Microstrip/Patch Antennas Intr

Cheatsheet Content

### Introduction Microstrip or patch antennas were developed in the early '70s for spatial and aeronautics applications. #### Advantages - Light weight and low volume - Low fabrication cost (mass production with PCB techniques) - Easily integrated with printed circuits - Conformability - Good for mobile wireless applications #### Disadvantages - Low efficiency - Narrow band ($B_r = 5\% \div 10\%$) - Low gain ($G = 6dB \div 9dB$) - Poor Polarization Purity ### Geometry A microstrip patch antenna consists of a radiating patch on one side of a dielectric substrate and a ground plane on the other side. - **Substrate Thickness (h):** $t \ll \lambda$, $h \ll \lambda$ (typically $0.003\lambda ### Examples Microstrip antennas are used in various applications: - **USB Key for 2.4GHz Communications:** Integrated into small form factor devices. - **Aircraft (F-35):** Conformal antennas for radar altimeters, UHF LOS, L-band communications, and GPS. - **Clothing:** Wearable antennas for communication (e.g., firefighter suits, life vests). ### Radiation Mechanism #### Transmission Line Model The patch can be seen as a microstrip transmission line with length $L$ and width $W$, with the two ends connected to open circuits. - **Open Circuit:** $\rho_L = 1$ (Total reflection) - **Wave Propagation:** Waves travel inside the line and reflect at the ends. - **Round-trip Length:** $2L$ - **Phase Shift:** After a round-trip, the wave returns to the starting point out-of-phase by $4\pi L / \lambda$. #### Self-Sustainable Wave A self-sustainable wave appears if the wave comes back to the starting point with the same phase, except for a multiple of $2\pi$. $$ \frac{4\pi}{\lambda} L = m2\pi \Rightarrow L = m \frac{\lambda}{2} $$ where $m$ is an integer. If the patch length $L$ is a multiple of half the wavelength of the wave inside the patch, a standing wave appears under the patch. $$ f = \frac{v_p}{\lambda} = \frac{m}{2L} v_p $$ The higher $m$, the higher the frequency $f$. Usually, $m=1$ is considered for fundamental mode operation, leading to a half-wavelength resonance. #### Ideal vs. Real Radiation - **Ideal Case:** In the ideal case, with perfect reflection ($\rho=1$) at the ends, no radiation occurs because power cannot escape outside the patch. - **Real Case (Edge Effects):** In a real patch, edge effects cause radiation. The fields at the edges act as magnetic dipoles. #### Edge Effects as Magnetic Dipoles - **Equivalence Theorem:** $\vec{M}_s = \vec{E} \times \hat{n} = \vec{E}_{tan} \times \hat{n}$ - **Radiation:** These magnetic dipoles radiate. - **Hertzian Dipole:** Electric current density $\vec{J}$. - **Magnetic Dipole:** Magnetic current density $\vec{J}_m$. - $\nabla \times \vec{E} = -j\omega\mu_0\vec{H} + \vec{J}_m$ - $\nabla \times \vec{H} = j\omega\epsilon\vec{E}$ ### Radiation Pattern #### Ideal Infinite Ground Plane An array of two magnetic dipoles radiates. Due to the ground plane, equivalent dipoles only radiate in the upper half-space. The pattern shows linear polarization. #### Real Finite Ground Plane In real patch antennas, the ground plane is of finite extent, leading to **back lobes** in the radiation pattern. ### Patch Antenna Design Two main steps: 1. **Design the geometrical parameters of the patch radiator.** * Rectangular patch (Length $L$ and Width $W$) 2. **Design the input impedance of the antenna.** * Matching with the feeding transmission line (usually $50\Omega$) #### Design of the Patch: Patch Length - Usually, the specification is in terms of working frequency $f_r$. - According to the Transmission Line Model (TLM), the resonance condition is $L = \lambda/2$. $$ L = \frac{v_p}{2f_r} $$ #### Design of the Patch: Additional Length (Edge Effects) - Resonance condition *should* be $L = \lambda/2$. - However, due to edge effects, the actual physical length is slightly shorter than the electrical length. - The patch *appears* longer to the EM wave. - There is an effective length $L_{eff}$: $$ L_{eff} = L + 2\Delta L $$ where $L$ is the physical length and $\Delta L$ is the additional length due to fringing fields. - **Empirical Formula for $\Delta L$**: $$ \frac{\Delta L}{h} = 0.412 \frac{(\epsilon_{eff} + 0.3)(W/h + 0.264)}{(\epsilon_{eff} - 0.258)(W/h + 0.8)} $$ where $\epsilon_{eff}$ is the effective permittivity of a microstrip with width $W$: $$ \epsilon_{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12 \frac{h}{W}}} $$ #### Design of the Patch: Working Frequency By imposing $L_{eff} = \lambda/2$: $$ f_r = \frac{c}{2(L + 2\Delta L)\sqrt{\epsilon_{eff}}} $$ Since $\lambda = \frac{c}{f_r \sqrt{\epsilon_{eff}}}$, then: $$ L = \frac{c}{2f_r \sqrt{\epsilon_{eff}}} - 2\Delta L $$ In addition, for good radiation, the width $W$ can be shown to be: $$ W = \frac{c}{2f_r \sqrt{\frac{\epsilon_r + 1}{2}}} $$ #### Design Example (Calculations) Given a substrate RT/duroid 5880 with $\epsilon_r=2.2$ and $h=0.1588$ cm, working at $f=10$ GHz: 1. **Calculate Patch Width (W):** $$ W = \frac{c}{2f_r \sqrt{\frac{\epsilon_r + 1}{2}}} = \frac{3 \times 10^8 \text{ m/s}}{2 \times 10 \times 10^9 \text{ Hz} \sqrt{\frac{2.2 + 1}{2}}} = 1.186 \text{ cm} = 11.9 \text{ mm} $$ 2. **Calculate Effective Permittivity ($\epsilon_{eff}$):** $$ \epsilon_{eff} = \frac{2.2 + 1}{2} + \frac{2.2 - 1}{2} \frac{1}{\sqrt{1 + 12 \frac{0.1588}{1.186}}} = 1.972 $$ 3. **Calculate Extra Length ($\Delta L$):** $$ \frac{\Delta L}{h} = 0.412 \frac{(\epsilon_{eff} + 0.3)(W/h + 0.264)}{(\epsilon_{eff} - 0.258)(W/h + 0.8)} $$ $$ \Delta L = 0.1588 \times 0.412 \frac{(1.972+0.3)(1.186/0.1588 + 0.264)}{(1.972-0.258)(1.186/0.1588 + 0.8)} = 0.081 \text{ cm} $$ 4. **Calculate Patch Length (L):** $$ L = \frac{c}{2f_r \sqrt{\epsilon_{eff}}} - 2\Delta L = \frac{3 \times 10^8 \text{ m/s}}{2 \times 10 \times 10^9 \text{ Hz} \sqrt{1.972}} - 2 \times 0.081 \text{ cm} = 0.906 \text{ cm} = 9.1 \text{ mm} $$ Resulting patch dimensions: $L = 9.1 \text{ mm}$, $W = 11.9 \text{ mm}$. ### Feeding Feeding consists of: - How to excite the standing wave under the patch. - How to transfer active power from the transmission line connecting antenna and transceiver to the standing wave. - How to excite the standing wave so that the antenna appears to the line with a certain impedance. - How to design the input impedance of the antenna. #### Feeding Techniques 1. **Probe (Coaxial Cable Feed)** * **Mechanism:** External conductor connected to ground plane, inner conductor connected to patch. The inner conductor passing through the dielectric is called the probe. * **Advantages:** Simple, directly compatible with coaxial cables, easy to obtain input match by adjusting feed position. * **Disadvantages:** Significant probe (feed) radiation for thicker substrates, significant probe inductance for thicker substrates (limits bandwidth), not easily compatible with arrays. 2. **Microstrip Line Feed** * **Mechanism:** Microstrip line directly connected to the patch. The shape of the edge where it connects is modified for matching. * **Advantages:** Simple, allows for planar feeding, easy to use with arrays, easy to obtain input match. * **Disadvantages:** Significant line radiation for thicker substrates, for deep notches, patch current and radiation pattern may show distortion. 3. **Proximity-coupled Feed (Non-contacting)** * **Mechanism:** The microstrip line is placed close to the patch, but not directly connected. * **Advantages:** Allows for planar feeding, less line radiation compared to microstrip feed (line closer to ground plane), can allow for higher bandwidth (no probe inductance, so substrate can be thicker). * **Disadvantages:** Requires multilayer fabrication, alignment is important for input match. 4. **Gap-coupled Feed (Non-contacting)** * **Mechanism:** A gap exists between the microstrip line and the patch. * **Advantages:** Allows for planar feeding, can allow for a match even with high edge impedances where a notch might be too large (e.g., with high permittivity substrate). * **Disadvantages:** Requires accurate gap fabrication, requires full-wave design. 5. **Aperture-coupled Patch (ACP)** * **Mechanism:** Microstrip line on one substrate, patch on another, coupled through an aperture (slot) in the common ground plane. * **Advantages:** Allows for planar feeding, FTL is isolated from patch radiation, higher bandwidth possible (probe inductance eliminated, allowing for thick substrate), double-resonance can be created, allows use of different substrates to optimize antenna and feed-circuit performance. * **Disadvantages:** Requires multilayer fabrication, alignment is important for input match. ### Input Impedance #### Ideal Case: Line with Open Loads - A patch can be modeled as a transmission line with open circuits at both ends. - When connecting a feeding transmission line (FTL), the patch is effectively divided into two parts, both with a pure reactive input impedance. The FTL always sees a pure reactive impedance. #### Real Case: At the Two Ends - **Active Power Radiation:** Little real load ($G$). - **Reactive Power Storage:** Little reactive load ($B$). - More reactive electric field leads to a capacitive load. #### Equivalent Circuit The patch can be represented as a parallel RLC circuit at resonance. At the two ends: - $G_1 = G_2$ - $B_1 = B_2$ Where: - **Conductance ($G_1$):** $$ G_1 = \frac{W}{120\lambda_0} \left(1 - \frac{1}{24} \left(\frac{2\pi h}{\lambda_0}\right)^2 \right) $$ - **Susceptance ($B_1$):** $$ B_1 = \frac{W}{120\lambda_0} \left(1 - 0.636\ln\left(\frac{2\pi h}{\lambda_0}\right) \right) $$ ($\lambda$ is wavelength in patch, $\lambda_0$ is wavelength in free space.) #### Input Admittance Assuming the external transmission line is connected to one end (end 1): - The input admittance is $Y_i = G_1 + jB_1 + Y_{12}$. - Since the line length $L \approx 0.45\lambda \div 0.49\lambda$ (i.e., $\Delta L \ll L$), then $Y_{12} = G_{12} + jB_{12} \approx G_1 - jB_1$. - Therefore, $Y_i = G_1 + jB_1 + G_1 - jB_1 = 2G_1$. The input resistance $R_i$ at the edge of the patch (where $x_0=0$) is: $$ R_i(0) = \frac{1}{2G_1} $$ Typical values for $R_i(0)$ are $150 \div 300 \Omega$. #### Connecting Away from the Edge If the connection is $x_0$ far from end 1: $$ R_i(x_0) = \frac{1}{2G_1} \cos^2\left(\frac{\pi}{L}x_0\right) $$ This shows the resistance varies with position, being maximum at the edge. #### Input Impedance and Bandwidth - **Matching at one frequency:** This implies a certain bandwidth. - **Thicker substrates:** Give wider bandwidth. #### Substrate Effects Several features depend on the substrate: $$ L = \frac{c}{2f_r \sqrt{\epsilon_{eff}}} $$ - The higher the permittivity ($\epsilon_r$), the smaller the antenna size ($L$). - **Directivity:** Increases with $\epsilon_r$ and $d/\lambda_0$. - **Bandwidth:** Decreases with increasing $\epsilon_r$. - **Efficiency:** Decreases with increasing $\epsilon_r$. #### Matching - Usually, the characteristic impedance of the feeding line is $50\Omega$. - It is not possible to feed the patch at one edge if $R_i(0) > 50\Omega$. - Connection patch-feeding line inside the patch is used for matching. This is achieved by adjusting the feed position $x_0$. #### Optimization for Desired Resonance 1. Vary the length $L$ first until you find the value that gives an input reactance of zero at the desired frequency. 2. Then adjust the feed position $x_0$ to make the real part of the input impedance $50\Omega$ at this frequency.