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

ABSTRACT
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 Sparameters, effective permeability, loss rate, bandwidth, etc. The circuit and electromagnetic simulation results show an excellent agreement as well as that of theory.

KEYWORD
Metamaterial , Split Ring Resonator , Negative Permeability.

Ⅰ. Introduction
The realization of media having negative permittivity and permeability, so called lefthanded metamaterial, became feasible after 1991 when Pendry et al. [1] 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 lefthanded characteristics [25]. 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 [1]. 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 lefthanded materials.
Ⅱ. Novel Modeling of a Ring Resonator (SRRs)
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 zdirection with the electric and magnetic fields oriented in the x and ydirections, respectively. The radius of the loop is
r and the radius of the ring isr_{ring} . The side length of the unit cell isa .C is the value of the chip capacitor. It is inserted for resonance of the ring resonator which has some inductanceL originally. The total resistanceR of the ring resonator is given bywhere
R_{r} is the radiation resistance,R_{l} is the ohmic resistance, andR_{L} is an additionally loaded resistance which is 0 originally, but can be added to help the resonator to resonate weakly. WhenR_{L} →∞, the current cannot flow on the resonator and the resonator becomes inactive. AsR_{L} is varied, theQ value of the resonator is changed and a wider operating bandwidth of the negative permeability may be achieved. As the magnetic fieldH _{0} is passes through the ring resonator, a voltageV_{emf} will be induced as shown in Eq. (2) based on Faraday’s law.Accordingly, the current
I can be easily determined as Eq. (3) simply by dividingV_{emf} with the resonator impedanceZ (ω) (series sum ofR ,L , andC ).The magnetization
M and the relative effective permeability can be obtained as (4) and (5), respectively.where
ω is angular frequency,ω _{0} is the resonant frequency,is magnetic dipole moment
Q is the value ofQ factor written as Eq. (6),χ_{m} is magnetic susceptibility, andμ _{0} is the permeability in free space (μ _{0}=4π ×10^{―7}[H/m]).The parameter
L in Eq. (6) indicates the selfinductance 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 crosssectional shape of a specific transmission line, we can obtain the effective inductance of the transmission line.For an instance,
g for the parallel plate waveguide transmission line with widthW and heighth is given byh /W . Now, the series impedance of the unit cell including the ring resonator can be expressed aswhere
d is 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 withL_{eq}·d ,C_{eq}·d and a parallelG' ,C' ,L' resonator in the series branch. The RH transmission line may be alternatively represented byZ _{c} andkd . The total series impedance of the equivalent circuit in Fig. 2 can be written asThe value of parameters
G' ,C' ,L' in Eq. (9) can be obtained as Eq. (10) by comparing the Eqs. (8) and (9).Ⅲ. Simulation Results and Discussion
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 (0.005λ_{0} ). The radius of the loop (r ) and the ring (r_{ring} ) are 5 cm and 1 mm, respectively. The ohmic resistanceR_{l} is 0.058 Ω and the radiation lossR_{r} is negligible (R_{L} =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 EMsimulated magnitude and phase of the Sparameters for the ring resonator when the TEM wave propagates in the zdirection as depicted in Fig. 1. The circuit based on Fig. 2 and the EMsimulated results show excellent agreement. The extracted real and imaginary parts of the effective permeability based on the Sparameter [6] 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 theQ values which can be controlled by varying the extra resistanceR_{L} . For this case, the ring resonators have been designed to have the specific value for effective permeability (μ_{eff} = ―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 (
η_{l} is 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
Q is lowered by inserting extra resistanceR_{L} ’s. Furthermore, the widest operating bandwidth of the effective permeability is achieved when the resonator has the specific value ofQ (Q =26.8,R_{L} =0.8 Ω). The widest bandwidth is only effective when the effective permeability is designed to have the value of ―1. The circuit and EMsimulated results are again in excellent agreement.Ⅳ. Conclusion
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 narrowbanded 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 Sparameters: (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).