Simplified Modeling of Ring Resonators and Split Ring Resonators Using Magnetization
- Author: Jeon Dongho, Lee Bomson
- Organization: Jeon Dongho; Lee Bomson
- Publish: Journal of electromagnetic engineering and science Volume 13, Issue2, p134~136, 30 June 2013
This paper examines various aspects of the electromagnetic responses of the ring resonator located in the transverse electromagnetic cell. In addition, an equivalent circuit for the ring resonator is proposed and analyzed based on the electromagnetic phenomenon of the resonator. The equivalent circuit was simply modeled based on the concept of magnetization. A method for achieving a wider operating bandwidth of the negative permeability is provided. The ring resonator with its resonant frequency of 13.56 MHz was designed and its characteristics were examined in terms of S-parameters, effective permeability, loss rate, bandwidth, etc. The circuit and electromagnetic simulation results show an excellent agreement as well as that of theory.
Metamaterial , Split Ring Resonator , Negative Permeability.
The realization of media having negative permittivity and permeability, so called left-handed metamaterial, became feasible after 1991 when Pendry et al.  proposed a new method that used an array of thin wires and split ring resonators (SRRs). A number of approaches followed in many aspects in an effort to realize similar left-handed characteristics [2-5]. However, most of these showed high losses and narrow operating bandwidth (having a very sharp slope near the resonant frequency). In fact, the problem of SRRs has seldom been approached seriously from an engineering viewpoint, although many trials have been made. Basically, SRRs are similar to the ring resonator in their characteristics. In this paper, we model the ring resonator (SRRs) starting from the definition of a magnetic dipole moment and leading to a useful equivalent circuit. The mechanism of SRRs is explained with more familiar terms than in . In addition, we provide a method of realizing the negative permeability over a large bandwidth. It is believed that the presented modeling will provide significant convenience and flexibility for the realization of the left-handed materials.
We have modeled the ring resonator using an equivalent circuit. The dimensions of the ring resonator and its orientation with respect to the given transverse electromagnetic (TEM) wave are depicted in Fig. 1. The wave travels in the z-direction with the electric and magnetic fields oriented in the x- and y-directions, respectively. The radius of the loop is
rand the radius of the ring is rring. The side length of the unit cell is a. Cis the value of the chip capacitor. It is inserted for resonance of the ring resonator which has some inductance Loriginally. The total resistance Rof the ring resonator is given by
Rris the radiation resistance, Rlis the ohmic resistance, and RLis an additionally loaded resistance which is 0 originally, but can be added to help the resonator to resonate weakly. When RL→∞, the current cannot flow on the resonator and the resonator becomes inactive. As RLis varied, the Qvalue of the resonator is changed and a wider operating bandwidth of the negative permeability may be achieved. As the magnetic field H0 is passes through the ring resonator, a voltage Vemfwill be induced as shown in Eq. (2) based on Faraday’s law.
Accordingly, the current
Ican be easily determined as Eq. (3) simply by dividing Vemfwith the resonator impedance Z(ω) (series sum of R, L, and C).
Mand the relative effective permeability can be obtained as (4) and (5), respectively.
ωis angular frequency, ω0 is the resonant frequency,
is magnetic dipole moment
Qis the value of Qfactor written as Eq. (6), χmis magnetic susceptibility, and μ0 is the permeability in free space ( μ0=4 π×10―7[H/m]).
Lin Eq. (6) indicates the self-inductance of the proposed ring resonator. Now, we want to devise an equivalent circuit for the structure in Fig. 1. By multiplying the derived relative effective permeability with the geometrical factor ( g) which is determined for the cross-sectional shape of a specific transmission line, we can obtain the effective inductance of the transmission line.
For an instance,
gfor the parallel plate waveguide transmission line with width Wand height his given by h/ W. Now, the series impedance of the unit cell including the ring resonator can be expressed as
dis the physical length of the transmission line unit cell. A close examination of (8) leads to the equivalent circuit depicted in Fig. 2. The equivalent circuit consists of a right handed (RH) transmission line with Leq·d, Ceq·dand a parallel G', C', L'resonator in the series branch. The RH transmission line may be alternatively represented by Zc and kd. The total series impedance of the equivalent circuit in Fig. 2 can be written as
The value of parameters
G', C', L'in Eq. (9) can be obtained as Eq. (10) by comparing the Eqs. (8) and (9).
The ring resonator used in the electromagnetic (EM) simulation is made of copper and is designed at 13.56 MHz. The unit cell size (
a) is 12 cm (0.005 λ0). The radius of the loop ( r) and the ring ( rring) are 5 cm and 1 mm, respectively. The ohmic resistance Rlis 0.058 Ω and the radiation loss Rris negligible ( RL=0 Ω assumed). The value of capacitance for the resonance is 533 pF. Fig. 3(a) and (b) show the circuit (based on Fig. 2) and EM-simulated magnitude and phase of the S-parameters for the ring resonator when the TEM wave propagates in the z-direction as depicted in Fig. 1. The circuit based on Fig. 2 and the EM-simulated results show excellent agreement. The extracted real and imaginary parts of the effective permeability based on the S-parameter  and theory (4) are shown in Fig. 4(a) and (b), respectively. The excellent agreement validates this modeling. In Fig. 5(a), the real part of the effective permeability is shown with respect to the Qvalues which can be controlled by varying the extra resistance RL. For this case, the ring resonators have been designed to have the specific value for effective permeability ( μeff= ―1) of 13.56 MHz (each case has different resonant frequencies).
Fig. 5(b) shows the loss rate of the ring resonator with different values of
Q. The loss rate is defined as Eq. (11).
Each case shows that the highest loss rate is shown at the resonant frequency, but loss rates are very low at the operating frequency of 13.56 MHz (
ηlis less than 2.5% for each case).
Comparison of Figs. 4(a) and 5(a) shows that the resonance occurs less sensitively as the value of
Qis lowered by inserting extra resistance RL’s. Furthermore, the widest operating bandwidth of the effective permeability is achieved when the resonator has the specific value of Q( Q=26.8, RL=0.8 Ω). The widest bandwidth is only effective when the effective permeability is designed to have the value of ―1. The circuit and EM-simulated results are again in excellent agreement.
The effective permeability of a ring resonator (or SRR) has been formulated based on the concept of magnetization. Its equivalent circuit has also been proposed and analyzed with necessary comparisons. The circuit and EM simulated results are in excellent agreement. The drawback of the narrow-banded SRRs can be significantly ameliorated based on the proposed modeling and formulations. With this modeling, the problem of synthesizing the effective medium can be engineered more systematically.
[Fig. 1.] Structure of a ring resonator.
[Fig. 2.] Equivalent circuit of ring resonator.
[Fig. 3.] Simulated S-parameters: (a) magnitude, (b) phase. EM =electromagnetic.
[Fig. 4.] Extracted effective permeability at 13.56 MHz: (a) real, (b) imaginary. EM=electromagnetic.
[Fig. 5.] Circuit/electromagnetic (EM)-simulated μeff and ηl with respect to RL. (a) Re[ μeff](varing RL from 0.15 to 500 Ω), (b) Loss rate, ηl (varing RL from 0.15 to 500 Ω).