Series Magnetic Circuits

Series magnetic circuits are those circuits in which the magnetic flux $\Phi$ remain same throughout the circuit.
Magnetic circuit problems, are basically of two types. In one type, $\Phi$ is given, and the impressed $mmf=NI$ must be computed. This is the type of problem encountered in the design of motors, generators, and transformers. In the other type, NI is given, and the flux $\Phi$ of the magnetic circuit must be found. This type of problem is encountered primarily in the design of magnetic amplifiers and is more difficult since the approach is "hit or miss."
As indicated in earlier discussions, the value of $\mu$ will vary from point to point along the magnetization curve. This eliminates the possibility of finding the reluctance of each "branch" or the "total reluctance" of a network, as was done for electric circuits where R had a fixed value for any applied current or voltage. If the total reluctance could be determined, $\Phi$ could then be determined using the ohm's law analogy for magnetic circuits.
For magnetic circuits, the level of B or H is determined from the each other using the B-H curve, and $\mu$ is seldom calculated unless asked for.
An approach frequently employed in the analysis of magnetic circuits is the table method. Before a problem is analyzed in detail, a table is prepared listing in the extreme left-hand column the various sections of the magnetic circuit. The columns on the right are reserved for the quantities to be found for each section. In this way, the individual doing the problem can keep track of what is required to complete the problem and also of what the next step should be. After a few examples, the usefulness of this method should become clear. In each example, the magnitude of the magnetomotive force is to be determined.
Example 1: For the series magnetic circuit of Fig. 1:
a. Find the value of $I$ required to develop a magnetic flux of $\Phi = 4 \times 10^{-4}$ Wb.
b. Determine $\mu$ and $\mu_r$ for the material under these conditions.
For example 1.
Solution: The magnetic circuit can be represented by the system shown in Fig. 2(a). The electric circuit analogy is shown in Fig. 2(b). Analogies of this type can be very helpful in the solution of magnetic circuits. Table 1 is for part (a) of this problem. The table is fairly trivial for this example, but it does define the quantities to be found.
Fig. 2: (a) Magnetic circuit equivalent and
(b) electric circuit analogy.
a. The flux density B is $$ \begin{split} B &= {\Phi \over A} \\ &= {4 x 10^{-4} Wb \over 2 \times 10^{-3} m^2}\\ &= 2 x 10^{-1} = 0.2T \end{split} $$ Using the B-H curves of given in the previous section, we can determine the magnetizing force H: $$H (cast steel) = 170 \text{At/m}$$ Applying Ampere's circuital law yields $$\begin{split} NI &= Hl \\ I &=Hl/N = {170 \times 0.16 \over 600t} \\ I &= 68mA \end{split} $$ b. The permeability of the material can be found using: $$\begin{split} B &= \mu H \\ \mu &= {B \over H} = {0.2 T \over 170 \text{At/m}}\\ \mu &= {B \over H} = {1.176 T \times 10^{-3} \text{Wb/Am}} \end{split} $$ and the relative permeability is $$\begin{split} \mu_r &= {\mu \over \mu_o} \\ &= {1.176 T \times 10^{-3} \over 4 \pi \times 10^{-7}} = 935.83\\ \end{split} $$