From 041de14e3c2d3b1c2ed51b0c776dc907ac4abbce Mon Sep 17 00:00:00 2001 From: BlitherBoom812 Date: Fri, 31 May 2024 15:05:47 +0800 Subject: [PATCH] 20240531 --- source/_posts/Antenna.md | 248 +++++++++++++++++++++++- source/images/Antenna/1717133761533.png | Bin 0 -> 42989 bytes source/images/Antenna/1717133858173.png | Bin 0 -> 18103 bytes source/images/Antenna/1717133859909.png | Bin 0 -> 18103 bytes source/images/Antenna/1717133876692.png | Bin 0 -> 18103 bytes source/images/Antenna/1717134268805.png | Bin 0 -> 62977 bytes source/images/Antenna/1717134287481.png | Bin 0 -> 65363 bytes source/images/Antenna/1717134331603.png | Bin 0 -> 28836 bytes source/images/Antenna/1717134353772.png | Bin 0 -> 217979 bytes source/images/Antenna/1717134725130.png | Bin 0 -> 90985 bytes source/images/Antenna/1717134745100.png | Bin 0 -> 161692 bytes source/images/Antenna/1717134897852.png | Bin 0 -> 186708 bytes source/images/Antenna/1717134942873.png | Bin 0 -> 52956 bytes source/images/Antenna/1717134955235.png | Bin 0 -> 42450 bytes source/images/Antenna/1717135044895.png | Bin 0 -> 212570 bytes source/images/Antenna/1717135171215.png | Bin 0 -> 200608 bytes source/images/Antenna/1717135198414.png | Bin 0 -> 202282 bytes source/images/Antenna/1717135386351.png | Bin 0 -> 87985 bytes source/images/Antenna/1717135675789.png | Bin 0 -> 261708 bytes source/images/Antenna/1717135951201.png | Bin 0 -> 138134 bytes source/images/Antenna/1717136046473.png | Bin 0 -> 342628 bytes source/images/Antenna/1717136115015.png | Bin 0 -> 309205 bytes source/images/Antenna/1717137633424.png | Bin 0 -> 27026 bytes source/images/Antenna/1717137646795.png | Bin 0 -> 331846 bytes source/images/Antenna/1717138714524.png | Bin 0 -> 113360 bytes source/images/Antenna/1717138873279.png | Bin 0 -> 476265 bytes 26 files changed, 247 insertions(+), 1 deletion(-) create mode 100644 source/images/Antenna/1717133761533.png create mode 100644 source/images/Antenna/1717133858173.png create mode 100644 source/images/Antenna/1717133859909.png create mode 100644 source/images/Antenna/1717133876692.png create mode 100644 source/images/Antenna/1717134268805.png create mode 100644 source/images/Antenna/1717134287481.png create mode 100644 source/images/Antenna/1717134331603.png create mode 100644 source/images/Antenna/1717134353772.png create mode 100644 source/images/Antenna/1717134725130.png create mode 100644 source/images/Antenna/1717134745100.png create mode 100644 source/images/Antenna/1717134897852.png create mode 100644 source/images/Antenna/1717134942873.png create mode 100644 source/images/Antenna/1717134955235.png create mode 100644 source/images/Antenna/1717135044895.png create mode 100644 source/images/Antenna/1717135171215.png create mode 100644 source/images/Antenna/1717135198414.png create mode 100644 source/images/Antenna/1717135386351.png create mode 100644 source/images/Antenna/1717135675789.png create mode 100644 source/images/Antenna/1717135951201.png create mode 100644 source/images/Antenna/1717136046473.png create mode 100644 source/images/Antenna/1717136115015.png create mode 100644 source/images/Antenna/1717137633424.png create mode 100644 source/images/Antenna/1717137646795.png create mode 100644 source/images/Antenna/1717138714524.png create mode 100644 source/images/Antenna/1717138873279.png diff --git a/source/_posts/Antenna.md b/source/_posts/Antenna.md index 6837c0b7..f779406a 100644 --- a/source/_posts/Antenna.md +++ b/source/_posts/Antenna.md @@ -663,4 +663,250 @@ The E-pattern is in shadow. ![1715929531751](../images/Antenna/1715929531751.png) -![1715929546494](../images/Antenna/1715929546494.png) \ No newline at end of file +![1715929546494](../images/Antenna/1715929546494.png) + +## Antenna Array + +1-D Linear Array + +2-D Planar Array + +3-D Conformal Array + +Array Element +* Dipoles +* Loops +* Slots +* Microstrip antennas + +### Two-Element array + +![1717133761533](../images/Antenna/1717133761533.png) + +$$ +\begin{gathered} +\vec{E}_1= \hat{\theta}\frac{I\Delta z}{4\pi}j\omega\mu\frac{e^{-jkr_1}}{r_1}\cos\theta_1 \\ +\vec{E}_{2}= \hat{\theta}\frac{I\Delta z}{4\pi}j\omega\mu\frac{e^{-jkr_2}}{r_2}\cos\theta_2 +\end{gathered} +$$ + +Remarks: +* Two element; +* Towards Y axis; +* Along Z axis; +* Space: d; +* Uniform phase +and amplitude; +* Observe in 2D +(YZ-plane). + +Far field Approximation + +![1717133876692](../images/Antenna/1717133876692.png) + +$$ +\begin{aligned}&\vec{E}_{total}=\vec{E}_1+\vec{E}_2\\&=\hat{\theta}\frac{I\Delta z}{4\pi}j\omega\mu\cos\theta\frac1r\Bigg(e^{-jk(r-\frac d2\cos\theta)}+e^{-jk(r+\frac d2\cos\theta)}\Bigg)\end{aligned} +$$ + +$$ +\begin{aligned} +\vec{E}_{total}& =\vec{E}_1+\vec{E}_2=\hat{\theta}\frac{I\Delta z}{4\pi}j\omega\mu\cos\theta\frac{1}{r}\Bigg(e^{-jk(r-\frac{d}{2}\cos\theta)}+e^{-jk(r+\frac{d}{2}\cos\theta)}\Bigg) \\ +&=\hat{\theta}\frac{I\Delta z}{4\pi}j\omega\mu\cos\theta\frac{e^{-jkr}}{r}\Bigg(e^{jk\frac{d}{2}\cos\theta}+e^{-jk\frac{d}{2}\cos\theta}\Bigg) \\ +&=\hat{\theta}\underbrace{\frac{I\Delta z}{4\pi}j\omega\mu\cos\theta\frac{e^{-jkr}}r}_{\text{Element pattern}}\underbrace{2\cos\biggl[\frac12kd\cos\theta\biggr]}_{\text{Array Factor (AF)}} +\end{aligned} +$$ + +Remarks: +* Uniform phase and amplitude; +* AF is related to space (d); +* AF is with no relation with antenna type. + +$$ +AF{=}2\cos\left[\frac12kd\cos\theta\right]\quad kd{=}\frac{2\pi}\lambda d{=}2\pi\frac d\lambda +$$ + +![1717134268805](../images/Antenna/1717134268805.png) + +![1717134287481](../images/Antenna/1717134287481.png) + +### N-Element array + +![1717134353772](../images/Antenna/1717134353772.png) + +$$ +\begin{aligned}&AF=1+e^{jkd\cos\theta}+e^{j2kd\cos\theta}+\cdots+e^{j(N-1)kd\cos\theta}\\&=\sum_{n=1}^Ne^{j(n-1)kd\cos\theta}=\sum_{n=1}^Ne^{j(n-1)\Psi}\end{aligned} +$$ + +$$ +AF=1+e^{j\Psi}+e^{j2\Psi}+\cdots+e^{j(N-1)\Psi}=\frac{e^{jN\Psi}-1}{e^{j\Psi}-1}\\=\frac{e^{j\frac N2\Psi}\left(e^{j\frac N2\Psi}-e^{-j\frac N2\Psi}\right)}{e^{j\frac12\Psi}\left(e^{j\frac12\Psi}-e^{-j\frac12\Psi}\right)}=\frac{e^{j\frac N2\Psi}\sin\left(\frac N2\Psi\right)}{e^{j\frac12\Psi}\sin\left(\frac12\Psi\right)} +$$ + +Refenece Point at the end: + +$$ +AF=\frac{e^{j\frac N2\Psi}\sin\left(\frac N2\Psi\right)}{e^{j\frac12\Psi}\sin\left(\frac12\Psi\right)},\Psi=kd\cos\theta, +$$ + +Refenece Point at the center: + +$$ +AF=\frac{\sin\left(\frac N2\Psi\right)}{\sin\left(\frac12\Psi\right)},\Psi=kd\cos\theta, +$$ + +In Progreessive Phase Shift: + +$$ +\Psi=kd\cos\theta+\alpha +$$ + +$$ +AF=1+e^{j(kd\cos\theta+\alpha)}+e^{j2(kd\cos\theta+\alpha)}+\cdots+e^{j(N-1)(kd\cos\theta+\alpha)}\\=\sum_{n=1}^Ne^{j(n-1)(kd\cos\theta+\alpha)}=\sum_{n=1}^Ne^{j(n-1)\Psi}=\frac{e^{j\frac N2\Psi}\sin\left(\frac N2\Psi\right)}{e^{j\frac12\Psi}\sin\left(\frac12\Psi\right)} +$$ + +Normalized Array Factor: + +$$ +\left|f(\Psi)\right|=\left|\frac{\sin\left(\frac N2\Psi\right)}{N\sin\left(\frac12\Psi\right)}\right| +$$ + +Grating Lobe: + +$$ +\begin{aligned} +\theta\in\begin{bmatrix}0,\pi\end{bmatrix}\text{ or }\theta\in\begin{bmatrix}\theta_1,\theta_2\end{bmatrix}\text{, visible region} \\ +\text{In the visible region,} \\ +ifwehaveY= 0\mathrm{~and~}\Psi=2\pi. +\end{aligned} +$$ + +Avoid grating lobe: +1. Smaller d; +2. Smaller phase shift. + +$$ +1.\mathrm{~For~}\alpha=\pi\Rightarrow2kd<2\pi\Rightarrow d\mathrm{~/~}\lambda<\frac12\\2.\mathrm{~For~}\alpha=0\mathrm{~}\Rightarrow\mathrm{~k}d<2\pi\Rightarrow d\mathrm{~/~}\lambda<1 +$$ + +#### Broadside Array + +Maximum @ $\theta = 90\degree$ + +$$ +AF\boldsymbol{=}N@\boldsymbol{\theta}\boldsymbol{=}\boldsymbol{\pi}/2\quad\boldsymbol{\Psi}\boldsymbol{=}kd\cos\boldsymbol{\theta}\boldsymbol{+}\boldsymbol{\alpha}|_{\theta=\pi/2}\boldsymbol{=}0 +$$ + +![1717134725130](../images/Antenna/1717134725130.png) + +![1717134745100](../images/Antenna/1717134745100.png) + +#### End-fire Array + +![1717134897852](../images/Antenna/1717134897852.png) + +$$ +\begin{aligned} +&AF= N@\theta{=}0 &\Psi=kd+\alpha=2n\pi(n=0,\pm1,\pm2\ldots) \\ +&\text{or} \\ +&AF= N@\theta{=}\pi &\Psi=-kd+\alpha=2n\pi(n=0,\pm1,\pm2...) \\ +&\Psi=kd\cos\theta+\alpha=2\pi\cos\theta +\end{aligned} +$$ + +Bidirectional: + +![1717134942873](../images/Antenna/1717134942873.png) + +Unidirectional: + +![1717134955235](../images/Antenna/1717134955235.png) + +#### Phased Array + +![1717135044895](../images/Antenna/1717135044895.png) + +#### Non-uniform Array + +Side Lobe + +Uniform array: +* Universal pattern: N↑, SLL↓ +* With a limit of -13.3 dB +* No control of SL + +How to reduce SLL? + +Non-uniform excitation + +![1717135171215](../images/Antenna/1717135171215.png) + +#### Planar Array + +![1717135198414](../images/Antenna/1717135198414.png) + +Can be viewed as product of two linear array factors: + +$$ +AF=\sum_{i=1}^{M\times N}I_ie^{jk\hat{r}\cdot\vec{r}_i}\\ +AF_n(\theta,\phi)=\left\{\frac{\sin(\frac M2\psi_x)}{M\sin\frac{\psi_x}2}\right\}\left\{\frac{\sin(\frac N2\psi_y)}{N\sin\frac{\psi_y}2}\right\};\\\psi_x=kd_x\sin\theta\cos\varphi+\alpha_x\\\psi_y=kd_y\sin\theta\sin\varphi+\alpha_y +$$ + +### Applications + +#### Yagi-Uda Antenna + +Basic configuration: +* One driven element; +* Two parasitic elements or more + +![1717135386351](../images/Antenna/1717135386351.png) + +Remarks: +* Parasitic elements are excited by near-field coupling from the driven element; +* Proper design of parasitic elements for end fire radiation; +* In far field, the radiated waves from all the elements are in-phase. + +![1717135675789](../images/Antenna/1717135675789.png) + +![1717135951201](../images/Antenna/1717135951201.png) + +#### Helix Antenna + +![1717136046473](../images/Antenna/1717136046473.png) + +![1717136115015](../images/Antenna/1717136115015.png) + +## Travelling-Wave Antennas + +### Travelling wave & standing wave + +#### Long wire antennas + +![1717137633424](../images/Antenna/1717137633424.png) + +![1717137646795](../images/Antenna/1717137646795.png) + +Note: +Long wire antennas: “l” = Several wavelength +* One end for excitation; +* The other end for load (open, short, or matching); +* Transmission line with radiation. + +### Log-periodic Antennas + +Yagi-Uda: High Gain + +Log-periodic: Wide Bandwidth + +![1717138714524](../images/Antenna/1717138714524.png) + +Why: +1. Feed from smaller dipole element; +2. Feed out-of-phase with adjacent elements; +3. 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