Does a cable's impedance change depending on its length?

No, it does not change. The characteristic impedance is a property of the cross-section and not a property of length. A short piece of wire (e.g. 30 cm) and a complete cable of a certain length (300 meters) both have the same characteristic impedance. While losses, total capacitance and delay do change with length, the impedance itself remains constant. This is why it cannot be measured with an ohmmeter either, as that measures the copper conductor and not an electromagnetic wave.

How to convert propagation speed to relative permittivity?

Square the ratio of 100 to the propagation speed in percent. For a cable with a nominal value of 66 percent, the relative dielectric constant is approximately 2.30. For a cable with a nominal value of 82 percent, it is approximately 1.49. Conversely divide 100 by the square root of the relative dielectric constant. If you select "Enter your own values" in the list of relative dielectric constants, the calculator will accept either of the two values given on the datasheet.

Why are there 50 ohms and 75 ohms in coaxial cables?

For coaxial cables with air as the dielectric, the geometric structure that will handle maximum power is near 30 ohms, while the structure with least loss is near 77 ohms. Fifty ohms is approximately the geometric mean of these two values and therefore a compromise for transmission equipment. Seventy-five ohms is on the side of lower losses and so is the standard format for receive-only applications such as radio, cable TV, and satellite where no signals are sent and all loss in decibels is important.

Can you connect a 75 ohm cable to a 50 ohm system?

Generally yes. The mismatch will create a VSWR of about 1.5 and a loss of about 0.18 dB. This can be ignored in the receive path and is usually within acceptable limits for short runs. When actual power is being transmitted, effects may occur if the connectors themselves have differing impedances or reflections are disturbing sensitive measurements. Open the match test to check the exact values for your combination.

What diameter must the shielding have for a 50 ohm cable?

Multiply the diameter of the center conductor by a value from an exponential function that results from target impedance multiplied by the square root of the dielectric constant, divided by 59.96. For polyethylene with a resistance of 50 ohms this ratio is 3.49. So a conductor with a diameter of 1 mm requires shielding with an inner diameter of 3.49 mm. When design mode is selected, the calculator will perform calculations in both directions. It can either calculate the required shield for a given conductor or the required size of the conductor for a given shield.

Cable Impedance Calculator

Cable type and dielectric

What are you working out?

Coaxial cable

Twisted pair or two-wire line

Design a coax to a target impedance

Dielectric between the conductors

Pick the insulation between the conductors. Every value here is the material's relative permittivity, which is the only property of the dielectric the maths needs.

Geometry

Inner conductor diameter (d)

Shield inner diameter (D)

Only the ratio matters for the impedance, so millimetres and inches give the same answer as long as both fields use the same unit. The absolute size only shows up in the cut-off frequency.

Fill in whichever conductor you already have. The calculator returns the shield you would need for that conductor and, separately, the conductor you would need for that shield.

Optional checks

Work out the totals for a real cable run

Per-metre figures times the length you are actually going to install: total capacitance, total inductance and total propagation delay.

Check the match to my system impedance

Reflection coefficient, VSWR, return loss and mismatch loss against the 50 or 75 ohm system you are plugging into.

Work at an operating frequency

Wavelength inside the cable, the quarter-wave length for stubs and phasing lines, and how many electrical degrees long your run is.

Your numbers

Characteristic impedance (ohm)

Velocity of propagation (%)

Capacitance (pF per metre)

Delay (ns per metre)

Characteristic impedance (ohm)

Velocity of propagation: %
Conductor ratio
Permittivity used

Capacitance (pF/m)
Inductance (nH/m)
Delay (ns/m)

Capacitance (pF/ft)
Inductance (nH/ft)
Delay (ns/ft)

Cut-off frequency (GHz)

A ratio of 3.5 in a dielectric of 2.25 gives 50.08 ohm. The impedance does not change with length: a 1 metre patch lead and a 100 metre drum of the same cable are both this number.

For 50 ohm in this dielectric you need a diameter ratio of 3.5. That ratio is the whole answer: scale both diameters together and the impedance does not move, only the cut-off frequency does.

Above 28.27 GHz this cable stops behaving like a single transmission line, because a second waveguide mode can start to propagate inside it. In practice cable is used well below that: attenuation, not the cut-off, is what limits most runs.

Impedance curve and design table

Show the impedance curve

How the impedance moves as the conductor ratio changes, with and without the dielectric.

Show the design table

The ratio and the shield diameter you would need for each of the standard impedances.

What each standard impedance would take, for your dielectric and your conductor
Impedance (ohm)	Ratio needed	Shield or spacing (mm)	Capacitance (pF/m)	Inductance (nH/m)
50	3.493	3.493	100.07	250.2
60	4.486	4.486	83.39	300.2
75	6.529	6.529	66.71	375.3
93	10.243	10.243	53.8	465.3
100	12.204	12.204	50.03	500.3
120	20.127	20.127	41.7	600.4
300	1,817.454	1,817.454	16.68	1,501

Read the third column against the conductor already in the form. Notice how far the geometry has to stretch for the high impedances: that is why 300 ohm exists as open-wire ribbon and almost never as coax.

A cable does not have a single resistance value like a resistor. It has a characteristic impedance which is the ratio of voltage to current for a wave propagating along the conductor.

This value is determined by the shape of the conductors and the insulating material between them. It does not change with length and is independent of frequency outside the audio range. Also, it cannot be measured with a multimeter.

This calculation tool offers three options: the forward direction calculation for coaxial cable, the forward direction calculation for twisted pair cable and the reverse direction calculation starting from a given target impedance.

Impedance is not equal to resistance.

When a DC ohmmeter is connected across the ends of a coaxial cable, it will measure nearly zero ohms on the center conductor and nearly infinite ohms on the shield. Neither of these values match the 50 ohms printed on the jacket.

The characteristic impedance is a property of the wave. When a signal is fed into the cable, it does not know how long the cable is or what is connected at the far end. So the cable draws in current in a fixed ratio determined by its geometry.

For lines with low losses this ratio is the square root of the value obtained by dividing the quotient of series inductance per unit length divided by the parallel capacitance.

Z_0 = \sqrt{\dfrac{L}{C}}

The following text describes all the methods for determining L and C from geometric structure. It also explains why the four values per unit length that this calculator outputs are always consistent with each other, since they all come from the same source.

Formula for coaxial cable:

For coaxial cables d is the outside diameter of the center conductor, D is the inside diameter of the shield, i.e., the outside diameter of the dielectric. The Greek letters represent the relative permittivity of the dielectric.

Z_0 = \frac{59.96}{\sqrt{\varepsilon_r}} \times \ln\!\left(\frac{D}{d}\right) \qquad = \frac{138.06}{\sqrt{\varepsilon_r}} \times \log_{10}\!\left(\frac{D}{d}\right)

In textbooks you will find 60 and 138. These are rounded values. The exact constants are 59.9585 and 138.0595, which are obtained by dividing the impedance of free space by twice pi. So this calculator uses the exact values, so the displayed values can be about 0.07 percent lower than if they were rounded to 60.

Capacity, inductance and delay can all be derived from the same log scale.

C = \frac{55.63 \, \varepsilon_r}{\ln(D/d)} \ \text{pF/m} \qquad L = 200 \times \ln\!\left(\frac{D}{d}\right) \ \text{nH/m} \qquad t_d = 3.3356 \sqrt{\varepsilon_r} \ \text{ns/m}

Notice what is missing from the propagation delay formula: a geometric structure. The speed at which a signal travels through a cable is determined only by the dielectric, and not by the arrangement of the conductor.

Calculation example:

Assume we use a cable with a center conductor of 1 mm diameter and a shield inner diameter of 3.5 mm using solid polyethylene as the dielectric. The ratio is 3.5, the natural logarithm is 1.2528, the square root of 2.25 is 1.5.

The impedance is 50.08 ohms when you multiply 59.9585 by 1.2528 and divide it by 1.5. The capacitance is 99.92 pF per meter when you multiply 55.63 by 2.25 and divide it by 1.2528.

The inductance is 250.55 nH per meter if you multiply 200 by 1.2528. The propagation delay is 5.003 ns per meter if you multiply 3.3356 by 1.5. These are the values that textbooks give for polyethylene coaxial cable with an impedance of 50 ohms. It's reassuring to see this formula produce such results.

Formulas for twisted pair and abbreviations everyone uses.

In two-wire circuits with twisted pairs or open lines, there is no shielding. The diameter d of the conductor and the average distance s between the two conductors are decisive.

For accurate results for two parallel cylinders, use the inverse hyperbolic cosine of the ratio of distances.

Z_0 = \frac{119.92}{\sqrt{\varepsilon_r}} \times \operatorname{arcosh}\!\left(\frac{s}{d}\right)

Most network calculation tools offer a simpler formula where the inverse hyperbolic cosine is replaced by a logarithm.

Z_0 \approx \frac{120}{\sqrt{\varepsilon_r}} \times \ln\!\left(\frac{2s}{d}\right)

This simplification is only valid if the distance between the conductors is large compared to their diameter; as they get closer together, the error becomes larger. In twisted pairs, the spacing between the two wires is approximately the minimum achievable.

This calculator shows the exact result as the main value and a simplified version next to it so you can see which formula was used to calculate a certain value. The difference between both values is as follows:

Spacing ratio s/d	Exact (arcosh)	Shortcut (ln 2s/d)	Error
1.2	74.6 ohm	105.1 ohm	41 percent high
1.5	115.4 ohm	131.8 ohm	14 percent high
1.8	143.0 ohm	153.7 ohm	7.5 percent high
3	211.4 ohm	215.0 ohm	1.7 percent high
6	297.1 ohm	298.2 ohm	0.4 percent high

The values given are for air as the dielectric material. If another dielectric is used, each value in the table must be divided by the square root of its relative permittivity. This does not change the percentage error.

For real twisted pair cables there is another important point: Part of the electric field between the conductors travels through the insulation, while the rest goes through the surrounding air. Therefore, the actual relative dielectric constant lies somewhere between 1 and the value given on the datasheet for the insulating material.

If the cable's propagation velocity is known, enter that value instead. The effective relative dielectric constant is already included and this is why manufacturers publish this parameter.

Propagation speed, delay, dielectric

Propagation speed (sometimes referred to as velocity factor) is the rate at which a signal travels along a cable, expressed as a percentage of the speed of light in vacuum.

\text{VoP} = \frac{100}{\sqrt{\varepsilon_r}}\% \qquad \varepsilon_r = \left(\frac{100}{\text{VoP}}\right)^{2}

Because both representations contain the same information, this calculator accepts either input. You can select a material from the list and enter its relative permittivity or you can enter the propagation speed given on the datasheet.

Dielectric	Relative permittivity	Velocity of propagation	Delay (ns/m)
Air or air-spaced	1.00	100 percent	3.34
Foamed PTFE	1.40	84.5 percent	3.95
Foam polyethylene	1.50	81.6 percent	4.09
Solid PTFE (Teflon)	2.10	69.0 percent	4.83
Polypropylene	2.20	67.4 percent	4.95
Solid polyethylene	2.25	66.7 percent	5.00
PVC	3.30	55.0 percent	6.06

This is also the reason for the existence of expanded dielectrics. By filling plastics with gas, the relative permittivity approaches 1, which increases cable speed, reduces capacitance and minimizes losses.

In other words, to achieve the same impedance foam cables must have different geometric structures. When replacing with another cable one can easily be surprised by these differences.

Design a cable with an impedance of 50 or 75 ohms.

If you reverse the formula, this ratio can be determined directly from the target impedance.

\frac{D}{d} = \exp\!\left(\frac{Z_0 \sqrt{\varepsilon_r}}{59.96}\right)

Only the ratio remains constant. Even if you increase both diameters in the same proportion, the impedance does not change at all. So it can be that a line with the same resistance of 50 Ohms is either very thin, like a hair, or so stiff that a human being could stick his arm into it.

The parameters that change with the dimensions are the cut-off frequency and loss. The larger the dimensions, the lower the loss and the lower the cut-off frequency. The smaller the dimensions, the greater the loss but also the margin increases.

Why are they 50 and 75 ohms?

For coaxial cables with air dielectric there are three important ratios that do not agree.

If the ratio is 1.65, the power carrying capacity is maximum and it is about 30 ohms.
When the ratio is 2.72, the voltage strength and puncture resistance are maximized at about 60 ohms.
When the ratio is 3.59, the attenuation is minimum and is about 76.7 ohms.

50 ohms is approximately the geometric mean of the optimal current carrying capacity ratio and the optimal low-loss ratio, making it a compromise cable for transmission. 75 ohms is more in the range of low loss and thus used primarily in receive-only applications such as TV broadcast, cable television, and satellite downlinks.

93 Ohm is a cable with low capacitance and was often used in the cabling of older computers and measuring devices. On the other hand 300 Ohm is the classic ratio for parallel feed lines of open TV antennas. With coaxial cables it's not possible to realize this impedance sensibly, because the shielding would become extremely large.

cut-off frequency

Coaxial cable only works in a frequency range where pure TEM waves can propagate and up to the frequency at which the first waveguide mode is included. Above this frequency energy is transmitted along two paths simultaneously, and the cable no longer behaves like a simple transmission line.

f_c \approx \frac{c}{\sqrt{\varepsilon_r} \times \pi \times (D + d)/2}

The cutoff frequency depends on the dimensions, since the cutoff wavelength is approximately equal to the average circumference of the space between the conductors. For examples with 1 mm and 3.5 mm it is about 28 GHz.

In practice you will run out of available signal long before the cutoff frequency is reached. The attenuation in the conductor increases with the square root of the frequency, while the attenuation in the dielectric increases linearly with frequency. So most cable datasheets only specify a very small fraction of the cutoff frequency.

Matching, VSWR, Return Loss

If the impedance of the cable is different from that of the system then some of the wave will be reflected back at the junction. The reflection coefficient gives the amount of this reflection.

\Gamma = \frac{Z_0 - Z_s}{Z_0 + Z_s} \qquad \text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}

\text{Return loss} = -20 \log_{10}|\Gamma| \ \text{dB} \qquad \text{Mismatch loss} = -10 \log_{10}\!\left(1 - |\Gamma|^{2}\right) \ \text{dB}

The return loss indicates how much signal is returned. The mismatch loss indicates how much signal is not passed through at all and this is usually the value you really want to know.

If a 75 ohm cable is connected to a 50 ohm system there will be a VSWR of about 1.5 and a loss of about 0.18 dB, which is small enough that it can often be ignored in receive-only equipment but not when transmitting or measuring noise levels.

Reasons why these values may differ from reality:

Wound central conductor. The formula assumes a smooth cylinder, so a strand of wires will behave more like a slightly smaller conductor and raise the impedance value slightly.
Wrong measurement of the diameter. D stands for the inner diameter of the shield, not the outer diameter of the sheath. This is the most common cause that makes the result unusually high.
Foam tolerances. There are differences between production lots of foam dielectrics, and a 5 percent variation in the dielectric constant results in about a 2.5 percent change in impedance value.
The relative permittivity of a twisted pair cable. Since some of the electric field is in air, the effective relative permittivity is lower than that of the insulating material itself. If a published propagation velocity is given, this should be used.
Semi-flexible shielding and corrugated shielding. As corrugated or spiral shields do not have a clear internal diameter the results should be considered as estimates.

None of these factors affect the practical applicability of the formula. With only the measurements from a caliper and one piece of information from the datasheet, deviations of a few percent from the actual cable can be obtained. In most cases this is sufficient to judge whether a cable is suitable for a particular purpose.

This calculation tool models the system as an ideal lossless and homogeneous transmission line using non-magnetic dielectrics. It does not take into account attenuation, skin effect, dielectric losses, connector effects or relative permeability of magnetic materials. For applications where certification is required you should use actual manufacturer's measurements.

Frequently asked questions

Does a cable's impedance change depending on its length?: No, it does not change. The characteristic impedance is a property of the cross-section and not a property of length. A short piece of wire (e.g. 30 cm) and a complete cable of a certain length (300 meters) both have the same characteristic impedance. While losses, total capacitance and delay do change with length, the impedance itself remains constant. This is why it cannot be measured with an ohmmeter either, as that measures the copper conductor and not an electromagnetic wave.
How to convert propagation speed to relative permittivity?: Square the ratio of 100 to the propagation speed in percent. For a cable with a nominal value of 66 percent, the relative dielectric constant is approximately 2.30. For a cable with a nominal value of 82 percent, it is approximately 1.49. Conversely divide 100 by the square root of the relative dielectric constant. If you select "Enter your own values" in the list of relative dielectric constants, the calculator will accept either of the two values given on the datasheet.
Why are there 50 ohms and 75 ohms in coaxial cables?: For coaxial cables with air as the dielectric, the geometric structure that will handle maximum power is near 30 ohms, while the structure with least loss is near 77 ohms. Fifty ohms is approximately the geometric mean of these two values and therefore a compromise for transmission equipment. Seventy-five ohms is on the side of lower losses and so is the standard format for receive-only applications such as radio, cable TV, and satellite where no signals are sent and all loss in decibels is important.
Can you connect a 75 ohm cable to a 50 ohm system?: Generally yes. The mismatch will create a VSWR of about 1.5 and a loss of about 0.18 dB. This can be ignored in the receive path and is usually within acceptable limits for short runs. When actual power is being transmitted, effects may occur if the connectors themselves have differing impedances or reflections are disturbing sensitive measurements. Open the match test to check the exact values for your combination.
What diameter must the shielding have for a 50 ohm cable?: Multiply the diameter of the center conductor by a value from an exponential function that results from target impedance multiplied by the square root of the dielectric constant, divided by 59.96. For polyethylene with a resistance of 50 ohms this ratio is 3.49. So a conductor with a diameter of 1 mm requires shielding with an inner diameter of 3.49 mm. When design mode is selected, the calculator will perform calculations in both directions. It can either calculate the required shield for a given conductor or the required size of the conductor for a given shield.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

NIST: characteristic impedance of vacuum
The 376.730 ohm constant behind the 59.96 and 119.92 coefficients used here.
NIST: speed of light in vacuum
The exact defined value used for propagation delay and cut-off frequency.
Wikipedia: Coaxial cable
Construction, standard impedances and the origin of the 50 and 75 ohm choices.
Wikipedia: Characteristic impedance
The transmission-line derivation of the square root of L over C.
Wikipedia: Twin-lead and two-wire transmission lines
The arcosh form for parallel conductors and where the logarithmic shortcut breaks down.
Wikipedia: Standing wave ratio
Reflection coefficient, VSWR, return loss and mismatch loss definitions.