MODERN ANTENNA HANDBOOK by CONSTANTINE A.BALANIS ch. 6.1 – 6.2.3 Bang Ji Hun

Contents 6. Frequency-Independent Antennas: Spirals and Log-periodics - 6.1. Introduction - 6.2. Theory - 6.2.1 Babinet’s Principle in Electromagnetic Fields - 6.2.2 Self-Complementary Structures - 6.2.3 Principle of Similitude for Electromagnetic Fields

6. Frequency-Independent Antennas : Spirals and Log-periodics 6.1 Intorduction 6.2 : presents a relationship between structure of an antenna and the input impedance - 6.2.1 : Using Babinet’s principle in electromagnetic fields. - 6.2.2 : Input impedance of an antenna with a self-complementary structure is proved to independent of frequency. - 6.2.3 : on the basis of the principle of similitude in electromagnetic fields.

Frequency-independent antenna 6.2 Theory Frequency-independent antenna : If the characteristics of an antenna are not a function of frequency, the antenna is called a frequency independent antenna. - constant input impedance - constant radiation characteristics The input impedance is related to the antenna structure. - self-complementary structures (Babinet’s principle) - self-similar structures both satisfaction However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. Self similar Self complementary 𝜏 𝜏 𝜏 Log periodic toothed antenna Log periodic dipole array antenna

6.2.1 Babinet’s Principle in Electromagnetics Fields : The relationship between the E, H field of an arbitrarily shaped slot antenna and its complementary plate antenna. Then using Babinet’s principle, relationship between input impedance of the two antennas. (Mushiake, 1948) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. (6.1) 𝑍 𝜎 = intrinsic impedance 𝑍 𝑝𝑙𝑎𝑡𝑒 𝑍 𝑠𝑙𝑜𝑡 Figure 6.2 Complementary structure: (a) plate and (b) slot cut in an infinite sheet.

6.2.1 Babinet’s Principle in Electromagnetics Fields J and J’ in Figure 6.2a are symmetric electric currents with respect to the x-y plane and M and −M’ in Figure 6.2b are antisymmetric magnetic currents. (Fig. 6.2a and Fig. 6.2b are complementary structure) These current generate the electric and magnetic fields ( 𝐸 1 , 𝐸 2 , 𝐻 1 , 𝐻 2 ) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. duality assuming complementary infinite extent plate slot Image theory symmetric anti-symmetric Figure 6.2 Complementary structure: (a) plate and (b) slot cut in an infinite sheet.

6.2.1 Babinet’s Principle in Electromagnetics Fields Maxwell’s equation Boundary condition (a) (b) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. Figure 6.2 Complementary structure: (a) plate and (b) slot cut in an infinite sheet.

6.2.1 Babinet’s Principle in Electromagnetics Fields Assuming that (M, M’)=(J, J’), we transform (6.4a), (6.4b), and (6.5) to where Zσ is defined as ** 𝑗𝜔𝜇 = 𝑗𝜔𝜀+𝜎 ∗ 𝑗𝜔𝜇 𝑗𝜔𝜀+𝜎 = 𝑗𝜔𝜀+𝜎 ∗ Zσ 2 However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design.

6.2.1 Babinet’s Principle in Electromagnetics Fields Comparing Eqs. (6.2), (6.3a), and (6.3b) to Eqs. (6.9), (6.8a), and (6.8b), respectively Thus, Once the electric and magnetic fields are obtained for one of the structures, the fields for remaining structure can be calculated using Eq. (6.11).  Babinet’s principle in electromagnetic fileds 𝐻 1 = −𝐸 2 𝐸 1 = Zσ 2 * 𝐻 2 (Z > 0) 𝐻 1 = 𝐸 2 𝐸 1 = − Zσ 2 * 𝐻 2 (Z < 0) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design.

6.2.1 Babinet’s Principle in Electromagnetics Fields Mushiake’s original work (To find input impedance relationship btw. Complementary structures at feed point) - Arbitrarily shaped slot antenna shown in Figure 6.1a - Denote the voltage and the current at the feed point as V slot and I slot, respectively. - The Voltage source can be replaced with an equivalent magnetic current source, which surrounds a conducting plate connecting points u and v. However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. Equivalent Magnetic Current soruce Complementary Plate structure slot plate 𝑉 𝑠𝑙𝑜𝑡 𝐼 𝑠𝑙𝑜𝑡 Magnetic Current source 𝑉 𝑝𝑙𝑎𝑡𝑒 𝐼 𝑝𝑙𝑎𝑡𝑒 Figure 6.1 Slot antenna and its complementary plate antenna: (a) slot antenna with a voltage source, (b) slot antenna with a magnetic current source, and (c) complementary plate antenna with a voltage source.

6.2.1 Babinet’s Principle in Electromagnetics Fields Using Babinet’s principle in electromagnetic fields, we relate V slot and V plate to I plate and I slot, respectively: The impedances for the slot and plate are defined as Thus 𝑍 𝑠𝑙𝑜𝑡 ∗ 𝑍 𝑝𝑙𝑎𝑡𝑒 = 𝑉 𝑠𝑙𝑜𝑡 ∗ 𝑉 𝑝𝑙𝑎𝑡𝑒 𝐼 𝑠𝑙𝑜𝑡 ∗ 𝐼 𝑝𝑙𝑎𝑡𝑒 = 1 2 𝐼 𝑝𝑙𝑎𝑡𝑒 ∗ 𝑍 𝜎 2 2 𝐼 𝑠𝑙𝑜𝑡 𝐼 𝑠𝑙𝑜𝑡 ∗ 𝐼 𝑝𝑙𝑎𝑡𝑒 = 𝑍 𝜎 2 4 = ( 𝑍 𝜎 2 ) 2 However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. ∴

6.2.2 Self-Complementary Structure : A structure whose conducting areas are congruent to the non-conducting areas (free space) The input impedance for a self-complementary antenna can be calculated from Eq. (6.1). A relationship of Z slot =Z plate exists and hence Eq. (6.1) is However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. (6.16) Figure 6.3 Self-complementary antennas: (a) point-symmetric self-complementary antenna and (b) line-symmetric self-complementary antenna.

Verification Sinuous antenna : Self-Complementary structure - Design : using Ansys HFSS ADK. - Operating frequency : 0.8 GHz ~ 6 GHz 𝜙= (−1) 𝑝 ∙𝛼 𝑝 ∙sin 𝜋∙ln⁡( 𝑟 𝑅 𝑝 ) ln⁡( 𝜏 𝑝 ) ±𝛿 air 𝑅 𝑝+1 <r< 𝑅 𝑝 𝜏 𝑝 = 𝑅 𝑝+1 𝑅 𝑝 plate 𝑅 1 = 𝜆 𝐿 /4 𝛼 1 +𝛿 𝛿= 𝜆 4𝑟 − 𝛼 𝑝

Input impedance  Because practical antennas have finite structures. Reflection coefficient (S11)

6.2.2 Self-Complementary Structure Four port Self-Complementary antenna - Fig 6.4(a) : four star-connected sources excite the plate arms. - Fig 6.4(b) : four ring-connected sources excite the slot arms. * Mushiake finds that Z plate (and Z slot) for the four-port case differs from that for the two-port case [3]: However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. Complementary Plate structure Figure 6.4 Self complementary antenna with four ports: (a) star-connected sources, (b) ringconnected sources, and (c) star (solid line) and ring (dotted line) connections [Ref. 3].

6.2.2 Self-Complementary Structure Using N-polyphase alternating current theory : represents the relationships between the voltages and currents of the star and ring connections : Input impedance of n-port self-complementary antennas However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. 𝑍 𝑝𝑙𝑎𝑡𝑒 = 𝑍 0 /4 sin⁡( 𝜋 𝑛 )

6.2.3 Principle of Similitude for Electromagnetic Fields Figure 6.5 : shows two antenna systems. (operating at frequencies f 1 and f 2, with a current J0 flowing on each antenna.) Antenna system 1 : ε, μ, σ1, 𝑙1. antenna system 2 : ε, μ, σ2, 𝑙2. Note that antenna 2 is geometrically similar to antenna 1, with 𝑙2 ≡K𝑙1. Maxwell equation Antenna system 1 However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. (6.25) Antenna system 1 (6.26) Antenna system 2 (6.27) Antenna system 2 (6.28) Figure 6.5 Similar antennas: (a) antenna system 1 operating at f 1 and (b) antenna system 2 operating at frequency f 2.

6.2.3 Principle of Similitude for Electromagnetic Fields For antenna system 2, we adopt new coordinates (x, y, z ) that are K times larger than the original coordinates (x, y, z ). Under the new coordinates, we denote the electric and magnetic fields for antenna system 2 as E’ and H’, respectively. Then, Maxwell’s equations with a new operational element ∇’ are If the following conditions are satisfied, Kω2 = ω1 (6.31) Kσ2 = σ1 (6.32) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. (6.29) (6.30)

6.2.3 Principle of Similitude for Electromagnetic Fields Eqs. (6.29) and (6.30) can be written Equations (6.33) and (6.34) for antenna system 2 are of the same form as Eqs. (6.25) and (6.26) for antenna system 1, respectively. This means that the electric and magnetic fields for antenna systems 1 and 2 are similar to each other. (principle of similitude for electromagnetic fields) However, an analysis technique or a CAD tool, by itself, cannot generate an antenna design. Antenna system 1 Antenna system 2

Thank you for giving your attention