---
title: "TE field components; 1 GHz in 5 cm waveguide"
subject: "RF and Microwave Engineering"
chapter: 4
key: "semester_7/rf_and_microwave_engineering:te10-1ghz-5cm-waveguide"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/te10-1ghz-5cm-waveguide
site: BCT Engineering
---

# TE field components; 1 GHz in 5 cm waveguide

RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices

- Asked in: 2070 Bhadra, 7+3 marks
- Video: [Rectangular Waveguide (Basics, Modes, Cut-Off Frequency, Wave Impedance, Phase & Group Velocity)](https://www.youtube.com/watch?v=Y3szgmhuXeg) — Engineering Funda
- 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/te10-1ghz-5cm-waveguide
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4

## Question

Define expressions for various field components of a rectangular waveguide in TE mode. Show that a 1 GHz signal cannot propagate in TE10 mode in a rectangular waveguide with a wall separation of 5 cm.

## Short answer

TE10 mode fields: Ey = -jωμ a/π H0 sin(πx/a) e^{-jβz}, Hx and Hz nonzero. Cutoff frequency is fc=c/(2a)=3×10^8/(2×0.05)=3 GHz. Since the 1 GHz signal is below cutoff, the mode is evanescent and cannot propagate.

## Answer

#### Field components of a rectangular waveguide in TE mode
For a guide with inner dimensions a × b (a > b) and Ez = 0, the solution of ∇t²Hz + kc²Hz = 0 with ∂Hz/∂n = 0 on the walls gives (m, n = 0, 1, 2 …, not both zero):
```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
```
with
```text
 kc² = (mπ/a)² + (nπ/b)²,   β = √(ω²με - kc²)
 fc = (c/2)√((m/a)² + (n/b)²)
 Z_TE = ωμ/β = η/√(1 - (fc/f)²)
 λg = λ0/√(1 - (fc/f)²)
```
For the dominant TE10 mode (m = 1, n = 0): only Ey, Hx, Hz exist, and fc = c/(2a).

#### Can 1 GHz propagate in TE10 with a = 5 cm?
```text
 fc(TE10) = c/(2a) = (3 × 10⁸)/(2 × 0.05)
          = 3 × 10⁹ Hz = 3 GHz
```
Equivalently, λc = 2a = 10 cm while λ0 = c/f = 30 cm.

Since f = 1 GHz < fc = 3 GHz (λ0 > λc), β = √(ω²με − kc²) is imaginary: the field decays exponentially (evanescent) instead of propagating.

**Answer: the TE10 cut-off is 3 GHz, so a 1 GHz signal cannot propagate in this waveguide.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the TE10 cut-off frequency of a rectangular guide of broad dimension a?
   A. c/a
   B. c/(2a)
   C. 2c/a
   D. c/(2b)
   Answer: B. With m = 1, n = 0, fc = (c/2)(1/a) = c/(2a). (Key point: TE10 cut-off formula)

2. For a = 5 cm, what is fc(TE10)?
   A. 1 GHz
   B. 6 GHz
   C. 3 GHz
   D. 1.5 GHz
   Answer: C. fc = 3×10^8/(2×0.05) = 3 GHz. (Key point: Numerical cut-off)

3. Why does a 1 GHz signal fail to propagate?
   A. f < fc, so β becomes imaginary and the field is evanescent
   B. The guide is lossless
   C. f > fc
   D. Ez is nonzero
   Answer: A. Below cut-off β = √(ω²με − kc²) is imaginary, so the field decays exponentially. (Key point: Why no propagation)

4. Which field components exist for TE10?
   A. Ex, Hy, Ez
   B. Ey, Hx, Hz
   C. Ez, Hx, Hy
   D. Ex, Ey, Hz
   Answer: B. For TE10 only Ey, Hx and Hz are present. (Key point: TE10 field components)

## Related in RF and Microwave Engineering

- [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×
- [Dominant mode and 2.254 cm waveguide at 6 GHz](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Awaveguide-dominant-mode-2-254cm) — 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 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×
- [Air-filled 3 cm waveguide at 6 GHz; coupler parameters](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Awaveguide-3cm-6ghz-directional-coupler) — 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

## Related in other subjects

- [Single-mode range, air-filled 2×1 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-2x1cm-single-mode-range) — Electromagnetism, chapter 6
- [Cut-off and single-mode range, 4×2 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-4x2cm-cutoff-single-mode-range) — Electromagnetism, chapter 6
- [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×