Characteristic Impedance
Every transmission line has a single most important electrical property: its characteristic impedance, written as Z₀ (spoken as "Z nought" or "Z zero"). You encounter this number every time you handle coaxial cable — 50 ohms, 75 ohms, 93 ohms. But what does it actually mean? It is not resistance, not the DC resistance you would measure with a multimeter. It is something more fundamental, and understanding it explains why impedance matching matters and where SWR comes from.
Coaxial cable cross-section showing the two conductor diameters that determine characteristic impedance. The ratio D/d and the dielectric constant of the insulator completely determine Z₀.
View LargerWhat Characteristic Impedance Means
The characteristic impedance of a transmission line is the ratio of the voltage to the current in a traveling wave on that line, when no reflections are present. More precisely, if you launched a single forward-traveling wave down an infinitely long line (or a line terminated in a perfectly matched load at the far end), the ratio V/I at every point would equal Z₀.
This is quite different from resistance. Resistance dissipates energy — it converts electrical energy to heat. Characteristic impedance does not dissipate energy (in an ideal lossless line). It simply describes the relationship between the voltage and current amplitudes of the traveling wave. A 50-ohm line means that for every volt of wave voltage, the wave current is 1/50 = 0.02 amperes, regardless of where you measure it along the line.
The most intuitive way to think about Z₀ is as the "natural" impedance of the line for traveling waves. If you connect a source of some impedance Z_S to a line of characteristic impedance Z₀, and the load at the far end also equals Z₀, then maximum power flows from source to load with no reflections. The line is "transparent" — you cannot tell whether the feedline is 1 meter or 100 meters long, because the impedance seen at the source is always exactly Z₀.
Conversely, whenever a wave reaches a point where the impedance changes — either at the load or at a connector joining two cables with different Z₀ — some of the wave is reflected back toward the source. The greater the impedance mismatch, the greater the reflected wave, and the higher the SWR. This connection between Z₀, mismatches, and SWR is the core of transmission line theory.
Where Z₀ Comes From: The L/C Ratio
In the previous lesson you learned that a transmission line can be modeled as an infinite series of tiny inductors (L per unit length) and shunt capacitors (C per unit length). The characteristic impedance arises directly from the ratio of these two distributed parameters.
For a lossless transmission line, the characteristic impedance is:
- L = inductance per unit length (henries per meter)
- C = capacitance per unit length (farads per meter)
- Z₀ = characteristic impedance (ohms)
This formula reveals something important: Z₀ depends on the ratio of L to C, not on either quantity alone. If you have a line with high inductance per unit length and low capacitance per unit length, you get high Z₀. If inductance is low and capacitance is high, you get low Z₀.
The physical geometry of the transmission line determines L and C. A coaxial cable with a thin inner conductor and a large outer conductor has large spacing between them, which means high inductance (magnetic field fills more space) and low capacitance (more spacing between the "plates"). This combination gives a high Z₀. A coaxial cable with a thicker inner conductor and smaller outer conductor has lower inductance and higher capacitance, giving lower Z₀.
This is why you cannot achieve just any characteristic impedance from coax — you are constrained by what is geometrically and mechanically practical. Too large a ratio of outer to inner conductor diameter gives a mechanically weak center conductor. Too small a ratio forces the conductors too close together, reducing voltage breakdown capability and increasing capacitance. The useful range for practical coaxial cables spans roughly 30 ohms to 185 ohms.
Calculating Z₀ for Coaxial Cable
For a coaxial cable, the characteristic impedance depends on only three things: the diameter of the inner conductor, the inner diameter of the outer conductor, and the dielectric constant of the insulating material between them.
Z₀ = (138 / √εr) × log₁₀(D/d)
- D = inner diameter of outer conductor (any unit, must match d)
- d = outer diameter of inner conductor (same unit as D)
- εr = relative dielectric constant of the insulating material (dimensionless)
- log₁₀ = base-10 logarithm
Let us verify this with a real example. Standard RG-213 coaxial cable has an inner conductor diameter of 2.26 mm and an outer conductor inner diameter of 7.25 mm. The dielectric is solid polyethylene with εr = 2.26.
D/d = 7.25 / 2.26 = 3.208
log₁₀(3.208) = 0.5063
√εr = √2.26 = 1.503
Z₀ = (138 / 1.503) × 0.5063 = 91.8 × 0.5063 = 46.5 Ω
Close to the nominal 50 Ω specification (small deviations occur due to dimensional tolerances and the exact εr of the specific polyethylene formulation).
Notice the effect of the dielectric constant. For air (εr = 1.0, as in an air-spaced coaxial cable), the formula simplifies to 138 × log₁₀(D/d). A higher dielectric constant (denser plastic insulation) reduces Z₀ for the same physical geometry. This is why foam-insulated cables (lower εr than solid polyethylene) have slightly higher Z₀ for the same D/d ratio and need a slightly larger D/d to achieve 50 ohms.
The dielectric constant of common insulating materials used in coaxial cable:
| Dielectric Material | εr (approx.) | Velocity Factor | Typical Use |
|---|---|---|---|
| Air | 1.0 | 1.00 (100%) | Air-spaced coax, waveguide |
| Expanded (foam) polyethylene | 1.4–1.5 | 0.81–0.86 | LMR-400, RG-8/U foam, premium coax |
| Solid polyethylene | 2.26 | 0.66 | RG-213, RG-58, most standard coax |
| PTFE (Teflon) | 2.1 | 0.70 | High-temperature military coax, RG-142 |
| Solid PTFE | 2.04–2.08 | 0.695 | Semi-rigid coax, microwave assemblies |
Calculating Z₀ for Open-Wire Feeders
For a parallel open-wire transmission line — two conductors of diameter d separated by a center-to-center spacing s — the characteristic impedance formula is different:
Z₀ = 276 × log₁₀(2S/d)
- S = center-to-center spacing between conductors (any unit)
- d = conductor diameter (same unit as S)
- Valid when S >> d (spacing much greater than conductor diameter)
Two conductors of diameter 1.5 mm (AWG 15) spaced 50 mm apart center-to-center.
2S/d = 2 × 50 / 1.5 = 66.7
log₁₀(66.7) = 1.824
Z₀ = 276 × 1.824 = 503 Ω
This is close to the classic "600-ohm open-wire" feeder. Increasing the spacing or decreasing the conductor diameter increases Z₀; decreasing spacing or increasing conductor diameter decreases it.
For ladder line — parallel conductors with periodic plastic spacers in air windows — the formula is slightly modified by the fraction of the line that is in solid dielectric. Commercial 450-ohm ladder line uses conductors and spacing chosen to achieve exactly 450 ohms with its specific plastic-window construction. 300-ohm twin-lead achieves its impedance with a narrower spacing and the conductors partly embedded in plastic.
Why 50 Ohms? The History of a Standard
The choice of 50 ohms as the standard for amateur radio and most professional RF work was not arbitrary. It represents a deliberate engineering compromise between two conflicting requirements.
If you want the lowest loss in a coaxial cable at a given outer diameter, the optimum ratio of D/d is about 3.6, corresponding to 77 ohms (in air-filled coax). This is why 75-ohm cable is used for broadcast television distribution — it minimizes attenuation per unit length.
If you want the highest power handling capability at a given outer diameter (limited by the breakdown electric field strength in the dielectric), the optimum ratio is about 1.65, corresponding to 30 ohms. This is because a smaller center conductor means a lower electric field peak for a given voltage.
Neither 77 ohms nor 30 ohms is ideal for both requirements simultaneously. The geometric mean of these two optima, balancing minimum loss with reasonable power handling, falls at about 50 ohms. This compromise became the US military standard (MIL-C-17) for coaxial cable connectors and feedlines, and then became the de facto worldwide standard through the influence of US military specifications on the electronics industry after World War II.
The 75-ohm standard persists in broadcast and CATV (cable television) applications because those systems prioritize low loss over power handling — the signals are small and the runs are long. In amateur radio, 50 ohms is by far the dominant standard, and virtually all amateur radio transceivers, amplifiers, and antenna analyzers are designed to work with 50-ohm systems.
There is also a 93-ohm coaxial cable (RG-62) that was once widely used in computer networking (ARCNET). It is rarely encountered in amateur radio but occasionally appears in surplus equipment.
The distributed circuit model of a transmission line. Each infinitesimally small section contains a series inductor and shunt capacitor. The ratio L/C determines Z₀; the product L×C determines the wave velocity.
View LargerZ₀ Does Not Depend on Length
This point is worth repeating because it confuses many beginners: cutting a cable to a different length does not change its characteristic impedance. Characteristic impedance is a property of the cable's geometry — its cross-sectional dimensions and dielectric material. It is the same for every meter of a given cable type.
What does change with length is the impedance that the source "sees" when the line is terminated with a mismatched load. The impedance presented at the input of a mismatched transmission line varies with length in a periodic way, repeating every half wavelength. This is the basis of quarter-wave impedance transformers and stub matching — which you will learn about in lessons M13J and M13K.
But if you take 5 meters of RG-213 and connect it between a 50-ohm source and a 50-ohm load, then add another 5 meters to make 10 meters, the source still sees 50 ohms. The Z₀ of the cable has not changed, and a matched line looks like its characteristic impedance from either end regardless of length.
Practical Consequences of Characteristic Impedance
Understanding Z₀ has immediate practical consequences for every feedline decision you make at the station.
Matching coax to your transceiver
Your transceiver's antenna output is designed to deliver power into a 50-ohm load. Your feedline should be 50-ohm coax. Your antenna, at its resonant frequency, should present 50 ohms (or close to it) at the feedpoint. When all three match, maximum power flows from transmitter to antenna, SWR is low, and no antenna tuner is needed.
Mixing 50-ohm and 75-ohm cable
It is technically possible to use 75-ohm cable in a 50-ohm system, but the impedance mismatch at each junction creates reflections. The reflection coefficient at a 50-to-75 ohm junction is (75-50)/(75+50) = 0.2, corresponding to an SWR of (1+0.2)/(1-0.2) = 1.5:1. This is mild but not zero. For very short runs at HF, the practical effect is negligible. For long VHF/UHF runs where cable loss is significant, using the wrong impedance cable adds both junction reflections and any additional loss caused by the elevated SWR.
Choosing between 50-ohm cable types
When choosing between different 50-ohm cables (RG-58, RG-213, LMR-400), the characteristic impedance is the same — but the loss per unit length, power handling capability, and physical flexibility differ. These practical differences are covered in detail in lesson M13E.
Antenna feedpoint impedance
A simple half-wave dipole at its resonant frequency presents a resistive impedance at its feedpoint. For a dipole in free space this is approximately 73 ohms. A real dipole installed over imperfect ground or near structures may deviate significantly from this. A vertical monopole over a perfect ground plane has a feedpoint impedance of about 36 ohms. Neither value is a perfect match to 50-ohm coax, but in practice the SWR caused by these mismatches is low enough (less than 2:1) to cause minimal additional feedline loss at HF. This is why many HF antennas are fed directly with 50-ohm coax without any matching network.
Frequently Asked Questions
My antenna analyzer shows the feedpoint impedance as, say, 72 + j15 ohms. Which part is Z₀?
Neither. The feedpoint impedance (72 + j15 ohms in this example) is the load impedance — what the antenna presents to the feedline. The characteristic impedance Z₀ is a property of the feedline itself — 50 ohms for standard amateur radio coax, 450 ohms for common ladder line. The SWR on your coax results from the ratio of your feedline's Z₀ (50 ohms) to the load impedance (72 + j15 ohms in this case). You can use the SWR formula or an antenna analyzer to calculate the resulting SWR, which you will learn in lesson M13H.
Can I use 75-ohm TV coax (RG-6 or RG-11) for HF amateur radio?
Yes, it works, but with the caveat that you have a 1.5:1 SWR mismatch at the junction with any 50-ohm equipment. For receiving only, this is insignificant. For transmitting, the mismatch causes a small reflected power loss and most modern transceivers will handle it fine. The bigger practical issue is that 75-ohm coax uses different connectors (F-type, BNC 75-ohm) than standard amateur radio connectors (PL-259, N-type 50-ohm), so you need adapters. RG-6 is low-loss, inexpensive, and widely available, making it a worthwhile compromise for certain installations — particularly when running long cable runs where its lower loss compared to RG-213 partially compensates for the small impedance mismatch penalty.
If I connect two different 50-ohm cables together, does the junction affect the impedance?
Electrically, no — as long as both cables have the same characteristic impedance (50 ohms), there is no impedance mismatch at the junction and no reflections are created. Any losses at the junction come from the connector itself: contact resistance, imperfect shielding, or dielectric differences at the mating surfaces. A good quality PL-259 or N-type connector introduces less than 0.1 dB of loss at HF, which is negligible. At VHF and UHF, connector quality becomes more important because even small imperfections at the junction can cause reflections at those frequencies. Always use appropriate connectors for the frequency range — N-type for VHF/UHF, not PL-259.
Why is Z₀ expressed in ohms if it does not dissipate power?
Ohms is the unit for the ratio of voltage to current, regardless of whether the circuit element dissipates power. A pure inductor or capacitor also has impedance measured in ohms, even though they store rather than dissipate energy. The characteristic impedance Z₀ represents the V/I ratio of a traveling wave, which is a real (not complex) quantity for a lossless line. The fact that it has units of ohms does not mean it behaves like a resistance — it only behaves like a resistance from the source's point of view when the line is terminated in a load that equals Z₀. In that matched case, the source cannot tell the difference between driving a 50-ohm resistor and driving a 50-ohm transmission line terminated in 50 ohms.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.