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# Rectangular Waveguide on $H_{10}$ ($TE_{10}$) Mode (4.2 GHz, C-Band)
This repository contains engineering design files, analytical calculations, and 3D electromagnetic numerical simulations for a hollow rectangular waveguide operating in the fundamental $H_{10}$ ($TE_{10}$) mode at a center frequency of $f_0 = 4.2\text{ GHz}$ (C-band).
The project includes analytical mathematical derivations in PTC Mathcad, parametric 3D CAD models with a standard mounting flange (SolidWorks / STEP), and full-wave electromagnetic verification in CST Studio Suite.
---
## Project Structure
```text
rectangular-waveguide-4.2ghz/
├── cad/ # 3D CAD models (SolidWorks SLDPRT, STEP format)
├── calculations/ # Analytical calculation worksheets in PTC Mathcad (.xmcd)
├── docs/images/ # Engineering calculation sheets, 3D model renders, and simulation plots
└── simulation/ # Electromagnetic simulation project in CST Studio Suite (.cst)
```
---
## Analytical Calculations
Analytical calculation of the waveguide channel was performed in PTC Mathcad using Maxwell's electrodynamic equations for hollow metallic waveguides.
### 1. Input Parameters and Physical Constants
- **Center Frequency**: $f_0 = 4.2\cdot 10^9\text{ Hz}$ ($4.2\text{ GHz}$)
- **Speed of Light in Vacuum**: $c \approx 2.998\cdot 10^8\text{ m/s}$
- **Free-Space Wave Impedance**: $Z_0 = 120\pi \approx 376.73\ \Omega$
- **Permittivity of Free Space**: $\varepsilon_0 \approx 8.854\cdot 10^{-12}\text{ F/m}$
- **Conductor Material**: Aluminum alloy with electrical conductivity $\sigma = 35.7\cdot 10^6\text{ S/m}$
### 2. Geometry Dimensions and Single-Mode Bandwidth
Free-space wavelength:
$$\lambda_0 = \frac{c}{f_0} = 71.379\text{ mm}$$
To ensure single-mode propagation ($TE_{10}$) and prevent higher-order modes ($TE_{20}$, $TE_{01}$), the broad wall dimension $a$ was calculated:
- Minimum broad wall dimension: $a_{min} = \frac{\lambda_0}{2} = 35.69\text{ mm}$
- Selected broad wall dimension: $a_{eff} = 53.534\text{ mm}$
- Narrow wall dimension ($b = 0.5 \cdot a_{eff}$): $b = 26.767\text{ mm}$
### 3. Electrodynamic Properties
- **Cutoff Wavelength ($TE_{10}$)**: $\lambda_{crit} = 2 \cdot a_{eff} = 107.069\text{ mm}$
- **Cutoff Frequency ($TE_{10}$)**: $f_{crit} = \frac{c}{\lambda_{crit}} = 2.80\text{ GHz}$
- **Higher-Order Mode Threshold**: $f_{high} = 2 \cdot f_{crit} = 5.60\text{ GHz}$
- **Characteristic Wave Impedance ($TE_{10}$)**:
$$W_h = \frac{Z_0}{\sqrt{1 - \left(\frac{\lambda_0}{\lambda_{crit}}\right)^2}} = 505.787\ \Omega$$
- **Guide Wavelength**:
$$\lambda_B = \frac{\lambda_0}{\sqrt{1 - \left(\frac{\lambda_0}{\lambda_{crit}}\right)^2}} = 95.765\text{ mm}$$
- **Attenuation Constant (Conductor Loss)**: $\alpha_R = 35.974\cdot 10^{-3}\text{ dB/m}$ ($0.036\text{ dB/m}$)
- **Breakdown Power Handling (in air)**: $P_{max} = 605.595\text{ kW}$
---
## 3D CAD Modeling
The waveguide section was modeled in SolidWorks and verified in CST Studio Suite. The assembly features an internal rectangular duct of $53.534 \times 26.767\text{ mm}$ and an interface flange with 4 mounting holes and an alignment shoulder.
- **Boundary Conditions**: Open add space along the propagation axis, lossy metal wall boundaries corresponding to aluminum conductivity.
- **Excitation Ports**: Waveguide ports (Port 1 and Port 2) positioned at the waveguide terminals supporting the fundamental $TE_{10}$ mode.
---
## Simulation Results
Electromagnetic numerical analysis was performed across the $3.0\text{--}5.0\text{ GHz}$ frequency spectrum.
### 1. Voltage Standing Wave Ratio (VSWR)
At the design center frequency of $4.2\text{ GHz}$:
- $\text{VSWR}_1 = 1.00183$
- $\text{VSWR}_2 = 1.00183$
Across the entire $3.0\text{--}5.0\text{ GHz}$ sweep, the VSWR remains below $1.004$, confirming impedance matching within nominal tolerances.
### 2. Scattering Parameters (S-Parameters)
At the design center frequency of $4.2\text{ GHz}$:
- **Reflection Coefficient ($S_{11}, S_{22}$)**: $-60.77\text{ dB}$ / $-60.79\text{ dB}$ (return loss $> 60\text{ dB}$).
- **Transmission Coefficient ($S_{21}, S_{12}$)**: $-0.067\text{ dB}$ (minimal insertion loss over the component length).
### 3. Impedance Parameters (Z-Parameters)
At $4.2\text{ GHz}$:
- **Mutual Transfer Impedance ($Z_{21}, Z_{12}$)**: $509.47\ \Omega$.
The simulated mutual impedance $Z_{21} \approx 509.5\ \Omega$ demonstrates high convergence with the analytical wave impedance $W_h = 505.79\ \Omega$ (deviation below $0.73\%$, attributed to line length transformation and port boundary discretization).
---
## License
Copyright (c) 2026 Ilya Kornilov
This source describes Open Hardware and is licensed under the CERN-OHL-P v2.
You may redistribute and modify this source and make products using it under
the terms of the CERN-OHL-P v2 (https://cern.ch/cern-ohl).
This source is distributed WITHOUT ANY EXPRESS OR IMPLIED WARRANTY,
INCLUDING OF MERCHANTABILITY, SATISFACTORY QUALITY AND FITNESS FOR A
PARTICULAR PURPOSE. Please see the CERN-OHL-P v2 for applicable conditions.
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