#### What is in this article?:

- Designing Coupled Inductors
- Equating Reluctance and Circuit Models
- Uncoupled vs. Coupled Design

In the January issue of *Power Electronics Technology*, the operation and benefits of the coupled inductor multiphase (CIMP) topology were reviewed, and the equations for the output and phase-ripple currents were derived using a circuit model representation of the coupled inductor. In the circuit model, the coupled inductor is represented by a leakage inductance in each phase (L_{K}), a magnetizing inductance (L_{M}) and an ideal transformer with a 1-to-1 turns ratio (**Fig. 1**).

It was determined that the principle benefit of the CIMP topology was a significant reduction in phase-ripple current for a given transient response condition when compared to the uncoupled version of this topology. It was shown that this ripple reduction was dependant on the duty cycle (D), which is V_{OUT}/V_{IN}, and the coupling ratio, which is p = L_{M}/L_{K}. This reduction allows for improved efficiency or, conversely, one can improve the transient response using the CIMP topology and still maintain the same efficiency as in the uncoupled variety. It is clear that the CIMP topology relies on the design and implementation of the coupled inductor component. By leveraging the previous analysis, we can determine the relevant equations required to design and optimize the coupled inductor component.

### The Coupled Inductor

Any inductor or transformer with multiple windings is a coupled inductor. Therefore, hundreds of implementations exist. However, in the vast majority of these implementations, with the notable exception of the common-mode choke, the driving source is only applied to the primary — or main — winding, and the other windings are simply “dumb” followers mimicking the behavior of the first.

Although, in theory, these devices could be used in a CIMP application, they would not work well. The trick is to determine how to design and optimize a part that has the required magnetizing and leakage inductances, appropriate saturation and thermal characteristics in a package size that is suitable to the application. To do this, it is necessary to first identify some possible core structures and their corresponding reluctance models, and then use the reluctance model in conjunction with the coupled inductor circuit model to realize a complete solution.

Although there is an infinite number of possible core structures for a coupled inductor, the toroid and E-core designs are two that immediately come to mind as shown in **Figs. 2a** and **2b**. A reluctance model is the magnetic equivalent of an electric circuit model in which a magnetic force (analogous to voltage) drives a magnetic field or flux (analogous to current) through a reluctance path (analogous to resistance), and as such any electrical theorems or rules apply similarly.

The easiest way to determine a reluctance model for a coupled inductor is to remove one of the windings and then envision the various paths that magnetic flux could take to complete a closed loop back to the driving force (magnetic force from the remaining winding is equal to the number of turns times the driving current). Each flux path has an equivalent reluctance that is equal to the length of the flux path (l_{E}) divided by the cross-sectional area of the path (A_{E}) and the permeability of the material (µ), or R = l_{E} / (A_{E} × µ). The lower the reluctance for a given path, the greater the magnetic flux through that path will be.

Although it is possible to develop a model in which every flux path is identified, it is easier and just as accurate to identify the most likely paths and use these in the reluctance model. It is then possible to refine the model by adding the less likely paths back in, if desired. After identifying the most likely flux paths, the other winding is re-inserted into the reluctance model and the model is complete.

As can be seen in **Fig. 2a**, most of the flux Φ_{1} (red) created by the magnetic force, N × I1, travels through the high-permeability core (low reluctance), although there is some leakage flux that takes the shorter but higher reluctance path through the air. Although the flux leaks everywhere, all of these reluctance paths are in parallel and can be equated with a single path through the center of the core. The same effect happens with the flux Φ2 (yellow) from the N × I2 magnetic source. In **Fig. 2b**, intentional air gaps are inserted into the high-permeability core material to help direct the flux. Although there is flux that travels outside the core geometry, it is comparatively small and can be ignored.

As shown in **Fig. 2c**, although the structures in **Figs. 2a** and **2b** are physically different, the reluctance model for each is the same. It can be assumed that the construction of the inductor is symmetrical and, therefore, the reluctances R_{A} and R_{B} are equal and can be replaced with R. For each of the identified structures, the values of reluctances R and R_{C} will vary depending on the placement of the windings, the core material used, the size of any inserted air gaps and the physical size of the core. The trick now is to determine how reluctances R and R_{C} from the reluctance model relate to the magnetizing inductance and the series leakage inductances L_{M} and L_{K} of the circuit model.