--- title: "Rectangular waveguide TM mode field equations" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:rectangular-waveguide-tm-equations" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/rectangular-waveguide-tm-equations site: BCT Engineering --- # Rectangular waveguide TM mode field equations RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2081 Bhadra, 8 marks; 2081 Baisakh, 8 marks; 2076 Bhadra, 10 marks; 2074 Bhadra, 10 marks; 2073 Magh, 5 marks; 2072 Magh, 5 marks (asked 6 times) - Video: [Rectangular Wave guide | Propagation of TM waves | Part-1/3 | Microwave Engineering | Lec-09](https://www.youtube.com/watch?v=AfSZWBrNl90) — Education 4u - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/rectangular-waveguide-tm-equations - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Explain the field and characteristic equations of a rectangular waveguide working in TM mode. ## Short answer For TMmn, Hz=0 and Ez=E0 sin(mπx/a)sin(nπy/b)e^-jβz; transverse fields follow from Maxwell's equations. Cutoff fc=(c/2)√((m/a)^2+(n/b)^2), guide wavelength λg=λ0/√(1-(fc/f)^2), wave impedance Z_TM=η√(1-(fc/f)^2). Lowest TM mode is TM11. ## Answer A rectangular waveguide is a hollow metal pipe of inner width *a* (x-direction) and height *b* (y-direction), a > b, with the wave travelling along *z*. In a **TM (transverse magnetic) mode**, Hz = 0 and Ez ≠ 0. ```text y ^ b |+-----------------+ || | wave travels in +z || air (μ0, ε0) | (into the page) |+-----------------+ 0------------------> x a ``` #### Wave equation and solution For fields varying as e^(j(ωt − βz)), Ez satisfies ```text ∂²Ez/∂x² + ∂²Ez/∂y² + kc² Ez = 0 kc² = k² - β², k = ω√(με) ``` By separation of variables, Ez = X(x)Y(y) with X = A1 cos kx·x + A2 sin kx·x, Y = B1 cos ky·y + B2 sin ky·y. **Boundary conditions:** tangential E is zero on the walls, so Ez = 0 at x = 0, a and y = 0, b. This gives A1 = B1 = 0, sin kx·a = 0 and sin ky·b = 0: ```text kx = mπ/a, ky = nπ/b, m, n = 1, 2, 3 ... kc² = (mπ/a)² + (nπ/b)² ``` #### Field equations (TMmn) ```text Ez = E0 sin(mπx/a) sin(nπy/b) e^(-jβz) Ex = -(jβ/kc²)(mπ/a) E0 cos(mπx/a) sin(nπy/b) e^(-jβz) Ey = -(jβ/kc²)(nπ/b) E0 sin(mπx/a) cos(nπy/b) e^(-jβz) Hx = (jωε/kc²)(nπ/b) E0 sin(mπx/a) cos(nπy/b) e^(-jβz) Hy = -(jωε/kc²)(mπ/a) E0 cos(mπx/a) sin(nπy/b) e^(-jβz) Hz = 0 ``` Since Ez contains sin(mπx/a)·sin(nπy/b), **m = 0 or n = 0 gives zero field**. So TM00, TM10, TM01 do not exist; the lowest TM mode is **TM11**. #### Characteristic equations - **Propagation constant:** β = √(ω²με − kc²); the mode propagates only when ω²με > kc². - **Cut-off frequency:** ```text fc = (1/(2√(με))) √((m/a)² + (n/b)²) = (c/2) √((m/a)² + (n/b)²) (air) ``` - **Cut-off wavelength:** λc = 2/√((m/a)² + (n/b)²) - **Guide wavelength:** λg = λ0/√(1 − (fc/f)²) (longer than free-space λ0) - **Phase velocity:** vp = ω/β = c/√(1 − (fc/f)²) > c - **Group velocity:** vg = c·√(1 − (fc/f)²) < c, with vp·vg = c² - **Wave impedance:** ```text Z_TM = Ex/Hy = β/(ωε) = η·√(1 - (fc/f)²) (η = 377 Ω in air) ``` Z_TM is always **less than** η and becomes zero at cut-off. #### Example For an X-band guide (a = 2.286 cm, b = 1.016 cm), the TM11 cut-off is fc = (3×10⁸/2)√((1/0.02286)² + (1/0.01016)²) ≈ 16.2 GHz, much higher than the TE10 cut-off (6.56 GHz). This is why TM modes are not used as the operating mode in standard guides. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the lowest TM mode in a rectangular guide TM11 and not TM10? A. Ez is proportional to sin(m pi x/a) sin(n pi y/b), so m = 0 or n = 0 makes all fields zero B. TM10 has a cut-off of zero C. TM10 radiates out of the walls D. Ez is cosine-shaped so m and n start from 1 for convenience Answer: A. The boundary conditions give Ez with sine factors in both directions, so any zero index makes the field vanish. (Key point: Lowest TM mode) 2. The cut-off frequency of TMmn in an air-filled guide is A. (c/2) sqrt((m/a)^2 + (n/b)^2) B. (c/2)(m/a + n/b) C. c sqrt(m a + n b) D. (c/2) sqrt((a/m)^2 + (b/n)^2) Answer: A. From kc^2 = (m pi/a)^2 + (n pi/b)^2 and fc = kc c/(2 pi). (Key point: Cut-off frequency) 3. How does Z_TM compare with free-space impedance eta? A. Greater than eta B. Less than eta and zero at cut-off C. Equal to eta D. Infinite at cut-off Answer: B. Z_TM = eta sqrt(1 - (fc/f)^2), which is below eta and falls to zero at cut-off. (Key point: Wave impedance) 4. Which relation holds for phase and group velocities in the guide? A. vp x vg = c^2, vp > c, vg < c B. vp = vg = c C. vp < c and vg > c D. vp x vg = c Answer: A. vp = c/sqrt(1-(fc/f)^2) is above c, vg = c sqrt(1-(fc/f)^2) is below c, and the product is c^2. (Key point: Velocity relation) ## Close variants set in other exams - [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× ## Related in RF and Microwave Engineering - [Circular waveguide TM field strength and dominant mode](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acircular-waveguide-tm-field) — RF and Microwave Engineering, chapter 4 - [Circular waveguide TE mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acircular-waveguide-te-equations) — RF and Microwave Engineering, chapter 4, asked 2× - [Rectangular TM equations; TE10 vs TE20](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-tm-te10-te20) — 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 - [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 - [Rectangular waveguide cut-off and phase velocity](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-assumed-dimensions) — RF and Microwave Engineering, chapter 4 ## Related in other subjects - [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 - [Modes in rectangular waveguide](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amodes-supported-rectangular-waveguide) — Electromagnetism, chapter 6, asked 4×