# Theoretical Assessment of DC/DC Power Converters’ Basic Topologies. A Common Static Model

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## Abstract

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## Featured Application

**DC/DC converters design and teaching.**

## Abstract

## 1. Introduction

_{ON}, regarding the switching period T). This means that for the coupling DC generator → DC/DC converter → load, if δ control is made adaptive (under generator and/or load demand), then the resistance of the generator and the load can be matched so that the DC/DC converter allows maximum power transfer from generator to load. Moreover, in the case of renewable power sources, which are subjected to constant fluctuations as a result of environmental changes, the DC/DC converter enables the generator to work at maximum power. Ideally speaking, the relationships that allow load adaptation by DC/DC converters according to their δ are well-known [34,35]. The problem lies in the fact that these expressions are approximate and in many operating conditions undergo deviations that can make them practically inapplicable, since DC/DC power converters are strongly non-ideal systems. Therefore, the theoretical results expected as a consequence of the application of the expressions commonly found in literature are usually very far from experimental reality. This requires that reliable experimental implementation demands a previous step: careful and exhaustive simulations.

_{Vi})-dependent sources: a current source in its input and a voltage source in its output. At the ideal converter input is connected an equivalent circuit consisting of the generator (V

_{g}) and a series resistor (R

_{X}), which is aimed at concentrating all the losses of interest in the actual converter originated by the parasitic resistances of its components (active and passive). Of course, R

_{X}can be added to the generator internal resistance.

_{X}) and the load are connected. V

_{X}concentrates the losses in the actual converter due to the threshold voltage of its diode. Currents and voltages in the model are given by their mean values.

_{X}, V

_{X}and A

_{Vi}are calculated for each basic topology (i.e., boost, buck and buck-boost). From here, input resistance, voltage gain and actual efficiency for each of these three topologies are found. Finally, model quality is assessed.

## 2. Operating Modes of a DC/DC Converter. Definition of Parameters

_{L}through its inductor) of every DC/DC converter regardless of its topology are explained briefly. In addition, some parameters of interest will also be defined. The operating modes are:

_{X}= T, so:

_{X}< T, so:

_{g}(Figure 1 and Figure 2) during a switching period:

## 3. Boost Converter

_{S}for the power switch resistance in “on” state (closed). The leakage resistance of the power switch in “off” state (open) is so large that it can be considered an open circuit. R

_{D}is the forward (on) resistance of the freewheeling diode (the leakage resistance in off condition is so large that it can be considered an open circuit) and V

_{γ}is its threshold voltage. The capacitor is considered an ideal element, as its series resistance has a very low value and its leakage resistance is very large (for capacitors of certain quality) whereby it can be neglected in parallel with the load.

#### 3.1. Determination of the Generator-Supplied Current

**0**≤

**t**≤

**T**

_{ON}_{L}and integrating the result into (12):

_{1}:

**T**≤

_{ON}**t**≤

**T**

_{X}_{X}) in this time interval. At this point, if the current reaches 0 (DCM), it then keeps this value until new switch conduction (see Figure 2b).

_{L}and integrating the result into (12):

_{2}:

_{1}and substituting it in (27a):

_{1}and k

_{2}given by (18) and (25) respectively:

#### 3.2. Loss Resistance, Loss Voltage and Voltage Gain Determination

_{o}is given by

_{Vr}is the converter actual voltage gain. That is:

_{X}, V

_{X}and A

_{Vi}of the model of Figure 1. From it:

_{γ}, R

_{D}and R

_{S}given by the manufacturers of the converter components) of the model parameters (Figure 1) of a generic and real boost converter.

_{X}can be obtained as shows Figure 6.

#### 3.3. Input Resistance

_{L},

_{i},

_{X}, V

_{X}and A

_{Vi}are given by (45)–(47) respectively. Equation (50) can be approximated taking into account that usually $\left|{A}_{Vi}{V}_{g}\right|\gg \left|{V}_{X}\right|$, wherewith,

#### 3.4. Efficiency

_{Vr}by its value given in (43):

_{X}, V

_{X}and A

_{Vi}are given by (45)–(47) respectively.

#### 3.5. Conventional Approximate Analysis

_{X}and A

_{Vi}, (45) and (47) respectively, it is easy to get their common idealized CCM values [24,36,37,38,39]:

_{Vi}can be approximated to:

#### 3.6. Discontinuous Conduction Mode (DCM)

_{2}and substituting the result in (63b),

## 4. Buck Converter

#### 4.1. Determination of Generator-Supplied Current

**0 ≤ t ≤ T**

_{ON}**T**≤

_{ON}**t**≤

**T**

_{X}_{1}and k

_{2}values from (18) and (25) respectively, ΔV from (7) and considering (35),

#### 4.2. Loss Resistance, Loss Voltage and Voltage Gain Determination

_{X}, V

_{X}and A

_{Vi}according to the converter parameters given in Figure 1, we proceeded as follows. Comparing (44) and (74) the following is obtained:

_{vr}.

#### 4.3. Input Resistance

_{X}, V

_{X}and A

_{Vi}from (75)–(77), respectively, in (50).

#### 4.4. Efficiency

_{X}, V

_{X}and A

_{Vi}from (75) to (77), respectively, in (53).

#### 4.5. Conventional Approximate Analysis

#### 4.6. Discontinuous Conduction Mode (DCM)

_{2}and substituting the result in (92b),

_{1}and k

_{2}are given by (18) and (25) respectively.

## 5. Buck-Boost Converter

#### 5.1. Determination of Generator-Supplied Current

**0**≤

**t**≤

**T**

_{ON}_{g}as was done from (13) to (14):

**T**≤

_{ON}**t**≤

**T**

_{X}

_{1}and k

_{2}values from (18) and (25) respectively,

#### 5.2. Loss Resistance, Loss Voltage and Voltage Gain Determination

_{X}, V

_{X}and A

_{Vi}according to the converter parameters given in Figure 1, we are going to proceed as follows. Comparing (44) and (103), the following is obtained:

#### 5.3. Input Resistance

_{X}, V

_{X}and A

_{Vi}from (104)–(106), respectively, in (50):

#### 5.4. Efficiency

_{X}, V

_{X}and A

_{Vi}from (104)–(106), respectively, in (53):

#### 5.5. Conventional Approximate Analysis

#### 5.6. Discontinuous Conduction Mode (DCM)

_{2}and substituting the result in (117b),

## 6. Results

_{γ}, R

_{D}and R

_{S}. Regarding r, it has been obtained by measuring a 1 mH inductor made in the laboratory. For example, 10 V has been chosen for V

_{g}and 20 or 10 Ω (depending on the characteristics of the active components) for R

_{L}. Finally, we have used f = 10 kHz because it is usual in DC/DC power converters.

#### 6.1. Boost Converter

_{Vr}(introducing (45)–(47), with their respective values, into (43)) and the actual obtained by simulation (by the circuit of Figure 13).

#### 6.2. Buck Converter

#### 6.3. Buck-Boost Converter

## 7. Discussion

_{Vr}, R

_{i}and h. Authors arrive at these expressions from approximations from the beginning, specifically considering that the current through r is constant and equal to a mean value. In the developed model, you get to the same expressions (Section 3.5, Section 4.5 and Section 5.5), but without previous approximations and by the resolving of exact equations.

_{Vr}) of the actual boost converter. In fact, as you can see in Table 1, the (NMAE%) is negligible, only 0.45%. Figure 16 shows up the great influence of its non-idealities on an actual boost converter operation. Ideally, a boost converter presents no upper limit in the gain (55), so it can take anyone above 1. However, in the actual case, as it is well known, and the developed model is able to show, the gain is drastically modified. Although the conventional model error is also small (2.93%), it is 6.5 times greater than the developed model.

_{Vr}, R

_{i}and η. For the developed model, NMAE% is again negligible: 0.73%, 1.61% and 0.31% respectively. However, for the conventional model, NMAE% is 7.10%, 5.39% and 8.09% respectively. Again the differences are notable, and especially regarding efficiency, where the error is much worse than in the buck converter, specifically 26 times greater.

_{Vr}, R

_{i}and η. For the developed model, NMAE% is once again negligible: 1.4%, 1.98% and 0.84% respectively. In the case of the conventional model, NMAE% is: 7.57%, 5.87% and 8.65% respectively. Now, the efficiency error is more than 10 times greater in the conventional model regarding the developed one.

## 8. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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Model | Converter | A_{Vr} | R_{i} | h |
---|---|---|---|---|

Developed | Boost | 0.45% | 0.66% | 0.30% |

Buck | 0.73% | 1.61% | 0.31% | |

Buck-boost | 1.4% | 1.98% | 0.84% | |

Conventional | Boost | 2.93% | 2.19% | 2.69% |

Buck | 7.10% | 5.39% | 8.09% | |

Buck-boost | 7.57% | 5.87% | 8.65% |

**Table 2.**Summary of the parameters for each DC/DC converter (see Figure 1).

Parameter | Boost | Buck | Buck-boost |
---|---|---|---|

R_{X} | $\frac{1}{\frac{\delta -k\beta}{r+{R}_{S}}+\frac{{\delta}_{f}+k\beta}{r+{R}_{D}}}$ | $\frac{r+{R}_{S}}{\delta +{k}_{1}\beta}$ | $\frac{r+{R}_{S}}{\delta +{k}_{1}\beta}$ |

V_{X} | ${V}_{\gamma}$ | $\left[\frac{{k}_{2}\beta}{k\beta -\delta}\right]{V}_{\gamma}$ | $-{V}_{\gamma}$ |

A_{Vi} | $1+\frac{r+{R}_{D}}{r+{R}_{S}}\cdot \frac{\delta -k\beta}{{\delta}_{f}+k\beta}$ | $\frac{\delta +{k}_{1}\beta}{\delta -k\beta}$ | $\frac{\delta +{k}_{1}\beta}{{k}_{2}\beta}$ |

δ_{f-crit} | ${k}_{2}\mathrm{ln}\left[1-\frac{{V}_{g}}{\Delta V-{V}_{\gamma}}\theta \right]$ | ${k}_{2}\mathrm{ln}\left[1+\frac{\Delta V}{{V}_{0}+{V}_{\gamma}}\theta \right]$ | ${k}_{2}\mathrm{ln}\left[1-\frac{{V}_{g}}{{V}_{0}-{V}_{\gamma}}\theta \right]$ |

where: ${k}_{1}=\frac{f\cdot L}{r+{R}_{S}};{k}_{2}=\frac{f\cdot L}{r+{R}_{D}};k={k}_{2}-{k}_{1};\beta =\frac{\left(1-{e}^{{\gamma}_{1}}\right)\cdot \left(1-{e}^{-{\gamma}_{2}}\right)}{{e}^{{\gamma}_{1}}-{e}^{-{\gamma}_{2}}};{\gamma}_{1}=\frac{\delta}{{k}_{1}};{\gamma}_{2}=\frac{{\delta}_{f}}{{k}_{2}};{\delta}_{f}=\frac{{T}_{X}-{T}_{ON}}{T};\Delta V={V}_{g}-{V}_{o};\theta =\frac{{k}_{1}}{{k}_{2}}\left(1-{e}^{-{\gamma}_{1}}\right).$ | |||

For all converters: ${A}_{Vr}=\frac{{V}_{o}}{{V}_{g}}=\frac{{A}_{Vi}-\frac{{V}_{X}}{{V}_{g}}}{1+{A}_{Vi}{}^{2}\frac{{R}_{X}}{{R}_{L}}};$${R}_{i}=\frac{{R}_{X}+\frac{1}{{A}_{Vi}{}^{2}}{R}_{L}}{1-\frac{{V}_{X}}{{A}_{Vi}{V}_{g}}};$$\eta =\frac{1-\frac{{V}_{X}}{{A}_{Vi}{V}_{g}}}{1+{A}_{Vi}{}^{2}\frac{{R}_{X}}{{R}_{L}}}$ |

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**MDPI and ACS Style**

Enrique, J.M.; Barragán, A.J.; Durán, E.; Andújar, J.M.
Theoretical Assessment of DC/DC Power Converters’ Basic Topologies. A Common Static Model. *Appl. Sci.* **2018**, *8*, 19.
https://doi.org/10.3390/app8010019

**AMA Style**

Enrique JM, Barragán AJ, Durán E, Andújar JM.
Theoretical Assessment of DC/DC Power Converters’ Basic Topologies. A Common Static Model. *Applied Sciences*. 2018; 8(1):19.
https://doi.org/10.3390/app8010019

**Chicago/Turabian Style**

Enrique, Juan Manuel, Antonio Javier Barragán, Eladio Durán, and José Manuel Andújar.
2018. "Theoretical Assessment of DC/DC Power Converters’ Basic Topologies. A Common Static Model" *Applied Sciences* 8, no. 1: 19.
https://doi.org/10.3390/app8010019