2064 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL, 47, NO Il, NOVEMBER 1999 Although the surface exhibits high impedance, it is not actually devoid of current. (If there were no current, electro- magnetic waves would be transmitted right through the ground plane. )However, the resonant structure provides a phase shift, thus. the image currents in the surface reinforce the currents in the antenna, instead of canceling them To the left-hand side of the light line in Fig. 7, we can determine the frequency range over which the radiation effi- ciency is high by using a circuit model, in which the antenna is modeled as a current source. The textured surface is modeled as an LC circuit in parallel with the antenna, and the radiation into free space is modeled as a resistor with a value of po/Eo/cos(0)= 377 S/cos(0). The amount of power dissipated in the resistor is a measure of the radiation efficiency of the antenna The maximum power dissipated in istor occurs at Fig 8. Reflection phase of the high-impedance surtace, calculated using the the LC resonance frequency of the ffective surface the surface reactance crosses through frequencies, or at very high frequencies, the current is shunted reflected waves. If the surface has low impedance, such as through the inductor or the capacitor, and the power flowing in the case of a good conductor, the ratio of electric field to to the resistor is reduced. It can be shown that the frequencies magnetic field is small. The electric field has a node at the where the radiation drops to half of its maximum value occur surface, and the magnetic field has an antinode. Conversely, when the magnitude of the surface impedance is equal to the for a high impedance surface, the electric field has an antinode impedance of free space. For normal radiation, we have the at the surface, while the magnetic field has a node. Another following equation term for such a surface is an artificial "magnetic conductor Recent work involving grounded frequency selective surfaces has also been shown to mimic a magnetic conductor [37] 2LC=7 However, these structures do not possess a complete surface- This can be solved for w to yield the following equation wave bandgap, since they lack the vertical conducting vias, which interact with the vertical electric field of tm surface (23) waves Typical parameters for a two-layer ground plane are 2 nH2 For typical geometries, L is usually on the order of I nH, of inductance, and 0.05 pF2 of capacitance. For these values, and C is in the range of 0.05-10 pF. With these values, the the reflection phase is plotted in Fig 8. At very low frequen- terms involving 1/rC are much smaller than the 1/LC cies, the reflection phase is T, and the structure behaves like terms, so we will eliminate them. This approximation yields an ordinary flat metal surface. The reflection phase slopes the following expression for the edges of the operating band downward, and eventually crosses through zero at the reso- nance frequency. Above the resonance frequency, the phase u=Wb11士 to -T. The phase falls within T/2 and -T/2 when the ude of the surface impedance exceeds the impedance The resonance frequency is wb=1/VLC, and Zo=VL/C rather than out-of-phase, and antenna elements may lie directly is the characteristic impedance of the LC circuit. With the adjacent to the surface without being shorted out parameters for L, C, and n given above, Zo is usually significantly smaller than n. Thus, the square root can be expanded in the following approximation (1±1z An antenna lying parallel to the textured surface will see the impedance of free space on one side, and the impedance The two frequencies designated by the t signs delimit the of the ground plane on the other side. Where the textured range over which an antenna would radiate efficiently or surface has low impedance, far from the resonance frequency, such a surface. The total bandwidth is roughly equal to the antenna current is mirrored by an opposing current in the characteristic impedance of the surface divided by the the surface. Since the antenna is shorted out by the nearby impedance of free space conductor, the radiation efficiency is very low. Within the forbidden bandgap near resonance, the textured surface has much higher impedance than free space, so the antenna is not shorted out. In this range of frequencies, the radiation This is also the bandwidth over which the reflection coefficient ency is high falls between +r/2 and -T/2, and image currents are more2064 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 11, NOVEMBER 1999 Fig. 8. Reflection phase of the high-impedance surface, calculated using the effective surface impedance model. reflected waves. If the surface has low impedance, such as in the case of a good conductor, the ratio of electric field to magnetic field is small. The electric field has a node at the surface, and the magnetic field has an antinode. Conversely, for a high impedance surface, the electric field has an antinode at the surface, while the magnetic field has a node. Another term for such a surface is an artificial “magnetic conductor.” Recent work involving grounded frequency selective surfaces has also been shown to mimic a magnetic conductor [37]. However, these structures do not possess a complete surface￾wave bandgap, since they lack the vertical conducting vias, which interact with the vertical electric field of TM surface waves. Typical parameters for a two-layer ground plane are 2 nH of inductance, and 0.05 pF of capacitance. For these values, the reflection phase is plotted in Fig. 8. At very low frequen￾cies, the reflection phase is , and the structure behaves like an ordinary flat metal surface. The reflection phase slopes downward, and eventually crosses through zero at the reso￾nance frequency. Above the resonance frequency, the phase returns to . The phase falls within and when the magnitude of the surface impedance exceeds the impedance of free space. Within this range, image currents are in-phase, rather than out-of-phase, and antenna elements may lie directly adjacent to the surface without being shorted out. C. Radiation Bandwidth An antenna lying parallel to the textured surface will see the impedance of free space on one side, and the impedance of the ground plane on the other side. Where the textured surface has low impedance, far from the resonance frequency, the antenna current is mirrored by an opposing current in the surface. Since the antenna is shorted out by the nearby conductor, the radiation efficiency is very low. Within the forbidden bandgap near resonance, the textured surface has much higher impedance than free space, so the antenna is not shorted out. In this range of frequencies, the radiation efficiency is high. Although the surface exhibits high impedance, it is not actually devoid of current. (If there were no current, electro￾magnetic waves would be transmitted right through the ground plane.) However, the resonant structure provides a phase shift, thus, the image currents in the surface reinforce the currents in the antenna, instead of canceling them. To the left-hand side of the light line in Fig. 7, we can determine the frequency range over which the radiation effi- ciency is high by using a circuit model, in which the antenna is modeled as a current source. The textured surface is modeled as an circuit in parallel with the antenna, and the radiation into free space is modeled as a resistor with a value of . The amount of power dissipated in the resistor is a measure of the radiation efficiency of the antenna. The maximum power dissipated in the resistor occurs at the resonance frequency of the ground plane, where the surface reactance crosses through infinity. At very low frequencies, or at very high frequencies, the current is shunted through the inductor or the capacitor, and the power flowing to the resistor is reduced. It can be shown that the frequencies where the radiation drops to half of its maximum value occur when the magnitude of the surface impedance is equal to the impedance of free space. For normal radiation, we have the following equation: (22) This can be solved for to yield the following equation: (23) For typical geometries, is usually on the order of 1 nH, and is in the range of 0.05–10 pF. With these values, the terms involving are much smaller than the terms, so we will eliminate them. This approximation yields the following expression for the edges of the operating band: (24) The resonance frequency is , and is the characteristic impedance of the circuit. With the parameters for , and given above, is usually significantly smaller than . Thus, the square root can be expanded in the following approximation: (25) The two frequencies designated by the signs delimit the range over which an antenna would radiate efficiently on such a surface. The total bandwidth is roughly equal to the characteristic impedance of the surface divided by the impedance of free space (26) This is also the bandwidth over which the reflection coefficient falls between and , and image currents are more