Transmission Lines and RF Wave Propagation

Introduction

At low frequencies, interconnections are often represented by ideal wires or lumped resistances, inductances and capacitances. This approximation becomes unreliable when the propagation time along a conductor is not negligible compared with the signal period or transition time. Voltage and current then vary with both position and time, and the interconnection must be treated as a distributed electromagnetic system—a transmission line.

Transmission-line theory connects circuit analysis with Maxwell’s equations. It is central to RF amplifiers, antennas, filters, mixers, high-speed digital links, microwave measurements and electromagnetic compatibility. Reflections caused by impedance discontinuities can reduce delivered power, distort modulation, increase device stress and create unintended radiation.

Learning Objectives

After studying this chapter, the reader should be able to:

• derive and apply the telegrapher’s equations;
• calculate characteristic impedance and propagation constant;
• analyze incident and reflected waves using a consistent coordinate convention;
• determine return loss, reflected power, standing-wave ratio and input impedance;
• distinguish phase velocity, group velocity and signal velocity;
• evaluate conductor and dielectric loss, dispersion and distortion;
• compare coaxial, stripline, microstrip and coplanar transmission lines; and
• apply matching principles to practical RF interconnections.

Distributed-Circuit Representation

A uniform transmission line is represented by four per-unit-length parameters:

• R: series resistance, in Ω/m;
• L: series inductance, in H/m;
• G: shunt conductance, in S/m;
• C: shunt capacitance, in F/m.

The parameters describe a differential section of length Δz. The series impedance is approximately (R + jωL)Δz, and the shunt admittance is approximately (G + jωC)Δz.

The distributed model is required when the line is electrically long. A common engineering guideline is to use transmission-line analysis when the line length exceeds approximately λ/10, where λ is the wavelength in the medium. For broadband or digital signals, the relevant wavelength corresponds to the highest significant spectral component, not merely the clock frequency.

Transmission-line theory normally assumes a uniform cross-section and a dominant transverse electromagnetic (TEM) or quasi-TEM mode. When transverse dimensions become comparable to wavelength, higher-order modes and full-wave electromagnetic effects must be considered.

Telegrapher’s Equations

Let z increase from the generator toward the load, and define current I(z,t) as positive in the +z direction. Applying Kirchhoff’s voltage and current laws to a differential section gives

∂V/∂z = −RI − L∂I/∂t,

∂I/∂z = −GV − C∂V/∂t.

For sinusoidal steady state with the phasor convention e^(jωt), these become

dV/dz = −(R + jωL)I,

dI/dz = −(G + jωC)V.

Differentiating the first equation and substituting the second produces the voltage wave equation:

d²V/dz² = γ²V,

where

γ = √[(R + jωL)(G + jωC)] = α + jβ.

Similarly,

d²I/dz² = γ²I.

Here α is the attenuation constant in Np/m and β is the phase constant in rad/m. The general solutions are

V(z) = V⁺e^(−γz) + V⁻e^(γz),

I(z) = (1/Z₀)[V⁺e^(−γz) − V⁻e^(γz)],

where V⁺ represents the wave traveling toward the load and V⁻ represents the wave traveling toward the generator. The characteristic impedance is

Z₀ = √[(R + jωL)/(G + jωC)].

Characteristic impedance is the voltage-to-current ratio of a single traveling wave. It is not generally equal to the DC resistance of the line.

Lossless Line

For an ideal lossless line, R = 0 and G = 0. Therefore,

Z₀ = √(L/C),

γ = jω√(LC),

β = ω√(LC),

vₚ = ω/β = 1/√(LC).

For a homogeneous TEM line, LC = με, so vₚ = 1/√(με). In a nonmagnetic dielectric, vₚ ≈ c/√εᵣ.

Low-Loss Approximation

When R ≪ ωL and G ≪ ωC,

Z₀ ≈ √(L/C),

β ≈ ω√(LC),

α ≈ (R/2)√(C/L) + (G/2)√(L/C).

Equivalently, for a nearly real characteristic impedance,

α ≈ R/(2Z₀) + GZ₀/2.

The first term represents conductor loss and the second dielectric leakage or loss.

Worked Example: Low-Loss Line Parameters

A line has R = 0.8 Ω/m, L = 250 nH/m, G = 20 μS/m and C = 100 pF/m at 1 GHz.

The low-loss conditions are satisfied because

ωL ≈ 1571 Ω/m ≫ 0.8 Ω/m,

ωC ≈ 0.628 S/m ≫ 20 μS/m.

Thus,

Z₀ ≈ √(250 nH/m ÷ 100 pF/m) = 50 Ω.

The phase constant is

β ≈ 2π(10⁹)√[(250 × 10⁻⁹)(100 × 10⁻¹²)]

= 31.42 rad/m.

The phase velocity is

vₚ = ω/β = 2.00 × 10⁸ m/s.

The attenuation constant is

α ≈ 0.8/(2 × 50) + (20 × 10⁻⁶)(50)/2

= 0.0085 Np/m.

Since 1 Np = 8.686 dB,

α ≈ 0.0738 dB/m.

Reflection at a Load

A wave is fully absorbed only when the load impedance equals the characteristic impedance. At the load, the voltage reflection coefficient is

ΓL = (ZL − Z₀)/(ZL + Z₀),

where ZL is the load impedance. This expression applies to complex impedances as written. For a passive load connected to a lossless line with real Z₀, |ΓL| ≤ 1.

Important special cases are:

• ZL = Z₀: ΓL = 0, matched load;
• ZL = 0: ΓL = −1, short circuit;
• ZL → ∞: ΓL = +1, open circuit;
• purely reactive load on a lossless line: |ΓL| = 1.

Spatial Reference Convention

To avoid ambiguity, define x as distance measured from the load toward the generator. Thus, the load is at x = 0, and a point l toward the generator is at x = +l.

With voltage amplitudes referenced at the load,

V(x) = V₀⁺e^(γx) + V₀⁻e^(−γx),

I(x) = (1/Z₀)[V₀⁺e^(γx) − V₀⁻e^(−γx)].

Since ΓL = V₀⁻/V₀⁺, the local reflection coefficient is

Γ(x) = ΓL e^(−2γx).

For a lossless line,

Γ(x) = ΓL e^(−j2βx).

Consequently, the magnitude of Γ is constant along a lossless line, while its phase changes by −2βx. On a lossy line, |Γ(x)| decreases toward the generator by e^(−2αx).

Worked Example: Complex Load Reflection

A 50 Ω lossless line is terminated by

ZL = 70 − j20 Ω.

The load reflection coefficient is

ΓL = [(70 − j20) − 50]/[(70 − j20) + 50]

= (20 − j20)/(120 − j20).

Multiplying numerator and denominator by 120 + j20 gives

ΓL = (2800 − j2000)/14800

= 0.1892 − j0.1351.

Therefore,

|ΓL| = 0.2325,

∠ΓL ≈ −35.5°.

The return loss is

RL = −20 log₁₀|ΓL| = 12.67 dB.

The voltage standing-wave ratio is

VSWR = (1 + |ΓL|)/(1 − |ΓL|)

= 1.61.

For a lossless line with real Z₀, the reflected-to-incident power ratio is

Pᵣ/Pᵢ = |ΓL|² = 0.0541.

Thus, approximately 5.41% of the incident power is reflected.

Standing Waves and VSWR

The total voltage is the superposition of incident and reflected waves. For a lossless line,

|V(x)| = |V₀⁺| |1 + ΓL e^(−j2βx)|.

Constructive and destructive interference produces voltage maxima and minima:

Vmax = |V₀⁺|(1 + |ΓL|),

Vmin = |V₀⁺|(1 − |ΓL|).

Hence,

VSWR = Vmax/Vmin = (1 + |Γ|)/(1 − |Γ|).

Adjacent voltage maxima, or adjacent minima, are separated by λ/2. A maximum and its nearest minimum are separated by λ/4.

Return loss and mismatch loss are distinct quantities. Return loss measures reflection:

RL = −20 log₁₀|Γ|.

Mismatch loss measures the reduction in delivered power caused by reflection:

ML = −10 log₁₀(1 − |Γ|²).

These power relations assume a real reference impedance and consistent power-wave definitions. For complex reference impedances, ordinary voltage-wave coefficients do not always provide power ratios directly.

Input Impedance of a Terminated Line

At a distance l from the load toward the generator, the input impedance is

Zin(l) = Z₀ [ZL + Z₀ tanh(γl)]/[Z₀ + ZL tanh(γl)].

For a lossless line, γ = jβ and tanh(jβl) = j tan(βl), giving

Zin(l) = Z₀ [ZL + jZ₀ tan(βl)]/[Z₀ + jZL tan(βl)].

This formula shows that a transmission line transforms impedance as a function of electrical length.

Important Special Cases

For a short-circuited lossless line,

Zin = jZ₀ tan(βl).

For an open-circuited lossless line,

Zin = −jZ₀ cot(βl).

For l = λ/4,

Zin = Z₀²/ZL.

The last result is the quarter-wave impedance transformation and is valid for a lossless line of exactly one-quarter wavelength, or an odd multiple thereof. A quarter-wave transformer matching two real positive resistances R1 and R2 requires

Zt = √(R1R2).

It does not directly match an arbitrary complex load without first removing or transforming the reactive part.

Worked Example: Impedance Transformation

A 50 Ω lossless line of length λ/8 is terminated in 100 Ω. Since βl = π/4 and tan(π/4) = 1,

Zin = 50(100 + j50)/(50 + j100).

After complex division,

Zin = 40 − j30 Ω.

Although the load is purely resistive, the input impedance is complex because the line is not an integer multiple of λ/2.

Phase Velocity, Group Velocity and Dispersion

The phase velocity is the velocity of a constant-phase point on a sinusoidal wave:

vₚ = ω/β.

For a modulated signal or pulse, the group velocity is

vᵍ = dω/dβ = 1/(dβ/dω).

If β is proportional to ω, phase and group velocities are equal and the line is nondispersive. If β is nonlinear with frequency, different frequency components experience different delays, causing waveform distortion.

Group delay through a line of length l is

τg = l(dβ/dω).

A constant group delay shifts a waveform in time without changing its shape. Variation of group delay over signal bandwidth causes phase distortion and can degrade error-vector magnitude, eye opening and pulse fidelity.

Group velocity should not automatically be interpreted as the velocity of information in strongly dispersive or anomalous media. Causality constrains the propagation of a new physical disturbance, even where calculated group velocity exceeds c or becomes negative.

Sources of Dispersion

Material dispersion occurs when ε or μ varies with frequency. Geometric or waveguide dispersion occurs when the mode propagation constant depends on cross-sectional geometry. Microstrip is dispersive because part of its field travels in the substrate and part in air. Rectangular waveguide is inherently dispersive because

β = √(k² − kc²),

where kc is the mode cutoff wavenumber.

For an ideal lossless TEM line in a frequency-independent homogeneous dielectric, β = ω√(με), and the line is nondispersive.

Loss and Distortion

Conductor Loss

Finite conductivity causes current crowding near conductor surfaces. The skin depth is

δ = √[2/(ωμσ)],

where σ is conductivity. Surface resistance is approximately

Rs = 1/(σδ) = √(πfμ/σ).

Thus, conductor attenuation generally increases with approximately √f in the classical skin-effect region. Surface roughness can increase effective resistance at millimetre-wave frequencies.

Dielectric Loss

A practical dielectric is often characterized by complex permittivity

εc = ε′ − jε″,

and loss tangent

tan δd = ε″/ε′.

For low-loss TEM structures, dielectric attenuation is approximately proportional to frequency and tan δd. The exact coefficient depends on field distribution and on how much electromagnetic energy resides in the dielectric.

Radiation and Discontinuity Loss

Open structures such as microstrip and coplanar waveguide may radiate at bends, gaps, launches and transitions. Connectors, vias and package interconnects can also excite unwanted modes. These effects are not always represented adequately by uniform RLGC parameters.

Distortionless-Line Condition

A line is theoretically distortionless when

R/L = G/C.

Under this Heaviside condition,

α = √(RG),

β = ω√(LC),

Z₀ = √(L/C).

Attenuation is then frequency independent and phase is linear with frequency. Practical RF lines rarely satisfy this condition over broad bandwidths, although equalization can compensate for some loss and dispersion.

Common Transmission-Line Structures

Coaxial Line

A coaxial cable has an inner conductor of radius a and an outer conductor with inner radius b. For a homogeneous dielectric,

C = 2πε/ln(b/a),

L = (μ/2π)ln(b/a),

Z₀ = (1/2π)√(μ/ε) ln(b/a).

For nonmagnetic material,

Z₀ ≈ [60/√εᵣ]ln(b/a) Ω.

Coaxial line supports a dominant TEM mode, provides strong shielding and is widely used for test equipment, antenna feeds and inter-module connections. At sufficiently high frequency, higher-order modes can propagate.

Stripline

Stripline places a conductor between two ground planes within a homogeneous dielectric. It supports a true TEM mode and offers good shielding and predictable impedance. Its disadvantages include fabrication complexity, dielectric loss and limited accessibility for components.

Microstrip

Microstrip consists of a conductor above a ground plane on a dielectric substrate. Fields occupy both air and dielectric, so propagation is quasi-TEM and weakly dispersive.

For negligible conductor thickness, a commonly used first-order estimate is

εeff ≈ (εᵣ + 1)/2 + [(εᵣ − 1)/2](1 + 12h/w)^(−1/2),

where w is trace width and h is substrate thickness. Approximate impedance formulas are

Z₀ ≈ [60/√εeff] ln(8h/w + w/4h), for w/h ≤ 1,

and

Z₀ ≈ 120π/{√εeff[w/h + 1.393 + 0.667 ln(w/h + 1.444)]}, for w/h ≥ 1.

These formulas are design approximations. Accurate work must include conductor thickness, copper roughness, solder mask, anisotropy, dispersion and fabrication tolerance.

Coplanar Waveguide

Coplanar waveguide places the signal conductor and adjacent ground conductors on the same surface. It supports convenient shunt grounding, compact transitions and easy integration with active devices. Ground continuity, slotline modes and via placement require careful control.

Waveguide

Metallic waveguide is not a two-conductor TEM line. It supports TE and TM modes above cutoff. It offers low loss and high power handling at microwave and millimetre-wave frequencies but is bulky and dispersive. Transmission-line concepts remain useful if mode-specific propagation constants and wave impedances are used.

Applied Case: A 2.4 GHz Load Interface

Background

This illustrative case considers a 2.4 GHz radio module connected through a nominal 50 Ω interconnect to an RF front-end input. De-embedded measurement gives

ZL = 68 − j16 Ω.

Problem

The mismatch must be quantified and a narrowband matching approach proposed.

Analysis

The reflection coefficient is

ΓL = [(68 − j16) − 50]/[(68 − j16) + 50]

= (18 − j16)/(118 − j16).

Rationalizing the denominator gives

ΓL = (2380 − j1600)/14180

≈ 0.1678 − j0.1128.

Therefore,

|ΓL| ≈ 0.2022,

RL ≈ 13.88 dB,

VSWR ≈ 1.51,

Pᵣ/Pᵢ ≈ 4.09%.

The mismatch may be acceptable for some receivers, but a power amplifier could require a closer match for efficiency, output power or reliability.

Approach and Solution

One idealized narrowband solution first cancels the load reactance with a series reactance of +j16 Ω. At 2.4 GHz, the corresponding ideal inductance is

L = 16/[2π(2.4 × 10⁹)] ≈ 1.06 nH.

The remaining 68 Ω resistance can then be transformed to 50 Ω using a quarter-wave section with

Zt = √(50 × 68) ≈ 58.3 Ω.

If the effective relative permittivity is approximately 3.0, the guided quarter wavelength is

l = c/[4f√εeff] ≈ 18.0 mm.

Outcome

In an ideal model, the network presents 50 Ω at 2.4 GHz. In production, the inductor’s parasitic capacitance, pad inductance, microstrip dispersion and dielectric tolerance shift the match. Electromagnetic simulation and vector network analyzer verification are therefore required.

Learning Points

• A moderate impedance mismatch can be quantified directly through Γ, return loss and VSWR.
• A quarter-wave transformer alone is not sufficient for a complex load.
• Reference-plane control and de-embedding are essential.
• A narrowband match may perform poorly away from its design frequency.

Measurement and Design Practice

A vector network analyzer measures scattering parameters. For a one-port device referenced to Zref,

S11 = Γ = (Zin − Zref)/(Zin + Zref).

Calibration moves the measurement reference plane to a defined location and removes systematic errors. Fixture de-embedding then removes launches or adapters not belonging to the device under test.

Time-domain reflectometry identifies discontinuities by mapping reflection delay to distance. Spatial resolution depends on measurement bandwidth and propagation velocity.

Good RF design practice includes:

• maintaining controlled impedance through traces, connectors and vias;
• minimizing abrupt width changes and poorly grounded bends;
• specifying substrate properties at the operating frequency;
• using electromagnetic simulation for electrically large discontinuities;
• applying tolerance and temperature analysis rather than relying only on nominal values;
• validating designs with calibrated measurements; and
• recording reference planes and de-embedding procedures.

Data-driven optimization and machine-learning surrogate models can accelerate geometry exploration, but their predictions remain constrained by training data and electromagnetic model quality. Final designs still require physical verification.

Challenges, Governance and Contemporary Developments

At millimetre-wave frequencies, packages, bond wires, vias and connectors become distributed structures. Surface roughness, dielectric anisotropy and manufacturing variation can dominate performance. Integrated transmission lines increasingly use substrate-integrated waveguide, slow-wave coplanar structures and on-chip microstrip or coplanar lines.

Professional RF practice also involves regulatory and ethical responsibilities. Unintended radiation must comply with electromagnetic compatibility requirements, while transmitters must satisfy spectrum allocations, emission masks and human-exposure limits. Misleading specification of return loss, cable loss or measurement uncertainty can result in unsafe or noncompliant products. Reproducible calibration, traceable measurements and honest reporting are therefore engineering obligations.

Chapter Summary

Transmission lines are distributed systems characterized by R, L, G and C per unit length. The telegrapher’s equations lead to the propagation constant γ = α + jβ and characteristic impedance Z₀. Impedance discontinuities produce reflected waves described by Γ, return loss, reflected power and VSWR. With distance measured from the load toward the generator, a point l toward the generator is x = +l and Γ(l) = ΓL e^(−2γl).

Terminated lines transform impedance according to their characteristic impedance and electrical length. Quarter-wave transformation is especially useful but directly matches only real resistances. Phase velocity describes sinusoidal phase progression, while group velocity and group delay govern modulated signals. Frequency-dependent loss and dispersion cause attenuation and waveform distortion. Coaxial line, stripline, microstrip, coplanar waveguide and metallic waveguide offer different combinations of loss, shielding, manufacturability, dispersion and power capability. Reliable RF implementation requires calibrated measurement, electromagnetic modeling, tolerance analysis and regulatory compliance.

Key Terms

Attenuation constant: The real part α of the propagation constant, measured in Np/m.

Characteristic impedance: The voltage-to-current ratio of a single traveling wave on a uniform line.

Dispersion: Frequency dependence of propagation velocity or phase constant.

Distributed circuit: A system whose electrical parameters are continuously distributed along its length.

Electrical length: Physical length expressed as phase shift, usually βl.

Group delay: The derivative of transmission phase with respect to angular frequency.

Group velocity: The velocity dω/dβ associated with a narrowband modulation envelope.

Mismatch loss: Reduction in delivered power caused by reflection.

Phase constant: The imaginary part β of the propagation constant, in rad/m.

Phase velocity: The velocity ω/β of a constant-phase point.

Propagation constant: The complex quantity γ = α + jβ.

Quasi-TEM mode: A mode whose fields are approximately transverse but occur in an inhomogeneous dielectric environment.

Reflection coefficient: Ratio of reflected voltage-wave amplitude to incident voltage-wave amplitude.

Return loss: The logarithmic reflection measure −20 log₁₀|Γ|.

Skin depth: Depth at which field magnitude falls to 1/e of its surface value in a conductor.

Standing wave: A spatial interference pattern formed by oppositely traveling waves.

Telegrapher’s equations: Coupled differential equations governing voltage and current on a transmission line.

VSWR: Ratio of maximum to minimum standing-wave voltage magnitude.

Review Questions

Short Answer Questions

1. Define characteristic impedance and distinguish it from DC resistance.

2. State the physical meanings of α and β.

3. Why is microstrip described as a quasi-TEM structure?

4. What is the relationship between reflection coefficient magnitude and reflected power on a lossless line with real Z₀?

5. Why is calibration reference-plane selection important in VNA measurements?

Long Answer Questions

1. Derive the telegrapher’s equations and obtain the voltage and current wave solutions.

2. Explain standing-wave formation and derive the relationship between VSWR and |Γ|.

3. Derive the input impedance of a terminated lossless transmission line and discuss open-, short- and quarter-wave cases.

4. Compare coaxial line, stripline, microstrip, coplanar waveguide and metallic waveguide for RF implementation.

5. Critically discuss conductor loss, dielectric loss, radiation loss and dispersion in broadband RF interconnects.

Analytical / Application Questions

1. A 75 Ω lossless line is terminated by 40 + j30 Ω. Calculate ΓL, return loss, VSWR and reflected power percentage.

2. A 50 Ω line is terminated in 25 Ω and has length 0.15λ. Determine its input impedance and explain the result on an impedance-transformation basis.

3. Design an ideal narrowband matching approach between a 50 Ω source and a load of 80 − j25 Ω at 1.8 GHz. State all assumptions and identify practical parasitic effects that would alter the design.

Multiple Choice Questions with Answers

Question 1

Which condition most directly indicates that an interconnection should be analyzed as a transmission line?

A. Its resistance is less than 1 Ω

B. Its length is a significant fraction of the relevant wavelength

C. Its load is purely resistive

D. Its conductor is made of copper

Correct Answer: B. Its length is a significant fraction of the relevant wavelength

Question 2

For a uniform transmission line, the propagation constant is

A. √[(R + jωL)(G + jωC)]

B. √[(R + jωC)/(G + jωL)]

C. R + G + jω(L + C)

D. √(L/C)

Correct Answer: A. √[(R + jωL)(G + jωC)]

Question 3

A 50 Ω lossless line is terminated by 70 − j20 Ω. What is the magnitude of the load reflection coefficient?

A. 0.135

B. 0.189

C. 0.233

D. 0.411

Correct Answer: C. 0.233

Question 4

Distance x is measured from the load toward the generator. For a lossless line, the local reflection coefficient is

A. ΓL e^(−j2βx)

B. ΓL e^(jβx)

C. ΓL e^(−jβx)

D. ΓL e^(j2βx)

Correct Answer: A. ΓL e^(−j2βx)

Question 5

What is the VSWR when |Γ| = 0.20?

A. 1.20

B. 1.25

C. 1.50

D. 2.00

Correct Answer: C. 1.50

Question 6

Group velocity is correctly defined as

A. ωβ

B. β/ω

C. dβ/dω

D. dω/dβ

Correct Answer: D. dω/dβ

Question 7

An ideal quarter-wave transformer directly matches 50 Ω to 98 Ω. What characteristic impedance is required?

A. 49 Ω

B. 70 Ω

C. 74 Ω

D. 148 Ω

Correct Answer: B. 70 Ω

Question 8

Which relation is the Heaviside condition for a distortionless line?

A. R/L = G/C

B. R/C = G/L

C. RG = LC

D. R + G = L + C

Correct Answer: A. R/L = G/C

Question 9

Why is microstrip normally classified as quasi-TEM rather than exact TEM?

A. It has no ground conductor

B. Its fields occupy both air and dielectric

C. It can carry only DC current

D. Its characteristic impedance is always complex

Correct Answer: B. Its fields occupy both air and dielectric

Question 10

A 50 Ω line is terminated by 68 − j16 Ω. Approximately what is |ΓL|?

A. 0.113

B. 0.168

C. 0.202

D. 0.320

Correct Answer: C. 0.202