## Modeling Coils in the AC/DC Module

##### Walter Frei March 3, 2016

One of the most common uses of the AC/DC Module is for modeling electromagnetic coils and the interactions with their surroundings. Today, we will look into one of the key concepts to keep in mind when modeling coils: closing the current loop. If your work involves modeling coils, you will need a complete understanding of this topic.

### How to Model a Basic Coil in COMSOL Multiphysics

Let’s start with a simple example of a conductive wire that is bent into a loop and connected to a constant voltage source, a battery (shown below). Due to the voltage difference, a current will flow through the wire. The magnitude and direction of this current throughout the conductor can be computed from Ohm’s law and the Charge conservation equation, along with a set of boundary conditions.

*A very simple electromagnetic coil connected to a DC voltage source.*

For this single wire, we can consider a grounded boundary condition at one end, meaning an electric potential of zero, and a higher electric potential on the other end. Current cannot flow in or out of the wire anywhere else, so the electric insulation condition is applicable on the remaining boundaries. This problem can be solved using the finite element method as implemented in the AC/DC Module in COMSOL Multiphysics.

As a consequence of the computed current flow, a magnetic field arises and surrounds the wire. This is a vector field, having both a magnitude and direction, and can be computed from AmpĂ¨re’s law. We are interested in learning how to model this magnetic field and how it interacts with other objects.

For the purpose of learning coil modeling, we won’t concern ourselves with exactly what is happening within the source itself. Instead, we will just assume that a device exists that provides a constant voltage, or a constant current. We also won’t concern ourselves with the wires between the coil and the source, but assume that they are electrically insignificant. Based upon these two assumptions, we will say that a reasonable computational model of a coil might look like the image below, which shows a single wire with a single turn as well as the surrounding magnetic field due to the current flow.

*A computational model of a coil with one turn. The current flow (black arrows) in the wire results in a magnetic field (colored arrows) in the surrounding air region.*

There are actually a couple of other assumptions that have gone into solving the above model. First, we can see that there is a cylinder around the coil representing the air domain. This is the computational domain within which we are solving for the magnetic field. This is a finite-sized domain, yet the magnetic field will actually extend infinitely far away from the coil. The field intensity will drop off with the inverse of distance to a straight wire, so the field magnitude will be quite small far away from the coil. Although it will never be precisely zero, we can reasonably truncate our modeling domain to a finite-sized space.

By choosing a finite modeling region around the coil, we are assuming that we are only interested in the fields in this region. This choice of finite-sized region introduces another assumption at the boundaries. We need to consider some boundary conditions along the boundaries of the cylinder or whatever surrounding domain shape we choose. We will discuss the various possible boundary conditions, their physical interpretations, and why to use each one at length in a later blog post, but for this particular case, we will consider a boundary condition that represents a perfect electrical conductor.

For the physical interpretation of this, we assume that our coil sits inside of a cylindrical metal container. As a consequence of the boundary condition, the current, which is flowing along the wire from one end to the other, will flow back along the surface of the modeling domain as shown in the image below.

*The arrows show the current flowing through the wire and back along the surfaces truncating the modeling domain.*

This brings us to one of the most basic points in coil modeling: the concept of a closed current path.

### An Important Modeling Condition: Closed Current Paths

Whenever you are modeling a coil, or any magnetic field excited by a current flow, you must have a closed path (loop) of current. Your models must satisfy this condition to be valid. If you would try to impose a current flowing through an open circuit coil, that would be equivalent to implying that the current (electrons) instantaneously moves from one end of the open coil to the other, and this would violate Maxwell’s equations. The image below illustrates one such invalid coil model. Since the coil wire does not extend to the boundary, there is no continuous path along which the current can flow.

*An invalid coil model. The current-carrying wires do not reach the boundary, so there is no way for the current to loop back and no current can flow.*

### Closing the Current Path Within the Model

We can take an alternative approach to our coil model by revisiting some of our assumptions. One of the assumptions that we made in the above example is that the source is outside of our modeling domain. We can, however, alter our model in a way that the source is within our modeling space.

In the image below, one such representative model is illustrated. A closed coil is shown with a small region representing the source that feeds the coil. This source (shown in green in the image below) is entirely within the modeling domain and closes the current path. It is also assumed that the wires feeding the coil are unimportant for our modeling needs and it is assumed that a closed torus is a reasonable approximation of the coil.

*A coil model with the source inside of the modeling domain. The source is the small green section. The coil has a current return path entirely within the modeling domain. The current flow (black arrows) in the wire results in a magnetic field (colored arrows) in the surrounding air region.*

An interesting consequence of this approach is that the boundaries of the modeling domain no longer need to provide a current return path. This opens up the possibility for alternative boundary conditions on the exterior of the modeling domain, which can be used to model an insulative surrounding or to approximate an infinite space rather than an enclosing conductive container. We will address the various appropriate ways to truncate a modeling domain in depth at a later point.

### Closing the Current Path in Two-Dimensional and Axisymmetric Models

For 2D and axisymmetric models, we also need to concern ourselves with the current return path, so let’s take a quick look at the valid configurations for these cases. An axisymmetric model is appropriate to use when the situation that you want to address is invariant, or nearly so, around an axis of rotation. The image below shows one such 3D geometry, and the associated axisymmetric 2D model. All fields are assumed to be invariant around the axis of symmetry, so the formulation itself takes care of closing the current path, thus we only need to model a 2D domain in the positive *rz*-plane.

*A 3D rotationally invariant coil can be modeled in the axisymmetric 2D plane.*

If the coil is very elongated, with negligible end effects such as those seen in motors, then it is also possible to reduce the model down to a simplified 2D model with two different choices for closing the current path. It is possible to model either a cross section of the entire coil, the wires carrying current in opposite directions, or only one half of the coil with a symmetry condition at the centerline.

*An elongated 3D coil model. If the end effects are neglected, this can be modeled in a 2D plane.*

When both turns of the coil are modeled, one turn carries current perpendicularly into the 2D modeling plane, and the other turn carries the current perpendicularly out of the 2D modeling plane. When only one turn is modeled, a symmetry condition must be used at the centerline that permits current to flow back in the opposite direction. The boundary of the 2D modeling plane closes the current path. These two cases are illustrated below.

*Two different approaches to modeling an elongated coil in the 2D plane. The current flowing in the positive and negative out-of-plane directions (gray arrows) flows along either the coil or the symmetry boundary condition.*

### Concluding Thoughts on Using the AC/DC Module for Coil Modeling

We have started to look at coil modeling here by introducing the concept of the current loop. If you are modeling a coil, or indeed doing almost any modeling work with currents and the resultant magnetic fields, you will always need to keep this concept in mind. The next topic that we will look at is the question of the appropriate boundary conditions for our computational model, so stay tuned. In the meantime, have a look at these resources:

- Read more about how you can use the AC/DC Module on the COMSOL Blog
- Learn about modeling computational electromagnetics with the AC/DC Module in under three minutes. Watch the video
- If you want to use COMSOL Multiphysics and the AC/DC Module for your coil modeling, please contact us

## Comments

Francisco VallejosJuly 1, 2016 11:48 amThe instructions PDF for this model are available for download?

yp zhaoJuly 7, 2016 3:03 pmFor “Closing the Current Path Within the Model”, if I do not want to use 2-D equivalent model, how to set appropriate boundary condition or excitation to make the current flowing in the 3-D coil with a small gap?

Thanks!

yp zhaoJuly 7, 2016 4:09 pmFor the coil, if it is stranded and no eddy-current is allowed there, we can apply a source current density there. But if the coil is solid where eddy-current can appear, then it is not clear how to treat the small gap region. It is helpful to explain more about this point.

Thanks!

Michael BurnsJanuary 6, 2017 10:20 amNot a very useful article.

Bin SunMarch 11, 2017 3:39 pmgive me a good understanding about coil. Thanks

Bin SunMarch 11, 2017 3:42 pmCould you please write more blogs about cable and electromagnetic field? Super interesting and useful. Thanks