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I have been trying to understand the expression of |s21| for the resonator mentioned in this article. For a circuit similar to the drawn below, in which R1 is used as the temperature sensitive element, the transmittance at resonance is given by $$ |S21| = 2\kappa\frac{G_0}{G_1+G_0} $$ where $$\kappa = \frac{C_2C_3}{C_2^2 + C_3^2}$$ $$G_0 = 4\pi^2({C_2^2 + C_3^2})Z_0f_0^2$$ I couldn't obtain these expressions on my own. Can somebody explain me the logic behind this derivation ?

schematic

simulate this circuit – Schematic created using CircuitLab

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2 Answers 2

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An author of the paper offers an explanation in his PhD dissertation, page 85.

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One method is impedance ratio divider.

schematic

simulate this circuit – Schematic created using CircuitLab

However an easier method is probably Admittance or Conductance ratios.

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  • \$\begingroup\$ I've already tried this method but it doesn't simplify to the expression above. \$\endgroup\$
    – Krlngc
    Jul 23, 2018 at 7:33
  • \$\begingroup\$ Perhaps matrix methods will work. but beyond me. Here's what it can look like goo.gl/PbzFHr Not too common to me and R1 does nothing useful. You dont show G1, but similar I expect. \$\endgroup\$ Jul 23, 2018 at 16:13
  • \$\begingroup\$ or closer values goo.gl/CDLFWE or even higher Q goo.gl/tC3fNh \$\endgroup\$ Jul 23, 2018 at 16:20
  • \$\begingroup\$ Strange L, must be in the equation but isnt shown and they say neglect G1 << G0 as the dominant resonance is determined by L1C1 get Q from high ratios of C1/C2 and C1/C3 slightly shifts resonance determined by L1C1. So my conclusion is same as yours ??? It does not seem correct . publications.lib.chalmers.se/records/fulltext/213291/… for others to read \$\endgroup\$ Jul 23, 2018 at 16:33

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