Transmission Lines and Guided-Wave Structures

Introduction and context

A transmission line is a distributed electromagnetic system that transfers energy and information between separated locations. Unlike a lumped circuit, its voltage and current vary with position because the physical length is not negligible compared with wavelength. At microwave frequencies, conductors, dielectrics, waveguides, and resonant structures must be analysed directly from Maxwell’s equations.

This chapter connects the telegrapher’s-equation model of transmission lines with the full-field description of guided waves. It covers reflection, impedance transformation, matching, Smith-chart methods, transient behaviour, rectangular waveguides, modal orthogonality, cutoff, power flow, and attenuation.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Derive and interpret the telegrapher’s equations and their solutions.
2. Distinguish phase velocity, group velocity, characteristic impedance, propagation constant, and reflection coefficient.
3. Analyse lossy and lossless lines without applying lossless formulas outside their domain.
4. Design basic impedance-matching networks and interpret Smith-chart constructions.
5. Explain reflection lattices and transient reflections.
6. Derive the modal fields of a rectangular waveguide and identify TE/TM restrictions.
7. Calculate cutoff frequency, guide wavelength, phase/group velocity, power, and attenuation.
8. Select suitable guided structures for practical high-frequency systems.

1. Distributed-parameter transmission lines

For a uniform line, let z increase from source toward load. Per-unit-length parameters are resistance R (Ω/m), inductance L (H/m), conductance G (S/m), and capacitance C (F/m). The distributed circuit contains series impedance R dz + jωL dz and shunt admittance G dz + jωC dz.

Figure: Source, distributed RLCG line, and load with the coordinate origin at the source end.

Applying Kirchhoff’s laws to an elemental length and taking the limit gives the phasor telegrapher’s equations, with time dependence e^{jωt}:

The propagation constant and characteristic impedance are

where α is attenuation in nepers per metre and β is phase constant in radians per metre. Square-root branches must be selected consistently with a passive line and a forward wave that decays as it travels in the +z direction.

The general solution is

For a lossless line, R = G = 0, γ = jβ, β = ω√(LC), and Z₀ = √(L/C), which is real and positive. The phase velocity is vₚ = 1/√(LC).

For a low-loss line, where R ≪ ωL and G ≪ ωC,

Here Z₀ in the approximation is the lossless characteristic impedance. The first term represents conductor loss and the second dielectric loss.

2. Reflection, impedance and standing waves

At z = ℓ, the load condition is V(ℓ)/I(ℓ) = Zᴸ. The load reflection coefficient, defined using the line’s characteristic impedance, is

For a general lossy line, Z₀ may be complex, and Γ is generally complex even for a resistive load. The familiar statement |Γ| < 1 for every passive load is guaranteed for a real positive reference impedance, such as a lossless line, or more generally when the reference impedance and power-wave definition satisfy the required passivity conditions. It must not be asserted indiscriminately for an arbitrary complex Z₀ on a lossy line.

At a distance d measured from the load toward the source, the reflection coefficient is

For a lossless line, |Γ| is constant and its phase changes by −2βd. The input impedance at distance ℓ from the load is

For a lossless line this reduces to

A quarter-wave transformer is a special lossless case. If the load resistance Rᴸ and source-side resistance Rₛ are real and positive, a quarter-wave section has

and length λ/4, where λ is the wavelength on that section.

VSWR and power

For a lossless line with real positive Z₀, the voltage standing-wave ratio is

and the return loss is RL = −20 log₁₀|Γ| dB. The same VSWR formula is not generally valid when Z₀ is complex on a lossy line; in that situation one should use travelling-wave amplitudes, power waves, or directly evaluate maxima and minima from the actual V(z).

For a lossless line, the time-average power is

and the delivered load power is Pᴸ = (1 − |Γᴸ|²)|V⁺(ℓ)|²/(2Z₀). For a lossy line, power decreases with z and must be calculated using the complex fields or voltage-current product P(z) = ½ Re[V(z)I*(z)].

3. Worked reflection and matching example

A 50 Ω lossless line is terminated in Zᴸ = 100 − j50 Ω.

The load reflection coefficient is

Thus |Γᴸ| = √(0.4² + 0.2²) = 0.447. Therefore, VSWR = 1.447/0.553 ≈ 2.62 and return loss ≈ 6.99 dB.

For a single shunt-stub match, normalize the load: zᴸ = 2 − j1. Moving toward the generator by d gives y(d) = 1/z(d). Choose d so that Re[y(d)] = 1; the stub then supplies susceptance −Im[y(d)]. This is the standard Smith-chart construction: rotate toward the generator on a constant-|Γ| circle, locate the unit-conductance intersections, and add a shunt stub whose normalized susceptance cancels the remaining susceptance. The stub length is obtained from its open- or short-circuit stub admittance. The chart is graphical, but the condition Re[y] = 1 is analytical and provides an unambiguous design criterion.

4. Smith chart and reflection lattice

The Smith chart is a conformal plot of normalized impedance z = Z/Z₀ or admittance y = Y/Y₀ for a real reference impedance Z₀. Constant-resistance circles, constant-reactance arcs, and constant-|Γ| circles permit rapid transformations. It is valid as a direct impedance chart under the usual real-reference convention; lossy lines require a lossy-line chart or numerical propagation using γ.

A reflection lattice is particularly useful for pulses. Suppose a lossless 50 Ω line has a 100 Ω load. The load reflection coefficient is Γᴸ = (100−50)/(100+50) = 1/3. If the source resistance is 50 Ω, Γˢ = 0, so the first incident step reflects at the load with amplitude one-third and is absorbed at the source on its return. If the one-way delay is T, the source launches a step at t = 0, the load changes at t = T, and the returning wave reaches the source at t = 2T, where it produces no further reflection. A source mismatch would generate repeated reflections with successive amplitudes multiplied by ΓˢΓᴸ.

5. Guided waves from Maxwell’s equations

In a uniform guide along z, fields are represented as transverse modal fields multiplied by e^(−jβz) under e^{jωt} convention. In a homogeneous medium, k = ω√(με), and

where k꜀ is the cutoff wavenumber. Above cutoff, β is real. Below cutoff, write β = −jα, with α = √(k꜀² − k²) > 0. Then e^(−jβz) = e^(−αz), which decays in +z. Choosing β = +jα would produce e^(+αz), a growing solution, and is inappropriate for a forward decaying field in a passive semi-infinite guide.

The cutoff frequency is f꜀ = k꜀/(2π√(με)). In an air-filled guide, v = c/√(εᵣμᵣ) and λ = v/f.

For a propagating mode,

Thus vₚv_g = v². In a nondispersive ideal guide, phase velocity may exceed c without violating relativity because information and energy are associated with group velocity and causal signal fronts.

6. Rectangular waveguide modes

Consider a perfectly conducting rectangular guide with cross-section 0 < x < a and 0 < y < b, extending along z. Perfect-conductor boundary conditions require tangential electric field to vanish on the walls.

For TE modes, E_z = 0 and H_z ≠ 0. The longitudinal magnetic field is

with m,n non-negative integers, not both zero. The cutoff wavenumber is

For TM modes, H_z = 0 and E_z ≠ 0. The longitudinal electric field is

where m,n are positive integers. Therefore TM modes cannot have m = 0 or n = 0. TE modes may have one index zero, but TE₀₀ is excluded because it gives no nontrivial transverse field in a simply connected hollow rectangular guide.

The dominant mode is TE₁₀ because it has the lowest cutoff: f꜀,10 = v/(2a). For a > b, the next competing cutoff is usually TE₂₀ or TE₀₁; single-mode operation requires operation above f꜀,10 and below the next-mode cutoff, with practical margin for discontinuities.

For TE modes, the modal wave impedance is

For TM modes,

These are not ordinary lumped impedances; they relate transverse electric and magnetic field amplitudes for a particular mode.

Field reconstruction and orthogonality

For a mode with longitudinal field ψ_z, transverse fields follow from Maxwell’s equations. With e^{jωt} and e^(−jβz),

For TM modes, corresponding expressions are obtained by interchanging the longitudinal electric and magnetic roles. The transverse eigenfunctions satisfy orthogonality integrals over the cross-section. For distinct eigenfunctions ψ_p and ψ_q,

under the appropriate PEC boundary conditions. Orthogonality allows an arbitrary field to be expanded as a sum of independently weighted modes and makes modal power and coupling calculations tractable. Degenerate modes require orthogonalization within the degenerate subspace.

7. Power flow in a waveguide

The time-average power is obtained from the Poynting vector:

For a propagating mode with real β, P is positive in the direction of propagation when the field amplitudes are selected consistently. Below cutoff, β = −jα and the ideal guide supports an evanescent field. Its net longitudinal time-average power is zero in a uniform lossless below-cutoff region, although reactive electric and magnetic energy exists. Power can be transferred through a finite evanescent region by tunnelling, but it is not carried as a propagating mode.

Numerical waveguide example

An air-filled rectangular guide has a = 2.286 cm and b = 1.016 cm, the common WR-90 dimensions. For TE₁₀,

f꜀,10 = c/(2a) = 3.00×10⁸/(2×0.02286) ≈ 6.56 GHz.

At f = 10 GHz,

β = k√[1 − (f꜀/f)²], where k = 2πf/c ≈ 209.44 rad/m.

Since (6.56/10)² ≈ 0.430, β ≈ 209.44√0.570 ≈ 158.1 rad/m. The guide wavelength is λ_g = 2π/β ≈ 39.7 mm. The phase velocity is vₚ = 2πf/β ≈ 3.97×10⁸ m/s and the group velocity is v_g = c²/vₚ ≈ 2.27×10⁸ m/s.

8. Conductor and dielectric attenuation

Real waveguides attenuate because finite conductivity produces surface current loss and dielectric loss produces volumetric dissipation. For a good conductor, surface resistance is

where σ is conductivity. The conductor power loss per unit length is

and the conductor attenuation is

where P is transmitted modal power. The factor follows because power varies approximately as e^(−2α_c z), while field amplitude varies as e^(−α_c z).

For a dielectric with small loss tangent tanδ = ε''/ε', the dielectric loss per unit length is

and

For a homogeneous TEM line, the low-loss dielectric contribution is approximately α_d ≈ β tanδ/2. In a waveguide, the electric-field filling factor and modal distribution must be included; therefore the TEM expression should not be transferred blindly to TE or TM guides.

Attenuation is commonly specified in dB/m: α_dB = 8.686 α when α is in nepers/m. Surface roughness, joints, bends, mode conversion, radiation leakage, and dielectric nonuniformity can add excess loss.

9. Comparison of guided structures

10. Applied case: selecting a radar feed structure

Background. An illustrative radar subsystem must deliver several kilowatts in pulses near 10 GHz from a solid-state power-combining unit to an antenna. The interconnect is 0.8 m long and must tolerate high peak electric field.

Problem. A microstrip implementation is attractive for integration but may suffer dielectric heating, conductor loss, and limited peak-voltage handling. A coaxial cable offers shielding but may have higher loss and power limits at this frequency.

Analysis. The WR-90 guide has a TE₁₀ cutoff near 6.56 GHz and operates at 10 GHz. The next cutoff is approximately TE₂₀ at 13.12 GHz or TE₀₁ at 14.75 GHz, so a substantial single-mode interval exists, subject to transitions and tolerances. At 10 GHz, λ_g is approximately 39.7 mm, so bends and junctions must be designed relative to the guided wavelength. Finite conductivity and flange discontinuities must be included in the loss budget. Mode-converting transitions are assessed using S-parameters rather than only static impedance.

Approach and solution. Use rectangular waveguide for the high-power section, a calibrated waveguide-to-coax transition near the source, and a low-reflection antenna transition. Verify peak field against breakdown limits, thermal rise against duty cycle, and return loss over the operating band. Use full-wave simulation to identify higher-order-mode excitation at bends and junctions, then confirm the design with vector-network-analyser measurements and high-power testing.

Outcome. The waveguide solution provides superior power handling and lower loss than an equivalent small coaxial path, while the transition regions become the main integration risk.

Learning points. Structure selection is a multi-objective problem involving cutoff margin, attenuation, power density, mechanical tolerance, manufacturability, mode purity, and measurement uncertainty. A nominal impedance match alone does not guarantee acceptable electromagnetic performance.

11. Design practice, limitations and contemporary directions

Good practice includes defining the phasor convention, coordinate direction, reference impedance, and mode normalization before calculation. Use lossless formulas only when R = G = 0 or as explicitly stated approximations. Check dimensions, limiting cases, and power conservation. For high-speed digital links, combine frequency-domain S-parameters with time-domain reflectometry and causal, passive macromodels. For microwave packaging, electromagnetic co-simulation is increasingly combined with optimization and machine learning, but AI-generated designs require independent Maxwell-equation verification, measurement correlation, and traceability.

Regulatory and safety considerations include RF exposure, arcing and breakdown, electromagnetic compatibility, shielding, leakage, and safe operation of high-power sources. Design documentation should record materials, surface finishes, tolerances, calibration standards, uncertainty, and acceptance criteria.

Chapter Summary

Transmission lines are distributed electromagnetic systems described by R, L, G, and C. Their telegrapher’s equations produce the propagation constant γ and characteristic impedance Z₀. Reflection depends on the load and reference impedance; VSWR and the simple power relation involving |Γ| are lossless-line results with a real positive Z₀ and must not be generalized to arbitrary lossy lines. Smith charts provide efficient real-reference matching methods, while reflection lattices explain pulse behaviour and repeated reflections.

Guided structures arise from Maxwell’s equations and support discrete TE, TM, TEM, or hybrid modes. In rectangular guides, TE₁₀ is normally dominant, TM indices must both be nonzero, and TE₀₀ is not a valid hollow-guide mode. Cutoff determines propagation, while phase and group velocities describe dispersion. Modal orthogonality enables field expansion and power calculations. Conductor and dielectric attenuation follow from surface and volume dissipation and must be evaluated using the actual modal fields. Engineering selection requires simultaneous consideration of loss, power, bandwidth, mode purity, transitions, manufacturability, and safety.

Key Terms

Transmission line: Distributed structure carrying electromagnetic energy.

Telegrapher’s equations: Differential voltage-current equations for a uniform line.

Propagation constant: γ = α + jβ, describing attenuation and phase variation.

Characteristic impedance: Ratio of forward-wave voltage to current.

Reflection coefficient: Complex ratio of reflected to incident wave amplitude.

VSWR: Maximum-to-minimum standing-wave voltage ratio for its stated applicability conditions.

Return loss: −20 log₁₀|Γ|, expressed in decibels.

Smith chart: Graphical mapping of normalized impedance or admittance.

Reflection lattice: Space-time diagram of travelling-wave reflections.

Cutoff frequency: Frequency separating evanescent and propagating modal behaviour.

TE mode: Mode with E_z = 0 and H_z ≠ 0.

TM mode: Mode with H_z = 0 and E_z ≠ 0.

TEM mode: Mode with both longitudinal electric and magnetic fields zero.

Dominant mode: Mode with the lowest cutoff frequency.

Guide wavelength: Axial wavelength λ_g = 2π/β.

Phase velocity: Speed of constant phase along the guide.

Group velocity: Derivative dω/dβ associated with energy and signal transport in the ideal guide.

Modal orthogonality: Integral property allowing distinct eigenmodes to be independently expanded.

Surface resistance: Effective AC resistance of a good conductor’s surface.

Evanescent field: Below-cutoff field decaying exponentially along the guide.

Review Questions

Short Answer Questions

1. State the telegrapher’s equations and define R, L, G, and C.
2. Why is the simple VSWR formula restricted to a real positive reference impedance?
3. Distinguish TE, TM, and TEM modes.
4. Explain why β = −jα is selected for a below-cutoff field decaying in +z under e^(−jβz).
5. Define modal orthogonality and state one practical consequence.

Long Answer Questions

1. Derive the input impedance of a uniform transmission line and obtain its lossless form.
2. Derive the cutoff frequencies and field restrictions of a rectangular waveguide.
3. Explain conductor and dielectric attenuation using Poynting’s theorem.
4. Discuss Smith-chart matching and compare series and shunt-stub procedures.
5. Derive the phase and group velocities of a hollow waveguide and interpret their product.

Analytical / Application Questions

1. A 75 Ω lossless line is terminated in 25 + j50 Ω. Calculate Γ, |Γ|, VSWR, and return loss.
2. A rectangular air-filled guide has a = 30 mm and operates at 8 GHz in TE₁₀. Calculate cutoff frequency, β, and guide wavelength.
3. A pulse travels on a lossless line with one-way delay 12 ns. The load reflection coefficient is 0.4 and the source reflection coefficient is −0.25. Determine the first three returning-wave amplitudes relative to the incident pulse and their arrival times at the source.

Multiple Choice Questions with Answers

Question 1

For a uniform lossless line with real positive characteristic impedance, which expression gives VSWR in terms of reflection-coefficient magnitude?

A. (1+|Γ|)/(1−|Γ|)

B. (1−|Γ|)/(1+|Γ|)

C. 1−|Γ|²

D. |Γ|/(1−|Γ|)

Correct Answer: A. (1+|Γ|)/(1−|Γ|)

Question 2

Using the e^(−jβz) convention, which choice represents a below-cutoff field that decays as z increases?

A. β = +jα, α > 0

B. β = −jα, α > 0

C. β = α, α > 0

D. β = 0

Correct Answer: B. β = −jα, α > 0

Question 3

Which mode is normally dominant in a hollow rectangular waveguide with a > b?

A. TE₁₀

B. TE₀₀

C. TM₁₀

D. TM₁₁

Correct Answer: A. TE₁₀

Question 4

For rectangular-waveguide TM modes, which index restriction applies?

A. m and n must both be zero

B. m may be zero but n must be positive

C. n may be zero but m must be positive

D. m and n must both be positive

Correct Answer: D. m and n must both be positive

Question 5

A 50 Ω lossless line has a 100 Ω resistive load. What is the magnitude of the load reflection coefficient?

A. 0.20

B. 0.333

C. 0.50

D. 0.667

Correct Answer: B. 0.333

Question 6

For an ideal lossless waveguide, what is the relation between phase and group velocities?

A. vₚ + v_g = c

B. vₚ/v_g = 1 always

C. vₚv_g = v²

D. vₚv_g = 0

Correct Answer: C. vₚv_g = v²

Question 7

Which formula gives the cutoff wavenumber of a rectangular-guide mode?

A. k_c = 2πab

B. k_c²=(mπ/a)²+(nπ/b)²

C. k_c=ωμ

D. k_c=β²−k²

Correct Answer: B. k_c²=(mπ/a)²+(nπ/b)²

Question 8

The conductor attenuation of a mode is obtained from conductor loss per unit length P′c and transmitted power P as:

A. α_c=P′_c/P

B. α_c=2P/P′_c

C. α_c=P′_c/(2P)

D. α_c=P′_cP

Correct Answer: C. α_c=P′_c/(2P)

Question 9

For a 50 Ω line terminated in 100 − j50 Ω, the calculated load reflection coefficient is:

A. 0.4 − j0.2

B. 0.2 + j0.4

C. −0.4 + j0.2

D. 0.5 − j0.5

Correct Answer: A. 0.4 − j0.2

Question 10

In a shunt-stub match using normalized admittance, the stub is attached after moving to a point where:

A. the normalized resistance is zero

B. the normalized conductance is one

C. the normalized reactance is one

D. the reflection coefficient is necessarily zero before the stub is attached

Correct Answer: B. the normalized conductance is one