.. _integration-radomes-a212: ============================ Radome and mechanical design ============================ A radome is the product housing or plastic cover in front of the radar that protects it from mechanical impact and weather, see :numref:`fig_transmissionreflection_a212`. When designed correctly, the radome is effectively transparent to the radar signal and thus not impacting the radiated performance. Transmission and reflection =========================== When an electromagnetic (EM) wave hits the interface between two :term:`dielectrics`, such as plastic and air, :term:`reflection` and :term:`refraction` occur at the boundary. Inside the :term:`dielectric`, the propagation speed and wavelength change depending on the material’s :term:`permittivity`. If the wavelength in free space is :math:`\lambda_{0}`, 5 mm at 60 GHz, the wavelength inside the material is :math:`\lambda=\frac{\lambda_{0}}{\sqrt{\varepsilon_r}}`, where :math:`\varepsilon_r` is the relative :term:`dielectric constant` of the material. In the case of :term:`linearly polarized` propagation at :term:`normal incidence`, the Fresnel equations relate the ratio between the incident and transmitted fields: .. math:: :label: eq_transmissionreflection_a212 r=\frac{Z_{2} - Z_{1}}{Z_{2} + Z_{1}}, \quad t=\frac{2Z_{2}}{Z_{2}+Z_{1}}, where :math:`Z_{1}` and :math:`Z_{2}` are the corresponding wave impedances. The :term:`transmission` and :term:`reflection` factors are also related by :math:`t = 1 + r`. In most cases, we have non-magnetic materials so that :math:`Z = \sqrt{\frac{\mu_{r}}{\varepsilon_r}} = \frac{1}{\sqrt{\varepsilon_r}}`. An important special case is :term:`normal incidence` between air and a :term:`dielectric`. The reflection coefficient then simplifies to .. math:: :label: eq_airdielectricreflection_a212 r=\frac{1 - \sqrt{\varepsilon_r}}{1 + \sqrt{\varepsilon_r}} From this expression, we can see that the materials with a higher :term:`dielectric constant` produce stronger reflections of the incidence signal. Conductors, such as metal, have a very high :term:`dielectric constant` (:math:`\varepsilon_r \rightarrow \infty`) and thus produce a strong :term:`reflection`. Most polymers have a :term:`dielectric constant` in the range 2–4, and therefore give a weaker :term:`reflection`. Radomes ======= Radome thickness ---------------- Let :math:`d_1` be the distance to the radome, :math:`d_2` the thickness, :math:`r_1` and :math:`r_2` the reflection coefficients, and :math:`t_1` and :math:`t_2` the corresponding transmission coefficients according to :numref:`fig_transmissionreflection_a212`. As an incident wave hits the first interface, a :term:`reflection` :math:`r_1`, and a :term:`transmission` :math:`t_1` occur. Note that :math:`r_1 < 1`, so the reflected wave is out of phase with the incident wave. At the second interface, another :term:`reflection` :math:`r_2` and :term:`transmission` :math:`t_2` take place. With air on both sides of the :term:`dielectric`, note from :eq:`eq_transmissionreflection_a212` that :math:`r_1 = -r_2`. The total reflection coefficient :math:`\Gamma_1` then becomes [#f1]_: .. math:: :label: eq_reflectioncoeff_a212 \Gamma_{1}=\frac{r_{1} \left(1 - e^{-2jkd_{2}}\right)}{1 - r_{1}^{2} e^{-2jkd_{2}}} where :math:`k = 2\pi/\lambda` is the wave number in the material. Two notable special cases follow: - For :math:`d_{2} = m\frac{\lambda}{2}, \; m = 1,2,...` we have :math:`e^{-2jk d_{2}}=1` and :math:`\Gamma_{1} = 0`. - For :math:`d_{2} = m\frac{\lambda}{4}, \; m = 1,3,5,...` we have :math:`e^{-2jk d_{2}}=-1` and :math:`\Gamma_{1} = \frac{2r_{1}}{1+r_{1}^{2}}`. The optimum radome thickness occurs when the :term:`dielectric` is perfectly :term:`reflectionless`, i.e. when the thickness is a multiple of half a wavelength. This can also be understood from the fact that the round trip of the wave inside the radome introduces a 360° phase shift, thereby cancelling the reflected wave :math:`r_{1}`. .. _fig_transmissionreflection_a212: .. figure:: /images/hw_images/radome_transmission_and_reflection.svg :align: center :width: 95% :term:`Transmitted` and :term:`reflected` signals from a half-wavelength radome. Secondary :term:`reflections` have been omitted for simplicity. As an example, if the material is polycarbonate with relative :term:`permittivity` :math:`\varepsilon_r=2.75`, the optimal radome thickness becomes: .. math:: :label: eq_exthicknesscalc_a212 d_{2}=m\frac{\lambda}{2} = m\frac{\lambda_{0}}{2\sqrt{\varepsilon_r}} = m\frac{5}{2\sqrt{2.75}} \approx m \times 1.51 \:\mathrm{mm}, m = 1,2,... If the :term:`dielectric` thickness is an odd multiple of :math:`\frac{\lambda}{4}`, maximum :term:`reflection` occurs. The relative :term:`dielectric constant` of some more materials can be found in :numref:`tab_common_material_a212`. Dielectric constant of common materials ======================================= The :term:`dielectric constant` of some common polymers at 60 GHz is presented in :numref:`tab_common_material_a212`. .. _tab_common_material_a212: .. table:: Relative :term:`permittivity` of common materials [#f2]_ :align: center :widths: auto ==================================== ===================================== Material Re(:math:`\varepsilon`) at 60 GHz ==================================== ===================================== Acrylic glass 2.5 Alumina 9.3 Fused Quartz 3.8 MACOR 5.5 PEEK 3.12 [#f3]_ PMMA 2.6 [#f4]_ Polycarbonate (PC) 2.75 [#f4]_ [#f5]_ Polyethylene (PE) 2.3 [#f5]_ [#f6]_ [#f7]_ [#f8]_ Polypropylene (PP) 2.2 [#f5]_ [#f8]_ Polystyrene (Rexolite) 2.5 PTFE (Teflon) 2.05 [#f5]_ ==================================== ===================================== .. rubric:: Footnotes .. [#f1] S. J. Orfanidis, *Electromagnetic Waves and Antennas*, New Jersey: Rutgers University, 2016. .. [#f2] S. R. Artem Boriskin, *Aperture Antennas for Millimeter and Sub-Millimeter Wave Applications*, Springer, 2017. .. [#f3] N. C. F. T. V. et al., "Complex Dielectric Permittivity of Engineering and 3D-Printing Polymers at Q-Band," *J. Infrared Milli Terahz Waves*, vol. 39, pp. 1140–1147, 2018. .. [#f4] B. L. G. et al., "A Comparison of Measurements of the Permittivity and Loss Angle of Polymers in the Frequency Range 10 GHz to 90 GHz," *2021 15th European Conference on Antennas and Propagation (EuCAP)*, pp. 1–5, 2021. .. [#f5] Y. O. K. F. M. M. K. S. Yuka Hasegawa, "Complex Permittivity Spectra of Various Insulating Polymers at Ultrawide-Band Frequencies," *Electrical Engineering in Japan*, vol. 198, no. 3, pp. 11–18, 2017. .. [#f6] J. W. Lamb, *Miscellaneous Data on Materials for Millimetre and Submillimetre Optics*. .. [#f7] Salski et al., "Complex Permittivity of Common Dielectrics in 20–110 GHz Frequency Range Measured with a Fabry–Perot Open Resonator." .. [#f8] Afsar et al., "A New 60 GHz Open-Resonator Technique for Precision Permittivity and Loss-Tangent Measurement."