Rectangular Waveguide TM_m,n Calculator

Rectangular Waveguide TM_m,n Calculator Tutorial: Step-by-Step Analysis of TM-Mode Guide Wavelength Problems

1. How the Rectangular Waveguide TM_m,n Calculator Works

Calculator inputs and outputs

Cutoff frequency and cutoff wavelength

TM_m,n modes have no axial magnetic field (H_z = 0). Their axial electric field varies as sin(mπx/a)·sin(nπy/b), so it must vanish on all four walls. That is why neither index can be zero and TM₁₁ is the lowest TM mode.^[1][2] The boundary conditions fix the cutoff wavenumber:

The calculator converts k_c to the cutoff frequency it reports in MHz:

The cutoff wavelength depends only on the cross-section, not on the filling. It is the longest wavelength (measured in the filling material) that the mode can carry:^[3][4]

Guide wavelength and the frequency-dependent outputs

Above cutoff the mode propagates with phase constant β = √(k² − k_c²). Its guide wavelength — the distance between equal-phase points along the guide — is always longer than the wavelength in the filling material:^[1][3]

Note that the TM wave impedance is lower than η and falls to zero at cutoff, the opposite of TE modes, whose impedance rises toward infinity at cutoff.^[1]

Attenuation

The calculator separates wall (conductor) loss and filling (dielectric and magnetic) loss and adds them. Conductor loss uses the surface resistance of the selected metal:^[1][2]

2. Tutorial 1: Find the TM₁₁ Guide Wavelength in WR-90

Problem: An air-filled, copper WR-90 waveguide (inside 0.900 × 0.400 in) is driven at 17.1 GHz, above its recommended band. Does the TM₁₁ mode propagate, and if so, what are its guide wavelength, half-wavelength and quarter-wavelength? These are the calculator's default values, so you can follow along without typing.

1. Select the dimension units. Under Select a Choice for Waveguide Dimensions, choose Inches. The unit applies to both a and b; outputs are always in SI units.
2. Describe the filling material. Enter relative permittivity 1.0, permittivity loss tangent 0.0, relative permeability 1.0 and permeability loss tangent 0.0 for air.
3. Enter the inside dimensions. Waveguide Dimension "a" = 0.9, Waveguide Dimension "b" = 0.4. Always use inside dimensions; the flange or outside size gives a wrong cutoff.
4. Enter the mode indices. m = 1, n = 1. The calculator requires both to be nonzero because TM_m0 and TM_0n do not exist.
5. Read the cutoff frequency. The calculator reports 16,145.09 MHz. Check it by hand with Equation 2:

   $$f_{c,11}=\frac{2.99792\times {10}^8}{2}\sqrt{{\left(\frac{1}{0.02286}\right)}^2+{\left(\frac{1}{0.01016}\right)}^2}=\mathrm{16,145.1}\text{MHz}$$

   f_c,11 = (2.99792×10^8 / 2)·√[(1/0.02286)² + (1/0.01016)²] = 16,145.1 MHz

   The matching cutoff wavelength from Equation 3 is λ_c = 18.57 mm.
6. Enter the operating frequency in hertz. In the frequency box type 17.1E9. The frequency-dependent outputs only make sense when f > f_c; here 17.1 GHz is 5.9% above cutoff, so TM₁₁ propagates.
7. Select the wall metal. Choose Annealed Copper (5.8E7 S/m). The metal affects only attenuation, never wavelength.
8. Read the wavelength results. The calculator returns the values below.

   Calculator output: WR-90, air-filled, annealed copper, TM₁₁ at 17.1 GHz

   Output	Calculator value	In engineering units
   Cutoff frequency fc	16,145.086 MHz	16.145 GHz
   Characteristic impedance ZTM	124.131 Ω	124.1 Ω
   Guide wavelength λg	0.0532077 m	53.21 mm
   Guide half-wavelength λg/2	0.0266039 m	26.60 mm
   Guide quarter-wavelength λg/4	0.0133019 m	13.30 mm
   Propagation constant β	118.088 rad/m	6,766°/m
   Phase velocity vp	9.0985 × 108 m/s	3.035c
   Group velocity vg	9.8780 × 107 m/s	0.3295c
   Conductor / dielectric / total attenuation	0.4269 / 0 / 0.4269 dB/m	0.427 dB/m

9. Verify the guide wavelength by hand. The free-space wavelength is λ = c/f = 17.532 mm. Substitute into Equation 4:

   $${\lambda }_g=\frac{17.532\text{mm}}{\sqrt{1-{\left(\frac{16.145}{17.1}\right)}^2}}=\frac{17.532\text{mm}}{0.32950}=53.21\text{mm}$$

   λ_g = 17.532 mm / √(1 − (16.145/17.1)²) = 17.532 / 0.32950 = 53.21 mm

   The same factor 0.32950 gives Z_TM = 376.73 × 0.32950 = 124.13 Ω, v_g = 0.3295c and v_p = c/0.3295 = 3.035c, all matching the calculator.
10. Check which other modes share the guide. TM₁₁ is never alone. Its cutoff ratio to the dominant TE₁₀ mode is √(1 + (a/b)²) = 2.462, so at 17.1 GHz WR-90 also carries TE₁₀, TE₂₀, TE₀₁ and TE₁₁:^[1]

    Air-filled WR-90 mode cutoffs (Equation 2)

    Mode	fc (GHz)	At 17.1 GHz
    TE10	6.557	Propagates
    TE20	13.114	Propagates
    TE01	14.754	Propagates
    TE11 / TM11	16.145	Propagates
    TE30	19.671	Cut off
    TE21 / TM21	19.740	Cut off
    TE31 / TM31	24.589	Cut off

    TE_mn and TM_mn with the same nonzero indices are degenerate: same cutoff, same λ_g, different field patterns and different impedance. Use the Rectangular Waveguide TE_m,n Calculator for the TE partners.

3. Tutorial 2: Find the Frequency for a Required Guide Wavelength

Problem: A slotted-line or probe spacing in the same WR-90 guide is fixed so that the TM₁₁ mode must have λ_g = 40.00 mm. At what frequency does this happen? The calculator works forward from frequency, so solve Equation 4 for f and then confirm with the calculator.

1. Get the cutoff from the calculator. Keep the Tutorial 1 inputs (a = 0.9 in, b = 0.4 in, m = n = 1, air). The calculator gives f_c = 16,145.09 MHz.
2. Rearrange the guide-wavelength equation. From 1/λ² = 1/λ_g² + 1/λ_c²:^[3][4]

   $$f=\sqrt{f_c^2+{\left(\frac{v}{{\lambda }_g}\right)}^2}=\sqrt{{\left(16.145\right)}^2+{\left(\frac{299.792}{40.00}\right)}^2}\text{GHz}=17.800\text{GHz}$$

   f = √[f_c² + (v/λ_g)²] = √[16.145² + (299.792/40.00)²] GHz = 17.800 GHz (v in mm·GHz, λ_g in mm)

3. Confirm with the calculator. Enter 17.8E9 in the frequency box. The calculator returns λ_g = 0.0399986 m (40.00 mm), λ_g/2 = 20.00 mm and λ_g/4 = 10.00 mm, with Z_TM = 158.63 Ω and β = 157.09 rad/m.
4. Check the sensitivity. Running the calculator at 17.1, 17.8 and 18.0 GHz gives λ_g = 53.21, 40.00 and 37.67 mm. A 4% frequency change near cutoff moves λ_g by 25%, which is why TM-mode wavelength measurements need a tightly controlled source frequency.

4. Tutorial 3: Guide Wavelength in a Dielectric-Filled Waveguide

Problem: The WR-90 guide is completely filled with PTFE (ε_r = 2.1, tan δ_e = 0.0002, representative values). Find the new TM₁₁ cutoff, the guide wavelength at 13 GHz, and how much the filling adds to the loss.

1. Enter the filled-guide inputs. Units = Inches, relative permittivity = 2.1, permittivity loss tangent = 0.0002, relative permeability = 1.0, permeability loss tangent = 0.0, a = 0.9, b = 0.4, m = n = 1.
2. Read the scaled cutoff. The calculator returns 11,141.17 MHz. Every cutoff divides by √(ε_rμ_r) = 1.4491:

   $$f_c\left(\text{filled}\right)=\frac{f_c\left(\text{air}\right)}{\sqrt{{\epsilon }_r{\mu }_r}}=\frac{\mathrm{16,145.1}\text{MHz}}{1.4491}=\mathrm{11,141.2}\text{MHz}$$

   f_c(filled) = f_c(air) / √(ε_r·μ_r) = 16,145.1 MHz / 1.4491 = 11,141.2 MHz

   The cutoff wavelength λ_c = 18.57 mm is unchanged; it is now measured in the PTFE.
3. Enter the frequency and metal. Frequency = 13E9, Annealed Copper.
4. Read and verify the wavelength. The calculator returns λ_g = 0.0308825 m (30.88 mm). By hand, use the wavelength in the filling: λ = c/(f√ε_r) = 15.914 mm and √(1 − (11.141/13)²) = 0.51529, so λ_g = 15.914/0.51529 = 30.88 mm. Using the free-space wavelength here would be the most common error.
5. Read the impedance and velocities. Z_TM = 133.96 Ω (η = 376.73/1.4491 = 259.97 Ω, times 0.51529), β = 203.45 rad/m, v_p = 4.015 × 10⁸ m/s, v_g = 1.066 × 10⁸ m/s. Their product equals v² = (c/1.4491)².
6. Read the loss split. Conductor = 0.2380 dB/m, dielectric = 0.6655 dB/m, total = 0.9035 dB/m. Check the dielectric term with Equation 7: k = 394.83 rad/m, so α_d = 394.83² × 0.0002 / (2 × 203.45) = 0.07662 Np/m = 0.6655 dB/m.

5. Tutorial 4: Higher-Order Mode Wavelength (TM₂₁ in WR-284)

Problem: An S-band WR-284 guide (inside 2.840 × 1.340 in) is excited at 7 GHz, well above its band, through an asymmetric probe that can launch TM₂₁. Find the TM₂₁ guide wavelength so that a mode filter or absorber can be placed at the right spacing.

1. Enter the geometry and mode. Inches, air (1.0 / 0 / 1.0 / 0), a = 2.84, b = 1.34, m = 2, n = 1.
2. Read the cutoff. The calculator returns 6,055.36 MHz. The m = 2 index halves the effective broad-wall width, so λ_c = 2/√((2/a)² + (1/b)²) = 49.51 mm.
3. Enter the frequency and metal. Frequency = 7E9, Annealed Copper.
4. Read the wavelength results. λ_g = 0.0853678 m (85.37 mm), λ_g/2 = 42.68 mm, λ_g/4 = 21.34 mm, Z_TM = 189.00 Ω, β = 73.60 rad/m, total attenuation 0.0574 dB/m.
5. Verify. λ = 42.827 mm; √(1 − (6.0554/7)²) = 0.50168; λ_g = 42.827/0.50168 = 85.37 mm.
6. Repeat for neighboring modes. Change m and n to map every TM mode that propagates at 7 GHz. For WR-284, TM₁₁ cuts off at 4.870 GHz and TM₂₁ at 6.055 GHz, so both propagate at 7 GHz with different guide wavelengths; TM₃₁ and TM₁₂ are cut off.

6. Tutorial 5: Use λ_g/2 to Size a TM_11p Cavity

Problem: Make a TM₁₁₁ resonator from a length of WR-90 shorted at both ends, resonant at 17.1 GHz.

1. Use the half-wavelength output. A cavity resonates when its length holds p half guide-wavelengths of the TM_m,n waveguide mode.^[1][2] With the Tutorial 1 inputs, the calculator's λ_g/2 = 0.0266039 m, so for p = 1 the length is ℓ = 26.60 mm.
2. Check the resonant frequency.

   $$\ell =p\frac{{\lambda }_g}{2},f_{mnp}=\frac{c}{2\sqrt{{\epsilon }_r{\mu }_r}}\sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2+{\left(\frac{p}{\ell }\right)}^2}$$

   ℓ = p·λ_g/2; f_mnp = [c / (2√(ε_r·μ_r))]·√[(m/a)² + (n/b)² + (p/ℓ)²]

   With a = 22.86 mm, b = 10.16 mm and ℓ = 26.604 mm, Equation 12 gives exactly 17.100 GHz.
3. Note the p = 0 case. TM₁₁₀ has no variation along the length, so it resonates at the TM₁₁ cutoff, 16.145 GHz, for any ℓ. Keep that in mind when a cavity has an unexpected resonance near the calculator's f_c.

7. Common Input Mistakes

8. Rectangular Waveguide TM_m,n Equation Summary

Need help with TM-mode suppression, a waveguide transition, a resonant cavity or a mode converter? Contact SiberSci, LLC. Related tools: Rectangular Waveguide TE_m,n Calculator, Rectangular Waveguide Calculator, Circular Waveguide Calculator, Wavelength Calculator and Skin Depth Calculator.

9. References

1. D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ: Wiley, 2012. Sec. 1.7, "Plane Waves in a Good Conductor"; Sec. 3.1, "General Solutions for TEM, TE, and TM Waves"; Sec. 3.3, "Rectangular Waveguide"; Sec. 6.3, "Rectangular Waveguide Cavity Resonators."
2. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. Hoboken, NJ: Wiley, 2012. Ch. 8, "Rectangular Cross-Section Waveguides and Cavities."
3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992 (reissued IEEE Press/Wiley, 2001). Ch. 3, "Transmission Lines and Waveguides."
4. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994. Ch. 8, "Waveguides with Cylindrical Conducting Boundaries."
5. N. Marcuvitz, Waveguide Handbook, MIT Radiation Laboratory Series, vol. 10. New York: McGraw-Hill, 1951; reprinted London: IEE/Peter Peregrinus, 1986. Ch. 2, "Transmission-Line Modes."
6. Electronic Industries Association, EIA Standard RS-261-B, Rectangular Waveguides (WR3 to WR2300). Washington, DC: EIA, 1979. (WR-90: 0.900 × 0.400 in; WR-284: 2.840 × 1.340 in.)
7. National Institute of Standards and Technology, "CODATA Value: speed of light in vacuum," c = 299,792,458 m/s (exact). https://physics.nist.gov/cgi-bin/cuu/Value?c

Tutorial by Dr. Scott Best, SiberSci, LLC. All calculator values were produced with the SiberSci Rectangular Waveguide TM_m,n Calculator and checked by hand with c = 299,792,458 m/s.