The magnetic flux фg that flows through the air gap in the structure of Figure 4.2, is a portion of the total flux generated by the current flowing in the coil which surrounds the core material.
This total flux ф, which flows through the coil enclosed cross sectional area, is produced by the coil current and is proportional to both the magnitude of the coil current i and the total number of turns N of wire that makes up the coil. The product of N and i is referred to as the magnetomotive force, or mmf, of the coil. The proportionality factor in the relationship between the total flux produced by the coil current and the coil mmf, has units of flux (measured in webers) per unit mmf (measured in ampere-turns).
Since the total flux ф is proportional to a force term – the mmf – a simple analogy can be made between a magnetic circuit and an electrical circuit. In a DC electrical circuit the current i flows due to an electromotive force (emf) E. Ohm’s Law for an electrical circuit states that
where R is the resistance of the circuit to the flow of current. A “Magnetic Ohm’s Law” is then
where R is the resistance of the magnetic circuit to the flow of flux. This resistance to flux flow R has been assigned a special name to differentiate it from resistance to current flow. We refer to it as the reluctance of the magnetic circuit.
The reluctance of a magnetic circuit will be approximately constant as long as the flux density in any one portion of the circuit is below the saturation flux density for that portion of the circuit. Ferromagnetic materials become saturated with magnetic flux at density levels of approximately 1-2 Teslas = 1-2 webers/(meter)2. At density levels near this saturation value, the effective reluctance of the material rises rapidly. At density levels below the saturation value, the reluctance of ferromagnetic elements is far below that of comparable sized elements constructed of non-magnetic materials.
To construct a representation of a simple lumped magnetic circuit for the magnetic circuit breaker structure of Figure 4.2, we must remember that the total flux created by the coil to be made up of two components: a gap component фg which flows through the coil, the core structure, the armature and the gap, and a leakage component фᵜ which flows through the coil, a portion of the core structure and a leakage air path. Figure 4.4a illustrates the two components flowing in their physical paths.
Figure 4.4b illustrates an electrical equivalent “magnetic circuit” for the device. Here, the coil mmf is shown as a DC voltage source of magnitude Ni, and the magnetic reluctances of the different flux paths are shown as equivalent resistances.
The reluctance portion of the core which carries both the leakage flux and the gap flux is labeled Rc; the reluctance of the air path portion of the leakage flux path is labeled Rᵜ; the reluctance of the core and armature portion of the gap flux path is labeled Rca; and the gap reluctance is termed Rg. The actual values of the reluctances Rc, Rᵜ, Rca and Rg are determined by the effective cross sectional areas of the respective flux paths, the effective lengths of the respective flux paths, and the magnetic permeabilities of the respective flux paths. If the path material is a ferromagnetic material, such as iron, the path reluctance will also be a function of the level of flux density within the path, if the density level is near or above the saturation value.
In general, for any given path, we have for the path reluctance
where Lp is the effective length of the path, Ap is the effective cross sectional area of the path, and µp is the magnetic permeability of path medium. If the path medium is ferromagnetic, at high levels of path flux, фp, flow, we have µp = µp (фp). Since the gap flux and the leakage flux are both air paths, both path permeabilities equal the permeability of free space µo – which is a strict constant, and not a function of path flux level.
For the gap we have approximately
where θg is the angular opening of the gap measured with the armature at its held position. If we define
we then have
Now from simple DC circuit analysis of the circuit of Figure 4.4b, we have
or, if we normalize the circuit reluctance as “seen” from the gap, Rca + Rᵜ Rc /(Rᵜ + Rc)to the maximum gap reluctance Rgmax, and let this ratio be termed the frame reluctance fraction m, we have
Since the core and armature are constructed with magnetic materials, we have in general,
This last result suggests that the frame reluctance fraction m is a small quantity. At high levels of core or armature saturation, however, these approximations become less accurate and the value of m will grow. Some devices are constructed with a “time-engineered” value of Rc. For the “time-engineered” devices, the core flux path reluctance Rc is high for a portion of their operating time. Thus the above approximations are also invalid.