Insights into an Angular-Motion Electromechanical-Switching Device: Characteristics, Behavior, and Modeling

Abstract

While extensive research has addressed electromechanical systems interacting with power electronic converters, most studies lack a unified modeling framework that simultaneously captures converter switching behavior, nonlinear dynamics, and linearized control-oriented representations. In particular, the dynamic interaction between two-level full-bridge converters and angular-motion electromechanical switching devices (EMDs) is often simplified or abstracted, thereby limiting control system design and frequency-domain analysis. This work presents a comprehensive dynamic modeling methodology for an angular-motion EMD driven by a full-bridge dc-dc converter. The modeling framework includes (i) a detailed nonlinear switching model, (ii) an averaged nonlinear model suitable for control design, and (iii) a small-signal linearized model for deriving transfer functions and evaluating system stability. The proposed models are rigorously validated through time-domain simulations and Bode frequency analysis, confirming both theoretical equilibrium points and dynamic characteristics such as resonant frequencies and phase margins. The results demonstrate strong consistency across the modeling hierarchy and reveal critical features—such as ripple-induced resonance and low-frequency coupling—that are essential for robust controller design. This framework established a foundational tool for advancing the control and optimization of electromechanical switching systems in high-performance applications.

1. Introduction

An electromechanical device represents the convergence of Maxwell’s electromagnetic theory and Newton’s classical mechanics, yielding a unified framework of dynamic equations. This theoretical synthesis characterizes electromechanical devices as integrated dynamic systems in which electrical and mechanical domains are intricately coupled through field interactions and motion-induced forces [1].

Concurrently, power electronic converters play an essential role in transforming fixed electrical input sources into regulated output signals tailored to specific application requirements. These converters enable efficient modulation of voltage, frequency, phase, and waveform characteristics, thereby enabling precise control across a broad range of electromechanical loads [2].

The interplay between electromechanical systems (EMSs) and power converters constitutes a fundamental interface that bridges the electrical and mechanical energy domains [3,4,5]. This integrated platform is central to a wide range of applications, including high-precision actuation, energy harvesting, automotive propulsion, and robotic manipulation.

Although the literature offers a wealth of analytical and simulation-based models for EMSs operating in conjunction with power converters [6,7,8,9,10,11,12], a significant gap remains in the development of switching models and their nonlinear-averaged and small-signal linear equivalents, particularly within integrated dynamic modeling frameworks. For instance, Reference [6] addresses the modeling and simulation of a hybrid rubber-tire gantry crane incorporating electromechanical subsystems and renewable energy integration, but it omits low-level switching behavior. In [7], the modeling and control of a coupled electromechanical system for ocean-wave energy recovery are developed using a linear permanent magnet generator. Although synchronization strategies are examined, the treatment of switching dynamics remains abstract. Reference [8] employs the Lagrangian formalism for EMS simulation, offering a powerful variational modeling framework capable of capturing energy interactions, constraints, and generalized coordinates. However, its application is largely limited to generalized electric machines and does not explicitly address switching converter dynamics. Reference [9] develops a MATLAB-Simulink^®, version R2019b model of an electromechanical system incorporating hydraulic and pneumatic components, including dc motors and actuators. While it effectively demonstrates system-level response analysis, it does not capture converter-switching phenomena. In [10], nonlinearities such as dry friction and elastic backlash are emphasized in servo electromechanical systems, leading to the rejection of conventional PID control in favor of nonlinear compensators. However, the power electronics interface is again simplified.

The study in [11] addresses actuation systems for aerospace and defense applications. It provides both linear and nonlinear control perspectives using motion-based equations but lacks detailed converter dynamics. Similarly, Reference [12] investigates elastic energy storage with synchronous machines, focusing on the mechanical-to-electrical transduction chain without exploring discrete-time switching effects. A distinct shift is observed in [13], which proposes a dynamic parameter adaptation method to improve the modeling of valve-switching behavior in semiconductor converters. This method accounts for inverse current phenomena and enhances simulation accuracy through real-time inductance and resistance adjustment.

In [14], an advanced brushless dc motor drive is presented featuring an LLC resonant dc-dc converter, SiC MOSFETs, and a nonlinear adaptive sliding controller. This integrated control strategy achieves superior harmonic rejection and energy efficiency, particularly in electric vehicle and automation applications. However, it focuses on general-purpose motors rather than angular motion-switching devices. Moreover, Reference [15] examines dc-dc converters in plasma arc applications. By modeling negative-resistance behavior and system instability, it highlights the importance of detailed small-signal and frequency-domain analysis but does not consider mechanical feedback.

Recent works have attempted to narrow this gap. For example, Reference [16] proposes a hybrid dynamical model for bistable electromagnetic actuators, combining equilibrium and stability analysis with an H-bridge drive. The authors in [17,18] design adaptive converter networks and high-bandwidth current control schemes for arrays of electromagnetic micro-actuators, integrating converter–actuator behavior. Reference [19] explores solenoid-based actuators with unknown parameters, developing a two-degree-of-freedom adaptive control framework based on an averaged model. In [20], buck-boost converters for capacitive actuator loads, using nonlinear averaging to define soft-charging strategies, are studied. Additionally, Reference [21] develops a comprehensive model of a permanent magnet dc motor driven by a full H-bridge converter, applied to a bidirectional conveyor system. The approach integrates state-space modeling with a feedforward control scheme and an extended state observer to enhance disturbance rejection and velocity tracking.

Despite these developments, a unified, full-order model that incorporates (i) switching dynamics, (ii) nonlinear averaging, and (iii) linearization around equilibrium points for angular-motion electromechanical switching devices (EMDs) driven by two-level full-bridge converters has not been reported in the literature. Therefore, this article presents a fully integrated modeling framework for an angular-motion EMD driven by a two-level full-bridge dc-dc converter. The contribution is threefold:

• It derives the full nonlinear switching model of the EMD-converter system, capturing detailed coupling between converter states and mechanical motion;
• It develops a nonlinear averaged model to extract meaningful dynamic behavior under high-frequency switching operations;
• It introduces small-signal linearization via Jacobian-based analysis, enabling transfer function derivation and closed-loop stability assessment in both time and frequency domains.

This comprehensive modeling methodology enables seamless system identification, controller design, and stability validation under switching regimes. Overall, the primary objective of this study is to establish a comprehensive mathematical and simulation framework for angular-motion EMDs coupled with two-level full-bridge dc-dc converters. This framework supports detailed stability and frequency-domain analysis using switching, averaged, and small-signal models and serves as a foundation for the future development of advanced control strategies for high-performance electromechanical systems.

The article is organized as follows. Section 2 introduces the topic and presents the topology and description of the device under study. In Section 3, the system modeling is developed. Section 4 and Section 5 present the stability analysis and frequency-domain analysis of the device, respectively, and finally, Section 6 and Section 7 show the simulation results and conclusions of the study.

2. Electromechanical-Switching Device Topology

The configuration of the EMD is shown in Figure 1 and comprises two integral stages: the EMS and the two-level full-bridge dc-dc converter (2LC).

The EMS consists of a torsion rod connected to a flywheel, with the connection facilitated by an L-type magnetic circuit. The torsion rod is characterized by torsional stiffness (k) and torsional damping (b) due to the gearing, and the flywheel provides polar inertia (J). The dynamic characteristics of the system include angular position θ(t) and its time derivatives, including angular velocity dθ(t)/dt and angular acceleration d²θ(t)/dt².

The magnetic assembly includes a single-loop inductor with constant resistance R and whose inductance L(θ(t)) varies linearly with θ(t): L(θ(t)) = A_L + B_L · θ(t), where A_L and B_L are constants. In particular, the driving coil is connected to the dynamic variables, i.e., voltage v_o(t) and current i_o(t).

The 2LC is powered by a DC power supply characterized by input voltage v_dc(t), internal resistance R_dc, and the dc current i_dc(t). This power source is fed into a DC link supported by a capacitor C_dc. In this context, dynamic variables are expressed as voltage v_Cdc(t) and current i₂(t). The full-bridge switching network interfaces with the dc link and is configured with two switches per leg, namely Q_x, where x ∈ {1, 2, 3, 4}.

Similar to a servomechanism [22], the EMD controls torsion rod angular positioning θ(t) by controlling v_o(t). The current i_o(t) induces a magnetic field in the L-type structure, which facilitates the coupling with the flywheel [23]. This coupling generates an electromagnetic torque T_e(t). The voltage v_o(t), under the control of the 2LC, is the final control variable of the EMS. The modulation of 2LC is controlled by the regulation of switches (Q_x, where x ∈ {1, 2, 3, 4}) over a given switching period T_s.

3. Modeling of the EMD

The derivation of the switching, nonlinear-averaged, and linear small-signal models of the EMD is presented based on the configuration shown in Figure 1.

3.1. Switching Model of the EMD

For the convenience of practical analysis, the model begins by isolating the EMS by virtually decoupling it from the 2LC. Applying Kirchhoff’s voltage law to the magnetic circuit yields:

$${\mathit{v}}_{\mathrm{o}}\left(\mathit{t}\right)=\mathit{R}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)+{\displaystyle \frac{\mathrm{d}\left(\mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)\right)}{\mathrm{d}\mathit{t}}}$$

(1)

Expanding the derivative term gives the following:

$${\mathit{v}}_{\mathrm{o}}\left(\mathit{t}\right)=\mathit{R}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)+{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)·{\displaystyle \frac{\partial \mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\partial \mathsf{\theta }\left(\mathit{t}\right)}}·{\displaystyle \frac{\mathrm{d}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\mathrm{d}\mathit{t}}}+\mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)·{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}$$

(2)

and multiplying both sides by i_o(t) yields the instantaneous electric power of the magnetic circuit P_e(t) which is given as follows:

$${\mathit{P}}_{\mathrm{e}}\left(\mathit{t}\right)={\mathit{v}}_{\mathrm{o}}\left(\mathit{t}\right)·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)=\mathit{R}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)+{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)·{\displaystyle \frac{\partial \mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\partial \mathsf{\theta }\left(\mathit{t}\right)}}·{\displaystyle \frac{\mathrm{d}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\mathrm{d}\mathit{t}}}+\mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)·{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}$$

(3)

The magnetic energy W_g(t) in the circuit is defined by

$${\mathit{W}}_{\mathrm{g}}\left(\mathit{t}\right)={\displaystyle \frac{1}{2}}·\mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)$$

(4)

according to [23,24]. The magnetic power P_g(t) is expressed as follows:

$${\mathit{P}}_{\mathrm{g}}\left(\mathit{t}\right)={\displaystyle \frac{\mathrm{d}{\mathit{W}}_{\mathrm{g}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{1}{2}}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)·{\displaystyle \frac{\partial \mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\partial \mathsf{\theta }\left(\mathit{t}\right)}}·{\displaystyle \frac{\mathrm{d}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\mathrm{d}\mathit{t}}}+\mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)·{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}$$

(5)

Considering P_e(t) and P_g(t), the power balance of the magnetic circuit is expressed as P_e(t) = P_m(t) + P_R(t) + P_g(t), where P_m(t) and P_R(t) = R∙i_o²(t) denote the mechanical power and energy dissipation through R, respectively. The expression for P_m(t) is then given by

$${\mathit{P}}_{\mathrm{m}}\left(\mathit{t}\right)={\displaystyle \frac{1}{2}}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)·{\displaystyle \frac{\partial \mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\partial \mathsf{\theta }\left(\mathit{t}\right)}}·{\displaystyle \frac{\mathrm{d}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\mathrm{d}\mathit{t}}}$$

(6)

For rotating devices, the mechanical power P_m(t) is defined as the product of angular velocity dθ(t)/dt and the mechanical torque T_m(t), expressed as P_m(t) = T_m(t)∙dθ(t)/dt.

Assuming a lossless magnetic circuit, T_m(t) = T_e(t), where T_e(t) is the electromagnetic torque derived as follows:

$${\mathit{T}}_{\mathrm{e}}\left(\mathit{t}\right)={\displaystyle \frac{1}{2}}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)·{\displaystyle \frac{\partial \mathit{L}\left(\mathsf{\theta }\left(\mathit{t}\right)\right)}{\partial \mathsf{\theta }\left(\mathit{t}\right)}}$$

(7)

Applying the torque balance to the torsion rod results in its dynamic expression, known as the torsional equation of motion [25], given as follows:

$$\mathit{J}·{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}{\mathit{t}}^2}}+\mathit{b}·{\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}+\mathit{k}·\mathsf{\theta }\left(\mathit{t}\right)={\mathit{T}}_{\mathrm{e}}\left(\mathit{t}\right)$$

(8)

By substituting the linear inductance L(θ(t)) into (2) and (8), updated versions of the dynamic model of the EMS are defined as follows:

$$\left\{\begin{array}{c}\mathit{J}·{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}{\mathit{t}}^2}}+\mathit{b}·{\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}+\mathit{k}·\mathsf{\theta }\left(\mathit{t}\right)={\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)\\ {\mathit{B}}_{\mathrm{L}}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)·{\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}+\left({\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·\mathsf{\theta }\left(\mathit{t}\right)\right)·{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}={\mathit{v}}_{\mathrm{o}}\left(\mathit{t}\right)-\mathit{R}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)\end{array}\right.$$

(9)

For the 2LC, the application of Kirchhoff’s voltage law leads to the dynamic expression:

$${\displaystyle \frac{\mathrm{d}{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{{\mathit{v}}_{\mathrm{dc}}\left(\mathit{t}\right)-{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)-{\mathit{R}}_{\mathrm{dc}}·{\mathit{i}}_2\left(\mathit{t}\right)}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}$$

(10)

which relates the input source to the dc link.

This equation results in the following set of expressions:

$$\left[\begin{array}{c}{\mathit{v}}_{\mathrm{o}}\left(\mathit{t}\right)\\ {\mathit{i}}_2\left(\mathit{t}\right)\end{array}\right]=\left[\begin{array}{cc}2·\left(\mathit{s}\left(\mathit{t}\right)-0.5\right)& 0\\ 0& 2·\left(\mathit{s}\left(\mathit{t}\right)-0.5\right)\end{array}\right]·\left[\begin{array}{c}{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)\\ {\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)\end{array}\right]$$

(11)

which relates the EMS to the 2LC. From here, the switching function s(t) is given by

$$\mathrm{s}\left(\mathit{t}\right)=\left\{\begin{array}{l}1, {\mathrm{Q}}_2 \mathrm{and} {\mathrm{Q}}_3 \mathrm{are} \mathrm{close} \mathrm{and} {\mathrm{Q}}_1 \mathrm{and} {\mathrm{Q}}_4 \mathrm{are} \mathrm{open}\\ \hspace{1em}0, \mathrm{otherwise} \end{array}\right.$$

(12)

The following expression:

$$\left\{\begin{array}{c}\mathit{J}·{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}{\mathit{t}}^2}}+\mathit{b}·{\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}+\mathit{k}·\mathsf{\theta }\left(\mathit{t}\right)={\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{\mathit{i}}_{\mathrm{o}}}^2\left(\mathit{t}\right)\\ {\mathit{B}}_{\mathrm{L}}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)·{\displaystyle \frac{\mathrm{d}\mathsf{\theta }\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}+\left({\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·\mathsf{\theta }\left(\mathit{t}\right)\right)·{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}=2·\left(\mathit{s}\left(\mathit{t}\right)-0.5\right)·{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)-\mathit{R}·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)\\ \begin{array}{c}{\displaystyle \frac{\mathrm{d}{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{{\mathit{v}}_{\mathrm{dc}}\left(\mathit{t}\right)-{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}-{\displaystyle \frac{2·{\mathit{R}}_{\mathrm{dc}}·\left(\mathit{s}\left(\mathit{t}\right)-0.5\right)·{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}\end{array}\end{array}\right.$$

(13)

is obtained by substituting (11) into (9) and (10), leading to the EMD switching model. Due to its inherent switching behavior, (13) is not directly applicable to linear control theory [22,26,27,28]. Therefore, an averaged model is required to address the discrete nature of the system as follows:

3.2. Nonlinear-Averaged Model of the EMD

Classical control theory is formulated for continuous systems and, therefore, requires the application of an averaging operator to transform discrete control variables (the switching functions) in (13) into continuous control variables (averaged switching functions or duty ratios) [26].

Averaging involves calculating the average values of the system variables over a switching period T_s. The averaging operator is defined as

$${⟨\mathit{x}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}={\displaystyle \frac{1}{{\mathit{T}}_{\mathrm{s}}}}·{\int }_{\mathit{t}-{\mathit{T}}_{\mathrm{s}}}^{\mathit{t}}\mathit{x}\left(\tau \right)·\mathrm{d}\tau$$

(14)

Here, <x(t)>_Ts represents any function. To minimize differences between actual and averaged values, especially for ac variables, a high switching frequency is essential, exceeding the frequency of the ac system (i.e., magnetic system) by a significant margin.

Substituting (14) into (13) yields the averaged model, which is defined as follows:

$$\left\{\begin{array}{c}\mathit{J}·{\displaystyle \frac{{\mathrm{d}}^2{⟨\mathsf{\theta }\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{\mathrm{d}{\mathit{t}}^2}}+\mathit{b}·{\displaystyle \frac{\mathrm{d}{⟨\mathsf{\theta }\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{\mathrm{d}\mathit{t}}}+\mathit{k}·{⟨\mathsf{\theta }\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}={\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{⟨{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}^2\\ {\mathit{B}}_{\mathrm{L}}·{⟨{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}·{\displaystyle \frac{\mathrm{d}{⟨\mathsf{\theta }\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{\mathrm{d}\mathit{t}}}+\left({\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{⟨\mathsf{\theta }\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}\right)·{\displaystyle \frac{\mathrm{d}{⟨{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{\mathrm{d}\mathit{t}}}=\\ =2·\left(\mathit{d}\left(\mathit{t}\right)-0.5\right)·{⟨{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}-\mathit{R}·{⟨{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}\\ {\displaystyle \frac{\mathrm{d}{⟨{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{{⟨{\mathit{v}}_{\mathrm{dc}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}-{⟨{\mathit{v}}_{{\mathrm{C}}_{\mathrm{dc}}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}-{\displaystyle \frac{2·{\mathit{R}}_{\mathrm{dc}}·\left(\mathrm{d}\left(\mathit{t}\right)-0.5\right)·{⟨{\mathit{i}}_{\mathrm{o}}\left(\mathit{t}\right)⟩}_{{\mathit{T}}_{\mathrm{s}}}}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}\end{array}\right.$$

(15)

Here, <s(t)>_Ts = d(t) is the duty ratio of the 2LC, which ranges continuously from 0 to 1. The nonlinear averaged model in (15), with second-order derivatives, can be simplified to a system of first-order differential equations based on state variables (SVs). The SVs are defined as x₁ = <i_o(t)>_Ts, x₂ = <v_Cdc(t)>_Ts, x₃ = <θ(t)>_Ts, and x₄ = <dθ(t)/dt>_Ts. The input variables are u₁ = <v_dc(t)>_Ts and u₂ = d(t).

Substituting these variables into the model in (15) yields a new version of the nonlinear-averaged system model, which is presented as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\mathit{x}}_1}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{1}{{\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{x}}_3}}·\left(2·\left({\mathit{u}}_2-0.5\right)·{\mathit{x}}_2-{\mathit{B}}_{\mathrm{L}}·{\mathit{x}}_1·{\mathit{x}}_4-\mathit{R}·{\mathit{x}}_1\right)\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_2}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{1}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}·\left({\mathit{u}}_1-{\mathit{x}}_2-2·{\mathit{R}}_{\mathrm{dc}}·\left({\mathit{u}}_2-0.5\right)·{\mathit{x}}_1\right)\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_3}{\mathrm{d}\mathit{t}}}={\mathit{x}}_4\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_4}{\mathrm{d}\mathit{t}}}={\displaystyle \frac{1}{\mathit{J}}}·\left({\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{\mathit{x}}_1}^2-\mathit{k}·{\mathit{x}}_3-\mathit{b}·{\mathit{x}}_4\right)\end{array}\right.$$

(16)

3.3. Small-Signal Linear Model of the EMD

The model presented in (16) is linearized using the Taylor expansion series technique, which requires finding the equilibrium points (EPs) of the system. This is achieved by setting the derivative expressions with respect to time in (16) to zero. The steady-state model of the EMD is then defined as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{{\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{x}}_3}}·\left(2·\left({\mathit{U}}_2-0.5\right)·{\mathit{X}}_2-{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_1·{\mathit{X}}_4-\mathit{R}·{\mathit{X}}_1\right)=0\\ {\mathit{U}}_1-{\mathit{X}}_2-2·{\mathit{R}}_{\mathrm{dc}}·\left({\mathit{U}}_2-0.5\right)·{\mathit{X}}_1=0\\ {\mathit{X}}_4=0\\ {\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{\mathit{X}}_1}^2-\mathit{k}·{\mathit{X}}_3-\mathit{b}·{\mathit{X}}_4=0\end{array}\right.$$

(17)

In this system, capital letters represent the steady-state variables for both state and input variables. Solving the system yields the expressions for the EPs, which are essentially the values of the state variables at steady-state defined as follows:

$$\left\{\begin{array}{c}{\mathit{X}}_1={\displaystyle \frac{2·{\mathit{U}}_1·\left({\mathit{U}}_2-0.5\right)}{\mathit{R}+4·{\mathit{R}}_{\mathrm{dc}}·{\left({\mathit{U}}_2-0.5\right)}^2}}\\ {\mathit{X}}_2={\displaystyle \frac{\mathit{R}·{\mathit{U}}_1}{\mathit{R}+4·{\mathit{R}}_{\mathrm{dc}}·{\left({\mathit{U}}_2-0.5\right)}^2}}\\ {\mathit{X}}_3={\displaystyle \frac{4·{\mathit{B}}_{\mathrm{L}}·{\mathit{U}}_1·{\left({\mathit{U}}_2-0.5\right)}^2}{2·\mathit{k}·{\left(\mathit{R}+4·{\mathit{R}}_{\mathrm{dc}}·{\left({\mathit{U}}_2-0.5\right)}^2\right)}^2}}\\ {\mathit{X}}_4=0\end{array}\right.$$

(18)

The influence of the EPs is primarily driven by the dynamics of the 2LC, highlighting its critical role in the proper operation of the EMS.

With the EPs determined, the next step is to apply the Taylor series. The dynamic equations in (16) are grouped into a vector of functions defined as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{\mathit{f}}_1\left({\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{u}}_1,{\mathit{u}}_2\right)={\displaystyle \frac{1}{{\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{x}}_3}}·\left(2·\left({\mathit{u}}_2-0.5\right)·{\mathit{x}}_2-{\mathit{B}}_{\mathrm{L}}·{\mathit{x}}_1·{\mathit{x}}_4-\mathit{R}·{\mathit{x}}_1\right)\\ =\end{array}\\ \begin{array}{c}{\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{u}}_1,{\mathit{u}}_2\right)={\displaystyle \frac{1}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}·\left({\mathit{u}}_1-{\mathrm{x}}_2-\left.2·{\mathit{R}}_{\mathrm{dc}}·\left({\mathit{u}}_2-0.5\right)·{\mathit{x}}_1\right)\right.\\ {\mathit{f}}_3\left({\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{u}}_1,{\mathit{u}}_2\right)={\mathit{x}}_4\end{array}\\ {\mathit{f}}_4\left({\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{u}}_1,{\mathit{u}}_2\right)={\displaystyle \frac{1}{\mathit{J}}}·\left({\displaystyle \frac{1}{2}}·{\mathit{B}}_{\mathrm{L}}·{{\mathit{x}}_1}^2-\mathit{k}·{\mathit{x}}_3-\mathit{b}·{\mathit{x}}_4\right)\end{array}\right.$$

(19)

The EPs and the function vector are used to linearize the nonlinear dynamic equations in (16). The small-signal linear model is then described as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}\widehat{\mathbf{x}}}{\mathrm{d}\mathit{t}}}={\mathbf{A}}_{\mathbf{ss}}·\widehat{\mathbf{x}}+{ \mathbf{B}}_{\mathbf{ss}}·\widehat{\mathbf{u}}\\ \widehat{\mathbf{y} }={\mathbf{C}}_{\mathbf{ss}}·\widehat{\mathbf{x}}+{\mathbf{D}}_{\mathbf{ss}}·\widehat{\mathbf{u}}\end{array}\right.$$

(20)

where $\widehat{\mathbf{x}}$ = [x₁, x₂, x₃, x₄]^T is the small signal vector of the state variables, û = [u₁, u₂]^T is the small signal vector of the input variables, and ŷ = $\widehat{\mathbf{x}}$ is the vector of the output variables. Moreover, {$\widehat{\mathbf{x}}$, ŷ} ∈ {R⁴} and û ∈ {ℝ²}.

The matrices A_ss = (∂$\widehat{\mathbf{f}}$/∂$\widehat{\mathbf{x}}$)|_ep, B_ss = (∂$\widehat{\mathbf{f}}$/∂û)|_ep, C_ss = (∂ŷ/∂$\widehat{\mathbf{x}}$)|_ep, and D_ss = (∂ŷ/∂û)|_ep are derived from the Jacobian matrices evaluated at the EPs vector defined as ep = [X₁ X₂ X₃ X₄]. Also, the vector $\widehat{\mathbf{f}}$ groups the dynamic functions described in (19). Moreover, {ep^T, $\widehat{\mathbf{f}}$} ∈ {ℝ⁴}. A_ss, B_ss, C_ss, and D_ss represent the state, input, output, and direct-transmission matrices of the system, respectively. Symbolically, {A_ss, C_ss} ∈ $ℳ$_4×4 {K} and {B_ss, D_ss} ∈ $ℳ$_4×2 {K}. The expressions of the A_ss and B_ss matrices are given by

$$\begin{gathered} {\mathbf{A}}_{\mathbf{ss}}=\left[\begin{array}{cc}\begin{array}{c}-{\displaystyle \frac{\mathit{R}+{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_4}{{\mathit{A}}_{\mathrm{L}}+{ \mathit{B}}_{\mathrm{L}}·{\mathit{X}}_3}}\\ -{\displaystyle \frac{2·\left({\mathit{U}}_2-0.5\right)}{{\mathit{C}}_{\mathrm{dc}}}}\\ 0\\ {\displaystyle \frac{{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_1}{\mathit{J}}}\end{array}& \begin{array}{c}-{\displaystyle \frac{2·\left({\mathit{U}}_2-0.5\right)}{{\mathit{C}}_{\mathrm{dc}}}}\\ -{\displaystyle \frac{1}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}\\ 0\\ 0\end{array}\end{array}\right. \\ \left.\begin{array}{cc}\begin{array}{c}{\displaystyle \frac{{\mathit{X}}_1·{\mathit{X}}_4·{{\mathit{B}}_{\mathrm{L}}}^2+\left(\mathit{R}·{\mathit{X}}_1-2·{\mathit{X}}_2·{\mathit{B}}_{\mathrm{L}}·\left({\mathit{U}}_2-0.5\right)\right)}{{\left({\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_3\right)}^2}}\\ 0\\ 0\\ -{\displaystyle \frac{\mathit{k}}{\mathit{J}}}\end{array}& \begin{array}{c}-{\displaystyle \frac{{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_1}{{\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_3}}\\ 0\\ 1\\ -{\displaystyle \frac{\mathit{b}}{\mathit{J}}}\end{array}\end{array}\right] \\ {\mathbf{B}}_{\mathbf{ss}}=\left[\begin{array}{cc}\begin{array}{c}0\\ {\displaystyle \frac{1}{{\mathit{R}}_{\mathrm{dc}}·{\mathit{C}}_{\mathrm{dc}}}}\\ 0\\ 0\end{array}& \begin{array}{c}{\displaystyle \frac{2·{\mathit{X}}_2}{{\mathit{A}}_{\mathrm{L}}+{\mathit{B}}_{\mathrm{L}}·{\mathit{X}}_3}}\\ -{\displaystyle \frac{2·{\mathit{X}}_1}{{\mathit{C}}_{\mathrm{dc}}}}\\ 0\\ 0\end{array}\end{array}\right] \end{gathered}$$

(21)

Also, C_ss and D_ss are identity and zero matrices, respectively.

Finally, the small-signal analysis assumes that any given variable g(t) can be represented as the sum of its steady-state value G and a small-signal component ĝ; therefore ĝ = g(t) − G, where G is significantly larger than ĝ (G >> ĝ). This assumption simplifies the analysis of the linearized behavior around the steady state [22,28,29].

4. Representation of the EMD in Transfer Functions

When designing control systems to ensure optimal EMD performance, it becomes imperative to establish an EMD model in the Laplace complex s-frequency domain. Although it is common practice to transition models from the time domain to the s-domain, it is equally viable and sometimes advantageous to utilize time domain models within state-space representations for control algorithm design [22,28,29].

The Laplace transform provides a method of representing system dynamics in the s-domain, where complex numbers facilitate the analysis of frequency response and stability. This representation is particularly effective in designing controllers capable of handling different frequency components and disturbances in the system [22,28,29]. Conversely, state-space representations are recognized as a robust tool for control system design due to their precise and intuitive ability to describe system behavior. In the state-space form, a system is represented by a set of first-order differential equations that define the relationships between the system’s state variables and inputs [22,28,29]. As explained in [22,28,29], the s-domain representation of the model in (20) is formulated as follows:

$${\displaystyle \frac{\mathbf{Y}\left(\mathrm{s}\right)}{\mathbf{U}\left(\mathrm{s}\right)}}={ \mathbf{C}}_{\mathbf{ss}}·{\left(\mathit{s}·{\mathbf{I}}_{4\times 4}-{\mathbf{A}}_{\mathbf{ss}}\right)}^{-1}·{\mathbf{B}}_{\mathbf{ss}}+{\mathbf{D}}_{\mathbf{ss}}$$

(22)

In this expression, Y(s), U(s), and I_4×4 denote the complex output and input vectors of the EMD model and the identity matrix of dimensions 4 × 4, respectively. These are defined as Y(s) = [Y₁(s), Y₂(s), Y₃(s), Y₄(s)]^T and U(s) = [U₁(s), U₂(s)]^T. Furthermore, Y_i(s) and U_j(s), where i ∈ {1, 2, 3, 4} and j ∈ {1, 2}, denote the s-domain transformations of the variables ${\widehat{\mathrm{x}}}_{\mathrm{i}}$ and û_j, respectively. Symbolically, Y(s) ∈ {₵⁴} and U(s) ∈ {₵²}.

Formulating the complex vector Equation (22) results in four transfer functions (TFs) per output for a total of eight TFs for the EMD. To maintain clarity and manage analytical complexity, each of these TFs is computed using the parameters of the EMD listed in

Table 1. EMD parameters.

Parameters	Values
AL	0.05
BL	2·AL
J	10 kgm2/rad
b	25 Nms/rad
k	10 Nm/rad
R	100 mΩ
Rdc	100 mΩ
Cdc	900 μF

. This approach enables an efficient evaluation of system behavior while avoiding unnecessary analytical burden.

Considering (20) and utilizing the superposition principle, the TFs of the system are formulated as shown in (22). Defining the TFs as Y_i(s)⁄U_j(s)|_Uj(s)=0 = G_ij(s), where i ∈ {1, 2, 3, 4} and j ∈ {1, 2}, captures the essence of the system behavior and response in the frequency domain. These transfer functions provide critical insight into the behavior of the EMD under various conditions and serve as indispensable tools for control system design and analysis. The TFs of the EMD are shown in Figure 2.

5. EMD Stability Analysis

Based on the time domain model of (20), this representation captures the dynamics of the EMD in the context of a set of first-order differential equations. Time domain stability analysis traditionally involves an examination of the eigenvalues included in the state matrix. If all of the eigenvalues have negative real components, the resulting system is classified as stable, indicating that small perturbations around the equilibrium state will attenuate over time and return the system to its steady-state operating configuration [22,28,29]. Conversely, the presence of any eigenvalue with a positive real component implies instability, which means that deviations will increase over time and initiate a departure from the equilibrium state [30]. This exposition emphasizes the importance of examining the stability of the EPs in the effective operation of the EMD.

Focusing on the s-domain model of (23), the dynamics of the EMD are represented by Laplace transforms, a framework conducive to frequency-domain analysis. In evaluating s-domain stability, it is imperative to evaluate the poles of the TFs emanating from the model. Sustainably structured s-domain systems have poles characterized by negative real elements, indicating that the system responses decay over time in response to disturbances. In contrast, systems characterized by instability exhibit poles with positive real components, implying that the system response intensifies over time and moves toward unbounded outcomes [22,29]. In both cases, the importance of stability analysis lies in its ability to highlight the behavior of the EMD around the equilibrium state, thereby providing valuable insights into the design of control systems. Stability, a key objective, ensures reliability and predictability of performance, while instability may require additional control measures or parameter adjustments to achieve stability [22,29].

5.1. EMD Stability Analysis

When the EPs defined in (18) are evaluated in terms of the parameters listed in Table 1, the following values are obtained: X₁ = I_o = 105.8824 A, X₂ = V_Cdc = 17.6471 V, X₃ = Θ = 56.0554°, and X₄ = dΘ/dt = 0 rad/s. This result reflects the marginal stability of the evaluated equilibrium points due to the presence of the eigenvalue λ₂ on the imaginary axis, while the remaining eigenvalues λ₁, λ₃, and λ₄ exhibit negative real parts satisfying the standard stability condition Re{λᵢ} < 0 [22,29].

5.2. Stability Analysis in S-Domain

To examine the stability within the s-domain, it is important to examine the stability of the TFs described in (22). Accordingly, the zeros for each TF are calculated and listed in

Table 2. Zeros of the TFs defined in (23).

TF	Zeros
G11(s)	−2.2478 + j·21.0506
	−2.2478 − j·21.0506
	−0.0045
G21(s)	−2
	−0.5
G31(s)	−
G41(s)	0
G12(s)	−67.1460
	−9.2217
	−0.1195
G22(s)	−4.9513·104
	−2
	0
G32(s)	−4.9505·104
G42(s)	−4.9505·104
	0

, where λ₂, being zero, does not strictly satisfy this criterion. Taking these observations into account, it can be concluded that the EMD exhibits A_ss; the matrix A_ss is obtained and shown as follows:

$${\mathbf{A}}_{\mathbf{ss}}=\left[\begin{array}{cccc}\begin{array}{c}-5·10^4\\ 12\\ 0\\ 0\end{array}& \begin{array}{c}-3·10^3\\ -2\\ 0\\ 0.176\end{array}& \begin{array}{c}0\\ 0\\ 0\\ -1\end{array}& \begin{array}{c}0\\ -2.506·10^3\\ 1\\ -2.5\end{array}\end{array}\right]$$

(23)

The methodology used to study the stability of EPs within the EMD is based on the calculation of eigenvalues (λ) via the characteristic polynomial, defined as follows:

P(λ) = det[λ∙I_4×4 − A_ss]

(24)

By solving for P(λ) = 0, the values of λ determined are λ₁ = −5.0∙10⁴, λ₂ = 0, and λ_3,4 = −2.61 ± j∙21.1. Based on the eigenvalues obtained, the system exhibits marginal stability [22,29]. In particular, all four outputs Y_i(s), share a common characteristic equation, which consequently imparts an identical set of poles to all TFs.

This result essentially means that the system response to inputs or disturbances will gradually attenuate over time and eventually converge to its equilibrium state.

Based on the results of the comprehensive stability analysis, which includes the calculation of λ from the matrix A_ss as well as the examination of the zeros and poles inherent in the TFs, a definitive conclusion can be drawn: the EMD system emphasizes stability.

6. EMD Frequency Analysis

The EMD frequency responses are examined by performing a detailed analysis of the Bode plots, including both magnitude and phase response plots, applied to the transfer functions defined as follows [22,28,29]:

$$\left\{\begin{array}{c}{\left.{\displaystyle \frac{Y_1\left(\mathrm{s}\right)}{{\mathrm{U}}_1\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_2\left(\mathrm{s}\right)=0}={\displaystyle \frac{50·10^3·{\mathrm{s}}^3+2.25·10^5·{\mathrm{s}}^2+2.241·10^7·\mathrm{s}+10^5}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_2\left(\mathrm{s}\right)}{{\mathrm{U}}_1\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_2\left(\mathrm{s}\right)=0}={\displaystyle \frac{6·10^5·{\mathrm{s}}^2+1.5·10^6·\mathrm{s}+6·10^5}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_3\left(\mathrm{s}\right)}{{\mathrm{U}}_1\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_2\left(\mathrm{s}\right)=0}={\displaystyle \frac{1.5·10^5}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_4\left(\mathrm{s}\right)}{{\mathrm{U}}_1\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_2\left(\mathrm{s}\right)=0}={\displaystyle \frac{1.5·10^5}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_1\left(\mathrm{s}\right)}{{\mathrm{U}}_2\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_1\left(\mathrm{s}\right)=0}=-{\displaystyle \frac{1.765·10^3·{\mathrm{s}}^3+1.35·10^7·{\mathrm{s}}^2+1.109·10^8·\mathrm{s}+1.306·10^7}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_2\left(\mathrm{s}\right)}{{\mathrm{U}}_2\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_1\left(\mathrm{s}\right)=0}={\displaystyle \frac{4.235·10^3·{\mathrm{s}}^3+2.097·10^8·{\mathrm{s}}^2+5.241·10^8·\mathrm{s}+2.096·10^8}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_3\left(\mathrm{s}\right)}{{\mathrm{U}}_2\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_1\left(\mathrm{s}\right)=0}={\displaystyle \frac{747.4·\mathrm{s}+3.7·10^7}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\\ {\left.{\displaystyle \frac{{\mathrm{Y}}_4\left(\mathrm{s}\right)}{{\mathrm{U}}_2\left(\mathrm{s}\right)}}\right|}_{{\mathrm{U}}_1\left(\mathrm{s}\right)=0}={\displaystyle \frac{747.4·\mathrm{s}+3.7·10^7}{{\mathrm{s}}^4+50·10^4·{\mathrm{s}}^3+2.614·10^5·{\mathrm{s}}^2+2.25·10^7·\mathrm{s}+1.36·10^5}}\end{array}\right.$$

(25)

Their utility extends to many aspects of frequency analysis, aiding in the identification of resonant frequencies, bandwidths, and cutoff frequencies for filter design [31] and control system formulation [32,33]. In addition, they clarify signal gain or attenuation at various frequencies, proving indispensable in applications such as signal processing [34,35], communications [36], and mechanical vibrations [37,38]. They also show the interaction between the system phase response and its stability, which has fundamental implications for control system design. In addition, Bode plots allow the evaluation of system stability, the fine-tuning of parameters to achieve optimal responses, the design of compensators to meet specific performance requirements, and even the estimation of transfer functions from empirical data. The simulation results regarding the TFs plots that illustrate the behavior of the system described in (22) presented in this study include Bode frequency domain, providing a variety of insights [22,28,29].

7. Simulation Results

The simulations were performed using MATLAB-Simulink, with the parameter values listed in Table 1. The input conditions for the converter were standardized, with a 24 V supply and a steady-state duty ratio of 0.8 pu at a f_s of 10 kHz. It is also important to note that all SVs were initialized to zero initial conditions, and the simulations were conducted in the absence of any external disturbances.

The simulations were performed using MATLAB-Simulink, with the parameter values listed in Table 1. The input conditions for the converter were standardized, with a 24 V supply and a steady-state duty ratio of 0.8 pu at a f_s of 10 kHz. It is also important to note that all SVs were initialized to zero initial conditions, and the simulations were conducted in the absence of any external disturbances.

The simulation results are visually presented in Figure 3a–d, each illustrating the dynamics of different SVs: i_osw(t), v_Cdc(t), θ(t), and dθ(t)/dt, and the angular acceleration d²θ(t)/dt². Each variable is shown in two forms: their switching versions, i_o(t)_swm, v_Cdc(t)_swm, θ(t)_swm, dθ(t)/dt_swm, and d²θ(t)/dt²_swm, extracted from model (13) and shown in blue, and their averaged versions, i_o(t)_avm, v_Cdc(t)_avm, θ(t)_avm, dθ(t)/dt_avm, and d²θ(t)/dt²)_avm, extracted from model (16) and shown in red.

Analysis of Figure 3a clearly indicates that the steady-state current i_o(t), in both its switched and averaged forms, converges to approximately 106 A. This result confirms the accuracy of the calculations derived from the expressions in (18). Similarly, the results of Figure 3b provide strong evidence by confirming that the steady-state voltage v_Cdc(t) stabilizes at approximately 18 V, which is consistent with the value predicted by (18).

Moreover, Figure 3c provides a crucial insight by showing that the steady-state angular position θ(t) is approximately 56°, effectively validating the calculations presented in (18). This consistency between calculated and observed values further reinforces the accuracy of the developed models.

Referring to Figure 3d,e, the angular velocity dθ(t)/dt and the angular acceleration d²θ(t)/dt² of the torsion rod are shown. It is important to note that both the angular velocity and acceleration profiles exhibit the expected behavior, initially peaking and subsequently converging to zero values. Such agreement with the predictions derived from (18) emphasizes the reliability, accuracy, and precision of the developed models in capturing the complex dynamics of the EMD system. Moreover, an examination of the results presented reveals the presence of high-frequency components around the f_s within each dynamic response. In particular, there is a noticeable high-frequency ripple in both the voltage waveform v_Cdc(t) and the current waveform i_o(t) around the f_s. This observation highlights the effect of switching operations on the system dynamics. The appearance of these high-frequency components provides essential insight into the transient behavior of the system. The oscillations at f_s are inherent to the switching nature of the converter and are attributable to the rapid transitions between different operating states. The observed ripples in both the voltage and current profiles represent the dynamic interaction between the switching action of the converter and the fundamental electromechanical processes.

While this high-frequency behavior is inherent and expected due to the switching nature of the converter, it is critical to consider it in system design and control strategies. Mitigating these high-frequency components, often through filtering or control techniques, is essential to ensure stable and efficient operation. In summary, the identification of high-frequency oscillations at f_s in both voltage and current dynamics reinforces the need to address these transients for the robust performance of the electromechanical system.

Figure 4a–d show the Bode plots of TFs G₁₁(s) = I_o(s)/V_dc(s), G₂₁(s) = V_Cdc(s)/V_dc(s), G₃₁(s) = Θ(s)/V_dc(s), and G₄₁(s) = w(s)/V_dc(s), respectively. w(s) is the Laplace transform of dθ(t)/dt.

In Figure 4a, the Bode plot of G₁₁(s) shows that its gain remains constant at unity over a significant frequency range. Specifically, G₁₁(s) exhibits a unity gain (0 dB) and a 0° phase at low frequencies. At approximately 8 kHz, the cutoff frequency f_c is observed, characterized by a −3 dB drop in gain. G₁₁(s) behaves similarly to a low-pass filter, attenuating frequencies beyond f_c. Its phase response varies gradually, reflecting a tendency toward non-oscillatory behavior [22,28,29]. In particular, at very low frequencies, a small phase shift of −3.52° occurs at 3.07 Hz, followed by a transition to 4.27° at 4.85 Hz, and finally, a return to the original phase. As the frequency increases, the phase converges to −90°. Figure 4b shows the frequency response of G₂₁(s) at −16 dB gain and 0° phase at system start-up. At very low frequencies, a resonant response appears with a resonant frequency (f_r) of 3.07 Hz and a peak gain of 5.28 dB. This resonant frequency matches the inherent natural frequency of the system associated with the input V_Cdc(s) [22,28,29]. The associated bandwidth (BW) is narrow, at 2.06 Hz, emphasizing the gradual response of the system to behavioral changes due to the low f_r. It is worth noting that G₂₁(s) manifests two cutoff frequencies, f_c1 at 2.21 Hz and f_c2 at 4.27 Hz, implying distinct behavioral transitions within different frequency ranges.

Examining the phase of G₂₁(s), its gradual transition indicates minimal oscillatory tendencies. The phase margins associated with f_c1 and f_c2 are PM₁ = 116° and PM₂ = 248°, respectively. These substantial phase margin values indicate the robust stability of G₂₁(s) [22,28,29].

Finally, Figure 4c,d show the frequency response of G₃₁(s) and G₄₁(s), respectively. As can be seen, both TFs exhibit identical frequency responses, with initial gain and phase values of 1.15 dB and −95.3°, respectively. G₃₁(s) and G₄₁(s) exhibit resonant dynamics at a very low f_r, specifically at 3.07 Hz, along with a narrow BW of 0.86 Hz, consistent with their gradual response to change [22,28,29].

The phase behavior of G₃₁(s) and G₄₁(s) exhibits a smooth transition without abrupt changes, indicating their non-oscillatory nature [22,28,29]. Notably, three phase shifts are observed: after f_r, the phase shifts from −90° to −270°, and in the higher frequency range, it extends to −360°.

In addition, Figure 5a–d show Bode plots for the transfer functions G₁₂(s) = I_o(s)/D(s), G₂₂(s) = V_Cdc(s)/D(s), G₃₂(s) = Θ(s)/D(s), and G₄₁(s) = w(s)/D(s), respectively.

In Figure 5a, the TF G₁₂(s) exhibits an initial gain and phase of 14.3 dB and 195°, respectively. Specifically, G₁₂(s) exhibits resonant behavior at a f_r = 3.07 Hz, characterized by a BW = 3.05 Hz and a peak gain of 3.06 dB. Because of its low f_r value, G₁₂(s) responds gradually to changes. The phase dynamics of G₁₂(s) show a smooth but highly multi-phase transition as the input frequency increases.

In Figure 5b, G₂₂(s) shows resonant behavior similar to G₁₂(s), with both the f_r and BW close to those of G₁₂(s). Notably, G₂₂(s) exhibits a gain-dependent f_c at 593 Hz, with a corresponding phase that provides insight into its high stability. This is reinforced by minimal phase changes and a phase margin of approximately 270°.

Figure 5c,d show G₃₂(s) and G₄₂(s), for which the dynamics are identical, with initial gain and phase values of −13.6 dB and −98.4°, respectively. Resonant characteristics emerge, sharing similar f_r and BW values with G₁₂(s). The phase dynamics of G₃₂(s) and G₄₂(s) show limited phase transitions, validating their non-oscillatory nature.

In summary, the analysis of the Bode plots provides valuable insight into the frequency response and stability characteristics of the EMD system, particularly in relation to the selected f_s of 10 kHz. The Bode plots of the system TFs illustrate its behavior over different frequency ranges and provide insight into its gain, phase, resonant frequencies, and stability margins.

In particular, the resonances and cutoff frequencies seen in the Bode plots have significant implications for the system performance. The resonant frequencies, such as those exhibited by G₂₁(s) and G₃₁(s), are influenced by the natural dynamics of the system, which can be enhanced or dampened by the f_s value. In addition, the cutoff frequencies of various transfer functions, such as those in G₁₁(s), G₂₁(s), and G₂₂(s), emphasize the system’s ability to attenuate or transmit certain frequency components, introduced into the system by switching dynamics [39,40,41]. This, in turn, can affect the observed resonances, phase shifts, and stability margins shown in the Bode plots [40,41]. The impact of the f_s value on switching losses, electromagnetic-emission interference, control algorithms, thermal management, and component selection further amplifies its importance in shaping the system behavior revealed by the Bode analysis [40,41]. Therefore, a comprehensive understanding of the Bode plots, combined with careful consideration of the chosen f_s, enables an integrated approach to designing and optimizing the performance, stability, and efficiency of the EMD system over a range of frequencies.

8. Conclusions

This comprehensive analysis has highlighted the intricate dynamics, stability, and frequency response of an electromechanical switching device, offering valuable insights for control and design efforts. The development of a comprehensive model incorporating both electrical and mechanical elements via differential equations and state variables formed the foundation of this study.

Through the derivation of transfer functions and subsequent Bode plots, the complex interactions between inputs and outputs were analyzed, revealing critical features such as resonant and cutoff frequencies. The rigorous stability analysis, conducted using both time and s-domain perspectives, confirmed the stability of the system, a cornerstone for ensuring consistent and reliable performance.

Additionally, MATLAB/Simulink simulations served as a robust validation tool, capturing the high-frequency dynamics that dominate real-world applications. As a result, this study not only deepens the understanding of electromechanical switching devices but also provides a practical foundation for the development of robust control strategies. These strategies, tailored to resonance and phase management while carefully considering the effects of switching frequency, hold promise for improving the performance and efficiency of such systems.

Looking ahead, this research will lay the groundwork for several lines of development. Future studies could address the fine-tuning of control algorithms to mitigate the effects of high-frequency switching dynamics, thereby improving system efficiency. Moreover, the integration of advanced materials and components could be explored to optimize the performance and reliability of the EMD. Finally, these findings may facilitate their application across various fields, from renewable energy systems to precision machinery, where electromechanical switching devices play a pivotal role in achieving operational excellence and sustainability.