Propagation constant, loss & distortion

For a uniform line, γ²=(R′+jωL′)(G′+jωC′). Write γ=α+jβ with the passive +z branch chosen so α≥0 and, for positive frequency on an ordinary line, β≥0. Expanding both sides separates real and imaginary parts.

Define A=R′G′−ω²L′C′ and B=ω(R′C′+L′G′). Then α²−β²=A and 2αβ=B. Because |γ²|=√(A²+B²), α and β can be written explicitly without first computing a complex square root numerically.

These exact formulas assume the RLGC parameters used at that frequency are known. They do not imply R′,L′,G′,C′ are constants over an arbitrary bandwidth.

When R′≪ωL′ and G′≪ωC′, factor the dominant reactive product from γ² and expand the square root to first order. The phase constant remains β≈ω√(L′C′), while attenuation is the sum of two positive first-order contributions.

Using the lossless characteristic impedance Z₀≈√(L′/C′), the conductor term is α_c≈R′/(2Z₀) and the dielectric/leakage term is α_d≈G′Z₀/2. Their units are inverse metres.

This split is physically useful but model-dependent. At RF, conductor R′ commonly changes with frequency through skin/proximity effects, while dielectric loss often makes G′ frequency-dependent. The formulas are evaluated locally in frequency, not with one universal constant α.

A forward amplitude varies as exp(−αz), so after length ℓ the amplitude ratio is exp(−αℓ). Power is quadratic in amplitude and therefore decays as exp(−2αℓ) for a single traveling wave on a uniform passive line.

The logarithmic loss in decibels is 20log₁₀ of an amplitude ratio or 10log₁₀ of a power ratio. Both give the same positive attenuation number 8.686αℓ dB because the factor of two in power is matched by 10 versus 20.

Report whether a specification is amplitude, voltage, power, insertion loss or attenuation constant. Writing '3 dB amplitude loss' without defining the intended ratio invites a factor-of-two mistake.

For a monochromatic component, phase velocity is v_p=ω/β. A length ℓ contributes phase −βℓ, so the phase delay written as βℓ/ω equals ℓ/v_p.

For a narrowband modulated waveform, the envelope delay is governed by the slope of phase with angular frequency: τ_g=ℓ dβ/dω and group velocity v_g=(dβ/dω)⁻¹. This derivative must include the actual frequency dependence of the line parameters, not only the explicit ω in a simplified formula.

If β is exactly proportional to ω over the signal band, phase and group velocities coincide and the phase response is linear. If β bends with frequency, spectral components acquire different delays and phase distortion can result.

A length ℓ has forward transfer factor H(ω)=exp[−γ(ω)ℓ]=exp[−α(ω)ℓ]exp[−jβ(ω)ℓ]. A waveform is reproduced at the output as only a scaled and delayed copy when the magnitude factor is frequency-independent across its relevant band and the phase is affine in ω.

Equivalently, α should be constant across the band while β should have the form β(ω)=ωτ/ℓ plus at most a constant phase offset. If α varies, the spectrum is reshaped in magnitude. If dβ/dω varies, different frequency components suffer different delay.

No real signal has infinite bandwidth, so 'distortionless' is always meaningful over a specified band and within an error tolerance.

Suppose the per-unit-length parameters are frequency-independent and satisfy R′/L′=G′/C′=k_H. Then Z′=L′(k_H+jω) and Y′=C′(k_H+jω). Their product is a perfect square.

The passive propagation constant becomes γ=√(L′C′)(k_H+jω). Therefore α=k_H√(L′C′)=√(R′G′), independent of frequency, while β=ω√(L′C′) is exactly linear in frequency. At the same time Z₀=√(L′/C′) is real and frequency-independent.

This is the Heaviside distortionless condition. It is exact inside the constant-RLGC model; it does not guarantee a physical cable remains distortionless when R′,L′,G′,C′ themselves change with frequency.

The constant-RLGC model is a local model, not a promise that four numbers describe a line from DC to millimetre waves. Conductor resistance rises when current crowds near surfaces; skin effect alone gives an approximate √f trend in simple good-conductor regimes, with proximity and roughness adding geometry-dependent corrections.

Dielectric loss is often represented over a limited band by a loss tangent tanδ, giving approximately G′≈ωC′tanδ for a quasi-TEM line when that model is appropriate. Permittivity and therefore C′ can also disperse with frequency. Internal inductance may decrease as current redistributes.

At sufficiently high frequency, higher-order modes, radiation, conductor dispersion or non-quasi-TEM effects can invalidate the one-dimensional line model itself. Then refining R′(ω) is not enough; the mode model must also be revisited.

Reuse R′=10 Ω/m, L′=250 nH/m, G′=2.0 mS/m and C′=100 pF/m at 100 MHz. The lossless Z₀ is 50 Ω. The low-loss estimate gives α_c=10/(2·50)=0.100 Np/m and α_d=(0.002·50)/2=0.050 Np/m, so α≈0.150 Np/m.

The exact calculation gives α≈0.14998 Np/m and β≈3.14199 rad/m, while the low-loss phase estimate is β≈π rad/m. The total attenuation is about 1.303 dB/m.

Over 2 m, the amplitude factor is exp(−0.29996)≈0.7408 and the power factor is about 0.5488. This power decrease is not an additional loss mechanism: it is the square of the same field-amplitude attenuation.

1. Expand γ² and identify A and B in γ²=A+jB. Derive α²−β²=A and 2αβ=B.
2. Starting from the low-loss assumptions, derive α≈(R′/2)√(C′/L′)+(G′/2)√(L′/C′).
3. A line has α=0.12 Np/m and length 3 m. Compute amplitude ratio, power ratio and loss in dB.
4. Explain why a 180° phase shift on a lossless line does not imply attenuation.
5. For β(ω)=aω+bω² with small b, derive τ_g/ℓ and show why group delay depends on frequency.
6. State the magnitude and phase conditions for a line section to reproduce a waveform as only a scaled and delayed copy.
7. Prove the Heaviside result γ=√(L′C′)(k_H+jω) when R′/L′=G′/C′=k_H.
8. With L′=250 nH/m,C′=100 pF/m,R′=10 Ω/m, what G′ is required by the exact Heaviside condition? Find α.
9. At 100 MHz for the chapter line, calculate α_c, α_d and their fraction of total first-order attenuation.
10. Why can a line satisfy the Heaviside relation at one frequency yet still distort a broadband pulse in a real implementation?

Chapter 05 will close RF-00 with transmission-line power flow: complex power from line phasors, forward/backward power, loss per unit length, energy velocity links in the lossless limit and practical power normalization before RF-01 introduces reflection coefficients and standing waves.