Hierarchical Electromagnetics
EmCAD – Cloud-native EM Simulator for Circuit-Level Modeling
Model Order Reduction
Projection on Nodal Voltage Subspace
The core of EmCAD’s reduction technique is a projection of the circuit matrices onto a reduced subspace of nodal voltages. This transformation produces a much smaller system that preserves the electromagnetic behavior of the original discretized model.
When the subspace is built using nodal solutions excited from all external ports at a given frequency, the reduced circuit response matches the original system exactly at that frequency — providing frequency-wise interpolation.
Early Version: Superposition of Frequency Bases and Orthonormalization via Resonant Modes
Initially, the reduced basis was assembled by combining local bases computed at multiple sample frequencies. This approach provided exact interpolation at each frequency, but often resulted in redundancies among the basis vectors.
To eliminate redundancy and improve numerical stability, the final basis was obtained by solving an eigenvalue problem that extracts the natural resonant modes of the structure. These are computed under the assumption that all external ports are terminated with open-circuit conditions, which isolates the intrinsic dynamics of the passive domain.
These modes form an orthonormal set that serves both as a compact representation and as a physically meaningful basis for projection. Their use guarantees accurate modeling of internal resonances without interference from port terminations.
Final Evolution: Direct Resonance-Based Basis
Using the resonant modes as a basis leads to (near-)diagonal internal matrices in the reduced circuit, which significantly simplifies simulation and reduces computational cost.
Having identified resonance as the key to model efficiency, a direct strategy was introduced: skip the frequency sampling phase entirely and compute the resonance modes of the original circuit upfront. This streamlined the workflow while preserving high accuracy.
Correction of High-Frequency Mode Truncation
Truncating the basis inevitably discards some high-frequency resonant modes. While negligible in spectral terms, their cumulative effect may cause a slight offset between the reduced and full model responses.
Since these modes behave inductively at operating frequencies and are effectively in series, their net effect can be modeled using a series-inductive network. In the simplest case of a single-port system, this reduces to a single series inductance. However, for multiport structures, the correction takes the form of a full symmetric matrix of mutual inductances between ports.
This correction network is extracted from the original model by evaluating the system response at a carefully chosen frequency far from major resonances, and it is added in series to the reduced model to restore accuracy.