$ABCD$-matrix analysis [@Pozar], for example. One then gets

$$\mathrm{Im}\left[Z_{12}\left(\omega\right)\right]=\frac{C_{c}^{2}L_{r}\omega/\left(C_{q}+C_{c}\right)}{C_{q}\left(1-\omega^{2}/\omega_{r}^{2}\right)+C_{c}\left(1-2\omega^{2}/\omega_{qr}^{2}-\omega^{2}/\omega_{r}^{2}\right)}$$

where $\omega_{r}=1/\sqrt{L_{r}C_{r}}$ and $\omega_{qr}=1/\sqrt{L_{r}C_{q}}$. We note that in actual devices $C_{q}\ll C_{r}$ hence $\omega_{r}\ll\omega_{qr}$. We can then neglect the term $-2\omega^{2}/\omega_{qr}^{2}$ appearing in the denominator compared to the term $-\omega^{2}/\omega_{r}^{2}$ such that

$$\mathrm{Im}\left[Z_{12}\left(\omega\right)\right]\cong\frac{C_{c}^{2}L_{r}\omega}{\left(C_{q}+C_{c}\right)^{2}\left(1-\omega^{2}/\omega_{r}^{2}\right)}$$

Noting also $C_{c}\ll C_{q}$ we have

$$\begin{aligned}
\mathrm{Im}\left[Z_{12}\left(\omega\right)\right] & \cong & \frac{C_{c}^{2}L_{r}\omega}{C_{q}^{2}\left(1-\omega^{2}/\omega_{r}^{2}\right)}\nonumber \\
 & = & \frac{1}{2}\frac{C_{c}^{2}L_{r}\omega_{r}\omega}{C_{q}^{2}}\left(\frac{1}{\omega_{r}-\omega}+\frac{1}{\omega_{r}+\omega}\right)\label{eq:ImZ12-exp}\end{aligned}$$

hence by Eq.

$$\begin{aligned}
J^{\left(Z\right)} & =-\frac{1}{8}\frac{\sqrt{\omega_{1}\omega_{2}}}{\sqrt{L_{1}L_{2}}}\frac{C_{c}^{2}L_{r}\omega_{r}}{C_{q}^{2}}\left(\frac{1}{\omega_{r}-\omega_{1}}+\frac{1}{\omega_{r}-\omega_{2}}+\right.\nonumber \\
 & +\left.\frac{1}{\omega_{r}+\omega_{1}}+\frac{1}{\omega_{r}+\omega_{2}}\right)\label{eq:JZ}\end{aligned}$$

where we used the superscript $Z$ to indicate the application of the impedance $J$-coupling formula in Eq. .

If we interpret the first two terms inside the paranthesis in Eq. as the RWA-terms we can write

$$\begin{aligned}
J_{RWA}^{\left(Z\right)} & =- & \frac{1}{8}\frac{\sqrt{\omega_{1}\omega_{2}}}{\sqrt{L_{1}L_{2}}}\frac{C_{c}^{2}L_{r}\omega_{r}}{C_{q}^{2}}\left(\frac{1}{\omega_{r}-\omega_{1}}+\frac{1}{\omega_{r}-\omega_{2}}\right)\nonumber \\
 & = & \left(\frac{\omega_{1}\omega_{2}}{\omega_{r}^{2}}\right)\frac{g_{1}g_{2}\left(\omega_{1}+\omega_{2}-2\omega_{r}\right)}{2\left(\omega_{1}-\omega_{r}\right)\left(\omega_{2}-\omega_{r}\right)}\nonumber \\
 & = & \left(\frac{\omega_{1}\omega_{2}}{\omega_{r}^{2}}\right)J\label{eq:JZ-RWA}\end{aligned}$$

We note here that the standard expression for the exchange coupling $J$ in Eq. is obtained with a RWA; this is why we only kept the first two terms inside the paranthesis in Eq. and defined $J_{RWA}^{\left(Z\right)}$ in Eq. .

We now compare the formulas obtained above in Fig. with the following set of realistic parameter values $g_{1}=g_{2}=100\,MHz$, $\omega_{1}=2\pi(4.90\,GHz)$ and $\omega_{2}=2\pi\left(5.10\,GHz\right)$, $\delta_{1}=\delta_{2}=-340\,MHz$.

![\[fig:compare-formulas-plot\]Comparison of $J$-coupling expressions $J^{\left(Z\right)},$ $J_{RWA}^{\left(Z\right)}$, $J$ and $J$+$J_{0}$ for bus frequency $f_{r}$ ranging from $5.5\,GHz$ to 