# A Building-Block Approach to State-Space Modeling of DC-DC Converter Systems

## Abstract

**:**

## 1. Introduction

## 2. Modular Modeling Approach

#### 2.1. Model for Passive Components

#### 2.2. Model for Controlled Converters and Converter Systems

#### 2.3. Model for Controllers

#### 2.4. Making Connections

## 3. Adding Control Loops to Converter Models

#### 3.1. Average Current Mode

#### 3.2. Voltage Mode or Outer Voltage Loops

## 4. Connecting Converter Models

## 5. Summary of the Building-Block Modeling Approach

#### 5.1. Building Block Types

#### 5.2. Connecting Building Blocks

- Connecting converters with passive components or other converters. Since both building blocks for passive elements as given by (1) and building blocks for converters as given by (2) are two-port models, the same connection operation can be used for all these cases. The resulting model (10) is obtained by applying (13).
- Creating control loops. Connecting a controller, that is, a building block type as given by (3), to a converter is done by creating an open-loop model (4) using (5). For closing the control loop, the model’s $\mathit{A}$ matrix must be adapted using (6), with a feedback gain matrix $\mathit{K}$ chosen depending on the controlled variable, for example, one of (7) or (8) for current or voltage mode control.

#### 5.3. Frequency-Domain Analysis

## 6. Examples

#### 6.1. Buck Converter with Multiloop (I2) Control

- As the starting point to build up the complete model, the two-port state-space model for the PCM buck converter from Reference [34] is being used. The example is parameterized as given in the following. Operating point: ${V}_{\mathrm{in}}=24\mathrm{V}$, ${V}_{\mathrm{out}}=12\mathrm{V}$, ${I}_{\mathrm{out}}=2.4\mathrm{A}$. Power stage: ${f}_{\mathrm{Switch}}=50\mathrm{k}\mathrm{Hz}$, $L=100\mathsf{\mu}\mathrm{H}$, $C=100\mathsf{\mu}\mathrm{F}$. Peak current mode controller: external ramp of $1.5\mathrm{A}$ per switching cycle.
- For the I2 current loop, an additional controller for the average inductor current is being added. A Type 1 controller with ${K}_{\mathrm{i}}=20$,000 is employed, cf. Appendix C, which is connected to the state-space model as described in Section 3.1.
- For the outer voltage control loop, a Type 2 controller with zero at $0.3\mathrm{k}\mathrm{Hz}$, pole at $25\mathrm{k}\mathrm{Hz}$ and ${K}_{\mathrm{i}}=3000$ is being used, cf. Appendix C. It is connected to the model as described in Section 3.2.

#### 6.2. Boost Converter with Input Filter

- A linearized state-space model for a boost converter in CCM as given in Appendix B.2. The example is parameterized as given in the following. Operating point: ${V}_{\mathrm{in}}=10\mathrm{V}$, ${V}_{\mathrm{out}}=24\mathrm{V}$, ${I}_{\mathrm{out}}=1.2\mathrm{A}$. Power stage: ${f}_{\mathrm{Switch}}=100\mathrm{k}\mathrm{Hz}$, $L=20\mathsf{\mu}\mathrm{H}$, $C=220\mathsf{\mu}\mathrm{F}$.
- A load resistance (model from Appendix A.1) corresponding to the operating point is connected at the converter output using Section 4.
- A Type 3 controller (cf. Appendix C) is being employed for the voltage control loop, with zeros placed at $10\mathrm{k}\mathrm{Hz}$, poles at $0.1\mathrm{k}\mathrm{Hz}$ and $50\mathrm{k}\mathrm{Hz}$, and ${K}_{\mathrm{i}}=10$. It is connected to the model as described in Section 3.2.
- Finally, an LC filter model (cf. Appendix A.2) accounting for both an input filter and wiring is connected at the input of the converter model using Section 4, with $L=5\mathsf{\mu}\mathrm{H}$, ${r}_{\mathrm{L}}=50\mathrm{m}\mathsf{\Omega}$, and $C=1\mathsf{\mu}\mathrm{F}$.

#### 6.3. Series-Connected Boost and Buck Converter Stages

- Reuse the final model from Section 6.2 (voltage-controlled boost converter including input filter) but without the load resistance, since the buck converter will constitute the new load to the boost stage.
- Reuse the final model from Section 6.1 (voltage-controlled buck converter with underlying I2 current control).
- Connect the buck converter model at the output of the boost converter, as described in Section 4.

## 7. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Building Blocks: Passive Component Models

#### Appendix A.1. Resistive Load

#### Appendix A.2. LC Filter

## Appendix B. Building Blocks: Converter Models

#### Appendix B.1. Available Converter Models

#### Appendix B.2. Example: Boost Converter

## Appendix C. Building Blocks: Controller Models

**Table A1.**Transfer function ${G}_{\mathrm{C}}\left(s\right)$ and state-space model matrices ${\mathit{A}}_{\mathrm{C}}$, ${\mathit{B}}_{\mathrm{C}}$, ${\mathit{C}}_{\mathrm{C}}$ according to (3) for controllers known as Type 1, 2, and 3.

Type 1 | Type 2 | Type 3 |
---|---|---|

$G}_{\mathrm{C}}\left(s\right)=\frac{{K}_{\mathrm{i}}}{s$ | $G}_{\mathrm{C}}\left(s\right)=\frac{{K}_{\mathrm{i}}}{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{1+{T}_{\mathrm{z}}s}{1+{T}_{\mathrm{p}}s$ | $G}_{\mathrm{C}}\left(s\right)=\frac{{K}_{\mathrm{i}}}{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{(1+{T}_{\mathrm{z}1}s)(1+{T}_{\mathrm{z}1}s)}{(1+{T}_{\mathrm{p}1}s)(1+{T}_{\mathrm{p}2}s)$ |

${\mathit{A}}_{\mathrm{C}}=0$ | ${\mathit{A}}_{\mathrm{C}}=\left(\begin{array}{cc}0& 0\\ 1& -\frac{1}{{T}_{\mathrm{p}}}\end{array}\right)$ | ${\mathit{A}}_{\mathrm{C}}=\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& -\frac{1}{{T}_{\mathrm{p}1}{T}_{\mathrm{p}2}}\\ 0& 1& -\frac{{T}_{\mathrm{z}1}+{T}_{\mathrm{z}2}}{{T}_{\mathrm{p}1}{T}_{\mathrm{p}2}}\end{array}\right)$ |

${\mathit{B}}_{\mathrm{C}}={K}_{\mathrm{i}}$ | ${\mathit{B}}_{\mathrm{C}}=\left(\begin{array}{c}\frac{{K}_{\mathrm{i}}}{{T}_{\mathrm{p}}}\\ \frac{{K}_{\mathrm{i}}{T}_{\mathrm{z}}}{{T}_{\mathrm{p}}}\end{array}\right)$ | ${\mathit{B}}_{\mathrm{C}}=\left(\begin{array}{c}\frac{{K}_{\mathrm{i}}}{{T}_{\mathrm{p}1}{T}_{\mathrm{p}2}}\\ \frac{{K}_{\mathrm{i}}\left({T}_{\mathrm{z}1}+{T}_{\mathrm{z}2}\right)}{{T}_{\mathrm{p}1}{T}_{\mathrm{p}2}}\\ \frac{{K}_{\mathrm{i}}{T}_{\mathrm{z}1}{T}_{\mathrm{z}2}}{{T}_{\mathrm{p}1}{T}_{\mathrm{p}2}}\end{array}\right)$ |

${\mathit{C}}_{\mathrm{C}}=1$ | ${\mathit{C}}_{\mathrm{C}}=\left(\begin{array}{cc}0& 1\end{array}\right)$ | ${\mathit{C}}_{\mathrm{C}}=\left(\begin{array}{ccc}0& 0& 1\end{array}\right)$ |

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**Figure 1.**Generic two-port network. Currents are defined as flowing into the input and output ports.

**Figure 2.**Two-port network description consisting of three subsystems: input filter, converter (with a control input $\mathit{ctl}$), and the actual load.

**Figure 3.**Interfaces of the two-port network models with control inputs when adding a controller to a converter. (

**a**) Open-loop case. (

**b**) Closed-loop case.

**Figure 4.**Series connection of two two-port networks (source and load). The control inputs ${\mathit{ctl}}^{\mathrm{S}/\mathrm{L}}\left(t\right)$ are optional and only present if source or load contain at least one converter, respectively.

**Figure 5.**Impact of a reference step $2.4\mathrm{A}\to 2.0\mathrm{A}$ at $t=0\mathrm{s}$ on the inductor current of the buck converter from example Section 6.1 with closed I2-current loop.

**Figure 6.**Reference step $13\mathrm{V}\to 12\mathrm{V}$ at $t=0\mathrm{s}$ of the buck converter from example Section 6.1 with closed voltage loop and underlying I2 current control.

**Figure 7.**Impact of a 25% load step $2.4\mathrm{A}\to 3.0\mathrm{A}$ at $t=0\mathrm{s}$ on the output voltage of the buck converter from example Section 6.1 with closed voltage loop.

**Figure 8.**Input and output impedance of the boost converter from example Section 6.2, with closed voltage loop, both with input filter (solid lines) and without input filter (dashed lines).

**Figure 9.**Impact of a 100% load step $1.2\mathrm{A}\to 2.4\mathrm{A}$ at $t=0\mathrm{s}$ on the output voltage of the boost converter from example Section 6.2 with closed voltage loop, both with input filter (solid lines) and without input filter (dashed lines). For better comparability, the circuit simulation results were filtered (moving average) to remove the switching ripple voltage.

**Figure 10.**Two-port view of a series connection of boost and buck converter stages. In both stages, the filter elements L and C are being modeled with equivalent series resistances (ESR).

**Figure 11.**Impact of a 25% output load step $2.4\mathrm{A}\to 3.0\mathrm{A}$ at $t=0\mathrm{s}$ of the series-connected buck stage from example Section 6.3 on the intermediate output voltage of the boost stage. For better comparability the circuit simulation result was filtered (moving average) to remove the switching ripple voltage.

Type | Equation | Definition |
---|---|---|

Passive components | (1) | $\begin{array}{cc}\hfill \dot{\mathit{x}}\left(t\right)& \hfill =\mathit{A}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{x}\left(t\right)+\left(\begin{array}{cc}{\mathit{B}}_{1}& {\mathit{B}}_{2}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}{v}_{\mathrm{in}}\left(t\right)\\ {i}_{\mathrm{out}}\left(t\right)\end{array}\right)\hfill \\ \hfill \left(\begin{array}{c}{i}_{\mathrm{in}}\left(t\right)\\ {v}_{\mathrm{out}}\left(t\right)\end{array}\right)& \hfill =\left(\begin{array}{c}{\mathit{C}}_{1}\\ {\mathit{C}}_{2}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{x}\left(t\right)+\left(\begin{array}{cc}{D}_{11}& {D}_{12}\\ {D}_{21}& {D}_{22}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}{v}_{\mathrm{in}}\left(t\right)\\ {i}_{\mathrm{out}}\left(t\right)\end{array}\right)\hfill \end{array}$ |

Controlled converters | (2) | $\begin{array}{cc}\hfill \dot{\mathit{x}}\left(t\right)& \hfill =\mathit{A}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{x}\left(t\right)+\left(\begin{array}{ccc}{\mathit{B}}_{1}& {\mathit{B}}_{2}& {\mathit{B}}_{3}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}{v}_{\mathrm{in}}\left(t\right)\\ {i}_{\mathrm{out}}\left(t\right)\\ \mathit{ctl}\left(t\right)\end{array}\right)\hfill \\ \hfill \left(\begin{array}{c}{i}_{\mathrm{in}}\left(t\right)\\ {v}_{\mathrm{out}}\left(t\right)\end{array}\right)& \hfill =\left(\begin{array}{c}{\mathit{C}}_{1}\\ {\mathit{C}}_{2}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{x}+\left(\begin{array}{ccc}{D}_{11}& {D}_{12}& {D}_{13}\\ {D}_{21}& {D}_{22}& {D}_{23}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}{v}_{\mathrm{in}}\left(t\right)\\ {i}_{\mathrm{out}}\left(t\right)\\ \mathit{ctl}\left(t\right)\end{array}\right)\hfill \end{array}$ |

Controllers | (3) | $\begin{array}{cc}\hfill {\dot{\mathit{x}}}_{\mathrm{C}}\left(t\right)& \hfill ={\mathit{A}}_{\mathrm{C}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{x}}_{\mathrm{C}}\left(t\right)+{\mathit{B}}_{\mathrm{C}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}e\left(t\right)\hfill \\ \hfill u\left(t\right)& \hfill ={\mathit{C}}_{\mathrm{C}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{x}}_{\mathrm{C}}\left(t\right)+{\mathit{D}}_{\mathrm{C}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}e\left(t\right)\hfill \end{array}$ |

**Table 2.**Transfer functions than can be obtained from the converter (system) model (2), cf. Section 2.2, at any modeling stage (e.g., with or without controller).

Name | Definition | ||
---|---|---|---|

Control-to-output ^{†} | ${G}_{\mathrm{co}}\left(s\right)$ | $=\frac{{v}_{\mathrm{out}}\left(s\right)}{\mathit{ctl}\left(s\right)}$ | $={\mathit{C}}_{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(s\mathit{I}-\mathit{A}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{B}}_{3}+{D}_{23}$ |

Output impedance | ${Z}_{\mathrm{out}}\left(s\right)$ | $=\frac{{v}_{\mathrm{out}}\left(s\right)}{{i}_{\mathrm{out}}\left(s\right)}$ | $={\mathit{C}}_{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(s\mathit{I}-\mathit{A}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{B}}_{2}+{D}_{22}$ |

Input admittance | ${Y}_{\mathrm{in}}\left(s\right)$ | $=\frac{{i}_{\mathrm{in}}\left(s\right)}{{v}_{\mathrm{in}}\left(s\right)}$ | $={\mathit{C}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(s\mathit{I}-\mathit{A}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{B}}_{1}+{D}_{11}$ |

Forward voltage gain | ${G}_{\mathrm{v}}\left(s\right)$ | $=\frac{{v}_{\mathrm{out}}\left(s\right)}{{v}_{\mathrm{in}}\left(s\right)}$ | $={\mathit{C}}_{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(s\mathit{I}-\mathit{A}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{B}}_{1}+{D}_{21}$ |

Reverse current gain | ${G}_{\mathrm{i}}\left(s\right)$ | $=\frac{{i}_{\mathrm{in}}\left(s\right)}{{i}_{\mathrm{out}}\left(s\right)}$ | $={\mathit{C}}_{1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\left(s\mathit{I}-\mathit{A}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathit{B}}_{2}+{D}_{12}$ |

^{†}If there is more than one control input (after connecting several converters to one model), the control-to-output transfer function for the k-th control input is obtained using the ${\mathit{B}}_{2+k}$ column of the $\mathit{B}$ matrix and ${D}_{2,(2+k)}$ instead of ${\mathit{B}}_{3}$ and ${D}_{23}$.

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Herbst, G.
A Building-Block Approach to State-Space Modeling of DC-DC Converter Systems. *J* **2019**, *2*, 247-267.
https://doi.org/10.3390/j2030018

**AMA Style**

Herbst G.
A Building-Block Approach to State-Space Modeling of DC-DC Converter Systems. *J*. 2019; 2(3):247-267.
https://doi.org/10.3390/j2030018

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Herbst, Gernot.
2019. "A Building-Block Approach to State-Space Modeling of DC-DC Converter Systems" *J* 2, no. 3: 247-267.
https://doi.org/10.3390/j2030018