Analytical Framework and Component Optimization for Minimizing Harmonic Distortion in DC-DC Small-Signal Converters

1. Introduction

Switching DC–DC converters are widely used in power supplies, renewable energy systems, communication electronics, and defense applications. Despite their advantages in efficiency and compactness, these converters inherently generate harmonic distortion due to their switching operation and nonlinear dynamics. In small-signal operation, the output voltage typically exhibits an AC ripple superimposed on the DC level, with significant contributions from low-order harmonics. Among these, the second harmonic is particularly problematic: it can induce oscillatory behavior in low-frequency systems, cause perceptible flicker effects in lighting, and introduce distortion in sensitive audio or radar equipment (Wang et al., 2023; Shen et al., 2024).

Traditional approaches to harmonic suppression rely on passive filtering, employing large inductors or high-capacitance components to attenuate ripple (Park et al., 2021). While effective, such methods increase system size, weight, and cost, often involving empirical tuning rather than predictive design. More advanced strategies include active compensation schemes (Zhang & Ruan, 2018; Li et al., 2024) and harmonic current reduction in PFC or resonant converter stages (Ruan et al., 2022a; Ruan et al., 2022b). Although these solutions can reduce distortion, they generally require additional circuitry or involve complex optimization processes.

Recent developments in modeling frameworks have focused on harmonic-state-space (HSS) and multifrequency small-signal approaches (Motwani et al., 2022; Mei et al., 2020; Li et al., 2022). These models provide powerful analysis tools but are mathematically intensive and less practical for straightforward component design. Other works have explored building-block state-space modeling (Herbst, 2019), advanced resonant modeling (Corti et al., 2025), and harmonic mitigation in rectifier and inverter front-ends (Du et al., 2021; Grosso et al., 2023). However, few offer closed-form, design-oriented metrics directly linking component values to harmonic attenuation.

This paper addresses this gap by introducing the concept of a harmonic reduction coefficient (Δ), derived from small-signal transfer functions using perturbation theory and separation-of-variables analysis. Unlike previous works, the proposed Δ-based framework directly quantifies the relationship between passive component values and harmonic suppression effectiveness. The main contributions are as follows:

1) A unified modeling framework for Buck, Boost, and Buck–Boost converters operating in CCM, extending earlier small-signal and resonant converter analyses (Yang et al., 1992; Tian et al., 2016; Vyapari, 2022).

2) A new design-oriented metric: the harmonic reduction coefficient Δ, enabling predictive suppression of low-order harmonics.

3) Closed-form analytical expressions for resonant frequency and Δ in major converter topologies.

4) Validation through MATLAB/Simulink simulation showing significant reduction of THD, complementing recent control-based suppression strategies (Escobar et al., 2021).

2. Theoretical Framework and Modeling

Harmonic suppression in converters has been studied from multiple perspectives. Passive filtering remains the most conventional solution (Park et al., 2021), though it is constrained by size and cost limitations. Active strategies, such as one-cycle control and harmonic current injection, have been successfully applied in single-phase inverters and DC-link converters (Zhang & Ruan, 2018; Li et al., 2024), while multi-stage architectures with resonant converters offer partial suppression of second-harmonic current (Ruan et al., 2022a; Ruan et al., 2022b).

On the modeling side, harmonic-state-space (HSS) techniques (Motwani et al., 2022; Mei et al., 2020; Li et al., 2022) and building-block small-signal approaches (Herbst, 2019) have deepened understanding of converter dynamics, but often at the expense of design tractability. Recent works on advanced resonant converter modeling (Corti et al., 2025; Tian et al., 2016) provide insights into harmonic propagation but lack a unified analytical metric for harmonic reduction. Reviews of multipulse rectifier and three-phase converter systems further highlight the importance of harmonics in large-scale applications (Du et al., 2021; Grosso et al., 2023).

In contrast, the present work develops a simplified yet rigorous mathematical framework that captures the essential harmonic behavior of Buck, Boost, and Buck–Boost converters. The proposed harmonic reduction coefficient (Δ) directly links circuit parameters to harmonic attenuation, filling the gap between detailed modeling and practical design guidelines.

Traditional approaches for characterizing the AC behavior of DC–DC converters often rely on detailed transfer function analysis, such as state-space averaging [1]. While accurate, many of these models are mathematically complex and cumbersome for routine engineering applications [2,4,5]. In contrast, the approach proposed herein emphasizes simplicity and analytical clarity, drawing on the methodology developed in this paper, which enables practical prediction of harmonic behavior with high accuracy.

The proposed mathematical framework is designed to be generalizable across various DC–DC converters and adaptable to custom design constraints.

However, the methods of large inductors or capacitors are not always cost-effective and often involve over-dimensioning. The core objective of this research is to derive analytically optimized component values that simultaneously meet system specifications and undesirable harmonics, particularly the second harmonics.

The underlying methodology employs perturbation theory in conjunction with a separation-of-variables approach. This framework divides circuit elements into two categories: fast-response elements (e.g., switches, diodes, and resistors) and slow-response elements (e.g., inductors and capacitors). Perturbations are then applied to the slow variables to derive a comprehensive transfer matrix. This matrix encapsulates the system’s small-signal behavior and serves as the foundation for calculating harmonic suppression characteristics.

This analytical technique is validated through implementation in MATLAB and Simulink. The simulation results confirm that the derived component values not only suppress the second harmonic effectively but also minimize total harmonic distortion (THD) without resorting to excessive filtering.

This method partitions circuit variables into:

- **Fast-response**: Resistors, diodes

- **Slow-response**: Filter inductors, capacitors(as output voltage), input voltage, duty cycle switch.

The slow response variables (as mentioned earlier and oscillating components) will be the ${\mathrm{V}}_{\mathrm{i}\mathrm{n}},\mathrm{D},{\mathrm{I}}_{\mathrm{L}}$ and ${\mathrm{V}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$, this paper will use ${\mathrm{V}}_{\mathrm{i}\mathrm{n}}$ as ${\mathrm{V}}_{\mathrm{d}}$ and ${\mathrm{V}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ as ${\mathrm{V}}_{\mathrm{x}}$ for simplicity.

Perturbation theory is applied to slow-response variables:

$$\Delta \mathrm{F}={\displaystyle \frac{\partial \mathrm{F}}{\partial {\mathrm{V}}_{\mathrm{d}}}}\Delta {\mathrm{V}}_{\mathrm{d}}+{\displaystyle \frac{\partial \mathrm{F}}{\partial {\mathrm{V}}_{\mathrm{x}}}}\Delta {\mathrm{V}}_{\mathrm{x}}+{\displaystyle \frac{\partial \mathrm{F}}{\partial {\mathrm{I}}_{\mathrm{L}}}}\Delta {\mathrm{I}}_{\mathrm{L}}+{\displaystyle \frac{\partial \mathrm{F}}{\partial \mathrm{D}}}\Delta \mathrm{D}$$

(1)

Traditional approaches for characterizing the AC behavior of DC–DC converters often rely on detailed transfer function analysis, such as state-space averaging [1]. While accurate, many of these models are mathematically complex and cumbersome for routine engineering applications [2,4,5]. In contrast, the approach proposed herein emphasizes simplicity and analytical clarity, drawing on the methodology developed in this paper, which enables practical prediction of harmonic behavior with high accuracy.

The proposed mathematical framework is designed to be generalizable across various DC–DC converters and adaptable to custom design constraints.

This yields the small-signal CCM transfer matrix:

$$\left[\begin{array}{c}\Delta {\mathrm{I}}_{\mathrm{d}}\\ \Delta {\mathrm{I}}_{\mathrm{x}}\\ \Delta {\mathrm{V}}_{\mathrm{L}}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}& {\mathrm{a}}_{14}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}& {\mathrm{a}}_{24}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}& {\mathrm{a}}_{34}\end{array}\end{array}\right]\left[\begin{array}{c}\Delta {\mathrm{V}}_{\mathrm{d}}\\ \Delta {\mathrm{V}}_{\mathrm{x}}\\ \Delta {\mathrm{I}}_{\mathrm{L}}\\ \Delta D\end{array}\right]$$

(2)

And a general small signal model as shown in Figure 1

This approach enables explicit derivation of matrix elements per converter topology, following the formalism established by Orugani [3] and Keong [18].

The averaged small signal equivalent circuit is used to relate the variations in output voltage to changes in input voltage and duty cycle. Based on Kirchhoff’s current and voltage laws, the following relationships for a simplified converter model can be written as:

As for convenience, we denote $\Delta {\mathrm{V}}_{\mathrm{x}}=\Delta {\mathrm{V}}_0$ and by Figure 2, the derived Equations are:

$$\Delta {\mathrm{I}}_{\mathrm{x}}=\Delta {\mathrm{I}}_{\mathrm{c}}+\Delta {\mathrm{I}}_0=\mathrm{s}\mathrm{C}\Delta {\mathrm{V}}_0+{\displaystyle \frac{{\mathrm{V}}_0}{\mathrm{R}}}$$

(3)

$$\Delta {\mathrm{V}}_{\mathrm{L}}=\mathrm{s}\mathrm{L}\Delta {\mathrm{I}}_{\mathrm{L}}$$

(4)

Substituting into the general transfer matrix form Equation (2), the complete small-signal transfer function can be obtained:

$$\begin{gathered} \left[\mathrm{s}\mathrm{C}+{\displaystyle \frac{1}{\mathrm{R}}}-\left({\mathrm{a}}_{22}+{\displaystyle \frac{{\mathrm{a}}_{23}{\mathrm{a}}_{32}}{\mathrm{s}\mathrm{L}-{\mathrm{a}}_{33}}}\right)\right]\Delta {\mathrm{V}}_0= \\ \left({\mathrm{a}}_{21}+{\displaystyle \frac{{\mathrm{a}}_{23}{\mathrm{a}}_{31}}{\mathrm{s}\mathrm{L}-{\mathrm{a}}_{33}}}\right)\Delta {\mathrm{V}}_{\mathrm{d}}+\left({\mathrm{a}}_{24}+{\displaystyle \frac{{\mathrm{a}}_{23}{\mathrm{a}}_{34}}{\mathrm{s}\mathrm{L}-{\mathrm{a}}_{33}}}\right)\Delta \mathrm{D} \end{gathered}$$

(5)

Solving for $\Delta \mathrm{D}=0$ the output to input voltage small-signal transfer function is:

$$(\Delta V\_0)/(\Delta V\_d )={\displaystyle \frac{{\mathrm{a}}_{21}\left(\mathrm{s}\mathrm{L}-{\mathrm{a}}_{33}\right)+{\mathrm{a}}_{23}{\mathrm{a}}_{31}}{{\mathrm{s}}^2\mathrm{L}\mathrm{C}+\mathrm{s}\left(\frac{\mathrm{L}}{\mathrm{R}}-\mathrm{L}{\mathrm{a}}_{22}-\mathrm{C}{\mathrm{a}}_{33}\right)-\frac{{\mathrm{a}}_{33}}{\mathrm{R}}+{\mathrm{a}}_{22}{\mathrm{a}}_{33}-{\mathrm{a}}_{23}{\mathrm{a}}_{32}}}$$

(6)

This mathematical formulation forms the basis for evaluating harmonic distortion characteristics and designing component values for suppression of second harmonic effects in the following section, as for simplicity, we will express the output to input voltage perturbation as a frequency response function $\mathrm{F}\left(\mathsf{\omega }\right)$ due its dependence on a frequency.

3. Harmonic Analysis of Converter Topologies

3.1. Buck Converter Small-Signal Analysis in CCM

The Buck converter, operating in continuous conduction mode (CCM), serves as the foundational case for small-signal analysis. Utilizing state averaging and perturbation theory, the transfer matrix is derived by analyzing the dynamic behavior of both slow and fast variables. The circuit is decomposed into two parts: an averaged equivalent model and a fast-dynamic circuit. Perturbations are introduced for input voltage, inductor current, duty cycle, and output voltage. These perturbations are systematically applied to formulate the full transfer matrix. This matrix captures the system’s dynamic behavior around the nominal operating point and is foundational for analyzing frequency-dependent gain and harmonic suppression.

Classical converter in Figure 3. As follows:

An equivalent fast dynamic model is:

Figure 4. Fast dynamic Buck model. A presentation of perturbed variables of the circuit.

Figure 4. Fast dynamic Buck model. A presentation of perturbed variables of the circuit.

State variables ${\mathrm{V}}_{\mathrm{L}}$, ${\mathrm{I}}_{\mathrm{d}}$, ${\mathrm{I}}_{\mathrm{x}}$ are obtained through averaging

$${\mathrm{V}}_{\mathrm{L}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{D}\mathrm{T}}\mathrm{d}\mathrm{t}\left({\mathrm{V}}_{\mathrm{d}}-{\mathrm{V}}_{\mathrm{x}}\right)-{\displaystyle \frac{1}{\mathrm{T}}}{\int }_{\mathrm{D}\mathrm{T}}^{\mathrm{T}}\mathrm{d}\mathrm{t}{\mathrm{V}}_{\mathrm{x}}=={\mathrm{V}}_{\mathrm{d}}\mathrm{D}-{\mathrm{V}}_{\mathrm{x}}$$

(7)

$${\mathrm{I}}_{\mathrm{d}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{D}\mathrm{T}}\mathrm{d}\mathrm{t}{\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{L}}\mathrm{D}\mathrm{ }$$

(8)

$${\mathrm{I}}_{\mathrm{x}}={\mathrm{L}}_{\mathrm{L}}$$

(9)

Perturbation analysis yields:

Perturbation of an Input voltage.

Figure 5. Input voltage perturbation. A perturbation of an input voltage variable and the calculated values as represented in a set of Equations (10).

Figure 5. Input voltage perturbation. A perturbation of an input voltage variable and the calculated values as represented in a set of Equations (10).

$$\Delta {\mathrm{V}}_{\mathrm{L}}^{\left(1\right)}=\Delta {\mathrm{V}}_{\mathrm{d}}\mathrm{D}$$

$$\Delta {\mathrm{I}}_{\mathrm{x}}^{\left(1\right)}=0$$

(10)

$$\Delta {\mathrm{I}}_{\mathrm{d}}^{\left(1\right)}=0$$

Figure 6. Inductor current perturbation. A perturbation of an inductor variable and the calculated values as represented in a set of Equations (11).

Figure 6. Inductor current perturbation. A perturbation of an inductor variable and the calculated values as represented in a set of Equations (11).

$$\Delta {\mathrm{I}}_{\mathrm{L}}^{\left(2\right)}=0$$

$$\Delta {\mathrm{I}}_{\mathrm{d}}^{\left(2\right)}=\Delta {\mathrm{I}}_{\mathrm{L}}\mathrm{D}$$

(11)

$$\Delta {\mathrm{I}}_{\mathrm{x}}^{\left(2\right)}=0$$

Figure 7. Duty cycle ratio perturbation. A perturbation of a duty cycle variable and the calculated values as represented in a set of Equations (12).

Figure 7. Duty cycle ratio perturbation. A perturbation of a duty cycle variable and the calculated values as represented in a set of Equations (12).

$$\Delta {\mathrm{V}}_{\mathrm{L}}^{\left(3\right)}={\mathrm{V}}_{\mathrm{d}}\Delta \mathrm{D}$$

$$\Delta {\mathrm{I}}_{\mathrm{d}}^{\left(3\right)}={\mathrm{I}}_{\mathrm{L}}\Delta \mathrm{D}$$

(12)

$$\Delta {\mathrm{I}}_{\mathrm{x}}^{\left(3\right)}=0$$

Figure 8. Output voltage perturbation. A perturbation of an output voltage variable and the calculated values as represented in a set of Equations (13).

Figure 8. Output voltage perturbation. A perturbation of an output voltage variable and the calculated values as represented in a set of Equations (13).

$$\Delta {\mathrm{V}}_{\mathrm{L}}^{\left(4\right)}={\mathrm{V}}_{\mathrm{d}}\Delta \mathrm{D}$$

$$\Delta {\mathrm{I}}_{\mathrm{d}}^{\left(4\right)}=0$$

(13)

$$\Delta {\mathrm{I}}_{\mathrm{x}}^{\left(4\right)}=0$$

As presented by Equations (10), (11), (12), and (13), the final composite transfer matrix is:

$$\left[\begin{array}{c}\Delta {\mathrm{I}}_{\mathrm{d}}\\ \Delta {\mathrm{I}}_{\mathrm{x}}\\ \Delta {\mathrm{V}}_{\mathrm{L}}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}0& 0& \mathrm{D}& {\mathrm{I}}_{\mathrm{L}}\\ 0& 0& 1& 0\\ \mathrm{D}& -1& 0& {\mathrm{V}}_{\mathrm{d}}\end{array}\end{array}\right]\left[\begin{array}{c}\Delta {\mathrm{V}}_{\mathrm{d}}\\ \Delta {\mathrm{V}}_{\mathrm{x}}\\ \Delta {\mathrm{I}}_{\mathrm{L}}\\ \Delta D\end{array}\right]$$

(14)

Using the transfer function derived earlier, Equation (6), and substituting the matrix elements from the Buck converter Equation (14), the simplified transfer function becomes:

$${\displaystyle \frac{\Delta {\mathrm{V}}_0}{\Delta {\mathrm{V}}_{\mathrm{d}}}}={\displaystyle \frac{\mathrm{D}}{{\mathrm{s}}^2\mathrm{L}\mathrm{C}+\mathrm{s}\left(\frac{\mathrm{L}}{\mathrm{R}}\right)+1}}$$

(15)

To analyze the magnitude of the transfer function in the frequency domain, as was mentioned earlier, the frequency response function $\mathrm{F}\left(\mathsf{\omega }\right)$ will be then:

$$\left|\mathrm{F}\left(\mathsf{\omega }\right)\right|={\displaystyle \frac{\mathrm{D}}{{\sqrt{{\left(1-\mathrm{L}\mathrm{C}{\mathsf{\omega }}^2\right)}^2+\left(\frac{\mathsf{\omega }\mathrm{L}}{\mathrm{R}}\right)}}^2}}$$

(16)

The maximum value of the function occurs at the resonant angular frequency given by:

$$\mathsf{\omega }=\sqrt{{\displaystyle \frac{1}{\mathrm{L}\mathrm{C}}}-{\displaystyle \frac{1}{2{\mathrm{R}}^2{\mathrm{C}}^2}}}$$

(17)

At this frequency, the transfer function's magnitude reaches:

$$\left|\mathrm{F}\left({\mathsf{\omega }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\right|={\displaystyle \frac{\mathrm{D}\mathrm{R}}{\sqrt{\frac{\mathrm{L}}{\mathrm{C}}\left(1-\frac{\mathrm{L}}{4{\mathrm{C}\mathrm{R}}^2}\right)}}}$$

(18)

3.2. Boost Converter Small-Signal Analysis in CCM

As was performed in the previous section, the same principle will be implemented here.

State variables ${\mathrm{V}}_{\mathrm{L}}$, ${\mathrm{I}}_{\mathrm{d}}$, ${\mathrm{I}}_{\mathrm{x}}$ are obtained through averaging

Figure 9. Classical Boost converter topology. The usual and well-known Boost converter topology for a voltage increase in a DC-DC converter.

Figure 9. Classical Boost converter topology. The usual and well-known Boost converter topology for a voltage increase in a DC-DC converter.

And a fast dynamic model is:

Figure 10. Fast dynamic Boost model. A presentation of perturbed variables of the circuit.

Figure 10. Fast dynamic Boost model. A presentation of perturbed variables of the circuit.

Which yields the following results:

$${\mathrm{V}}_{\mathrm{L}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{D}\mathrm{T}}{\mathrm{V}}_{\mathrm{d}}\mathrm{d}\mathrm{t}+{\displaystyle \frac{1}{\mathrm{T}}}{\int }_{\mathrm{D}\mathrm{T}}^{\mathrm{T}}\left({\mathrm{V}}_{\mathrm{d}}-{\mathrm{V}}_{\mathrm{x}}\right)\mathrm{d}\mathrm{t}=$$

$${\mathrm{V}}_{\mathrm{d}}+{\mathrm{V}}_{\mathrm{x}}\left(\mathrm{D}-1\right)$$

(19)

$${\mathrm{I}}_{\mathrm{x}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{T}}{\mathrm{i}}_{\mathrm{x}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_{\mathrm{D}\mathrm{T}}^{\mathrm{T}}{\mathrm{I}}_{\mathrm{L}}dt={\mathrm{I}}_{\mathrm{L}}\left(1-\mathrm{D}\right)$$

(20)

$${\mathrm{I}}_{\mathrm{d}}={\mathrm{I}}_{\mathrm{L}}$$

(21)

Perturbation analysis yields:

ΔI_d= ΔI_L

ΔI_x= (1-D)ΔI_L -I_LΔD

(22)

ΔV_L=ΔV_d -(1-D)ΔV_x+V_x ΔD

The composite transfer matrix is:

$$\left[\begin{array}{c}\Delta {\mathrm{I}}_{\mathrm{d}}\\ \Delta {\mathrm{I}}_{\mathrm{x}}\\ \Delta {\mathrm{V}}_{\mathrm{L}}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}0& 0& \mathrm{D}& {\mathrm{I}}_{\mathrm{L}}\\ 0& 0& (1-\mathrm{D})& -{\mathrm{I}}_{\mathrm{L}}\\ \mathrm{D}& -1(1-\mathrm{D})& 0& {(\mathrm{V}}_{\mathrm{d}}+{\mathrm{V}}_{\mathrm{x}})\end{array}\end{array}\right]\left[\begin{array}{c}\Delta V_d\\ \Delta V_x\\ \Delta I_L\\ \Delta D\end{array}\right]$$

(23)

Using Equations (6) and (23), the transfer function will be:

$${\displaystyle \frac{\Delta V_0}{\Delta V_d}}={\displaystyle \frac{1}{\left(1-D\right)}}\left({\displaystyle \frac{1}{{\mathrm{s}}^2\frac{\mathrm{L}\mathrm{C}}{{\left(1-\mathrm{D}\right)}^2}+\mathrm{s}\frac{\mathrm{L}}{\mathrm{R}{\left(1-\mathrm{D}\right)}^2}+1}}\right)$$

(24)

And the magnitude in the frequency domain is:

$$\left|F\left(\omega \right)\right|={\displaystyle \frac{1}{\left(1-D\right)}}{\displaystyle \frac{D}{\sqrt{{\left(1-\frac{LC{\omega }^2}{{\left(1-D\right)}^2}\right)}^2+{\left(\frac{\omega L}{R{\left(1-D\right)}^2}\right)}^2}}}$$

(25)

The resonant frequency at which this function reaches its peak is given by:

$$\omega =\sqrt{{\displaystyle \frac{{\left(1-\mathrm{D}\right)}^2}{\mathrm{L}\mathrm{C}}}-{\displaystyle \frac{1}{2{\mathrm{R}}^2{\mathrm{C}}^2}}}$$

(26)

At this frequency, the peak value of the transfer function is:

$$F\left({\omega }_{max}\right)={\displaystyle \frac{\mathrm{R}}{\sqrt{\frac{\mathrm{L}}{\mathrm{C}}\left(1-\frac{\mathrm{L}}{4{\mathrm{R}}^2\mathrm{C}{\left(1-\mathrm{D}\right)}^2}\right)}}}$$

(27)

3.3. Buck-Boost Converter Small-Signal Analysis in CCM

As was performed in the previous section, the same principle will be implemented here.

State variables ${\mathrm{V}}_{\mathrm{L}}$, ${\mathrm{I}}_{\mathrm{d}}$, ${\mathrm{I}}_{\mathrm{x}}$ are obtained through averaging

Figure 11. Classical Buck-Boost converter topology. The usual and well-known Buck-Boost converter topology for a voltage increasing/decreasing of DC-DC converter.

Figure 11. Classical Buck-Boost converter topology. The usual and well-known Buck-Boost converter topology for a voltage increasing/decreasing of DC-DC converter.

And a fast dynamic model is:

Figure 12. Fast dynamic Buck-Boost model. A presentation of perturbed variables of the circuit.

Figure 12. Fast dynamic Buck-Boost model. A presentation of perturbed variables of the circuit.

Which yields the following results:

$${\mathrm{V}}_{\mathrm{L}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{D}\mathrm{T}}{\mathrm{V}}_{\mathrm{d}}\mathrm{d}\mathrm{t}+{\displaystyle \frac{1}{\mathrm{T}}}{\int }_{\mathrm{D}\mathrm{T}}^{\mathrm{T}}\left({\mathrm{V}}_{\mathrm{d}}-{\mathrm{V}}_{\mathrm{x}}\right)\mathrm{d}\mathrm{t}=$$

$${\mathrm{V}}_{\mathrm{d}}+{\mathrm{V}}_{\mathrm{x}}\left(\mathrm{D}-1\right)$$

(28)

$${\mathrm{I}}_{\mathrm{x}}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_0^{\mathrm{T}}{\mathrm{i}}_{\mathrm{x}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}={\displaystyle \frac{1}{\mathrm{T}}}{\int }_{\mathrm{D}\mathrm{T}}^{\mathrm{T}}{\mathrm{I}}_{\mathrm{L}}\mathrm{d}\mathrm{t}=$$

$${\mathrm{I}}_{\mathrm{L}}\left(1-\mathrm{D}\right)$$

(29)

$${\mathrm{I}}_{\mathrm{d}}={\mathrm{I}}_{\mathrm{L}}$$

(30)

Perturbation analysis yields:

ΔI_d= ΔI_L

ΔI_x= (1-D)ΔI_L -I_LΔD

(31)

ΔV_L=ΔV_d -(1-D)ΔV_x+V_x ΔD

The composite transfer matrix is:

$$\left[\begin{array}{c}\Delta {\mathrm{I}}_{\mathrm{d}}\\ \Delta {\mathrm{I}}_{\mathrm{x}}\\ \Delta {\mathrm{V}}_{\mathrm{L}}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}0& 0& \mathrm{D}& {\mathrm{I}}_{\mathrm{L}}\\ 0& 0& (1-\mathrm{D})& -{\mathrm{I}}_{\mathrm{L}}\\ \mathrm{D}& -1(1-\mathrm{D})& 0& {(\mathrm{V}}_{\mathrm{d}}+{\mathrm{V}}_{\mathrm{x}})\end{array}\end{array}\right]\left[\begin{array}{c}\Delta {\mathrm{V}}_{\mathrm{d}}\\ \Delta {\mathrm{V}}_{\mathrm{x}}\\ \Delta {\mathrm{I}}_{\mathrm{L}}\\ \Delta D\end{array}\right]$$

(32)

Using Equation (6), (24), the transfer function will be:

$${\displaystyle \frac{\Delta {\mathrm{V}}_0}{\Delta {\mathrm{V}}_{\mathrm{d}}}}={\displaystyle \frac{1}{\left(1-\mathrm{D}\right)}}\left({\displaystyle \frac{1}{{\mathrm{s}}^2\frac{\mathrm{L}\mathrm{C}}{{\left(1-\mathrm{D}\right)}^2}+\mathrm{s}\frac{\mathrm{L}}{\mathrm{R}{\left(1-\mathrm{D}\right)}^2}+1}}\right)$$

(33)

And the magnitude in the frequency domain is:

$$\left|\mathrm{F}\left(\mathsf{\omega }\right)\right|={\displaystyle \frac{1}{\left(1-\mathrm{D}\right)}}{\displaystyle \frac{\mathrm{D}}{\sqrt{{\left(1-\frac{\mathrm{L}\mathrm{C}{\mathsf{\omega }}^2}{{\left(1-\mathrm{D}\right)}^2}\right)}^2+{\left(\frac{\mathsf{\omega }\mathrm{L}}{\mathrm{R}{\left(1-\mathrm{D}\right)}^2}\right)}^2}}}$$

(34)

The resonant frequency at which this function reaches its peak is given by:

$$\mathsf{\omega }=\sqrt{{\displaystyle \frac{{\left(1-\mathrm{D}\right)}^2}{\mathrm{L}\mathrm{C}}}-{\displaystyle \frac{1}{2{\mathrm{R}}^2{\mathrm{C}}^2}}}$$

(35)

At this frequency, the peak value of the transfer function is:

$$\mathrm{F}\left({\mathsf{\omega }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)={\displaystyle \frac{\mathrm{R}}{\sqrt{\frac{\mathrm{L}}{\mathrm{C}}\left(1-\frac{\mathrm{L}}{4{\mathrm{R}}^2\mathrm{C}{\left(1-\mathrm{D}\right)}^2}\right)}}}$$

(36)

4. Design Examples and Simulation Validation

4.1. Buck Converter Harmonic Analysis

As a practical validation of the theory, consider a converter with the following parameters and under the CCM conditions for

$\mathrm{L}>{\displaystyle \frac{\mathrm{R}\left(1-\mathrm{D}\right)}{2{\mathrm{ƒ}}_{\mathrm{s}}}}$:

Design example: ${\mathrm{V}}_{\mathrm{d}}=28\mathrm{V};{\mathrm{V}}_0=14\mathrm{V};\mathrm{D}={\displaystyle \frac{{\mathrm{V}}_0}{{\mathrm{V}}_{\mathrm{d}}}}=0.5;\mathrm{R}=10\mathsf{\Omega };$${\mathrm{ƒ}}_{\mathrm{s}}=100\mathrm{K}\mathrm{H}\mathrm{z};\mathrm{L}=30\mathsf{\mu }\mathrm{H};\mathrm{C}=68\mathsf{\mu }\mathrm{F}$

Substituting these values into Equation (17), the resonant frequency is calculated as:

$\mathsf{\omega }=6912{\displaystyle \raisebox{1ex}{$\mathrm{r}\mathrm{a}\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}\mathrm{e}\mathrm{c}$}\right.}$ or the frequency as $\mathrm{ƒ}=1100\mathrm{H}\mathrm{z}$. This matches the peak in the MATLAB plot simulation in Figure 13.

The plot in Figure 13 shows the transfer function magnitude in decibels ($20\mathrm{l}\mathrm{o}\mathrm{g}\left[\mathrm{F}\left(\mathsf{\omega }\right)\right])$ as a function of frequency, with the peak occurring at approximately 1100 Hz and

$\mathrm{F}\left(\mathsf{\omega }\right)\approx 7.60\mathrm{d}\mathrm{B}$ which is very consistent with the theory. At 50Hz (or 100Hz for the second harmonic), the gain is substantially lower, indicating limited attenuation.

As a Simulink simulation, consider a simple Buck converter circuit as an AC disturbed signal of $\Delta \mathrm{V}=0.5\mathrm{V}$ applied to the DC working voltage of V=28V in Figure 14.

The input signal Figure is:

The FFT analyzes the THD of the voltage input and output transfer function.

The THD in both input and output voltage is the same and small, but it was obvious by examining Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, where the disturbed voltage is almost pure sign of 50Hz frequency (except switching frequency that induces a small harmonics), and parameters of the components have been chosen to satisfy $1100\mathrm{H}\mathrm{z}$ peak frequency, which is very far from the working frequency (50Hz).

To effectively suppress the second harmonic, the design must shift the resonant (knee) frequency to be smaller than the target frequency (second harmonic frequency). This is accomplished by selecting new component values such that the knee frequency will be smaller than the second harmonic, enabling attenuation from the resonant frequency to the target frequency. The reduction coefficient Δ in decibels is defined by comparing the gain at the Target (second harmonic) frequency to the DC (ω=0) gain:

$$\mathsf{\Delta }=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{\left(1-\mathrm{L}\mathrm{C}{\mathsf{\omega }}^2\right)}^2+{\left({\displaystyle \frac{\mathsf{\omega }\mathrm{L}}{\mathrm{R}}}\right)}^2\right]$$

(37)

Given a desired suppression (e.g., 39.9 dB) and design parameters such as

${\mathrm{ƒ}}_{\mathbf{s}}=100\mathbf{K}\mathbf{H}\mathbf{z};$

${\mathbf{V}}_{\mathbf{i}\mathbf{n}}=24\mathbf{V};\mathbf{R}=20\mathsf{\Omega };\mathbf{D}=0.5;\mathbf{L}=20\mathbf{m}\mathbf{H}$

The required capacitor value for a resonant frequency of 10Hz is calculated as

C1 = 12.64mF. Simulations confirm the theoretical prediction by showing the reduced gain at 10Hz to target a 100Hz frequency. As appears in the response plot in Figure 19

Figure 20. Transfer function Plot as a dependence of frequency for L=20mH and.

Figure 20. Transfer function Plot as a dependence of frequency for L=20mH and.

C=12.64mF as shown for the second harmonic at 100Hz, the transfer function has a value of -45dB, a reduction of 39.9 dB from a saturation line.

Alternatively, due to the nonlinearity of Equation (17), the same target frequency can also be achieved at ${\mathrm{C}}_2=25\mathsf{\mu }\mathrm{F}$ but in this case, the reduction gain will have the value of

$\mathsf{\Delta }=0.166\mathrm{d}\mathrm{B}.$ A MATLAB plot simulation of the second case is:

C=$25\mu F$ As shown for the second harmonic at 100Hz, the transfer function has a value of -6.18dB. A reduction of 0.16 dB from a saturation line.

Both conFigure urations represent the target angular frequency of $\omega =62.83{\displaystyle \raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$sec$}\right.}$ but with different attenuation.

For the analysis, consider a realistic design with a full bridge rectifier and a capacitor filter of $150\mu F$ to reduce high ripple and the parameters as in the previous design (${\mathrm{ƒ}}_{\mathrm{s}}=100\mathrm{K}\mathrm{H}\mathrm{z};{\mathrm{V}}_{\mathrm{i}\mathrm{n}}=24\mathrm{V};\mathrm{R}=20\mathsf{\Omega };\mathrm{D}=0.5;\mathrm{L}=20\mathrm{m}\mathrm{H}$), although without the filter capacitor, we will get very high input harmonics, the output results of the harmonics will still be very satisfactory and highly reduced

Using the simulation structure (Figure 21), we can get results that characterize system behavior Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27. The first design with ${\mathrm{C}}_1=12.64\mathrm{m}\mathrm{F}$ and the reduction coefficient of $\mathsf{\Delta }=39.9\mathrm{d}\mathrm{B}$

The FFT analysis will then be:

And for the second case design with ${\mathrm{C}}_2=25\mathsf{\mu }\mathrm{F}$

The presentation of the design with nominated customized component values was resolved mathematically by the reduction coefficient of the specific cost-effective design, which reduced THD.

The same methodological framework is applied to the Boost and Buck–Boost converters. The classical averaged Equations are derived for each topology, and the small-signal transfer matrix is constructed through the perturbation of key system variables. The derived transfer functions demonstrate topological dependence on passive component values and modulation parameters, such as duty cycle.

In both Boost and Buck–Boost cases, component sensitivity significantly influences the location and amplitude of resonance peaks, particularly near the system’s knee frequency. These peaks correspond to heightened harmonic content, especially at low-order harmonics such as the second harmonic.

4.2. Boost Converter Harmonic Analysis

To quantify harmonic suppression at a specific frequency, such as the second harmonic (typically 100Hz if the base is 50Hz), we define a reduction coefficient $\mathsf{\Delta }$ in decibels:

$$\mathsf{\Delta }=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{\left(1-{\displaystyle \frac{\mathrm{L}\mathrm{C}{\mathsf{\omega }}^2}{{\left(1-\mathrm{D}\right)}^2}}\right)}^2+{\left({\displaystyle \frac{\mathsf{\omega }\mathrm{L}}{\mathrm{R}{\left(1-\mathrm{D}\right)}^2}}\right)}^2\right]$$

(38)

Note that the DC gain (saturation line) in Boost converter is $\mathrm{F}\left(\mathsf{\omega }=0\right)={\displaystyle \frac{1}{\left(1-\mathrm{D}\right)}}$

Design example:

To reduce the second harmonic, consider the following component values:

Input voltage: ${\mathrm{V}}_{\mathrm{d}}=12\mathrm{V}$

Output voltage: ${\mathrm{V}}_0=24\mathrm{V}$

Duty cycle: $\mathrm{D}={\displaystyle \frac{{\mathrm{V}}_0-{\mathrm{V}}_{\mathrm{d}}}{{\mathrm{V}}_0}}=0.5$

Load resistance: $\mathrm{R}=20\mathsf{\Omega }$

Switching frequency: ${\mathrm{ƒ}}_{\mathrm{s}}=100\mathrm{K}\mathrm{H}\mathrm{z}$

Inductor: $\mathrm{L}=20\mathrm{m}\mathrm{H}$ under CCM condition L$>{\displaystyle \frac{\mathbf{R}\mathbf{D}{\left(1-\mathbf{D}\right)}^2}{2{\mathrm{ƒ}}_{\mathbf{s}}}}$

The aim of the suppression of $\mathsf{\Delta }=39.44\mathrm{d}\mathrm{B}$ which is suitable for a resonant frequency of 10Hz. By using Equation (38), the calculation of the required capacitor is:

${\mathrm{C}}_1=3\mathrm{m}\mathrm{F}$

Alternatively, the second choice of the capacitor by the same resonant frequency as calculated by Equation (26) is:

${\mathrm{C}}_2=103.37\mathsf{\mu }\mathrm{F}$ However, the harmonic suppression with ${\mathrm{C}}_2$ is significantly weaker, as the MATLAB plot simulation shows for both cases, Figure 28 and Figure 29:

$$\mathit{\Delta }=10.58dB$$

As indicated in Figure 29 and calculated by Equation (38), the reduction with the ${\mathrm{C}}_2$ design is $\mathsf{\Delta }=10.58\mathrm{d}\mathrm{B}$

For the Boost analysis, consider a model with a full bridge rectifier but without a filter capacitor (unnecessary due to the inductor filtering).

The input voltage in both cases will be considered the same due to an insignificant difference.

The FFT analysis will be:

With the aim of suppressing $\mathsf{\Delta }=39.44\mathrm{d}\mathrm{B}$

With the suppression of $\mathsf{\Delta }=10.58\mathrm{d}$ This analysis highlights the critical role of the tradeoffs between the inductor and the capacitor in shaping harmonic rejection, guided by the reduction coefficient $\mathsf{\Delta }$ as shown in Figure 30,Figure 31 and Figure 32.

4.3. Buck-Boost Converter Harmonic Analysis

As with previous converters, the reduction coefficient $\mathsf{\Delta }$ defines the suppression level compared to the saturation line at $\mathrm{F}\left(\mathsf{\omega }=0\right)={\displaystyle \frac{\mathrm{D}}{\left(1-\mathrm{D}\right)}}$ and at a specific frequency (second harmonic):

$$\mathsf{\Delta }=10{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{\left(1-{\displaystyle \frac{\mathrm{L}\mathrm{C}{\mathsf{\omega }}^2}{{\left(1-\mathrm{D}\right)}^2}}\right)}^2+{\left({\displaystyle \frac{\mathsf{\omega }\mathrm{L}}{\mathrm{R}{\left(1-\mathrm{D}\right)}^2}}\right)}^2\right]$$

(39)

Design example:

To reduce the second harmonic, consider the following component values:

Input voltage: ${\mathrm{V}}_{\mathrm{d}}=24\mathrm{V}$

Output voltage: ${\mathrm{V}}_0=-6\mathrm{V}$

Duty cycle: $\mathrm{D}={\displaystyle \frac{{\mathrm{V}}_0}{{\mathrm{V}}_0-{\mathrm{V}}_{\mathrm{d}}}}=0.2$

Load resistance: $\mathrm{R}=20\mathsf{\Omega }$

Switching frequency: ${\mathrm{ƒ}}_{\mathrm{s}}=100\mathrm{K}\mathrm{H}\mathrm{z}$

Inductor: $\mathrm{L}=20\mathrm{m}\mathrm{H}$ under CCM condition L$>{\displaystyle \frac{\mathbf{R}{\left(1-\mathbf{D}\right)}^2}{2{\mathrm{ƒ}}_{\mathbf{s}}}}$ To achieve a reduction coefficient $\mathsf{\Delta }=39.8\mathrm{d}\mathrm{B}$ which is once more suitable for ƒ=10Hz, the calculation of the required capacitor yields: ${\mathrm{C}}_1=8\mathrm{m}\mathrm{F}$ and an alternative conFigure uration with the same target frequency but reduced attenuation yields ${\mathrm{C}}_2=39.25\mathsf{\mu }\mathrm{F}$.

The MATLAB plot simulations are shown in Figure 33 and Figure 34:

And a reduced coefficient for ${\mathrm{C}}_2$ is

$\mathsf{\Delta }=0.89\mathrm{d}\mathrm{B}$

The next Simulink results (Figure 35, Figure 36 and Figure 37) are shown when using Buck-Boost small signal model with a full bridge rectifier and a C=150μF filter to reduce the ripple:

As it was observed, the $C_1$ conFigure uration achieves deep second harmonic suppression while the $C_2$ produces limited attenuation despite having the same resonant frequency, which was determined by Equation (35), and the reduction coefficient $\mathsf{\Delta }$ of the Equation (39).