Reduced-Order Nonlinear Envelope Modeling and Simulation of Resonant Inverter Driving Series Resistor–Inductor–Capacitor Load with Time-Varying Component Values

Abstract

Envelope modeling is an efficient way to obtain the large-signal amplitude and phase dynamics of fast-varying sinusoidal signals required for, e.g., resonant frequency tracking or energy transfer rate regulation in power converters. In addition, the method eliminates fast-varying parameters from the model so that the simulation time and memory requirements are reduced. This paper reveals the envelope-modeling process of a capacitor-powered resonant inverter feeding a time-varying series RLC load, often employed in pulsed-power applications. Such an arrangement is nontrivial since the system does not reach a steady state within a single pulse duration. Furthermore, model order reduction is carried out without performing linearization due to large variations in the expected operation point. As a result, a reduced-order nonlinear envelope model is derived and validated by simulations. Both the proposed modeling method and the derived model aim to simplify the challenging task of feedback controller design.

1. Introduction

The need for resonant converters is prominent in the field of induction heating in the metal industry [1], domestic induction cooking [2], and pulsed-power plasma applications [3]. Magnetic levitation systems for a variety of transportation solutions [4], wireless power supplies/chargers for electric vehicles [5,6], cellphones [7], and biomedical implants [8] are also based on resonant conversion. A typical resonant power delivery system utilizes an inverter (or DC–AC converter [9,10]) to create an alternating-current (AC) flow through a coil for the generation of alternating magnetic flux, inducing a voltage across a secondary coil or workpiece [11,12]. The equivalent load “seen” by the inverter may be represented by a series resistor–inductor (RL) arrangement, as shown in Figure 1a [13]. In this model, the energy transferred to the load is represented by the equivalent resistance R, while the equivalent inductance L reflects the magnetic behavior of the system. Typical resonant converters operate at relatively high switching frequencies ω_s (several tens of rad/s up to several Mrad/s). As a result, the equivalent inductance impedance jω_sL limits the amount of power transferred to the load by increasing the impedance of the equivalent load “seen” by the inverter to |Z_eq,a(ω_s)| = (R² + (ω_sL)²)^0.5.

To overcome this limitation, a common solution is adding a resonant capacitor C in series with the coil, as shown in Figure 1b [14]. As a result, the equivalent load impedance “seen” by the inverter becomes |Z_eq,b(ω_s)| = (R² + (ω_sL − 1/ω_sC)²)^0.5 so that the inductance contribution is offset by that of the capacitance. If | ω_sL − 1/ω_sC| < ω_sL, then |Z_eq,b(ω_s)| < |Z_eq,a(ω_s)|, and the power transferred to the load by the same inverter increases.

Another typical challenge in the process of resonant power delivery is the time-varying nature of equivalent resistance and/or inductance due to, e.g., workpiece state-of-matter changes in induction heating applications and load/coupling coefficient variations in wireless power transfer applications. In addition, the value of the resonant capacitor may also drift due to temperature variations, high-current-flow-induced mechanical stresses, and degradation [15]. As a result, the equivalent load impedance “seen” by the inverter varies (relatively slowly) during the process. Consequently, a feedback control system is employed to counteract these variations by adjusting the inverter’s operational state (output voltage and/or frequency) accordingly. This is successfully accomplished in long processes, such as metal processing and battery charging. However, the above-mentioned challenge is critical in pulsed-power applications [16], such as hydrogen fusion. In fusion systems, magnetic levitation and induction heating are simultaneously carried out during a process in which the hydrogen is rapidly heated to a high temperature by an intensive short-time (millisecond-order) power burst for gas-to-plasma conversion. In case energy delivery maximization is the goal, the impedance “seen” by the inverter should be minimized by keeping the system at resonance (forcing ω_sL − 1/ω_sC = 0, as shown in Figure 1b). Meanwhile, if it is desired to stabilize the power delivered to the load at a certain reference value, then the current (and, hence, |Z_eq,b(ω_s)|) should be regulated. However, the values of RLC network parameters are expected to vary due to the above-mentioned reasons. Therefore, the feedback control system must react very quickly (within the fraction of the power burst duration) to these changes by varying the operational frequency to restore the desired operation point.

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 $v\left(t\right)=V_{M}sin\left({\omega }_{s}t\right)$ 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 $i\left(t\right)=I_M\left(t\right)sin\left(\int \omega \left(t\right)dt\right)$, as shown in Figure 2. In a steady state, the linear time-invariant nature of the system imposes $i^{ss}\left(t\right)=I_M^{ss}sin\left({\omega }_{s}t+{\phi }^{ss}\right)$, with the constants $I_M^{ss}$ = V_M/|Z_eq,b(ω_s)|, ${\phi }^{ss}=-arg{Z}_{eq,b}\left({\omega }_s\right)$, and $\omega ={\omega }_s$. However, the instantaneous current magnitude and frequency vary during the transient period so that $I_M\left(t\right)\ne I_M^{ss}$ and $\omega \left(t\right)\ne {\omega }_s$. Meanwhile, the instantaneous current frequency may be expressed as ${\omega \left(t\right)=\omega }_s+∆\omega \left(t\right)$. Hence, the current flowing through the network may be expressed as $i\left(t\right)=I_M\left(t\right)sin\left({\omega }_{s}t+\phi \left(t\right)\right)$ with $\phi \left(t\right)=\int ∆\omega \left(t\right)dt$ 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) ≈ 0.5V_MI_M(t)cos(φ(t)). In case energy delivery maximization is the goal, then $\int I_M\left(t\right)\cos \left(\phi \left(t\right)\right)dt$ should be maximized by the controller over the power burst period. Alternatively, $I_M\left(t\right)\cos \left(\phi \left(t\right)\right)$ 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). Another method to achieve a low-order envelope model based on the energetic coupling between the system inductance and capacitance was proposed in [22]. However, it is only valid in the close vicinity of the resonance. Consequently, a nonlinear reduced-order model capable of accurately capturing the system behavior under large signal variations is necessary. It must be emphasized that due to the extremely high current flow required for relatively short periods, pulsed-power applications in general, and fusion inverters in particular, are typically powered by pre-charged capacitor banks rather than constant-voltage sources [16]. Consequently, the inverter input voltage should be considered as an additional non-controllable state variable, increasing the overall system order while imposing the operational frequency to serve as the only controllable input to the system. When frequency control is adopted, operation in the inductive region is usually preferable as it allows attaining zero voltage switching, alleviating converter switch stresses and losses, thus enhancing the longevity and increasing the efficiency of the system [23].

Table 1. Modeling methods comparison.

Method	Valid in Steady State	Valid in Transients
Classical phasor-domain analysis [17]	Yes	No
Linearized reduced-order envelope modeling [21]	Yes	Small variations only
Energetic coupling envelope modeling [22]	No	Near resonance only
Proposed nonlinear reduced-order envelope model	No	Yes

summarizes the modeling methods presented above, indicating their validity range and applicability.

Taking all the above into account, the nonlinear envelope-modeling process of a capacitor-powered resonant inverter driving series RLC load with time-varying component values is presented in this paper. First, a full-order envelope model is derived. Then, model order reduction is carried out under the assumption of a slow-varying amplitude derivative and phase (SVADP), as proposed in [21], yet without linearization, resulting in a novel nonlinear reduced-order envelope model, valid both at the resonance frequency and off-resonance. In order to demonstrate the validity of the proposed approach, the device-level switched model of the capacitor-powered resonant inverter driving series RLC load with time-varying component values and the corresponding reduced-order envelope model are simulated concurrently, and close matching between the outcomes is demonstrated under different operating conditions.

The rest of this paper is organized as follows. The system under consideration and basic assumptions are introduced in Section 2. The derivation of the full-order nonlinear system envelope model is carried out in Section 3. The proposed model reduction process is revealed in Section 4 and verified by simulations in Section 5. This paper is concluded in Section 6.

2. System Under Consideration and Basic Assumptions

A full-bridge inverter [24,25] feeding a generalized series-connected resistor–inductor–capacitor (RLC) network with time-varying component values [26], as shown in Figure 3, is considered in this work. Due to the nature of pulsed-power applications, the pre-charged DC-side capacitance C_in functions as the only system energy source within a single pulse period. The values of the RLC network components are assumed to vary in time according to

$$\begin{array}{l}R(t)\hspace{0.33em}=\hspace{0.33em}{R}_0+\Delta R(t)\hspace{0.17em}\left[\Omega \right]\hspace{0.33em}\\ L(t) \hspace{0.33em}=\hspace{0.33em}{L}_0+\Delta L(t)\hspace{0.17em}\left[\mathrm{H}\right]\hspace{0.33em}\\ C(t) \hspace{0.33em}=\hspace{0.33em}{C}_0+\Delta C(t)\hspace{0.17em}\left[\mathrm{F}\right]\hspace{0.33em}\end{array},$$

(1)

where the first (zero-indexed) right-hand side term denotes the corresponding constant nominal value, and the second right-hand term signifies the matching time-varying quantity. In addition, the practical relative rate of change in R(t), L(t), and C(t) is significantly lower than the angular converter switching frequency ω_s(t). Consequently, the RLC network component values would be considered constant over a single switching period T_s = 1/f_s [s] with f_s(t) [Hz] = ω_s(t)/2$\pi$. The DC-side capacitance C_in is assumed to be constant and known, pre-charged to an initial voltage of v_in(0) = V₀ prior to the pulse creation at t = 0. During operation, the DC-side capacitance discharges while providing energy to the resistance R(t). Nevertheless, the C_in is typically high enough to assume that the average value of v_in(t) remains unchanged during a single switching period T_s. The converter switches operate in a complementary fashion with a duty cycle of 50% so that their output voltage v(t) may be approximated by the following square-wave within a single switching period:

$$v(t)=\left\{\begin{array}{l}{v}_{in}(t),\hspace{0.33em}0\le t\le {\displaystyle \frac{T_s(t)}{2}}\hfill \\ -v_{in}(t),\hspace{0.33em}{\displaystyle \frac{ T_s(t)}{2}}\le t\le T_s(t)\hfill \end{array}\right.,$$

(2)

where switch parasitics are neglected for simplicity. In addition, note that the converter switching frequency may vary in general, yet the corresponding relative rate of change is significantly lower than ω_s(t).

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):

$$v(t)=\underset{v_R(t)}{\underbrace{R(t)i(t)}}+v_L(t)+v_C(t).$$

(3)

The voltages across the inductance and capacitance are related to the AC-side current i(t) as

$$v_L(t)={\displaystyle \frac{ d\left(L(t)i(t)\right)}{dt}}={\displaystyle \frac{ dL(t)}{dt}}i(t)+{\displaystyle \frac{di(t)}{dt}}L(t),$$

(4)

and

$$i(t)={\displaystyle \frac{ d\left(C(t)v_C(t)\right)}{dt}}={\displaystyle \frac{ dC(t)}{dt }}{v}_C(t)+{\displaystyle \frac{d{v}_C(t)}{dt}}C(t),$$

(5)

respectively. Meanwhile, v(t) may be described (cf. (2)) by the following Fourier series:

$$\begin{array}{cc}v(t)={\displaystyle \sum _n}{A}_n\sin \left({\displaystyle \int {\omega }_s(t)dt+{\theta }_n}\right),& A_n={\displaystyle \frac{4v_{in}}{n\pi }}\sin \left({\displaystyle \frac{n\pi }{2}}\right).\end{array}$$

(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

$$\begin{array}{cc}v(t) \approx v_1(t)=A_1(t)\sin \left({\theta }_s(t)+{\theta }_1\right),& {\theta }_s(t)={\displaystyle \int {\omega }_s(t)dt.}\end{array}$$

(7)

with (where the corresponding phase is used as the reference hereafter)

$$\begin{array}{cc}{A}_1(t)={\displaystyle \frac{4v_{in}(t)}{\pi }},& {\theta }_1=0.\end{array}$$

(8)

Consequently, the general expressions for the AC-side capacitor voltage and current are given by

$$v_C(t)=V_M(t)\sin \left({\theta }_s(t)+{\phi }_v(t)\right)=V_M(t)\cos \left({\phi }_v(t)\right)\sin \left({\theta }_s(t)\right)+V_M(t)\sin \left({\phi }_v(t)\right)\cos \left({\theta }_s(t)\right)$$

(9)

and

$$i(t)=I_M(t)\sin \left({\theta }_s(t)+{\phi }_i(t)\right)=I_M(t)\cos \left({\phi }_i(t)\right)\sin \left({\theta }_s(t)\right)+I_M(t)\sin \left({\phi }_i(t)\right)\cos \left({\theta }_s(t)\right),$$

(10)

respectively [29], where V_M(t) and I_M(t) stand for the corresponding magnitudes, and φ_v(t) and φ_i(t) symbolize the corresponding phases, denoting

$$\begin{array}{cc}{V}_M(t)\cos \left({\phi }_v(t)\right)\triangleq v_C^s(t), & V_M(t)\sin \left({\phi }_v(t)\right)\triangleq v_C^c(t)\end{array},$$

(11)

and

$$\begin{array}{cc}{I}_M(t)\cos \left({\phi }_i(t)\right)\triangleq i^s(t), & I_M(t)\sin \left({\phi }_i(t)\right)\triangleq i^c(t)\end{array},$$

(12)

Equations (9) and (10) may be rewritten as

$$v_C(t)=v_C^s(t)\sin \left({\theta }_s(t)\right)+v_C^c(t)\cos \left({\theta }_s(t)\right)$$

(13)

and

$$i(t)=i^s(t)\sin \left({\theta }_s(t)\right)+i^c(t)\cos \left({\theta }_s(t)\right),$$

(14)

respectively. The time-domain derivatives of (13) and (14) are then, respectively, given by

$${\displaystyle \frac{d{v}_C(t)}{dt}}=\left[{\displaystyle \frac{d{v}_C^s(t)}{dt}}-{\omega }_s(t)v_C^c\left(t\right)\right]\sin \left({\theta }_s(t)\right)+\left[{\displaystyle \frac{d{v}_C^c(t)}{dt}}+{\omega }_s(t)v_C^s\left(t\right)\right]\cos \left({\theta }_s(t)\right),$$

(15)

and

$${\displaystyle \frac{di(t)}{dt}}=\left[{\displaystyle \frac{d{i}^s(t)}{dt}}-{\omega }_s(t)i^c(t)\right]\sin \left({\theta }_s(t)\right)+\left[{\displaystyle \frac{d{i}^c(t)}{dt}}+{\omega }_s(t)i^s(t)\right]\cos \left({\theta }_s(t)\right).$$

(16)

Combining (13)–(16) with (3)–(5) and (7) yields

$$\begin{array}{l}{\displaystyle \frac{4v_{in}(t)}{\pi }}\sin \left({\theta }_s(t)\right)=\left(R(t)+{\displaystyle \frac{dL(t)}{dt}}\right)\left[i^s(t)\sin \left({\theta }_s(t)\right)+i^c(t)\cos \left({\theta }_s(t)\right)\right]\\ +L(t)\left[{\displaystyle \frac{d{i}^s(t)}{dt}}-{\omega }_s(t)i^c(t)\right]\sin \left({\theta }_s(t)\right)+L(t)\left[{\displaystyle \frac{d{i}^c(t)}{dt}}+{\omega }_s(t)i^s(t)\right]\cos \left({\theta }_s(t)\right)\\ +v_C^s(t)\sin \left({\theta }_s(t)\right)+v_C^c(t)\cos \left({\theta }_s(t)\right)\end{array}$$

(17)

and

$$\begin{array}{l}{i}^s(t)\sin \left({\theta }_s(t)\right)+i^c(t)\cos \left({\theta }_s(t)\right)={\displaystyle \frac{dC(t)}{dt}}\left[v_C^s(t)\sin \left({\theta }_s(t)\right)+v_C^c(t)\cos \left({\theta }_s(t)\right)\right]+\\ C(t)\left[\left({\displaystyle \frac{d{v}_C^s(t)}{dt}}-{\omega }_s(t)v_C^c(t)\right)\sin \left({\theta }_s(t)\right)+\left({\displaystyle \frac{d{v}_C^c(t)}{dt}}+{\omega }_s(t)v_C^s(t)\right)\cos \left({\theta }_s(t)\right)\right],\end{array}$$

(18)

respectively. Separating (17) and (18) into sine and cosine terms gives

$$\begin{array}{l}{\displaystyle \frac{4v_{in}(t)}{\pi }}\sin \left({\theta }_s(t)\right)=\\ \left(R(t)+{\displaystyle \frac{dL(t)}{dt}}\right)i^s(t)\sin \left({\theta }_s(t)\right)\hspace{0.17em}+L(t)\left({\displaystyle \frac{d{i}^s(t)}{dt}}-{\omega }_s(t)i^c(t)\right)\sin \left({\theta }_s(t)\right)+v_C^s(t)\sin \left({\theta }_s(t)\right),\end{array},$$

(19)

$$\begin{array}{l}0=\\ \left(R(t)+{\displaystyle \frac{dL(t)}{dt}}\right)i^c(t)\cos \left({\theta }_s(t)\right)+L(t)\left({\displaystyle \frac{d{i}^c(t)}{dt}}+{\omega }_s(t)i^s(t)\right)\cos \left({\theta }_s(t)\right)+v_C^c(t)\cos \left({\theta }_s(t)\right),\end{array}$$

(20)

and

$$i^s(t)\sin \left({\theta }_s(t)\right)=C(t)\left({\displaystyle \frac{d{v}_C^s(t)}{dt}}-{\omega }_s(t)v_C^c(t)\right)\sin \left({\theta }_s(t)\right)+{\displaystyle \frac{dC(t)}{dt}}{v}_C^s(t)\sin \left({\theta }_s(t)\right),$$

(21)

$$i^c(t)\cos \left({\theta }_s(t)\right)=C(t)\left({\displaystyle \frac{d{v}_C^c(t)}{dt}}+{\omega }_s(t)v_C^s(t)\right)\cos \left({\theta }_s(t)\right)+{\displaystyle \frac{dC(t)}{dt}}{v}_C^c(t)\cos \left({\theta }_s(t)\right),$$

(22)

respectively. Equating the coefficients at both sides of each equation, a set of four envelope dynamics equations with corresponding general initial conditions is established as [30,31]

$$\begin{array}{cc}{\displaystyle \frac{d{i}^s(t)}{dt}}={\omega }_s(t)i^c(t)+{\displaystyle \frac{1}{L(t)}}\left[{\displaystyle \frac{4v_{in}(t)}{\pi }}-\left(R(t)+{\displaystyle \frac{dL(t)}{dt}}\right)i^s(t)-v_C^s(t)\right], & i^s(0)=I_0^s,\end{array}$$

(23)

$$\begin{array}{cc}{\displaystyle \frac{d{i}^c(t)}{dt}}=-{\omega }_s(t)i^s(t)-{\displaystyle \frac{1}{L(t)}}\left[\left(R(t)+{\displaystyle \frac{dL(t)}{dt}}\right)i^c(t)+v_C^c(t)\right],& i^c(0)=I_0^c\end{array}$$

(24)

$$\begin{array}{cc}{\displaystyle \frac{d{v}_C^s(t)}{dt}}={\displaystyle \frac{1}{C(t)}}{i}^s(t)+{\omega }_s(t)v_C^c(t)-{\displaystyle \frac{dC(t)}{dt}}{v}_c^c(t), & v_C^s(0)=V_{C0}^s,\end{array}$$

(25)

$$\begin{array}{cc}{\displaystyle \frac{d{v}_C^c(t)}{dt}}={\displaystyle \frac{1}{C(t)}}{i}^c(t)-{\omega }_s(t)v_C^s(t)-{\displaystyle \frac{dC(t)}{dt}}{v}_C^c(t),& v_C^c(0)=V_{C0}^c.\end{array}$$

(26)

Moreover, the following holds, according to (11) and (12):

$$V_M(t)=\sqrt{{\left(v_C^s(t)\right)}^2+{\left(v_C^c(t)\right)}^2},$$

(27)

$$I_M(t)=\sqrt{{\left(i^s(t)\right)}^2+{\left(i^c(t)\right)}^2},$$

(28)

$${\phi }_v(t)=\arctan \left({\displaystyle \frac{v_C^c(t)}{v_C^s(t)}}\right),$$

(29)

$${\phi }_i(t)=\arctan \left({\displaystyle \frac{i^c(t)}{i^s(t)}}\right).$$

(30)

In order to find the relation between the AC-side envelope dynamics described by (23)–(26) and the DC-side inverter voltage, note that the DC-side current and instantaneous power are given by

$$i_{in}(t)=C_{in}{\displaystyle \frac{d{v}_{in}(t)}{dt}}$$

(31)

and

$$p_{in}(t)=v_{in}(t)i_{in}(t)=v_{in}(t)C_{in}{\displaystyle \frac{d{v}_{in}(t)}{dt}},$$

(32)

respectively. In case of lossless conversion,

$${\overline{p}}_{in}(t)={\overline{p}}_{out}(t)$$

(33)

with

$$\overline{x}(t)={\displaystyle \frac{1}{T_s}}{\displaystyle \underset{t}{\overset{t+T_s}{\int }}x(u)du}$$

(34)

denote the average value of x(t) over T_s. The average output and input converter power are given by (cf. (10))

$${\overline{p}}_{out}(t)=R(t){\left({\displaystyle \frac{I_M(t)}{\sqrt{2}}}\right)}^2$$

(35)

and

$${\overline{p}}_{in}\left(t\right)={\overline{v}}_{in}\left(t\right)C_{in}{\displaystyle \frac{d{\overline{v}}_{in}\left(t\right)}{dt}},$$

(36)

respectively. Equations (35) and (36) yield the input voltage dynamics given by

$${\displaystyle \frac{d{\overline{v}}_{in}\left(t\right)}{dt}}=-{\displaystyle \frac{R\left(t\right)}{2{\overline{v}}_{in}\left(t\right)C_{in}}}{\left(I_M\left(t\right)\right)}^2, \hspace{0.17em}{\overline{v}}_{in}\left(0\right)=V_0$$

(37)

with the corresponding general initial condition. The set of state-space Equations (23)–(26) and (37) and output Equations (27)–(30) form a full (fifth-order) envelope model of the system [32]. The corresponding block diagram is depicted in Figure 4.

4. Envelope Model Order Reduction

The system under consideration (cf. Figure 3) includes a single AC-side capacitor and a single AC-side inductor, indicating second-order AC-side dynamics. Considering the DC-side capacitor, third-order overall system dynamics are expected. Nonetheless, a fifth-order envelope model was derived in the preceding section. This stemmed from the nature of the sine/cosine components’ separation process adopter for the envelope dynamics capturing process, leading to system order augmentation. Therefore, original (third) model order restoration would be beneficial. The fourth-order set of equations describing the AC-side model given by (23)–(26) may be represented by the following general form:

$${\displaystyle \frac{d\mathit{x}(t)}{dt}}=\mathit{A}(t)\mathit{x}(t)+\mathit{B}(t)u(t),$$

(38)

where

$$\mathit{x}(t)={\left(\begin{array}{cc}\begin{array}{ccc}{i}^c(t)& i^s(t)& v_C^c(t)\end{array}& v_C^s(t)\end{array}\right)}^T,$$

(39)

$$\mathit{A}(t)=\left(\begin{array}{cccc}-{\displaystyle \frac{R(t)+\frac{dL(t)}{dt}}{L(t)}}& -{\omega }_s(t)& -{\displaystyle \frac{1}{L(t)}}& 0\\ {\omega }_s(t)& -{\displaystyle \frac{R(t)+\frac{dL(t)}{dt}}{L(t)}}& 0& -{\displaystyle \frac{1}{L(t)}}\\ {\displaystyle \frac{1}{C(t)}}& 0& -{\displaystyle \frac{\frac{dC(t)}{dt}}{C(t)}}& -{\omega }_s(t)\\ 0& {\displaystyle \frac{1}{C(t)}}& {\omega }_s(t)& -{\displaystyle \frac{\frac{dC(t)}{dt}}{C(t)}}\end{array}\right),$$

(40)

$$\begin{array}{cc}\mathit{B}(t)={\left(\begin{array}{cc}\begin{array}{ccc}0& {\displaystyle \frac{4}{\pi L(t)}}& 0\end{array}& 0\end{array}\right)}^{\mathrm{T}},& u(t)={\overline{v}}_{in}(t)\end{array}.$$

(41)

The order of the system (38) is dictated by the size of matrix A(t). In order to reduce it, the elimination of some state equations is needed. Note that the set given in (23)–(26) may also be expressed as

$$\left(\begin{array}{c}{\displaystyle \frac{d{\mathit{x}}_1(t)}{dt}}\\ {\displaystyle \frac{d{\mathit{x}}_2(t)}{dt}}\end{array}\right)=\left(\begin{array}{cc}{\mathit{A}}_{11}(t)& {\mathit{A}}_{12}(t)\\ {\mathit{A}}_{21}(t)& {\mathit{A}}_{22}(t)\end{array}\right)\left(\begin{array}{c}{\mathit{x}}_1(t)\\ {\mathit{x}}_2(t)\end{array}\right)+\left(\begin{array}{c}{\mathit{B}}_1(t)\\ {\mathit{B}}_2\end{array}\right)u(t),$$

(42)

where

$$\begin{array}{cc}{\mathit{x}}_1(t)={\left(\begin{array}{cc}{i}^c(t)& i^s(t)\end{array}\right)}^T,& {\mathit{x}}_2(t)={\left(\begin{array}{cc}{v}_C^c(t)& v_C^s(t)\end{array}\right)}^T,\end{array}$$

(43)

$$\begin{array}{cc}{\mathit{A}}_{11}(t)=\left(\begin{array}{cc}-{\displaystyle \frac{R(t)+\frac{dL(t)}{dt}}{L(t)}}& -{\omega }_s(t)\\ {\omega }_s(t)& -{\displaystyle \frac{R\left(t\right)+\frac{dL(t)}{dt}}{L(t)}}\end{array}\right), & {\mathit{A}}_{12}(t)=\left(\begin{array}{cc}-{\displaystyle \frac{1}{L(t)}}& 0\\ 0& -{\displaystyle \frac{1}{L(t)}}\end{array}\right),\end{array}$$

(44)

$$\begin{array}{cc}{\mathit{A}}_{21}(t)=\left(\begin{array}{cc}{\displaystyle \frac{1}{C(t)}}& 0\\ 0& {\displaystyle \frac{1}{C(t)}}\end{array}\right),& {\mathit{A}}_{22}(t)=\left(\begin{array}{cc}-{\displaystyle \frac{\frac{dC(t)}{dt}}{C(t)}}& -{\omega }_s(t)\\ {\omega }_s(t)& -{\displaystyle \frac{\frac{dC(t)}{dt}}{C(t)}}\end{array}\right)\end{array},$$

(45)

$${\mathit{B}}_1(t)=\left(\begin{array}{c}0\\ {\displaystyle \frac{4}{\pi L(t)}}\end{array}\right), {\mathit{B}}_2=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

(46)

The main interest is in the DC-side-to-AC-side power transfer by means of the AC-side current, i.e., the AC-side current-related state variables x₁(t) are the main interest, while the AC-side capacitor voltage-related state variables x₂(t) are of secondary importance. Hence, in case a linear function F(t) satisfying

$${\displaystyle \frac{d{\mathit{x}}_2(t)}{dt}}=\mathit{F}(t){\displaystyle \frac{d{\mathit{x}}_1(t)}{dt}}$$

(47)

exists, substituting it into (42) would yield

$${\mathit{x}}_2(t)={\mathit{A}}_{22}^{-1}(t)\left(\mathit{F}(t){\displaystyle \frac{d{\mathit{x}}_1(t)}{dt}}-{\mathit{A}}_{21}(t){\mathit{x}}_1(t)-{\mathit{B}}_{2}u(t)\right).$$

(48)

Combining (48) with (42) would eliminate the AC-side capacitor voltage-related state variables x₂(t) from (42), reducing it to be a function of the state variable vector of interest x₁(t) only:

$${\displaystyle \frac{d{\mathit{x}}_1(t)}{dt}}=\hspace{0.17em}{\left(\mathit{I}-{\mathit{A}}_{12}(t){\mathit{A}}_{22}(t)\mathit{F}(t)\right)}^{-1}\left[\left({\mathit{A}}_{11}(t)-{\mathit{A}}_{12}(t){\mathit{A}}_{22}^{-1}(t){\mathit{A}}_{21}(t)\right){\mathit{x}}_1(t)+\left({\mathit{B}}_1(t)-{\mathit{A}}_{12}(t){\mathit{A}}_{22}^{-1}(t){\mathit{B}}_2\right)u(t)\right].$$

(49)

In order to obtain the function F(t), a methodology assuming the slow-varying amplitude’s derivative and phase (SVADP) proposed in [21] is adopted hereafter. In the system under consideration, the SVADP is adopted by assumed that the capacitor voltage phase and derivative of the capacitor voltage magnitude are slow-varying functions of time. The derivative of (9) is given by

$${\displaystyle \frac{d{v}_C(t)}{dt}}={\displaystyle \frac{d{V}_M(t)}{dt}}\sin \left({\theta }_s(t)+{\phi }_v(t)\right)+V_M(t)\left({\omega }_s(t)+{\displaystyle \frac{d{\phi }_v(t)}{dt}}\right)\cos \left({\theta }_s(t)+{\phi }_v(t)\right),$$

(50)

reducing to

$${\displaystyle \frac{d{v}_C(t)}{dt}}\approx V_M(t){\omega }_s(t)\cos \left({\theta }_s(t)+{\phi }_v(t)\right),$$

(51)

under the SVADP assumption. Reformulating (51) according to (13) yields

$${\displaystyle \frac{d{v}_C(t)}{dt}}\approx v_C^s(t){\omega }_s(t)\cos \left({\theta }_s(t)\right)-v_C^c(t){\omega }_s(t)\sin \left({\theta }_s(t)\right).$$

(52)

Further combining this with (5) while adopting the basic assumptions given in Section 2 yields

$$i(t)\approx C(t){\displaystyle \frac{d{v}_C(t)}{dt}}=C(t){\omega }_s(t)v_C^s(t)\cos \left({\theta }_s(t)\right)-C(t){\omega }_s(t)v_C^c(t)\sin \left({\theta }_s(t)\right).$$

(53)

Reformulating (53) according to (14) yields

$$v_C^s(t)\cos \left({\theta }_s(t)\right)-v_C^c(t)\sin \left({\theta }_s(t)\right)={\displaystyle \frac{1}{C(t){\omega }_s(t)}}\left[i^c(t)\cos \left({\theta }_s(t)\right)+i^s(t)\sin \left({\theta }_s(t)\right)\right].$$

(54)

Separating into sine and cosine terms and equating the coefficients on both sides of each equation gives

$$\begin{array}{cc}{v}_C^s(t)={\displaystyle \frac{1}{C(t){\omega }_s(t)}}{i}^c(t),& v_C^c(t)=-{\displaystyle \frac{1}{C(t){\omega }_s(t)}}{i}^s(t)\end{array}.$$

(55)

Applying the derivative to (55) results in

$${\displaystyle \frac{d{v}_C^s(t)}{dt}}=-{\displaystyle \frac{\left(\frac{dC(t)}{dt}{\omega }_s(t)+\frac{d{\omega }_s(t)}{dt}C(t)\right)}{{\left(C(t){\omega }_s(t)\right)}^2}}{i}^c(t)+{\displaystyle \frac{1}{C(t){\omega }_s(t)}}{\displaystyle \frac{d{i}^c(t)}{dt}}$$

(56)

and

$${\displaystyle \frac{d{v}_C^c(t)}{dt}}={\displaystyle \frac{\left(\frac{dC(t)}{dt}{\omega }_s(t)+\frac{d{\omega }_s(t)}{dt}C(t)\right)}{{\left(C(t){\omega }_s(t)\right)}^2}}{i}^s(t)-{\displaystyle \frac{1}{C(t){\omega }_s(t)}}{\displaystyle \frac{d{i}^s(t)}{dt}},$$

(57)

respectively. Applying the SVADP assumption, (56) and (57) reduce to

$${\displaystyle \frac{d{v}_C^c(t)}{dt}}=-{\displaystyle \frac{1}{C(t){\omega }_s(t)}}{\displaystyle \frac{d{i}^s(t)}{dt}},$$

(58)

$$\hspace{0.17em}\hspace{0.17em} {\displaystyle \frac{d{v}_C^s(t)}{dt}}={\displaystyle \frac{1}{C(t){\omega }_s(t)}}{\displaystyle \frac{d{i}^c(t)}{dt}}.$$

(59)

Comparing (58) with (47), the corresponding function F(t) is obtained as

$$\mathit{F}(t)=\left(\begin{array}{cc}0& -{\displaystyle \frac{1}{C(t){\omega }_s(t)}}\\ {\displaystyle \frac{1}{C(t){\omega }_s\left(t\right)}}& 0\end{array}\right).$$

(60)

Combining (43)–(46) with (49) and (60) yields a set of two differential equations related to the AC-side current state variables:

$$\begin{array}{l}{\displaystyle \frac{d{i}^c(t)}{dt}}=\\ -{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s^2(t)-\frac{1}{C(t)L(t){\omega }_s^2(t)}\right)i^s(t)+C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}{i}^c(t)}{1+C(t)L(t){\omega }_s^2(t)}},\end{array}$$

(61)

$$\begin{array}{l}{\displaystyle \frac{d{i}^s(t)}{dt}}=\\ {\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s^2(t)}\right)i^c(t)-C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}{i}^s(t)}{1+C(t)L(t){\omega }_s^2(t)}}+{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left(\frac{4{\overline{v}}_{in}(t)}{\pi L(t)}\right)}{1+C(t)L(t){\omega }_s^2(t)}}.\end{array}$$

(62)

The state Equations (37), (61), and (62) and the output Equations (28) and (30) form a reduced (third-order) envelope model of the system. The corresponding block diagram is depicted in Figure 5.

It should be emphasized that state variables i^c(t) and i^s(t) neither have any physical meaning nor can be directly measured. It is, thus, proposed to perform a change of variables so that the measurable quantities (the current magnitude and phase) serve as the state variables. According to (30),

$$\tan \left({\phi }_i(t)\right)={\displaystyle \frac{i^c(t)}{i^s(t)}}.$$

(63)

The corresponding derivative is then

$${\displaystyle \frac{d}{dt}}\tan \left({\phi }_i(t)\right)={\displaystyle \frac{\left(\frac{d{i}^c(t)}{dt}{i}^s(t)-\frac{d{i}^s(t)}{dt}{i}^c(t)\right)}{{\left(i^s(t)\right)}^2}},$$

(64)

Merging (63) and (64) yields

$${\displaystyle \frac{d}{dt}}\tan \left({\phi }_i(t)\right)={\displaystyle \frac{1}{i^s(t)}}\left({\displaystyle \frac{d{i}^c(t)}{dt}}-\tan \left({\phi }_i(t)\right){\displaystyle \frac{d{i}^s(t)}{dt}}\right).$$

(65)

Combining (63) and (65) with (61) and (62) yields

$$\begin{array}{l}{\displaystyle \frac{d}{dt}}\tan \left({\phi }_i(t)\right)=\\ -{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s(t)}\right)\left(1+{\tan }^2\left({\phi }_i(t)\right)\right)}{C\left(t\right)L\left(t\right){\omega }_s^2\left(t\right)+1}}-{\displaystyle \frac{\frac{4{\overline{v}}_{in}(t)}{\pi L(t)}C(t)L(t){\omega }_s^2(t)\tan \left({\phi }_i(t)\right)}{i^s(t)\left(C(t)L(t){\omega }_s^2(t)+1\right)}}.\end{array}$$

(66)

The derivative of (28) is given by

$${\displaystyle \frac{d{I}_M(t)}{dt}}={\displaystyle \frac{i^s(t)\frac{d{i}^s(t)}{dt}+i^c(t)\frac{d{i}^c(t)}{dt}}{\sqrt{{\left(i^s(t)\right)}^2+{\left(i^c(t)\right)}^2}}}.$$

(67)

Combining (63), (65), and (67) with (61) and (62) yields

$$\begin{array}{l}{\displaystyle \frac{d{I}_M(t)}{dt}}=\\ {\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s^2(t)}\right)\tan \left({\phi }_i(t)\right)-C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}}{\frac{1}{i^s(t)}\left(C(t)L(t){\omega }_s^2(t)+1\right)\sqrt{1+{\tan }^2\left({\phi }_i(t)\right)}}}+\\ +{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left(\frac{4{\overline{v}}_{in}(t)}{\pi L(t)}\right)\frac{1}{i^s(t)}}{\frac{1}{i^s(t)}\left(C(t)L(t){\omega }_s^2(t)+1\right)\sqrt{1+{\tan }^2\left({\phi }_i(t)\right)}}}-\\ -{\displaystyle \frac{\tan \left({\phi }_i(t)\right)\left[C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s^2(t)}\right)+C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}\tan \left({\phi }_i(t)\right)\right]}{\frac{1}{i^s(t)}\left(C(t)L(t){\omega }_s^2(t)+1\right)\sqrt{1+{\tan }^2\left({\phi }_i(t)\right)}}}.\end{array}$$

(68)

Meanwhile, the following relation holds according to (28) and (63):

$$i^s(t)=\sqrt{{\displaystyle \frac{I_M^2(t)}{1+{\tan }^2\left({\phi }_i(t)\right)}}}.$$

(69)

Substituting (69) into (66) and (68) and rearranging yields a modified reduced (third-order) nonlinear envelope model of the system, where subscript “r” is added to the state variables for the sake of differentiation from the full-order model counterparts:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\tan \left({\phi }_{ir}(t)\right)=& \\ & -{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s(t)}\right)\left(1+{\tan }^2\left({\phi }_{ir}(t)\right)\right)}{C(t)L(t){\omega }_s^2(t)+1}}\hfill \\ & -{\displaystyle \frac{\sqrt{\frac{1+{\tan }^2\left({\phi }_{ir}(t)\right)}{I_{Mr}^2(t)}}·\frac{4{\overline{v}}_{inr}(t)}{\pi L(t)}C(t)L(t){\omega }_s^2(t)\tan \left({\phi }_{ir}(t)\right)}{C(t)L(t){\omega }_s^2(t)+1}}, \tan \left({\phi }_{ir}(0)\right)=\tan \left({\phi }_0\right)\hfill \end{array}$$

(70)

$$\begin{array}{l}{\displaystyle \frac{d{I}_{Mr}(t)}{dt}}=\\ {\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s(t)}\right)\tan \left({\phi }_{ir}(t)\right)-C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}}{\left(C(t)L(t){\omega }_s^2(t)+1\right)\frac{\left(1+{\tan }^2\left({\phi }_{ir}(t)\right)\right)}{I_{Mr}(t)}}}\\ +{\displaystyle \frac{C(t)L(t){\omega }_s^2(t)\left(\frac{4{\overline{v}}_{inr}(t)}{\pi L(t)}\right)\sqrt{1+{\tan }^2\left({\phi }_{ir}(t)\right)}}{\left(C(t)L(t){\omega }_s^2(t)+1\right)\left(1+{\tan }^2\left({\phi }_{ir}(t)\right)\right)}}-\\ -{\displaystyle \frac{\tan \left({\phi }_{ir}(t)\right)C(t)L(t){\omega }_s^2(t)\left({\omega }_s(t)-\frac{1}{C(t)L(t){\omega }_s(t)}\right)}{\left(C(t)L(t){\omega }_s^2(t)+1\right)\frac{\left(1+{\tan }^2\left({\phi }_{ir}(t)\right)\right)}{I_{Mr}(t)}}}\\ -{\displaystyle \frac{{\tan }^2\left({\phi }_{ir}(t)\right)C(t)L(t){\omega }_s^2(t)\frac{R(t)}{L(t)}}{\left(C(t)L(t){\omega }_s^2(t)+1\right)\frac{\left(1+{\tan }^2\left({\phi }_{ir}(t)\right)\right)}{I_{Mr}(t)}}}, I_{Mr}(0)=I_{M0}\end{array}$$

(71)

$${\overline{v}}_{in,r}(t)C_{in}{\displaystyle \frac{d{\overline{v}}_{in,r}(t)}{dt}}=R\left(t\right){\left({\displaystyle \frac{I_{Mr}(t)}{\sqrt{2}}}\right)}^2, \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{v}}_{inr}(0)=V_{in0}$$

(72)

with matching general initial conditions. The resulting block diagram is shown in Figure 6.

5. Verification

In order to validate the derived modified reduced envelope model, the circuit shown in Figure 3 was simulated using the PSIM 2025 software concurrently using the component-level circuit and the equations of the modified reduced envelope model. The corresponding values of the system parameters are summarized in

Table 2. System parameter values.

Parameter	Value	Unit
V0	87	V
Cin	80	mF
f1	500	Hz
IM0	0	A
tan(φ0)	0	----
R0	61	mΩ
L0	4.6	μH
C0	8.58	μF

. The simulations were carried out with the RLC network parameters, varying in time according to (cf. (1)):

$$\begin{array}{ccc}\Delta R(t)=R_1\sin \left(2\pi f_{1}t\right),& \Delta L(t)=L_1\sin \left(2\pi f_{1}t\right),& \Delta C(t)=C_1\sin \left(2\pi f_{1}t\right) .\end{array}$$

(73)

The nominal resonant frequency of the system is defined as

$$f_0={\displaystyle \frac{1}{2\pi \sqrt{C_0{L}_0}}}=25.3\hspace{0.17em}[\mathrm{kHz}].$$

(74)

Six simulations were carried out (referred to as Sim 1–Sim 6 in

Table 3. Varying parameter values.

Parameter	Simulation Parameters
	Sim 1	Sim 2	Sim 3	Sim 4	Sim 5	Sim 6
fs [Hz]	1.05f0	0.95f0	f0	f0	f0	f0
L1 [H]	0	0	0	0.05L0	0.05L0	0.05L0
C1 [F]	0	0	0	0	0.05C0	0.05C0
R1 [Ω]	0	0	0	0	0	0.05R0

, summarizing the corresponding parameter variations), with φ_v taken as the reference phase and the φ_i of the switched device-level model obtained using the method proposed in [33].

In the first three simulations, the parameter values of the RLC network components remained nominal and constant. The system behavior was then captured for an operational frequency above resonance (Sim 1, Figure 7; φ_i < 0), below resonance (Sim 2, Figure 8; φ_i > 0), and at resonance (Sim 3, Figure 9; φ_i = 0). For clarity, the zoomed waveforms of the DC-side converter input voltage, AC-side output converter voltage, and AC-side current at, above, and below resonance are depicted in Figure 10. In the next three simulations, the responses of the system operating at the resonant frequency to variations in the inductance only (Sim 4, Figure 11), inductance and capacitance (Sim 5, Figure 12), and all three parameters (Sim 5, Figure 13) were obtained. It is well-evident that the modified nonlinear reduced-order model output accurately followed the envelope behavior of the switched component-level circuit in all examined cases. Consequently, the proposed model may be adopted for establishing a feedback controller aiming to regulate the AC-side magnitude and/or phase.

6. Conclusions

This paper suggested a process of reduced-order nonlinear dynamic envelope modeling applied to a capacitor-powered resonant inverter feeding a time-varying series RLC load, often employed in pulsed-power applications. Such arrangements are particularly challenging due to the fact that their operation time is lower than the corresponding system settling time, imposing significant large-signal operation point variations, thus making classical linearization-based reduced models invalid. The derived third-order model (attained by applying an SVADP assumption, which also imposes the proposed methodology limitation) was proven to accurately represent the highly nonlinear time-varying behavior of the fifth-order model during transients, where conventional simplified models fall short. Future work on this subject will focus on a feedback controller design based on the derived nonlinear model of the system and the investigation of a possible negative interaction between the controller and the plant (see, e.g., [34]). In addition, a high-current resonant inverter prototype is currently being developed, so a feedback control algorithm designed according to the proposed methodology is expected to be validated experimentally in the near future.