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 a cable or read a high-speed design guide — 50 ohms for test leads, 75 ohms for video, 90 ohms for USB, 100 ohms for Ethernet. 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 termination and impedance matching matter and where reflections come 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.

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What 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 cable 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. This connection between Z₀, mismatches, and reflections 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:

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.

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 ε_r = 2.26.

Notice the effect of the dielectric constant. For air (ε_r = 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 ε_r 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:

Calculating Z₀ for Parallel Pairs and PCB Traces

For a parallel two-wire transmission line in air — two conductors of diameter d separated by a center-to-center spacing S — the characteristic impedance formula is different:

Commercial twisted pair is built to a specified impedance: Cat 5e and Cat 6 Ethernet cable is 100 Ω differential, and the data pair in a USB cable is 90 Ω. 300-ohm twin-lead, once used for TV antennas, achieves its impedance with wider spacing and the conductors embedded in a flat plastic ribbon.

Circuit-board traces (microstrip)

A trace on the outer layer of a circuit board, running over a solid ground plane, is a microstrip transmission line. Its impedance depends on the trace width w, the trace thickness t, the height h of the trace above the ground plane, and the dielectric constant ε_r of the board material. A widely used approximation (IPC-2141) is:

Why 50 Ohms? The History of a Standard

The choice of 50 ohms as the standard for test equipment and most 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 video, broadcast and CATV (cable television) applications because those systems prioritize low loss over power handling — the signals are small and the runs are long. Signal generators, oscilloscopes with 50 Ω inputs, spectrum analyzers, frequency counters and network analyzers almost all use 50 Ω, so 50 Ω BNC and SMA cables are the everyday currency of the electronics bench.

There is also a 93-ohm coaxial cable (RG-62) that was once widely used in computer networking (ARCNET and early IBM terminals). It is rarely encountered today but occasionally appears in older or 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.

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Z₀ 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 the quarter-wave impedance transformer, which you will meet later in this module.

But if you take 5 meters of RG-58 and connect it between a 50-ohm signal generator and a 50-ohm load, then add another 5 meters to make 10 meters, the generator 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 cable, connector and board layout decision you make.

Matching cables to test equipment

A signal generator's output is designed to drive a 50-ohm load through 50-ohm cable. When the cable is 50 Ω and the far end is terminated in 50 Ω — a 50 Ω oscilloscope input, or a feed-through terminator on a 1 MΩ input — the signal arrives with the amplitude and shape the generator intended. Leave the far end unterminated and the generator's output voltage doubles at the open end, and every fast edge reflects back down the cable.

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: 20% of the incident voltage is reflected, corresponding to a VSWR (voltage standing wave ratio) of (1+0.2)/(1−0.2) = 1.5:1. For short runs and slow signals the practical effect is negligible. For fast edges and long runs, the reflections show up as steps and ringing on the waveform. The two kinds of BNC connector look almost identical, so check the cable marking.

Choosing between 50-ohm cable types

When choosing between different 50-ohm cables (thin RG-174 and RG-316, standard RG-58, low-loss RG-213 and LMR-400), the characteristic impedance is the same — but the loss per unit length, flexibility, size and price differ. Thin cables are fine for short internal connections; long runs at high frequencies need lower-loss, thicker cable.

Board traces and data cables

High-speed interfaces specify the impedance of the whole path, not just the cable: USB requires 90 Ω differential traces from the connector to the chip, Ethernet 100 Ω, HDMI 100 Ω, and many single-ended high-speed signals 50 Ω. A cable of the right impedance plugged into a board whose traces are the wrong width still produces reflections at the connector. This is why high-speed board layouts specify trace widths and spacing, and why a USB or HDMI link that works with a short cable can fail with a long one when the board is marginal.