⚡ Advanced Circuit Simulator

A Rigorous, High-Precision DC Circuit Analysis Tool using Modified Nodal Analysis (MNA)

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📌 Table of Contents

1. Overview
2. Key Features
3. Mathematical Foundations
4. Numerical Methods
5. Workflow
6. Example Analysis
7. Verification & Validation
8. Advanced Extensions
9. Future Roadmap

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📖 Overview

The Advanced Circuit Simulator is designed to perform accurate, stable, and verifiable DC circuit analysis using Modified Nodal Analysis (MNA). It’s not just a black-box solver — it’s an educational and professional tool that lets you:

• See the full equation system your circuit produces
• Understand matrix assembly from physical components
• Verify your results against KCL, KVL, and power conservation
• Extend to AC, nonlinear, and transient analyses

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🚀 Key Features

• 🔍 Accurate MNA Implementation for DC steady-state analysis
• 📐 Union-Find Node Detection for correct topology mapping
• ⚙️ LU Decomposition with Partial Pivoting for stable solving
• 📊 Post-Processing: per-component currents & power
• ✅ Automatic Verification of physical conservation laws
• 📈 Residual Norm & Condition Number reporting
• 🔄 Ground Node Reduction for efficiency
• 🛠 Educational Mode showing intermediate matrices

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📚 Mathematical Foundations

Problem Setup Given a circuit with:

• $n$ non-ground nodes
• $m$ voltage sources

The MNA formulation is:

$$ \begin{bmatrix} G & B \ B^T & 0 \end{bmatrix} \begin{bmatrix} V \ I \end{bmatrix}

\begin{bmatrix} I_s \ V_s \end{bmatrix} $$

Where:

• $G$: Conductance matrix (Ohm’s law contributions)
• $B$: Voltage source incidence matrix
• $I_s$: Current injection vector
• $V_s$: Voltage source vector

Dimension: $(n+m) \times (n+m)$

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🔢 Numerical Methods

The simulator is numerically robust thanks to careful preprocessing and solving strategies.

1. Ground Node Reduction

We select one node as reference (ground) and remove its row and column from the MNA system.

• Why?

  • Ground voltage is fixed ($V_g = 0$) → no need to solve for it
  • Reduces size from $(n+m)$ to $(n+m-1)$
  • Improves conditioning by removing a trivial equation

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2. LU Decomposition with Partial Pivoting

We solve:

$$ A_{red} \cdot x_{red} = z_{red} $$

Using the factorization:

$$ P \cdot A_{red} = L \cdot U $$

• $L$: Lower triangular (1’s on diagonal)
• $U$: Upper triangular
• $P$: Row permutation matrix (from pivoting)

Partial Pivoting:

• Finds the largest pivot element in the current column
• Swaps rows to place it on the diagonal
• Prevents division by small numbers
• Reduces round-off error growth

Algorithm (Doolittle with Pivoting):

1. For column $k$:

   • $p = \arg\max_{i \ge k} |A[i,k]|$

   • Swap rows $k$ and $p$ in $A$, update $P$

   • For rows $i > k$:

     $$ L[i,k] = \frac{A[i,k]}{A[k,k]} $$

     Update:

     $$ A[i,j] = A[i,j] - L[i,k] \cdot A[k,j] $$

2. After factorization:

   • Forward substitution: $L \cdot y = P \cdot z_{red}$
   • Backward substitution: $U \cdot x_{red} = y$

Complexity:

• $O((n+m)^3)$ worst-case for dense systems
• Reduced by ground removal

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3. Residual & Accuracy Checks

After solving:

$$ r = |A_{red} x_{red} - z_{red}|_2 $$

• Goal: $r \leq 10^{-10}$
• Large residual → possible ill-conditioning or singularity

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4. Condition Number Analysis

The simulator computes:

$$ \kappa(A) = |A| \cdot |A^{-1}| $$

• High $\kappa(A)$ → sensitive to rounding & component tolerances
• Helps detect unstable circuits before trusting results

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🛠 Workflow

graph TD;
    A[Component & Wire Input] --> B[Node Identification (Union-Find)]
    B --> C[MNA Matrix Assembly (G, B, I_s, V_s)]
    C --> D[Ground Node Reduction]
    D --> E[LU Decomposition & Solve]
    E --> F[Post-Processing (Currents & Powers)]
    F --> G[Verification (KCL, KVL, Power)]

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💡 Example Analysis — Voltage Divider

Circuit:

• Vs = 12 V
• R1 = 1 kΩ
• R2 = 2 kΩ

Results:

Quantity	Value
V₂	8 V
I	4 mA
P_R1	16 mW
P_R2	32 mW
P_Vs	-48 mW

✅ KCL/KVL & Power Conservation verified.

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✅ Verification & Validation

• KCL: Currents at every node sum to ≈ 0
• KVL: Loop voltages sum to ≈ 0
• Power: Total supplied = total consumed
• Residual Norm: Reports how close numerical solution is to exact
• Condition Number: Alerts on stability risk

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🔬 Advanced Extensions

• 🌀 AC Phasor Analysis
• 📉 Nonlinear Components with Newton-Raphson
• 🎯 Sensitivity & Monte Carlo parameter analysis
• ⚡ Sparse Matrix Solver for large-scale systems

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📜 License

MIT License — free to use, modify, and distribute.

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