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6. Appendices
Magnetic field description of SMES
As implied in the name, the stored energy can be calculated from the total magnetic field:
1 2
E = ∫ B dV ,
2
µ 0
where the field B is integrated over all space, and µ 0 is the permeability. Note that the stored
energy depends on the square of the magnetic field. Thus, the size of the storage device can be
reduced considerably by increasing the magnetic field.
Structural requirements of SMES
The various figures in this chapter of SMES devices do not show the structural characteristics of
the coils. The interaction between the current in the coil and the magnetic field produce an
outward force that must be contained by structural material. This material provides an opposing
force that is associated with its internal strain. A detailed set of calculations is required for each
coil design to provide adequate structure and to position it properly. There is, however, a
fundamental relationship between the mechanical stored energy in a system and the minimum
quantity of material required for its support. This relationship is referred to the virial theorem,
and it applies equally to magnetic energy storage, kinetic energy storage (flywheels) and
compressed air energy storage (CAES). The fundamental equation is:
∫
r r
x ⋅ F dV = −
∫
r
x ⋅ ∇ S dV ,
where x r is the distance moved, F r is the force that causes the motion, and S is the elastic strain
tensor. These can be solved to produce an expression with validity for all mechanical systems.
The straightforward expression of the formulation is given in the following relationship:
κ ⋅ m ⋅σ
m ⋅σ
E = κ ⋅σ
⋅ V = ≥ ,
ρ ρ
where κ is a number that is related to the geometry and is greater than or equal to 1, V is the
volume of structural material, σ is the working stress in the material, and ρ is the density of the
material.
The application of this relation to a SMES system depends on its size. In a solenoid, for
example, the thickness of the superconductor needed to produce a certain field is essentially
constant, independent of the size of the coil and the total energy stored. In practice, the copper
and superconductor in small coils, e.g. those storing a few megajoules, require no additional
structure. Larger coils with the same field—and thus the same thickness of superconductor—
that have diameters of 2 meters or more, almost always require additional structure for support.
SMES Page 27