Inductance Matrices
At the simplest level, inductance is proportional to the energy stored in the magnetic field when current flows:
where Um is the energy stored in the magnetic field, L is the inductance, and I is the current flowing in the circuit. Q3D Extractor uses the eddy current simulator to compute the inductance between two lines by simulating the magnetic field that arises when a current is applied and then computing the energy stored in the field.
The solver uses the following equation to solve for inductance in terms of Um:
To compute the elements in the inductance matrix for an n conductor transmission line, Q3D Extractor performs a sequence of field simulations. In each field simulation, 1 amp is allowed to flow in a single conductor, and no current is allowed to flow in any other conductor.
The energy stored in the magnetic field associated with the inductance between two conductors is given by the following relationship:
where:
- Uij is the energy stored in the magnetic field linking conductor i with conductor j.
- Bi is the magnetic flux density produced when one amp is allowed to flow through conductor i and no current flows in the other conductors.
- Hi is the magnetic field produced when one amp is allowed to flow through conductor j and no current flows in the other conductors.
The inductance coupling conductors i and j is therefore:
