Transmission Lines Q&A

A transmission line is a guided structure that carries RF energy from source to load while controlling impedance, minimising radiation and managing reflections. It maintains a defined geometry between two or more conductors so that the characteristic impedance is constant along the length.

At low frequencies a simple wire works because the wavelength is far larger than the wire — voltage and current are essentially uniform. As frequency rises, wavelength shrinks to the size of the conductor and three problems emerge:

• Standing waves — voltage and current vary dramatically along the line, causing hot-spots and cold-spots
• Radiation — an uncontrolled conductor radiates like an antenna, wasting power and causing interference
• Reflections — impedance discontinuities reflect energy back toward the source, causing signal distortion and potential component damage in power amplifiers

Characteristic impedance Z₀ is the ratio of voltage to current for a wave travelling along an infinitely long line, or equivalently on a finite matched line producing no reflections. It is a property of the line's cross-sectional geometry and the dielectric material.

Z₀ depends on: conductor geometry (spacing, width, diameter), dielectric constant εr, dielectric thickness.

Z₀ does NOT depend on: line length, frequency (for lossless TEM lines), signal level, load impedance.

• 50 Ω — RF/microwave standard: compromise between max power (~30 Ω) and min loss (~77 Ω) for air coax
• 75 Ω — broadcast/cable TV: minimum loss over long cable runs is more important than power transfer
• 100 Ω differential — high-speed digital (Ethernet, USB, PCIe)
• 300 Ω — twin-lead TV antenna feedline (still found on older installations)

The telegrapher's equations model a transmission line as a distributed network — infinitely many tiny lumped elements cascaded along the length. They describe how voltage and current vary along the line and over time.

Physical meaning of each parameter (per unit length):

• R (Ω/m) — series resistance of conductors. Increases with frequency due to skin effect (∝√f). Represents conductor loss.
• L (H/m) — series inductance. Stores energy in the magnetic field between conductors. Nearly constant with frequency for TEM lines.
• G (S/m) — shunt conductance. Represents dielectric loss (current leaking through the insulator). Increases with frequency (∝f·tanδ). Often negligible for low-loss substrates.
• C (F/m) — shunt capacitance. Stores energy in the electric field between conductors. Nearly constant with frequency.

For a lossless line (R=0, G=0): Z₀ = √(L/C) and phase velocity vp = 1/√(LC) = c/√εr.

VSWR (Voltage Standing Wave Ratio) is the ratio of the maximum to minimum voltage along a transmission line when standing waves are present due to reflections. It is always ≥ 1.

Key VSWR values:

• VSWR = 1.0 — perfect match. ZL = Z₀ exactly. No reflections. All incident power absorbed by the load.
• VSWR = 1.5 — return loss 14 dB, 4% power reflected. Typical specification for base station antennas and connectors.
• VSWR = 2.0 — return loss 9.5 dB, 11% power reflected. Marginal — acceptable only for non-critical applications.
• VSWR = ∞ — total reflection. Either open circuit (|Γ|=1, ∠Γ=0°) or short circuit (|Γ|=1, ∠Γ=180°).

Return Loss (RL) is how many dB of the incident signal is reflected back. It is always a positive number expressed in dB.

Mismatch Loss (ML) is the power lost to reflection that never reaches the load — the power that would have been delivered if the match were perfect.

Quick reference table:

• RL = 6 dB → VSWR = 3.0 → |Γ| = 0.50 → 25% power reflected → ML = 1.25 dB
• RL = 10 dB → VSWR = 1.93 → |Γ| = 0.32 → 10% power reflected → ML = 0.46 dB
• RL = 14 dB → VSWR = 1.50 → |Γ| = 0.20 → 4% power reflected → ML = 0.18 dB
• RL = 20 dB → VSWR = 1.22 → |Γ| = 0.10 → 1% power reflected → ML = 0.04 dB
• RL = 30 dB → VSWR = 1.07 → |Γ| = 0.03 → 0.1% power reflected → ML = 0.004 dB

At the load (z=0), voltage and current must be continuous. The total voltage is the sum of the forward (+) and backward (−) travelling waves; total current is their difference (normalised by Z₀):

Special cases:

• ZL = Z₀ → Γ = 0 (matched, no reflection)
• ZL = 0 (short) → Γ = −1 (full reflection, voltage inverted)
• ZL = ∞ (open) → Γ = +1 (full reflection, voltage doubled)
• ZL = jX (pure reactive) → |Γ| = 1 (full reflection, but phase shifts)

Γ is a complex number in general. On a Smith chart, the magnitude |Γ| is the distance from the centre (50 Ω point) and the angle is the phase of the reflection coefficient.

The general voltage and current on a lossless line are:

Critical special cases — memorise these for interviews:

• l = λ/4 (βl = 90°, tan→∞): Zin = Z₀²/ZL — impedance inverter. High impedance becomes low and vice versa.
• l = λ/2 (βl = 180°, tan=0): Zin = ZL — line reproduces the load regardless of Z₀. Used to move an impedance to a more convenient location.
• ZL = 0 (short circuit): Zin = jZ₀ tan βl — purely reactive, cycles from 0 to +j∞ to 0 to −j∞ as l goes from 0 to λ/2.
• ZL = ∞ (open circuit): Zin = −jZ₀ cot βl — purely reactive, cycles from +j∞ to 0 to −j∞ as l goes from 0 to λ/2.

For a short-circuit (SC) stub: Zin = jZ₀ tan(βl). Cycling through length:

Practical applications:

• λ/4 SC stub as RF choke: Presents infinite impedance at the design frequency while providing a DC path to ground. Used in bias injection networks for transistor amplifiers.
• SC stub for matching: Place a shunt SC stub at distance d from the load where the real part of the load admittance equals Y₀, then set stub length to cancel the imaginary part.
• Band-stop filter: A series λ/4 OC stub presents a short circuit at the design frequency, blocking signal flow.

Step 1: Calculate transformer impedance.

Step 2: Find microstrip trace width for 70.71 Ω on Rogers 4350B using Hammerstad-Jensen synthesis. For εr=3.66, h=0.762 mm, t=35 μm → W ≈ 0.96 mm.

Step 3: Calculate effective permittivity. εeff ≈ 2.85 for this geometry.

Step 4: Calculate physical λ/4 length.

Step 5: Verify bandwidth. Single-section λ/4 transformer bandwidth (for return loss >10 dB) ≈ 20–30% of f₀ = 1–1.5 GHz centred at 5 GHz.

Single-stub matching places a shunt transmission line stub at distance d from the load. The distance d is chosen so that the real part of the admittance at that point equals Y₀ = 1/Z₀. The stub length is then set to cancel the remaining imaginary part.

Step 1: Normalise the load. yL = Y₀/YL = Z₀/ZL = 50/(25−j30) = (25+j30)/(25²+30²) × 50 = 50/(25−j30) = 0.55 + j0.66

Step 2: Find distance d such that Re(yin) = 1. From admittance transformation formula:

Step 3: Find susceptance at the stub junction for solution 1. bin ≈ −0.98. Stub must supply +0.98 normalised susceptance.

Step 4: For a short-circuit stub, b_stub = −cot(βl) = +0.98 → βl = arctan(1/0.98) → l ≈ 0.126λ.

The Smith chart is a graphical tool for transmission line analysis that maps the complex impedance plane to the complex reflection coefficient (Γ) plane. Every impedance corresponds to a unique point inside a circle of radius 1.

Structure of the Smith chart:

• Horizontal axis: Real axis. Centre = 50 Ω (z=1 normalised). Left edge = 0 Ω (short). Right edge = ∞ Ω (open).
• Constant resistance circles: Family of circles all passing through the right edge (z=∞). Labelled with normalised resistance r = R/Z₀.
• Constant reactance arcs: Family of arcs. Upper half = inductive (+jX). Lower half = capacitive (−jX).

Reading z = 1 − j1:

1. Find the resistance circle r=1 — it passes through the centre of the chart.
2. Find the reactance arc x=−1 in the lower half (capacitive).
3. The intersection is the point z=1−j1. On the outer scale, read the wavelength scale for stub calculations.

Key Smith chart operations:

• Moving toward generator (clockwise rotation): distance corresponds to wavelengths toward generator (WTG)
• Shunt element: move along constant conductance circles
• Series element: move along constant resistance circles
• λ/4 rotation: moves a point to its "opposite" — short becomes open and vice versa

A balun (balanced-unbalanced transformer) converts between a balanced transmission line (equal and opposite currents in both conductors, e.g. twin-lead) and an unbalanced line (one conductor is ground, e.g. coaxial cable).

Why you need it: A coaxial cable fed directly into a dipole antenna creates an unbalanced feed — current flows on the outside of the outer conductor, causing radiation from the feed cable and distorting the radiation pattern. A balun forces equal and opposite currents in the two dipole arms.

Types:

• λ/4 sleeve balun (choke balun): A metal sleeve over the coax outer conductor, λ/4 long and short-circuited at the bottom, presents high impedance to the outer conductor current at the operating frequency.
• Ferrite bead balun: Ferrite rings threaded over the coax add inductance to the outer conductor, blocking common-mode current. Works over a wider bandwidth than the λ/4 sleeve.
• 180° hybrid (rat-race) coupler: Used at microwave frequencies — the Σ and Δ ports provide the balanced-to-unbalanced conversion.

α (attenuation constant, Np/m or dB/m): How quickly the wave amplitude decays along the line. α = αc + αd where αc is conductor loss (∝√f due to skin effect) and αd is dielectric loss (∝f·tanδ). Both increase with frequency — this is why high-frequency PCB design is difficult.

β (phase constant, rad/m): How quickly the phase of the wave changes along the line. β = 2π/λ = ω/vp. The wavelength on the line is λ = 2π/β, shorter than free-space wavelength by 1/√εeff.

Phase velocity (vp = ω/β): The speed at which the phase of a single-frequency carrier wave propagates. For a TEM line: vp = c/√εr. For a waveguide: vp = c/√(1−(fc/f)²) > c above cutoff.

Group velocity (vg = dω/dβ): The speed at which the envelope of a modulated signal (a group of frequencies) propagates. This carries the actual information.

Signal velocity: In practice equal to group velocity for well-behaved (non-anomalous) dispersion.

Can vp exceed c? Yes — in waveguide above cutoff. This does NOT violate relativity because vp only describes the phase relationship of a single sinusoidal wave, which carries no information. The signal velocity (group velocity) is always ≤ c. A single-frequency sinusoid is infinite in time and cannot carry information.

Dispersion means that different frequency components of a signal travel at different phase velocities, causing the signal pulse to spread out and distort over distance. A non-dispersive line has vp independent of frequency.

• Coaxial cable (TEM mode): Non-dispersive — vp = c/√εr is constant with frequency. Ideal for wideband signals. (Higher modes are dispersive, but these don't propagate below the cutoff of the first higher mode.)
• Rectangular waveguide: Highly dispersive — vp = c/√(1−(fc/f)²) varies strongly with frequency. Phase velocity → ∞ as f → fc, then slowly approaches c as f >> fc. Never used for wideband pulse transmission.
• Microstrip: Slightly dispersive — εeff increases slowly with frequency due to field confinement in the dielectric. Negligible for narrow-band signals but significant for wideband designs above ~10 GHz.
• Stripline: Essentially non-dispersive (TEM mode in homogeneous dielectric). Preferred over microstrip for wideband or high-precision designs.

At DC, current flows uniformly through the entire conductor cross-section. At high frequencies, electromagnetic induction pushes current to the surface. Skin depth δs is the depth at which current density falls to 1/e ≈ 37% of its surface value.

Standard PCB copper thickness: 1 oz = 35 μm, 0.5 oz = 17.5 μm. The trace must be several skin depths thick for low loss. At 1 GHz, 35 μm copper is ~17 skin depths — adequate. At 100 GHz, 35 μm copper is ~167 skin depths — fine, but surface roughness of 0.5–2 μm (typical for standard PCB copper) is now comparable to δs, causing additional roughness loss.

Loss tangent (tan δ, also written tan δ or DF = dissipation factor) is the ratio of the imaginary part to the real part of the complex permittivity of the dielectric material. It represents how much RF energy is absorbed and converted to heat in the dielectric.

Since αd ∝ f × tanδ, dielectric loss increases linearly with frequency. For a given substrate, there is a crossover frequency above which dielectric loss exceeds conductor loss:

• FR4: tanδ = 0.020 → crossover ~3 GHz → FR4 is poor above 5 GHz
• Rogers 4350B: tanδ = 0.0037 → crossover ~10–15 GHz → excellent to 30+ GHz
• Rogers 5880: tanδ = 0.0009 → crossover ~30+ GHz → suitable for mmWave (77 GHz automotive radar)
• Air/vacuum: tanδ = 0 → no dielectric loss → used in waveguide for high-power and ultra-low-loss applications

Total loss = conductor loss αc + dielectric loss αd. For a 50 Ω line on FR4 at 5 GHz:

This is significant — a single long PCB trace from PA output to antenna connector can easily add 0.5–1 dB of loss, reducing transmitted power by 10–20%. This is why RF engineers minimise transmission line length and use low-loss substrate for high-frequency designs.

Microstrip: Signal trace on top surface of PCB, ground plane on bottom. Fields exist in both the dielectric and the air above — quasi-TEM mode. The effective permittivity εeff is between 1 (air) and εr (substrate).

Stripline: Signal trace buried between two ground planes inside the PCB stackup. Fully enclosed in dielectric — pure TEM mode. εeff = εr exactly.

In microstrip, the electromagnetic field exists partly in the substrate (εr > 1) and partly in the air above the trace (εr = 1). The signal effectively "sees" a weighted average of both dielectrics. This average is called the effective permittivity εeff.

Consequences:

• λ on the line = λ₀/√εeff, shorter than free space but longer than if fully embedded in εr
• Electrical length of a physical trace depends on the trace width (because W changes εeff)
• At 50 Ω on FR4 (εr=4.4), εeff ≈ 2.7 → λ is 61% of free space wavelength
• εeff increases slightly with frequency (dispersion) because the field becomes more confined to the substrate at higher frequencies

Coplanar waveguide (CPW) has the signal conductor and both ground planes on the same layer — the signal trace is flanked by two ground conductors with a gap s on each side. The characteristic impedance is controlled by the trace width W and gap width s.

Key advantages of CPW over microstrip:

• No via needed for ground connections: Both grounds are on the same layer — ideal for monolithic microwave integrated circuits (MMICs) where ground vias are difficult or expensive
• Easier to connect shunt components: Shunt capacitors and resistors connect directly from signal trace to the adjacent ground — no long via required
• Lower dispersion: More of the field is in the substrate, less in air → εeff closer to εr and less frequency-dependent
• Lower radiation: Ground planes on either side reduce radiation from the signal line

When microstrip is preferred: When you need to easily route ground plane under the line, when component footprints require it, or when the PCB stackup makes CPW impractical.

Microstrip can support unwanted modes in addition to the intended quasi-TEM signal mode. These spurious modes degrade circuit performance by coupling energy away from the signal path.

Surface wave modes (TM₀, TE₁): The dielectric substrate can guide waves laterally along its surface. Above the surface wave cutoff frequency fs = (c/4h)/√(εr−1), energy couples into surface waves and propagates away from the circuit. This is especially problematic for antenna feed networks and phased arrays where surface waves cause unintended mutual coupling between elements.

Substrate modes (parallel-plate): If the substrate is too thick, it acts as a dielectric waveguide, trapping energy between the top and bottom surfaces. Cutoff: fc = c/(2h√εr).

Prevention strategies:

• Use thin substrate: Thinner h raises the surface wave cutoff frequency. At 60 GHz, h should be < 0.25 mm on Rogers 5880.
• Via fence/via wall: Row of vias alongside the trace at intervals ≤ λ/10 suppresses parallel-plate modes by stitching the ground planes together.
• Use substrate with lower εr: Lower εr raises the surface wave cutoff and reduces the surface wave problem.
• Conductor-backed CPW: The ground planes on either side suppress transverse modes effectively.

The dominant mode in rectangular waveguide is the TE₁₀ mode. It has the lowest cutoff frequency of all modes and is therefore the first to propagate when frequency is raised from zero. All practical waveguide systems operate in TE₁₀ to ensure single-mode propagation.

Why TE₁₀ has no Eₓ field: The TE₁₀ mode has only Ey, Hx and Hz field components. There is no electric field in the x-direction (direction of broad wall). This makes it easy to excite with a vertical probe or loop antenna aligned with the Ey direction.

Waveguide has lower loss than coaxial for three reasons:

1. No dielectric loss (air-filled): Standard metal waveguide is air-filled, so G = 0 and there is no dielectric loss at all. All loss comes from conductor walls. Air-filled coax also eliminates dielectric loss, but requires expensive mechanical support structures (air-spaced coax).

2. Larger conductor area: Waveguide has a much larger cross-section than coaxial cable at the same frequency. Conductor loss ∝ 1/(surface area carrying current) — the bigger the waveguide, the lower the loss per unit length.

3. No centre conductor: In coaxial cable, the inner conductor is small and carries most of the current — its surface resistance dominates total loss. Waveguide has no inner conductor, so loss is spread across the large inner wall area.

TDR (Time Domain Reflectometry) sends a fast rise-time step pulse down a transmission line and measures the reflected signal over time. Since reflections from impedance discontinuities travel at a known velocity (~c/√εeff), the arrival time of each reflection gives the location of the discontinuity. The amplitude and shape of the reflection give the nature of the discontinuity.

Signature of common discontinuities:

• Impedance too high (wide gap/narrow trace): Positive step in reflected voltage — looks like Γ > 0
• Impedance too low (capacitive pad/wide trace): Negative step — Γ < 0
• Open circuit: Reflected step = +1× incident (doubled voltage)
• Short circuit: Reflected step = −1× incident (cancelled voltage)
• Capacitor (via pad, connector): Initial negative spike then recovery — capacitor looks like a short initially, then charges to open
• Inductor (bond wire, via): Initial positive spike then recovery — inductor looks like open initially, then acts as short at DC

Both split RF power between ports, but they serve fundamentally different purposes:

Power Divider (e.g. Wilkinson): A three-port network designed to split power equally (or in a fixed ratio) between two output ports. The input port and both output ports are all matched to Z₀. Output ports are isolated from each other. Used when you want to feed two antenna elements, two amplifiers or two circuits with the same signal.

Directional Coupler: A four-port network where most power flows through (Port 1→Port 2) and only a small fraction is coupled to a monitoring port (Port 3). Port 4 is isolated and terminated. Used for power monitoring, VSWR sensing, and signal sampling without interrupting the main signal path.

A Wilkinson divider uses two λ/4 transmission line arms plus a resistor between the output ports to achieve three simultaneous properties that a T-junction cannot: matched input, matched outputs AND isolation between outputs.

T-junction problem: A simple 3-way wire junction splits power but (1) causes reflections at the input because the output impedances are in parallel (two 50 Ω = 25 Ω at the junction, mismatching the input), and (2) provides zero isolation between the two output ports — a signal at output 1 appears directly at output 2.

How the isolation resistor works: When the divider is balanced (equal signals at both outputs), there is zero voltage across R — no current flows, zero power dissipated. When one output has a different signal (combiner mode or fault), current flows through R and absorbs the imbalance, preventing it reaching the other output port.

Mason's rule (Signal Flow Graph method) calculates the transfer function of a complex interconnected network by writing equations for signal flow paths and loops. It is extensively used in RF amplifier analysis, feedback amplifier design, and coupled cavity filter design.

For a two-port S-parameter network with source and load reflections (Γs, ΓL):

Mason's rule is particularly useful for showing why stability conditions matter — when a loop gain = 1, the denominator Δ = 0 and gain goes to infinity (oscillation).

Group delay is the derivative of phase with respect to angular frequency: τg = −dφ/dω. It represents how long a group of frequencies (an information-carrying signal) is delayed passing through a component.

Why flat group delay matters: If group delay varies with frequency, different frequency components of a modulated signal arrive at different times at the receiver. This is called group delay distortion and causes:

• Pulse broadening — a sharp edge becomes a slow ramp
• Inter-symbol interference (ISI) in digital systems
• Ringing and pre-cursors in time-domain signals
• Bit error rate increase in communications systems

Filter comparison for group delay flatness:

• Bessel (Thomson): Maximally flat group delay — specifically designed to have constant τg across the passband. Best for pulse transmission. Worst stopband attenuation.
• Butterworth: Moderate group delay variation — acceptable for many applications.
• Chebyshev: Significant group delay peaking near the passband edge — not suitable for applications requiring phase linearity.
• Elliptic: Worst group delay — severe distortion near band edge. Best stopband attenuation but only used when amplitude selectivity is critical and phase is irrelevant.

In a differential pair, two traces carry equal and opposite signals. Mode conversion occurs when asymmetries in the pair cause some of the differential signal to convert to common-mode (both traces moving in the same direction). This is characterised by the Mixed-Mode S-parameter Scd (common-mode output from differential input).

Causes of mode conversion:

• Length mismatch: Even 0.1 mm difference in trace length causes a phase offset — the two signals no longer cancel perfectly and a common-mode component appears
• Impedance mismatch between the two traces: If one trace is wider than the other (e.g. due to copper fill nearby), it has a different single-ended impedance, causing asymmetric reflection
• Via asymmetry: Differential vias placed at different positions relative to ground planes present different parasitic capacitances to each trace
• Asymmetric ground plane cuts: Gaps or splits in the ground plane under one trace but not the other

Minimisation strategies:

• Maintain tight length matching — PCB design rules typically require < 5–10 mil (0.13–0.25 mm) skew for GHz interfaces
• Keep differential pairs tightly coupled and symmetric about their centreline
• Route differential pairs over solid, unbroken ground planes
• Match via placement and pad geometry exactly for both traces

A λ/4 open-circuit stub looks like a short circuit at the design frequency. This is used as a band-stop (notch) filter — the stub is connected in series with the main line, and presents a short circuit (blocks signal) at the resonant frequency while passing all other frequencies.

Insertion loss is how much signal is lost going through a component (Port 1 to Port 2). Return loss is how much signal is reflected back from the input (Port 1 to Port 1). Both are defined as positive numbers expressed in dB, but insertion loss being large is bad (signal lost) while return loss being large is good (small reflection).

The electric and magnetic fields of the signal interact with the dielectric material in the substrate. The dielectric slows the wave by storing energy in electric polarisation — the molecules in the dielectric respond to the oscillating field, momentarily absorbing and re-emitting energy. This creates an equivalent "slower" wave velocity. The higher εr, the more the molecules interact and the slower the wave.

This is why PCB trace delays are calculated as 1/vp = √εeff/c ≈ 54–70 ps/cm for typical FR4 microstrip — a 1 ns delay requires ~14–18 cm of trace at 1 GHz effective frequency.

A via (vertical interconnect access) is a drilled and plated hole through a PCB that connects conductors on different layers. At RF frequencies, vias are far from ideal — they introduce significant parasitic inductance and capacitance.

Parasitic elements of a signal via:

• Inductance (the dominant effect): L ≈ 0.2 nH for a typical through-hole via in a 1.6mm board. At 5 GHz, this is jωL ≈ j6.3 Ω — a significant series impedance.
• Capacitance from pad to ground: C ≈ 0.1–0.5 pF depending on pad size and clearance to ground. At 5 GHz, this is 1/(jωC) ≈ 60–320 Ω.
• Self-resonance: The via resonates when its inductance and capacitance resonate. Above self-resonance, the via acts as a capacitor and blocks signal.

Minimisation strategies:

• Use the smallest possible drill diameter (lower inductance)
• Use back-drilled vias (remove the stub below the connection point)
• Place ground vias immediately adjacent to the signal via
• Use microvias (blind/buried vias) for the smallest inductance
• For >20 GHz, model each via with 3D EM simulation and de-embed it from measurements

ZL = 75 Ω is real — no reactive part to cancel. Normalised: yL = Y₀/YL = Z₀/ZL = 50/75 = 0.667.

Step 1: Find distance d from load where Re(yin) = 1 (normalised).

Step 2: Find Im(y_d) to determine stub susceptance needed.

Step 3: Short-circuit stub length. Normalised susceptance of SC stub: b = cot(βl) = 0.707 → βl = arccot(0.707) = 54.74° → l = 54.74/360 × λ

Step 4: Convert to physical dimensions on FR4 at 3 GHz.