W0 là tần số cộng hưởng góc, U là năng lượng được lưu trữ trên mỗi chu kỳ, và W là công suất trung bình mất trên mỗi chu kỳ. Ba tổn thất chính liên quan đến mạch microstrip dẫn thiệt hại, tổn thất điện môi, và thiệt hại bức xạ. Tổng Q-yếu tố, Q0, có thể được thể hiện như là 1 1 1 1 = + + Q0 Qc Qd QR (6,11) | DISPERSION DIELECTRIC CONSTANT AND Q-FACTOR MEASUREMENTS 143 FIGURE Resonator frequency response. 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 1111 Q0 Qc Qd Qr where Qc Qd and Qr are the individual Q-values associated with the conductor 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 radiated 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 shows a typical resonator frequency response. The loaded Q of the resonator is Ql -W-w 1 w 2 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 144 MEASUREMENT APPLICATIONS USING RING RESONATORS Q0 1 _ 10-L 20 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 resonator 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 .
