Microwave Ring Circuits and Related Structures phần 5 doc

where w0 is the angular resonant frequency, U is the stored energy per cycle,

and W is the average power lost per cycle. The three main losses associated

with microstrip circuits are conductor losses, dielectric losses, and radiation

losses. The total Q-factor, Q0, can be expressed as

(6.11)

where Qc, Qd, and Qr are the individual Q-values associated with the conduc￾tor, dielectric, and radiation losses, respectively [14].

For ring and linear resonators of the same length, the dielectric and

conductor losses are equal and therefore Qc and Qd are equal.The power radi￾ated, Wr, is higher for the linear resonator. This results in a lower Qr for

the linear resonator relative to the ring. We can conclude that because Qc and

Qd are equal for the two resonators, and that Qr is higher for the ring, that the

ring resonator has a higher Q0.

The unloaded Q, Q0, can also be determined by measuring the loaded Q￾factor, QL, and the insertion loss of the ring at resonance. Figure 6.3 shows a

typical resonator frequency response. The loaded Q of the resonator is

(6.12)

where w0 is the angular resonant frequency and w1 - w2 is the 3-dB bandwidth.

Normally a high QL is desired for microstrip measurements. A high QL

requires a narrow 3-dB bandwidth, and thus a sharper peak in the frequency

response. This makes the resonant frequency more easily determined.

The unloaded Q-factor can be calculated from

QL = -

w

w w

0

1 2

1111

Q QQQ 0 cdr

=++

DISPERSION, DIELECTRIC CONSTANT, AND Q-FACTOR MEASUREMENTS 143

FIGURE 6.3 Resonator frequency response.

(6.13)

where L is the insertion loss in dB of the ring at resonance [2]. Because the

ring resonator has a higher Q0 and lower insertion loss than the linear res￾onator, it will also have a higher loaded Q, QL. Therefore the ring resonator

has a smaller 3-dB bandwidth and sharper resonance than the linear resonator.

This also makes the ring more desirable for microstrip measurements.

Troughton recognized the disadvantages associated with using the linear

resonators for measurements and introduced the ring resonator in 1969 [1].

He proposed that the unknown effects of either open- or short-circuit cavity

terminations could be avoided by using the ring in dispersion measurements.

The equation to be used to calculate dispersion can be found by combining

Equations (6.1) and (6.4) to yield

(6.14)

Any ill effect introduced by the ring that might falsify the measured value

of wavelength or dispersion can be reduced by correctly designing the circuit.

There are five sources of error that must be considered:

a. Because the transmission line has a curvature, the dispersion on the ring

may not be equal to the straight-line dispersion.

b. Field interactions across the ring could cause mutual inductance.

c. The assumption that the total effective length of the ring can be calcu￾lated from the mean radius.

d. The coupling gap may cause field perturbations on the ring.

e. Nonuniformities of the ring width could cause resonance splitting.

To minimize problems (a) through (d) only rings with large diameters

should be used. Troughton used rings that were five wavelengths long at the

frequency of interest. A larger ring will result in a larger radius of curvature

and thus approach the straight-line approximation and diminish the effect of

(a). The large ring will reduce (b) and the effect of (d) will be minimized

because the coupling gap occupies a smaller percentage of the total ring. The

effect of the mean radius, (c), can be reduced by using large rings and narrow

line widths.

An increased ring diameter will also increase the chance of variations in the

line width, and the possibility of resonance splitting is increased. The only way

to avoid resonance splitting is to use precision circuit processing techniques.

Troughton used another method to diminish the effect of the coupling gap.

An initial gap of 1 mil was designed. Using swept frequency techniques, Q￾factor measurements were made.The gap was etched back until it was obvious

that the coupling gap was not affecting the frequency.

e

p eff f nc

fr ( ) = Ê

Ë

ˆ

2 ¯

2

Q QL

0 L 20 1 10 = ( ) - -

144 MEASUREMENT APPLICATIONS USING RING RESONATORS

The steps Troughton used to measure dispersion can be summarized as

follows:

1. Design the ring at least five wavelengths long at the lower frequency of

interest.

2. Minimize the effect of the coupling gap by observing the Q-factor and

etching back the gap when necessary.

3. Measure the resonant frequency of each mode.

4. Apply Equation (6.4) to calculate eeff.

5. Plot eeff versus frequency.

This technique was very important when it was introduced because of the

very early stage that the microstrip transmission line was in. Because it was in

its early stage, there had been little research that resulted in closed-form

expressions for designing microstrip circuits. This technique allowed the fre￾quency dependency of eeff to be quickly measured and the use of microstrip

could be extended to higher frequencies more accurately.

6.3 DISCONTINUITY MEASUREMENTS

One of the most interesting applications of the ring is its use to characterize

equivalent circuit parameters of microstrip discontinuities [3, 12]. Because

discontinuity parameters are usually very small, extreme accuracy is needed

and can be obtained with the ring resonator.

The main difficulty in measuring the circuit parameters of microstrip dis￾continuities resides in the elimination of systematic errors introduced by the

coaxial-to-microstrip transitions. This problem can be avoided by testing dis￾continuities in a resonant microstrip ring that may be loosely coupled to test

equipment. The resonant frequency for narrow rings can be approximated

fairly accurately by assuming that the structure resonates if its electrical length

is an integral multiple of the guided wavelength.When a discontinuity is intro￾duced into the ring, the electric length may not be equal to the physical length.

This difference in the electric and physical length will cause a shift in the res￾onant frequency. By relating the Z-parameters of the introduced discontinu￾ity to the shift in the resonance frequency the equivalent circuit parameters

of the discontinuity can be evaluated.

It has also been explained that the TMn10 modes of the microstrip ring are

degenerate modes.When a discontinuity is introduced into the ring, the degen￾erate modes will split into two distinct modes. This splitting can be expressed

in terms of an even and an odd incidence on the discontinuity. The even case

corresponds to the incidence of two waves of equal magnitude and phase. In

the odd case, waves of equal magnitude but opposite phase are incident from

both sides. Either mode, odd or even, can be excited or suppressed by an

appropriate choice of the point of excitation around the ring.

DISCONTINUITY MEASUREMENTS 145