# Amperes Circuital Law

Amperes circuital law states that the algebraic sum of the rises and drops of the mmf around a closed loop of a magnetic circuit is equal to zero; that is, the sum of the rises in mmf equals the sum of the drops in mmf around a closed loop.
As mentioned in the introduction to this chapter, there is a broad similarity between the analyses of electric and magnetic circuits. This has already been demonstrated to some extent for the quantities in Table 1. If we apply the "cause" analogy to Kirchhoff's voltage law ( $\sum V=0$), we obtain the following: $$\sum mmf = 0 \tag{1}$$ Equation (1) is referred to as Ampere's circuital law. When it is applied to magnetic circuits, sources of mmf are expressed by the equation $$mmf = NI \tag{2}$$ The equation for the mmf drop across a portion of a magnetic circuit can be found by applying the relationships $$mmf = \Phi R \tag{3}$$ where $\Phi$ is the flux passing through a section of the magnetic circuit and $R$ is the reluctance of that section. The reluctance, however, is seldom calculated in the analysis of magnetic circuits. A more practical equation for the mmf drop is $$mmf = Hl \tag{4}$$ where $H$ is the magnetizing force on a section of a magnetic circuit and $l$ is the length of the section. As an example of Eq. (1), consider the magnetic circuit appearing in Fig. 1 constructed of three different ferromagnetic materials. Applying Ampere's circuital law, we have $$\sum mmf = 0$$ $$+NI - H_{ab} l_{ab} - H_{bc} l_{bc} - H_{ca} l_{ca} = 0$$ $$NI = H_{ab} l_{ab} - H_{bc} l_{bc} - H_{ca} l_{ca}$$
Fig. 1: Series magnetic circuit of three different materials.