--- id: fund-02 title: "The Basic Spark Circuit Model" section: "Fundamentals" difficulty: "beginner" estimated_time: 25 prerequisites: ["fund-01"] objectives: - Understand what capacitance represents physically - Distinguish between mutual capacitance (C_mut) and shunt capacitance (C_sh) - Learn the empirical 2 pF/foot rule for spark capacitance - Draw the correct circuit topology for a Tesla coil spark - Identify the topload port as the measurement reference tags: ["capacitance", "circuit-topology", "C_mut", "C_sh", "measurement"] --- # The Basic Spark Circuit Model ## Introduction A spark isn't just a resistor - it's a complex structure with multiple electrical properties. Understanding how to model a spark as a circuit with the correct topology is essential for analyzing Tesla coil performance. ## What is Capacitance Physically? **Definition:** Capacitance (C) is the ability to store electric charge for a given voltage: ``` Q = C × V Units: Farads (F), typically pF (10⁻¹² F) for Tesla coils ``` **Physical picture:** - Electric field between two conductors stores energy - Higher field → more stored energy → more capacitance - Capacitance depends on geometry, NOT on voltage **For parallel plates:** ``` C = ε₀ × A / d where ε₀ = 8.854×10⁻¹² F/m (permittivity of free space) A = plate area (m²) d = separation distance (m) ``` **Key insight:** Capacitance increases with: - Larger conductor area (more field lines) - Smaller separation (stronger field concentration) ## Self-Capacitance vs Mutual Capacitance **Self-capacitance:** Capacitance of a single conductor to infinity (or ground) - Topload has self-capacitance to ground - Depends on size and shape - Toroid: C ≈ 4πε₀√(D×d) where D = major diameter, d = minor diameter **Mutual capacitance:** Capacitance between two conductors - Energy stored in field between them - Both conductors at different potentials - Can be positive or negative in matrix formulation **For Tesla coils with sparks:** - **C_mut:** mutual capacitance between topload and spark channel - **C_sh:** capacitance from spark to ground (shunt capacitance) ## Shunt Capacitance and the 2 pF/Foot Rule Any conductor elevated above ground has capacitance to ground. **For vertical wire above ground plane:** ``` C ≈ 2πε₀L / ln(2h/d) where L = wire length h = height above ground d = wire diameter ``` **For Tesla coil sparks:** Empirical rule based on community measurements: ``` C_sh ≈ 2 pF per foot of spark length Examples: 1 foot (0.3 m) spark: C_sh ≈ 2 pF 3 feet (0.9 m) spark: C_sh ≈ 6 pF 6 feet (1.8 m) spark: C_sh ≈ 12 pF ``` This rule is surprisingly accurate (±30%) for typical Tesla coil geometries. ### Worked Example: Estimating C_sh **Given:** A 2-meter (6.6 foot) spark **Find:** Estimated shunt capacitance **Solution:** ``` C_sh ≈ 2 pF/foot × 6.6 feet C_sh ≈ 13.2 pF ``` **Refined estimate using cylinder formula:** Assume spark is vertical cylinder: - Length L = 2 m - Diameter d = 2 mm (typical for bright spark) - Height above ground h = L/2 = 1 m (average height) ``` C ≈ 2πε₀L / ln(2h/d) C ≈ 2π × 8.854×10⁻¹² × 2 / ln(2×1/0.002) C ≈ 1.112×10⁻¹⁰ / ln(1000) C ≈ 1.112×10⁻¹⁰ / 6.91 C ≈ 16 pF ``` The empirical rule (13 pF) and formula (16 pF) agree reasonably well. ## Why Sparks Have TWO Capacitances A spark channel is a conductor in space with: 1. **Proximity to the topload** → mutual capacitance C_mut 2. **Proximity to ground/environment** → shunt capacitance C_sh **Both exist simultaneously** because the spark interacts with multiple conductors. **Analogy:** A wire near two metal plates - Capacitance to plate 1: C₁ - Capacitance to plate 2: C₂ - Both must be included in the circuit model ![Field lines showing C_mut and C_sh](assets/field-lines-capacitances.png) **Field line visualization:** - **C_mut field lines:** Connect topload surface to spark channel - Start on topload outer surface - End on spark channel surface - Concentrated near base of spark - These store mutual electric field energy - **C_sh field lines:** Connect spark to remote ground - Start on spark surface - Radiate outward to walls, floor, ceiling - Distributed along entire spark length - These store shunt field energy **Key observation:** The same spark channel participates in BOTH capacitances! This is why we need a specific circuit topology. ## The Correct Circuit Topology ``` Topload (measurement reference) | [C_mut] ← Mutual capacitance between topload and spark | +---------+--------- Node_spark | | [R] [C_sh] ← Shunt capacitance spark-to-ground | | GND ------------ GND ``` **Equivalent description:** - C_mut and R in parallel - That parallel combination in series with C_sh - All connected between topload and ground **Why this topology?** 1. C_mut couples topload voltage to spark 2. R represents plasma resistance (where power is dissipated) 3. C_sh provides current return path to ground 4. Current through R must also flow through either C_mut or C_sh (series connection) ## Where is "Ground" in a Tesla Coil? **Earth ground:** Actual connection to soil/building ground **Circuit ground (reference):** Arbitrary 0V reference point **For Tesla coils:** - Primary circuit: Chassis/mains ground is reference - Secondary base: Usually connected to primary ground via RF ground - **Practical ground:** Floor, walls, nearby objects, you standing nearby - **Measurement ground:** Choose ONE point as 0V reference (usually secondary base) **Important:** "Ground" in spark model means "remote return path" - could be walls, floor, strike ring, or actual earth. ## The Topload Port **Definition:** The two-terminal measurement point between topload and ground where we characterize impedance and power. ``` Port definition: Terminal 1: Topload terminal (high voltage) Terminal 2: Ground reference (0V) ``` **All impedance measurements reference this port:** - Z_spark: impedance looking into spark from topload - Z_th: Thévenin impedance of coil at this port - V_th: Open-circuit voltage at this port **Not the same as:** - V_top / I_base (includes displacement currents from entire secondary) - Any two-point measurement along the secondary winding We'll explore why V_top/I_base is incorrect in a later lesson. ## Worked Example: Drawing the Complete Circuit **Given:** - Spark is 3 feet long - FEMM analysis gives C_mut = 8 pF (between topload and spark) - Assume R = 100 kΩ - Estimate C_sh using empirical rule **Task:** Draw complete circuit diagram **Solution:** Step 1: Calculate C_sh ``` C_sh ≈ 2 pF/foot × 3 feet = 6 pF ``` Step 2: Draw topology ``` Topload (V_top) | [C_mut = 8 pF] | +-------- Node_spark | | [R = 100 kΩ] [C_sh = 6 pF] | | GND -------- GND ``` Step 3: Alternative representation showing parallel/series structure ``` Topload | +---- [C_mut = 8 pF] ----+ | | +---- [R = 100 kΩ] ------+ Node_spark | [C_sh = 6 pF] | GND ``` This is the basic lumped model for a Tesla coil spark. ![3D geometry to circuit schematic translation](assets/geometry-to-circuit.png) ## Key Takeaways - Capacitance stores energy in electric fields, depends on geometry - **C_mut:** mutual capacitance between topload and spark - **C_sh:** shunt capacitance from spark to ground, approximately **2 pF/foot** - Both capacitances exist simultaneously on the same conductor - **Correct topology:** (R || C_mut) in series with C_sh - **Topload port:** measurement reference between topload and ground - Ground means "remote return path" in this context ## Practice {exercise:fund-ex-02} **Problem 1:** Draw the circuit for a spark with: L = 5 feet, C_mut = 12 pF (from FEMM), R = 50 kΩ. Label all component values. **Problem 2:** A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark length using the empirical rule? **Problem 3:** A 4-foot spark is formed. Estimate C_sh using the empirical rule. If the topload has C_topload = 30 pF unloaded, what is the total system capacitance with the spark? (Hint: Consider how C_mut and C_sh combine in the circuit.) --- **Next Lesson:** [Admittance Analysis](03-admittance-analysis.md)