Transmission Lines Q&A

A transmission line is a structure designed to guide electromagnetic energy from one point to another while minimising radiation, reflections, and losses. It consists of two or more conductors with a defined geometry that maintains a constant characteristic impedance along its length.

At low frequencies, a simple wire works fine because the wavelength is much larger than the wire length, so voltage and current are essentially uniform along it. At high frequencies, the wavelength becomes comparable to or smaller than the circuit dimensions. This means:

• Voltage and current vary significantly along the line (standing waves form)
• The wire acts as an antenna and radiates energy
• Impedance mismatches cause reflections that degrade signal integrity
• Parasitic capacitance and inductance become dominant

Transmission line theory (governed by the telegrapher's equations) treats the line as a distributed-parameter circuit, properly accounting for these effects.

Characteristic impedance Z₀ is the ratio of voltage to current in a wave travelling along an infinitely long transmission line (or a matched, reflectionless line). It is a property of the line's physical geometry and materials — not its length.

Z₀ depends on the geometry (trace width, spacing, conductor diameter) and the dielectric material (εr). It does NOT depend on the line length or the frequency (for a lossless line).

• Microstrip: wider trace → lower Z₀; thicker substrate → higher Z₀
• Coaxial: Z₀ = (60/√εr) · ln(D/d)
• Common standards: 50 Ω (RF/microwave), 75 Ω (cable TV), 100 Ω (Ethernet diff. pair)

VSWR (Voltage Standing Wave Ratio) is the ratio of the maximum to minimum voltage amplitude along a transmission line that has reflections. It quantifies how well a load is matched to the line.

• VSWR = 1.0: Perfect match — no reflections, all power delivered to load. ZL = Z₀.
• VSWR = 2.0: |Γ| = 0.333 — about 11% of power reflected, 89% delivered. Return loss ≈ 9.5 dB. Acceptable in many systems.
• VSWR = ∞: Total reflection — open circuit or short circuit. ZL = 0 or ∞.

In practice, a VSWR below 1.5:1 is considered good for most RF applications. Cellular base station antennas typically require VSWR ≤ 1.5:1.

For a lossless line of length l, characteristic impedance Z₀, terminated in ZL, the input impedance looking into the line is:

This is one of the most important equations in RF engineering. It shows that the impedance you see looking into a transmission line depends on: the load impedance, the line impedance, and the electrical length βl.

Key special cases:

• l = λ/4 (quarter-wave): Zin = Z₀² / ZL — the line is an impedance inverter
• l = λ/2 (half-wave): Zin = ZL — the line repeats the load impedance
• ZL = 0 (short): Zin = jZ₀·tan(βl) — purely reactive, ranges from 0 to ±∞
• ZL = ∞ (open): Zin = −jZ₀·cot(βl) — purely reactive

The reflection coefficient Γ (Gamma) is the ratio of the reflected wave voltage to the incident wave voltage at a discontinuity or load. It is a complex number with magnitude 0 ≤ |Γ| ≤ 1.

Relationship to S11: S11 IS the reflection coefficient at port 1, measured when port 2 is terminated in the reference impedance Z₀. In other words, S11 = Γ at the input port under matched output conditions.

In practice, S11 is measured by a vector network analyser (VNA) which measures both magnitude and phase of the reflected wave. The result is displayed on a Smith Chart or as |S11| in dB (= Return Loss).

• Return Loss (dB) = −20·log₁₀(|Γ|) = −20·log₁₀(|S11|)
• Mismatch Loss (dB) = −10·log₁₀(1 − |Γ|²)

For a short-circuit termination (ZL = 0), the input impedance formula gives Zin = jZ₀·tan(βl). This is purely imaginary (reactive), cycling between inductive and capacitive behaviour as the length changes.

• l = 0: Zin = 0 (short — same as the load)
• l = λ/8: Zin = +jZ₀ (inductive reactance = +Z₀)
• l = λ/4: Zin = ∞ (open circuit! — impedance inversion)
• l = 3λ/8: Zin = −jZ₀ (capacitive reactance = −Z₀)
• l = λ/2: Zin = 0 (short again — repeats every λ/2)

The key insight: a λ/4 short-circuit stub looks like an open circuit at the design frequency. This is widely used to create RF chokes, resonators, and band-stop filters on PCBs without using lumped components.

The telegrapher's equations describe voltage and current as functions of position and time along a transmission line. They treat the line as a distributed network of infinitesimal RLGC sections.

For a lossless line (R = G = 0), these simplify to wave equations with phase velocity vp = 1/√(LC) and characteristic impedance Z₀ = √(L/C).

The general solution gives forward (+z) and backward (−z) travelling waves, leading to the standing wave pattern when both exist simultaneously.

A single-section quarter-wave transformer matches two real impedances by choosing an intermediate impedance Z₁ = √(Z_S · Z_L) and making the section exactly λ/4 long at the centre frequency.

Step 1 — Find the matching impedance:

Step 2 — Find the physical length:

Step 3 — Find the trace width: Use the microstrip synthesis calculator (Synthesis mode, Z₀ = 70.71 Ω, h = 1.6 mm, εr = 4.4) to get the trace width. Result: W ≈ 1.86 mm.

Limitations: This matching only works over a narrow bandwidth around the design frequency. For broadband matching, use multi-section transformers (Chebyshev or binomial tapers).

The propagation constant γ (gamma) is a complex number that describes how a wave changes in both amplitude and phase as it travels along the line.

The voltage wave solution is: V(z) = V⁺e^(−γz) + V⁻e^(+γz)

• V⁺e^(−γz): forward wave, decays as e^(−αz), phase shifts as e^(−jβz)
• V⁻e^(+γz): reflected wave, grows toward the source

Attenuation sources: α = αc + αd where αc is conductor loss (skin effect, ∝√f) and αd is dielectric loss (∝f·tanδ). At microwave frequencies, dielectric loss often dominates on PCBs.

The skin effect is the tendency of alternating current to concentrate near the surface of a conductor at high frequencies, reducing the effective cross-sectional area carrying current and increasing resistance.

Since conductor resistance ∝ 1/δs ∝ √f, the series resistance R of the line increases as √f with frequency. This means conductor loss increases as √f (in dB/m).

Practical implications:

• Copper plating must be at least 3–5 skin depths thick (≥ 6–10 μm at 1 GHz)
• Surface roughness becomes critical when Ra ≈ δs, causing significant additional loss
• Silver and gold plating improve conductivity slightly but mainly prevent oxidation
• At mmWave (60+ GHz), δs ≈ 0.2 μm — surface finish quality is critical

Return loss (RL) is the ratio in dB of the incident power to the reflected power at a port. A higher return loss means less power is reflected — i.e., a better match.

Common values to memorise:

• RL = 6 dB → VSWR ≈ 3.0 → |Γ| = 0.5 → 25% power reflected
• RL = 10 dB → VSWR ≈ 1.93 → |Γ| = 0.316 → 10% reflected
• RL = 14 dB → VSWR ≈ 1.5 → |Γ| = 0.2 → 4% reflected
• RL = 20 dB → VSWR ≈ 1.22 → |Γ| = 0.1 → 1% reflected
• RL = 40 dB → VSWR ≈ 1.02 → |Γ| = 0.01 → 0.01% reflected

Real-world example: A cellular base station antenna spec typically requires RL ≥ 14 dB (VSWR ≤ 1.5:1) to prevent reflected power from damaging the power amplifier and to ensure efficient radiation.

These three velocities describe how different aspects of a wave propagate along the line:

• TEM line (microstrip, coax): vp = vg = c/√εeff — no dispersion, both equal
• Waveguide (TE/TM modes): vp > c (phase velocity exceeds c), vg < c, vp·vg = c²/εr. No information travels faster than c — this is not a paradox.
• Dispersive lines: vp varies with frequency → pulse distortion (broadening)

Waveguide group velocity represents actual energy/signal propagation: vg = c·√(1−(fc/f)²)/√εr, which approaches zero as frequency approaches cutoff fc. Phase velocity vp = c/(√εr · √(1−(fc/f)²)) exceeds c but carries no information.