A number of procedures are available for the analysis of electromagnetic forming. At the most comprehensive end of the scale, fully three-dimensional models of electromechanical coil-workpiece-die interactions exist that require significant amounts of time to run on large computers. While models of this sort would be required in the careful design of advanced components and forming operations, these models lend little practical insight. Simple models are still the best for understanding how trends in coil and/or bank design or workpiece configuration may affect system performance. In the following a brief quantitative description of electromagnetic forming is provided. The equations provided are sufficient to allow ballpark design of EM forming systems. Also, the equations provided are all easily manipulated and solved. The primary limitation of this system of equations is that they assume that the workpiece does not move in the time of the calculation and static pressures under this assumption are calculated and average uniform pressures are calculated. That is, both spatial and temporal response are simplified. Iterative models that consider motion, changes in circuit properties and further changes in pressure are required to go beyond this level. This requires numerical solutions and with this an intuitive feel of what the equations offer is lost.

The primary electromagnetic forming circuit is a simple coupled capacitor, inductor, resistor circuit which is commonly known as an LRC circuit. The capacitors are charged to an initial voltage Vo, this corresponds to a stored energy of , where C is the capacitance of the bank. When the fast-action switches (usually ignitrons) are closed to complete the circuit it responds with a current vs. time behavior of a classic damped sine wave. If resistance is sufficiently low (damping is minimal), the peak current is:

where L is the inductance of the circuit (which will be discussed later). In actuality, resistance will reduce the value of the peak current by some amount (typically to about 80% of that shown above). Also the ringing frequency of a simple L-C circuit can be expressed as:

where w is expressed in radians/second. The pressures produced in electromagnetic forming are produced by the electrical pulse of very high current being run through a coil that is placed in close proximity to a conductive workpiece. The current in the coil produces a transient magnetic field about it. The change in field induces currents in the workpiece. The currents in the workpiece and coil are in opposite directions and as a result they repel each other. Another way of analyzing this phenomenon simply, is by realizing that if the current frequency is sufficiently high, the field is excluded from the workpiece except for within the skin depth, d, which can be calculated as:

where µo is the magnetic permeability of free space and s is the conductivity of the material. If the current frequency is sufficiently high, d is much less than the thickness of the sheet and the field is effectively excluded from the workpiece. If this condition is not met, the field leaks through and the magnetic pressure is reduced (and more advanced analysis methods are needed). Thus, this technique works well only when conductive materials are used with high frequency pulses. As a result, systems require low resistance, inductance, and low capacitance is also desired (which reduces stored energy at a fixed voltage). For this reason charging voltages are typically fairly high (above 5kV).

If the field is effectively excluded from the workpiece then the 'magnetic pressure', Pm that acts on the workpiece can be calculated as:

where H is the electromagnetic field intensity. H is expressed in Amp•turns/length. Thus it very simply follows from the geometry of the actuator (whether planar or cylindrical) and the current, I. It is relatively straightforward to calculate flux densities for simple wound solenoids. Generally it is proportional to the number of turns per length and the current that runs through it. There is an engineering trade-off as, if the number of turns increases, the inductance of the system rises, reducing ringing frequency. This reduces the peak current and will allow more flux to leak through the workpiece[5].

To look at this a little more closely, one needs to consider the behavior of the LRC circuit in a little more depth. The resistance R, of the circuit is composed of resistances from both the bank and buswork itself, Rbank, and that of the coil, Rcoil. At the frequencies of interest the current travels on the surfaces of conductors so for both the components in the bank and the coil the resistance of the conducting path in the condition where the skin depth is less than the thickness of the conductor (per unit length) is:

Here, P is the perimeter length of the conductor (e.g., the circumference of the wire that makes up the solenoid). And of course, Rcircuit=Rbank+Rcoil. One thing that this demonstrates, is that for materials with lower intrinsic conductivity the skin depth is greater. The resistance of an actuator made of a low intrinsic conductivity material can be made smaller by using thick conductors (i.e. both P and d are large). Thus, relatively short, but massive actuators made of materials such as carbon steel may be practical.

Simple methods also exist that may be used to estimate the inductance of a solenoid or a coil. On a first order the inductance of a current carrying body is related to the volume it fills with magnetic flux. Thus when a workpiece excludes flux from a volume, it lowers the inductance of the coil that produces the flux. For a simple N-turn solenoid that has an internal area Ao and an amount Ai of this is excluded by enclosing a conductive workpiece, the coil inductance can be estimated as:

where l is the length of the coil. Note that as the workpiece collapses (reducing Ai in this case) the inductance of the coil decreases (but only modestly -- by a factor of 1/2 at most). Again the capacitors themselves and buswork have inductance themselves and this must be factored into the total circuit inductance. These terms all add in a simple linear way yielding in the simplest case Lcircuit=Lbank+Lcoil.

The response of LRC circuits is treated well in most texts on circuits. The characteristic equation can be written as:

Where the natural frequency w is defined previously and zeta is the damping term defined as:

Where R and L can be calculated as outlined above. If z>1 the system is termed to be overdamped and all of the energy stored in the capacitor is essentially dissipated after the first half-cycle. If z<1 the system is underdamped and is well approximated by a exponentially-damped sine function. For electromagnetic forming systems values between about 0.7 and 0.3 are desirable as this keeps peak currents high giving forming efficiency and provides damping reduces ring-back that decreases capacitor life. The solution to the characteristic equation is given by:

where

Through these procedures one can fairly readily make predictions of the maximum force that an electromagnetic forming system can exert on a workpiece. The calculations for inductance and resistance for a single rather simple element such as a coil are relatively simple and straightforward. It is difficult to analytically predict the system inductance and capacitance for a capacitor bank with its buswork and necessary electrical connections. This must be determined experimentally. Possibly the simplest way of doing this is by simply charging the bank to a low voltage, shorting the bank and measuring the resulting current vs. time trace. The equations already presented can then be used to solve for Lbank and Rbank. These values are additive with those of the coil. Also, the coil capacitance and resistance can be measured experimentally as a function of frequency with frequency response analysis equipment.

In the case where the system is underdamped, the voltage time behavior can be approximated as:

where the exponential decay term, a, can be expressed as:

Using these equations with experimentally measured voltage-time traces gives a relatively simple method of determining the resistance and capacitance of discharge circuits (with or without coils to find system or bank values). Gregg Fenton carried this out for the Ohio State capacitor bank with one of eight capacitors in use and with his coil the system resistance and inductance were found to be 63.9 mW and 1.91 µH, respectively. Without a coil in place these values dropped to 46 Wm and 99.2 nH.

The procedures outlined here are very valuable in rough design of systems (and in fact, the author estimates that over 90% of the forming systems operating have been designed with analysis no more complicated than this). However, one must bear in mind that once workpieces begin to move or if spatial force distributions are required, the situation becomes significantly more complicated. Inductance is now a function of time and detailed simulations require iterative numerical simulations. Examples of these more robust calculations have been offered elsewhere [20,25,26,39].

28. F. A. Nichols, Acta Metall., 28, 663 (1980).

29. P.S. Follansbee, U.F. Kocks, Acta Metall., v. 36, pp. 81, 1988.