Introduction to Transmission Lines
In Part 1 we described impedance as a property of components. At radio frequencies, the wires that connect components also have an impedance of their own. When a cable or PCB trace is a noticeable fraction of a wavelength long, a signal takes time to travel along it, and the voltage and current vary from point to point. Such a connection has to be treated as a transmission line, characterised above all by its characteristic impedance Z₀.
Characteristic impedance is the reference against which every load is judged. It is the "Z₀" that appears in the reflection coefficient (Part 3) and the number every Smith chart is normalised to (Part 7).
This article covers what the Smith chart needs. The telegrapher's equations, wave propagation, line losses, stripline and the full microstrip design procedure are covered in our dedicated transmission line series. Open the Transmission Line Theory series →
What Is a Transmission Line?
A transmission line is any pair of conductors that guides an electromagnetic wave from one place to another while keeping its cross-section the same along its length: a coaxial cable, a twisted pair, a microstrip trace over a ground plane, a stripline buried in a PCB. The rule of thumb for when a connection must be treated as a line is:
Electrically, each short length Δz of line has a series inductance L′Δz (from the magnetic field around the current) and a shunt capacitance C′Δz (from the electric field between the conductors), plus small loss terms R′Δz and G′Δz. L′ and C′ are per-unit-length values set entirely by the line's geometry and materials.
Definition of Characteristic Impedance
The characteristic impedance Z₀ of a transmission line is the ratio of voltage to current in a single wave travelling in one direction along it. Equivalently, it is the input impedance of an infinitely long line — or of any length of line terminated in Z₀.
Note that Z₀ is not a resistance you could measure with an ohmmeter — a lossless line dissipates no power. It describes how the line relates voltage to current for a travelling wave, and therefore what load it "wants" to see to avoid reflections.
Relationship Between Voltage and Current
On a line, the total voltage and current at any point are the sum of a forward wave (+) and a reflected wave (−). Each wave individually obeys V/I = Z₀, but because the reflected current flows the opposite way, the signs differ:
Characteristic Impedance of Coaxial Cables
Characteristic Impedance of Microstrip Lines
A microstrip line is a copper strip of width W on a substrate of thickness h and permittivity ε_r, over a ground plane. Part of its field is in the substrate and part in the air above, so the wave sees an effective permittivity ε_eff between 1 and ε_r. A widely used closed-form result (Hammerstad) for W/h ≥ 1 is:
50-Ohm and 75-Ohm Transmission Systems
Why 50 Ω? For an air-filled coaxial line with a fixed outer diameter, the conductor loss is lowest when b/a ≈ 3.59 (Z₀ ≈ 77 Ω), while the peak power before the air breaks down is highest when b/a ≈ 1.65 (Z₀ = 30 Ω). Fifty ohms is a compromise between low loss and high power handling, and became the standard for RF test equipment, transmitters and most RF circuits. Seventy-five ohms, close to the low-loss optimum, became the standard where signals are weak and power is low: video, cable TV and satellite receivers.
| System | Z₀ | Typical uses |
|---|---|---|
| RF / microwave | 50 Ω | Test equipment, radios, Wi-Fi, cellular, radar, most RF ICs and connectors (SMA, N, BNC) |
| Video / broadband | 75 Ω | Cable TV, satellite LNBs, SDI video, F-connectors |
| Differential digital | 85–100 Ω diff. | USB, Ethernet, PCIe, HDMI (pairs of coupled lines) |
| Antenna feed lines | 300 / 450 Ω | Balanced twin-lead and ladder line for HF antennas |
Factors Affecting Characteristic Impedance
- Conductor spacing. Moving the conductors apart lowers C′ and raises L′, so Z₀ rises (larger b/a in coax, thicker substrate h in microstrip).
- Conductor size. A wider strip or fatter inner conductor raises C′ and lowers L′, so Z₀ falls.
- Dielectric constant. Higher ε_r raises C′ and lowers Z₀ in proportion to 1/√ε_r (fully filled lines) or 1/√ε_eff (microstrip).
- Surroundings. Nearby copper, solder mask, enclosures or a second trace add capacitance and lower Z₀ — why coplanar and coupled lines need their own formulas.
- Frequency. For low-loss lines Z₀ is nearly constant, but microstrip ε_eff rises slowly with frequency (dispersion), and at low frequencies the loss terms R′ and G′ make Z₀ complex.
- Manufacturing. Etching tolerance on W, laminate ε_r tolerance and copper thickness typically give ±5–10% on PCB lines, which is why controlled-impedance boards are specified and tested.
Worked Examples
Z₀ and Velocity From L′ and C′
Problem: A cable datasheet lists L′ = 250 nH/m and C′ = 100 pF/m. Find Z₀ and the propagation velocity.
The velocity factor of 0.67 = 1/√2.25 tells us the dielectric is solid polyethylene.
Designing a 75 Ω Coaxial Cable
Problem: What diameter ratio b/a gives 75 Ω with foamed polyethylene (ε_r = 1.5), and with solid polyethylene (ε_r = 2.25)?
Foam dielectric allows a thicker inner conductor for the same outer size, which lowers loss — why RG-6 satellite cable uses foam.
Where 50 Ω Comes From
Problem: For air-dielectric coax, find Z₀ at the minimum-loss ratio b/a = 3.591 and at the maximum-power ratio b/a = √e = 1.649. What is their geometric mean?
Fifty ohms sits close to that compromise and was adopted as the standard; the exact choice was partly historical, but the physics explains why it is near 50 and not 10 or 200.
50 Ω and 75 Ω Microstrip on FR-4
Problem: Find the trace widths for 50 Ω and 75 Ω microstrip on 1.6 mm FR-4 (ε_r = 4.4), and the effective permittivity of each.
| Target Z₀ | W | W/h | ε_eff | Formula used |
|---|---|---|---|---|
| 50 Ω | 3.08 mm | 1.93 | 3.33 | Hammerstad, W/h ≥ 1 |
| 75 Ω | 1.43 mm | 0.89 | 3.15 | Hammerstad, W/h < 1 |
Raising Z₀ by half needs a trace less than half as wide, because Z₀ depends roughly logarithmically on W/h. The narrower 75 Ω line has slightly lower ε_eff since more of its field is in air.
75 Ω Cable on 50 Ω Equipment
Problem: A 75 Ω TV cable is accidentally used between a 50 Ω transmitter and a 50 Ω antenna. What reflection coefficient does the transmitter see at the junction?
Two such junctions exist, and their reflections add or cancel depending on the cable's electrical length — so the input impedance swings with frequency. Parts 3–5 show how to calculate exactly this.
Summary
- A connection longer than about λ/10 must be treated as a transmission line.
- Characteristic impedance is the voltage-to-current ratio of a travelling wave: Z₀ = √(L′/C′) for a low-loss line.
- For coax Z₀ = (60/√ε_r) ln(b/a); for microstrip it depends on W/h and ε_r through ε_eff.
- 50 Ω balances low loss and high power; 75 Ω minimises loss for low-power video and broadband systems.
- Z₀ is set by geometry and materials, not by length.
Now that we have a reference impedance, Part 3: Understanding Load Impedance and Reflection Coefficient asks what happens when a line is terminated in something other than Z₀.
Put It Into Practice
Use the NEXT ANTENNAS calculators to apply the ideas from this article to real designs.