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id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
femm-workflow	FEMM Electrostatic Workflow for Spark Capacitance Extraction	established	spark-physics.txt: Part 7.2 (lines 458-477), Part 8.2 (lines 559-572), Part 6 (lines 389-438)	[lumped-model distributed-model circuit-topology capacitive-divider field-thresholds equations-and-bounds open-questions]	[C_mut = \|C_12\| from Maxwell matrix C_sh = C_22 - \|C_12\| Partial capacitance transformation E_tip from FEMM field solution C_sh validation: 2 pF/foot rule]	[FEMM Finite Element Method Maxwell capacitance matrix partial capacitance electrostatic simulation self-capacitance mutual capacitance mesh refinement axisymmetric Dirichlet boundary]	[femm-geometry-setup-lumped.png femm-geometry-setup-distributed.png field-lines-capacitances.png femm-field-plot-example.png electric-field-enhancement.png maxwell-matrix-extraction.png partial-capacitance-transformation.png capacitance-matrix-heatmap.png]	[femm-lumped-extraction.md]	[How does mesh quality near the spark tip affect E_tip computation accuracy, and what is the optimal element size at the tip? Can FEMM axisymmetric simulations capture non-vertical spark geometries, or is 3D FEA required for curved/angled sparks? What is the systematic error of modeling the spark as a perfect cylinder versus a tapered or irregular channel? How should multiple breakout points be handled in a single FEMM simulation?]

FEMM Electrostatic Workflow for Spark Capacitance Extraction

FEMM (Finite Element Method Magnetics) is an open-source finite element analysis program that, despite its name, handles electrostatic problems with high accuracy. It is the primary tool for extracting the capacitance values that populate the lumped-model and distributed-model. This document covers the complete FEMM workflow: geometry setup, meshing, solution, matrix extraction, sign convention handling, field computation, and validation. FEMM also provides the electric field at the spark tip, which is needed for growth prediction in the field-thresholds and energy-and-growth analyses.

1. FEMM Fundamentals

1.1 What FEMM Computes

For electrostatic problems, FEMM solves Laplace's equation (nabla squared V = 0) in air with boundary conditions defined by conductor potentials and far-field grounding. From the solution, FEMM computes:

Potential field V(r, z) everywhere in the domain
Electric field E = -grad(V) at any point
Charge on each conductor Q_i = integral of epsilon_0 * E_n dA over the conductor surface
Capacitance matrix C[i,j] relating charges to voltages: Q_i = sum_j C[i,j] * V_j

1.2 Problem Type and Symmetry

Problem type: Electrostatic (DC). Although Tesla coils operate at 50-400 kHz, the wavelength (750-6000 m) is vastly larger than the spark geometry (1-5 m), so the quasi-static approximation is excellent. Capacitance values extracted at DC are accurate at operating frequency.

Symmetry: Use axisymmetric (r-z) geometry whenever possible. A vertical spark emerging from a centered toroidal topload has cylindrical symmetry, reducing the 3D problem to 2D. This reduces computation time by orders of magnitude and improves accuracy for a given mesh density.

When 3D is needed: Horizontal sparks, sparks from off-center breakout points, or sparks near asymmetric grounded objects cannot exploit axisymmetry and require full 3D FEA (not available in FEMM; use other tools like Elmer or COMSOL).

1.3 Coordinate System

In FEMM's axisymmetric mode:

r-axis: Radial distance from the axis of symmetry (horizontal)
z-axis: Vertical position (typically z = 0 at ground plane)
All geometry is drawn in the r >= 0 half-plane
FEMM automatically revolves it about the z-axis

2. Geometry Setup

2.1 Topload

Model the topload as a toroid in the r-z plane. A toroid of major diameter D_major and minor diameter D_minor centered at height h is represented by its cross-sectional circle:

Circle center: (r_center, z_center)
r_center = D_major/2 - D_minor/2
z_center = h
Circle radius = D_minor/2

Draw the right half of this circle (r >= 0) using arc segments. Close the contour along the axis if needed. Assign the topload surface to Conductor Group 1.

2.2 Spark Channel

For the lumped model (single cylinder): Model the entire spark as one vertical cylinder extending downward from the bottom of the topload. Key parameters:

Diameter: 1 mm for burst mode analysis, 3 mm for QCW analysis (nominal values; the weak logarithmic dependence of capacitance on diameter makes the exact choice non-critical)
Length: The target spark length L
Gap: Leave a 0.1-0.5 mm gap between the topload surface and the top of the spark cylinder for numerical stability

In the r-z plane, the cylinder is a thin rectangle from (0, z_base) to (r_spark, z_tip), where r_spark = d/2 and z_tip = z_base - L. Assign the spark surface to Conductor Group 2.

For the distributed model (n segments): Divide the cylinder into n equal sections, each of length L_seg = L/n. Leave 0.1 mm gaps between segments. Assign each segment to its own Conductor Group (2 through n+1). See distributed-model for the segment numbering convention.

2.3 Ground Plane

Model the ground plane as a horizontal line from (0, 0) to (R_boundary, 0), where R_boundary is the outer boundary radius. Assign Dirichlet boundary condition V = 0 to this line.

2.4 Outer Boundary

Create a rectangular boundary enclosing all geometry:

Radial extent: R_boundary = 3 to 10 times the maximum dimension (topload diameter or spark length, whichever is larger)
Vertical extent: From well below the spark tip to well above the topload

Assign V = 0 (Dirichlet) or a mixed boundary condition to the outer boundary. The boundary must be far enough that C_sh changes by less than 5% when the boundary is moved 50% farther.

2.5 Material Properties

Assign the material "Air" with relative permittivity epsilon_r = 1.0 to all regions outside the conductors. The conductors themselves are equipotential surfaces (boundary conditions, not material regions).

2.6 Mesh Control

Critical near the spark channel: The thin spark cylinder (1-3 mm diameter) requires fine mesh elements. Set the mesh element size near the spark to be no larger than the spark diameter. For a 2 mm spark, use 2 mm maximum element size.

Near the topload: Element size of 5-10 mm is typically sufficient.

Far field: Coarse mesh is acceptable (50-100 mm elements). The far field contributes little to the capacitance between nearby conductors.

Mesh quality check after generation:

No extremely elongated triangles (aspect ratio below 10:1)
Fine mesh near conductors with smooth transition to coarse mesh
Total element count: typically 15,000-50,000 for lumped models, 30,000-100,000 for distributed models

3. Solution and Matrix Extraction

3.1 Running the Solver

FEMM solves the Laplace equation iteratively. Check:

Convergence: Final residual below 1e-8
Iteration count: Typically 3-10 iterations for well-conditioned problems
No warnings about poor mesh quality

3.2 Visual Verification

Before extracting numbers, visually inspect the solution:

Topload should be at the specified potential (uniform color on surface)
Spark should be at a lower, uniform potential (floating conductor acquires a potential determined by coupling)
Ground plane should be at V = 0
Field lines should emerge from the topload, with some terminating on the spark and others reaching ground
No anomalous hot spots or discontinuities

3.3 Extracting the Maxwell Capacitance Matrix

FEMM Circuit Properties dialog provides:

Voltage of each conductor (specified or computed)
Charge on each conductor
Capacitance matrix elements

For the lumped model (2x2 matrix):

       [Topload]  [Spark]
[Top]  [ C_11      C_12  ]
[Spk]  [ C_21      C_22  ]

For the distributed model ((n+1) x (n+1) matrix):

       [Top]  [Seg1] [Seg2] ... [Segn]
[Top]  [C_00  C_01   C_02  ... C_0n ]
[Seg1] [C_10  C_11   C_12  ... C_1n ]
[Seg2] [C_20  C_21   C_22  ... C_2n ]
...
[Segn] [C_n0  C_n1   C_n2  ... C_nn ]

3.4 Sign Convention: Maxwell vs. Circuit

Maxwell capacitance matrix convention:

C_ii > 0: Self-capacitance (total charge on conductor i when i is at 1V and all others are grounded)
C_ij < 0 for i != j: Mutual coupling (charge induced on conductor i when j is at 1V and all others are grounded). The negative sign reflects that positive voltage on j induces negative charge on i.

Circuit element convention:

All capacitances are positive values

Conversion for the lumped model:

C_mut = |C_12|                (take absolute value of negative off-diagonal)
C_sh = C_22 - |C_12|         (total self-cap minus mutual coupling)

Conversion for the distributed model: Use the partial capacitance transformation (see Section 4 below).

Warning: Mixing conventions is the most common source of errors in this workflow. Always write out the absolute value signs explicitly and verify that all circuit element capacitances are positive.

4. Partial Capacitance Transformation

4.1 Purpose

The Maxwell matrix contains negative off-diagonal elements that cannot be directly implemented as capacitors in SPICE. The partial capacitance transformation produces an equivalent network with only positive elements.

4.2 Transformation Formulas

From the Maxwell matrix C_maxwell, compute:

Capacitance between node i and node j (for i != j):

C_branch[i,j] = -C_maxwell[i,j] = |C_maxwell[i,j]|

This is a positive capacitor connected between nodes i and j.

Capacitance from node i to ground:

C_ground[i] = C_maxwell[i,i] + sum_{j != i} C_maxwell[i,j]
            = C_maxwell[i,i] - sum_{j != i} |C_maxwell[i,j]|

This is the residual capacitance to the implicit ground node. It should be non-negative for a valid matrix. If it is slightly negative (numerical noise), it indicates that the conductor is almost entirely coupled to other conductors with negligible direct coupling to ground.

4.3 Network Implementation

The resulting circuit has:

One capacitor C_branch[i,j] between each pair of nodes (i,j)
One capacitor C_ground[i] from each node to ground
All values are positive

For n+1 conductors, this produces up to n*(n+1)/2 branch capacitors plus n+1 ground capacitors. For n = 10 (11 conductors), this is up to 55 branch capacitors plus 11 ground capacitors. In practice, many branch capacitors are negligibly small and can be omitted.

5. Electric Field Computation

5.1 Tip Field for Growth Prediction

FEMM computes the electric field magnitude at any point in the domain. For spark growth analysis (see field-thresholds and energy-and-growth), the critical quantity is the electric field at the spark tip.

Procedure:

Set topload to the peak operating voltage V_top
Include the spark at its current length L
Solve the electrostatic problem
Query E_tip = |E| at the tip of the spark cylinder

The tip field includes geometric enhancement:

E_tip = kappa * E_average

where kappa = 2 to 5 is the field enhancement factor due to the small radius of curvature at the spark tip. FEMM automatically captures this enhancement if the mesh is sufficiently fine near the tip.

5.2 Calibration of E_propagation

The field threshold for sustained spark propagation is determined by combining FEMM with experimental observation:

Run the coil with known drive conditions
Measure the final (stalled) spark length L_stall
From SPICE simulation, determine V_top at the time of stall
In FEMM, set up the topload at V_top with a spark of length L_stall
Compute E_tip at the stall point
This E_tip equals E_propagation for this coil/environment

Typical result: E_propagation = 0.4 to 1.0 MV/m, depending on altitude, humidity, and channel condition.

5.3 Accuracy Considerations

FEMM field accuracy near the tip: The field at a sharp geometric feature (like the end of a thin cylinder) is the hardest quantity to compute accurately with FEA. The field diverges as the radius of curvature approaches zero. In practice:

Round the tip of the spark cylinder with a hemispherical cap of radius equal to half the spark diameter
Refine the mesh to at least 5 elements across the hemisphere
Report E_tip averaged over the hemisphere surface, not at a single point

Overall capacitance accuracy: FEMM capacitance extraction is typically accurate to +/-10% for well-meshed problems. With careful mesh refinement and boundary testing, +/-5% is achievable.

6. Practical Workflow Summary

6.1 Lumped Model Extraction

1. Create FEMM geometry: topload + single spark cylinder + ground plane
2. Set topload to V = 1V (test voltage)
3. Set spark as floating conductor
4. Mesh and solve
5. Extract 2x2 Maxwell capacitance matrix
6. Compute: C_mut = |C_12|, C_sh = C_22 - |C_12|
7. Validate: C_sh within factor 2-3 of (2 pF/foot * L)
8. Calculate: R = 1/(omega * (C_mut + C_sh)), clip to [1 kilohm, 100 megohm]
9. Build SPICE netlist

6.2 Distributed Model Extraction

1. Create FEMM geometry: topload + n spark segments + ground plane
2. Set topload to V = 1V (test voltage)
3. Set all segments as floating conductors
4. Mesh and solve
5. Extract (n+1)x(n+1) Maxwell capacitance matrix
6. Verify: symmetry, positive diagonals, negative off-diagonals
7. Transform to partial capacitances for SPICE implementation
8. Compute per-segment: C_total[i] = sum |C[i,j]| for j != i
9. Calculate: R[i] = 1/(omega * C_total[i]), clip to position-dependent bounds
10. Build SPICE network with partial capacitances and resistances

6.3 Parametric Studies

FEMM simulations are fast enough (seconds to minutes) to enable parametric sweeps:

Spark length variation: Run for L = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 m to build lookup tables of C_mut(L), C_sh(L)
Topload size variation: Compare small, medium, and large toploads to understand the effect on C_mut/C_sh ratio (and hence on the phase constraint from circuit-topology)
Ground plane distance: Vary the ground plane height to assess environmental sensitivity of C_sh
Spark diameter: Verify the weak logarithmic dependence of capacitance on diameter

7. Common Mistakes and Troubleshooting

7.1 Errors in Setup

Mistake	Symptom	Fix
Spark touching topload (no gap)	Matrix extraction fails or gives anomalous values	Insert 0.1-0.5 mm gap
Boundary too close	C_sh varies >5% when boundary moved	Increase to 5-10x max dimension
Mesh too coarse near spark	Poor convergence or C values change with refinement	Refine mesh to spark diameter
Wrong conductor assignment	Off-diagonal elements have wrong sign or magnitude	Verify conductor groups

7.2 Errors in Extraction

Mistake	Symptom	Fix
C_sh = C_22 + C_12 (wrong formula)	Conceptual error; coincidentally gives correct result	Always use C_sh = C_22 - abs(C_12)
Forgetting absolute value	Negative C_mut (impossible)	Take abs() of all off-diagonal elements
Units mismatch	R_opt off by orders of magnitude	FEMM uses cm internally; convert to SI for formulas
Non-symmetric matrix	Indicates poor convergence or bug	Re-mesh, refine, check boundary conditions

7.3 Validation Failures

Issue	Likely cause	Action
C_sh > 5 * (2 pF/ft * L)	Boundary too close; extra grounded objects	Move boundary; check geometry
C_sh < 0.2 * (2 pF/ft * L)	Spark shielded by topload; ground too far	Physical; not necessarily wrong
C_mut < 1 pF	Spark too far from topload; gap too large	Check gap size and topload model
Negative C_ground[i] after partial transform	Numerical noise in matrix	Add +0.1 pF to diagonal

8. Connection to Other Topics

Key Relationships

Serves: lumped-model -- FEMM provides the 2x2 matrix from which C_mut and C_sh are extracted for the lumped model.
Serves: distributed-model -- FEMM provides the (n+1)x(n+1) matrix that defines the entire capacitive network of the distributed model.
Computes: field-thresholds -- FEMM computes E_tip for a given V_top and L, enabling growth prediction and E_propagation calibration.
Informs: capacitive-divider -- The voltage distribution along the spark (visible in FEMM's potential plot) directly shows the capacitive divider effect.
Depends on: circuit-topology -- The physical topology (C_mut || R in series with C_sh) motivates what quantities to extract from FEMM.
Validates against: equations-and-bounds -- All extracted capacitances and derived resistances must fall within documented physical ranges.

Worked Example

The femm-lumped-extraction.md worked example demonstrates the complete workflow for a 30 cm x 8 cm toroid with a 1.8 m spark at 200 kHz, including mesh convergence testing, boundary sensitivity analysis, and parametric studies.

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