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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
model-03	Distributed Model Theory	Advanced Modeling	advanced	40	[model-01 model-02]	[Understand when and why distributed models are necessary Master nth-order segmentation strategy and circuit topology Learn the trade-offs between lumped and distributed approaches Apply distributed models to long sparks and research applications]	[distributed-model segmentation nth-order circuit-topology]

Distributed Model Theory

The distributed spark model divides the spark into multiple segments, each with its own resistance and capacitance network. This captures spatial variations in current, voltage, and plasma properties along the spark length.

Why Distributed Models?

Limitations of Lumped Models

Lumped models treat the entire spark as a single element, which fails to capture:

1. Current distribution along spark

Base: Full current (directly coupled to topload)
Middle: Reduced current (capacitive shunting)
Tip: Much lower current (weak coupling, high shunt)

Lumped model: Assumes uniform current everywhere (wrong!)

2. Voltage distribution

Actual: Non-linear voltage drop due to distributed capacitance
Lumped: Assumes simple voltage divider (oversimplified)

Capacitive divider effects occur at EACH point along spark

3. Base vs tip physical differences

Base properties:
- Hot plasma (continuously heated)
- Well-coupled to topload
- Low resistance (leader regime)
- High current density

Tip properties:
- Cool plasma (sporadic heating)
- Weakly coupled
- High resistance (streamer regime)
- Low current density

Lumped model: Single R averages this out (loses physics!)

4. Leader/streamer transitions

Long sparks: Base forms leader, tip remains streamer
Different physics: Different R, different behavior
Lumped R: Cannot represent this transition zone

5. Very long sparks (>3 m)

Distributed effects dominate
Single lumped R is poor approximation
Error: Can be factor of 2-5 in current distribution

When to Use Distributed Models

Use distributed when:

Spark length > 1-2 meters
- Spatial variations become significant
- Base-to-tip differences critical
Current distribution matters
- Measuring actual current profile along spark
- Validating against detailed experimental data
- Understanding leader formation dynamics
Research applications
- Physics investigations
- Leader/streamer transition studies
- Publication-quality results
Extreme parameters
- Very low frequency (λ comparable to L)
- Very high voltage (breakdown physics critical)
- Unusual geometries (horizontal, branched)

Stick with lumped when:

Quick design iterations
- Impedance matching studies
- Component selection
- Performance estimates
Short sparks (<1 m)
- Uniform properties adequate
- Computational efficiency critical
Engineering estimates
- ±20% accuracy sufficient
- Fast turnaround needed

Computational trade-off:

Lumped model: <1 second
Distributed (n=10): ~10-30 seconds
Distributed (n=20): ~1-5 minutes

Speedup factor: 600-18000×

Use distributed only when benefits justify cost!

Segmentation Strategy

Dividing the Spark

Equal-length segments:

n = number of segments (typically 5-20)
L_segment = L_total / n

Segment numbering:
  i = 1: Base (connected to topload)
  i = 2, 3, ..., n-1: Middle sections
  i = n: Tip (furthest from topload)

Example: 2.4 m spark, n=6 segments

L_segment = 2.4 / 6 = 0.4 m each

Segment 1 (base): z = 0 to -0.4 m
Segment 2: z = -0.4 to -0.8 m
Segment 3: z = -0.8 to -1.2 m
Segment 4: z = -1.2 to -1.6 m
Segment 5: z = -1.6 to -2.0 m
Segment 6 (tip): z = -2.0 to -2.4 m

Why Equal Lengths?

Advantages:

1. Simple FEMM geometry
   - Uniform cylinder sections
   - Easy to script/automate

2. Uniform discretization
   - No bias toward any region
   - Straightforward convergence analysis

3. Easy implementation
   - Regular array indexing
   - Simple matrix structure

4. Standard practice
   - Literature comparisons
   - Validated approach

Non-uniform segmentation possible:

Alternative: Finer near tip (where R changes rapidly)

Example: Geometric progression
  L[i] = L_base × ratio^(i-1)

Benefits: Better captures tip physics with fewer segments

Drawbacks:
  - More complex FEMM setup
  - Harder to interpret results
  - Diminishing returns for extra complexity

Recommendation: Use equal lengths unless specific research need

Choosing n (Number of Segments)

Convergence vs computational cost:

n = 1: Lumped model (fastest, least accurate for long sparks)
n = 5: Coarse distributed (captures main trends)
n = 10: Standard distributed (good balance)
n = 20: Fine distributed (research quality)
n = 50: Overkill (no improvement, much slower)

Rule of thumb:

L < 1 m: Use lumped (n=1)
L = 1-2 m: n = 5-10
L = 2-4 m: n = 10-15
L > 4 m: n = 15-20

Convergence test: Double n, check if results change <10%
If yes: Original n sufficient
If no: Use higher n

Practical limitations:

FEMM: (n+1)×(n+1) matrix, scales as O(n²)
SPICE: Network complexity, scales as O(n²-n³)
Optimization: R sweep, scales as O(n)

Total time ≈ t_FEMM × n² + t_SPICE × n² + t_optimize × n

Diminishing returns beyond n ≈ 20

Circuit Topology

Per-Segment Components

Each segment i has:

1. Resistance R[i]

Physical meaning: Plasma resistance of that segment
Units: Ohms (typically kΩ to MΩ)
Variable: To be optimized
Expectation: Monotonically increasing from base to tip

2. Mutual capacitances C[i,j]

Coupling to:
  - Topload (j=0)
  - All other segments (j=1 to n, j≠i)

Extracted from FEMM (n+1)×(n+1) matrix

Expectation:
  - Stronger coupling to nearby segments
  - Weaker coupling to distant segments
  - C[i,j] decreases with |i-j|

3. Shunt capacitance to ground

Included in capacitance matrix diagonal
NOT a separate component in circuit

C[i,i] (diagonal) represents self-capacitance
Includes ground coupling implicitly

Network Structure

Full distributed network:

Topload (node 0, V_top)
   |
   +---[C[0,1]]---+
   |              |
   +---[C[0,2]]---|---+
   |              |   |
   +---[C[0,3]]---|---|---+
   |              |   |   |
   ...            |   |   |
                  |   |   |
                [R[1]] |   |
                  |   |   |
              Node 1  |   |
                  |   |   |
              [C[1,2]]|   |
              [C[1,3]]|---|
                  |   |   |
                [R[2]] |   |
                  |   |   |
              Node 2  |   |
                  |   |   |
              [C[2,3]]|---|
                  |   |   |
                [R[3]] |   |
                  |   |   |
              Node 3  |   |
                  |   |   |
                  |   |   |
                 GND GND GND
                 (implicit in C matrix)

Matrix representation:

For n=3 segments + topload (4×4 matrix):

         [0]    [1]    [2]    [3]
[0]  [  C₀₀    C₀₁    C₀₂    C₀₃  ]  Topload
[1]  [  C₁₀    C₁₁    C₁₂    C₁₃  ]  Segment 1 (base)
[2]  [  C₂₀    C₂₁    C₂₂    C₂₃  ]  Segment 2
[3]  [  C₃₀    C₃₁    C₃₂    C₃₃  ]  Segment 3 (tip)

Plus resistances:
R[1], R[2], R[3] (one per segment)

Total unknowns: 3 R values (n in general)

Complexity Analysis

For n segments:

Capacitance matrix: (n+1)×(n+1) = n² + 2n + 1 elements
Due to symmetry: (n+1)(n+2)/2 unique values

Resistances: n values

Circuit nodes: n+1 (including topload)

SPICE equations: O(n²) for capacitance network
                 O(n) for resistances

Total complexity: O(n²) dominated by capacitance couplings

Physical Expectations

Resistance Distribution

Expected profile:

R[1] < R[2] < R[3] < ... < R[n]

Monotonically increasing from base to tip

Typical values at 200 kHz:

Base (segment 1):
  R[1] ≈ 5-20 kΩ
  Hot leader, well-coupled
  High current, low resistance

Middle (segments 2 to n-1):
  R[i] ≈ 10-100 kΩ
  Transition region
  Moderate coupling

Tip (segment n):
  R[n] ≈ 100 kΩ - 10 MΩ
  Cool streamer, weakly coupled
  Low current, high resistance

Total resistance:

R_total = Σ R[i]

Expected: 50-500 kΩ at 200 kHz for 2-3 m spark

Compare to lumped: Should be similar order of magnitude
If factor >5 different: Check model carefully

Capacitance Patterns

Mutual capacitance C[i,j] (i≠j):

Nearby segments: Larger |C[i,j]|
  Example: |C[2,3]| > |C[2,5]|

Distant segments: Smaller |C[i,j]|
  Example: |C[1,10]| << |C[1,2]|

Topload coupling: Decreases with distance
  |C[0,1]| > |C[0,2]| > ... > |C[0,n]|

Self-capacitance C[i,i] (diagonal):

Positive (always)
Includes shunt to ground
Typically: 5-15 pF per segment

Total shunt: Σᵢ (C[i,i] - |C[i,0]|) ≈ 2 pF/ft × L_total
(Approximate, factor of 2-3 variation acceptable)

Current Distribution

Expected behavior:

|I[1]| > |I[2]| > ... > |I[n]|

Current decreases from base to tip

Physical reason:

Capacitive shunting at each segment:
- Some current diverts to ground through C_sh
- Less current reaches next segment
- Accumulates along spark length

Weak coupling at tip:
- High R, low current naturally
- Capacitive shunting reduces current further
- Tip current can be 10-50× lower than base

Validation:

After simulation, plot I[i] vs position
Should be monotonically decreasing
If not: Check R distribution, C matrix

Voltage Distribution

Expected behavior:

V[1] > V[2] > ... > V[n]

Voltage decreases from base to tip

But NOT linear!

Simple resistor chain: ΔV = I × R (linear)

Distributed spark: Capacitive divider at each point
  - Voltage "leaks" to ground through shunt capacitance
  - Non-linear profile
  - Steeper drop near base (high current)
  - Flatter near tip (low current)

Lumped vs Distributed Comparison

Equivalent Impedance

Both models should give similar Z_spark at topload:

Lumped: Z = R + 1/(jωC_total)
Distributed: Z = [complex network impedance]

At topload port, similar order of magnitude
Difference: Typically 10-30% for well-designed models

If very different (factor >2):

Check:
1. Total resistance: Σ R[i] vs R_lumped
2. Total capacitance: C_total_distributed vs C_mut + C_sh
3. Matrix extraction errors
4. Convergence of n (try higher n)

Power Dissipation

Lumped:

P_total = 0.5 × I² × R

Single power value

Distributed:

P[i] = 0.5 × I[i]² × R[i]
P_total = Σ P[i]

Can see where power is dissipated:
- Base: High current, moderate R → high power
- Middle: Moderate current and R → moderate power
- Tip: Low current, high R → low power (often <10% of base)

Insight from distributed model:

Most power dissipated in base 1/3 of spark
Tip contributes little to total power
But tip electric field critical for growth!

This explains why:
- Short sparks easier (more efficient power coupling)
- Long sparks harder (tip poorly coupled)
- QCW benefits (maintains hot base channel)

Worked Example: 3-Segment Model

Given:

Total spark: 1.5 m
Divide into n = 3 equal segments
Each segment: 0.5 m

Segment locations:

Segment 1 (base): z = 0 to -0.5 m
Segment 2 (middle): z = -0.5 to -1.0 m
Segment 3 (tip): z = -1.0 to -1.5 m

Expected capacitance matrix (example values):

         [0]     [1]     [2]     [3]
[0]  [  30.0    -9.0    -3.5    -1.5  ]  pF
[1]  [  -9.0    14.0    -3.0    -1.0  ]
[2]  [  -3.5    -3.0    10.5    -2.5  ]
[3]  [  -1.5    -1.0    -2.5     8.0  ]

Properties:
✓ Symmetric
✓ Diagonal positive
✓ Off-diagonal negative
✓ Nearby segments more strongly coupled

Expected resistance distribution:

R[1] = 30 kΩ   (base, hot)
R[2] = 60 kΩ   (middle, moderate)
R[3] = 150 kΩ  (tip, cool)

Total: 240 kΩ

Monotonically increasing ✓

Circuit implementation:

Convert capacitance matrix to SPICE (see next lesson)
Add resistances R[1], R[2], R[3]
Simulate to get currents and voltages

Expected results (qualitative):

If V_topload = 1 V (test):

I[1] ≈ 15 μA   (base current)
I[2] ≈ 8 μA    (middle current, ~50% of base)
I[3] ≈ 3 μA    (tip current, ~20% of base)

V[1] ≈ 0.8 V   (base voltage)
V[2] ≈ 0.5 V   (middle voltage)
V[3] ≈ 0.2 V   (tip voltage, non-linear drop!)

P[1] ≈ 7 μW    (base power, 50% of total)
P[2] ≈ 4 μW    (middle power, 30%)
P[3] ≈ 3 μW    (tip power, 20%)

Key Takeaways

Distributed models divide spark into n segments, capturing spatial variations in current, voltage, and resistance
Use when: sparks >2 m, current distribution needed, research applications, extreme parameters
Segmentation: equal-length segments, n = 5-20 typical, convergence test by doubling n
Circuit topology: (n+1)×(n+1) capacitance matrix plus n resistances, O(n²) complexity
Physical expectations: R monotonically increasing, current decreasing, voltage non-linear, power concentrated at base
Trade-off: 1000-2000× slower than lumped, use only when benefits justify computational cost
Validation: Compare to lumped model (similar Z_spark), check physical trends (I, V, R distributions)
Next steps: FEMM extraction for n-segment geometry (Lesson 4), resistance optimization (Lesson 5)

Practice

{exercise:model-ex-03}

Next Lesson: FEMM Extraction for Distributed Models

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