Characteristic Impedance Is Not a Resistor
Characteristic Impedance Is Not a Resistor
Why 50-ohm coax can look exactly like a 50-ohm resistor at its input—even though no such resistor exists inside the cable.
“Fifty-ohm coax” sounds as though the cable contains 50 Ω of resistance. An ohmmeter says otherwise: open at one end, short at the other, and only a fraction of an ohm along the copper. Both descriptions are correct because they answer different questions.
A resistor establishes a voltage-current relation at its terminals and converts real power into heat.
Z0 = V+/I+ for the forward mode supported by the line.
It depends on Z0, line length, loss, frequency and the termination.
The central distinction: a resistor absorbs energy where it sits. An ideal transmission line guides electromagnetic energy away from the source. A matched load—or the ever-expanding fields of an ideal infinite line—accounts for the real power accepted at the input.
Where 50 Ohms Comes From
A transmission line is a distributed structure. Every small length contributes series inductance and resistance, plus shunt capacitance and dielectric leakage. Those quantities are specified per unit length:
-
R′— series conductor resistance in Ω/m; -
L′— series inductance in H/m; -
G′— shunt dielectric conductance in S/m; -
C′— shunt capacitance in F/m.
Engineer’s corner: the full definition
Z0 = √[(R′ + jωL′) / (G′ + jωC′)]
For a passive line, choose the square-root branch associated with forward power flow and attenuation away from the source. In general Z0 can be complex and frequency-dependent.
For a low-loss line at the frequency of interest:
Z0 ≈ √(L′/C′)
For a homogeneous, non-magnetic coax:
Z0 ≈ (60/√εr) ln(b/a)
Here a is the centre-conductor radius and b is the inside radius of the shield.
This makes characteristic impedance a property of geometry, dielectric and frequency-dependent material behaviour—not a hidden 50-ohm deposit. Real coax is designed so that Z0 is close to a real 50 Ω over its intended band.
Use primes. R′, L′, G′ and C′ are per-unit-length parameters. They are not the total resistance, inductance, leakage or capacitance of the cable.
The Thought Experiment That Makes It Click
Connect a step source to a long 50 Ω cable. The wave begins moving down the line with voltage and current related by V+/I+ = Z0. Until a reflection returns, the source has no causal information about the far-end termination.
source
load
The source establishes the forward-wave ratio set by the cable’s Z0.
The load absorbs it, reflects it, or does both. T is the one-way propagation delay.
Any reflected wave reaches the input and changes the voltage-current relation seen there.
During that first round trip, even a finite cable terminated in an open or short presents its characteristic impedance to a sufficiently fast measurement. That is the basis of a time-domain reflectometer. A practical displayed plateau also depends on source impedance, step rise time, instrument bandwidth, calibration plane and fixture response.
Open, Short and Matched: The Three Revealing Cases
ΓL = +1
Voltage reflects with the same polarity. At the open load, voltage doubles ideally while total current becomes zero.
ΓL = −1
Voltage reflects with opposite polarity. At the short, total voltage is zero while current doubles ideally.
ΓL = 0
No load reflection returns. A 50 Ω line terminated in 50 Ω continues to look like 50 Ω at its input.
ΓL = (ZL − Z0) / (ZL + Z0)
The magnitude tells us how much wave amplitude is reflected; the angle tells us its phase. For a passive load referenced to a real, positive Z0, 0 ≤ |ΓL| ≤ 1.
Why the Matched Line Looks Like a Resistor
On an ideal lossless line terminated in ZL = Z0, the forward wave is absorbed by the load and no reflected wave returns. The source therefore sees a real input impedance equal to Z0, regardless of line length.
For a real-valued Z0 and RMS wave quantities:
P+ = Vrms2/Z0 = Irms2Z0
This simple form requires a real Z0. For complex Z0, calculate average power from Re{V I*}.
A load equal to a complex Z0 is reflectionless at that junction; that is not automatically the same as a conjugate power match to a separate source. That real power is not mysteriously burned along an ideal cable. It flows in the electromagnetic fields between the conductors and is dissipated in the matched load. A real cable adds some conductor and dielectric loss, but those losses are represented by R′ and G′; they are not what “50 Ω characteristic impedance” means.
An infinite lossless line is the subtle case
An ideal infinite line also has a real driving-point impedance Z0, although it contains no load and dissipates no power. The source continually launches energy into electromagnetic fields occupying an ever-longer portion of the line. No reflection ever returns because the wave never reaches an end.
This is a mathematical idealisation, but it proves an important point: a real input impedance does not always mean local heat dissipation at the input. It can represent one-way energy transport into a region that has not yet returned a wave.
Characteristic Impedance Is Not Input Impedance
Z0 belongs to the propagation mode of the uniform line. Zin is the terminal ratio measured at a particular position after forward and reflected waves combine.
Lossless-line input impedance
Zin = Z0[ZL + jZ0tan(βl)] / [Z0 + jZLtan(βl)]
| Termination or length | Ideal lossless result | Meaning |
|---|---|---|
ZL = Z0 |
Zin = Z0 for any length |
The load absorbs the travelling wave without reflection. |
| Open circuit | Zin = −jZ0cot(βl) |
An open line can look capacitive, inductive, open or short depending on length. |
| Short circuit | Zin = jZ0tan(βl) |
A shorted line can also synthesize a frequency-dependent reactance. |
| Quarter wavelength | Zin = Z02/ZL |
The line inverts/transforms the load impedance. |
| Half wavelength | Zin = ZL |
The load impedance repeats at the input. |
This is why “the coax is 50 ohms” and “the radio sees 50 ohms” are not equivalent statements. The first describes the line. The second describes the complete line-and-load system at a stated frequency and measurement plane.
What an Ohmmeter Actually Shows
| Ohmmeter connection | Typical DC result | Why |
|---|---|---|
| Centre to shield, far end open | Open circuit | The dielectric blocks DC; no closed conduction path exists. |
| Centre to shield, far end shorted | Low resistance | The meter sees the centre and shield conductor resistances around the loop. |
| Centre to shield, far end terminated in 50 Ω | About 50 Ω | The meter sees the terminator plus small conductor resistances—not the cable’s Z0. |
| Centre conductor end to end | Low resistance | This is the copper’s DC resistance. |
Safety: discharge long cables before connecting sensitive instruments. A disconnected cable is a capacitor and can retain charge from static electricity or previous testing.
Three Bench Experiments That Make It Visible
Test open, shorted and 50-ohm-terminated coax. This separates DC continuity from characteristic impedance.
Measure the same three loads through a known cable. Watch open and short transform with electrical length while the matched load stays near the centre of the Smith chart.
Observe the initial Z0 plateau, then the load reflection after the round-trip delay. Cable faults appear at their propagation delay.
When Must a Wire Be Treated as a Transmission Line?
There is no single magic boundary. The line model becomes useful when propagation delay is no longer negligible compared with the waveform’s period or transition time.
- For a sinusoid, a common first warning is a physical length around one-tenth of a wavelength in the medium.
- For pulses or digital edges, compare the one-way delay with rise/fall time—not merely with the repetition frequency.
- High-impedance nodes, narrow tolerances or resonant structures can require transmission-line treatment at even shorter fractions.
At 30 MHz, a 2-metre coax jumper can already be a substantial electrical length after velocity factor is included. At 1 kHz, the same cable is usually safe to treat as an ordinary interconnect.
Why the Wrong Mental Model Survives
Ham-radio shorthand compresses several different quantities into the word “impedance”:
| Quantity | What it describes | Typical question |
|---|---|---|
| Resistance | Real voltage-current relation associated with energy conversion or radiation. | Where is accepted power going? |
Load impedance ZL
|
Terminal ratio at the end of the line. | What is connected to the cable? |
Characteristic impedance Z0
|
Forward-wave voltage-current ratio of the line’s mode. | What wave does this geometry support? |
Input impedance Zin
|
Terminal ratio at the chosen measurement plane. | What does the transmitter or analyzer see here? |
A durable mental model: the line launches a wave according to Z0; the load decides the reflection; line length and loss determine how the returning wave combines at the input.
The Takeaway
A 50-ohm cable contains distributed resistance, inductance, dielectric conductance and capacitance—but it does not contain a hidden 50-ohm resistor. Its characteristic impedance is the voltage-to-current ratio of a travelling wave supported by its geometry and materials.
A matched line looks resistive because energy leaves the source and is not reflected back. In a finite system, the matched load absorbs it; in the ideal infinite-line thought experiment, the fields occupy an ever-growing length. Real cable loss is a separate effect.
Once characteristic impedance, load impedance and input impedance are kept separate, most of the apparent contradictions disappear.
Technical references
- MIT OpenCourseWare: Guided Electromagnetic Waves — propagation, characteristic impedance and reflections.
- Rutgers University: Transmission Lines — telegrapher equations, input impedance, reflection coefficient and Smith-chart relationships.
Mini-FAQ
- Why does an ohmmeter not show 50 ohms on 50-ohm coax? It measures DC conduction. Characteristic impedance describes a travelling-wave voltage-current ratio.
- Why does a matched line look like a resistor? The load absorbs the forward wave without reflection, so the line’s input equals its characteristic impedance.
- What does the source see before a reflection returns? It sees the line’s characteristic impedance, because information about the distant load has not yet propagated back.
- Is characteristic impedance always a real number? No. The full lossy-line expression can be complex and frequency-dependent; practical RF coax is designed to be close to a real nominal value across its working band.
- Is characteristic impedance the same as input impedance? No. Input impedance also depends on termination, electrical length, loss, frequency and measurement position.
- Does real coax dissipate power? Yes. Conductor and dielectric losses dissipate power, but those losses are distinct from the meaning of characteristic impedance.