How to estimate the Q of spiral inductors using a planar EM simulator, Part 1 of 2

John M. Dunn and Vladimir Veremey of AWR Corp explains how to estimate the quality factor (Q) of spiral inductors using a planar EM simulator.

Electromagnetic (EM) simulators are commonly used to model spiral inductors in various planar technologies: for example, silicon and gallium arsenide chips, as well as the module and board level. The most common figure of merit for an inductor is its quality factor, or as it is commonly called, the Q.  Unfortunately, accurately getting the Q for an inductor is a difficult task for an EM simulator.  Planar simulators have a difficult time correctly calculating the resistive loss of the metal.  They rely on impedance boundary conditions, the accuracy of which is suspect at the corners of the conductors.  A second problem is that the mesh on the surface of the conductor must be extremely fine, leading to an impractically large mesh.  3D EM simulators have similar problems.  The interior of the metal must be excessively meshed to model the resistive loss accurately.  Second, most finite element simulators, the most popular 3D simulator, iteratively refine the mesh until the problem converges.  Unfortunately, unless care is taken, the simulator can mistakenly determine it has converged; yet, the calculated value of the Q still has significant error.

The method we demonstrate here has a number of distinct advantages.  First, a planar simulator can be used with normal mesh densities, resulting in a more realistic simulation time.  The only other required simulation tool is a cross-sectional transmission line solver that can accurately give the electrical, transmission line parameters for coupled lines.  Such tools are available at very low cost, and are extremely fast to run.  Second, because of the speed of the simulation, the designer can quickly adjust design parameters to get the desired response.  Tuning and optimization studies become realistic.

The technique works by taking advantage of the fact that most spiral inductors can be viewed as straight sections of coupled lines tied together at the ends.  Of course this is somewhat technology and geometry dependent. The current in spirals mostly flows in

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