## 1 Introduction
Note that the discussion above is based on phasor-domain analysis, where the impedances are inductive and are mathematically expressed as functions of the operating frequency. Nevertheless, phasor-domain analysis is only valid for systems operating in a steady state (i.e., at constant magnitudes and phases) with constant-frequency sinusoidal excitation. Nonetheless, in short-pulsed-power applications (e.g., hydrogen fusion systems), the power burst duration is lower than the settling time of the system, so classical phasor-domain analysis is irrelevant. As an example, consider a constant-magnitude constant-frequency sinusoidal voltage \(v(t)=V_{M}sin(\omega_{s}t)\) applied to a series RLC network (temporarily considering non-varying parameter values for brevity), as shown in Figure 1b at \(t=0\). The current flowing through the network would then be given by \(i(t)=I_{M}(t)sin(\int\omega(t)t)\), as shown in Figure 2. In a steady state, the linear time-invariant nature of the system imposes \(i^{ss}(t)=I_{M}^{ss}sin(\omega_{s}t+\varphi^{ss})\), with the constants \(I_{M}^{ss}=V_{M}/\mid Z_{eq,b}(\omega_{s})\mid\), \(\varphi^{ss}=-argZ_{eq,b}(\omega_{s})\), and \(\omega=\omega_{s}\). However, the instantaneous current magnitude and frequency vary during the transient period so that \(I_{M}(t)\
eq I_{M}^{ss}\) and \(\omega(t)\
eq\omega_{s}\). Meanwhile, the instantaneous current frequency may be expressed as \(\omega(t)=\omega_{s}+\Delta\omega(t)\). Hence, the current flowing through the network may be expressed as  
Figure 1: Series capacitor addition to RL load to allow resonant operation. \(i(t)=I_{M}(t)sin(\omega_{s}t+\varphi(t))\) with \(\varphi(t)=\int\Delta\omega(t)t\) so that the varying frequency is translated into a constant frequency with a varying phase, as shown in Figure 2. In practice, the inverter is switched off (in the given example) at 1 ms, while the system would have reached a steady state only around 2.75 ms. Consequently, classical phasor analysis considering a steady state is irrelevant for analyzing the system behavior within the power burst duration period (as mentioned above). The switching-period-averaged power delivered to the RLC network in the example above is \(P(t)\approx 0.5V_{M}I_{M}(t)\text{cos}(\varphi(t))\). In case energy delivery maximization is the goal, then \(\int I_{M}(t)\text{cos}(\varphi(t))dt\) should be maximized by the controller over the power burst period. Alternatively, \(I_{M}(t)\text{cos}(\varphi(t))\) should be regulated by the controller over the power burst period if power stabilization is desired. In order to design suitable controllers for either case or both, a dynamic model linking system inputs with an instantaneous current magnitude and phase is necessary. It was demonstrated in [17] that phasor-domain analysis may also be applied to systems with slow-varying (rather than constant) magnitudes and phases. The approach is better known as \"envelope modeling\" [18], originating from the fact that the average power delivered to a load by a sinusoidal current is only determined by its magnitude (envelope) and phase, as shown by the average power equation above [19]. As a result, switching-frequency behavior may be disregarded since it is irrelevant for the average power transfer process (disregarding switching losses). Unfortunately, the major drawback of the envelope-modeling approach is probably the inherent system order augmentation (imposed by splitting each AC-side state variable into the corresponding magnitude and phase). The resulting high-order dynamic model of the plant complicates the controller design process so that model order reduction is beneficial [20]. Traditionally, reduced-order models [21] rely on linearization techniques to achieve simplification. However, such approaches are suitable for steady-state operating systems with small variations around the operation point while struggling to correctly capture the behavior of systems with significantly varying operation points (such as short-time pulsed-power applications).