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TEC-ELEC48-06 · Lesson 6 · 12.5 clock hours · theory

Transmission Line Theory: Impedance, SWR, and Reflections

Objective. Students explain impedance matching, calculate standing-wave ratio (SWR) from a reflected-power or return-loss measurement, and describe the effects of impedance mismatch on transmitted power and equipment.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by an industry-certified professional or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

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What Is a Transmission Line?

A transmission line is any structure that guides electromagnetic energy from one point to another—coaxial cable, twin-lead, waveguide, or printed microstrip. In RF work, the most common type is coaxial cable. Every transmission line has a characteristic impedance (Z₀), a fixed value set by its physical dimensions and the dielectric material between conductors. Common values in telecommunications are 50 Ω (two-way radio, test equipment) and 75 Ω (cable TV, video). Characteristic impedance is not the same as DC resistance; it is determined by the ratio of inductance per unit length to capacitance per unit length and remains essentially constant regardless of line length.

Why Impedance Matching Matters

For maximum power transfer, the impedance of the source, the transmission line, and the load must all match. When a load impedance (Z_L) equals Z₀, every watt of forward power is absorbed by the load—nothing is sent back. When Z_L ≠ Z₀, some energy is reflected back toward the source. These reflections combine with the forward wave to create a standing wave on the line. Standing waves cause voltage and current to peak at fixed points along the cable. High voltage peaks can break down dielectric insulation; high current peaks cause localized heating. Transmitter finals and amplifier stages are designed for a specific load impedance; excessive reflections can cause them to overheat, reduce efficiency, or trigger protective shutdown circuits.

The Reflection Coefficient (Γ)

The reflection coefficient Γ (Greek capital gamma) is the ratio of the reflected voltage wave to the forward voltage wave at the load:

Γ = (Z_L − Z₀) / (Z_L + Z₀)

Γ is dimensionless and ranges from −1 to +1 in magnitude. A perfect match gives Γ = 0 (no reflection). An open circuit (Z_L = ∞) gives Γ = +1; a short circuit (Z_L = 0) gives Γ = −1. In both extremes, 100% of power is reflected.

Standing-Wave Ratio (SWR)

SWR (also written VSWR when referring to voltage) describes the severity of the standing wave pattern:

SWR = (1 + |Γ|) / (1 − |Γ|)

SWR is always ≥ 1. An SWR of 1.0:1 is a perfect match. Practical systems aim for SWR ≤ 1.5:1 or better. A two-way radio base station manufacturer may specify maximum SWR of 2.0:1 before the transmitter folds back power. SWR by itself does not tell you where the mismatch is—it only tells you how bad it is.

Return Loss

Return loss (RL) expresses reflected power in decibels relative to forward power:

RL (dB) = −20 × log₁₀(|Γ|)

Return loss is always stated as a positive number of dB when using this convention. A higher return loss means less reflected power and a better match. A perfect match has infinite return loss. Typical acceptable return loss in an antenna system is 14 dB or better (corresponding to SWR ≈ 1.5:1).

To find the percentage of power reflected:

% Reflected Power = |Γ|² × 100

You can also go directly from RL:

% Reflected Power = 10^(−RL/10) × 100

Effects on Transmitted Power

Not all reflected power is lost—some re-reflects at the source and re-enters the line—but in practice, mismatch always reduces delivered power and can stress equipment. At SWR 2.0:1, roughly 11% of forward power is reflected; at SWR 3.0:1, about 25%. Beyond SWR 6.0:1, reflected power exceeds 50%.

Common Causes of Mismatch in the Field

A sudden rise in SWR on a base station antenna system almost always points to a physical fault: a corroded or improperly installed connector, a pinched or kinked cable that has damaged the dielectric, water ingress at a connector or at a crack in the jacket, or physical damage to the antenna element. The test sequence is to start at the transmitter and work outward—check the short patch cable and connector at the radio first, then the feedline run, then the antenna.

Safety Notes

Before connecting or disconnecting any cable at a transmitter or base station:

Key terms8

Characteristic Impedance (Z₀)The impedance a transmission line presents to a traveling wave, determined by its physical construction; 50 Ω and 75 Ω are the most common values in telecom work.
Reflection Coefficient (Γ)The ratio of the reflected voltage wave amplitude to the forward voltage wave amplitude at a load; ranges from 0 (perfect match) to 1 in magnitude (total reflection).
Standing-Wave Ratio (SWR)A dimensionless ratio ≥ 1 that describes the severity of standing waves on a transmission line caused by impedance mismatch; 1.0:1 is a perfect match.
Return LossThe ratio of reflected power to forward power, expressed as a positive dB value; higher return loss means less reflection and a better match.
Impedance MismatchA condition where load impedance differs from the transmission line's characteristic impedance, causing partial reflection of the forward traveling wave.
Forward PowerThe RF power traveling from the source toward the load along a transmission line.
Reflected PowerThe portion of forward power that is sent back toward the source due to an impedance mismatch at the load or along the line.
Lockout/Tagout (LOTO)An OSHA-required safety procedure that isolates and de-energizes equipment and secures it against accidental re-energization before maintenance is performed.

Worked examples4

1. A 50 Ω coaxial line is terminated in a 150 Ω load. Find Γ, SWR, return loss, and percentage of reflected power.
  1. Step 1 – Find Γ: Γ = (Z_L − Z₀) / (Z_L + Z₀) = (150 − 50) / (150 + 50) = 100 / 200 = 0.5
  2. Step 2 – Find SWR: SWR = (1 + |Γ|) / (1 − |Γ|) = (1 + 0.5) / (1 − 0.5) = 1.5 / 0.5 = 3.0
  3. Step 3 – Find return loss: RL = −20 × log₁₀(0.5) = −20 × (−0.301) = 6.02 dB
  4. Step 4 – Find % reflected power: |Γ|² × 100 = 0.5² × 100 = 0.25 × 100 = 25%
Answer: Γ = 0.5, SWR = 3.0:1, Return Loss ≈ 6.0 dB, 25% of power is reflected.
2. A technician's SWR meter reads 1.5:1. What is the reflection coefficient and the return loss?
  1. Step 1 – Solve for |Γ| from SWR: |Γ| = (SWR − 1) / (SWR + 1) = (1.5 − 1) / (1.5 + 1) = 0.5 / 2.5 = 0.2
  2. Step 2 – Find return loss: RL = −20 × log₁₀(0.2) = −20 × (−0.699) = 13.98 dB ≈ 14.0 dB
  3. Step 3 – Find % reflected power: |Γ|² × 100 = 0.04 × 100 = 4%
Answer: |Γ| = 0.2, Return Loss ≈ 14.0 dB, approximately 4% of power is reflected.
3. An antenna system has a measured return loss of 20 dB. Find SWR and percentage of reflected power.
  1. Step 1 – Find |Γ| from return loss: |Γ| = 10^(−RL/20) = 10^(−20/20) = 10^(−1) = 0.1
  2. Step 2 – Find SWR: SWR = (1 + 0.1) / (1 − 0.1) = 1.1 / 0.9 ≈ 1.22
  3. Step 3 – Find % reflected power: |Γ|² × 100 = 0.01 × 100 = 1%
Answer: SWR ≈ 1.22:1, 1% of power is reflected — an excellent match.
4. A 50 Ω line is terminated in a short circuit (Z_L = 0 Ω). Find Γ, SWR, and return loss.
  1. Step 1 – Find Γ: Γ = (0 − 50) / (0 + 50) = −50 / 50 = −1; |Γ| = 1
  2. Step 2 – Find SWR: SWR = (1 + 1) / (1 − 1) = 2 / 0 → undefined (approaches infinity); by convention SWR = ∞
  3. Step 3 – Find return loss: RL = −20 × log₁₀(1) = −20 × 0 = 0 dB
  4. Step 4 – Interpret: 0 dB return loss means all forward power is reflected; SWR = ∞ indicates total mismatch.
Answer: |Γ| = 1, SWR = ∞, Return Loss = 0 dB — 100% of power is reflected.

The math in this lesson2

Learners compute reflection coefficient, SWR and percent reflected power from load and line impedances.
A-SSE.1F-LE.4
Brush up in the math halls at newSchoolWorks.com:
In class: Work through a guided derivation of SWR from reflection coefficient (Γ) and convert between SWR, return loss (dB), and percentage of reflected power using a provided formula sheet and ten practice problems. · Use an SWR bridge connected to a signal generator and a variable load to deliberately create mismatches of known value; students measure SWR, record results, and plot SWR vs. load resistance on a graph. · Analyze three technician trouble-ticket scenarios (high SWR alarm on a two-way radio base station): students identify likely causes (bad connector, cable kink, antenna damage) and propose a logical test sequence to isolate the fault.
Materials: SWR formula sheet, scientific calculators, SWR/return-loss bridge, signal generator, variable resistive load decade box, coaxial barrel connectors, trouble-ticket scenario handouts
Check: Ten SWR conversion problems scored (80% to pass); lab measurement results recorded correctly and SWR plotted within ±0.2 of calculated expected value for each test load.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

ETA CET — RF and transmission line theory
1. A 50 Ω transmission line feeds an antenna with Z_L = 100 Ω. What is the reflection coefficient |Γ|?
ETA CET — RF and transmission line theory
2. Which SWR value indicates a perfect impedance match between a transmission line and its load?
ETA CET — RF and transmission line theory
3. A measured return loss of 6 dB corresponds to what percentage of reflected power?
ETA CET — Safety and workplace practices
4. A technician notices a high SWR alarm on a VHF base station. What is the safest FIRST action before touching any connectors on the back of the transmitter?
ETA CET — RF and transmission line theory
5. The SWR on a 50 Ω antenna system is measured at 2.0:1. Approximately what percentage of forward power is reflected?
ETA FOI/FOT — Antenna and RF system troubleshooting
6. Which of the following physical faults is MOST likely to cause a sudden increase in SWR on a base-station antenna system that was previously operating normally?
ETA CET — RF measurements and test equipment
7. Return loss is expressed as a positive dB value. A higher return loss indicates:
ETA CET — RF and transmission line theory
8. A transmission line is terminated in an open circuit. What is the SWR?
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