# what is temperature coefficient of resonant frequency?

I am trying to find the definition for temperature coefficient of resonant frequency, (TCF),

Even for articles in google scholar.

Fortunately, I found that TCF is

$$T_f=\frac{1}{f_0}\frac{\Delta f_0}{\Delta T}$$

Any reference?

## Concerning the meaning of "temperature coefficient of [measurement]".

Generally, when it appears without additional adjectives, the "temperature coefficient" of anything refers to the fractional slope of a linear fit to that quantity as a function of temperature.

In the introductory class we often introduce

• A linear coefficient of thermal expansion, in which the length of some object is a function of temperature $$L(T) = L_0 \left[ 1 + \alpha (T - T_0) \right ]\,.$$ Here $L_0$ is the length at the reference temperature $T_0$, and $\alpha$ is the coefficient and is tabulated.

• Similarly in basic circuits we define the temperature coefficients of resistance and resistivity \begin{align*} R(T) &= R_0 \left[1 + \alpha (T - T_0)\right] \\ \rho(T) &= \rho_0 \left[1 + \alpha (T - T_0)\right] \,, \end{align*} with similar meanings (the two $\alpha$s that appear here are the same to leading order, but can differ if examined to high precision, see the comment by Massimo Ortolano for the hairy details).

You'll see in the examples above that $\alpha$ is a very common symbol for such parameters, though $\beta$ and $\gamma$ also appear when there might be more than one such coefficient in a problem.

In principle there can be higher degree coefficients needed. You might see something like $$X(T) = X_0 \left[ 1 + \alpha (T - T_0) + \beta (T - T_0)^2 \right ]\,,$$ when more precision is needed. Here $\beta$ would be a "quadratic temperature coefficient of [quantity represented by $X$]". You could add a cubic coefficient with a term like ${}+ \gamma(T - T_0)^3$ inside the brackets.

It is no accident that the form of these expressions is reminiscent of a Taylor series; they come about exactly from approximating the desired quantity by a polynomial near some reference value.

The resonant frequency of some system depends on various physical parameters (possibly including both length and resistivity). So if you have measured the resonant frequency at, say, $20^\circ C$, but need to use the device at $25^\circ C$, you may need to assume a different resonant frequency. Having the temperature coefficient of that quantity will allow you to conveniently calculate the adjusted value.
Now, to compare the above to the expression you give in the question we need to transform the above. Starting a 'generic' temperature coefficient expression: \begin{align*} X(T) &= X_0 \left[ 1 + \alpha (T - T_0) \right] \\ X(T) &= X_0 \left[ 1 + \alpha \Delta T \right] \\ X(T) &= X_0 + X_0 \alpha \Delta T \\ X(T) - X_0 &= X_0 \alpha \Delta T \\ \Delta X &= X_0 \alpha \Delta T \\ \frac{\Delta X }{\Delta T} \frac{1}{X_0} = \alpha \,. \end{align*} So now we are able to interpret the symbols above: $f_0$ is the resonance frequency at the reference temperature (equivalent to $X(T)$), $\Delta f_0$ is the change in the resonance frequency ($\Delta X$) and $\Delta T$ is the temperature difference between your working environment and the reference temperature. Finally that leaves $T_f$ as the temperature coefficient (a nasty bit of notation when you are using $T$ for temperature in my opinion).
It is also worth unpacking a difference in the meaning of the $_0$ subscript in my examples and in the expression you found. In my examples it means "evaluated at the reference temperature", but it the expression you found it means "resonance"; the expression you found doesn't need an explicit notation for "at reference temperature" because all the uses of that symbol are buried in the $\Delta$ expressions.