Inductance Calculations

Calculating inductance for standard geometries is a core skill tested in GATE electromagnetics. The three most important configurations are the solenoid, the toroid, and the coaxial cable. Each has a closed-form formula derived from Ampere's law and the definition L = N·Φ / I. Mastering these derivations builds confidence for handling unfamiliar configurations.

Core Concept Explanation

Every inductance calculation follows the same three-step procedure: first apply Ampere's law to find H and then B inside the geometry, then compute the total flux Φ through the relevant cross-section, and finally use L = N·Φ / I. The geometry determines which coordinate system to use and where the field is confined.

Solenoid Inductance

A solenoid is a tightly wound helix of wire with N turns, length l, and cross-sectional area A. For an ideal solenoid (l >> radius), the magnetic field inside is uniform and axial: B = μ₀·N·I / l. Outside the solenoid the field is essentially zero. The flux through one turn is Φ = B·A = μ₀·N·I·A / l. Total flux linkage λ = N·Φ = μ₀·N²·I·A / l. Dividing by I gives L = μ₀·N²·A / l. Using turn density n = N/l, this is often written L = μ₀·n²·V where V = A·l is the volume.

Toroid Inductance

A toroid is a solenoid bent into a closed ring (doughnut shape). Applying Ampere's law along a circular path of radius r inside the core gives H·(2πr) = N·I, so H = N·I / (2πr). The field varies with r inside the core. For a rectangular cross-section toroid with inner radius a, outer radius b, and height h, the flux is found by integrating: Φ = ∫(a to b) μ₀·H·h·dr = μ₀·N·I·h·ln(b/a) / (2π). With N turns, L = μ₀·N²·h·ln(b/a) / (2π). The key advantage of the toroid is that the field is entirely confined inside the core, so there is no external fringing flux and no EMI radiation.

Coaxial Cable Inductance

A coaxial cable has a solid inner conductor of radius a and a hollow outer conductor of inner radius b. Current I flows inward on the inner conductor and returns outward on the outer conductor. By Ampere's law, the magnetic field exists only in the annular region a < r < b: B = μ₀·I / (2πr). The flux per unit length between the conductors is Φ/l = ∫(a to b) B·dr = μ₀·I·ln(b/a) / (2π). Since the coaxial cable is a single-turn structure (N = 1), inductance per unit length is L/m = μ₀·ln(b/a) / (2π). Typical coaxial cables have L/m in the range of 200–300 nH/m.

Mathematical Expression

All three formulas share the same mathematical structure: they involve μ₀ (or μ₀μr for non-air cores), a geometric factor involving dimensions, and N² for wound structures. The logarithm ln(b/a) appears whenever the field is non-uniform and integration over a radial cross-section is needed. For rectangular or circular uniform-field structures, the integration simply gives the area A, and no logarithm appears.

Practical Understanding

In RF transformers and inductors, toroidal cores are preferred because the closed magnetic path prevents flux leakage and reduces interference with nearby components. In high-frequency transmission lines, the coaxial cable formula determines the distributed inductance per unit length, which along with capacitance per unit length sets the characteristic impedance Z0 = √(L/C).

For power inductors, solenoids wound on ferrite cores achieve inductance values in the millihenry to henry range despite compact dimensions. If a ferromagnetic core with relative permeability μr is used, all inductance formulas simply replace μ₀ with μ = μ₀·μr.

Given:
Toroid: N = 200 turns, inner radius a = 4 cm = 0.04 m, outer radius b = 6 cm = 0.06 m
Height h = 2 cm = 0.02 m, core: air (μ₀ = 4π×10⁻⁷ H/m)

Why this formula applies:
Toroid with rectangular cross-section uses the logarithmic formula.

Formula:
L = μ₀·N²·h·ln(b/a) / (2π)

Substitution:
L = (4π×10⁻⁷) × (200)² × (0.02) × ln(0.06/0.04) / (2π)

Calculation:
ln(1.5) ≈ 0.405
L = (4π×10⁻⁷) × 40000 × 0.02 × 0.405 / (2π)
L = (4π×10⁻⁷ / 2π) × 40000 × 0.02 × 0.405
L = (2×10⁻⁷) × 40000 × 0.0081
L = (2×10⁻⁷) × 324
L = 648×10⁻⁷ = 6.48×10⁻⁵ H

Final Answer: L ≈ 64.8 μH for the air-core toroid.

Exam Tip: For toroid problems, always check if the cross-section is circular or rectangular. The formula L = μ₀·N²·h·ln(b/a)/(2π) applies to rectangular cross-sections. For a thin toroid where b-a << (a+b)/2, ln(b/a) ≈ (b-a)/r_avg, and the toroid formula reduces to the solenoid formula.

Comparison of Geometries

Mechanism Comparison Points

• Solenoid: Uniform field inside, near-zero outside. L = μ₀N²A/l. Valid when l >> radius. Field escapes at ends (fringing).
• Toroid: Non-uniform radial field (B ∝ 1/r). Entirely confined inside the core. L involves ln(b/a) due to radial integration.
• Coaxial cable: Same 1/r field as toroid but expressed as inductance per unit length. N = 1, so L/m = μ₀·ln(b/a)/(2π).
• Adding a ferromagnetic core with permeability μr multiplies all formulas by μr. Iron core (μr ≈ 1000) gives 1000× more inductance.
• For GATE: the coaxial inductance and characteristic impedance are closely linked. Z0 = (1/2π)·√(μ/ε)·ln(b/a).

Quick Revision

• Solenoid: L = μ₀N²A/l = μ₀n²Al. Scales as N², A, 1/l.
• Toroid (rectangular section): L = μ₀N²h·ln(b/a)/(2π). Key term: ln(b/a).
• Coaxial (per unit length): L/m = μ₀·ln(b/a)/(2π). Same form as toroid/2πN².
• All formulas scale with μ = μ₀μr. Ferrite core multiplies L by μr.
• Trap: Toroid formula assumes rectangular cross-section. Circular cross-section toroid needs different integral.
• Coaxial cable inductance does NOT depend on length explicitly; it is per unit length already.
• For thin toroid (b-a << r_avg), toroid formula reduces to solenoid: ln(b/a) ≈ (b-a)/r_avg gives L ≈ μ₀N²A/l_eff.