Relationship between Q factor and complex resonant frequency: investigations using RLC series circuit

Transcription

1 ETTER IEICE Electronics Exress, Vol.14, No.1, 1 5 Relationshi between Q factor and comlex resonant frequency: investigations using RC series circuit Takuichi Hirano a) School of Environment and Society, Tokyo Institute of Technology, 1 1 O-okayama, Meguro-ku, Tokyo , Jaan a) hirano.t.aa@m.titech.ac.j Abstract: It is well known that Q factor can be calculated from comlex resonant frequency of a non-excitation roblem. However, two definitions are used to obtain Q factor from comlex resonant frequency. One definition uses the real art of the comlex frequency while the other uses an absolute value from the numerator of the Q factor. The meaning and difference of the two definitions is investigated using an RC series circuit, and the findings are resented in this article. Keywords: comlex resonant frequency, non-excitation, Q factor Classification: Electromagnetic theory References [1] E. I. Green: The story of Q, Am. Sci. 43 (1955) 584. [] D. M. Pozar: Microwave Engineering (Wiley, Hoboken, NJ, 005) 3rd ed. 66. [3] T. Kuroda: Wireless ower transfer, J. IEICE 93 (010) 964 (Jaanese). [4] Secial Section on Novel Develoment of Short Distance Wireless Power Transmission Technology, J. IEICE 95 (01) 33 (Jaanese). [5] T. Ohira: What in the world is Q, IEEE Microw. Mag. 17 (016) 4 (DOI: /MMM ). [6] T. Ohira: Every time we meet, my honey Q brings her new facet, J. IEICE 99 (016) 856 (Jaanese). [7] R. E. Collin: Foundations for Microwave Engineering (Wiley, New York, 1991) nd ed [8] K. Mizuno: A hysical meaning of the Q-factor, J. IEICE 99 (016) 1191 (Jaanese). [9] M. Tsuji, et al.: Analytical and exerimental considerations on the resonant frequency and the quality factor of dielectric resonators, IEEE Trans. Microw. Theory Techn. 30 (198) 195 (DOI: /TMTT ). [10] M. Tsuji, et al.: On the comlex resonant frequency of oen dielectric resonators, IEEE Trans. Microw. Theory Techn. 31 (1983) 39 (DOI: /TMTT ). Coyedited November 10, 017 1

2 IEICE Electronics Exress, Vol.14, No.1, Introduction Q factor [1] is an imortant value for evaluating the quality of inductors or caacitors, designing filters [], wireless ower transfer systems [3, 4], and so on. There are various exressions to define the Q factor [5, 6]. One exression is based on comlex resonant frequency, wherein there are two tyes of definitions. This study investigates the difference between the two definitions using a simle RC series circuit model. Various definitions of Q The Q factor for a well-known RC series circuit shown in Fig. 1(a) [] can be obtained as R ¼ 1! 0 CR ¼ 1 rffiffiffiffi : ð1þ R C where! 0 ¼ 1= C is the resonant angular frequency. At!0, the ratio ji=vj ¼ j1=zj holds eak value, where I and V are the current and voltage, resectively. In addition, Z ¼ R þ j! þ 1 j!c : ðþ The second definition for the Q factor is obtained using bandwidth! ¼!! 0, where ω is the angular frequency such that ji=vj ¼j1=Zj becomes 1= ffiffiffi against the eak value. ð3þ! The Q factor, according to [, 3], can be defined as 1 dz Z d! : ð4þ The definition of Q factor using energy [, 7, 8], which can be alied generally to any cavities, is exressed as U du dt where U is the electromagnetic energy stored in the resonator. We now introduce the comlex resonant frequency, ð5þ! c ¼! r þ j! i ð6þ to take energy dissiation into account in time harmonic scenario (e j! ct ¼ e j! rt e! it ). The comlex resonant frequency is normally introduced in eigenmode analysis for electromagnetic cavities. Therefore, the Q factor can be exressed [, 7] as Coyedited November 10, 017 Q ¼! r : ð7þ! i However, in [9, 10], a different exression with Eq. (6) was also roosed.

3 IEICE Electronics Exress, Vol.14, No.1, 1 5 (a) Excitation scenario Fig. 1. RC series circuit. (b) Non-excitation scenario Q ¼ j! cj ð8þ! i The reason for the definition of Eq. (8) does not seem to be clear. The values of Eq. (7) and Eq. (8) are obviously different. The difference between Eq. (7) and Eq. (8) is discussed in the following two sections using the RC series circuit model shown in Fig Definitions of resonance in mathematical style Fig. 1(a) shows the excitation scenario, wherein the circuit is excited by voltage source V. The Q factor for this circuit can be calculated using (1), (3), and (4). Fig. 1(b) illustrates the non-excitation roblem and the equation for this circuit is ZI ¼ 0 in mathematical style. Eq. (9) is an indeterminate equation. Z must be zero so that I exist without the voltage source. By solving Z ¼ 0,! c can be obtained. It is emhasized that the definition of resonance is different between Eq. (9) and Eq. (1). In (1), the real valued angular frequency! 0 is searched to maximize ji=vj. 4 Q factor evaluation by comlex resonant frequency for RC series circuit ð9þ Coyedited November 10, 017 Substituting Eq. () in Z ¼ 0, and solving for the comlex resonant angular frequency, we get sffiffi 1! c ¼ C R þ j R ð10þ by choosing the aroriate solution such that Re½! c Š > 0. It is clear that! c ¼! 0 ¼ 1= C when R ¼ 0. Substituting Eq. (10) into Eq. (7) yields 3

4 IEICE Electronics Exress, Vol.14, No.1, 1 5 sffiffiffi Q ¼ 1 R C R ð11þ Substituting Eq. (10) into Eq. (8) yields Q ¼ 1 rffiffiffiffi ð1þ R C which is unexectedly identical with Eq. (1). Moreover, j! c j¼1= C ¼!0 using Eq. (10) can also be derived. Comlex angular frequency (! c ) and Q factor with R are shown in Fig. by fixing ¼ 1 H and C ¼ 1 F(! 0 ¼ 1 rad/s). In Fig. (a),! r is overlaed with! 0 when the Q factor >10. In Fig. (b), the Q factor defined by Eq. (7) also overlas (a) Comlex angular frequency Coyedited November 10, 017 (b) Q factor Fig.. Comlex angular frequency and Q factor with R. ( = 1H, C = 1F) 4

5 IEICE Electronics Exress, Vol.14, No.1, 1 5 the Q factor defined by Eq. (1), Eq. (3), Eq. (4), and Eq. (8). The difference of (real art of) resonant frequencies and Q factors are both 0.15% when Q factor by Eq. (8) is 10. The resonators are normally designed with a Q factor >10, so the very small difference (0.15%) illustrated in the grahs is negligible. 5 Conclusion The two definitions used to obtain Q factor from comlex resonant frequency are resented, and the difference between the two definitions were investigated using an RC series circuit model. It was found that the resonant angular frequency corresonds to! 0 ¼ 1= C incidentally when Q factor is defined using the absolute value of comlex resonant frequency in the numerator. However, it was also found that the difference between the two definitions is negligible (0.15% when Q ¼ 10) when Q factor is greater than 10. Acknowledgments This work was suorted in art by MEXT/JSPS KAKENHI (Grant number /17K06418). Coyedited November 10, 017 5