plasma-expert/context/circuit-topology.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
circuit-topology	Fundamental Circuit Topology and Phase Constraints	established	spark-physics.txt: Part 1 (lines 11-72), Part 11 (lines 736-751)	[power-optimization thevenin-method coupled-resonance capacitive-divider lumped-model distributed-model femm-workflow equations-and-bounds]	[Input admittance Y Re{Y} and Im{Y} decomposition Impedance phase angle phi_Z Fundamental phase constraint phi_Z_min Capacitance ratio r]	[mutual capacitance shunt capacitance admittance impedance phase angle topological constraint phasor susceptance conductance]	[complex-plane-admittance.png phase-angle-visualization.png phase-constraint-graph.png admittance-vector-addition.png]	[]	[How does the phase constraint shift if C_mut becomes frequency-dependent at very high frequencies? What is the exact crossover geometry (topload size vs. spark length) where r = 0.207? How do proximity effects from nearby grounded objects alter the effective C_sh and thus r?]

Fundamental Circuit Topology and Phase Constraints

This document establishes the foundational circuit model for Tesla coil sparks. Every subsequent analysis in the knowledge graph -- power optimization, Thevenin extraction, lumped and distributed modeling -- builds on the topology and admittance relationships derived here. The central result is a topological phase constraint that limits the impedance angle the spark can present to the resonant circuit, independent of plasma physics.

1. The Basic Spark Circuit Model

1.1 Physical Origin of the Two Capacitances

FEMM electrostatic analysis of a Tesla coil with an extended spark channel reveals two distinct capacitances at the topload connection point:

• Mutual capacitance (C_mut): The capacitive coupling between the spark channel and the topload. This is the path through which displacement current flows from the topload into the spark plasma. C_mut depends on the topload geometry, spark channel shape, and their relative orientation. For a typical toroidal topload with a spark emerging from its edge, C_mut ranges from roughly 3 to 15 pF depending on topload size and spark length.

• Shunt capacitance (C_sh): The capacitance from the spark channel to ground (and to all other grounded or low-potential objects in the environment). Empirically, C_sh scales approximately linearly with spark length at ~2 pF per foot (~6.6 pF per meter). This scaling holds because longer sparks present more conductor length to the surrounding environment.

1.2 Circuit Topology

The two capacitances, together with the spark channel resistance R, form the following topology at the topload node:

Topload ---[C_mut || R]--- Spark tip
   |                          |
   |                       [C_sh]
   |                          |
  GND ---------------------- GND

Reading this circuit:

• C_mut and R are in parallel between the topload node and the spark tip node. The parallel combination represents the fact that current can flow from topload to spark either through the capacitive coupling (displacement current through C_mut) or through the resistive plasma channel (conduction current through R).
• C_sh connects the spark tip to ground, representing the distributed capacitance of the spark channel to its environment.
• The topload itself connects to ground through the Tesla coil secondary (not shown here; that is the source impedance).

This is NOT a simple series or parallel RLC. The topology is a bridged-T or pi-network, and this specific arrangement is what creates the phase constraint discussed below.

1.3 Phasor Convention

All phasor quantities in this framework use peak values, not RMS. Power formulas therefore include the factor of 0.5:

P = 0.5 * Re{V * I*}

where I* denotes the complex conjugate of I. This convention is consistent throughout all topics in the knowledge graph.

2. Admittance Analysis

2.1 Definitions

At angular frequency omega = 2pif, define:

• G = 1/R : conductance of the spark channel [siemens]
• B_1 = omega * C_mut : susceptance due to mutual capacitance [siemens] (positive, capacitive)
• B_2 = omega * C_sh : susceptance due to shunt capacitance [siemens] (positive, capacitive)

Note that B_1 and B_2 are defined as positive quantities (the conventional "capacitive susceptance" magnitude). The imaginary part of the admittance of a capacitor C is +jomegaC in the Y-domain.

2.2 Input Admittance at Topload

The admittance looking into the spark circuit from the topload node (with ground as the return) is computed by combining the parallel combination (G + jB_1) in series with jB_2:

Y = ((G + jB_1) * jB_2) / (G + j(B_1 + B_2))

Derivation: The impedance of the parallel (C_mut || R) branch is Z_parallel = 1/(G + jB_1). The impedance of C_sh is Z_sh = 1/(jB_2). The total impedance from topload to ground is Z_total = Z_parallel + Z_sh. The total admittance is Y = 1/Z_total. Inverting:

Y = 1 / [1/(G + jB_1) + 1/(jB_2)]
  = (G + jB_1) * jB_2 / [(G + jB_1) + jB_2]
  = ((G + jB_1) * jB_2) / (G + j(B_1 + B_2))

2.3 Real and Imaginary Parts

Multiplying numerator and denominator by the conjugate of the denominator:

Real part (conductance component):

Re{Y} = G * B_2^2 / (G^2 + (B_1 + B_2)^2)

Imaginary part (susceptance component):

Im{Y} = B_2 * [G^2 + B_1*(B_1 + B_2)] / (G^2 + (B_1 + B_2)^2)

Verification of limiting cases:

• R -> infinity (G -> 0): Re{Y} -> 0, Im{Y} -> B_1*B_2/(B_1 + B_2). This is the series combination of two capacitances, as expected (no conduction, pure capacitive divider).

• R -> 0 (G -> infinity): Re{Y} -> B_2^2/G -> 0 (approaches short at topload, all current bypasses C_sh). More carefully: Y -> jB_2, since the short across C_mut || R removes C_mut and leaves only C_sh.

• C_sh -> 0 (B_2 -> 0): Y -> 0. No path to ground through the spark; the circuit is open.

2.4 Admittance and Impedance Phase Angles

The admittance phase angle is:

theta_Y = atan(Im{Y} / Re{Y})

The impedance phase angle, which is what is typically measured and discussed in Tesla coil literature, is the negative of the admittance phase:

phi_Z = -theta_Y = atan(-Im{Y} / Re{Y})

Sign convention: A purely capacitive load has phi_Z = -90 degrees. A purely resistive load has phi_Z = 0 degrees. The spark load always has phi_Z between -90 degrees and 0 degrees (capacitive side), because the circuit contains only capacitors and a resistor (no inductance).

Important: When Tesla coil builders discuss "matching to -45 degrees" or "the impedance angle," they are referring to phi_Z, not theta_Y.

3. The Fundamental Phase Constraint

3.1 Derivation

The impedance phase angle phi_Z depends on R (equivalently, on G = 1/R). As R varies from 0 to infinity, phi_Z traces a curve. There exists a minimum achievable impedance phase angle (maximum negative value) that depends only on the ratio of capacitances:

phi_Z_min = -atan(2 * sqrt(r * (1 + r)))

where r = C_mut / C_sh

Derivation sketch: Setting d(phi_Z)/dG = 0, the condition for extremum yields G_opt = omega * sqrt(C_mut * (C_mut + C_sh)), which corresponds to R_opt_phase = 1/(omega * sqrt(C_mut * (C_mut + C_sh))). Substituting back gives the minimum phase expression above.

3.2 The -45 Degree Impossibility

Setting phi_Z_min = -45 degrees and solving:

atan(2 * sqrt(r * (1 + r))) = 45 degrees
2 * sqrt(r * (1 + r)) = 1
4 * r * (1 + r) = 1
4r^2 + 4r - 1 = 0
r = (-4 + sqrt(16 + 16)) / 8 = (-4 + 4*sqrt(2)) / 8 = (sqrt(2) - 1) / 2 ~ 0.207

Critical insight: When r >= 0.207, achieving phi_Z = -45 degrees is mathematically impossible, regardless of the value of R. This is a topological constraint imposed by the circuit structure, not a limitation of plasma physics or any material property.

3.3 Practical Implications

For typical Tesla coil geometries:

Topload / Spark Configuration	Approximate r = C_mut/C_sh	phi_Z_min
Large topload, short spark	1.0 - 2.0	-55 to -70 deg
Medium topload, medium spark	0.5 - 1.0	-50 to -55 deg
Small topload, long spark	0.2 - 0.5	-45 to -50 deg

Since most practical configurations have r > 0.207, the -45 degree "matched" condition is almost never achievable. This explains why real sparks typically present impedance angles in the -55 to -75 degree range.

3.4 The "R approximately equals |X_c|" Myth

Tesla coil literature often states that spark resistance approximately equals the magnitude of the capacitive reactance: R ~ |X_c|. This relationship does emerge approximately from the power optimization (see power-optimization), but it does NOT imply that -45 degrees is achievable. The approximate equality arises because R_opt_power = 1/(omega * C_total) ~ 1/(omega * C_sh) when C_mut and C_sh are comparable, and 1/(omega * C) is the reactance magnitude. The phase angle at R_opt_power, however, is typically -55 to -75 degrees, not -45 degrees.

4. Effect of Secondary Losses

4.1 Parasitic Conductance

Real Tesla coil secondaries have losses: wire resistance, dielectric losses in the coil form, corona losses, and radiation. These appear as a parallel conductance G_sec on the source side (topload-to-ground), in addition to the spark circuit.

4.2 Impact on Phase Constraint

The additional parallel conductance G_sec increases the real part of the total admittance seen by the source but does NOT change the spark circuit's fundamental phase constraint. The spark still cannot present an impedance angle better than phi_Z_min. The secondary losses simply add a real (resistive) load in parallel with the spark's complex load. The total phase angle of the combined load will actually be closer to zero (more resistive), but this is because power is being wasted in the secondary, not because the spark is better matched.

Practical note: When measuring the total Q of a loaded Tesla coil, the measured Q reflects both secondary losses and spark loading. Separating the two requires the thevenin-method or careful ringdown analysis.

5. Frequency Dependence

5.1 How Admittance Scales with Frequency

Since B_1 = omega * C_mut and B_2 = omega * C_sh, both susceptances scale linearly with frequency. The admittance components Re{Y} and Im{Y} therefore have non-trivial frequency dependence. However, the phase constraint phi_Z_min depends only on the ratio r = C_mut/C_sh, which is frequency-independent (assuming frequency-independent capacitances). Thus:

• The minimum achievable phase angle does not change with frequency.
• The resistance value that achieves the minimum phase (R_opt_phase) does change with frequency (it is inversely proportional to omega).
• The resistance value that maximizes power (R_opt_power) also changes with frequency.

5.2 Relevance to Frequency Tracking

When a spark loads the Tesla coil, the resonant frequency shifts (see coupled-resonance). As frequency changes, B_1 and B_2 change proportionally, which shifts R_opt_power and R_opt_phase. However, because r is fixed, phi_Z_min is unaffected. The spark must re-optimize its resistance to the new R_opt_power at the new operating frequency.

6. Connection to Other Topics

Key Relationships

• Derives from: FEMM electrostatic analysis (physical measurement of C_mut and C_sh)
• Enables: power-optimization (R_opt_power and R_opt_phase are computed from the admittance expressions derived here)
• Enables: thevenin-method (the spark circuit topology defines what Z_load looks like to the Thevenin equivalent)
• Enables: lumped-model (the lumped model IS this circuit, with FEMM-extracted capacitance values)
• Constrains: coupled-resonance (the phase constraint limits how "resistive" the spark can look, affecting power transfer)
• Extended by: distributed-model (the distributed model generalizes this single-section topology to n sections)
• Extended by: capacitive-divider (the voltage division at the spark tip is a direct consequence of this topology)

Summary of Key Results

1. The spark circuit is a bridged-T network with C_mut || R in series with C_sh.
2. The input admittance Y has closed-form real and imaginary parts in terms of G, B_1, B_2.
3. The impedance phase angle phi_Z is bounded by phi_Z_min = -atan(2sqrt(r(1+r))).
4. For r >= 0.207 (almost all practical configurations), -45 degrees is impossible.
5. Secondary losses do not relax the phase constraint.
6. The constraint is topological (circuit structure), not physical (plasma properties).