Magnetic circuit

This article tries to introduce PMDC motor’s magnetic circuit by the most acquainted concepts. It also tries to introduce some characteristics and parameters of the motors by means of magnetic circuit.

1. Fundamental knowledge of magnet
The scope of magnet is rather wide. It is too extensive to be discussed. Magnetic field is the medium for the motor to realize energy exchange (magnetic field & current are the 2 basic conditions for the motor to work). So we will start with the magnetic field. 
First of all, let’s get to know some necessary concepts. For better understanding, we adopt narrowly defined concepts that are conditionally correct or, correct in most cases.
(1) Magnetic flux and magnetic flux density
The space near an energized conductor or a permanent magnet is full of magnetic field. Every point in the magnetic field has magnitude and argument. We usually illustrate the distribution of magnetic field by means of magnetic line of force (figure 1). Hereby we introduce two physical quantities by meaning of magnetic line of force.

Magnetic flux is expressed by the quantity of magnetic line of force. The total amount of magnetic line of force in the space near the energized conductor is called total magnetic flux. Its unit is Mx or Wb. 1 Wb=108  Mx Magnetic flux density B is the quantity of magnetic line of force in unit area. In figure 1, if Φ is magnetic line of force (magnetic flux) that goes through shadow area S, we can then get B=Φ/S. Unit: GS or T, 1T=10⁴ GS
(2) Magnetic field and magnetic field density.
Figure 2 is similar to a commonly used magnetizing experimental facility.  
Cut off a section of one homogeneous permanent ring magnet. There is the air gap with length Lδ, at its upper end. We assemble a coil with turns of W to its lower end. When we apply electricity to the coil, there is magnetic flux in the ring magnet and the air gap Lδ. We define “I*W” as Magnetomotive force (F), unit: Gb or A and we have:
1A＝0.4π Gb
We know that Ampère's circuital law can be expressed as：
F=∮H*dL＝∑(H_i*△L_i)＝H₁△L₁+H₂△L₂+•••+H_n△L_n+•••
Here H is the magnetic field density. Its physical meaning can be expressed in the following analysis. If we put back the cut magnet section in figure 2, we will see a special magnetic path with total length of L. In a homogeneous magnetic circuit H is the same everywhere, therefore
F＝∑(H_i•△L_i) ＝ H•L, H＝F/L(A/M)
Narrowly understanding, magnetic field strength H is the magnetic potential on unit length in the magnetic circuit.
(3) Remanence, coercive force, magnetization curve and magnetic permeability
As per figure 2, we put back the cut magnet forming a homogeneous magnetic circuit. When we increase the current continuously from”0”, the magnetic flux in the magnetic circuit increases continuously (figure 3, section “og”) until it reaches the knee-point g. Magnetic flux Φ increases linearly with magnetic potential F(W*I).
“o-g” is basically a straight line. After it reaches point g, the magnetic circuit is gradually saturated. The increasing speed of Φ reduces to almost zero when it reaches point a. Starting from this point, F is reduced simultaneously when we reduce the current. It doesn’t decrease along the curve “a-g-o”. It declines along “a-b”. When F=0, magnetic flux is not zero (see point b), we call the magnetic flux at point b remanence and mark it as Φ. When we change the direction of current in the coil (F is negative), the value of Φ decreases gradually until it reaches point c. At this point, remanence Φ=0, magnetic potential F at this point is marked as Fc. We call it coercive force. Similarly, the curve goes back along the curve “c→d→e→f→a”.
                           

We call figure 3 magnetization curve. It is expressed by function: Φ=f(F). The fact that Φ and F are influenced by the physical dimensions doesn’t help to understand and compare the magnetic materials. So we divide the magnetic flux Φ, the ordinate in figure 3 by S, the cross-sectional area of the magnetic circuit and get magnetic flux density B. We also divide the magnetic potential F, the abscissa in figure 3 by L, the length of the magnetic circuit and get magnetic field strength H. The magnetization curve in figure 3 is therefore turned into figure 4, magnetization curve B=f(H). Physically, to a homogeneous magnetic path, it shows the capability of unit magnetic potential generating unit magnetic flux. Such magnetizing capability of the material has nothing to do with the dimensions. It is therefore widely applied in engineering.
Usually, the remanence (B_r) and coercive force H that we see in the engineering manual come from B=f(H) curve.
We need to introduce the concept of permeability (μ) here. μ＝B/H, B=μ*H, this is a very important formula 
Because of the saturation in ferrite magnet, μ is a varied value. In vacuum (air can also be deemed roughly as vacuum), μ is a constant, marked as μ₀, μ₀ ＝4π•10^-7 (H/m)
(4) Hysteresis, hard magnetic material and soft magnetic material
In the above mentioned magnetization curve, when H=0, B=Br≠0. The variation of magnetic flux density B is slower than the variation of magnetic field strength H. We name it Hysteresis. In figure 3 and 4, the magnetization curve a→b→c→d→e→f is also called hysteresis curve.
If the hysteresis curve of the material is very wide, the Br and Hc value is relatively large, we call such material hard magnetic material. The permanent magnet belongs to such material.
If the hysteresis curve of the material is very narrow, the Br and Hc value is relatively small, we call such material soft magnetic material. Silicon steel, galvanized steel and cold rolled steel belong to such material.