Computing the Inductance of a Straight Wire
A question that comes up occasionally is whether or not you can compute the inductance of a single straight wire. This seemingly simple question actually does not really have an answer, and gives us the opportunity to talk about a very interesting point when solving Maxwell’s equations. Anybody working in the field of computational electromagnetics should have an understanding of this key concept, as it will help you properly set up and interpret models involving magnetic fields.
What is the Inductance of a Single Straight Wire?
Before we get to this concept, let’s look at the question again: What is the inductance of a single straight wire? To begin to address this, we consider the energy-based definition of inductance:
The inductance, L, is defined in terms of the current, I, flowing through the system, and Wm, the magnetic energy density is proportional to B2. The magnetic field, B, exists within and outside of the wire. Within the wire, it increases linearly outwards from the centerline, and outside of the wire it falls off as 1/r, where r is the radius. So, to compute the magnetic field and the magnetic energy distributions we build a model based on Maxwell’s equations, in this case Ampère’s Law. A current source is applied to one end of the wire while the other end is grounded. The tangential magnetic flux density is constrained to zero on the exterior boundaries of the model.
We can build this model in COMSOL Multiphysics using the Magnetic and Electric Fields physics interface of the AC/DC Module, which will solve for the electric fields and currents in the wire, as well as the magnetic fields in the wire and surrounding air. The Terminal boundary condition will excite the structure, and automatically compute the inductance using the above formula. The Ground boundary condition at the other end of the wire provides a current sink, and the Magnetic Insulation boundary condition sets the tangential magnetic flux density to zero.
Now, the fields extend infinitely around the wire, but the magnetic energy density falls off as 1/r2, so we might think that it is sufficient to study increasing radii of our modeling domain, and that the inductance will converge. If we try this, however, the inductance as a function of radius will look like this:
As the radius increases, the inductance also increases. No matter how large we make the modeling domain, the inductance for this model of a straight wire does not converge!
Not Only Solving for the Inductance
Things are looking rather grim if we can’t even compute this simple case correctly, right? Well, actually, we aren’t just solving for the inductance of the wire in this model. You are always solving for the inductance of the system, and that has to include the current return path. The governing Maxwell’s equations are formulated under the assumption that current (electrons) can be neither created nor destroyed. In fact, the Magnetic Insulation boundary conditions provide this current return path. We can see this by plotting the current in the volumes as well as the surface currents:
The red arrows in the figure above show that the current is flowing along the wire, and also flowing along the Magnetic Insulation boundary conditions as a surface current. The Magnetic Insulation boundary condition can be thought of as representing a material with infinite conductivity along which current can flow unimpeded. The current path in the above figure is solenoidal, meaning the current is flowing in a closed path. There is a conductive path not only from the terminal to ground boundary conditions, through the wire, but also from the ground boundary condition back to the terminals, along the boundaries.
This concept of a solenoidal path for the current is the key point here. Whenever you are setting up any kind of magnetic fields model, you must ensure that current can flow in a closed loop. You don’t have a choice: this is required by Maxwell’s equations. You also have to be aware that this current return path will affect the results. In the above model, we are not only computing the inductance due to the current flowing along the wire, we are also considering the effect of the current flowing back along the magnetic insulation boundary conditions.
When setting up any kind of magnetic fields model, you have to model a complete solenoidal current path, and the inductance that you compute from such a model is of the entire current loop.
So that is why you can’t compute the inductance of a single straight wire!
Of course, as long as the system you are analyzing is solenoidal, with current flowing in a loop, you can correctly compute the inductance.
To see examples that compute the mutual inductance between circular loops of wire and compares against analytic solutions, please see:
- COMSOL Now
- Today in Science