Table 3 Calculation examples of propagation constant γ, reflection coefficient Γ, and microwave penetration depth δ.
From: Microwave resonance detection method for hidden crack depth in rock and cementitious mortar
Name | Calculation formula | Granite material parameters | Calculated value of granite material G1 | Calculated value of granite material G2 | Cement mortar material parameters | Calculated value of cement mortar material C1 | Calculated value of cement mortar material C2 |
|---|---|---|---|---|---|---|---|
Conductivity σ (S/m) | 10−8–10−6 | 10−8 | 10−6 | 10−3–10−2 | 10−3 | 10−2 | |
Permeability of vacuum µ0 (H/m) | µ0 = 4π × 10−7 | 1.256 × 10−6 | 1.256 × 10−6 | 1.256 × 10−6 | 1.256 × 10−6 | 1.256 × 10−6 | 1.256 × 10−6 |
Relative permeability of sample µr | \(\mu_{r}=\mu/\mu_{0}\) | 1.04 | 1.04 | 1.04 | 1.04 | 1.04 | 1.04 |
Magnetic permeability of sample µ (H/m) | 1.306 × 10−6 | 1.306 × 10−6 | 1.306 × 10−6 | 1.306 × 10−6 | |||
Vacuum dielectric constant ε0 (F/m) | 8.854 × 10−12 | 8.854 × 10−12 | 8.854 × 10−12 | 8.854 × 10−12 | 8.854 × 10−12 | 8.854 × 10−12 | |
Relative permittivity of sample εr | \(\varepsilon_{r}=\varepsilon/\varepsilon_{0}\) | 4–9 | 5 | 5 | 4–40 | 4 | 40 |
Permittivity of sample ε (F/m) | 4.427 × 10−11 | 4.427 × 10−11 | 3.542 × 10−11 | 3.542 × 10−10 | |||
Angular frequency ω (Hz) | \(\omega = 2\pi f\) | 2.217 × 10−10 | 2.217 × 10−10 | 2.217 × 10−10 | 2.217 × 10−10 | ||
Propagation constant γ | \(\gamma = \sqrt{j\omega \mu \sigma}\) | 0.017 | 0.170 | 5.381 | 17.017 | ||
Wave impedance Z (Ω) | \(Z = \frac{E}{H} = \frac{\gamma}{j \omega \varepsilon} = \sqrt{j\omega \mu/ \sigma}\) | 1.702 × 106 | 1.702 × 105 | 5.381 × 103 | 1.701 × 103 | ||
Wave impedance of sample Z2 (Ω) | \(Z_{2} = \sqrt{\mu/ \varepsilon}\) | 171.772 | 171.772 | 192.047 | 60.731 | ||
Air dielectric wave impedance Z1 (Ω) | \(Z_{1} = \sqrt{\mu_{0}/ \varepsilon_{0}}\) | 376.635 | 376.635 | 376.635 | 376.635 | ||
Reflection coefficient Γ | \(\varGamma = \frac{Z_2 - Z_1}{Z_2 + Z_1}\) | − 0.374 | − 0.374 | − 0.325 | − 0.722 | ||
Microwave penetration depth d (mm) | \(d = \sqrt{2/(\mu \omega \sigma)}\) | 83,107 | 8311 | 263 | 83 |
