--- title: "Rectangular waveguide TE mode field equations" subject: "RF and Microwave Engineering" chapter: 4 key: "semester_7/rf_and_microwave_engineering:rectangular-waveguide-te-equations" source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/rectangular-waveguide-te-equations site: BCT Engineering --- # Rectangular waveguide TE mode field equations RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices - Asked in: 2080 Chaitra, 10 marks; 2079 Chaitra, 10 marks; 2077 Chaitra, 8 marks; 2074 Magh, 4 marks; 2080 Bhadra, 4 marks (asked 5 times) - Video: [Rectangular Waveguide | Propagation of waves | Part-1/3 | Microwave Engineering | Lec-06](https://www.youtube.com/watch?v=yKI8mLwGVjM) — 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-te-equations - Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4 ## Question Define the EM field equations and characteristic parameters of a rectangular waveguide working on TE mode (derive the TEmn field equations). ## Short answer For TEmn, Ez=0 and Hz=H0 cos(mπx/a)cos(nπy/b)e^-jβz; transverse fields derived from Hz. Cutoff fc=(c/2)√((m/a)^2+(n/b)^2), guide wavelength λg=λ0/√(1-(fc/f)^2), wave impedance Z_TE=η/√(1-(fc/f)^2). Dominant mode is TE10 with fc=c/2a. ## Answer In a rectangular waveguide (inner dimensions a × b, a > b, propagation along *z*), a **TE (transverse electric) mode** has Ez = 0 and Hz ≠ 0. All other field components are derived from Hz. ```text y b +-----------------+ | air (μ, ε) | propagation along +z +-----------------+ 0 a -> x ``` #### Derivation With fields varying as e^(j(ωt − βz)), Hz satisfies the reduced wave equation ```text ∂²Hz/∂x² + ∂²Hz/∂y² + kc² Hz = 0, kc² = ω²με - β² ``` Separation of variables, Hz = X(x)·Y(y): ```text X = A1 cos kx·x + A2 sin kx·x Y = B1 cos ky·y + B2 sin ky·y, kx² + ky² = kc² ``` The transverse fields come from Maxwell's curl equations: ```text Ex = -(jωμ/kc²) ∂Hz/∂y Hx = -(jβ/kc²) ∂Hz/∂x Ey = (jωμ/kc²) ∂Hz/∂x Hy = -(jβ/kc²) ∂Hz/∂y ``` **Boundary conditions** (tangential E = 0 on walls): Ex = 0 at y = 0, b → ∂Hz/∂y = 0 there; Ey = 0 at x = 0, a → ∂Hz/∂x = 0 there. This gives A2 = B2 = 0 and ```text kx = mπ/a, ky = nπ/b, m, n = 0, 1, 2 ... (not both zero) ``` #### TEmn field equations ```text Hz = H0 cos(mπx/a) cos(nπy/b) e^(-jβz) Hx = (jβ/kc²)(mπ/a) H0 sin(mπx/a) cos(nπy/b) e^(-jβz) Hy = (jβ/kc²)(nπ/b) H0 cos(mπx/a) sin(nπy/b) e^(-jβz) Ex = (jωμ/kc²)(nπ/b) H0 cos(mπx/a) sin(nπy/b) e^(-jβz) Ey = -(jωμ/kc²)(mπ/a) H0 sin(mπx/a) cos(nπy/b) e^(-jβz) Ez = 0 ``` Because Hz uses cosines, one index may be zero (TE10, TE01, TE20 exist), but TE00 gives no field. #### Characteristic parameters - **Cut-off wave number:** kc = √((mπ/a)² + (nπ/b)²) - **Cut-off frequency:** fc = (c/2)√((m/a)² + (n/b)²) (air-filled) - **Cut-off wavelength:** λc = 2/√((m/a)² + (n/b)²); for TE10, λc = 2a - **Phase constant:** β = √(k² − kc²) = k√(1 − (fc/f)²) - **Guide wavelength:** λg = λ0/√(1 − (fc/f)²) - **Phase velocity:** vp = c/√(1 − (fc/f)²); **group velocity:** vg = c√(1 − (fc/f)²); vp·vg = c² - **Wave impedance:** ```text Z_TE = Ex/Hy = ωμ/β = η/√(1 - (fc/f)²) (η = 377 Ω) ``` Z_TE is always **greater than** η. #### Dominant TE10 mode With m = 1, n = 0 (and a > b), TE10 has the lowest cut-off fc = c/2a: ```text Hz = H0 cos(πx/a) e^(-jβz) Hx = (jβa/π) H0 sin(πx/a) e^(-jβz) Ey = -(jωμa/π) H0 sin(πx/a) e^(-jβz) ``` Only Ey, Hx and Hz exist; E is maximum at the centre (x = a/2) and zero at the side walls. **Example:** for a = 2.286 cm at 10 GHz, fc = 6.56 GHz, λg = 3.98 cm and Z_TE ≈ 500 Ω. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What boundary condition on Hz does zero tangential E on the walls lead to? A. Hz = 0 on all walls B. The normal derivative of Hz is zero on the walls, giving cosine functions C. Hz is maximum at the centre D. Hz is infinite at the walls Answer: B. Ex = 0 gives dHz/dy = 0 at y = 0, b and Ey = 0 gives dHz/dx = 0 at x = 0, a, so Hz is a product of cosines. (Key point: Boundary conditions) 2. Which TE modes exist? A. m, n from 1 only B. m, n from 0, but not both zero (TE10, TE01, TE20 exist) C. Only m = n D. TE00 is the dominant mode Answer: B. Hz has cosines, so one index can be zero; only TE00 gives no field. (Key point: Allowed indices) 3. Which is the dominant mode of a rectangular guide (a > b) and its cut-off? A. TM11, fc = c/(2a) B. TE10, fc = c/(2a) C. TE01, fc = c/(2b) D. TE10, fc = c/a Answer: B. TE10 has the lowest cut-off, fc = c/2a (lambda_c = 2a). (Key point: Dominant mode) 4. Z_TE compared with eta is A. Equal B. Less than eta C. Greater than eta, equal to eta/sqrt(1-(fc/f)^2) D. Zero at cut-off Answer: C. Z_TE = omega mu/beta = eta/sqrt(1-(fc/f)^2), always above eta and infinite at cut-off. (Key point: TE wave impedance) ## Close variants set in other exams - [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× ## Related in RF and Microwave Engineering - [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× - [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 field components; 1 GHz in 5 cm waveguide](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Ate10-1ghz-5cm-waveguide) — RF and Microwave Engineering, chapter 4 - [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 - [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 modes and critical parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-modes-parameters) — RF and Microwave Engineering, chapter 4 ## Related in other subjects - [EM wave in dielectric from Maxwell's equations](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-em-wave-dielectric-velocity) — Engineering Physics, chapter 9, asked 2×