Corrections to “Temperature Correction for Cylindrical Cavity Perturbation Measurements” [Jun 17 2153-2161] (2017)

In the above paper <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , it has come to the attention of the authors that a typographical error exists in (16) and (17). The error, while small in syntax, has a large impact on the understanding behind the explained technique. Reference <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1, eq. (15)]</xref> states that by subtracting the fractional frequency shifts of the sample perturbed and the unperturbed case, any resultant fractional frequency shift is caused by the temperature dependent properties of the sample, if any. Now, the printed and erroneous <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1, eq. (16)]</xref> should be <disp-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">\begin{equation*} \varepsilon _{1}( T )\approx -2\left [{ \frac {f_{s}( T )-f_{0}(0)}{f_{0}( 0)} }\right ]\frac {V_{c}}{V_{s}}G_{\mathrm {mnp}}+1. \end{equation*} </tex-math></disp-formula> We note that (16) incorrectly states that the unperturbed frequency ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{0}$ </tex-math></inline-formula> ) is a function of temperature. This is incorrect since all temperature dependence in the unperturbed state is already addressed in <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1, eq. (15)]</xref> . Additionally, the fractional frequency shift is normally taken with respect to the reference unperturbed frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{0}$ </tex-math></inline-formula> as opposed to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{s}$ </tex-math></inline-formula> ; however, since the relative difference in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{s}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{0}$ </tex-math></inline-formula> is small at gigahertz frequencies, this is not a large issue. Reference <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1, eq. (16)]</xref> should be printed as <xref ref-type="disp-formula" rid="deqn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1)</xref> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{s}( T )=f_{s}( {0})\times ( 1+(({\Delta f_{s}( T )})/({f_{s}( 0)})) )$ </tex-math></inline-formula> is the temperature dependent frequency while the erroneous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{0}( T )$ </tex-math></inline-formula> has been replaced with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{0}(0)$ </tex-math></inline-formula> , the unperturbed frequency taken at a reference temperature. All temperature dependence is addressed in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{s}( T )$ </tex-math></inline-formula> and if there was no temperature dependence, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{s}( T )$ </tex-math></inline-formula> would be a constant and the original cavity perturbation approximation would apply. Similarly, <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1, eq. (17)]</xref> should be printed as <disp-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">\begin{equation*} \varepsilon _{2}( T )\approx \left [{ \frac {1}{Q_{s}( T )}-\frac {1}{Q_{0}( 0)} }\right ]\frac {V_{c}}{V_{s}}G_{\mathrm {mnp}} \end{equation*} </tex-math></disp-formula> where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({1}/({Q_{s}( T)}))=(({{\text {BW}}_{s}(0)})/({f_{s}( T )}))\times ( 1+(({\Delta {\text {BW}}_{s}( T )})/({\text {BW}_{s}{(0)}})) )$ </tex-math></inline-formula> is the reciprocal of the temperature dependent quality factor and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{0}( T )$ </tex-math></inline-formula> has been replaced with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{0}( {0} )$ </tex-math></inline-formula> which is the unperturbed quality factor at a reference temperature.