1. Hanyang University 1/16 Antennas & RF Devices Lab. MODERN ANTENNA HANDBOOK by CONSTANTINE A.BALANIS ch. 6.2.4 – 6.3.2 Kim Sung Peel

2. Hanyang University 2/16 Contents - 6.2.4 Self-Similar Structures 6.3 Analysis Techniques - 6.3.1 Analysis Based on Method of Moments - 6.3.2 Analysis Based on Finite-Difference Time-Domain Method

3. Hanyang University 3/16 6.2.4 Self-Similar Structures Consider an antenna whose size is increased(or decreased) by a factor K. →If the resulting structure is the same as the original structure, the original antenna structure is called a self-similar structure. → An antenna that has a self-similar structure is called a self-similar antenna. Electric and magnetic fields for antenna system (a) and (b) are similar to each other.

4. Hanyang University 4/16 6.2.5 Transformation of Radial Distance to Angles Original antenna: New antenna: →If the antenna conductors are infinite, change of scale is equivalent to a rotation [4]: where C is the rotation angle around the z-axis. (K depends on C)

5. Hanyang University 5/16 6.2.5 Transformation of Radial Distance to Angles ① About C: Using, Eq. (6.42) is written or →A general solution to Eq. (6.44) :

6. Hanyang University 6/16 6.2.5 Transformation of Radial Distance to Angles 6.3 Analysis Techniques Impedance changes when the infinite structure is made finite. The truncation of the infinite structure causes the radiation pattern and gain to different from those obtained for the infinite structure. Generally, it is difficult to analytically determine the effects of the finite structure on the impedance, radiation pattern, and gain. →This difficulty is overcome using numerical methods. There are two analysis techniques. ① Method of Moments(MoM): suitable for analyzing an arbitrarily shaped finite-length wire antenna ② Finite-difference time-domain method(FDTDM): suitable for analyzing a finite-size plate antenna with and without conducting cavity.

7. Hanyang University 7/16 6.3.1 Analysis Based on Method of Moments(MoM) s’: a point on the wire I(s’): the current at point s’ Radiation field is suitable for linearly polarized(LP) antenna. This form can be transformed to a form suitable for a circularly polarized(CP) antenna: RHCP wave component LHCP wave component →Axial ratio:

8. Hanyang University 8/16 6.3.1 Analysis Based on Method of Moments(MoM) After transforming the radiation field into, using, →the absolute gain: The gains relative to right-hand and left-hand circularly polarized isotropic antennas are written respectively as: According to axial ratio:

9. Hanyang University 9/16 6.3.1 Analysis Based on Method of Moments(MoM) For obtaining the current, we derive the electric field generated from the current and use the boundary condition that the tangential component of the electric field on the perfectly conducting wire surface is zero. → The boundary condition for the antenna shown in Figure 6.6: where is the tangential component of an incident electric field on the wire. To obtain the current I(s’) in Eq. (6.50), the MoM is employed with,

10. Hanyang University 10/16 6.3.1 Analysis Based on Method of Moments(MoM) Eq. (6.50) becomes

11. Hanyang University 11/16 6.3.2 Analysis Based on Finite-Difference Time-Domain Method(FDTDM) FDTDM is used to obtain the electric and magnetic fields within an analysis space that surrounds the antenna. →These fields are used to calculate the antenna characteristics. Cartesian coordinates: (x, y, z) are denoted by integers (i, j, k), based on the definition Its derivative with respect to x are expressed as(central finite difference)

12. Hanyang University 12/16 6.3.2 Analysis Based on Finite-Difference Time-Domain Method(FDTDM) Derivatives of the E- and H-fields with respect to time t are expressed as(central finite difference) → → These are transformed to → → → →

13. Hanyang University 13/16 6.3.2 Analysis Based on Finite-Difference Time-Domain Method(FDTDM) Finite-difference form for the x-component of Eq.(6.62): → The two other components of the electric field and the three components of the magnetic field are similarly formulated. The solutions to Maxwell’s equations, E and H, are obtained by iterating Eq.(6.61) & (6.62) with n=1, 2,… until these fields become constant. ☞ This method is FDTDM

14. Hanyang University 14/16 6.3.2 Analysis Based on Finite-Difference Time-Domain Method(FDTDM) The current along the antenna conductor, I(t), can be calculated by integrating the magnetic field obtained with the FDTDM around the antenna conductor. (Ampere’s law: ) The voltage between two points, V(t), can be calculated by taking product of the electric field obtained with the FDTDM and the distance between the two points.

15. Hanyang University 15/16 6.3.2 Analysis Based on Finite-Difference Time-Domain Method(FDTDM) The radiation field is calculated using the equivalence theorem(Fourier transform)[12]. For this, ① the time-domain electric- and magnetic-current densities on the closed surface of the analysis space are defined as and respectively. r’: position vector from the coordinate origin to the point located on the closed surface. : the outward unit vector normal to the closed surface. ② These current densities are Fourier-transformed to (Frequency domain) ③ Vectors N and L are defined as ∴ Thus the input impedance and the radiation field are calculated.

16. Hanyang University 16/16 Thank you for your attention

17. Hanyang University 17/16 Weierstrass function