Transmission Line Theory

// 01 — The Model

The Distributed Circuit Model

At low frequencies a wire is just a wire — its inductance and capacitance can be ignored. But at RF and microwave frequencies, the wavelength becomes comparable to physical dimensions. A 10 cm trace at 1 GHz is λ/3 long. The voltage and current vary along the line and cannot be treated as lumped elements.

A transmission line is modelled as an infinite cascade of infinitesimal sections, each with distributed parameters per unit length:

Distributed Parameters per Unit Length

R' — series resistance (Ω/m) — conductor and skin-effect losses
L' — series inductance (H/m) — magnetic energy stored around conductors
G' — shunt conductance (S/m) — dielectric leakage losses
C' — shunt capacitance (F/m) — electric energy between conductors

For a lossless line: R' = 0, G' = 0 (ideal conductors, perfect dielectric)

Rule of thumb: Treat a trace as a transmission line when its length exceeds λ/20 at the highest frequency. At 1 GHz in FR4 (ε_eff≈3.5), λ≈160 mm — so traces longer than 8 mm must be treated as transmission lines.

Microstrip Calculator — find λ for your substrate → Coaxial Line Parameters →

// 02 — Telegrapher's Equations

Telegrapher's Equations

Applying Kirchhoff's laws to an infinitesimal section dz gives two coupled differential equations for voltage V(z,t) and current I(z,t) along the line. For time-harmonic signals (e^jωt):

Telegrapher's Equations — Phasor Form

dV/dz = −(R' + jωL') · I(z) ≡ −Z' · I(z)
dI/dz = −(G' + jωC') · V(z) ≡ −Y' · V(z)

Combining: d²V/dz² = γ² · V(z) where γ = α + jβ = √(Z'·Y')

α — attenuation constant (Np/m) · β — phase constant (rad/m)
β = ω√(L'C') for lossless line (= 2π/λ)

General Solution

General Voltage and Current on the Line

V(z) = V⁺·e^−γz + V⁻·e^+γz
I(z) = (V⁺/Z₀)·e^−γz − (V⁻/Z₀)·e^+γz

V⁺ = forward (incident) wave · V⁻ = reverse (reflected) wave · z=0 at load

Physical meaning: For a matched load V⁻ = 0 — only one wave on the line, no standing wave. Mismatch creates both forward and reflected waves.

// 03 — Characteristic Impedance

Characteristic Impedance Z₀

Z₀ is the ratio of voltage to current for a single travelling wave. It is set entirely by geometry and material — not by frequency (for lossless lines).

Characteristic Impedance

Z₀ = √(Z'/Y') → √(L'/C') for lossless line (purely real)

Why 50 Ω? Compromise between min. loss (~77 Ω in coax) and max. power (~30 Ω). Industry standard since WWII.

Z₀ for common line types

Line Type	Z₀ Formula	Key Parameters
Coaxial	Z₀ = (60/√ε_r)·ln(b/a)	a = inner radius, b = outer radius
Microstrip	Z₀ = (87/√(ε_r+1.41))·ln(5.98h/(0.8W+t))	h = height, W = width, t = thickness
Stripline	Z₀ = (60/√ε_r)·ln(4b/(0.67π(0.8W+t)))	b = plate separation
Two-Wire	Z₀ = (120/√ε_r)·ln(D/d)	D = spacing, d = wire diameter
CPW	Z₀ = (30π/√ε_eff)·K'(k)/K(k)	K = elliptic integral

Microstrip Z₀ → Coaxial Z₀ → Stripline → CPW → Two-Wire →

// 04 — Propagation

Propagation & Phase Velocity

Phase Constant, Wavelength and Velocity

β = 2π/λ = ω/v_p = ω√(L'C') (lossless, rad/m)
v_p = c/√ε_eff · λ = λ₀/√ε_eff · λ₀ = 300 mm / f[GHz]
Velocity factor: Vf = 1/√ε_eff

Example — Wavelength on different substrates at 2.4 GHz

λ₀ = 300/2.4 = 125.0 mm in free space

Medium	ε_eff	Vf	λ at 2.4 GHz	λ/4
Air / PTFE coax	1.00	1.000	125.0 mm	31.25 mm
PTFE (ε_r=2.1)	2.10	0.690	86.3 mm	21.6 mm
FR4 microstrip 50 Ω	3.22	0.557	69.6 mm	17.4 mm
Rogers RO4003 50 Ω	2.82	0.596	74.4 mm	18.6 mm
GaAs MMIC	7.15	0.374	46.8 mm	11.7 mm

Verification FR4: λ = 125/√3.22 = 125/1.794 = 69.7 mm ✓

Compute ε_eff and λ for microstrip →

// 05 — Reflection & Standing Waves

Reflection & Standing Waves

Reflection and Standing Waves

Γ_L = (Z_L−Z₀)/(Z_L+Z₀) · VSWR = (1+|Γ|)/(1−|Γ|)
|V|_max = |V⁺|(1+|Γ|) · |V|_min = |V⁺|(1−|Γ|) · P_del = P_inc·(1−|Γ|²)

Z_L=Z₀ → Γ=0, VSWR=1 · Z_L=0 (SC) → Γ=−1 · Z_L=∞ (OC) → Γ=+1

Worked Example — Z₀ = 50 Ω, Z_L = 150 Ω, P_inc = 1 W

Example 1 — Standing wave on a mismatched line

1

Reflection coefficient: Γ = (150−50)/(150+50) = 100/200 = +0.500

2

VSWR: (1+0.5)/(1−0.5) = 3.0 : 1

3

Return loss: −20·log₁₀(0.5) = 6.02 dB · Mismatch loss: −10·log₁₀(0.75) = 1.25 dB

4

Power delivered: 1.0 × (1−0.25) = 0.75 W (25% reflected)

5

Voltage envelope: |V|_max = 1.50 V at load · |V|_min = 0.50 V at λ/4 from load

✓ Γ=0.5, VSWR=3:1, RL=6.0 dB, 750 mW delivered from 1 W. Voltage maximum at the load (Γ real positive).

VSWR / Return Loss Calculator → Γ Reflection Calculator → Plot Γ on Smith Chart →

// 06 — Impedance Transformation

Impedance Transformation

Input Impedance of a Lossless Transmission Line

Z_in(ℓ) = Z₀ · [Z_L + j·Z₀·tan(βℓ)] / [Z₀ + j·Z_L·tan(βℓ)]

βℓ = (2π/λ)·ℓ = electrical length in radians · θ = 360°·ℓ/λ in degrees

Key insight: A line of length ℓ rotates Γ_L by 2βℓ clockwise on the Smith chart. One full revolution = λ/2 — not λ (factor of 2 because wave travels to load AND back).

Z_in vs length — Z₀ = 50 Ω, Z_L = 100 Ω

Z_in vs electrical length — verified

Length ℓ	βℓ	tan(βℓ)	Z_in	Character
0	0°	0	100 + j0 Ω	= Z_L
λ/16	22.5°	0.4142	82.3 + j12.9 Ω	Inductive
λ/8	45°	1.000	40.0 − j30.0 Ω	Capacitive
λ/6	60°	1.732	19.1 − j22.9 Ω	Capacitive
λ/4	90°	∞	25.0 + j0 Ω	Real — QWT
3λ/8	135°	−1.000	40.0 + j30.0 Ω	Inductive
λ/2	180°	0	100 + j0 Ω	= Z_L again

λ/8: Z_in=50·(100+j50)/(50+j100)=50·(10000−j7500)/12500=40−j30 Ω ✓ · λ/4: Z_in=Z₀²/Z_L=2500/100=25 Ω ✓

Worked Example — Z_L = 30+j40 Ω, λ/8 line

Example 2 — Complex load transformation to pure real

1

Num: Z_L+jZ₀·tan45° = (30+j40)+j50 = 30+j90

2

Den: Z₀+jZ_L·tan45° = 50+j(30+j40) = 50−40+j30 = 10+j30

3

Ratio: (30+j90)/(10+j30). Num×(10−j30)=300−j900+j900+2700=3000+j0. |den|²=1000. Ratio=3+j0

4

Z_in = 50 × 3.0 = 150 + j0 Ω — purely real. Ready for a QWT.

✓ λ/8 line cancels the reactance of Z_L=30+j40 Ω, leaving Z_in=150 Ω (real). This pre-transformation step enables a QWT to be applied — see Impedance Matching.

Trace this rotation on Smith Chart → Impedance Matching Techniques →

// 07 — Special Line Lengths

Special Line Lengths

λ/4 Transformer — Real Impedance Inversion

Quarter-Wave Impedance Inversion

Z_in = Z₀² / Z_L · For matching: choose Z₀ = √(Z_source·Z_load)

Z_L=25 Ω → Z_in=100 Ω · Z_L=100 Ω → Z_in=25 Ω · Z_L=j50 → Z_in=−j50 Ω

Inductive↔Capacitive inversion: A λ/4 line converts SC→OC and OC→SC. Inductors become capacitors and vice versa — used intentionally in filter and matching network design.

λ/2 Line — Impedance Repeat

Half-Wave Line — Impedance Unchanged

At ℓ=λ/2, tan(180°)=0: Z_in = Z_L (regardless of Z₀ — transparent line, repeats every λ/2)

Half-wave resonator at 5.0 GHz on RO4003 (ε_eff=2.82): λ/2 = 300/(2×5×√2.82)/2 = 17.9 mm

Summary of special lengths

Length	tan(βℓ)	Z_in (general)	SC load	OC load
0	0	Z_L	0	∞
λ/8	1	Z₀(Z_L+jZ₀)/(Z₀+jZ_L)	+jZ₀	−jZ₀
λ/4	∞	Z₀²/Z_L	∞ (OC)	0 (SC)
3λ/8	−1	Z₀(Z_L−jZ₀)/(Z₀−jZ_L)	−jZ₀	+jZ₀
λ/2	0	Z_L	0	∞

// 08 — Stubs as Reactive Elements

Stubs as Reactive Elements

An open-circuit or short-circuit stub presents a purely imaginary impedance — acting as an inductor or capacitor depending on its length. Stubs replace chip components at GHz frequencies where parasitics degrade lumped element performance.

Short-Circuit Stub — Z_L = 0

Short-Circuit Stub Input Impedance

Z_SC = j·Z₀·tan(βℓ)
0<ℓ<λ/4 → +jX (inductive) · ℓ=λ/4 → ∞ (OC) · λ/4<ℓ<λ/2 → −jX (capacitive)

Open-Circuit Stub — Z_L = ∞

Open-Circuit Stub Input Impedance

Z_OC = −j·Z₀·cot(βℓ)
0<ℓ<λ/4 → −jX (capacitive) · ℓ=λ/4 → 0 (SC) · λ/4<ℓ<λ/2 → +jX (inductive)

PCB preference: OC stubs are preferred in microstrip — no via to ground required. SC stubs need a via which adds inductance (~0.66 nH for h=1.6 mm, d=0.3 mm — see Example 8).

Worked Examples — Stub as Lumped Element

Example 3 — SC stub as 8 nH inductor at 1.0 GHz, FR4 (Vf=0.57)

1

X_L = 2π×10⁹×8×10⁻⁹ = 50.27 Ω

2

tan(βℓ) = 50.27/50 = 1.0054 → βℓ = arctan(1.0054) = 45.15°

3

λ = 0.57×300 = 171 mm → ℓ = (45.15/360)×171 = 21.4 mm

✓ 21.4 mm SC stub ≈ 8 nH at 1.0 GHz on 50 Ω FR4 microstrip.

Example 4 — OC stub as 1.5 pF capacitor at 2.4 GHz, FR4 (Vf=0.57)

1

X_C = 1/(2π×2.4×10⁹×1.5×10⁻¹²) = 44.2 Ω

2

cot(βℓ) = 44.2/50 = 0.884 → βℓ = arctan(1/0.884) = 48.5°

3

λ = 0.57×125 = 71.25 mm → ℓ = (48.5/360)×71.25 = 9.60 mm

✓ 9.60 mm OC stub ≈ 1.5 pF at 2.4 GHz. No via — preferred for PCB.

Example 5 — SC stub >λ/4 as 2.0 pF capacitor at 1.0 GHz, coax (Vf=0.66)

1

X_C = 1/(2π×10⁹×2×10⁻¹²) = 79.6 Ω

2

tan(βℓ) = −79.6/50 = −1.592 → βℓ = 180°−57.9° = 122.1° (between λ/4 and λ/2)

3

λ = 0.66×300 = 198 mm → ℓ = (122.1/360)×198 = 67.1 mm

✓ 67.1 mm SC stub ≈ 2.0 pF at 1.0 GHz (capacitive region). An OC stub achieves same in 9.6 mm — OC always preferred.

Stub reactance summary — Z₀ = 50 Ω, f = 1.0 GHz, Vf = 0.66, λ = 198 mm

Stub	Length	βℓ	Reactance	Equiv.	Type
SC	λ/16 = 12.4 mm	22.5°	+j20.7 Ω	3.30 nH	Inductive
SC	λ/8 = 24.75 mm	45°	+j50.0 Ω	7.96 nH	Inductive
OC	λ/16 = 12.4 mm	22.5°	−j120.7 Ω	1.32 pF	Capacitive
OC	λ/8 = 24.75 mm	45°	−j50.0 Ω	3.18 pF	Capacitive
SC	3λ/8 = 74.25 mm	135°	−j50.0 Ω	3.18 pF	Capacitive
OC	3λ/8 = 74.25 mm	135°	+j50.0 Ω	7.96 nH	Inductive

Microstrip stub dimensions → Single/Double Stub Matching → Plot stub on Smith Chart →

// 09 — Lossy Lines

Lossy Lines & Attenuation

Attenuation — Low-Loss Approximation

α_c = R'/(2Z₀) (conductor, Np/m) · α_d = (π·f·ε_r·tan δ)/(c·√ε_eff) (dielectric, Np/m)
α_total (Np/m) × 8.686 = dB/m · Skin depth: δ_s = √(1/(πfμσ)) — at 1 GHz copper: 2.09 μm

Example 6 — FR4 microstrip at 5 GHz: ε_r=4.3, tan δ=0.020, W=3 mm, h=1.6 mm

1

δ_s at 5 GHz = 1/√(π×5×10⁹×4π×10⁻⁷×5.8×10⁷) = 0.934 μm (skin effect fully developed)

2

α_d = π×5×10⁹×4.3×0.020/(3×10⁸×1.794) = 21.8 dB/m

3

α_c ≈ 8.7×R_s/(Z₀×W) = 8.7×0.0184/(50×0.003) ≈ 9.3 dB/m

4

Total: 21.8+9.3 = ~31 dB/m. A 50 mm trace loses ~1.55 dB at 5 GHz.

⚠ FR4 dielectric loss dominates above 1 GHz. Rogers RO4003 (tan δ=0.0027) gives ~4 dB/m dielectric loss at 5 GHz — 6× lower.

FR4 at mmWave: FR4 tan δ≈0.020 makes it unusable above ~6 GHz for RF. Rogers RO4003C (0.0027), RO4350B (0.0037), Taconic TLX (0.0019) are common alternatives. Dielectric loss doubles with each doubling of frequency.

// 10 — Practical Design