30 ANALYSIS AND MODELING OF RING RESONATORS 1 G 2 G l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1,2=0 V 1 V 2 I 1 I 2 I V (a) 1 G 2 G l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1,2=0 V 1 V 2 I 1 I 2 I V r (b) FIGURE 2.15 The configurations of one-port (a) square and (b) annular ring resonators [10]. TABLE 2.4 A Comparison of Table 2.3 and the Theoretical Results from (upper) the Transmission Line Method and (lower) the Magnetic-Wall Model Frequency Error (%) Circuit n = 1 n = 2 1 0.79 0.60 2 0.28 0.49 3 0.56 0.28 Frequency Error (%) Circuit n = 1 n = 2 1 0.78 0.89 2 0.07 0.37 2 0.63 0.38 is considered to be a transmission line. z 1 and z 2 are the coordinates corre- sponding to sections l 1 and l 2 , respectively. The ring is fed by the source voltage V at somewhere with z 1,2 < 0. The positions of the zero point of z 1,2 and the voltage V are arbitrarily chosen on the ring. For a lossless transmission line, the voltages and currents for the two sections are given as follows: (2.65a) (2.65b) where V + o e -jbz 1,2 is the incident wave propagating in the +z 1,2 direction, V + o G 1,2 (0)e jbz 1,2 is the reflected wave propagating in the -z 1,2 direction, G 1,2 (0) is the reflection coefficient at z 1,2 = 0, and Z 0 is the characteristic impedance of the ring. When a resonance occurs, standing waves set up on the ring. The shortest length of the ring resonator that supports these standing waves can be obtained from the positions of the maximum values of these standing waves. These positions can be calculated from the derivatives of the voltages and currents in Equation (2.65). The derivatives of the voltages are (2.66) Letting , the reflection coefficients can be found as G 1,2 (0) = 1 (2.67) Substituting G 1,2 (0) = 1 into Equation (2.65), the voltages and currents can be obtained as (2 .68a) (2.68b) Based on Equation (2.68), the absolute values of voltage and current stand- ing waves on each section l 1 and l 2 are shown in Figure 2.16. Inspecting Figure 2.16, the standing waves repeat for multiples of l g /2 on the each section of the ring. Thus, to support standing waves, the shortest length of each section on the ring has to be l g /2, which can be treated as the fundamental mode of the ring. For higher order modes, for n = 1, 2, 3, . . . (2.69) ln g 12 2 , = l Iz jV Z z o o 12 12 12 2 ,, , sin () =- () + b Vz V z o12 12 12 2 ,, , cos () = () + b ∂ () ∂ = = Vz z z 12 12 12 0 12 0 ,, , , ∂ () ∂ =- - () () + - Vz z jV e e o jz jz 12 12 12 12 12 12 0 ,, , , ,, b bb G Iz V Z ee o o jz jz 12 12 12 12 12 0 ,, , ,, () =- () () + - bb G Vz Ve e o jz jz 12 12 12 12 12 0 ,, , ,, () =+ () () + - bb G TRANSMISSION-LINE MODEL 31 where n is the mode number. Therefore, the total length of the square ring resonator is l = l 1 + l 2 = nl g (2.70) or in terms of the annular ring resonator with a mean radius r as shown in Figure 2.15b, l = nl g = 2pr (2.71) Equation (2.70) shows a general expression for frequency modes and may be applied to any configuration of microstrip ring resonators, including those shown in [28, 29]. 2.4.7 An Error in Literature for One-Port Ring Circuit In [11], one- and two-port ring resonators show different frequency modes. For a one-port ring resonator, as shown in Figure 2.17a, the frequency modes are given as n = 1, 2, 3, . . . (2.72a) (2.72b) f nc r o e ff = 4pe 2 2 p l rn g = 32 ANALYSIS AND MODELING OF RING RESONATORS l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1,2=0 V 1 V 2 I 1 I 2 I V 22 () Vz 22 () Iz -z 2 11 () Vz 11 () Iz 2 g - l g l - -z 1 -z 1 =0 g l - 2 g - l -z 2 =0 FIGURE 2.16 Standing waves on each section of the square ring resonator [10]. where f o is the resonant frequencies. For the two-port ring resonator, as shown in Figure 2.17b, the frequency modes are n = 1, 2, 3, . . . (2.73a) (2.73b) However, in Section 2.4.6, the one-port ring resonator has the same frequency modes given in Equation (2.71) as those of the two-port ring resonator given in Equation (2.73a). The results can also be investigated by EM simulation performed by the IE3D electromagnetic simulator based on the method of moment [30]. The ring resonators in Figure 2.17 are designed at fundamental f nc r o eff = 2pe 2plrn g = TRANSMISSION-LINE MODEL 33 Y F X : max V : 0 = I : 0 = V : max I (a) Y F X : max V : 0 = I : 0 = V : max I (b) FIGURE 2.17 Simulated electrical current standing waves for (a) one- and (b) two- port ring resonators at n = 1 mode [10]. mode at 2 GHz with dielectric constant e r = 10.2 and thickness h = 50 mil. As seen from the simulation results in Figure 2.17, both exhibit the same electri- cal current flows, which are current standing waves. Therefore, both one- and two-port ring resonators have the same frequency modes as given in Equa- tions (2.71) or (2.73a). 2.4.8 Dual Mode The dual mode is composed of two degenerate modes or splitting resonant frequencies that may be excited by perturbing stubs, notches, or asymmetrical feed lines. The dual mode follows from the solution of Maxwell’s equations for the magnetic-wall model of the ring resonator in Equations (2.3)–(2.5) and (2.8)–(2.10). However, the ring resonator with a perturbing stub or notch at F=45°, 135°, 225°, or 315° generates the dual mode only for odd modes. Inspecting Equations (2.3)–(2.5) and (2.8)–(2.10), they cannot explain why the dual mode only happens for odd modes instead of even modes when the ring resonator has a perturbing stub or notch at F=45°, 135°, 225°, or 315°. Also, the magnetic-wall model cannot explain the dual mode of the ring resonator with complicate boundary conditions. This dual-mode phenomenon may be explained more simply and more generally using the transmission-line model of Section 2.4.6, which describes the ring resonator as two identical l g /2 res- onators connected in parallel. As seen in Figure 2.17, two identical current standing waves are established on the ring resonator in parallel. If the ring does not have any perturbation and is excited by symmetrical feed lines, two identical resonators are excited and produce the same frequency response, which overlap each other. However, if one of the l g /2 resonators is perturbed out of balance with the other, two different frequency modes are excited and couple to each other. To investigate the dual-mode behavior, a perturbed square ring resonator is simulated in Figure 2.18. The perturbed square ring designed at fundamental mode of 2 GHz is fabricated on a RT/Duroid 6010.2 e r = 10.2 substrate with a thickness h = 25 mil. Figure 2.18 shows the simulated electric currents on the square ring res- onator with a perturbing stub at F=45° for the n = 1 and the n = 2 modes. For the n = 1 mode, one of l g /2 resonators is perturbed so that the two l g /2 resonators do not balance each other. Thus, two splitting different resonant frequencies are generated. Figures 2.18a and 2.18b show the simulated elec- trical currents for the splitting resonant frequencies. Figure 2.19 illustrates the measured S21 confirming the splitting frequencies for the n = 1 mode around 2 GHz. Furthermore, for the n = 2 mode, Figure 2.18c shows the perturbing stub located at the position of zero voltage, which is a short circuit. Therefore, the perturbed stub does not disturb the resonator and both l g /2 resonators balance each other without frequency splitting. Measured results in Figure 2.19 has confirmed that the resonant frequency at the n = 2 mode of 4 GHz is not affected by the perturbation. 34 ANALYSIS AND MODELING OF RING RESONATORS 2.5 RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C The basic operation of the ring resonator based on the magnetic-wall model was originally introduced by Wolff and Knoppik [1]. In addition, a simple mode chart of the ring was developed to describe the relation between the physical ring radius and resonant mode and frequency [4].Although the mode RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C 35 Input Output : max V : 0= I : 0 = V : max I (a) : max V : 0 = I : 0= V : max I Input Output (b) Input Output : max V : 0 = I I : 0 = V : max (c) FIGURE 2.18 The simulated electrical currents of the square ring resonator with a perturbed stub at F=45° for (a) the low splitting resonant frequency of n = 1 mode and (b) high splitting resonant frequency of mode n = 1, and (c) mode n = 2 [10]. chart of the magnetic-wall model has been studied extensively, it provides only a limited description of the effects of the circuit parameters and dimensions. A further study on a ring resonator using the transmission-line model was introduced in Section 2.4. The transmission-line model used a T-network in terms of equivalent impedances to analyze a ring circuit. Although this model could predict the behavior of a ring resonator well, it could not provide a straightforward circuit view, such as equivalent lumped elements G, L, and C for the ring circuit. 2.5.1 Equivalent Lumped Elements for Closed- and Open-Loop Microstrip Ring Resonators [12] As seen in Figure 2.20, the two-port network with an open circuit at port 2 (i 2 = 0) models a one-port network to find the equivalent input impedance through ABCD matrix and Y parameters operations [31]. The ring resonator is divided by input and output ports on arbitrary posi- tions of the ring with two sections l 1 and l 2 to form a parallel circuit. For this parallel circuit, the overall Y parameters are given by (2.74) By setting i 2 to zero, the input impedance Z ic of the ring can be found as follows: Y Y Y Y Yl l Yl l Yl l Yl l o o o o 11 12 12 22 12 2 12 12 È Î Í ˘ ˚ ˙ = () + () [] - () + () [] È Î Í - () + () [] () + () [] ˘ ˚ ˙ coth coth csc csc csc cot cot gg gg gg ggcsch h hh hh 1 36 ANALYSIS AND MODELING OF RING RESONATORS 12345 Frequency (GHz) -80 -60 -40 -20 Magnitude (dB) S21 1n = 2n = FIGURE 2.19 The measured results for modes n = 1 and 2 of the square ring resonator with a perturbed stub at F=45° [10]. (2.75) where l g = l/2 = l g /2. Using some assumptions and derivations for al g and bl g [12], the input impedance Z ic can be approximated as (2.76) For a general parallel G, L, C circuit, the input impedance is [32] (2.77) Comparing Equation (2.76) with Equation (2.77), the conductance of the equivalent circuit of the ring is (2.78a) and the capacitance of the equivalent circuit of the ring is (2.78b) GZ co o = pw GlZ Z cgogo ==2aal Z GjC i = + 1 2 Dw Z Zl jl ic ag go @ + 2 1 a pwawD Z Y YY YY Z jl l lj l ic o gg gg = - = + () () () + () 22 11 22 12 21 2 1 tanh tan tanh tan ab ab RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C 37 Port 1 i 1 v 1 Port 2 i 2 v 2 w l 1 l 2 l=l 1 +l 2 Z ic = v 1 i 1 i 2 =0 i 1 v 1 Z ic = v 1 i 1 i 2 =0 Z ic 1 v 1 i l 2 l 1 l=l 1 +l 2 w i 2 v 2 Port 1 Port 2 FIGURE 2.20 The input impedance of two-port network of the closed-loop ring resonator [12]. (Permission from IEEE.) The inductance of the equivalent circuit of the ring can be derived from and is given by (2.78c) where G c , L c , and C c stand for the equivalent conductance, inductance, and capacitance of the closed-loop ring resonator. Figure 2.21 shows the equiva- lent lumped element circuit of the ring in terms G c , L c , and C c . Moreover, the unloaded Q of the ring resonator can be found from Equation (2.78) and the unloaded Q is (2.79) Figure 2.22a shows the configuration of open-circuited l g /2 microstrip ring resonators with annular and U shapes. As seen in Figure 2.22a, l 3 is the phys- QCG uc o c c g ==wpal LC coc = 1 2 w w 0 1= LC cc 38 ANALYSIS AND MODELING OF RING RESONATORS co c C L 2 1 w = oo c Z C w p = ic Z o g c Z G al = FIGURE 2.21 Equivalent elements G c , L c , and C o of the closed-loop ring [12]. (Permission from IEEE.) i 1 v 1 w i 2 v 2 l 3 C f C f C g w v 1 i 1 i 2 v 2 l 3 C f C f C g (a) oo o C L 2 1 w = oo o Z C w p 2 = io Z o g o Z G 2 la = (b) FIGURE 2.22 Transmission-line model of (a) the open-loop ring resonator and (b) its equivalent elements G o , L o , and C o [12]. (Permission from IEEE.) ical length of the ring, C g is the gap capacitance, and C f is the fringe capaci- tance caused by fringe field at the both ends of the ring. The fringe capaci- tance can be replaced by an equivalent length Dl [33]. Considering the open-end effect, the equivalent length of the ring is l 3 + 2Dl = l g /2 = l g for the fundamental mode. Through the same derivations in Section 2.5.1, the input impedance Z io of the open-loop ring can be approximated as (2.80) Comparing Equation (2.80) with Equation (2.77), the conductance, capaci- tance, and inductance of the equivalent circuit of the ring are (2.81) The equivalent circuit in terms of G o , L o , and C o is shown in Figure 2.22b. Moreover, the unloaded Q of the ring is given by (2.82) Inspecting the equivalent conductances, capacitances, and inductances of the closed- and open-loop ring resonators in Equations (2.78) and (2.81), the relations of the equivalent lumped elements G, L, C between these two rings can be found as follows: (2.83a) (2.83b) In addition, observing the Equations (2.79) and (2.82), the unloaded Q of the closed- and open-loop ring resonators are equal, namely Q uc = Q uo for the same attenuation constant (2.84) Equations (2.83a) and (2.84) sustain for the same losses condition of the closed- and the open-loop ring resonator. In practice, the total losses for the closed- and the open-loop ring resonator are not the same. In addition to the dielectric and conductor losses, the open-loop ring resonator has a radia- tion loss caused by the open ends [34]. Thus, total losses of the open-loop ring are larger than that of the closed-loop ring. Under this condition, Equations (2.83a) and (2.84) should be rewritten as follows: Q uc > Q uo and G c < 2G o (2.85) C C and L L co cc ==22 GG co = 2 for the same attenuation constant QCG uo o o o g ==wpal G Z C Z and L C ogoo o o oo == =al p w w22 1 2 ,, Z Zl jl io og gg = + a pwaw1 D RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C 39 [...]... Effect Square Ring U-Shape Open-loop Ring 2. 45 Ơ 10-3 0.508 2. 43 Ơ 10-3 0 .25 6 2. 45 Ơ 10-3 0.508 2. 43 Ơ 10-3 0 .25 6 4.17 2. 08 4.17 2. 08 1. 52 3.04 1. 52 3.04 2. 38 Ơ 10-3 0.495 2. 43 Ơ 10-3 0 .25 3 2. 36 Ơ 10-3 0.49 2. 42 Ơ 10-3 0 .25 2 4 .25 2. 12 4 .22 2. 1 1.55 3.1 1.54 3.07 Microstrip Dispersion When a radio frequency (RF) wave propagates down a microstrip line, both longitudinal and transverse currents are excited... (dB) Measured 3-dB Bandwidth BW3dB,mens (MHz) Measured Loaded Q Measured Unloaded Q Calculated Unloaded Q Open-loop Ring with the Curvature Effect Square Ring U-Shape Open-loop Ring 2 2 2 2 1.974 1.968 2. 03 2. 03 35.83 35.48 35.48 33.4 20 .5 21 20 .5 21 96 .29 97.87 93.65 95.36 96.99 93 .21 97.71 99.38 93.65 95.38 97.46 93 .21 TABLE 2. 8 Equivalent Elements for the Parameters: er = 10 .2, h = 10 mil, t = 0.7... expressed as [ 42] where a d = 27 .3 w h Ê 1 2p ac = 8.68Rs1 M ấ h hU 1+ + ậ 2pZ o h w eff pw eff dB unit length (2. 99b) 1 2p Ê w h Ê 2 ac = 8.68Rs1 M ấ h hV + 1 + 2pZo h ậ weff pweff dB unit length (2. 99c) w h 2 ẩ weff ph ấ 8.68Rs1 weff h hV ac = + + 1 + ậ 2pZo h h weff weff pweff + 0.94 ẻ 2h -2 weff 2 ẩ weff ẽ ấ á Ơè + In 2pe + 0.94 dB unit length ậ 2h p ẻ ể h (2. 99d) 48 ANALYSIS AND MODELING... transformation: ẩD ẩY11 Y 12 = 1 B Y21 Y 22 Z0 -1 ẻ ẻB BC - AD B A B (2. 103) The Y-matrices for each half-section are then added, and the resultant matrix is transformed into an ABCD-matrix in accordance with the following transformation: DISTRIBUTED TRANSMISSION-LINE MODEL 49 FIGURE 2. 27 Flowchart for estimation of S-parameters -Y 22 -1 ẩA B = 1 ẩ ẻC D Y21 ẻ(Y12Y21 - Y11Y 22 ) -Y11 (2. 104) The overall... modes of a novel meander loop resonator, IEEE Microwave Guided Wave Lett., Vol 5, pp 3713 72, November 1995 [30] IE3D Version 9.16, Zeland Software Inc., Fremont, CA, July 20 02 [31] C.-C Yu and K Chang, Transmission-line analysis of a capacitively coupled microstrip -ring resonator, IEEE Trans Microwave Theory Tech., Vol 45, pp 20 1 820 24, November 1997 [ 32] D M Pozar, Microwave Engineering, 2nd ed New York:... ANALYSIS AND MODELING OF RING RESONATORS (a) (b) (c) (d) FIGURE 2. 23 Layouts of the (a) annular, (b) square, (c) open-loop with the curvature effect, and (d) U-shaped open-loop ring resonators [ 12] (Permission from IEEE.) 2. 5 .2 Calculated and Experimental Results To verify the calculations of the unloaded Q and G, L, C of the closed- and open-loop ring resonators [ 12] , four congurations of the closed- and. .. Open-loop Ring 2 2 2 2 1.963 1.964 1.977 1.983 Annular Ring Resonators Designed Resonant Frequency (GHz) Measured Resonant Frequency (GHz) Measured Insertion Loss Lmeas (dB) Measured 3-dB Bandwidth BW3dB,means(MHz) Measured Loaded Q Measured Unloaded Q Calculated Unloaded Q 32. 66 31.33 32. 3 33. 12 19 19.5 19 19.5 103. 32 105.78 103.35 100. 72 103.53 1 02. 41 104.05 106.64 103.35 101.69 103.98 1 02. 41 TABLE 2. 6... for openand close -ring microstrip resonators, IEEE Trans Microwave Theory Tech., Vol MTT- 32, pp 405409, April 1984 [8] K Chang, T S Martin, F Wang, and J L Klein, On the study of microstrip ring and varactor-tuned ring circuits, IEEE Trans Microwave Theory Tech., Vol MTT35, pp 128 8 129 5, December 1987 [9] G K Gopalakrishnan and K Chang, Bandpass characteristics of split-modes in asymmetric ring resonators,... capacitance for microstrip gaps and steps, IEEE Trans Microwave Theory Tech., Vol MTT -20 , pp 729 733, November 19 72 [23 ] A Ralston and P Rabinowitz, A First Course in Numerical Analysis, McGrawHill, New York, 1965 [24 ] P Silvester and P Benedek, Equivalent capacitance of microstrip open circuits, IEEE Trans Microwave Theory Tech., Vol MTT -20 , pp 511516, August 19 72 [25 ] R Garg and I J Bahl, Microstrip discontinuities,... 1.5763 P ( f ) = P1P2 [(0.1844 + P3 P4 )10 fh] (2. 87) with ẩ w 0. 525 P1 = 0 .27 488 + 0.6315 + - 0.065683e -8.7513w / h 20 (1 + 0.157 fh) h ẻ (2. 88) P2 = 0.33 622 [1 - e -0.03442er ] (2. 89) [ P3 = 0.0363e -4.6w / h 1 - e ( - fh /3.87) [ P4 = 1 + 2. 751 1 - e ( - er /15.916) 8 4.97 ] (2. 90) ] (2. 91) where f is the frequency in GHz; w and h are the microstrip width and height in cm, respectively; er is the relative . ANALYSIS AND MODELING OF RING RESONATORS 1 G 2 G l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1 ,2= 0 V 1 V 2 I 1 I 2 I V (a) 1 G 2 G l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1 ,2= 0 V 1 V 2 I 1 I 2 I V r (b) FIGURE 2. 15. z o 12 12 12 2 ,, , cos () = () + b ∂ () ∂ = = Vz z z 12 12 12 0 12 0 ,, , , ∂ () ∂ =- - () () + - Vz z jV e e o jz jz 12 12 12 12 12 12 0 ,, , , ,, b bb G Iz V Z ee o o jz jz 12 12 12 12 12 0 ,,. RING RESONATORS l 1 l 2 l=l 1 +l 2 -z 1 -z 2 z 1 ,2= 0 V 1 V 2 I 1 I 2 I V 22 () Vz 22 () Iz -z 2 11 () Vz 11 () Iz 2 g - l g l - -z 1 -z 1 =0 g l - 2 g - l -z 2 =0 FIGURE 2. 16 Standing waves on each