Electromagnetic waves and transmission lines

In short

This topic covers how time-varying electric and magnetic fields propagate as waves through free space, dielectrics and conductors, and how those same waves travel as guided signals along two-conductor structures such as coaxial cable, twisted pair, microstrip and overhead lines. Maxwell's equations give the field picture; the telegrapher's equations give the circuit picture, and the two are the same physics seen from different ends. It is a guaranteed scoring area in GATE EC (electromagnetics typically carries a steady 5-10 marks, mostly on plane waves, skin depth, reflection and line impedance), and it appears in ISRO, DRDO, BARC, BEL, ECIL and state PSU papers. In industry it decides whether a PCB trace, an antenna feeder, a radar waveguide run or a 5G RF front end works at all - asked at Qualcomm, Nokia, Ericsson, Samsung R&D, Tejas Networks, Astra Microwave, Data Patterns, SAMEER, Sterlite and Jio/Airtel network teams.

Explained

Maxwell's equations and the wave equation

• Gauss's law: ∇·D = ρv - electric flux diverges from charge. Gauss's law for magnetism: ∇·B = 0 - no magnetic monopoles, flux lines close on themselves.
• Faraday's law: ∇×E = −∂B/∂t - a changing magnetic field creates a circulating electric field.
• Ampere-Maxwell law: ∇×H = J + ∂D/∂t. The second term is displacement current density, Maxwell's addition; without it the equations are inconsistent with charge conservation and no wave exists.
• Constitutive relations: D = εE, B = μH, J = σE, with ε = ε0εr, μ = μ0μr.
• Taking the curl of Faraday's law and substituting Ampere's law gives the wave equation; in phasor form for a source-free medium ∇²E = γ²E, where γ² = jωμ(σ + jωε). The solution is a travelling wave e^−γz, so the fields propagate rather than act instantly.
• Boundary conditions: tangential E is continuous across any interface; tangential H jumps by the surface current; normal D jumps by the surface charge; normal B is continuous. On a perfect conductor, tangential E = 0 and normal B = 0.
• Poynting theorem: S = E × H (W/m²) is the instantaneous power flow per unit area; its closed-surface integral is the net power leaving the volume, which equals minus the rate of change of stored energy minus the ohmic dissipation inside.

Uniform plane waves: parameters, polarization and power

• A uniform plane wave has E and H constant in magnitude and direction over any plane perpendicular to the direction of travel; E, H and the propagation direction form a right-handed orthogonal set (a TEM wave). There is no field component along the direction of travel.
• Key quantities: propagation constant γ = α + jβ; attenuation constant α (Np/m); phase constant β (rad/m); wavelength λ = 2π/β; phase velocity vp = ω/β; intrinsic impedance η = E/H.
• In a lossless dielectric (σ = 0): α = 0, β = ω√(με), vp = c/√(μrεr), η = √(μ/ε) = 377/√(εr/μr) Ω. In free space η0 ≈ 377 Ω and vp = c ≈ 3 × 10⁸ m/s.
• Polarization is the path traced by the tip of E in a fixed plane: linear (two orthogonal components in phase), circular (equal amplitudes, 90° phase difference), elliptical (the general case). Circular polarization is specified as right- or left-hand; satellite TV and GNSS links use it because it is insensitive to receiver rotation and to Faraday rotation in the ionosphere.
• Group velocity vg = dω/dβ carries the information; in a non-dispersive medium vg = vp. Where β is not proportional to ω, different frequencies travel at different speeds - dispersion - and pulses spread.
• Average power density of a wave in a lossless medium = |E0|²/(2η) = η|H0|²/2 for peak amplitudes.

Lossy media, good conductors and skin effect

• Loss tangent tan δ = σ/(ωε) classifies a medium at a given frequency: tan δ ≪ 1 → low-loss dielectric; tan δ ≫ 1 → good conductor; comparable → quasi-conductor. The same material can be either, depending on frequency.
• Low-loss dielectric: α ≈ (σ/2)√(μ/ε), β ≈ ω√(με) - the wave travels almost as in a lossless medium but decays slowly.
• Good conductor: α = β = √(πfμσ); skin depth δ = 1/α = 1/√(πfμσ) is the depth at which the amplitude falls to 1/e (36.8%) of its surface value. Power falls to 1/e² in the same distance.
• δ falls as 1/√f: current crowds into a thin surface layer at high frequency, so AC resistance rises and only the surface finish and plating matter. This is why RF conductors are silver-plated and why hollow tubes work as well as solid rods.
• Surface resistance Rs = 1/(σδ) Ω per square; the intrinsic impedance of a good conductor is ηc = (1 + j)Rs, so E and H are 45° out of phase and the wave is heavily attenuated.
• In a perfect conductor (σ → ∞) δ = 0: fields do not penetrate, which is the basis of shielding, ground planes and waveguide walls.

Reflection and refraction at boundaries

• At normal incidence on a plane interface between media of intrinsic impedances η1 and η2: Γ = (η2 − η1)/(η2 + η1), τ = 2η2/(η1 + η2), and 1 + Γ = τ.
• Γ = −1 for a wave hitting a perfect conductor (total reflection with phase reversal, standing wave with a null at the surface); Γ = 0 when η2 = η1 (matched). Power reflected = |Γ|², transmitted = 1 − |Γ|².
• The incident and reflected waves add to a standing wave: maxima and minima repeat every λ/2, adjacent maximum and minimum are λ/4 apart.
• At oblique incidence, Snell's laws apply: θi = θr and √(ε1) sin θi = √(ε2) sin θt (for non-magnetic media, n1 sin θi = n2 sin θt). Two polarizations behave differently - perpendicular (TE, s) and parallel (TM, p) - with separate Fresnel coefficients.
• Brewster angle (parallel polarization only): tan θB = √(ε2/ε1); the reflected wave vanishes, which is how polarizing filters and low-reflection windows work.
• Total internal reflection occurs when going from dense to rarer medium beyond the critical angle sin θc = √(ε2/ε1) - the principle behind optical fibre and dielectric waveguides.

Transmission lines: distributed model and the telegrapher's equations

• A line must be treated as distributed when its length is comparable with the wavelength (a common rule of thumb: l > λ/10, or rise time shorter than twice the one-way delay). Below that it is just a lumped wire.
• Primary (per-unit-length) constants: R (Ω/m, conductor loss), L (H/m), G (S/m, dielectric leakage), C (F/m). They depend on geometry and materials, not on line length.
• Telegrapher's equations: −dV/dz = (R + jωL)I and −dI/dz = (G + jωC)V. Solving gives V(z) = V⁺e^−γz + V⁻e^+γz, a forward and a backward wave.
• Secondary constants: γ = α + jβ = √((R + jωL)(G + jωC)) and characteristic impedance Z0 = √((R + jωL)/(G + jωC)). For a lossless line Z0 = √(L/C) (purely real), β = ω√(LC), vp = 1/√(LC) and α = 0.
• Z0 is the ratio of voltage to current of a single travelling wave - it is the impedance an infinitely long (or perfectly matched) line presents, not a resistance that dissipates power on a lossless line.
• Velocity factor vp/c = 1/√(εr,eff): about 0.66 for solid-polyethylene coax, 0.8-0.86 for foam coax, about 0.95-0.99 for air lines.
• Distortionless (Heaviside) condition: R/L = G/C. Then α = √(RG) and β = ω√(LC), so attenuation is frequency-independent and vp is constant - no phase distortion. Historically achieved by loading coils (Pupin coils) on telephone trunks to raise L.
• Line types: two-wire open line (Z0 roughly 300-600 Ω), coaxial (50 Ω for instruments and RF, 75 Ω for cable TV and video), twisted pair (100 Ω differential in LAN cable, 120 Ω in some industrial buses), microstrip and stripline on PCBs, coplanar waveguide. 50 Ω is a practical compromise between lowest coax loss (about 77 Ω) and highest power handling (about 30 Ω).

Reflection, VSWR and impedance transformation on a line

• Load reflection coefficient: ΓL = (ZL − Z0)/(ZL + Z0), a complex number with |ΓL| ≤ 1 for passive loads. At distance l from the load, Γ(l) = ΓL e^−2γl - on a lossless line only the phase rotates, at 2βl.
• Three special loads: matched (ZL = Z0, Γ = 0, no reflection, all power absorbed); short circuit (Γ = −1); open circuit (Γ = +1). The last two reflect all power.
• VSWR S = Vmax/Vmin = (1 + |Γ|)/(1 − |Γ|), ranging from 1 (perfect match) to ∞ (total reflection). Return loss RL = −20 log10|Γ| dB; mismatch (reflection) loss = −10 log10(1 − |Γ|²) dB.
• Along the line Zmax = S·Z0 at a voltage maximum and Zmin = Z0/S at a voltage minimum; both are purely real there.
• Input impedance of a lossless line of length l: Zin = Z0 (ZL + jZ0 tan βl)/(Z0 + jZL tan βl). This is the single most examined transmission-line formula.
• Special lengths (lossless): l = λ/4 → Zin = Z0²/ZL (impedance inverter - a short looks like an open and vice versa); l = λ/2 → Zin = ZL (and repeats every λ/2); any length with ZL = Z0 → Zin = Z0.
• Stubs: a shorted line shorter than λ/4 looks inductive, an open line of the same length looks capacitive - so a length of line is a reactive component at microwave frequencies. Shorted stubs are preferred in practice (no radiation, DC path for grounding).
• Lossy line: Zin → Z0 as l → ∞ regardless of load, because the reflected wave dies out. Total attenuation in dB = 8.686 α(Np/m) × l, and a mismatched lossy line shows a *lower* VSWR at its input than at its load.
• Transients: a step launched on a line travels at vp, reflects at the load with ΓL and again at the source with ΓS; the resulting bounce diagram explains overshoot and ringing on fast digital lines and is the basis of TDR fault location (distance = vp × t/2).

Practical limits, standards and Indian practice

• Loss mechanisms: conductor loss (rises as √f through skin effect), dielectric loss (rises roughly linearly with f through tan δ), radiation and leakage from poor shielding, and connector/joint loss. Cable attenuation is quoted in dB/100 m at stated frequencies.
• Power handling is limited by conductor heating (average power) and by dielectric breakdown at voltage maxima (peak power) - at VSWR S the voltage maximum rises to (1 + |Γ|) times its matched value for the same incident power, so cable power ratings are derated by roughly the factor S and high VSWR is a real failure mode in transmitters.
• Impedance discontinuities - a bad connector, a via, a sharp bend, a change in PCB dielectric - each create their own reflection; for digital links this shows as eye-diagram closure and is characterised by S-parameters on a vector network analyser.
• Indian regulatory and test context: spectrum assignment and transmitter clearance are handled by the WPC Wing (and SACFA siting clearance) of the Department of Telecommunications; TRAI regulates telecom service; the Telecommunications Act, 2023 has replaced the Indian Telegraph Act, 1885 and is being brought into force in phases, so check the current notification status. Telecom equipment sold in India generally needs TEC MTCTE certification, and EMC/EMI testing follows CISPR/IEC-based Indian standards at labs such as ERTL/STQC and SAMEER; cables are specified to BIS/IEC series standards (for example the IEC 61196 family for RF coaxial cables) - verify the exact standard number in force for a given cable.

Key points

• Displacement current - Maxwell's ∂D/∂t term; without it there is no electromagnetic wave.
• TEM wave - E, H and direction of travel mutually perpendicular; no field along the propagation direction.
• Intrinsic impedance - η = √(μ/ε); 377 Ω in free space, 377/√εr in a non-magnetic dielectric.
• Phase velocity in a dielectric - vp = c/√εr; wavelength shrinks by the same factor.
• Poynting vector - S = E × H, direction of power flow, units W/m²; average = |E0|²/2η.
• Loss tangent - σ/(ωε) decides whether a material behaves as a dielectric or a conductor at that frequency.
• Skin depth - δ = 1/√(πfμσ); falls as 1/√f; 66 µm in copper at 1 MHz, about 2 µm at 1 GHz.
• Surface resistance - Rs = 1/(σδ); conductor loss in all RF structures traces back to it.
• Normal-incidence reflection - Γ = (η2 − η1)/(η2 + η1); Γ = −1 at a perfect conductor.
• Brewster angle - tan θB = √(ε2/ε1); zero reflection for parallel polarization only.
• Primary line constants - R, L, G, C per metre; secondary - Z0 and γ derived from them.
• Characteristic impedance (lossless) - Z0 = √(L/C), real, independent of length and frequency.
• Reflection coefficient - ΓL = (ZL − Z0)/(ZL + Z0); magnitude fixed on a lossless line, phase rotates 2βl.
• VSWR - (1 + |Γ|)/(1 − |Γ|); 1 means matched, ∞ means total reflection; measures mismatch, not loss.
• Line input impedance - Zin = Z0(ZL + jZ0 tan βl)/(Z0 + jZL tan βl); repeats every λ/2.
• Quarter-wave line - impedance inverter, Zin = Z0²/ZL; short becomes open, open becomes short.
• Stub reactance - short stub < λ/4 is inductive; open stub < λ/4 is capacitive.
• Distortionless condition - RC = LG gives frequency-independent α and constant vp.
• Distributed treatment needed - when line length exceeds roughly λ/10 or the signal rise time is shorter than twice the one-way delay.

Formulas and examples

Example 1 - plane wave in free space and in a dielectric. A 100 MHz uniform plane wave in air has peak E0 = 10 V/m.

• η0 = 377 Ω → H0 = 10/377 = 26.5 mA/m.
• λ = c/f = 3 × 10⁸/10⁸ = 3 m; β = 2π/3 = 2.094 rad/m; period T = 10 ns.
• Pav = E0²/(2η0) = 100/754 = 0.133 W/m².
• Now the same wave enters a lossless dielectric with εr = 4, μr = 1: η = 377/2 = 188.5 Ω, vp = 3 × 10⁸/2 = 1.5 × 10⁸ m/s, λ = 1.5 m. Frequency does not change; wavelength and velocity halve.

Example 2 - skin depth and surface resistance in copper (σ = 5.8 × 10⁷ S/m, μ = μ0).

• At 1 MHz: δ = 1/√(π × 10⁶ × 4π × 10⁻⁷ × 5.8 × 10⁷) = 66.1 µm.
• At 1 GHz: δ scales as 1/√f, so δ = 66.1/√1000 = 2.09 µm.
• Rs at 1 GHz = 1/(σδ) = 1/(5.8 × 10⁷ × 2.09 × 10⁻⁶) = 8.25 mΩ per square.
• Interpretation: at 1 GHz a plating thickness of about 5δ ≈ 10 µm carries essentially all the current, so a 1 mm thick bar is 99% dead metal.

Example 3 - reflection at a dielectric interface. A wave in air strikes a thick slab with εr = 4 at normal incidence.

• η1 = 377 Ω, η2 = 188.5 Ω → Γ = (188.5 − 377)/(188.5 + 377) = −188.5/565.5 = −0.333 (phase reversal).
• τ = 1 + Γ = 0.667; power reflected = |Γ|² = 11.1%; power transmitted = 88.9%.
• Standing-wave ratio in air = (1 + 0.333)/(1 − 0.333) = 2.0; adjacent minima are λ/2 apart.

Example 4 - a 50 Ω line with a mismatched load. A lossless coax has L = 250 nH/m and C = 100 pF/m, and feeds ZL = 100 + j50 Ω at 500 MHz.

• Z0 = √(250 × 10⁻⁹/100 × 10⁻¹²) = √2500 = 50 Ω; vp = 1/√(LC) = 1/√(2.5 × 10⁻¹⁷) = 2 × 10⁸ m/s (velocity factor 0.667, so εr,eff ≈ 2.25 - solid polyethylene).
• λ = 2 × 10⁸/5 × 10⁸ = 0.4 m; β = 2π/0.4 = 15.71 rad/m.
• ΓL = (100 + j50 − 50)/(100 + j50 + 50) = (50 + j50)/(150 + j50) = 0.447 ∠26.6°.
• VSWR = 1.447/0.553 = 2.62; return loss = −20 log10(0.447) = 7.0 dB; mismatch loss = −10 log10(1 − 0.2) = 0.97 dB, i.e. 80% of the incident power reaches the load.
• Impedance extremes on the line: Zmax = 2.62 × 50 = 131 Ω, Zmin = 50/2.62 = 19.1 Ω.
• Input impedance λ/8 from the load (βl = 45°, tan βl = 1): Zin = 50(100 + j50 + j50)/(50 + j(100 + j50)) = 50(100 + j100)/(j100) = 50 − j50 Ω.
• If instead the load were a short and l = λ/8: Zin = j50 tan 45° = +j50 Ω (inductive); an open-circuited λ/8 line gives −j50 Ω (capacitive).

Example 5 - cable loss budget and fault location. A 10 m RG-58 jumper (about 0.7 dB/m at 1 GHz - always use the maker's datasheet, as grades vary) carries 10 W from a transmitter into an antenna with VSWR 2.

• Cable attenuation = 0.7 × 10 = 7 dB → power at the far end = 10 × 10^−0.7 = 2.0 W.
• VSWR 2 → |Γ| = (2 − 1)/(2 + 1) = 0.333 → mismatch loss = −10 log10(1 − 0.111) = 0.51 dB, so about 1.78 W is actually accepted by the antenna.
• A TDR pulse on a long run of the same cable type (velocity factor 0.66, vp = 1.98 × 10⁸ m/s) returns after 480 ns: fault distance = 1.98 × 10⁸ × 480 × 10⁻⁹/2 = 47.5 m.

Where it is used

• Telecom and mobile networks: 50 Ω feeder and jumper runs from BTS radios to antennas on Jio, Airtel and Vi towers; VSWR and return-loss sweeps with a site analyser are routine acceptance tests, and a VSWR alarm is one of the commonest tower faults.
• PCB and high-speed digital design: controlled-impedance microstrip and stripline for DDR, PCIe, USB, HDMI and Ethernet - length matching, termination and via stubs are all transmission-line problems; roles in product companies and EMS houses in Bengaluru, Pune, Hyderabad and Noida.
• Defence, space and radar: BEL, DRDO labs (LRDE, RCI), ISRO/URSC, Astra Microwave and Data Patterns use feeders, couplers and stub networks whose design rests on Z0, VSWR and quarter-wave behaviour; skin depth governs plating choices and shielding.
• Power and instrumentation: travelling-wave fault location on EHV transmission lines and TDR/OTDR fault location on underground cable and fibre in state discoms and telecom networks.
• Test and measurement: vector network analysers, calibration kits and 50 Ω loads at Keysight/Rohde & Schwarz India, SAMEER, ERTL and NABL-accredited EMC labs - S-parameter measurement is reflection-coefficient measurement.
• Broadcast and cable TV: 75 Ω coax distribution for DD and local cable networks, where impedance mismatch shows up directly as ghosting and signal loss.

Interview questions

• Q: Why did Maxwell add the displacement current term? A: Ampere's original law contradicts charge conservation for time-varying fields and leaves a gap in a capacitor circuit; ∂D/∂t closes the loop and is what makes wave propagation possible.
• Q: What does the characteristic impedance of a lossless line physically mean? A: It is the ratio of voltage to current of a single travelling wave on that line, set purely by geometry and material as √(L/C) - it is independent of length and does not dissipate power.
• Q: Does a line with Z0 = 50 Ω dissipate power like a 50 Ω resistor? A: No. On a lossless line the power is carried away by the wave; it only looks resistive at the input because the energy never returns.
• Q: Why is 50 Ω the standard RF impedance? A: It is a compromise in coaxial cable between minimum attenuation (around 77 Ω) and maximum power handling (around 30 Ω), and it became the instrument and connector standard.
• Q: What does a VSWR of 2 mean in power terms? A: |Γ| = 1/3, so 11.1% of the incident power is reflected and about 89% is accepted - roughly 0.5 dB of mismatch loss.
• Q: Why does a quarter-wave line turn a short into an open? A: Zin = Z0²/ZL, so ZL = 0 gives Zin → ∞; physically the reflected wave returns 180° out of phase after the 2βl = 180° round trip and cancels the current at the input.
• Q: How does skin depth change with frequency, and why does it matter? A: δ ∝ 1/√f, so at GHz frequencies current flows in a layer a few microns thick, raising AC resistance and making surface finish, plating and roughness the dominant loss factors.
• Q: When must a wire be treated as a transmission line? A: When its length is a significant fraction of the wavelength (roughly λ/10 or more), or when the signal rise time is shorter than twice the one-way propagation delay.
• Q: What is the distortionless condition and why is it desirable? A: RC = LG; then α is frequency-independent and β ∝ ω, so all frequency components are attenuated equally and travel at the same speed, preserving pulse shape.
• Q: Can the reflection coefficient magnitude exceed one? A: Not for a passive load; |Γ| > 1 implies the load returns more power than it receives, which only happens with active or negative-resistance devices.
• Q: Why is a shorted stub preferred over an open stub in practice? A: It does not radiate from an open end, is mechanically robust, provides a DC ground path, and its effective length is not corrupted by end fringing capacitance.
• Q: What is the difference between phase velocity and group velocity? A: Phase velocity is the speed of a constant-phase point (ω/β); group velocity is the speed of the envelope or information (dω/dβ). They are equal only in a non-dispersive medium.

Common mistakes

• Treating Z0 as a resistance that absorbs power, or assuming it changes with line length.
• Confusing VSWR with loss - a high VSWR means reflection and local voltage stress, not necessarily heat in the cable; a lossy line can hide a bad load behind a deceptively low input VSWR.
• Forgetting that wavelength, not physical length, matters: using λ in free space instead of λ = c/(f√εr) on the line, so every βl and stub length comes out wrong.
• Using degrees and radians inconsistently in tan βl, or forgetting that βl = 2πl/λ.
• Applying the Brewster-angle formula to perpendicular polarization, where no such zero-reflection angle exists.
• Mixing nepers and decibels: 1 Np = 8.686 dB, and α in the exponential is in Np/m.
• Using peak values in Pav = |E|²/(2η) where RMS values were given (then the factor 2 must be dropped), and dropping the factor √εr when computing η in a dielectric.

Related topics

Smith chart and impedance matching · Antennas and wave propagation · Microwave engineering basics · RF circuit design basics · Signals and systems (Fourier and frequency response) · Optical fibre communication.