# Asymmetrical Interleaved DC/DC Switching Converters for Photovoltaic and Fuel Cell Applications—Part 2: Control-Oriented Models

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## Abstract

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## 1. Introduction

## 2. DCM Modeling Review

#### 2.1. Reduced-Order Models

#### 2.2. Full-Order Models

## 3. Modeling the AIDB Converter Using the Revised Averaging Method

**Figure 1.**Asymmetrical interleaved dual boost (AIDB) converter circuit and waveforms: (

**a**) electrical scheme; and (

**b**) discontinuous currents.

- Topology 1 (${S}_{B}$ and ${D}_{A}$ ON, ${S}_{A}$ and ${D}_{B}$ OFF): it is active in the first interval $(0\le t\le {d}_{1}\xb7T$, ${d}^{{}^{\prime}}={d}_{1})$ during ${d}_{1}\xb7T$, it begins when ${S}_{B}$ is turned ON and ${S}_{A}$ is turned OFF, and it ends when ${S}_{B}$ is turned OFF and ${S}_{A}$ is turned ON (Topology 2). In this topology ${i}_{B}$ increases while ${i}_{A}$ and ${i}_{AO}$ decrease;
- Topology 2 (${S}_{A}$ and ${D}_{B}$ ON, ${S}_{B}$ and ${D}_{A}$ OFF): it is active in the second interval $[{d}_{1}\xb7T<t\le ({d}_{1}+{d}_{2})\xb7T]$ during ${d}_{2}\xb7T$, it begins when ${S}_{B}$ is turned OFF and ${S}_{A}$ is turned ON, and it ends when ${D}_{B}$ is automatically turned OFF (Topology 3). In this topology ${i}_{B}$ decreases while ${i}_{A}$ and ${i}_{AO}$ increase;
- Topology 3 (${S}_{A}$ ON, ${S}_{B}$, ${D}_{A}$ and ${D}_{B}$ OFF): it is active in the third interval $[({d}_{1}+{d}_{2})\xb7T\le t<T]$ during ${d}_{3}\xb7T$, it begins when ${D}_{B}$ is automatically turned OFF by the DCM condition generated when ${i}_{B}={i}_{AO}$ (null current in ${D}_{B}$), and it ends when ${S}_{B}$ is turned ON and ${S}_{A}$ is turned OFF (Topology 1). In this topology ${i}_{B}$ and ${i}_{AO}$ are constant and equal while ${i}_{A}$ still increases.

#### 3.1. New Calculation Procedure for Duty Cycle ${d}_{2}$

#### 3.2. RAM Applied to the AIDB Using the New ${d}_{2}$ Value

**Figure 5.**AIDB Bode diagrams from circuital simulation and revised averaging method (RAM) simulation: (

**a**) ${i}_{A}$; (

**b**) ${i}_{B}$; (

**c**) ${i}_{AO};$ and (

**d**) ${v}_{AB}$.

**Figure 6.**AIDB Bode diagrams from circuital simulation and RAM simulation: output voltage ${v}_{o}$.

## 4. Improved Method for Modeling High-Order Converters in DCM

- Identify new variables that affect the converter dynamics using circuital analyses;
- Include in the topologies equations the new variable;
- Obtain the averaged equations from the modified topologies equations;
- Calculate the correct expression for the duty cycle ${d}_{2}$;
- Calculate the new variable in terms of the circuit states and inputs;
- Replace both the new variable and ${d}_{2}$ in the averaged equations;
- Calculate the steady-state and small-signal expressions.

#### 4.1. Modeling the AIDB Using the IAM

**Step 1**: From the AIDB scheme in Figure 1a, it is noted that the output current ${I}_{O}$ is obtained by adding two currents: the current in the inductor ${L}_{AO}$ and the current in the diode ${D}_{B}$. Therefore, to provide an accurate prediction of the AIDB output current, it is necessary to include the current in diode ${D}_{B}$ in the analytical model as the new variable ${i}_{DB}$. In this way, the error generated in RAM [8] by neglecting this current is avoided.**Step 2**: From the circuital topologies (Figure 3) it is noted that diode ${D}_{B}$ conducts in Topology 2. Therefore, the expressions in Equation (4) describing this topology are modified to include the current ${i}_{DB}$ as follows:$$\begin{array}{c}\phantom{\rule{6.0pt}{0ex}}{L}_{A}{\displaystyle \frac{d{i}_{A}}{dt}}={v}_{g}\\ \phantom{\rule{6.0pt}{0ex}}{L}_{B}{\displaystyle \frac{d{i}_{B}}{dt}}={v}_{g}-{v}_{o}\\ \phantom{\rule{6.0pt}{0ex}}{L}_{AO}{\displaystyle \frac{d{i}_{AO}}{dt}}={v}_{AB}\\ \phantom{\rule{6.0pt}{0ex}}{C}_{AB}{\displaystyle \frac{d{v}_{AB}}{dt}}=-{i}_{AO}\\ \phantom{\rule{6.0pt}{0ex}}{C}_{o}{\displaystyle \frac{d{v}_{o}}{dt}}={i}_{AO}+{i}_{DB}-{\displaystyle \frac{{v}_{o}}{R}}\end{array}$$**Step 3**: The new state space averaged system ($D>38\%$) including the new variable ${i}_{DB}$ is given in Equation (24), where ${\overline{i}}_{DB}$ represents the averaged value of ${i}_{DB}$:$$\left(\right)open="\{"\; close>\begin{array}{c}\phantom{\rule{6.0pt}{0ex}}{\displaystyle \frac{d{\overline{i}}_{A}}{dt}}={\displaystyle \frac{{v}_{g}-{\overline{v}}_{AB}{d}_{1}}{{L}_{A}}}\\ \phantom{\rule{6.0pt}{0ex}}{\displaystyle \frac{d{\overline{i}}_{B}}{dt}}={\displaystyle \frac{{v}_{g}}{{L}_{B}}}\left(\right)open="("\; close=")">{d}_{1}+{d}_{2}-{\displaystyle \frac{{\overline{v}}_{o}}{{L}_{B}}}{d}_{2}+{\displaystyle \frac{{v}_{g}-{\overline{v}}_{o}+{\overline{v}}_{AB}}{{L}_{B}+{L}_{AO}}}{d}_{3}\end{array}\phantom{\rule{6.0pt}{0ex}}{\displaystyle \frac{d{\overline{i}}_{AO}}{dt}}={\displaystyle \frac{{\overline{v}}_{AB}}{{L}_{AO}}}\left(\right)open="("\; close=")">{d}_{1}+{d}_{2}-{\displaystyle \frac{{\overline{v}}_{o}}{{L}_{AO}}}{d}_{1}+{\displaystyle \frac{{v}_{g}-{\overline{v}}_{o}+{\overline{v}}_{AB}}{{L}_{B}+{L}_{AO}}}{d}_{3}$$**Step 4**: The procedure to calculate the correct duty cycle ${d}_{2}$, for the AIDB converter, was described in Subsection 3.1 leading to Equation (14). Moreover, the duty cycle ${d}_{3}$ is calculated from the fundamental relation given in Equation (1).**Step 5**: Taking into account that diode ${D}_{B}$ only conducts in Topology 2 with ${i}_{DB}={i}_{B}-{i}_{AO}>0$, it describes a triangular waveform in ${i}_{DB}$ with a peak current equal to $\Delta {i}_{B}+\Delta {i}_{AO}$: since $d{i}_{B}/dt<0$ and $d{i}_{AO}/dt>0$ in Topology 2, the maximum current in ${D}_{B}$ (peak current) is achieved at the beginning of the interval $[{d}_{1}\xb7T,({d}_{1}+{d}_{2})\xb7T]$ when both ${i}_{B}$ and ${i}_{AO}$ exhibit their peak currents, i.e., $\Delta {i}_{B}$ and $\Delta {i}_{AO}$, respectively. Then, ${\overline{i}}_{DB}$ is calculated as in Equation (25), where substituting the expressions for $\Delta {i}_{B}$ and $\Delta {i}_{AO}$ obtained in Equation (13) and the expression of ${d}_{2}$ given in Equation (14), the ${\overline{i}}_{DB}$ expression given in Equation (26) is obtained:$${\overline{i}}_{DB}={\displaystyle \frac{\Delta {i}_{B}+\Delta {i}_{AO}}{2}}{d}_{2}$$$${\overline{i}}_{DB}=\left(\right)open="("\; close=")">{\displaystyle \frac{{v}_{g}}{{L}_{B}}}+{\displaystyle \frac{{\overline{v}}_{o}-{\overline{v}}_{AB}}{{L}_{AO}}}{\displaystyle \frac{{d}_{1}T}{2}}\left(\right)open="("\; close=")">{\displaystyle \frac{2\left(\right)open="("\; close=")">{\overline{i}}_{B}-{\overline{i}}_{AO}}{}\left(\right)open="("\; close=")">{\displaystyle \frac{{v}_{g}}{{L}_{B}}}+{\displaystyle \frac{{\overline{v}}_{o}-{\overline{v}}_{AB}}{{L}_{AO}}}{d}_{1}T}$$**Step 6**: Replacing Equations (1), (14) and (26) in Equation (24), the new averaged equations in terms of the duty cycle ${d}_{1}={d}^{{}^{\prime}}$ are:$$\begin{array}{c}\phantom{\rule{6.0pt}{0ex}}{\displaystyle \frac{d{\overline{i}}_{A}}{dt}}={\displaystyle \frac{{v}_{g}-{\overline{v}}_{AB}{d}_{1}}{{L}_{A}}}\\ \phantom{\rule{6.0pt}{0ex}}{\displaystyle \frac{d{\overline{i}}_{B}}{dt}}={\displaystyle \frac{2{\overline{i}}_{B}{L}_{B}{L}_{AO}^{2}\left(\right)open="("\; close=")">{a}_{1}}{+}+{\overline{v}}_{AB}{d}_{1}T{L}_{B}^{2}\left(\right)open="("\; close=")">{a}_{3}}+{\overline{v}}_{o}{d}_{1}T{L}_{B}{L}_{AO}\left(\right)open="("\; close=")">{a}_{4}+{d}_{1}T{V}_{g}\left(\right)open="("\; close=")">{a}_{5}\\ \left(\right)open="("\; close=")">-{V}_{g}{L}_{AO}-{L}_{B}{\overline{v}}_{o}+{L}_{B}{v}_{AB}{L}_{B}\left(\right)open="("\; close=")">{L}_{AO}+{L}_{B}\end{array}{d}_{1}T$$$$\begin{array}{c}\phantom{\rule{6.0pt}{0ex}}{a}_{1}=-{V}_{g}+{\overline{v}}_{o}+{\overline{v}}_{AB}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{2}={V}_{g}-{\overline{v}}_{o}-{\overline{v}}_{AB}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{3}={V}_{g}+{\overline{v}}_{AB}+{\overline{v}}_{o}{d}_{1}-2{\overline{v}}_{o}\\ \phantom{\rule{6.0pt}{0ex}}{a}_{4}={V}_{g}-{d}_{1}{V}_{g}+{d}_{1}{\overline{v}}_{AB}-{d}_{1}{\overline{v}}_{o}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{5}=\left(\right)open="("\; close=")">-{V}_{g}-{\overline{v}}_{AB}{L}_{B}{L}_{AO}-{\overline{v}}_{o}{L}_{B}^{2}-{\overline{v}}_{o}{d}_{1}{L}_{AO}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\end{array}\phantom{\rule{6.0pt}{0ex}}{a}_{6}=\left(\right)open="("\; close=")">-{V}_{g}+{\overline{v}}_{o}{L}_{AO}+{\overline{v}}_{AB}{L}_{B}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{7}=-{V}_{g}+{\overline{v}}_{o}\left(\right)open="("\; close=")">{d}_{1}+1& -{\overline{v}}_{AB}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{8}=-{\overline{v}}_{AB}+{\overline{v}}_{o}$$**Step 7**: The calculation of the expressions for both the steady-state and small-signal model are performed following the procedures given in Subsection 3.2, where the states vector is defined as $x={\left[{i}_{A}\phantom{\rule{0.277778em}{0ex}}{i}_{B}\phantom{\rule{0.277778em}{0ex}}{i}_{AO}\phantom{\rule{0.277778em}{0ex}}{v}_{AB}\phantom{\rule{0.277778em}{0ex}}{v}_{o}\right]}^{T}$. Again, the input and state variables of the model are represented in terms of steady-state and small signal components as in Equations (17) and (18).

#### 4.2. Simulation and Experimental Validation

**Figure 7.**AIDB Bode diagrams from circuital simulation and improved averaging method (IAM) simulation: (

**a**) ${i}_{A}$; (

**b**) ${i}_{B}$; (

**c**) ${i}_{AO}$; and (

**d**) ${v}_{AB}$.

**Figure 8.**AIDB Bode diagrams from circuital simulation and IAM simulation: output voltage ${v}_{o}$.

**Figure 9.**Experimental setup used to obtain the AIDB frequency response: (

**a**) block diagram; and (

**b**) test bench.

## 5. Other AIC Family Models Based on IAM

#### 5.1. The AIDBB Converter

#### 5.2. The AIDF Group

**Figure 12.**Electrical circuit of the isolated non-inveting asymmetrical interleaved dual flyback (AIDF) converter.

## 6. Application Example: AIDB Control Design

#### 6.1. The LQR Controller

#### 6.2. Simulation Results

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Arango, E.; Ramos-Paja, C.A.; Calvente, J.; Giral, R.; Serna-Garces, S.I.
Asymmetrical Interleaved DC/DC Switching Converters for Photovoltaic and Fuel Cell Applications—Part 2: Control-Oriented Models. *Energies* **2013**, *6*, 5570-5596.
https://doi.org/10.3390/en6105570

**AMA Style**

Arango E, Ramos-Paja CA, Calvente J, Giral R, Serna-Garces SI.
Asymmetrical Interleaved DC/DC Switching Converters for Photovoltaic and Fuel Cell Applications—Part 2: Control-Oriented Models. *Energies*. 2013; 6(10):5570-5596.
https://doi.org/10.3390/en6105570

**Chicago/Turabian Style**

Arango, Eliana, Carlos Andres Ramos-Paja, Javier Calvente, Roberto Giral, and Sergio Ignacio Serna-Garces.
2013. "Asymmetrical Interleaved DC/DC Switching Converters for Photovoltaic and Fuel Cell Applications—Part 2: Control-Oriented Models" *Energies* 6, no. 10: 5570-5596.
https://doi.org/10.3390/en6105570