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16g. Summary of analysis of a transmission line excited resonator using a circulator

**Keywords:** scattering, resonant, circulator, summary, coupling, matching, Smith chart, transmission line

Postings 3.16 through 3.16f concern the analysis of Fig. 35 of posting 3.15 using various methods of conventional circuit analysis. The results confirm the statements made in postings previous to these concerning coupling and *Q*'s. The postings represent a sampling of the methods a person might use to calculate various aspects of a resonant structure.

Below is a listing of various aspects covered in postings 3.16 through 3.16f.

- The simple electrical LRC circuit that is equivalent to the acoustical circuit of Fig. 35 of posting 3.15.
- Why a small coupling hole in an acoustical circuit is equivalent to a coupling inductor in the electrical analog.
- Equations for
*Q*_{0}and*Q*of an electrical LRC circuit._{L} - Yet another graphical way to present a resonance: as the real and imaginary parts of the impedance of the resonator as a function of frequency.
- An equation for the approximate resonant frequency of the LRC circuit with the coupling inductor and also for the value of the coupling inductor required for unity coupling.
- A polar plot of the complex impedance of an LRC resonator.
- Equations and graph of the resonance curve of the complete electrical circuit (LRC resonator plus driving source, its output resistance and the coupling inductor).
- Transmission line resonators.
- Qo of a transmission line resonator - equations and graphs.
- An electrical transmission line model for the acoustic circuit of Fig. 35.
- A Smith chart to graphically solve for the resonant frequency and required
*L*for unity coupling._{c} - Discussion of the dilemma of how an inductor can possibly be used to match impedances.
- Differential equation for the complete circuit responding to transients such as a sinusoidal burst: equations and numerical solutions and graphs.
- SPICE simulation of burst response, including a graph that shows agreement with the differential equation solution.

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