--- title: "Short note: TE and TM modes" subject: "Electromagnetism" chapter: 6 key: "semester_3/electromagnetism:te-tm-modes-rectangular-waveguide" source: https://www.bctengineer.com/q/semester_3/electromagnetism/te-tm-modes-rectangular-waveguide site: BCT Engineering --- # Short note: TE and TM modes Electromagnetism · Chapter 6: Wave guides - Asked in: 2076 Asoj, 3 marks; 2075 Chaitra, 3 marks (asked 2 times) - Video: [Origin of TE and TM Modes](https://www.youtube.com/watch?v=dYKGsJGPsDo) — Jordan Edmunds Chetty - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/te-tm-modes-rectangular-waveguide - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-6 ## Question Explain the transverse electric (TE) and transverse magnetic (TM) modes used in rectangular waveguides. ## Short answer TE modes (E_z = 0) have H_z = H_0 cos(mπx/a) cos(nπy/b) e^{-jβz}, wave impedance Z_TE = η/√(1-(f_c/f)²). TM modes (H_z = 0) have E_z = E_0 sin(mπx/a) sin(nπy/b) e^{-jβz}, Z_TM = η√(1-(f_c/f)²). Cut-off f_c = (c/2)√((m/a)²+(n/b)²). Dominant TE_10, lowest TM_11. ## Answer In a rectangular waveguide (a × b, a > b, wave travelling along +z) the fields cannot be TEM, so they are split into two families defined by which longitudinal field is zero. #### TE (transverse electric) modes The electric field has no component along the direction of travel: **E_z = 0**, H_z ≠ 0. ```text H_z = H₀ cos(mπx/a) cos(nπy/b) e^{−jβz} ``` - m = 0, 1, 2, ...; n = 0, 1, 2, ... (not both zero). - TE₁₀ is the dominant mode of the guide. - Wave impedance Z_TE = η/√(1 − (f_c/f)²), which is greater than η. #### TM (transverse magnetic) modes The magnetic field has no component along z: **H_z = 0**, E_z ≠ 0. ```text E_z = E₀ sin(mπx/a) sin(nπy/b) e^{−jβz} ``` - m = 1, 2, ...; n = 1, 2, ... (both at least 1, since E_z must be zero at the walls). - TM₁₁ is the lowest TM mode. - Wave impedance Z_TM = η√(1 − (f_c/f)²), which is less than η. #### Common relations ```text f_c = (c/2)√[(m/a)² + (n/b)²] β = β₀ √(1 − (f_c/f)²) λ_g = λ₀/√(1 − (f_c/f)²) v_p = c/√(1 − (f_c/f)²), v_g = c√(1 − (f_c/f)²) ``` where η = 377 Ω in air and v_p v_g = c². TE_mn and TM_mn with the same m, n have the same cut-off frequency. Below cut-off, β is imaginary and the mode does not propagate. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which condition defines a TE mode in a rectangular waveguide? A. E_z = 0 and H_z is not zero B. H_z = 0 and E_z is not zero C. E_z = H_z = 0 D. Both E_z and H_z are non-zero Answer: A. TE means no electric field along the direction of travel, so E_z = 0 while H_z is non-zero. (Key point: TE definition) 2. Why must both m and n be at least 1 for a TM mode? A. Because H_z must vanish at the walls B. Because E_z is proportional to sin(mπx/a) sin(nπy/b) and must be zero at the walls, so m = 0 or n = 0 would make all fields vanish C. Because TM modes have a lower cut-off than TE modes D. Because the wave impedance would otherwise exceed η Answer: B. E_z has sine variation and must be zero at the conducting walls; with m = 0 or n = 0 the field is identically zero. (Key point: TM mode indices) 3. How do the wave impedances of TE and TM modes compare with η above cut-off? A. Both are greater than η B. Both are less than η C. Z_TE is less than η and Z_TM is greater than η D. Z_TE is greater than η and Z_TM is less than η Answer: D. Z_TE = η/√(1 − (f_c/f)²) > η, while Z_TM = η√(1 − (f_c/f)²) < η. (Key point: Wave impedances) 4. What happens to a mode when the operating frequency is below its cut-off frequency? A. It propagates with a larger phase velocity B. It propagates as a TEM wave C. β becomes imaginary and the mode does not propagate D. Its guide wavelength becomes zero Answer: C. Below cut-off β is imaginary, so the field decays (evanescent) and the mode does not propagate. (Key point: Cut-off behaviour) ## Related in Electromagnetism - [Modes in rectangular waveguide](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amodes-supported-rectangular-waveguide) — Electromagnetism, chapter 6, asked 4× - [Define TM mode; TM11 cut-off 5×3 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atm-mode-define-cutoff-5x3-guide) — Electromagnetism, chapter 6 - [Why no TEM mode; first four modes](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-no-tem-four-modes) — Electromagnetism, chapter 6 - [Potential gradient gives electric field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apotential-gradient-gives-electric-field) — Electromagnetism, chapter 2 ## Related in other subjects - [Circular waveguide TE equations and degenerate modes](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acircular-te-rectangular-modes) — RF and Microwave Engineering, chapter 4 - [TE vs TM mode in circular waveguide](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acircular-waveguide-te-vs-tm) — RF and Microwave Engineering, chapter 4 - [Rectangular waveguide TE mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-te-equations) — RF and Microwave Engineering, chapter 4, asked 11× - [Rectangular waveguide TM mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-tm-equations) — RF and Microwave Engineering, chapter 4, asked 11×