Arbitrary Capacitance Matrix

Compute the capacitance matrix of N conducting spheres in an arbitrary spatial distribution using the method of mirror images.

This project is based on the numerical implementation I developed during my BSc thesis. It allows for the computation of the mutual capacitance matrix of multiple spherical conductors in a bounded region using a physically grounded and mathematically robust approach.

🚀 Installation

Install the package using pip:

pip install arbitrary-capacitance-matrix

Or for development:

git clone https://github.com/carlosperezespinar/arbitrary-capacitance-matrix.git
cd arbitrary-capacitance-matrix
pip install -e .

📖 Quick Start

import numpy as np
from capmatrix import compute_capacitance_matrix, Sphere

# Define two spheres
spheres = [
    Sphere(center=[0, 0, 0], radius=1e-9),      # 1 nm radius at origin
    Sphere(center=[3e-9, 0, 0], radius=1e-9)    # 1 nm radius, 3 nm away
]

# Compute capacitance matrix
C = compute_capacitance_matrix(spheres)
print("Capacitance matrix (F):")
print(C)

🔬 Background & Theory

The method of mirror images is a classic electrostatics technique to enforce boundary conditions by introducing fictitious "image charges." For a system of N conducting spheres, we recursively place image charges inside each sphere to satisfy the equipotential condition, then assemble the resulting potentials into the capacitance matrix C:

C_ij = 4 * π * ε₀ * R_i * Q_ij

where Q_ij is the total induced charge on sphere i when sphere j is held at unit potential and all others are grounded.

This approach handles:

• Arbitrary number (N) of spheres
• Random positions & radii
• No assumed symmetries or simplifying approximations

🖼 Images & Diagrams

1. Basic image-charge construction

Iterative placement of image charges inside each sphere to enforce constant potential.

2. Generalized vector formulation

Vector-based formula for computing the coordinates and strength of each new image charge from known charge positions.

🔍 Examples

Two Spheres

from capmatrix import compute_capacitance_matrix, Sphere

# Two identical spheres
radius = 1e-9  # 1 nm
separation = 3e-9  # 3 nm center-to-center

spheres = [
    Sphere([0, 0, 0], radius),
    Sphere([separation, 0, 0], radius)
]

C = compute_capacitance_matrix(spheres)

Three Spheres in a Line

# Three spheres with different radii
spheres = [
    Sphere([-6e-9, 0, 0], 2e-9),   # Left sphere
    Sphere([0, 0, 0], 1.5e-9),     # Center sphere  
    Sphere([6e-9, 0, 0], 1e-9)     # Right sphere
]

C = compute_capacitance_matrix(spheres, tolerance=1e-10, max_iterations=100)

See the examples/ folder for complete working examples:

• examples/two_spheres.py - Two sphere benchmark with analytical comparison
• examples/three_spheres.py - Three sphere configuration
• demo.py - Simple demonstration script

📋 API Reference

Classes

Sphere(center, radius)

Represents a conducting sphere.

• center: 3D coordinates [x, y, z]
• radius: Sphere radius (must be positive)

Charge(value, radial_distance, coordinates, sphere_index, iteration=0)

Represents a point charge in the system.

Functions

compute_capacitance_matrix(spheres, tolerance=1e-8, max_iterations=50)

Compute the capacitance matrix for a list of spheres.

Parameters:

• spheres: List of Sphere objects
• tolerance: Convergence tolerance (default: 1e-8)
• max_iterations: Maximum iterations (default: 50)

Returns: NxN numpy array of capacitances in Farads

🧪 Testing

Run the test suite:

pytest tests/

📚 References

1. Wasshuber, C. (1997). About Single-Electron Devices and Circuits. PhD Thesis, Technische Universität Wien. Online PDF – Chapter on method of images
2. Pérez Espinar, C. BSc Thesis (2019), Universitat de Barcelona

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.