Simulation of S-parameters of general multilayer boxed PCBs with the method of moments and the scattering matrix algorithm

This paper presents a numerically stable Method of Moments tool for simulating S-parameters of multilayer boxed PCBs by combining an S-matrix formalism to derive the complete dyadic Green's function with various basis functions to model both transverse and longitudinal currents.

The Gist

Imagine you are an architect designing a skyscraper. But instead of concrete and steel, your building is made of layers of plastic and thin sheets of copper, stacked like a sandwich. This is a Printed Circuit Board (PCB), the brain of almost every electronic device.

Before you actually build this skyscraper, you want to know: Will the electricity flow smoothly from the top floor to the bottom? Will it get stuck or bounce back in weird ways?

In the real world, you'd have to build a prototype, test it, and if it fails, tear it down and start over. That's expensive and slow. So, engineers use computer simulations to "test" the design virtually. This paper presents a new, smarter way to run those simulations.

Here is the breakdown of their method using simple analogies:

1. The Problem: A Noisy, Crowded Room

Imagine trying to hear a whisper in a room filled with echoes.

• The PCB is the room. It has many layers (dielectrics) and metal sheets (conductors).
• The Signal is the whisper (electricity).
• The Challenge: When electricity moves through these layers, it bounces off the walls and the metal sheets. To predict exactly how the signal behaves, you have to calculate how every single wave interacts with every other wave.

Traditionally, calculating these "echoes" (mathematically called Green's functions) is like trying to count every single grain of sand on a beach. It takes a huge amount of computing power and time, especially when the signal source and the listener are close together. The math gets messy, unstable, and slow.

2. The Solution: The "Scattering Matrix" (The Magic Mirror)

The authors propose a new way to handle these echoes using something called the S-matrix (Scattering Matrix) method.

Think of the PCB as a series of mirrors and windows stacked on top of each other.

• Old Way: You calculate the path of a light beam by tracing every single bounce off every single surface individually. It's tedious.
• The New Way (S-matrix): Instead of tracing every bounce, you treat each layer as a "black box" with a specific rulebook.
  • If a wave hits the top of Layer A, the rulebook tells you exactly how much bounces back and how much goes through to Layer B.
  • You combine the rulebooks of Layer A, Layer B, and Layer C to get the rulebook for the whole building.

This is like playing a game of "telephone" where you don't need to know the whole story; you just need to know how each person in the chain changes the message. By using these "rulebooks" (S-matrices), the math becomes much more stable and easier to calculate, even for complex, multi-layered structures.

3. The "Rooftops" and "Pulses" (The Building Blocks)

To simulate the electricity, the computer needs to break the metal sheets and wires into tiny pieces.

• Flat Metal Sheets: The authors use shapes that look like rooftops (a flat top with sloping sides) to represent the current flowing across the flat metal layers.
• Vertical Wires (Vias): PCBs often have tiny wires that punch through the layers to connect the top to the bottom. The authors use pulse and linear shapes (like a flat block or a ramp) to represent the current flowing up and down these wires.

They figured out the exact mathematical formulas to calculate how these "rooftops" and "pulses" interact with the "echoes" (the S-matrix rules). This allows the computer to build a giant equation that predicts the behavior of the whole board.

4. The Speed Boost (The Fast Fourier Transform)

Even with the new "rulebook" method, the computer still has to do millions of calculations.

• The Analogy: Imagine you have a massive spreadsheet where every cell needs to be filled. Doing it one by one takes forever.
• The Fix: The authors use a technique called FFT (Fast Fourier Transform). Think of this as a super-fast sorting machine. Instead of checking every single cell individually, the machine groups them in a clever way to find the answer almost instantly. This makes the simulation fast enough to be practical for real-world designs.

5. The Proof: Did It Work?

The authors tested their new method on two examples:

1. A Filter: A standard electronic component with three layers of plastic and six strips of metal. They compared their computer results with known data from other studies, and the numbers matched perfectly.
2. The Filter with Wires: They added two vertical wires (vias) connecting the layers. This is a harder problem because it involves current moving up and down, not just side-to-side. Their method handled this successfully, showing how the wires changed the signal.

The Bottom Line

This paper doesn't invent a new type of circuit board. Instead, it invents a better calculator for engineers.

By using a "rulebook" approach (S-matrix) to handle the complex echoes inside the board, and by using "rooftop" shapes to map the electricity, they created a simulation tool that is:

• More Stable: It doesn't crash or give weird numbers when things get complicated.
• More Intuitive: It's easier to understand and program than previous methods.
• Faster: It uses speed-boosting tricks to solve problems quickly.

This helps engineers design better, more reliable electronic devices without having to build as many physical prototypes.

Technical Summary

Technical Summary: Simulation of S-parameters of General Multilayer Boxed PCBs with the Method of Moments and the Scattering Matrix Algorithm

Problem Statement
The paper addresses the computational challenge of modeling shielded multilayer Printed Circuit Boards (PCBs) to simulate S-parameters prior to physical manufacturing. While PCBs are characterized by thin, highly conductive elements (metallization patches, strips, and vias) embedded within a stratified dielectric host, traditional volume-discretization methods are inefficient. The Method of Moments (MoM) applied to surface and volume integral equations is the preferred approach, as it confines discretization to the conductors. However, existing 2.5D MoM formulations for shielded structures face two primary difficulties:

1. Green's Function Complexity: Deriving the full dyadic Green's function for a layered waveguide that accounts for both transverse (horizontal) and longitudinal (vertical) currents is analytically cumbersome. Many existing works focus only on transverse currents or require complex derivations for specific layer counts.
2. Numerical Stability and Convergence: The Green's function in a shielded waveguide typically involves infinite double series that converge slowly, particularly when source and observation points are near each other. Furthermore, constructing the MoM matrix requires calculating overlap integrals between these Green's function components and basis functions, a process that often lacks a unified, stable, and intuitive algorithmic framework.

Methodology
The authors propose a unified framework combining the Method of Moments with the Scattering Matrix (S-matrix) formalism to solve the Electric Field Integral Equation (EFIE) for shielded PCBs.

• Green's Function Derivation via S-Matrix: Instead of deriving explicit infinite series for the Green's function, the authors express the field components in terms of aggregated scattering matrices. The layered waveguide is treated as a sequence of homogeneous dielectric layers and interfaces.

  • Elementary Matrices: The method defines elementary S-matrices for interfaces between dielectric layers, propagation through homogeneous layers, and boundary conditions (Perfect Electric Conductor or open waveguide).
  • Aggregated Matrices: Using the Redheffer star product, the authors construct three sets of aggregated S-matrices: "Above" ($S^A$), "Below" ($S^B$), and "Pair" ($S^P$) matrices. These represent the waveguide sections above, below, and between the source and observation points, respectively.
  • Stability: The formulation ensures numerical stability by expressing the solution in terms of exponential factors with positive length constants, avoiding the numerical instability often associated with evanescent modes in direct series summation.

• Basis Functions and Overlap Integrals:

  • Planar Metallizations: Currents on horizontal strips and patches are modeled using surface rooftop basis functions (full, up-ramp, and down-ramp half-rooftops) on a regular grid.
  • Vertical Interconnects (Vias): Currents on vertical vias are modeled using volume pulse and linear basis functions.
  • Analytical Integration: The paper provides explicit analytical expressions for the overlap integrals (reaction integrals) between the derived Green's function components and these basis functions. This includes integrals for source and observation points in the same layer, different layers, and for vias crossing multiple layers.

• Matrix Construction and Scaling:

  • The MoM linear system is constructed by projecting the integral equation onto the chosen basis functions.
  • To handle the mixed physical dimensions of surface currents (A/m) and volume currents (A/m²), the authors introduce a specific rescaling of the matrix elements and source vectors. This ensures the final system matrix has units of Ohms, the voltage vector has units of Volts, and the current vector has units of Amperes.

• Acceleration:

  • The paper acknowledges that the infinite series in the Green's function converge slowly. To accelerate the matrix filling and solution, the authors employ Fast Fourier Transform (FFT) techniques. By exploiting the regular grid structure, the matrix elements are expressed as sums of block Toeplitz and Hankel matrices, allowing matrix-vector multiplication to be performed in $O(n \log n)$ operations.

Key Contributions

1. Unified S-Matrix Formalism: The paper provides a transparent and general algorithm to construct the complete dyadic Green's function for arbitrary source and observation points in a multilayer shielded waveguide. The function is expressed solely in terms of three pre-calculated sets of S-matrices ($S^A, S^B, S^P$), simplifying implementation compared to traditional transmission-line or modal expansion approaches.
2. Comprehensive Basis Function Support: The work explicitly derives the overlap integrals required to model both horizontal metallizations (using rooftop functions) and vertical interconnects (using pulse and linear volume functions) within the same S-matrix framework.
3. Numerical Stability: The derived expressions are shown to be numerically stable for all layer configurations, including those with evanescent modes, due to the specific arrangement of exponential terms.
4. Implementation and Validation: The method is implemented in C++ and validated against existing literature and a modified test case.

Results
The authors validated the method using two numerical examples:

1. Filter with Three Dielectric Layers: A structure with six metal strips in a waveguide was simulated. The calculated S-parameters ($S_{11}$ and $S_{21}$) over the 1–4 GHz range showed excellent agreement with results from a previously published study (referenced as [82]), confirming the accuracy of the Green's function and overlap integral derivations.
2. Filter with Vertical Interconnects: The first example was modified to include two vertical vias connecting different metal layers. The simulation successfully demonstrated the method's capability to model the impact of these vertical interconnects on the S-parameters, validating the integration of volume basis functions with the S-matrix Green's function.

Significance and Claims
The paper claims that while the general problem of PCB modeling is not new, the specific contribution lies in the alternative solution method characterized by:

• Transparency and Elegance: The S-matrix approach offers a more intuitive and concise way to construct the Green's function compared to existing complex derivations.
• Generality: The method is applicable to arbitrary numbers of layers and can model both horizontal and vertical currents without requiring separate, distinct theoretical frameworks for each.
• Ease of Implementation: The algorithmic nature of the S-matrix construction (recursive combination of elementary matrices) makes the approach straightforward to implement in software.
• Extensibility: While the current implementation uses rooftop and pulse/linear bases, the authors note that the derived Green's function expressions can be combined with other basis functions (e.g., RWG, sinusoidal) to model objects of various shapes.

The authors maintain a modest stance, noting that the current implementation assumes uniform current distribution along the length of vias and thin metallization layers, but the method is readily extensible to non-uniform currents and finite thickness layers. The work does not claim to solve the problem of open (unshielded) media Green's functions, focusing strictly on the shielded (boxed) waveguide environment.