Coax TEM Mode Calculator

The Coax TEM Mode Calculator (Coax Calculator) provides engineers with an online tool to analyze Coax Configurations for their circuits. The Coax TEM Mode Calculator (Coax Calculator) is used to determine the TEM Mode guided wave properties in a Coax Transmission Line for the dimensions and material properties defined by the user.

The Coax TEM Mode Calculator (Coax Calculator) determines following characteristics of a Coax TEM Mode:

1Cutoff Frequency for the Coax TE_1,1 mode.
2Coax TEM Mode Characteristic Impedance.
3Coax TEM Mode Guide Wavelength.
4Coax TEM Mode Propagation Constant.
5Coax TEM Mode Phase Velocity.
6Coax TEM Mode Group Velocity.
7Coax TEM Mode Attenuation Constant due to Metal and Dielectric Loss.

Tutorial · RF & Microwave Transmission Lines

Coaxial TEM Mode Calculator Tutorial

By Dr. Scott Best, SiberSci, LLC · October 30, 2026

This tutorial walks through a complete analysis of a coaxial waveguide geometry with the SiberSci Coax TEM Mode Calculator. Using an SMA-size, PTFE-filled copper line at 10 GHz, each step shows what to enter, what the calculator returns, and the equation behind the result so the number can be checked by hand.

Open the Coax TEM Mode Calculator

1. What the Coax TEM Mode Calculator computes

A coaxial line supports a transverse electromagnetic (TEM) mode with no cutoff frequency. Given the conductor dimensions, the dielectric fill, the conductor metal, and the operating frequency, the calculator returns seven properties of that TEM mode:

TE_1,1 cutoff frequency — the upper frequency limit for single-mode (TEM-only) operation.
Characteristic impedance $Z_{0}$ in ohms.
Guide wavelength $λ_{g}$ .
Propagation constant $γ = α + j β$ .
Phase velocity $v_{p}$ .
Group velocity $v_{g}$ .
Attenuation constant from conductor (metal) loss $α_{c}$ and dielectric loss $α_{d}$ .

The equations below assume a non-magnetic dielectric ( $μ_{r} = 1$ ), non-magnetic conductors, and low loss ( $α ≪ β$ ), which covers nearly all practical RF and microwave coax.

2. Coaxial line geometry and notation

Figure 1. Coaxial cross-section.

d = 2 a

is the center-conductor outside diameter;

D = 2 b

is the outer-conductor inside diameter.

Symbol	Meaning	Units
$d = 2 a$	Outside diameter of the center conductor	m
$D = 2 b$	Inside diameter of the outer conductor	m
$ε_{r}$	Relative permittivity of the dielectric fill	—
$\tan δ$	Dielectric loss tangent	—
$σ$	Conductor conductivity	S/m
$f$	Operating frequency	Hz
$c$	Speed of light in vacuum, 299 792 458	m/s
$μ_{0}$	Permeability of free space, $\approx 4 π \times 10^{- 7}$	H/m
$η_{0}$	Impedance of free space, $μ_{0} c \approx 376.73$	Ω

3. Worked example: inputs

The example line is a semi-rigid, SMA-size coax: a 0.050-inch (1.27 mm) copper center conductor inside a 4.20 mm bore, filled with solid PTFE. It is analyzed at 10 GHz.

Input	Value	Notes
Center conductor diameter, $d$	1.27 mm	Radius $a = 0.635$ mm
Outer conductor inside diameter, $D$	4.20 mm	Radius $b = 2.10$ mm; $D / d = 3.307$
Relative permittivity, $ε_{r}$	2.1	Solid PTFE
Loss tangent, $\tan δ$	0.0002	Typical PTFE at microwave frequencies
Conductivity, $σ$	5.8 × 10⁷ S/m	Annealed copper, both conductors
Frequency, $f$	10 GHz

4. Step-by-step analysis

1Enter the conductor geometry

Open the Coax TEM Mode Calculator and enter the center-conductor outside diameter $d = 1.27$ mm and the outer-conductor inside diameter $D = 4.20$ mm. Confirm the unit selection matches the values typed. Only the ratio $D / d$ sets the impedance, but the absolute size sets the TE_1,1 cutoff and the conductor loss, so both diameters matter.

Equation 1

D/d = 4.20 mm / 1.27 mm = 3.307

2Enter the dielectric properties

Enter $ε_{r} = 2.1$ and $\tan δ = 0.0002$ for PTFE. The permittivity controls impedance, wavelength, velocity, and cutoff; the loss tangent controls only the dielectric attenuation. For air-spaced lines use $ε_{r} = 1.0$ and $\tan δ = 0$ .

3Enter the conductor conductivity

Enter $σ = 5.8 \times 10^{7}$ S/m for copper. The calculator uses it to find the surface resistance and conductor loss. Plating matters: if the center pin is silver- or gold-plated, use the conductivity of the plating, provided the plating is several skin depths thick at the operating frequency (see the Skin Depth Calculator).

4Enter the frequency and run the calculation

Enter $f = 10$ GHz and calculate. The free-space wavelength at this frequency, used in several of the steps below, is:

Equation 2

λ_0 = c / f = 2.9979×10^8 / (10×10^9) = 29.98 mm

5Read the characteristic impedance, $Z_{0}$

For the TEM mode, the characteristic impedance depends only on the diameter ratio and the dielectric:

Equation 3

Z_0 = η_0 / (2π·√ε_r) · ln(D/d) ≈ (59.96 / √ε_r) · ln(D/d) Ω

Pozar [1], §2.2; Ramo, Whinnery & Van Duzer [3]

Equation 4

Z_0 = (59.96 / √2.1) · ln(3.307) = 41.38 × 1.196 = 49.49 Ω

Result:

Z_{0} = 49.49 Ω

, within 1.1% of a 50 Ω system.

The same geometry gives the per-unit-length inductance and capacitance, useful for circuit models:

Equation 5

L′ = (μ_0 / 2π) · ln(D/d) = 239.2 nH/m; C′ = 2π·ε_0·ε_r / ln(D/d) = 97.68 pF/m

Pozar [1], §2.1; Collin [2]

As a check, $\sqrt{L' / C'} = 49.49 Ω$ .

6Check the TE_1,1 cutoff frequency

The TEM mode propagates at any frequency, but above the cutoff of the first higher-order mode, TE_1,1, energy can couple into a non-TEM field pattern and cause resonances and impedance errors. The widely used approximation sets the TE_1,1 cutoff wavelength in the dielectric equal to the mean circumference of the line:

Equation 6

λ_c ≈ π·(a + b)·√ε_r ⟹ f_c,TE11 ≈ c / (π·(a + b)·√ε_r) = 2c / (π·(D + d)·√ε_r)

Pozar [1], §3.5; Marcuvitz [5], §2.4

Equation 7

f_c = 2·(2.9979×10^8) / (π·(5.47×10^−3)·√2.1) = 24.08 GHz

Result:

f_{c}^{{TE}_{11}} = 24.08

GHz. At 10 GHz,

f / f_{c} = 0.42

, so the line is safely single-mode TEM.

Design margin: many designers keep the highest operating frequency at or below about 90–95% of the computed TE_1,1 cutoff to allow for dimensional tolerances, the approximation's error (a few percent for typical 50 Ω ratios), and discontinuities such as connectors and bends. The exact cutoff comes from the root of a Bessel-function equation [5], [6].

7Read the guide wavelength, $λ_{g}$

Because the TEM mode has no cutoff, the guide wavelength is simply the free-space wavelength scaled by the dielectric:

Equation 8

λ_g = c / (f·√ε_r) = λ_0 / √ε_r = 29.98 mm / 1.449 = 20.69 mm

Pozar [1], §2.1; Collin [2]

Result:

λ_{g} = 20.69

mm. A quarter-wave section of this line at 10 GHz is 5.17 mm long.

8Read the propagation constant, $γ = α + j β$

The phase constant gives the phase shift per unit length:

Equation 9

β = 2π / λ_g = 2π·f·√ε_r / c = 2π / 0.02069 m = 303.7 rad/m

Pozar [1], §2.1

That is 17 402°/m, or about 17.4° per millimeter of line at 10 GHz. Combining $β$ with the total attenuation from Step 10:

Result:

γ = 0.1164 + j 303.7 m^{- 1}

9Read the phase and group velocities

For TEM propagation in a non-dispersive dielectric, phase and group velocity are equal and independent of frequency:

Equation 10

v_p = ω / β = c / √ε_r; v_g = (dβ/dω)^−1 = c / √ε_r

Pozar [1], §2.1; Ramo, Whinnery & Van Duzer [3]

Equation 11

v_p = v_g = 2.9979×10^8 / √2.1 = 2.069×10^8 m/s; VF = v_p / c = 1 / √ε_r = 0.690

Result:

v_{p} = v_{g} = 2.069 \times 10^{8}

m/s; velocity factor 0.690; time delay 4.83 ns/m.

This equality is a key difference from hollow-waveguide modes, where $v_{p} > c$ , $v_{g} < c$ , and both change rapidly near cutoff (compare the Circular Waveguide Calculator).

10Read the attenuation constant

Conductor loss. First find the surface resistance and skin depth of the metal:

Equation 12

R_s = √(π·f·μ_0 / σ) = √(π·(10^10)·(4π×10^−7) / (5.8×10^7)) = 0.02609 Ω; δ_s = 1 / √(π·f·μ_0·σ) = 0.661 μm

Pozar [1], §1.7; Ramo, Whinnery & Van Duzer [3]

Then, with $η = η_{0} / \sqrt{ε_{r}} = 260.0 Ω$ , so $2 η \ln (b / a) = 621.9 Ω$ , $1 / a = 1575 m^{- 1}$ and $1 / b = 476.2 m^{- 1}$ :

Equation 13

α_c = R_s / (2η·ln(b/a)) · (1/a + 1/b) = (0.02609 / 621.9) · (1575 + 476.2) = 0.0860 Np/m

Pozar [1], §2.7 (Example 2.6) and §3.4; Collin [2]

Dielectric loss. For any TEM line:

Equation 14

α_d = k·tanδ / 2 = π·f·√ε_r·tanδ / c = π·(10^10)·(1.449)·(0.0002) / (2.9979×10^8) = 0.0304 Np/m

Pozar [1], §3.1

Total loss and conversion to decibels (1 Np = 8.686 dB):

Equation 15

α = α_c + α_d = 0.1164 Np/m = 1.011 dB/m

Result:

α_{c} = 0.747

dB/m,

α_{d} = 0.264

dB/m, total

α = 1.011

dB/m (about 30.8 dB per 100 ft).

Interpretation: conductor loss dominates here (74% of the total). Because

α_{c} \propto \sqrt{f}

and

α_{d} \propto f

, dielectric loss takes a larger share as frequency rises. The ideal smooth-conductor result is a lower bound; surface roughness, plating, and connector interfaces add loss in real assemblies.

5. Results summary table

SMA-size PTFE coax: $d = 1.27$ mm, $D = 4.20$ mm, $ε_{r} = 2.1$ , $\tan δ = 0.0002$ , $σ = 5.8 \times 10^{7}$ S/m, $f = 10$ GHz.

Calculator output	Equation	Value
TE_1,1 cutoff frequency	`2c / (π·(D + d)·√ε_r)`	24.08 GHz
Characteristic impedance	`(η_0 / (2π·√ε_r))·ln(D/d)`	49.49 Ω
Guide wavelength	`c / (f·√ε_r)`	20.69 mm
Phase constant	`2π·f·√ε_r / c`	303.7 rad/m
Propagation constant	`α + jβ`	0.1164 + j303.7 m⁻¹
Phase velocity	`c / √ε_r`	2.069 × 10⁸ m/s
Group velocity	`c / √ε_r`	2.069 × 10⁸ m/s
Conductor attenuation	`R_s·(1/a + 1/b) / (2η·ln(b/a))`	0.0860 Np/m (0.747 dB/m)
Dielectric attenuation	`π·f·√ε_r·tanδ / c`	0.0304 Np/m (0.264 dB/m)
Total attenuation	`α_c + α_d`	0.1164 Np/m (1.011 dB/m)

6. Using the calculator for design

Finding the diameter ratio for a target impedance

Inverting the impedance equation gives the required ratio directly:

Equation 16

D/d = exp(Z_0·√ε_r / 59.96)

Target $Z_{0}$	Dielectric	$ε_{r}$	Required $D / d$
50 Ω	Air	1.00	2.302
50 Ω	PTFE	2.10	3.348
50 Ω	Solid polyethylene	2.25	3.493

In practice, pick the nearest standard wire or tube size for one conductor, compute the other from the ratio, then enter both into the calculator to confirm $Z_{0}$ and check the TE_1,1 cutoff against the highest operating frequency.

A practical workflow

Choose the dielectric from mechanical, thermal, and loss requirements.
Compute $D / d$ for the target impedance.
Size the line so the TE_1,1 cutoff sits above the highest operating frequency with margin. Higher frequencies force smaller lines, which is why millimeter-wave coax and connectors are physically small.
Check the attenuation at the highest frequency. A smaller line raises $α_{c}$ , so cutoff and loss pull in opposite directions.
Use $λ_{g}$ and $β$ to size stubs, quarter-wave transformers, and phase-matched lengths.

7. Common input mistakes

Using the outer jacket diameter for $D$ . $D$ is the inside diameter of the outer conductor (equal to the dielectric outside diameter), not the cable's overall diameter.
Mixing radii and diameters. The impedance ratio is the same either way, but the TE_1,1 cutoff and conductor loss will be off by a factor of two if radii are entered where diameters are expected.
Unit mismatches. Confirm mm, inches, and GHz selections before reading results.
Using foamed-dielectric permittivity for solid PTFE (or vice versa). Foamed dielectrics have an effective $ε_{r}$ well below the solid value; use the manufacturer's velocity factor to back out $ε_{r, eff} = 1 / {VF}^{2}$ .
Operating near TE_1,1 cutoff. The approximation and tolerances both introduce error; leave margin.

8. References

D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: John Wiley & Sons, 2012. Ch. 1 (surface resistance, skin depth), Ch. 2 (TEM transmission line parameters, coax $L'$ , $C'$ , $Z_{0}$ , conductor loss), Ch. 3 (TEM dielectric loss, coaxial line higher-order modes and TE₁₁ cutoff approximation).
R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: IEEE Press / Wiley, 2001. Ch. 3 (TEM transmission lines and coaxial line attenuation).
S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: John Wiley & Sons, 1994. Chs. 5 and 8 (transmission lines, coaxial line fields, skin effect).
C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2012. Ch. 9 (coaxial line TEM, TE and TM modes).
N. Marcuvitz, Waveguide Handbook, MIT Radiation Laboratory Series, vol. 10. New York: McGraw-Hill, 1951; reprinted London: IEE / Peter Peregrinus, 1986. §2.4 (coaxial waveguide modes and cutoff wavelengths).
S. A. Schelkunoff, Electromagnetic Waves. New York: D. Van Nostrand, 1943. (Coaxial line higher-order mode cutoffs.)
E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018,” Rev. Mod. Phys., vol. 93, 025010, 2021. (Values of $c$ , $μ_{0}$ , $ε_{0}$ .)

Related SiberSci calculators

Coax TEM Mode Calculator
Skin Depth Calculator — verify plating thickness against $δ_{s}$
Wavelength Calculator
Circular Waveguide Calculator
Rectangular Waveguide Calculator

Need help with a coaxial design problem? Contact SiberSci, LLC.

© 2026 SiberSci, LLC. Equations are given for low-loss lines with non-magnetic materials and smooth conductors. Numerical values in the worked example were computed with $c = 299 792 458$ m/s and $μ_{0} = 4 π \times 10^{- 7}$ H/m.

Finally, SiberSci, LLC also provides visitors with access to Electromagnetic Engineering Calculators that are used for designing circuits, or exploring solutions to problems. Four transmission line calculators are provided that include a Circular Waveguide Calculator, Coax TEM Mode Calculator, a Rectangular Waveguide TE_m,n Calculator, and to complete this set a Rectangular Waveguide TM_m,n Calculator. The remaining three Electromagnetic Engineering Calculators are general purpose analysis tools. For instance, the Wavelength Calculator can be used to determine the length of a Monopole Antenna, while the Skin Depth Calculator is commonly used to determine the metal thickness for Microstrip, Stripline, and CPW transmission lines. The Skin Depth Calculator is also used to determine the metal thickness for electroplating cavity assemblies and waveguide assemblies. The final calculator is the Dielectric Heating Calculator, which is used to estimate the frequency-dependent power and electric field to heat a defined volume and mass of dielectric material in a defined time period. If you do not see the calculator that you need for your problem, then Contact Us to determine if we can provide you with this calculator.

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Coax TEM Mode Calculator FAQ

What does the SiberSci Coax TEM Mode Calculator compute?

From the coaxial line's dimensions and material properties, it computes seven quantities: the TE₁₁ higher-order mode cutoff frequency, TEM characteristic impedance, guide wavelength, propagation constant, phase velocity, group velocity, and attenuation constant from metal (conductor) and dielectric loss. It was developed by Dr. Scott Best of SiberSci, LLC for RF and microwave transmission line design.

How do you calculate the characteristic impedance of a coaxial cable?

Characteristic impedance of a coaxial line is Z₀ = (η₀ / 2π√ε_r)·ln(D/d) ≈ (59.96 / √ε_r)·ln(D/d) ohms, where D is the inside diameter of the outer conductor, d is the outside diameter of the center conductor, and ε_r is the dielectric's relative permittivity. Example: d = 1.27 mm, D = 4.20 mm, PTFE (ε_r = 2.1) gives Z₀ ≈ 49.5 Ω, close to a standard SMA line.

What D/d ratio gives a 50 ohm coaxial line?

Solve Z₀ = (59.96/√ε_r)·ln(D/d) for D/d = exp(Z₀·√ε_r / 59.96). For 50 Ω, an air-dielectric line needs D/d ≈ 2.30, a PTFE line (ε_r = 2.1) needs D/d ≈ 3.35, and a solid polyethylene line (ε_r ≈ 2.25) needs D/d ≈ 3.49. Use the calculator to iterate dimensions against standard tube and wire sizes.

What is the maximum frequency of a coaxial cable?

A coaxial line is limited by the onset of the TE₁₁ mode, whose cutoff is approximately f_c ≈ 2c / (π·(D + d)·√ε_r). Above that frequency, energy can couple into a non-TEM mode and cause resonances and impedance errors. For the PTFE-filled 4.20 mm / 1.27 mm example, TE₁₁ cutoff is about 24 GHz. Smaller coax and lower-ε_r dielectrics push this limit higher, which is why millimeter-wave connectors are physically tiny.

Why is 50 ohms the standard impedance for coaxial cable?

50 Ω is a compromise between two optima for air-dielectric coax: minimum conductor attenuation occurs at about 77 Ω (D/d ≈ 3.59), and maximum power handling before voltage breakdown occurs at about 30 Ω (D/d ≈ 1.65). 50 Ω sits between them, giving good loss and power performance, and became the RF test and measurement standard. 75 Ω, near the low-loss optimum, is common in video and cable TV systems where power is low.

What is the velocity factor of a coaxial cable?

Velocity factor is the propagation speed relative to the speed of light: VF = 1/√ε_r for a TEM line with a non-magnetic dielectric. Solid PTFE gives VF ≈ 0.69, solid polyethylene ≈ 0.66, and foamed or air-spaced dielectrics typically 0.80 to 0.90 or higher. The calculator reports phase velocity directly, which is needed for cutting phase-matched cables and delay lines to length.

Are phase velocity and group velocity the same in coaxial cable?

For the TEM mode in a lossless coaxial line with a non-dispersive dielectric, phase velocity and group velocity are equal (v = c/√ε_r) and independent of frequency, so pulses travel without dispersion. This is a key difference from waveguide modes, where phase velocity exceeds c, group velocity is below c, and both vary strongly near cutoff. In real cables, small differences arise from conductor skin effect and frequency-dependent dielectric properties.

How do you calculate the guide wavelength in a coaxial line?

In TEM mode, guide wavelength is λ_g = c / (f·√ε_r), the free-space wavelength divided by √ε_r. At 1 GHz in PTFE-filled coax, λ_g ≈ 207 mm, so a quarter-wave stub is about 52 mm long. Guide wavelength is used to size quarter-wave transformers, stubs, baluns, and phase-shifting sections.

How is attenuation calculated for a coaxial line?

Total attenuation is the sum of conductor and dielectric loss. Conductor loss is α_c = R_s·(1/a + 1/b) / (2η·ln(b/a)) Np/m, where a and b are the inner and outer radii, η = η₀/√ε_r, and R_s = √(πfμ₀/σ) is the surface resistance. Dielectric loss is α_d = π·f·√ε_r·tanδ / c Np/m. Multiply Np/m by 8.686 for dB/m. Conductor loss grows with √f and dielectric loss grows with f, so dielectric loss dominates at high frequencies.

What is the propagation constant of a coaxial transmission line?

The propagation constant is γ = α + jβ, where α is the attenuation constant (Np/m) and β = 2πf√ε_r/c is the phase constant (rad/m). β determines phase shift per unit length and α determines amplitude decay. Together they describe how a signal changes along the line and are the inputs needed for ABCD or S-parameter models of coaxial sections.

FAQ

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Coax TEM Mode Calculator

The Coax TEM Mode Calculator (Coax Calculator) determines following characteristics of a Coax TEM Mode:

Coaxial TEM Mode Calculator Tutorial

1. What the Coax TEM Mode Calculator computes

2. Coaxial line geometry and notation

3. Worked example: inputs

4. Step-by-step analysis

1Enter the conductor geometry

2Enter the dielectric properties

3Enter the conductor conductivity

4Enter the frequency and run the calculation

5Read the characteristic impedance, Z0

6Check the TE1,1 cutoff frequency

7Read the guide wavelength, λg

8Read the propagation constant, γ=α+jβ

9Read the phase and group velocities

10Read the attenuation constant

5. Results summary table

6. Using the calculator for design

Finding the diameter ratio for a target impedance

A practical workflow

7. Common input mistakes

8. References

Related SiberSci calculators

Do you need another Calculator for your design analysis, then chose one from the following list:

Electromagnetic Engineering Consulting Services offered:

Electromagnetic Engineering Calculators offered:

If you need help with your Coax TEM Mode Calculator design problem, send us a message.

Coax TEM Mode Calculator FAQ

What does the SiberSci Coax TEM Mode Calculator compute?

How do you calculate the characteristic impedance of a coaxial cable?

What D/d ratio gives a 50 ohm coaxial line?

What is the maximum frequency of a coaxial cable?

Why is 50 ohms the standard impedance for coaxial cable?

What is the velocity factor of a coaxial cable?

Are phase velocity and group velocity the same in coaxial cable?

How do you calculate the guide wavelength in a coaxial line?

How is attenuation calculated for a coaxial line?

What is the propagation constant of a coaxial transmission line?

Consulting Services

Contact

5Read the characteristic impedance, $Z_{0}$

6Check the TE_1,1 cutoff frequency

7Read the guide wavelength, $λ_{g}$

8Read the propagation constant, $γ = α + j β$