## 3 Full-Order Envelope Model Derivation  
The converter output voltage may be expressed as the sum of voltages across the individual RLC network elements (cf. Figure 3):  
\[v(t)=\underbrace{R(t)i(t)}_{v_{R}(t)}+v_{L}(t)+v_{C}(t). \tag{3}\]  
The voltages across the inductance and capacitance are related to the AC-side current \(i(t)\) as  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]  
Figure 3: System under consideration. \[i(t)=\frac{d(C(t)v_{\rm C}(t))}{dt}=\frac{dC(t)}{dt}v_{\rm C}(t)+\frac{dv_{\rm C}( t)}{dt}C(t), \tag{5}\]  
respectively. Meanwhile, \(v(t)\) may be described (cf. (2)) by the following Fourier series:  
\[v(t)=\sum\limits_{n}A_{n}\sin(\int\omega_{\rm s}(t)dt+\theta_{n}),\quad A_{n}= \frac{4\
u_{in}}{n\pi}\sin\Bigl{(}\frac{n\pi}{2}\Bigr{)}. \tag{6}\]  
In practical series resonant converters, the current is nearly-sinusoidal due to the high quality factor of the RLC network [27; 28] so that the first inverter output voltage harmonic only is relevant to the power transfer. The inverter output voltage is replaced for the sake of envelope modeling by its first harmonic given by  
\[v(t)\ \approx v_{1}(t)=A_{1}(t)\sin(\theta_{\rm s}(t)+\theta_{1}),\quad\theta_{ \rm s}(t)=\int\omega_{\rm s}(t)dt. \tag{7}\]  
with (where the corresponding phase is used as the reference hereafter)  
\[A_{1}(t)=\frac{4\
u_{in}(t)}{\pi},\quad\theta_{1}=0. \tag{8}\]  
Consequently, the general expressions for the AC-side capacitor voltage and current are given by  
\[v_{\rm C}(t)=V_{M}(t)\sin(\theta_{\rm s}(t)+\varphi_{v}(t))=V_{M}(t)\cos( \varphi_{v}(t))\sin(\theta_{\rm s}(t))+V_{M}(t)\sin(\varphi_{v}(t))\cos( \theta_{\rm s}(t)) \tag{9}\]  
and  
\[i(t)=I_{M}(t)\sin(\theta_{\rm s}(t)+\varphi_{i}(t))=I_{M}(t)\cos(\varphi_{i}( t))\sin(\theta_{\rm s}(t))+I_{M}(t)\sin(\varphi_{i}(t))\cos(\theta_{\rm s}(t)), \tag{10}\]  
respectively [29], where \(V_{M}(t)\) and \(I_{M}(t)\) stand for the corresponding magnitudes, and \(\varphi_{v}(t)\) and \(\varphi_{i}(t)\) symbolize the corresponding phases, denoting  
\[V_{M}(t)\cos(\varphi_{v}(t))\triangleq v_{\rm C}^{\rm s}(t),\quad\ V_{M}(t) \sin(\varphi_{v}(t))\triangleq v_{\rm C}^{\rm c}(t), \tag{11}\]  
and  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]0  
Equations (9) and (10) may be rewritten as  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]1  
and  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]2  
respectively. The time-domain derivatives of (13) and (14) are then, respectively, given by  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]3  
and  
\[v_{L}(t)=\frac{d(L(t)i(t))}{dt}=\frac{dL(t)}{dt}i(t)+\frac{di(t)}{dt}L(t), \tag{4}\]4Combining (13)-(16) with (3)-(5) and (7) yields  
\[ \begin{array}{l}\frac{4v_{in}(t)}{\pi}\sin(\theta_{s}(t))=\Big{(}R(t)+\frac{dL( t)}{dt}\Big{)}[\tilde{r}^{s}(t)\sin(\theta_{s}(t))+\tilde{r}^{c}(t)\cos(\theta_{s}(t))] \\ +L(t)\Big{[}\frac{d\tilde{r}^{s}(t)}{dt}-\omega_{s}(t)\tilde{r}^{c}(t)\Big{]} \sin(\theta_{s}(t))+L(t)\Big{[}\frac{d\tilde{r}^{c}(t)}{dt}+\omega_{s}(t) \tilde{r}^{s}(t)\Big{]}\cos(\theta_{s}(t))\\ +v^{s}_{\mathbb{C}}(t)\sin(\theta_{s}(t))+v^{c}_{\mathbb{C}}(t)\cos(\theta_{s }(t))\end{array}  \tag{17}\]  
and  
\[ \begin{array}{l}\tilde{r}^{s}(t)\sin(\theta_{s}(t))+\tilde{r}^{c}(t)\cos( \theta_{s}(t))=\frac{4\mathbb{C}(t)}{dt}\big{[}v^{s}_{\mathbb{C}}(t)\sin( \theta_{s}(t))+v^{c}_{\mathbb{C}}(t)\cos(\theta_{s}(t))\big{]}+\\ C(t)\Big{[}\Big{(}\frac{dv^{s}_{\mathbb{C}}(t)}{dt}-\omega_{s}(t)v^{c}_{ \mathbb{C}}(t)\Big{)}\sin(\theta_{s}(t))+\Big{(}\frac{dv^{c}_{\mathbb{C}}(t) }{dt}+\omega_{s}(t)v^{s}_{\mathbb{C}}(t)\Big{)}\cos(\theta_{s}(t))\Big{]}, \end{array}  \tag{18}\]  
respectively. Separating (17) and (18) into sine and cosine terms gives  
\[