Microwave Ring Circuits and Related Structures phần 2 doc

Therefore, the total length of the square ring 2.4.7 An Error in Literature for One-Port Ring Circuit In [11], one- and two-port ring resonators show different frequency modes.. This dua

2G

l 1

l 2 l=l 1 +l 2 -z 1

(a)

1G2G

l 1

l 2 l=l 1 +l 2 -z 1

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

is considered to be a transmission line z1and z2are the coordinates

corre-sponding to sections l1and l2, respectively The ring is fed by the source voltage

V at somewhere with z1,2 < 0 The positions of the zero point of z1,2 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+e -jbz

1,2 is the incident wave propagating in the +z1,2 direction,

V+G1,2(0)e jbz

1 , 2 is the reflected wave propagating in the -z1,2direction, G1,2(0) is

the reflection coefficient at z1,2= 0, and Z0is the characteristic impedance ofthe ring

When a resonance occurs, standing waves set up on the ring The shortestlength of the ring resonator that supports these standing waves can beobtained 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

Based on Equation (2.68), the absolute values of voltage and current

stand-ing waves on each section l1and l2are shown in Figure 2.16

Inspecting Figure 2.16, the standing waves repeat for multiples of lg/2 onthe each section of the ring Thus, to support standing waves, the shortestlength of each section on the ring has to be lg/2, which can be treated as thefundamental mode of the ring For higher order modes,

for n = 1, 2, 3, (2.69)

l1 2 n g

2, = l

o o

2, ( , )= - sin( , )

+b

where n is the mode number Therefore, the total length of the square ring

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 aregiven as

g

l-

where fois the resonant frequencies For the two-port ring resonator, as shown

in Figure 2.17b, the frequency modes are

V

: I= 0 : V= 0 : Imax

V

: I= 0 : V=0:

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 er= 10.2 and thickness h = 50 mil As

seen from the simulation results in Figure 2.17, both exhibit the same cal current flows, which are current standing waves Therefore, both one- andtwo-port ring resonators have the same frequency modes as given in Equa-tions (2.71) or (2.73a)

electri-2.4.8 Dual Mode

The dual mode is composed of two degenerate modes or splitting resonantfrequencies that may be excited by perturbing stubs, notches, or asymmetricalfeed lines The dual mode follows from the solution of Maxwell’s equationsfor 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 thedual mode only happens for odd modes instead of even modes when the ringresonator 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 resonatorwith complicate boundary conditions This dual-mode phenomenon may beexplained more simply and more generally using the transmission-line model

of Section 2.4.6, which describes the ring resonator as two identical lg/2 onators connected in parallel As seen in Figure 2.17, two identical currentstanding waves are established on the ring resonator in parallel If the ringdoes not have any perturbation and is excited by symmetrical feed lines, twoidentical resonators are excited and produce the same frequency response,which overlap each other However, if one of the lg/2 resonators is perturbedout of balance with the other, two different frequency modes are excited and couple to each other To investigate the dual-mode behavior, a perturbedsquare ring resonator is simulated in Figure 2.18 The perturbed square ringdesigned at fundamental mode of 2 GHz is fabricated on a RT/Duroid 6010.2

res-er = 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 lg/2resonators do not balance each other Thus, two splitting different resonantfrequencies 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 lg/2 resonatorsbalance 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

2.5 RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C

The basic operation of the ring resonator based on the magnetic-wall modelwas originally introduced by Wolff and Knoppik [1] In addition, a simplemode chart of the ring was developed to describe the relation between thephysical ring radius and resonant mode and frequency [4] Although the mode

RING EQUIVALENT CIRCUIT IN TERMS OF G, L, C 35

:Vmax

: I= 0 :V=0

(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 wasintroduced in Section 2.4 The transmission-line model used a T-network interms of equivalent impedances to analyze a ring circuit Although this modelcould 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

(i2= 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 secposi-tions l1and l2to form a parallel circuit For this

parallel circuit, the overall Y parameters are given by

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].

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

(2.79)

Figure 2.22a shows the configuration of open-circuited lg/2 microstrip ring

resonators with annular and U shapes As seen in Figure 2.22a, l3is the

phys-Q uc =wo C G c c =p alg

L c = 1wo2C c

w0=1 L C c c

c o c C

L 21

w

=

o o c Z C

w

p

=

ic Z

(a)

o o o

Z C

w p

2

io

Z

o g o

Z G

2

l a

=

(b)

FIGURE 2.22 Transmission-line model of (a) the open-loop ring resonator and (b) its

equivalent elements G , L , and C [12] (Permission from IEEE.)

ical length of the ring, C g is the gap capacitance, and C f is the fringe tance caused by fringe field at the both ends of the ring The fringe capaci-

capaci-tance can be replaced by an equivalent length Dl [33] Considering the open-end effect, the equivalent length of the ring is l3+ 2Dl = l g /2 = l gfor thefundamental mode

Through the same derivations in Section 2.5.1, the input impedance Z ioofthe open-loop ring can be approximated as

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 theclosed- and the open-loop ring resonator In practice, the total losses for theclosed- 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 ringare larger than that of the closed-loop ring Under this condition, Equations(2.83a) and (2.84) should be rewritten as follows:

(a) (b)

(c) (d)

FIGURE 2.23 Layouts of the (a) annular, (b) square, (c) open-loop with the

curva-ture 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

loop ring resonators [12], four configurations of the closed- and loop ring resonators as shown in Figure 2.23 were designed at the fundamen-tal mode of 2 GHz The ring resonators were fabricated for two differentdielectric constants: RT/Duriod 5870 with er = 2.33, h = 10 mil, and t = 0.7 mil

open-and RT/Duriod 6010.2 with er = 10.2, h = 10 mil, and t = 0.7 mil, where e ris the

relative dielectric constant, h is the substrate thickness, t is the foil thickness,

and D is the surface roughness

As seen in Tables 2.5 through 2.8, the measured unloaded Qs and

equiva-lent lumped elements of the closed- and open-loop rings show good ment with each other The largest difference between the measured and

agree-calculated unloaded Q shown in Table 2.7 for the closed-loop square ring

resonator is 5.7%

2.6 DISTRIBUTED TRANSMISSION-LINE MODEL

The transmission-line model described in Section 2.4 is straightforward andprovides reasonably accurate results for simple circuits at low frequencies Themethod lends itself to CAD implementation, and circuits loaded with solid-state devices and discontinuities along the rings can be analyzed However,the model is not accurate because the effects of the dispersive nature of the microstrip line and curvature of the ring resonator are neglected A moreaccurate distributed transmission-line model has been proposed to overcomethese problems [9, 35] The model includes the losses and can deal with mul-tiple devices, discontinuities, and feeds located at any place along the ring Thissection summarizes this method based on [35]

DISTRIBUTED TRANSMISSION-LINE MODEL 41

TABLE 2.5 Unloaded Qs for the Parameters: e r = 2.33, h = 10 mil, t = 0.7 mil,

w = 0.567 mm for a 60-ohms Line, D = 1.397 mm, and l g= 108.398 mm

Open-loop

TABLE 2.6 Equivalent Elements for the Parameters: er = 2.33, h = 10 mil, t = 0.7 mil,

w = 0.567 mm for a 60-ohms Line, D = 1.397 mm, and l g= 108.398 mm

TABLE 2.8 Equivalent Elements for the Parameters: er = 10.2, h = 10 mil, t = 0.7 mil,

w = 0.589 mm for a 30-ohms Line, D = 1.397 mm, and l g= 55.295 mm

TABLE 2.7 Unloaded Qs for the Parameters: e r = 10.2, h = 10 mil, t = 0.7 mil,

w = 0.589 mm for a 30-ohms Line, D = 1.397 mm, and l g= 55.295 mm

wave number and frequency is thus introduced, causing different frequencies

to propagate at different velocities This phenomenon is termed microstrip

(2.87)with

(2.88)

(2.89)(2.90)

(2.91)

where f is the frequency in GHz; w and h are the microstrip width and height

in cm, respectively; eris the relative dielectric constant of the substrate; and ee

is the static value of the effective dielectric constant, which is dependent on

the geometry of the microstrip In the limit f Æ 0, eeff( f ) Æ ee Here eeis givenby

(2.92)where

(2.93)

In the preceding equation, t denotes the thickness of the metal that constitutes

the microstrip line The accuracy of Equation (2.86) is better than 0.6% in the

range 0.1 £ w/h £ 100, and 1 £ er£ 20, and is valid up to about 60 GHz Thisequation spans a fairly wide variety of frequencies and dielectric substrates,hence e ( f ) can be evaluated very accurately.

h

h w

w h

w h h

w

w h

ÊË

ˆ

¯=+ÊË

ˆ

-ÊË

ˆ

+ÊË

ˆ

ÏÌÔ

Ó

ÔÔ

.

if if

-2

12

2.6.2 Effect of Curvature

A curved microstrip line can be modeled as a cascade of sections of microstriplines with chamfered bends Illustrated in Figure 2.24a is a typical bend in amicrostrip line for an arbitrary bend angle q ; also shown in the same figureare the reference planes that define the edges of the bend The equivalentcircuit of the bend, in the region restricted to the confines of the referenceplanes, is shown in Figure 2.24b For optimum chamfer, the ratio of the width

of the chamfered region b to the width of the microstrip line w is

approxi-mately 0.5 [38] In the equivalent-circuit representation of the bend, the

induc-tance L and capaciinduc-tance C represent the inducinduc-tance associated with the

discontinuity and the capacitance to ground, respectively Kirschning et al [39]derived an empirical closed-form expression to represent the equivalent

circuit of the bend For optimum chamfer, the capacitance C (pf) and tance L (nH) are given by

induc-(2.94)

(2.95)

where h and e rare the thickness in mm and the dielectric constant, tively, of the substrate; and q is the angle of the chamfer in degrees, and in the

respec-limit q Æ 180, C, L Æ 0 This reduces to the straight-line case; hence there are

no discontinuities These equations are in general valid for w/h and e rin the

ranges 0.2 £ w/h £ 6, and 2 £ e r £ 13 When 0.2 £ w/h £ 1, the accuracy of the

model is within 0.3% Since ring resonators are usually built on substrates with

dielectric constants greater than 2, and since w/h £ 1 for standard 50-W lines

on high dielectric-constant substrates, this model can be applied to accuratelymodel the curvature of conventional microstrip ring resonators A moredetailed account of the application of this model to the microstrip ring res-onator is presented in the following

2.6.3 Distributed-Circuit Model

The distributed ring circuit model is described in [9, 35] The basic microstripring resonator is illustrated in Figure 2.25 Power is coupled into and out ofthe resonator via two feed lines located at diametrically opposite points If thedistance between the feed lines and the resonator is large, then the couplinggap does not affect the resonant frequencies of the ring The resonator in thiscase is said to be “loosely coupled.” Loose coupling is a manifestation of thenegligibly small capacitance of the coupling gap If the feed lines are movedcloser to the resonator, however, the coupling becomes tight and the gapcapacitance becomes appreciable This causes the circuit’s resonant frequency

to deviate from the intrinsic resonant frequency of the ring Hence, to rately model the ring resonator, the capacitance of the coupling gap should beconsidered in conjunction with microstrip dispersion and curvature

accu-When the mean circumference of the ring resonator is equal to an integralmultiple of a guided wavelength, resonance is established This may beexpressed as

2pr = nl g for n = 1, 2, (2.96)

where r is the mean radius of the ring (i.e., r = (r i + r o)/2); lgis the guided

wave-length; and n is the mode number This relation is valid for the loose coupling

case, as it does not take into consideration the effect of the coupling gap

DISTRIBUTED TRANSMISSION-LINE MODEL 45

In order to apply the distributed transmission-line model, the mean radius

of the ring resonator must be known This may be estimated from Equation(2.96) as follows: For a given frequency and a dielectric material of knownthickness, the dimensions of a 50-W line are estimated from a commercially

available program called Linecalc [40]; the effective dielectric constant and

guided wavelength are estimated from Equations (2.86) through (2.93) Thevalue of the guided wavelength thus determined is substituted into Equation(2.96) to evaluate the mean radius of the ring Although the resonant fre-quencies of an ideal ring resonator are independent of the characteristicimpedance of the line that forms the closed loop, it is conventional to use lineswhose characteristic impedance corresponds to 50 W

The approach underlying the distributed transmission-line model is that the

ring is analyzed as a polygon of n sides.This is illustrated in Figure 2.26 wherein

the ring resonator is represented by a 16-sided polygon In actuality, however,

a 36-sided polygon was used, and it was found that any further increase in thenumber of sides did not improve the accuracy of the model The sides of thepolygon and the feed lines were modeled as sections of lossy microstrip trans-mission lines; the length of each side of the polygon was fixed to be on thirty-sixth of the ring’s mean circumference The discontinuities at the vertices ofthe polygon were modeled as optimally chamfered microstrip bends; for a 36-sided polygon, the bend angle q is 170° The gap between the feed lines andthe resonator is modeled in accordance with Hammerstad’s model [41] for themicrostrip gap Although this gap model is valid only for symmetric gaps, itwas successfully applied to the asymmetric gaps between the feed lines andthe resonator In actuality, the small curvature of the ring over the region cor-responding to the width of the feed lines makes the gaps appear symmetric.Further, when the ring is symmetrically excited, the maximum field points in

FIGURE 2.26 Distributed transmission-line model [9] (Permission from Electronics

Letters.)