# What is characteristic impedance?

The RF world is full of esoteric terminology, and not least of these is characteristic impedance, or Z_{0}. Most coaxial cables have a characteristic impedance of 50 or 75 ohms. If you want to know why 50 ohms was arrived at as a sort of standard for radio applications, you can read this post. In the simplest terms, the characteristic impedance is calculated by the formula on the right, taking the square root of the distributed inductance (nH per meter, for example) divided by the distributed capacitance (pF per meter, for example).

Properly defined, the characteristic impedance is:

The ratio of the amplitudes of the voltage and current of a single wave, propagating down the line.

A single wave meaning a wave traveling in one direction, with no reflected waves or other anomalies.

What’s in a name? Co-axial means that both conductors in the cable share the same axis, i.e: if you cut a piece of coaxial cable and look at the cut end, the radius of the both inner and outer conductors can be measured from the center line of the inner conductor.

It is readily understood that a piece of wire has an inherent inductance. It is also readily understood that there will be capacitance between the inner core and outer braid of a coaxial cable, or between the lines making up a ladder line. Therefore, a transmission line (ladder line, coaxial, microstrip, coplanar waveguide, etc) can be thought of as a series of inductors and capacitors, as illustrated to the right, starting with L1, C1, then L2, C2, and continuing on down the line until Ln, Cn at the end of the line. These are not real inductors or capacitors, but rather they are the distributed electrical properties of the transmission line (series inductance, shunt capacitance), and they oppose the change in voltage and current in the line. See this page on the distributed model for more information on that.

Imagine a 1 watt signal in a 50 ohm transmission line. This equates to a 1 volt peak. (power to voltage in a 50 ohm system: V_{peak}= 10^{P[}^{dBm]-10/20}).

We enable the signal, and it begins propagating via repelling electrons down the cable at (for sake of argument) the speed of light. It has covered 1 meter of cable In 3.3 nanoseconds. Behind the propagating wave front is a 1 volt potential difference, created by this wave, and it exists between the center conductor and the shield of the cable. This is effectively a capacitor with stored positive charge on the center conductor, and stored negative charge on the shield. The distributed inductance is also storing this energy in a magnetic field.

Ahead of the wave front, the cable is still not charged, and remains at a zero volt potential. However, every 3.3 nanoseconds the signal propagates another meter into the cable, it charges each “capacitor” and “inductor” to that 1 volt potential by drawing more charge from the signal source.

The characteristic impedance of a transmission line is directly related to the physical geometric construction of the transmission line elements, and the dielectric medium that exists between them. As such, characteristic impedance does not change with transmission line length.

For example, a twin lead or ladder line transmission line characteristic impedance is a function of the diameter of the cables used, and the distance between them. For a coaxial cable, the characteristic impedance is a function of the outer diameter of the inner core (**d2** below), the inner diameter of the outer shield (**d1** below), and the dielectric constant of the insulating foam, PTFE, or whatever is used between the two conductors (** k** below). It is easy to see now why coaxial cable has a minimum bend radius associated with its size, as crushed or kinked coaxial cable is a bad thing. There is an impedance step change wherever the damage occurs due to the dimensions between the two conductors being different.

Characteristic impedance can not be measured with a multimeter set to Ohms. I have seen people try to do this, looking for a result of 50 ohms. It doesn’t work that way. Characteristic impedance can also be called *instantaneous impedance* or *surge impedance*, since it is a measure of the cable’s response to a short, finite step, or pulse, as that pulse propagates down the line. Or, in other words, it is the instantaneous impedance that a pulse (or wave front) sees as it propagates down the line. The technical side of how characteristic impedance is determined (dV/dt, dI/dt , wave equations, etc) can get very complex very quickly, and is widely published, so no need to delve in quite so deep here.

### Experiment

Take a length of coaxial cable and a battery. Measure the maximum instantaneous current when you connect the battery to the coaxial cable (shield to negative, center conductor to positive, for example) as the battery charges the cable. It will happen very fast. You’ll probably need an oscilloscope with a differential probe, or some other sort of jig for measuring current spikes. Note the current value. From there, it is a simple Z_{o}=V_{bat}/I_{measured} to determine the characteristic impedance.

### Momentary Measurement

If you measured an infinitely long piece of 50 ohm coaxial cable (for example) with your super-unbelievably-fast multimeter on the ohms scale, and even if the signal from your multimeter was traveling at the speed of light down the cable, and there were no losses in the cable, you would indeed measure 50 ohms, since there is no end to this cable.

In a (very slightly) more practical example, lets take a coaxial cable which is 100 miles long. This coaxial cable uses a center insulator with a dielectric constant of 2.3, which translates to a velocity constant of 66% of the speed of light. Your multimeter’s signal would travel down the cable, and you’d measure 50 ohms for the time that the signal takes to propagate down the cable at 66% of the speed of light, which is 814 μs. After which, you’d measure an open circuit if the cable was open at the other end, or a short circuit if it was a short circuit at the other end.

For a 10 meter piece of this same cable, you would measure 50 ohms for only 0.05 μs (50 nanoseconds).

### Irrational 377 ohms

Characteristic impedance does not need a transmission line to exist. Free space has a characteristic impedance of 377 ohms.

Close permittivity and permeability approximations for free space:

Permittivity: epsilon_{o} __~__ [1 / (36 pi)] x 10^{-9} Farads per meter.

Permeability: mu_{o} __~__ (4 pi) x 10^{-7} Henries per meter.

Solving for free space characteristic impedance, we insert the above characteristics into the impedance equation:

Z_{o} = sqrt (mu_{o} / epsilon_{o}) (H / F)^{1/2}

Z_{o} = sqrt { (4 pi) x 10^{-7} / [1 / (36 pi)] x 10^{-9} } ohms

Z_{o} = sqrt { 36 x 4 x pi^{2} x 10^{2} } ohms

Z_{o} = { 6 x 2 x pi x 10 } ohms

Z_{o} = { 120 x pi } ohms

Z_{o} = **376.991118431 ohms**

Knowing what we know about impedance mismatch and energy loss, how do we match the 50 ohm characteristic impedance of the transmitter and coaxial cable to the 377 ohms characteristic impedance of free space? We use an impedance transformer, commonly known as an antenna.

### In Summary

Characteristic impedance is:

- The ratio of the amplitudes of the voltage and current of a single wave, propagating down the line.
- An impedance response to an instantaneous pulse, or wave front.
- Defined by the geometric dimensions of the transmission line.
- Not a DC property, so cannot be measured by a multimeter resistance function.

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