because magnetic permeability also exists outside materials (see vacuum permeability). Thus, there may be significant "leakage flux" in the space outside the magnetic cores, which must be taken into account but is often difficult to calculate. Most importantly, magnetic circuits are nonlinear; the reluctance in a magnetic circuit is not constant, as resistance is, but varies depending on the magnetic field. At high magnetic fluxes the ferromagnetic materials used for the cores of magnetic circuits saturate, limiting further increase of the magnetic flux through, so above this level the reluctance increases rapidly. In addition, ferromagnetic materials suffer from hysteresis so the flux in them depends not just on the instantaneous MMF but also on the history of MMF. After the source of the magnetic flux is turned off, remanent magnetism is left in ferromagnetic materials, creating flux with no MMF. Circuit laws Magnetic circuits obey other laws that are similar to electrical circuit laws. For example, the total reluctance of reluctances in series is: This also follows from Ampère's law and is analogous to Kirchhoff's voltage law for adding resistances in series. Also, the sum of magnetic fluxes into any node is always zero: This follows from Gauss's law and is analogous to Kirchhoff's current law for analyzing electrical circuits. Together, the three laws above form a complete system for analysing magnetic circuits, in a manner similar to electric circuits. Comparing the two types of circuits shows that: The equivalent to resistance R is the reluctance The equivalent to current I is the magnetic flux Φ The equivalent to voltage V is the magnetomotive Force F Magnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of Kirchhoff's Voltage Law (KVL) for pure source/resistance circuits. Specifically, whereas KVL states that the voltage excitation applied to a loop is equal to the sum of the voltage drops (resistance times current) around the loop, the magnetic analogue states that the magnetomotive force (achieved from ampere-turn excitation) is equal to the sum of MMF drops (product of flux and reluctance) across the rest of the loop. (If there are multiple loops, the current in each branch can be solved through a matrix equation—much as a matrix solution for mesh circuit branch currents is obtained in loop analysis—after which the individual branch currents are obtained by adding and/or subtracting the constituent loop currents as indicated by the