

Let us start from the system where inductor is executed as the multi-turn coil.

As it follows from the analysis in the part 3.4 the most perspective construction of all considered variants is the inductor system with two sheet workpieces which are situated symmetrically relative to the coil plane. The construction like that is characterized by the acting field maximum degree in the working zone and by the most efficiency of the electromagnetic source energy usage.

A scheme of the inductor system similar type is shown on Fig.3.15.

A brief description of the suggested construction is given below.

The coil parameters: R₁ and R₂ ó are the coil inner and external radiuses; w ó is a quantity of the turns; d₁ó is a spiral step; d₂ó is a width of the turn.

The insulating inserts (2) (thickness ) separate the working surfaces of the coil from the thin sheet workpieces (3) (thickness ). The last ones are placed on the inserts (4) (thickness ) which represent some plates from a hard metal with a low electrical conductivity. The main purpose of inserts consists in a heat-removal of the workpieces

The final elements of the suggested construction are the dielectric matrices (5). They can be massive enough, take the functions of the bandages and provide all inductor system stiffness.

The matrix for simultaneous stamping of several workpieces is shown separately on Fig.3.15c. Its working zones providing of an articles demanded form are situated diametrically underneath the inductor coil surface. A working zones quantity what can be made is fixed by the azimuth dimensions of the articles being stamped.

To illustrate in figures some electrical dynamics possibilities of the suggested construction it is necessary to concretize a production operation being executed.

Let considered inductor system is intended for stamping of some flat articles from a cooper foil with thickness . The article dimensions are: 

The dimensions of articles fix the dimensions of the working zones in the matrices and also the coil geometry.

The inductor system parameters can be taken such that:

• the coil inner and external radiuses  and  the turns quantity w = 5; the spiral step ; the turn width - ;
• the coil is cut from a sheet cooper with thickness ~m;
• the thickness of the insulating inserts between the sheet workpieces and the coils surfaces is chosen in limits 
• the metal inserts on the dielectric matrixes are made from steel plates with conductivity  and thickness ;
• the dimensions of the articles being stamped are such that the inductor system accepted geometry allows working of 4-8 articles simultaneously, this is 4-8 working zones with demanded profile should be cut in the matrix body.

For definiteness we will take the magnetic pulse plant made at the Kharkov State Polytechnical University. Its parameters: a capacity  a self-inductance  an energy stored an attenuation relative decrement 

Now we will calculate the inductance of the accepted inductor system. For this it is necessary defining of the tool inductance at the first turn.

A derivation of a rigorous dependence what can be found with attraction of the skin-effect condition is not expedient for the present problem. To evaluate this fundamental characteristic we will use the approximate equality of the current electromagnetic energy in the coil to the magnetic field energy in the inductor system working volume (we remind this is the space between the coil surfaces and the sheet workpieces planes!).

Thus, we have

where

is the inductance of the inductor system;

is the current in the coil;

is the magnetic field intensity in the space (volume) between the coil surfaces and the sheet workpieces planes;

is the volume of the space between the coil surfaces and the sheet workpieces planes, .

Now it can be found from the equality (3.111),

For assigned geometry we are finding that 

With allowance for the magnetic pulse installation self-inductance the total inductance of the discharging circuit will be equal to 

Now we will find the expected frequency,

The wave resistance of the discharging circuit is being determined by relationship:

Finally, the connection between the discharge current amplitude and the capacitor bank voltage will be directly proportional, this is

Further, we will execute some evaluations of the magnetic field uniformity degree in the working zone of the suggested inductor system.

For the frequency  we have . This value of satisfies to inequality (3.13) -

with execution of which the electromagnetic processes are absent in a radial direction. It means the exciting field represents the plane waves packet.

Let us check coordination of the accepted value  with the condition what fixes the acceptable magnitude of the insulating gap. This condition is defined by the inequality  in the expression (3.59). After substituting of  and in the inequality pointed out it can be got that  value is limited from above: 

Thus, assigned geometry of the inductor system allows dividing of the space and time dependencies in the function expression for the acting field intensity (3.59).

We will continue the evaluation of the high-mentioned dividing.

With help of the condition (3.60) we will compute that

.

The last estimate shows that in the suggested construction of the "two-sides" inductor system the space and time dependencies of the magnetic field intensity are divided fully and the field (in accordance with the formula (3.62)) will have the uniform distribution.

This is

The intensity amplitude will be connected with the capacitor bank voltage by the directly proportional dependence what can be got with help of the expressions (3.116) and (3.115):

We will finish the description of the inductor system suggested construction with the multi-turn coil by the estimate of the heat-removal efficiency with help of the steel insert with thickness . This  value accords to the thin-walled conductor condition since .

The steel insert usage allows increasing 1.25 times of the acting field intensity and not melting of the workpiece.

Let us pass to consideration of the inductor system of the identical purpose but with the paired current-carrying conductors.

A scheme of present construction (in accordance to the part 3.6 recommendations) is given on Fig.3.16.

1. the scheme of the strictly inductor system as an assembly;
2. the inductor with the paired current-carrying conductors;

We will suppose the suggested system is intended for stamping of the same as before articles (we keep in mind their final dimensions and the row material: the copper sheet with thickness ).

The assigned production operation can be realized in the inductor system with parameters according to the geometry on Fig.3.16: , .

Strictly inductor is made from the sheet metal ó the cooper with thickness ~ .

As before the insulating inserts thickness is chosen equaled to m. The steel plates-inserts with thickness  are placed on the matrix working surfaces. The inserts existence permits increasing of the acting magnetic field intensity  holding the acceptable temperature regime.

Let us compute the fundamental characteristic of the system being considered ó its inductance. For this parameter estimate we will use the approximate equality of the current electromagnetic energy in the inductor to the magnetic field energy in the volume between the surfaces of the current-carrying conductors and the sheet workpieces.

Identically to the expression (3.111) we will write that

where - is the volume of space between the surfaces of the current-carrying conductors and the sheet workpieces, ;

is the magnetic field intensity in the volume , in accordance with the total current law we have that ;

is the space volume between the surfaces of "periphery" current-carrying conductors and the sheet workpieces, ;

- is the magnetic field intensity in the volume - , in accordance with the total current law we have that .
 

 

Executing all necessary substitutions we will get the formula for inductance of the suggested construction with the paired current-carrying conductors:

The expression (3.119) can be simplified if to take in account that  for the system on Fig.3.16, then we are getting,

Further, we will compute the main characteristics of the inductor system with the paired current-carrying conductors identically to the multi-turn coil construction.

We will find that with help of the expression (3.120) after substituting of the inductor system assigned parameters.

The total inductance of the discharging circuit will be equal to  if the present inductor system is connected to the installation MIU-10 with capacity  and the self-inductance .

The frequency being expected can be calculated by the formula (3.113):

The wave resistance of the discharging circuit,

The connection between the current amplitude in discharge and the capacitor bank voltage,

The magnetic field intensity in the working zones of the inductor system being considered,

Now we will execute an estimate of the field uniformity possible degree in the working zones.

We will have that  for the frequency . This magnitude of the parameter  satisfies approximately to the inequality (3.65), execution of which fixes the plane electromagnetic waves excitation in the suggested inductor system.

We are stopping at these estimates. All other parameters can be computed by the formulas represented in the part 3.6. If it becomes necessary the suggested inductor system characteristics will have to be corrected in accordance with the calculations got results.

Finishing the description of the inductor systems variants for flat stamping of the thin-walled sheet workpieces we will mark the got dependencies and formulas both for the coil construction and for system with the paired current-carrying conductors carry out an illustrative character exclusively. The concrete calculated results (for instance, the currents and acting magnetic fields expressions) are interesting only when there is a real technical task as to a production operation. In this case at first a capacitor bank demanded voltage should be found according to the energy known magnitude what is necessary for the working process realization. After this a discharge current, a magnetic field intensity et cetera Ö have to be computed.

Several words should be said about the practical possibilities and some application area of the inductor systems suggested variants for the thin-walled metals working in the chapter conclusion.

The first, what should be distinguished under comparison of the considered constructions, it is the essential difference of the fundamental characteristics values. So, the inductance of the constructions with paired current-carrying conductors is several times less than identical parameter of the systems with the multi-turn coils. Accordingly, the higher frequencies can be achieved in the first than in the second case.

Where and why is it important?

We shall not enumerate the production operations the successful realization of which depends upon the acting field frequency characteristics so or differently. We will stop at an example what illustrates visually the defining role of the process frequency from the physical point of view. Question will be the through pictures cutting-out in the flat sheet workpieces about.

We shall remind the magnetic pressure is caused by appearance of the ponder-motor forces. They are the interaction forces of the external field with the curl currents induced in the conductor-workpiece. If a part shear has happened the curl currents distribution were being changed as consequence. It took place due to open-circuit of the electrical connections in some closed loops where these currents have been flowing. Present fact conducts to the ponder-motor forces decreasing and, ultimately, to abrupt falling down of the acting pressure.

It is obviously from some physical considerations this negative obstacle can be moved across if the magnetic field action will have an impact character. In this case some motion quantity pulse is being transmitted to the workpiece metal. In ideal it means the metal "elements" are getting the initial velocities but not the displacements. Then because of the mechanical processes inertia an appearance of the separate shears must not have an effect as to dynamics of the through picture cutting-out in the sheet workpiece.

Thus, the magnetic pulse method successful application is fixed by the acting field time characteristics for a cutting out realization to a large measure. In practice of the magnetic pulse metals working it means: the acting magnetic field frequency is higher the real situation is more adequate to that idealization where the ponder-motor forces impact takes place.

Let us pass to comparing and integrating of all got results.

We will formulate the advantages, the demerits and some application possible areas of the suggested constructions.

The inductor systems with the multi-turn coils:

the advantages ó the acting field uniformity high degree; the high productivity possibility at the expense of arrange several working zones on the matrix in a circle for simultaneous stamping of the articles according quantity; the high efficiency of the coil magnetic field energy usage;

the demerits ó the limited application caused by the relatively low frequencies of the field being excited; the constructive execution complicity of the current-outputs for the inductor connection to the magnetic pulse installation.

The inductor systems with the paired current-carrying conductors:

the advantages ó the realization possibility of the high frequency electrical dynamical processes; the current rectilinear way in the conductors over the inductor system working zone what leads to exciting of the rigorously plane-parallel magnetic fields and to achieving of the effective force action on the thin-walled sheet workpieces accordingly; the technical execution relative simplicity of the system in the whole including the current-outputs construction for connection to the magnetic pulse installation;

the demerits ó the acting field uniformity degree depends on the current cross distribution in the conductors over the system working zone to a large measure; the relatively low productivity (in comparison with the coil construction!).

So for instance, the acting field uniformity degree can be increased in the constructions with the paired current-carrying conductors at the expense of the parallel conductors more quantity usage over the working zone.

The inductance and the frequency characteristics of the systems with the multi-turn coils can be varied by special choice of the insulating gaps values (this is a distance to the workpieces) or by some other constructive suggestions. Probably, the different technical solutions of arising problems are possible. The undoubted statement is the following: all likely and feasible suggestions will have to be closely connected with the concrete construction intended for the execution of the rigorous determined production operation.

The application possible area of the suggested inductor systems follows from their analysis of the advantages and the demerits immediately. It is expediently to use the inductor systems with the multi-turn coils (if their possibilities allow) in the mass manufacture.

For the experimental and small manufactures a preference should be given to the constructions with the paired current-carrying conductors. Besides it can not forget the more wide range of these systems production possibilities what is caused by their high-frequency characteristics.