Basics
Crystal Oscillator
Resonant Frequency
We want to calculate the resonant frequency of a crystal, including \(C_0, R_m, L_m, C_m\). We will first calculate the total impedance. Then by definition, resonat frequency is the frequency such that the total impedance is purely real.
\[Z_0 = -\dfrac{j}{\omega C_0}\] \[Z_m = R_m + j\omega L_m - \dfrac{j}{\omega C_m}\] \[Z_p = \dfrac{Z_0 Z_m}{Z_0 + Z_m} = \dfrac{\dfrac{L_m}{C_0} - \dfrac{1}{\omega^2 C_0 C_m} - \dfrac{j R_m}{\omega C_0}}{R_m + j\left(\omega L_m - \dfrac{1}{\omega C_m} - \dfrac{1}{\omega C_0}\right)}\]The condition for \(Z_p\) to be purely real
\[\dfrac{-\dfrac{R_m}{\omega C_0}}{\dfrac{L_m}{C_0}-\dfrac{1}{\omega^2 C_0 C_m}} = \dfrac{\omega L_m - \dfrac{1}{\omega C_m} - \dfrac{1}{\omega C_0}}{R_m}\] \[-\dfrac{R_m^2}{\omega C_0} = \left(\dfrac{L_m}{C_0}-\dfrac{1}{\omega^2 C_0 C_m}\right) \left(\omega L_m - \dfrac{1}{\omega C_m} - \dfrac{1}{\omega C_0}\right)\] \[\omega^4 + \omega^2 \left(\dfrac{R_m^2}{L_m^2}-\dfrac{2}{L_m C_m} - \dfrac{1}{L_m C_0}\right) + \left(\dfrac{1}{L_m^2 C_m^2} + \dfrac{1}{L_m^2 C_m C_0}\right) = 0\]we can find two resonant frequency
\[\omega^2 = \dfrac{1}{2} \left\{\left(\dfrac{2}{L_m C_m} + \dfrac{1}{L_m C_0} - \dfrac{R_m^2}{L_m^2}\right) \pm\left[\left(\dfrac{R_m^2}{L_m^2}-\dfrac{2}{L_m C_m} - \dfrac{1}{L_m C_0}\right)^2 - 4\left(\dfrac{1}{L_m^2 C_m^2} + \dfrac{1}{L_m^2 C_m C_0}\right)\right]^{1/2}\right\}\]some more algebra
\[\begin{align} &\left(\dfrac{R_m^2}{L_m^2}-\dfrac{2}{L_m C_m} - \dfrac{1}{L_m C_0}\right)^2 - 4\left(\dfrac{1}{L_m^2 C_m^2} + \dfrac{1}{L_m^2 C_m C_0}\right)\\ =& \left( \dfrac{1}{L_m C_0}-\dfrac{R_m^2}{L_m^2} \right)^2 - \dfrac{4 R_m^2}{L_m^3 C_m} \end{align}\]We assume \(\left( \dfrac{1}{L_m C_0}-\dfrac{R_m^2}{L_m^2} \right)^2 \gg \dfrac{4 R_m^2}{L_m^3 C_m}\). For example, \(R_m=10\Omega, L_m=500uH, C_m=5fF, C_0=1pF\), that will be \(4 \times 10^{30} \gg 6.4 \times 10^{26}\). Then
\[\omega^2 \approx \left(\dfrac{1}{L_m C_m} + \dfrac{1}{2 L_m C_0} - \dfrac{R_m^2}{2 L_m^2}\right) \pm \left( \dfrac{1}{2 L_m C_0}-\dfrac{R_m^2}{2 L_m^2} \right)\] \[\omega_s \approx \dfrac{1}{\sqrt{L_m C_m}}\] \[\omega_a \approx \left(\dfrac{1}{L_m C_m} + \dfrac{1}{L_m C_0} - \dfrac{R_m^2}{L_m^2}\right)^{1/2}\]assume again \(\dfrac{1}{L_m C_0} \gg \dfrac{R_m^2}{L_m^2}\), for example, \(2\times 10^{15} \gg 4 \times 10^{8}\).
\[\begin{align} \omega_a \approx \sqrt{\dfrac{1}{L_m \left(C_m \Vert C_0\right)}} \end{align}\]or
\[\begin{align} \omega_a \approx \sqrt{\dfrac{1}{L_m C_m}} \left(1+\dfrac{C_m}{C_0}\right)^{1/2} \approx \sqrt{\dfrac{1}{L_m C_m}} \left(1+\dfrac{C_m}{2C_0}\right) \end{align}\]If there are load capacitance, the load capacitance \(C_L\) will be parallel with \(C_0\), thus
\[\omega_s \approx \dfrac{1}{\sqrt{L_m C_m}}\] \[\omega_a \approx \sqrt{\dfrac{1}{L_m \left(C_m \Vert \left(C_0+C_L\right)\right)}} \approx \sqrt{\dfrac{1}{L_m C_m}} \left(1+\dfrac{C_m}{2(C_0+C_L)}\right)\]depending on how the oscillator is designed, the oscillation frequency will be betwenn the two value
\[\omega_s \le \omega_{osc} \le \omega_a\]Equivalent Resistance
Let’s denote \(jX = j\omega L_m - \dfrac{j}{\omega C_m}\), and \(j X_{C0} = -\dfrac{j}{\omega C_0}\). The total impedence of the crystal
\[\begin{align} Z &= \dfrac{\left(R_m + jX\right)\left(j X_{C0}\right)}{R_m + j(X+X_{C0})}\\ &= \dfrac{R_m X_{C0}^2}{R_m^2 + \left(X+X_{C0}\right)^2} + \dfrac{jX_{C0}\left[R_m^2 + X\left(X+X_{C0}\right)\right]}{R_m^2 + \left(X+X_{C0}\right)^2} \end{align}\]The effective resistance is given by
\[R_e = \dfrac{R_m X_{C0}^2}{R_m^2 + \left(X+X_{C0}\right)^2}\]consider the frequency between \(\dfrac{1}{\sqrt{L_m C_m}} \le \omega \le \dfrac{1}{\sqrt{L_m C_m}} \left(\dfrac{C_m}{2(C_0+C_L)}\right)\). Since it is a narrow frequency range, \(X_{C0}\) stay almost constant. From \(\omega_s\) to \(\omega_a\), \(X\) will increase from 0 to some positive value. Since \(X_{C0}\) is negative, \(R_e\) will increase if we sweep the frequency from \(\omega_s\) to \(\omega_a\). At \(\omega_s\), \(X=0\)
\[R_e\vert_{\omega=\omega_s} = \dfrac{R_m X_{C0}^2}{R_m^2 + X_{C0}^2} \approx R_m \quad (\text{ usually } \vert X_{C0} \vert \gg R_m)\]on the other hand, when \(\omega=\omega_a\), we can use either of the two equations for \(\omega_a\), consider the one
\[\omega_a \approx \sqrt{\dfrac{1}{L_m \left(C_m \Vert \left(C_0+C_L\right)\right)}}\]then
\[\omega_a L_m = \dfrac{1}{\omega_a C_m} + \dfrac{1}{\omega_a (C_0+C_L)}\]and
\[X = \dfrac{1}{\omega_a\left(C_L+C_0\right)}\]then
\[\begin{align} R_e\vert_{\omega=\omega_a} &= \dfrac{R_m \left(\dfrac{1}{\omega_a C_0}\right)^2}{R_m^2 + \left(\dfrac{1}{\omega_a\left(C_L+C_0\right)} - \dfrac{1}{\omega_a C_0}\right)^2}\\ &= \dfrac{R_m X_{C0}^2}{R_m^2 + X_{C0}^2 \left(\dfrac{C_L}{C_L+C_0}\right)^2} \end{align}\]assume \(\left\vert X_{C0} \left(\dfrac{C_L}{C_L+C_0} \right)\right\vert \gg R_m\)
\[\begin{align} R_e\vert_{\omega=\omega_a} &\approx R_m \left(\dfrac{C_L+C_0}{C_L}\right)^2 \end{align}\]