Electromagnetics and Transmission Lines
Electromagnetics and Transmission Lines tests Maxwell's equations as working tools, boundary conditions, plane-wave propagation in lossy media, and transmission-line behaviour - reflection, standing waves, impedance matching and the Smith chart.
Field laws and boundaries
- ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t, ∇·D = ρ, ∇·B = 0. In a good conductor the skin depth is δ = √(2/(ωμσ)).
- Tangential E and normal B are continuous; normal D jumps by surface charge density and tangential H by surface current density.
- Intrinsic impedance η = √(μ/ε), with η₀ = 377 Ω in free space. Average power flow is S = ½Re(E × H*).
Transmission lines
- Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1 + |Γ|)/(1 − |Γ|).
- Input impedance: Z_in = Z₀(Z_L + jZ₀tanβl)/(Z₀ + jZ_Ltanβl). A quarter-wave section inverts the load, Z_in = Z₀²/Z_L; a half-wave section repeats it.
- Matching: a λ/4 transformer needs Z₀′ = √(Z₀Z_L) and works only at the design frequency. Stub matching cancels the line susceptance at one frequency.
Worked example
Z₀ = 50 Ω feeds Z_L = 100 + j50 Ω. Then Γ = (50 + j50)/(150 + j50) = 0.2 + j0.2 after multiplying by the conjugate. |Γ| = √0.08 = 0.283, so VSWR = 1.283/0.717 ≈ 1.79, and reflected power is |Γ|² = 8 %. With ∠Γ = 45°, the first voltage minimum lies at (θ + π)/(4π)·λ = 0.3125λ from the load toward the generator.
Traps
Confusing the load reflection coefficient with the value at a point along the line; using |Γ|² as the power delivered to the load; forgetting that Smith-chart rotation toward the generator is clockwise; applying a quarter-wave transformer to a complex load before cancelling the reactance; assuming skin depth marks a hard field cutoff rather than 1/e decay.