This article continues the discussion of how to use 2D and 3D numerical codes to find inductance. The first article covered the use of the energy method for simple systems with a single circuit. An example showed the application of static-field codes to estimate inductances of current-carrying rods in both the low and high-frequency limits. This article extends the discussion to multi-circuit systems like transformers. The emphasis is on theory. A third article will show how to set up and to interpret a practical three-dimensional simulation with Magnum.
For a complete description, we need to find mutual inductances between circuits as well as self-inductances of each circuit. Here, we shall use the term circuit to denote coils of continuous wire, usually consisting of a large number of parallel turns. We shall assume that current density is distributed almost uniformly over the cross section of multi-turn coils, without inquiring into details of the distribution within the wires (i.e., the wire self-inductance is small compared to the coil inductance).
To address the complexity of mutual inductances, we must have a clear definition for terms. In the following discussion, I have adopted material and notation from D.K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, 1989), Sects. 6.11 and 6.12. Suppose we have two circuits and Circuit 1 has drive current of I1 A. The quantity Λ12 is the total magnetic flux enclosed by the turns of Circuit 2. If Circuit 2 has N2 identical turns and φ12 is the flux enclosed by one turn, then Λ12 = N2*Φ12. The mutual inductance between Circuits 1 and 2 is defined as:
L12 = Λ12/I1.
The quantity L12 has units of Henries. According to Faraday's law, the voltage induced in Circuit 2 by a changing current in Circuit 1 is
V12 = L12*(dI1/dt).
In a three-dimensional magnetic-field calculation, it would be laborious to attempt to integrate flux over every turn of a circuit to evaluate mutual inductance. The procedure would also be quite inaccurate if coils are represented by filamentary current elements as in Magnum. As is often the case in numerical methods, the solutions lies in seeking an indirect method that relies on a volume integral. As in the case of self-inductance, we can use the field energy method. It is based on two facts that are proved in the Cheng reference and most texts on introductory electromagnetism. First, mutual inductances have the following symmetry property:
Lnm = Lmn.
The second relationship is that the total field energy of a system of N circuits may be written as a sum of products of mutual inductances and drive currents:
U = (1/2) ∑ ∑ Lmn*Im*In,
where the sums are taken over m and n. To illustrate the method, consider a two-circuit system like a transformer. We need three quantities for a complete characterization of the system inductances: L11, L22 and L12 (which equals L21). The procedure is to excite the circuits with unit drive currents and to determine the associated inductance from magnetic field energy integrals (performed automatically in MagView and PerMag). We can find the self-inductance of Circuit 1 by setting I1 = 1.0 A and I2 = 0.0 A. Designating the resulting field energy as U10, the self-inductance of Circuit 1 is:
L11 = 2*U10
Similarly, we run a second solution with I1 = 0.0 A and I2 = 1.0 A to determine the field energy U01 such that L22 = 2*U01. Finally, we find the mutual inductance M12 = L12 = L21 from a third calculation with I1 = 1.0 A and I2 = 1.0 A. Using the energy-sum equation, the mutual inductance is related to the known self-inductances and calculated field energy U11 by:
M12 = (1/2)*(2*U11 - L11 = L22)
The price to pay to find the three quantities is that we must run three Magnum solutions.
The method can be generalized to any number of circuits, although the amount of work increases rapidly. As an example, there are six independent quantities in a three circuit system: L11, L22, L33, L12, L13 and L23. Six Magnum runs are required to find six values of magnetic field energy. If we denote the exciting currents as a vector [I1,I2,I3]. then there is a base set of six vectors: [1,0,0], [0,1,0], [0,0,1], [1,1,0], [1,0,1] and [0,1,1]. First, we determine the three self-inductances. Then, we must solve three coupled linear equations to find M12, M13 and M23.
The next article discusses a Magnum calculation for a simple toroidal transformer that we can compare to analytic results. Emphasis will be on setup techniques and numerical accuracy. Use this link if you'd like more information about Magnum: http://www.fieldp.com/magnum.html
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