TELSIKS 2001, University of Nis, Yugoslavia (September 19-21, 2001) and MICROWAVE REVIEW 1 RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes K.L. Corum* and J.F. Corum** “Is there, I ask, can there be, a more interesting study than that of alternating currents.” Nikola Tesla, (Life Fellow, and 1892 Vice President of the AIEE) Abstract – By modeling a wire-wound coil as an anisotropically conducting cylindrical boundary, one may start from Maxwell’s equations and deduce the structure’s resonant behavior. Not only can the propagation factor and characteristic impedance be determined for such a helically disposed surface waveguide, but also its resonances, “self-capacitance” (so-called), and its voltage magnification by standing waves. Further, the Tesla coil passes to a conventional lumped element inductor as the helix is electrically shortened. Keywords – coil, helix, surface wave, resonator, inductor, Tesla. One of the more significant challenges in electrical science is that of constricting a great deal of insulated conductor into a compact spatial volume and determining the resulting structure’s electrical properties. The problem is not only of interest at very low frequencies, where the arrangement can be treated as a lumped element (an inductor), but also at frequencies where the current distribution over the structure is not uniform and the conventional lumped element assumption fails. A simple helical coil geometry is considered in the following note. (It is a common misconception that a full field analysis is necessary only at relatively high frequencies. In point of fact, it is essential whenever the current distribution over the structure is not uniform.) Adler, Chu and Fano have pointed out that there are three important characteristics of high Q linear systems: (1) their free oscillations are slightly damped; (2) Their input impedance has a rapid and radical variation around resonance; (3) Their field distributions possess characteristic space patterns that accompany the sharply selective behavior of the system. It is this third feature that is often “…overlooked in lumped networks and is of great importance in understanding distributed resonant systems.” A helically wound Tesla coil is just such a system, as will be shown in the following note. (The analysis is performed during the linear system – pre-discharge epoch, while the voltage rise phenomenon is occurring. The system has a nonlinear load during discharge, of course.) A uniform helix is described by its radius (r = a), its pitch (or turn-to-turn wire spacing "s"), and its pitch angle ψ, which is the angle that the tangent to the helix makes with a plane perpendicular to the axis of the structure (z). Geometrically, ψ = cot⁻¹(2πa/s). The wave equation is not separable in helical coordinates and there exists no rigorous solution of Maxwell's equations for the solenoidal helix. However, at radio frequencies a wire-wound helix with many turns per free-space wavelength (e.g., a Tesla coil) may be modeled as an idealized anisotropically conducting cylindrical surface that conducts only in the helical direction. The conductivity normal to the helical path is taken to be zero. (This cylindrical tube-shaped structure is the classic sheath helix model due to Ollendorf.) Formally, for time-harmonic fields (assume e^(jωt) time variation), the homogeneous vector Helmholtz equations ∇² E + k² E = 0 and ∇² H + k² H = 0 (for r ≠ a) (1), where k = ω(με)^(1/2) = 2π/λ = k₀(ε_r μ_r)^(1/2), must be solved subject to a set of appropriate boundary conditions. [The passage to quasistatic field theory and lumped circuit analysis occurs as the wavelength becomes infinite (or, equivalently, as c → ∞).] The Helmholtz equation is separable in eleven three-dimensional orthogonal coordinate systems so that product solutions may be formed. The helical coordinate system is not one of them. The helical structure supports propagation along the longitudinal (z) axis with traveling wave variations of the form e^(j(ωt - βz)), where β is to be determined. (We have temporarily neglected dissipation so that the propagation factor γ = α + jβ → jβ.) The fields may be decomposed into transverse and longitudinal components, there being both TE and TM modes present along this anisotropic wave guide. Further, the ∇ operator separates into a transverse and a longitudinal operator, that is, ∇ = ∇_t - (jβ)z, to give the Laplacian operator ∇² = ∇_t² - β², so that the D'Alembertian operator [∇² + k²] = [∇_t² + (k² - β²)], and Equation (1) separates into the operations: ∇_t² E_t - τ² E_t = 0 ∇_t² H_t - τ² H_t = 0 (2) on the transverse field components, and the operations: ∇_t² E_z - τ² E_z = 0 ∇_t² H_z - τ² H_z = 0 (3) on the longitudinal field components. In these expressions, the all important radial wave number, τ, is given by: τ² = -(γ² + k²) = β² - k² (4) The central issue for wave propagation on helical structures is the behavior of τ as the frequency and helix geometry (a, s, and ψ) are varied. Unlike the case of a TEM transmission line, and this is important, E_z ≠ 0 and H_z ≠ 0 on the helix wave guide. Furthermore, β ≠ k (where k² = ω²με is the plane wave propagation constant). From the cylindrical symmetry, the axial component of the internal (i) and external (o) time-harmonic fields will each be a superposition of forward propagating modes in the form of a product of radial, azimuthal, and axial functions of the form: E_z^(i,o)(r, φ, z) = [A_n^(i,o) I_n(τ_n r) + B_n^(i,o) K_n(τ_n r)] e^(j(ωt - nφ - β_n z)) (5) H_z^(i,o)(r, φ, z) = [C_n^(i,o) I_n(τ_n r) + D_n^(i,o) K_n(τ_n r)] e^(j(ωt - nφ - β_n z)) (6) where the eight constants A, B, C, D [both inside (superscript i) and outside (superscript o)] are to be determined for each mode from the boundary conditions and the driving source, and β_n is the propagation constant along the helix for the nth mode. There will also be a set of linearly superposed backward propagating modes, too, but, for the time being, we will consider only the forward propagating modes. The radial functions I_n(x) and K_n(x) are the modified Bessel functions of the first and second kind, respectively, of order n and argument x. For the case of interest, we have circular symmetry azimuthally around the solenoid, so we take only the mode with n = 0. (This has been called the helix transmission line T₀ mode.) In order to find the other field components inside and outside of the helix, one employs Ampere’s law and Faraday’s law to solve for E_r, E_φ, H_r, and H_φ in terms of derivatives of E_z and H_z. Consider the n = 0 mode (azimuthal symmetry in the fields). Physically, the fields must be finite at r = 0 and vanish as r → ∞. From the properties of the modified Bessel functions, this implies that A^o, B^i, C^o, D^i must all vanish. Using ∂I₀(τr)/∂r = τ I₁(τr) and ∂K₀(τr)/∂r = -τ K₁(τr) (7) and the fact that K₀(τr) diverges as r → 0, expressions for the fields within the helix (r ≤ a) can be found. Outside the helix (r ≥ a), I₀(τr) diverges as r → ∞, and expressions for the external fields are readily determined. However, these expressions are all in terms of the unknown constants A^i, B^o, C^i, D^o and the propagation constant τ. These constants and the propagation factor can be determined by imposing the anisotropic boundary conditions across the helical surface at r = a. There are three boundary conditions to be considered at the surface of the anisotropic sheath helix. One may summarize these as follows: (1) The Maxwell boundary conditions at the cylindrical surface make the internal and external axial components of field strength along (i.e., parallel to) the wire vanish as r → a: E(a) = 0. This may be written as E_||^i(a⁻) = E_||^o(a⁺) = 0 (8) (2) The components of the electric field strength normal to the wire are continuous across the cylindrical boundary as approached from within or without: E_⊥^i(a⁻) = E_⊥^o(a⁺) (9) (3) Finally, the components of the magnetic field strength along the wire are continuous across the cylindrical boundary: H_||^i(a⁻) = H_||^o(a⁺) (10) Imposing these boundary conditions on the expressions for the internal and external fields gives four equations in the four remaining unknown constants A^i, B^o, C^i, D^o and the propagation constant τ: A^i I₀(τa) sinψ - (jωμ/τ) C^i I₁(τa) cosψ = 0 (11) B^o K₀(τa) sinψ + (jωμ/τ) D^o K₁(τa) cosψ = 0 (12) A^i I₀(τa) cosψ + (jωμ/τ) C^i I₁(τa) sinψ = B^o K₀(τa) cosψ - (jωμ/τ) D^o K₁(τa) sinψ (13) C^i I₀(τa) sinψ + (jωε/τ) A^i I₁(τa) sinψ = D^o K₀(τa) sinψ - (jωε/τ) B^o K₁(τa) cosψ (14) These are four linear equations in five unknowns... *Kenneth L. Corum is at CPG Technologies, LLC, 104 River Rd., Plymouth, NH, 03264. **James F. Corum is Chief Scientist at the Institute for Software Research, Inc., 1000 Technology Drive, Suite 1110, Fairmont, WV 26554. © 2001 IEEE