Propagation Constant & Phase Velocity

Fig 1 — Two sinusoids at 1.00 and 1.02 GHz on a coax with velocity factor 0.66: their sum is a carrier (moving at the phase velocity) inside an envelope (moving at the group velocity), with a beat length of 9.9 m

v(z) at one instantenvelope 2 cos(Δβ z/2)

Introduction to Signal Propagation

A transmission line does two things to a signal: it delays it and it attenuates it. Part 3 showed that both effects are captured by one complex number, the propagation constant γ, and Part 4 examined the other key line parameter, the characteristic impedance. This article looks at γ in depth — what its real and imaginary parts mean, how to calculate them exactly and approximately, and the two different velocities that describe how fast a signal travels.

The distinction between those velocities matters in practice. The phase velocity sets the wavelength on the line and therefore the length of every quarter-wave transformer, stub and resonator. The group velocity sets how fast information — a pulse, a modulation envelope, a data bit — actually arrives. On a TEM line they are equal. On a waveguide or a dispersive microstrip they are not, and confusing them leads to wrong delays, wrong phase shifts and distorted signals.

What Is the Propagation Constant?

On any uniform line the forward wave has the form V(z) = V₀⁺e^−γz. Splitting γ into real and imaginary parts separates the two effects:

Propagation constant

γ = α + j β = \sqrt{(R^{'} + j ω L^{'}) (G^{'} + j ω C^{'})}

V_{0}^{+} e^{- γ z} = V_{0}^{+} e^{- α z} e^{- j β z}

e^−αz is a real, decaying envelope; e^−jβz has unit magnitude and only rotates the phase. Units: γ in m⁻¹, α in Np/m, β in rad/m.

Because γ depends only on the per-unit-length parameters and the frequency, it is a property of the line alone — like Z₀, it does not depend on the source, load or length. A line of length ℓ multiplies a forward wave by e^−γℓ, which is why the transmission coefficient S₂₁ of a matched line section is exactly e^−γℓ.

Attenuation Constant

The attenuation constant α measures how quickly the amplitude decays. Over a length ℓ the voltage and current fall by e^−αℓ and the power by e^−2αℓ, since power is proportional to the square of the amplitude.

Derivation — nepers, decibels and the factor 8.686

The natural logarithm of an amplitude ratio is measured in nepers (Np); the decibel is defined from the power ratio. For a matched line of length ℓ:

loss [Np] = \ln \frac{| V (0) |}{| V (ℓ) |} = α ℓ

loss [dB] = 10 \log_{10} \frac{P (0)}{P (ℓ)} = 10 \log_{10} e^{2 α ℓ} = 20 \log_{10} (e) α ℓ

Conversion

α [dB/m] = 8.686 α [Np/m]

1 Np = 8.686 dB. Datasheets quote loss in dB per metre, per 100 m or per 100 ft.

The attenuation is the sum of independent contributions from the conductors, the dielectric and radiation, α = α_c + α_d + α_r, which Part 8 analyses mechanism by mechanism. Because they add in Np/m (and therefore in dB/m), each mechanism can be calculated separately and summed.

Phase Constant

The phase constant β is the phase shift per metre. A forward wave at position z lags the input by βz radians, so β is the spatial counterpart of ω: the phase changes by 2π in one period in time and by 2π in one guided wavelength in space.

Phase constant, guided wavelength and electrical length

β = \frac{2 π}{λ_{g}}

θ = β ℓ = 2 π \frac{ℓ}{λ_{g}}

β = \frac{2 π f}{c} \sqrt{ε_{eff}} = k_{0} \sqrt{ε_{eff}}

For a TEM or quasi-TEM line β = k₀√εeff, where k₀ = 2π/λ₀ is the free-space wavenumber.

β is what makes transmission-line circuits work. Every impedance transformation depends on the electrical length βℓ: a quarter-wave transformer is βℓ = π/2, a half-wave resonator βℓ = π. Part 6 builds the whole idea of electrical length on β.

The Complex Propagation Constant: Exact Formulas

The square root in γ can be evaluated by any complex-arithmetic tool, but closed-form expressions for α and β are often more convenient and give insight into how each depends on the line parameters.

Derivation — exact α and β from R′, L′, G′, C′

Write γ² = Z′Y′ = A + jB with real A and B:

A = R^{'} G^{'} - ω^{2} L^{'} C^{'} B = ω (R^{'} C^{'} + G^{'} L^{'})

Since (α + jβ)² = (α² − β²) + j2αβ, equating real parts gives α² − β² = A, and taking magnitudes gives α² + β² = |γ²| = √(A² + B²). Adding and subtracting:

Exact attenuation and phase constants

α = \sqrt{\frac{\sqrt{A^{2} + B^{2}} + A}{2}} β = \sqrt{\frac{\sqrt{A^{2} + B^{2}} - A}{2}}

Valid for any line and any frequency. Both α and β are taken positive, so that e^−γz decays and lags in +z.

For a low-loss line (R′ ≪ ωL′, G′ ≪ ωC′) these reduce to the formulas derived in Part 3:

Low-loss line

α \approx \frac{R^{'}}{2 Z_{0}} + \frac{G^{'} Z_{0}}{2}

β \approx ω \sqrt{L^{'} C^{'}} [1 + \frac{1}{8} {(\frac{R^{'}}{ω L^{'}} - \frac{G^{'}}{ω C^{'}})}^{2}]

α_c = R′/(2Z₀) is the conductor loss and α_d = G′Z₀/2 the dielectric loss. The next correction to β is of second order in the loss ratios.

α and β of common lines

Line	Phase constant β	Attenuation (dominant terms)	Dispersion
Coax, dielectric εr	k₀√εr	α_c = R_s(1/a + 1/b)/(4πZ₀), α_d = (k₀√εr/2) tan δ	None (TEM)
Stripline	k₀√εr	α_c from incremental inductance; α_d = (k₀√εr/2) tan δ	None (TEM)
Microstrip	k₀√εeff(f)	α_c ≈ R_s/(Z₀w); α_d = k₀εr(εeff − 1) tan δ/[2√εeff(εr − 1)]	Weak, grows with f·h
Rectangular waveguide TE₁₀	√(k² − (π/a)²)	Wall loss only (no centre conductor), lowest of all at microwave frequencies	Strong near cutoff

For any line filled with a single dielectric, α_d = (β/2) tan δ: the dielectric loss per wavelength is fixed by the loss tangent alone, π tan δ nepers per guided wavelength. A substrate with tan δ = 0.02 (FR-4) therefore loses 0.55 dB per wavelength to its dielectric whatever the line geometry — the reason FR-4 becomes unusable for long resonant structures at microwave frequencies.

Phase Velocity

Follow a crest of the forward wave v(z, t) = |V₀⁺|e^−αzcos(ωt − βz). Its phase ωt − βz is constant, so dz/dt = ω/β. This is the phase velocity:

Phase velocity

v_{p} = \frac{ω}{β} = \frac{1}{\sqrt{L^{'} C^{'}}} = \frac{c}{\sqrt{ε_{eff}}}

λ_{g} = \frac{v_{p}}{f}

The velocity factor VF = v_p/c = 1/√εeff is printed on cable datasheets: 0.66 for solid PE, 0.69–0.70 for PTFE, 0.80–0.88 for foam.

The phase velocity is the speed of a pattern of constant phase in a single, endless sinusoid. Such a wave carries no information — it has been doing the same thing forever — so nothing prevents v_p from exceeding c, and in hollow waveguides it does. A signal, however, must start, stop or be modulated, and that requires a band of frequencies.

Group Velocity

The simplest signal with an "envelope" is the sum of two equal sinusoids at slightly different frequencies (Fig 1). Its envelope moves at a different speed from the carrier.

Derivation — group velocity from two tones

Let the two waves have frequencies ω ± Δω and phase constants β ± Δβ. Using cos x + cos y = 2 cos((x − y)/2) cos((x + y)/2):

\cos [(ω + Δ ω) t - (β + Δ β) z]

+ \cos [(ω - Δ ω) t - (β - Δ β) z]

= 2 \cos (Δ ω t - Δ β z) \cos (ω t - β z)

The second factor is the carrier, moving at ω/β = v_p. The first is the envelope; its points of constant phase move at Δω/Δβ, which becomes a derivative as Δω → 0:

Group velocity

v_{g} = \frac{d ω}{d β} = {(\frac{d β}{d ω})}^{- 1} = v_{p} - λ \frac{d v_{p}}{d λ}

The group velocity is the speed of the envelope, and hence of energy and information on a low-loss line. The second form uses v_p(λ) and shows that v_g < v_p when the phase velocity rises with wavelength.

The derivative dβ/dω has a direct practical meaning: it is the group delay per metre, τ_g = dβ/dω = 1/v_g. A network analyser measures group delay as −dφ/dω of S₂₁, and a filter or line with flat group delay passes a modulated signal without distortion.

For a TEM line, β = ω/v_p with constant v_p, so v_g = v_p and the line is non-dispersive. For other lines:

Hollow waveguide: β = √(k² − k_c²) gives v_p = c/√(1 − (f_c/f)²) > c and v_g = c√(1 − (f_c/f)²) < c, with v_pv_g = c² (air-filled).
Microstrip and CPW: εeff rises with frequency, so v_p falls slowly and v_g is a little lower still: v_g = c/[√εeff + f d√εeff/df].
RC line (low frequency): β ≈ √(ωR′C′/2), so v_p ∝ √f and v_g = 2v_p — strong dispersion, as on long telephone lines.

Relationship Between Frequency and Propagation Constant

The ω–β (dispersion) diagram plots frequency against phase constant and shows both velocities at a glance: the slope of a line from the origin to a point is the phase velocity ω/β, and the slope of the curve itself is the group velocity dω/dβ (Fig 2).

Fig 2 — ω–β diagram: an air TEM line and a PTFE coax are straight lines through the origin (v_p = v_g); WR-90 waveguide starts at its 6.56 GHz cutoff with zero slope (v_g = 0) and approaches the air line from above

air TEM line (v = c)PTFE coax (v = 0.69c)WR-90 TE₁₀

Fig 3 plots the two velocities against frequency. For WR-90 the phase velocity falls from infinity at cutoff towards c, while the group velocity rises from zero. For 50 Ω microstrip on 0.813 mm RO4003C the effect is much smaller but not negligible: by 20 GHz the group velocity is 3.7% below the phase velocity.

Fig 3 — Phase and group velocity (relative to c) for WR-90 waveguide and for 50 Ω microstrip on 0.813 mm RO4003C (Kirschning–Jansen dispersion)

WR-90 phase velocityWR-90 group velocity (dashed)microstrip phase velocitymicrostrip group velocity (dashed)

Propagation in Lossless Transmission Lines

On a lossless TEM line γ = jω√(L′C′): α = 0, β is proportional to frequency, and v_p = v_g = 1/√(L′C′) at every frequency. The line delays every component of a signal by exactly the same time, ℓ/v_p, and does not change its amplitude. Such a line is a perfect delay line: any waveform comes out of it unchanged except for the delay, as the d'Alembert solution of Part 3 showed.

The phase shift through the line, θ = βℓ = ωℓ/v_p, is a linear function of frequency with slope equal to the delay. This linear phase is what "non-dispersive" means in the frequency domain, and it is the reason true time-delay lines, rather than phase shifters, are used to steer wideband phased arrays without beam squint.

Propagation in Lossy Transmission Lines

Real lines add three effects:

Frequency-dependent attenuation. α_c ∝ √f and α_d ∝ f, so high frequencies are attenuated more than low ones. A digital pulse loses the sharpness of its edges; this is the dominant impairment on long PCB traces and cables at multi-gigabit rates and is corrected by equalisation.
Small changes to β. Loss adds a second-order correction to β and, through the internal inductance and the frequency dependence of εr, makes v_p slightly frequency dependent even on a coax.
Strong dispersion at low frequency. When R′ is comparable to ωL′ the line becomes an RC line in which both α and v_p vary as √f (Example 10).

Over most of the RF range the low-loss approximation is excellent, and the practical rule is simple: compute β (and the electrical length) as if the line were lossless, and compute α from R′/2Z₀ + G′Z₀/2.

Dispersion and signal distortion

A signal with bandwidth B is distorted by a line if its group delay or its attenuation changes appreciably across B. The group-delay spread of a length ℓ is

Group-delay spread across a band

Δ τ_{g} = ℓ [\frac{1}{v_{g} (f_{2})} - \frac{1}{v_{g} (f_{1})}] \approx ℓ \frac{d^{2} β}{d ω^{2}} 2 π B

τ_g = ℓ/v_g = ℓ dβ/dω. For a TEM line dτ_g/df = 0. A pulse of duration T is badly distorted once Δτ_g approaches T.

Waveguides are strongly dispersive near cutoff: in WR-90 the group delay per metre is 5.6 ns at 8.2 GHz but 3.9 ns at 12.4 GHz, so a wideband radar pulse that crosses the whole band is smeared by about 1.6 ns per metre of guide. Microstrip is far better but not perfect (Example 4). Coaxial cable and stripline are essentially non-dispersive, which is why they are used for true time-delay lines and wideband pulse distribution.

Frequency-dependent loss is the other source of distortion. On a lossy PCB trace the high-frequency content of a fast edge is attenuated more than the low-frequency content, so the edge slows and successive bits blur into each other (inter-symbol interference). High-speed serial links counter this with equalisers that boost the high frequencies — a direct application of the α(f) of a line.

Below Cutoff: Evanescent Waves

For a waveguide below its cutoff frequency, β = √(k² − k_c²) becomes imaginary. The propagation constant is then purely real: the field does not propagate but decays exponentially, without loss of power to heat — the energy is simply reflected. This evanescent field is the basis of the waveguide-below-cutoff attenuator and of the honeycomb vents in shielded rooms and microwave-oven doors.

Evanescent attenuation (f < f_c)

γ = α = \sqrt{k_{c}^{2} - k^{2}} = k_{c} \sqrt{1 - {(\frac{f}{f_{c}})}^{2}}

Far below cutoff α → k_c = π/a for TE₁₀, independent of frequency: 27.3 dB per guide width a. A hole whose length is three times its width therefore attenuates by about 80 dB.

Measuring the Propagation Constant

The cleanest way to measure γ uses two lines of the same cross-section but different lengths, ℓ₁ and ℓ₂, built with identical connectors or launches. The connectors contribute the same unknown error to both, so it cancels in the ratio:

Derivation — the two-line method

For a matched (or nearly matched) line, S₂₁ = A e^−γℓ, where A lumps together the two transitions. Dividing the two measurements removes A:

Two-line extraction of γ

\frac{S_{21}^{(2)}}{S_{21}^{(1)}} = e^{- γ (ℓ_{2} - ℓ_{1})} \Rightarrow γ = - \frac{\ln (S_{21}^{(2)} / S_{21}^{(1)})}{ℓ_{2} - ℓ_{1}}

The real part gives α (Np per metre of extra length), the phase gives β. Choose ℓ₂ − ℓ₁ between about 20° and 160° at every frequency of interest to avoid ambiguous phase and poor sensitivity; several line pairs cover a wide band (multiline TRL).

This is how laminate suppliers characterise dielectric loss and how high-speed designers extract the insertion loss per inch of a PCB stack-up. A time-domain alternative measures the delay difference between the two lines with a fast step, giving the group velocity directly.

Practical Propagation Constant Calculations

The examples use c = 2.998 × 10⁸ m/s.

Exact α and β of a Data-Cable Pair at 1 MHz

Problem: A twisted pair has R′ = 0.6 Ω/m (skin effect included), L′ = 0.55 µH/m, G′ = 2 µS/m and C′ = 50 pF/m. Find α and β at 1 MHz exactly, and compare with the low-loss formulas.

Step 1 — A and B (ω = 6.283 × 10⁶ rad/s)

A = R^{'} G^{'} - ω^{2} L^{'} C^{'} = 1.2 \times 10^{- 6} - 1.0857 \times 10^{- 3} = - 1.0845 \times 10^{- 3}

B = ω (R^{'} C^{'} + G^{'} L^{'}) = 6.283 \times 10^{6} (3.0 \times 10^{- 11} + 1.1 \times 10^{- 12})

= 1.954 \times 10^{- 4}

Step 2 — Exact α and β (√(A² + B²) = 1.1019 × 10⁻³)

α = \sqrt{\frac{1.1019 - 1.0845}{2} \times 10^{- 3}} = 2.955 \times 10^{- 3} Np/m

β = \sqrt{\frac{1.1019 + 1.0845}{2} \times 10^{- 3}} = 0.03306 rad/m

Step 3 — Low-loss formulas (Z₀ = √(L′/C′) = 104.9 Ω)

α \approx \frac{0.6}{2 \times 104.9} + \frac{2 \times 10^{- 6} \times 104.9}{2} = 2.965 \times 10^{- 3}

β \approx ω \sqrt{L^{'} C^{'}} = 0.03295

Answer: α = 2.96 × 10⁻³ Np/m (25.7 dB/km), β = 0.0331 rad/m

Here R′/(ωL′) = 0.17 — not very small — yet the low-loss α is still within 0.3% and β within 0.35%. The second-order correction factor 1 + (0.17)²/8 = 1.0036 accounts for almost all of the difference in β. The guided wavelength is 2π/β = 190 m and v_p = 1.90 × 10⁸ m/s.

Velocity Factor from a Phase Measurement

Problem: A VNA measures the unwrapped phase of S₂₁ through a matched 2.00 m cable as −1056° at 300 MHz. Find β, v_p and the velocity factor.

Phase constant and velocity

β = \frac{1056^{\circ} \times π / 180^{\circ}}{2.00} = 9.215 rad/m

v_{p} = \frac{2 π \times 3 \times 10^{8}}{9.215} = 2.045 \times 10^{8} m/s

Answer: β = 9.22 rad/m, v_p = 2.05 × 10⁸ m/s, VF = 0.68

A velocity factor of 0.68 suggests a PTFE (εr ≈ 2.1–2.2) dielectric. "Unwrapped" matters: the VNA displays phase between ±180°, so the total phase must be reconstructed by counting wraps from low frequency — a common source of errors of a whole multiple of 360°.

Phase and Group Velocity in WR-90

Problem: Find v_p and v_g in WR-90 (f_c = 6.557 GHz) at 10 GHz, and the time a radar pulse takes to travel through 3 m of guide.

Velocities (√(1 − 0.6557²) = 0.7550)

v_{p} = \frac{c}{0.7550} = 1.324 c v_{g} = 0.7550 c v_{p} v_{g} = c^{2}

Pulse delay

t = \frac{3}{0.7550 \times 2.998 \times 10^{8}} = 13.25 ns

Answer: v_p = 1.32c, v_g = 0.755c; the pulse takes 13.3 ns, not 10.0 ns as in free space

The pulse envelope travels at v_g, slower than light, while the phase pattern inside it slides forward faster than light. Radar range calibration must use the group delay of every waveguide run.

Group Velocity on Dispersive Microstrip

Problem: For 50 Ω microstrip on 0.813 mm RO4003C, the Kirschning–Jansen model gives √εeff = 1.6956 at 10 GHz, rising at d√εeff/df = 0.00333 per GHz. Find v_p, v_g and the group delay of a 100 mm line.

Phase and group index

n_{p} = \sqrt{ε_{eff}} = 1.6956

n_{g} = n_{p} + f \frac{d n_{p}}{d f} = 1.6956 + 10 \times 0.00333 = 1.7289

Velocities and delay

v_{p} = \frac{c}{1.6956} = 0.590 c

v_{g} = \frac{c}{1.7289} = 0.578 c

τ_{g} = \frac{0.1 \times 1.7289}{2.998 \times 10^{8}} = 577 ps

Answer: v_p = 0.590c, v_g = 0.578c, τ_g = 577 ps per 100 mm at 10 GHz

The group delay of the same 100 mm line is 558 ps at 1 GHz and 600 ps at 20 GHz. A broadband pulse spanning 1–20 GHz therefore spreads by about 40 ps over 100 mm — significant for 50 Gb/s signalling and for wideband radar.

Two-Tone Beat on a Coax

Problem: Tones at 1.00 and 1.02 GHz travel together on a coax with VF = 0.66 (Fig 1). Find the beat length (the spatial period of the envelope) and the envelope speed.

Phase constants and beat length (v_p = 1.979 × 10⁸ m/s)

β_{1} = \frac{2 π \times 10^{9}}{1.979 \times 10^{8}} = 31.76 rad/m

β_{2} = 32.39 rad/m

Λ = \frac{2 π}{β_{2} - β_{1}} = 9.89 m

Answer: Λ ≈ 9.9 m, and the envelope moves at v_g = v_p = 0.66c

On a non-dispersive coax Δω/Δβ = v_p, so the envelope and carrier move together. The beat length is simply v_p/Δf = 1.979 × 10⁸/(2 × 10⁷) = 9.9 m.

Nepers, Decibels and Delivered Power

Problem: A cable has α = 0.5 dB/m. Express α in Np/m and find the fraction of the input voltage and power that reaches the end of a 30 m run (matched).

Conversion and total loss

α = \frac{0.5}{8.686} = 0.0576 Np/m α ℓ = 1.727 Np = 15 dB

Voltage and power ratios

\frac{| V (ℓ) |}{| V (0) |} = e^{- 1.727} = 0.178 \frac{P (ℓ)}{P (0)} = e^{- 3.454} = 10^{- 1.5} = 0.032

Answer: α = 0.0576 Np/m; 17.8% of the voltage and 3.2% of the power arrive

15 dB of feeder loss wastes 97% of a transmitter's power and, in a receiver, adds directly to the noise figure. This is the arithmetic behind choosing a lower-loss cable or moving the radio to the antenna.

Extracting γ and εeff from S₂₁

Problem: A matched 0.5 m length of foam-dielectric cable has |S₂₁| = −2.3 dB and an unwrapped phase of −1500° at 2 GHz. Find α, β and the effective permittivity.

Since S₂₁ = e^−γℓ for a matched line

α = \frac{2.3}{8.686 \times 0.5} = 0.530 Np/m (4.6 dB/m)

β = \frac{1500 \times π / 180}{0.5} = 52.36 rad/m

Effective permittivity

ε_{eff} = {(\frac{β}{k_{0}})}^{2} = {(\frac{52.36}{41.92})}^{2} = 1.56

Answer: α = 0.53 Np/m (4.6 dB/m), β = 52.4 rad/m, εeff = 1.56 (VF = 0.80)

A VF of 0.80 is typical of foamed PE. In practice the line is rarely perfectly matched, so the extraction uses all four S-parameters (or two lines of different length) to remove the connector mismatches.

WR-90 as a Below-Cutoff Attenuator

Problem: Find the attenuation of the TE₁₀ mode in WR-90 (a = 22.86 mm) at 5 GHz, below its 6.56 GHz cutoff. How long a section gives 60 dB?

Evanescent decay constant

k_{c} = \frac{π}{0.02286} = 137.4 m^{- 1}

k = \frac{2 π \times 5 \times 10^{9}}{2.998 \times 10^{8}} = 104.8 m^{- 1}

α = \sqrt{{137.4}^{2} - {104.8}^{2}} = 88.9 Np/m = 7.72 dB/cm

Answer: 7.7 dB/cm; 60 dB needs about 78 mm

Unlike a resistive attenuator, the below-cutoff section absorbs nothing: it reflects the power. Its attenuation in dB is exactly proportional to length and very stable, which made it the standard precision attenuator for calibrating receivers.

Phase Matching Two Cables

Problem: Two 3.00 m cables feed the elements of a 5.8 GHz antenna array. Their velocity factors are 0.695 and 0.700. What phase difference do they introduce?

Electrical length of each cable, θ = 360° ℓ f / (VF·c)

θ_{1} = \frac{360 \times 3 \times 5.8 \times 10^{9}}{0.695 \times 2.998 \times 10^{8}} = 30 063^{\circ}

θ_{2} = 29 849^{\circ}

Answer: Δθ ≈ 215° — more than half a cycle

A 0.7% difference in VF over 58 wavelengths of cable produces a 215° phase error, enough to steer the beam completely wrongly. This is why array feed cables are bought as phase-matched sets, trimmed while being measured on a VNA.

Dispersion on a Long Telephone Line

Problem: On the telephone pair of Part 3 (R′ = 0.27 Ω/m, L′ = 0.6 µH/m, C′ = 50 pF/m) the exact γ gives v_p = 3.03 × 10⁷ m/s at 1 kHz and 5.17 × 10⁷ m/s at 3 kHz. Find the delays over 5 km and their difference.

Phase delays

τ_{1 kHz} = \frac{5000}{3.03 \times 10^{7}} = 165 μ s

τ_{3 kHz} = \frac{5000}{5.17 \times 10^{7}} = 97 μ s

Answer: 165 µs and 97 µs — a 68 µs spread across the voice band

Different parts of a speech signal arrive at noticeably different times, and the higher frequencies are also attenuated more. Both are consequences of the RC regime (v_p ∝ √f), and both were reduced by the loading coils of Part 3.

Summary

γ = α + jβ = √((R′ + jωL′)(G′ + jωC′)); a matched line section multiplies a wave by e^−γℓ.
α is the attenuation in Np/m (×8.686 for dB/m); β = 2π/λ_g is the phase shift per metre.
Exactly, α and β = √((√(A² + B²) ± A)/2) with A = R′G′ − ω²L′C′ and B = ω(R′C′ + G′L′). Low-loss: α ≈ R′/2Z₀ + G′Z₀/2, β ≈ ω√(L′C′).
Phase velocity v_p = ω/β; group velocity v_g = dω/dβ, the speed of envelopes, energy and information.
TEM lines: v_p = v_g (no dispersion). Waveguides: v_pv_g = c². Microstrip: v_g slightly below v_p. RC lines: v_g = 2v_p.
Below cutoff a waveguide mode is evanescent: γ = √(k_c² − k²) is real, and the field decays without dissipation.

β turns physical length into phase. Part 6: Guided Wavelength and Electrical Length develops that idea into guided wavelength, velocity factor and electrical length — the language of quarter-wave and half-wave lines.

📘 Go deeper: Waves in free space and in guides

Phase and group velocity have the same meaning for plane waves and for every guided mode. Open Maxwell's Equations series →

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Propagation Constant, Phase Velocity and Group Velocity