---
id: model-02
title: "FEMM Extraction for Lumped Models"
section: "Advanced Modeling"
difficulty: "advanced"
estimated_time: 45
prerequisites: ["model-01", "phys-08"]
objectives:
  - Master FEMM setup for two-body electrostatic problems (topload + spark)
  - Extract and interpret Maxwell capacitance matrices
  - Apply correct sign conventions for mutual and shunt capacitances
  - Validate extracted capacitances against empirical rules
tags: ["FEMM", "electrostatics", "capacitance-matrix", "extraction", "validation"]
---

# FEMM Extraction for Lumped Models

This lesson covers the detailed procedure for using FEMM (Finite Element Method Magnetics) to extract capacitances for lumped spark models. We'll focus on the two-body problem: topload and spark channel.

## The Maxwell Capacitance Matrix

### Mathematical Definition

FEMM outputs the **Maxwell capacitance matrix** [C] which relates charges to voltages:

```
[Q] = [C] × [V]

Where:
Q_i = charge on conductor i (coulombs)
V_i = potential of conductor i (volts)
[C] = capacitance matrix (farads)
```

### Matrix Properties

The Maxwell matrix has specific mathematical properties:

**1. Symmetry:**
```
C_ij = C_ji

Physical basis: Maxwell's equations are symmetric
Numerical check: |C_ij - C_ji| / |C_ij| < 0.01
```

**2. Diagonal elements positive:**
```
C_ii > 0    (self-capacitance)

Physical meaning: Charge required to raise conductor i to 1V
```

**3. Off-diagonal elements negative:**
```
C_ij < 0    for i ≠ j

IMPORTANT: This is the Maxwell convention!

Physical meaning: Negative charge induced on conductor j
when conductor i is at +1V (field lines terminate)
```

**4. Row sum equals zero:**
```
Σ_j C_ij = 0    for each row i

Conservation: Total charge to ground = 0 when far-field grounded
```

### Two-Body System

For topload (conductor 1) and spark (conductor 2), with ground implicit:

```
       [1]   [2]
[1]  [ C₁₁   C₁₂ ]
[2]  [ C₂₁   C₂₂ ]

Example values:
       [Top] [Spark]
[Top]  [ 30    -8   ] pF
[Spark][ -8    14   ] pF
```

**Interpretation:**

- **C₁₁ = 30 pF:** Topload self-capacitance (to infinity/ground at ∞)
- **C₂₂ = 14 pF:** Spark self-capacitance (to infinity)
- **C₁₂ = C₂₁ = -8 pF:** Mutual capacitance (negative per convention)

**Note:** These are NOT the circuit elements we need directly. Extraction required!

## FEMM Setup for Lumped Model

### Problem Type and Geometry

**Problem configuration:**
```
Type: Electrostatic, axisymmetric
Coordinates: r-z (cylindrical)
Frequency: 0 Hz (pure electrostatic)
Precision: 1e-8 (default)
```

**Geometry components:**

**1. Topload (Conductor 1):**
```
Typical: Toroid
- Major diameter: 20-50 cm
- Minor diameter: 5-15 cm
- Or sphere: radius 10-25 cm

Position: Origin at center
Material: Perfect conductor (grouped as Conductor 1)
```

**2. Spark channel (Conductor 2):**
```
Shape: Vertical cylinder
- Length: Target spark length (e.g., 1.5 m)
- Diameter: 1-3 mm (typical plasma channel)
- Position: Base at topload bottom, extending downward

Material: Perfect conductor (grouped as Conductor 2)

Note: Small gap (0.1 mm) between topload and spark base
      This is numerical convenience; results insensitive
```

**3. Ground plane:**
```
Position: Below spark tip
Distance: 10-20 cm below tip (or room floor distance)
Extent: Large radius (3-5× max dimension)

Boundary: V = 0 (Dirichlet condition)
```

**4. Outer boundary:**
```
Shape: Large cylindrical volume
Radius: 3-5× maximum geometry dimension
Height: Extends above and below structure

Boundary condition: V = 0 (or mixed, grounded at ∞)
```

**5. Medium:**
```
All regions: Air
ε_r = 1 (vacuum permittivity)
```

### Step-by-Step FEMM Procedure

**Step 1: Create geometry**

```
1. Draw toroid in r-z plane (right half only, axisymmetric)
   - Use arc and line segments
   - Close contour

2. Draw spark cylinder
   - Rectangle in r-z coordinates
   - r: [0, radius], z: [z_base, z_tip]

3. Draw ground plane
   - Horizontal line at z = z_ground
   - r: [0, r_max]

4. Draw outer boundary box
   - Enclose all geometry
   - Large enough to avoid boundary effects
```

**Step 2: Define materials**
```
Create air material block:
- Name: "Air"
- Relative permittivity: ε_r = 1
- Apply to all regions
```

**Step 3: Define conductors**

```
Property → Conductors → Add Conductors:

Conductor 1 (Topload):
- Select all topload surface nodes/segments
- Group: "1"
- Voltage: 1V (test voltage)

Conductor 2 (Spark):
- Select all spark cylinder surfaces
- Group: "2"
- Voltage: <In Group> (floating potential)

Ground plane:
- Boundary condition: V = 0 (not a separate conductor)
```

**Step 4: Mesh generation**

```
Mesh → Create Mesh

Automatic meshing with refinement near conductors:
- Typical element size: 1-5 mm near spark
- 10-50 mm in far field
- Total elements: 5,000-20,000 for lumped model

Check mesh quality visually (no overly elongated triangles)
```

**Step 5: Solve**

```
Analysis → Solve

Solver runs (typically <10 seconds for lumped model)

Check for convergence:
- Should converge in <100 iterations
- Final residual < 1e-8
- No warnings about poor mesh quality
```

**Step 6: Extract capacitance matrix**

```
View → Circuit Props

Output shows:
- Conductor properties (V, Q for each)
- Capacitance matrix [C]

Copy matrix values to spreadsheet or script
```

### Example FEMM Output

**Conductor properties:**
```
Conductor 1 (Topload):
  Voltage: 1.0000 V (fixed)
  Charge: 3.52e-11 C = 35.2 pC

Conductor 2 (Spark):
  Voltage: 0.2982 V (computed, floating)
  Charge: 1.68e-11 C = 16.8 pC
```

**Capacitance matrix [C]:**
```
       [1]      [2]
[1]  [ 35.2    -10.5  ] pF
[2]  [-10.5     16.8  ] pF
```

**Verify properties:**
```
✓ Symmetric: C₁₂ = C₂₁ = -10.5 pF
✓ Diagonal positive: C₁₁, C₂₂ > 0
✓ Off-diagonal negative: C₁₂, C₂₁ < 0
✓ Row sum: 35.2 + (-10.5) = 24.7 ≈ 0? NO - ground implicit!

Row sum ≠ 0 is OK: ground is not in matrix (infinite conductor)
```

## Extracting Circuit Elements

### Formula Derivation

**Goal:** Extract C_mut and C_sh for this circuit:

```
Topload ---[C_mut]--- Spark ---[C_sh]--- Ground
```

**C_mut (Mutual Capacitance):**

Mutual capacitance is the capacitance *between* topload and spark.

```
C_mut = |C₁₂| = |C₂₁|

Take absolute value of off-diagonal element

Why absolute?
- Circuit element capacitances are positive
- Maxwell convention uses negative for mutual coupling
- |C₁₂| converts to standard circuit convention
```

**Example:**
```
C₁₂ = -10.5 pF
C_mut = |-10.5| = 10.5 pF ✓
```

**C_sh (Shunt Capacitance to Ground):**

Shunt capacitance is spark-to-ground with topload present.

**Method 1: From row sum**

The charge on spark (row 2) with V₁=V_topload, V₂=V_spark is:
```
Q₂ = C₂₁ × V₁ + C₂₂ × V₂

Charge to ground = -(Q₂) assuming no other charges
But this includes charge from topload coupling!

Actual spark-to-ground capacitance:
C_sh = C₂₂ + C₂₁
     = C₂₂ - |C₁₂|    (since C₂₁ = C₁₂ < 0)
```

**Derivation:**
```
Consider: Topload grounded (V₁ = 0), spark at V₂ = 1V

Charge on spark: Q₂ = C₂₁ × 0 + C₂₂ × 1 = C₂₂
But part of this is coupled to topload!

Spark-to-actual-ground capacitance:
Total capacitance to ∞ = C₂₂
Minus coupling through topload = -C₂₁ = |C₁₂|
Net shunt: C_sh = C₂₂ - |C₁₂|
```

**Example:**
```
C₂₂ = 16.8 pF
C₁₂ = -10.5 pF
C_sh = 16.8 - 10.5 = 6.3 pF ✓
```

**Method 2: Direct measurement** (verification)

Run second FEMM simulation:
```
- Topload: V = 0 (grounded)
- Spark: V = 1V
- Ground: V = 0

Measure charge on spark → this is C_sh directly

Should match Method 1 result
```

### Sign Convention Summary

**CRITICAL: Understand the sign conventions!**

```
Maxwell Matrix:
  C_ij < 0 for i ≠ j (negative mutual elements)

Circuit Elements:
  All capacitances > 0 (positive values)

Conversion:
  C_mut = |C₁₂|           (absolute value)
  C_sh = C₂₂ - |C₁₂|      (subtract absolute value)
```

**Common error:** Using C₁₂ directly as C_mut without absolute value
**Result:** Negative capacitance in SPICE → error or nonsensical results

## Validation Checks

### 1. Matrix Symmetry

```
Check: |C₁₂ - C₂₁| / |C₁₂| < 0.01

If not symmetric:
- Mesh too coarse → refine near conductors
- Convergence issue → lower tolerance
- Geometry problem → check closed contours
```

### 2. Physical Value Ranges

**C_mut (Mutual):**
```
Expected: 5-20 pF for typical Tesla coil toploads

Too high (>30 pF): Check geometry (topload too large?)
Too low (<2 pF): Check geometry (spark too short/far?)
```

**C_sh (Shunt):**
```
Empirical rule: C_sh ≈ 2 pF/foot × L_spark

Example: L = 1.8 m = 5.9 ft
Expected: C_sh ≈ 2 × 5.9 = 11.8 pF

Acceptable range: 0.5× to 2.5× empirical prediction
```

**Why deviations occur:**
```
Higher than expected:
- Nearby ground objects (walls, floor close)
- Wide spark base (cone shape)
- Ground plane too close in simulation

Lower than expected:
- Elevated spark (no ground plane modeled)
- Thin diameter (<1 mm)
- Topload shielding effect strong
- Empirical rule may include mutual capacitance
```

**Important note for distributed models:**
When using distributed models (Part 4, Lesson 4), the total C_sh from summing all segments may differ from the 2 pF/foot rule by a larger factor. This is because:
- Matrix extraction method sums individual contributions
- Mutual couplings between segments affect total
- Distributed geometry changes field distribution
- Factor of 2-3 deviation is normal and acceptable
- Use FEMM value (more accurate for specific geometry)

### 3. Energy Conservation Check

```
Total energy stored should be conserved:

W = 0.5 × V^T × C × V

For V = [1, V₂]:
W = 0.5 × (C₁₁ + 2×C₁₂×V₂ + C₂₂×V₂²)

Check: Should be positive, finite
```

### 4. Ground Distance Sensitivity

**Test:** Vary ground plane distance, check C_sh

```
Ground at z = -2.0 m: C_sh = 6.8 pF
Ground at z = -3.0 m: C_sh = 6.2 pF
Ground at z = -5.0 m: C_sh = 6.0 pF

Expect: C_sh decreases as ground moves away
Convergence: <5% change when distance > 2× spark length
```

If C_sh changes significantly (>20%) with ground distance:
- Ground plane too close
- Move ground further away
- Or accept measured geometry (e.g., actual room)

## Worked Example: Complete Extraction

**Given:**
- Spark length: 1.8 m = 5.9 feet
- FEMM simulation output (see above)
- Operating frequency: 200 kHz

**FEMM capacitance matrix:**
```
       [1]      [2]
[1]  [ 35.2    -10.5  ] pF
[2]  [-10.5     16.8  ] pF
```

**Step 1: Extract C_mut**
```
C_mut = |C₁₂| = |-10.5| = 10.5 pF ✓
```

**Step 2: Extract C_sh**
```
C_sh = C₂₂ + C₂₁
     = C₂₂ - |C₁₂|
     = 16.8 - 10.5
     = 6.3 pF ✓
```

**Step 3: Validate C_sh**
```
Empirical prediction:
C_sh_predicted = 2 pF/ft × 5.9 ft = 11.8 pF

FEMM result:
C_sh_FEMM = 6.3 pF

Ratio: 6.3 / 11.8 = 0.53

This is LOWER than expected by factor ~2
```

**Analysis of discrepancy:**

**Possible explanations:**
```
1. Empirical rule assumes straight vertical spark
   - If spark is angled or curved: less capacitance
   - FEMM models idealized vertical cylinder

2. Empirical rule from community measurements
   - May include some C_mut in "measured" value
   - Difficult to separate mutual from shunt experimentally
   - Pure C_sh might be lower

3. Ground plane distance matters
   - FEMM: specific ground geometry (15 cm below tip)
   - Empirical rule: "typical" room (floor 1-2 m away)
   - Closer ground in measurements → higher C_sh

4. Diameter assumption
   - Thinner diameter → lower C_sh (logarithmic dependence)
   - C ∝ 1/ln(h/d), so d = 1 mm vs 3 mm changes C by ~30%
```

**Decision: Use FEMM value**
```
For modeling: Use C_sh = 6.3 pF (FEMM result)
Reason: More accurate for specific geometry
Empirical rule: Rough check only

Within factor of 2-3: Acceptable agreement
```

**Step 4: Calculate total capacitance**
```
C_total = C_mut + C_sh
        = 10.5 + 6.3
        = 16.8 pF
```

**Step 5: Calculate R_opt**
```
f = 200 kHz
ω = 2π × 200×10³ = 1.257×10⁶ rad/s

R_opt = 1/(ω × C_total)
      = 1/(1.257×10⁶ × 16.8×10⁻¹²)
      = 47.3 kΩ ✓

Within physical bounds (5-500 kΩ)
```

**Step 6: Build circuit**
```
SPICE netlist:
C_mut topload spark_n 10.5p
R_spark spark_n spark_r 47.3k
C_sh spark_r 0 6.3p
```

Ready for simulation!

## Common FEMM Errors and Troubleshooting

### Problem: Matrix not symmetric

**Symptoms:**
```
|C₁₂ - C₂₁| / |C₁₂| > 0.05
```

**Causes and fixes:**
```
1. Mesh too coarse
   → Refine mesh near conductors
   → Increase total element count

2. Poor convergence
   → Lower precision requirement (1e-9 or 1e-10)
   → Check mesh quality

3. Geometry errors
   → Verify all contours closed
   → Check no overlapping regions
```

### Problem: Negative C_sh

**Symptoms:**
```
C_sh = C₂₂ - |C₁₂| < 0
```

**Causes:**
```
This should NEVER happen physically!

1. Wrong extraction formula used
   → Double-check: C_sh = C₂₂ - |C₁₂|, not C₂₂ + C₁₂

2. FEMM simulation error
   → Check conductor assignments
   → Verify boundary conditions
   → Remake geometry from scratch

3. Conductors not properly grouped
   → Each conductor must be single contiguous group
```

### Problem: C_sh >> empirical rule (factor >5)

**Symptoms:**
```
C_sh = 50 pF for 1 m spark (expected: 6 pF)
```

**Causes:**
```
1. Ground plane too close
   → Move ground plane further away
   → Check z-coordinate

2. Spark diameter too large
   → Should be 1-3 mm, not 1-3 cm!
   → Check units

3. Multiple ground connections
   → Check only one ground boundary condition
```

### Problem: C_mut unreasonably large

**Symptoms:**
```
C_mut > 50 pF for medium toroid
```

**Causes:**
```
1. Topload size too large
   → Check diameter in correct units

2. Spark embedded in topload
   → Should have small gap (0.1-1 mm)

3. Scale error
   → Check all dimensions (cm? m? mm?)
```

## Best Practices

**1. Consistent units:**
```
Recommended: Centimeters throughout FEMM
- Easy to work with Tesla coil scales
- Avoid mixing mm/cm/m
- Output still in standard SI units
```

**2. Mesh refinement:**
```
Start coarse → check matrix → refine if needed

Adequate: Symmetry <1% error
Overkill: >50,000 elements for lumped model (slow, no benefit)
```

**3. Parametric studies:**
```
Vary one parameter at a time:
- Spark length: C_sh should scale linearly
- Ground distance: C_sh should saturate at large distance
- Diameter: C_sh logarithmic dependence (weak)

Check trends make physical sense
```

**4. Documentation:**
```
Save for each simulation:
- Geometry parameters (toroid size, spark length, ground position)
- Mesh statistics (elements, convergence)
- Raw matrix output
- Extracted C_mut, C_sh
- Validation checks

Build database for future reference
```

## Key Takeaways

- **Maxwell matrix** uses negative off-diagonals for mutual capacitance (standard convention)
- **Extraction formulas:** C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂| (absolute value critical!)
- **FEMM setup:** Axisymmetric, two conductors (topload at 1V, spark floating), ground boundary
- **Validation:** Check symmetry, C_sh ≈ 2 pF/ft ± factor 2, physical value ranges
- **Discrepancies:** FEMM more accurate than empirical rules for specific geometry
- **Common errors:** Wrong sign conversion, mesh too coarse, units mismatch, ground too close
- **Use FEMM values** in circuit model, not empirical estimates

## Practice

{exercise:model-ex-02}

---
**Next Lesson:** [Distributed Model Theory](03-distributed-model.md)