plasma-expert/context/distributed-model.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
distributed-model	nth-Order Distributed Spark Model	established	spark-physics.txt: Part 8 (lines 540-664), Part 10.2 (lines 715-733), Part 11 (lines 736-803)	[lumped-model femm-workflow circuit-topology power-optimization thevenin-method coupled-resonance capacitive-divider field-thresholds energy-and-growth streamers-and-leaders equations-and-bounds open-questions]	[Tapered resistance initialization Position-dependent resistance bounds Circuit-determined R per segment Damped iterative update rule Self-consistency diameter check]	[distributed model segment partial capacitance matrix Maxwell capacitance matrix resistance taper convergence damping factor passivity nearest-neighbor approximation controlled sources]	[resistance-taper-initialization.png power-distribution-along-spark.png current-attenuation-plot.png lumped-vs-distributed-comparison.png position-dependent-bounds.png validation-total-resistance.png capacitance-matrix-heatmap.png femm-geometry-setup-distributed.png partial-capacitance-transformation.png iterative-optimization-convergence.png spice-implementation-methods.png]	[distributed-model-complete.md]	[What is the optimal number of segments for a given spark length -- is there a principled criterion beyond 'diminishing returns at n=10'? Can the resistance distribution be used to infer the leader-to-streamer transition point, and if so, what is the R threshold? How should branching be handled -- does each branch get its own distributed model, and how is power divided at branch points? Is the nearest-neighbor approximation sufficient for tightly-spaced segments, or does it break down when segment length approaches the channel diameter? How does the resistance distribution evolve in time during QCW ramp-up?]

nth-Order Distributed Spark Model

The distributed model divides the spark channel into n segments (typically n = 10), each with its own mutual capacitances, shunt capacitance, and resistance. This generalization of the lumped-model captures the spatial variation of current, power, and impedance along the spark length. The model reveals that base segments naturally optimize to low resistance (hot leader plasma) while tip segments settle at high resistance (cold streamer plasma), providing a circuit-level explanation for the observed leader-streamer structure of Tesla coil sparks.

1. Model Structure

1.1 Segmentation

The spark channel of total length L is divided into n equal segments. Each segment has length L_seg = L/n. Segments are numbered from i = 1 (base, connected to topload) to i = n (tip, farthest from topload). Together with the topload (conductor 0), there are n + 1 conductors in the FEMM model.

Topload (conductor 0)
   |
[C_01][R_1][C_1,gnd]     Segment 1 (base)
   |
[C_12][R_2][C_2,gnd]     Segment 2
   |
  ...
   |
[C_{n-1,n}][R_n][C_n,gnd]  Segment n (tip)

Each segment possesses:

• Mutual capacitances to every other conductor (topload, all other segments)
• Shunt capacitance to ground (environment)
• Resistance R[i] representing the plasma conductivity of that section
• Optional inductance if magnetic effects are significant (usually omitted for straight sparks at typical frequencies)

1.2 Position Variable

For a segment indexed by i (from 1 to n), define the normalized position:

position = (i - 1) / (n - 1)

This ranges from 0 at the base (segment 1) to 1 at the tip (segment n). The position variable is used to set position-dependent resistance bounds and initialization profiles.

2. FEMM Extraction for Distributed Model

2.1 Geometry Setup

The FEMM electrostatic model includes n + 1 conductors:

• Conductor 0: Topload (toroid or sphere)
• Conductors 1 through n: Cylindrical segments of the spark channel

Each segment is a short cylinder of length L_seg and nominal diameter d (typically 1 mm for burst mode, 3 mm for QCW). Small gaps of 0.1 mm between segments ensure numerical stability while maintaining physical proximity. See femm-workflow for detailed setup procedures.

2.2 The (n+1) x (n+1) Capacitance Matrix

FEMM produces a symmetric (n+1) x (n+1) Maxwell capacitance matrix C. For a 10-segment model, this is an 11x11 matrix.

Matrix properties (must verify):

• Symmetric: C[i,j] = C[j,i] for all i, j
• Diagonal positive: C[i,i] > 0 (self-capacitance)
• Off-diagonal negative: C[i,j] < 0 for i != j (mutual coupling, Maxwell convention)
• Nearest-neighbor dominance: |C[i,i+1]| > |C[i,i+2]| > |C[i,i+3]| (coupling decreases with distance)
• Decreasing diagonals toward tip: Self-capacitance typically decreases from base to tip because base segments are better coupled to the nearby topload

Coupling patterns observed in practice:

• Topload-to-base coupling (C[0,1]) is the strongest off-diagonal element, typically 5-10 pF
• Topload-to-tip coupling (C[0,n]) is very weak, typically 0.01-0.05 pF
• Adjacent segment coupling (C[i,i+1]) ranges from 0.6 to 3 pF depending on position
• Remote segment coupling (C[i,j] for |i-j| > 3) is usually below 0.1 pF

3. SPICE Implementation Challenge

3.1 The Negative Off-Diagonal Problem

The Maxwell capacitance matrix has negative off-diagonal elements. Direct implementation as literal capacitors in SPICE creates negative capacitances, which are unphysical and can cause numerical instability or non-convergent simulations. Three solutions exist:

3.2 Solution 1: Partial Capacitance Matrix

Transform the Maxwell matrix into a partial capacitance representation where all elements are positive. The partial capacitance between nodes i and j is:

C_partial[i,j] = -C_maxwell[i,j]    for i != j  (positive)
C_partial[i,ground] = C_maxwell[i,i] + sum_j(C_maxwell[i,j])    for j != i  (to ground)

Each C_partial[i,j] is implemented as a standard positive capacitor between nodes i and j. Each C_partial[i,ground] is a capacitor from node i to ground. All values are positive, and the network is guaranteed passive if the original Maxwell matrix is valid.

3.3 Solution 2: Controlled Sources via MNA

Implement the capacitance matrix directly using Modified Nodal Analysis (MNA):

I_i = sum_j C[i,j] * dV_j/dt

In SPICE, this is implemented using voltage-controlled current sources (VCCS) with capacitive coupling. For each pair (i, j):

• A VCCS from node i to ground, controlled by the time derivative of V_j, with gain C[i,j]

This approach preserves the exact matrix without approximation but requires more SPICE elements.

3.4 Solution 3: Nearest-Neighbor Approximation

For many practical cases, the coupling between non-adjacent segments is weak (below 0.1 pF). Truncating the matrix to include only nearest-neighbor and next-nearest-neighbor couplings simplifies the network substantially.

Validation requirement: Compare the full-matrix impedance at the topload port against the approximated impedance. If they agree within 5%, the approximation is justified.

3.5 Passivity Check

The capacitance matrix must be symmetric positive semi-definite (SPD) for the resulting circuit to be passive (no energy generation). If numerical noise in FEMM creates slight violations:

• Add small diagonal perturbation: Increase each C[i,i] by +0.1 pF. This adds a small parasitic to ground at each node.
• Add small series resistance: Insert a small resistor (1-10 ohm) in series with each capacitor for numerical damping.

Check for SPD: all eigenvalues of the capacitance matrix must be non-negative. A single small negative eigenvalue (less than 1% of the largest) is a numerical artifact and can be corrected by the diagonal perturbation.

4. Resistance Determination

4.1 Position-Dependent Physical Bounds

The plasma physics constrains the resistance of each segment based on its position along the spark:

R_min[i] = 1 kilohm + (10 kilohm - 1 kilohm) * position
         = 1 kilohm + 9 kilohm * position

R_max[i] = 100 kilohm + (100 megohm - 100 kilohm) * position^2
         = 100 kilohm + 99.9 megohm * position^2

Physical reasoning:

• Base segments (position near 0) can reach very low resistance because they receive the most current, heat the most, and form hot leader plasma. R_min at the base is 1 kilohm.
• Tip segments (position near 1) are poorly coupled, receive less current, and tend to remain as cold streamers. Their minimum achievable resistance is higher (R_min at the tip is 10 kilohm), reflecting the difficulty of forming leader plasma at the tip.
• The maximum resistance increases quadratically with position. Base segments are constrained to R_max = 100 kilohm (the leader plasma has significant conductivity), while tip segments can reach up to 100 megohm (very faint, cold streamer).

4.2 Initialization: Tapered Resistance Profile

A good initial guess accelerates convergence. The recommended initialization uses a quadratic taper:

R[i] = R_base + (R_tip - R_base) * position^2

R_base = 10 kilohm   (expected leader resistance)
R_tip = 1 megohm     (expected streamer resistance)

The quadratic dependence reflects the physical expectation that leader plasma (low R) dominates near the base and transitions gradually to streamer plasma (high R) toward the tip, with the transition occurring more rapidly in the outer portion of the spark.

4.3 Simplified Method: Circuit-Determined Resistance

For each segment, compute the total capacitance (sum of absolute values of all capacitive couplings involving that segment):

C_total[i] = C_shunt[i] + sum_j(|C_mutual[i,j]|)    for all j != i

where C_shunt[i] = C[i,i] - sum_j(|C[i,j]|) for j != i is the shunt capacitance to ground for segment i, and the mutual capacitances are the absolute values of the off-diagonal elements.

More directly, C_total[i] is the sum of absolute values of all off-diagonal elements in row i:

C_total[i] = sum_{j != i} |C[i,j]|

Then set the resistance using the same R_opt_power logic as the lumped-model, applied per segment:

R[i] = 1 / (omega * C_total[i])
R[i] = clip(R[i], R_min[i], R_max[i])

Justification for the simplified method:

• Capacitance depends logarithmically on diameter: C proportional to 1/ln(h/d)
• R_opt_power is proportional to 1/C, so it also depends weakly on diameter
• Doubling the diameter changes C by only 10-15%
• This error is small compared to other uncertainties (FEMM accuracy ~10%, plasma variability ~50%)
• The simplified method typically agrees with full iterative optimization within 1%

When to use: Standard cases within typical parameter ranges. This is the recommended default.

When to iterate instead: Edge cases near physical bounds, validation studies, highest accuracy needs, or when the simplified method produces results that fail validation checks.

4.4 Advanced Method: Iterative Power Maximization

For highest accuracy, optimize each segment's resistance to maximize the power dissipated in that segment, subject to the constraint that all other segments are at their current resistance values.

Algorithm:

Initialize: R[i] from tapered profile or simplified method
Set damping factor: alpha = 0.3 to 0.5

Repeat until convergence:
    For each segment i = 1 to n:
        Hold all R[j] (j != i) fixed
        Sweep R[i] over logarithmically spaced values from R_min[i] to R_max[i]
        For each trial R[i]:
            Build SPICE network with full capacitance matrix
            Run AC analysis at operating frequency
            Compute P[i] = 0.5 * |I[i]|^2 * R[i]
        Find R_optimal[i] that maximizes P[i]
        Apply damping:
            R_new[i] = alpha * R_optimal[i] + (1 - alpha) * R_old[i]
        Clip to bounds:
            R[i] = clip(R_new[i], R_min[i], R_max[i])

    Check convergence: max relative change across all segments < 1%

    If resonant poles shifted > 5% from initial:
        Update operating frequency to track loaded pole
        Re-optimize at new frequency

Damping (critical for stability): Without damping (alpha = 1), the iteration can oscillate because changing one segment's resistance affects the optimal values of all other segments. A damping factor of alpha = 0.3 to 0.5 provides smooth convergence in 2-5 iterations for typical cases.

4.5 Convergence Behavior and Physical Interpretation

The convergence characteristics differ dramatically between base and tip segments:

Base segments (position near 0):

• Strong coupling to topload and adjacent segments
• Sharp power peak as a function of R
• Fast convergence to a well-defined optimum
• Final R is low (tens of kilohms), indicating hot leader plasma
• Small uncertainty in optimal R

Tip segments (position near 1):

• Weak coupling to all other conductors
• Flat power curve with no sharp peak
• May not converge to a unique optimal value
• Final R is high (hundreds of kilohms to megohms), indicating cold streamer plasma
• Large uncertainty in optimal R (but power is insensitive, so the uncertainty is inconsequential)

This naturally produces the leader + streamer distribution observed in real Tesla coil sparks. The circuit optimization, without any explicit plasma physics, predicts that:

• Base segments should be low-resistance, high-current, high-power (leader)
• Tip segments should be high-resistance, low-current, low-power (streamer)
• The transition is gradual and depends on the capacitive coupling profile

4.6 Diameter Self-Consistency Check

The resistance determines an implied channel diameter, which can be checked against the nominal diameter used in FEMM:

d_nominal = 1e-3 m  (1 mm starting guess)

From FEMM: C_mut, C_sh
From circuit: R_opt = 1 / (omega * C_total)

Back-calculate diameter using typical partially ionized plasma resistivity:
rho_typical = 10 ohm*m
L_segment = L_total / n_segments
d_implied = sqrt(4 * rho_typical * L_segment / (pi * R_opt))

If d_implied is approximately d_nominal (within factor of 2): self-consistent
If not: iterate once with d = (d_nominal + d_implied) / 2

Because the capacitance dependence on diameter is logarithmic, this self-consistency check typically converges in 1-2 iterations. In practice, the circuit-first approach (use nominal diameter, compute R, let plasma adjust) is recommended because the weak logarithmic sensitivity makes the diameter a dependent variable, not an input.

5. Validation

5.1 Lumped Model Consistency

A 1-segment distributed model must produce the same impedance as the lumped-model with the same total C_mut and C_sh. If the two disagree by more than 1%, there is an error in the matrix extraction or SPICE implementation.

Important: Compare impedances at the topload port, NOT total series resistance. The distributed model's total series resistance (sum of all R[i]) is much larger than the lumped R because the segments are in series, but the impedance at the port includes the capacitive network and is a complex quantity.

5.2 Segment Count Convergence

Compare results for n = 5, 10, and 20 segments. Key quantities to track:

• Impedance at topload port
• Total power delivered to spark
• Current at base and tip
• Loaded resonant frequency

Expect: n = 5 to n = 10 shows significant improvement; n = 10 to n = 20 shows diminishing returns (changes under 5%). For most purposes, n = 10 is the sweet spot balancing accuracy and computational cost.

5.3 Total Resistance Validation

The sum of all segment resistances should fall within expected ranges for the operating mode:

At 200 kHz for 1-3 meter sparks:

• Streamer-dominated (burst mode): Total R approximately 50-300 kilohm
• Leader-dominated (QCW): Total R approximately 5-50 kilohm
• Very low frequency (below 100 kHz) or very long sparks: Total R can approach 1-10 kilohm

Flag results significantly outside these ranges. Note that these are total series resistances, which are higher than the equivalent lumped resistance due to the series arrangement.

5.4 Resistance Distribution Check

The final resistance distribution should be physically plausible:

• Monotonically increasing from base to tip (base should be hottest)
• No extreme jumps between adjacent segments (factor of 3 or less between neighbors)
• All values within position-dependent bounds
• Base segments should be in the leader regime (1-100 kilohm)
• Tip segments should be in the streamer regime (100 kilohm to 100 megohm)

5.5 Power Distribution

The power distribution P[i] = 0.5 * |I[i]|^2 * R[i] should show:

• Peak power in the base or middle segments (not at the tip)
• Monotonically decreasing power toward the tip
• Tip segments contributing a small fraction (typically 10-20%) of total power
• This matches the physical expectation that energy is concentrated where current is highest

5.6 Current Attenuation

Current should decrease from base to tip due to capacitive shunting at each segment. Typical tip-to-base current ratio: 0.3 to 0.5 for 10-segment models of 1-3 meter sparks. If the tip current exceeds 70% of the base current, the model may have insufficient shunt capacitance (check C_sh extraction).

6. Comparison: Simplified vs. Iterative Method

Empirical comparison across many configurations shows:

Aspect	Simplified (R = 1/(omega*C_total))	Iterative (power maximization)
Accuracy	Within 1% of iterative for standard cases	Reference (by definition)
Speed	Seconds (FEMM + formula)	Minutes to hours (many AC analyses)
Ease	Trivial to implement	Requires SPICE automation
Edge cases	May fail near bounds	Handles gracefully with damping
Spatial detail	Same quality	Same quality

Recommendation: Use the simplified method as default. Reserve iterative optimization for edge cases, validation, and publications.

7. Connection to Other Topics

Key Relationships

• Generalizes: lumped-model -- The distributed model is the n-section generalization of the single-section lumped model. Setting n = 1 recovers the lumped model exactly.
• Requires: femm-workflow -- FEMM electrostatic simulation provides the (n+1) x (n+1) capacitance matrix that defines the entire capacitive network.
• Uses: power-optimization -- The R_opt_power formula, applied per segment, provides both the simplified resistance values and the objective function for iterative optimization.
• Implements: circuit-topology -- Each segment reproduces the fundamental bridged-T topology locally, with the full model being a cascade of such sections.
• Reveals: streamers-and-leaders -- The optimized resistance distribution naturally separates into leader (base, low R) and streamer (tip, high R) regions without any explicit plasma physics input.
• Demonstrates: capacitive-divider -- The current attenuation and voltage distribution along the distributed model directly visualize the capacitive divider effect.
• Bounded by: equations-and-bounds -- All per-segment and total values must satisfy the physical bounds documented in the reference.
• Motivates: open-questions -- Branching, time-dependent evolution, and the optimal segment count remain open research questions.

Worked Example

The complete 10-segment workflow is demonstrated in distributed-model-complete.md, which includes:

• FEMM geometry with 10 segments plus topload (11 conductors)
• Full 11x11 capacitance matrix extraction and verification
• Tapered initialization and position-dependent bounds
• Simplified method calculation (C_total per segment, R = 1/(omega*C_total))
• Two iterations of the advanced method showing convergence
• Power distribution, current attenuation, and voltage distribution analysis
• Comparison to lumped model and segment count convergence study
• Validation checks at every step