**1. Introduction**

Despite the fact that internal combustion engines (ICEs) have been a mature technology for the past 100 years, it is expected that electric vehicles (EVs) will break the monopoly of conventional vehicles using only ICEs because of their performance and superior fuel economy [1]. Due to the strict regulations on global warming and energy resources constraints, and on reducing fossil fuel prices as well as gas emissions, environmental awareness has led to a high interest in EVs as an alternative solution for further improvement compared to ICEs [2,3]. To increase the adoption of EVs, significant e fforts in terms of reducing the charging time are required. Consequently, to allow massive market penetration of EVs, the concept of ultrafast charging (UFC) requires more investigation. In this regard, several research studies targeting fast chargers and UFC for EVs have been provided in the literature [4–8]. UFC technology is a high power charging technology (≥ 400 kW) that can replace or substitute the ICE technology and can charge EVs' battery packs in ≤ 10 min [9–11].

Advanced power electronics converters are considered as key enabling technologies for realizing EV UFC, where high-power DC-DC converters are needed. The critical requirements for designing EV battery chargers are high e fficiency, low cost, high power density, and galvanic isolation [12]. Furthermore, one of the UFC stations' requirements is to design the DC-DC converters in a modular manner to o ffer easy maintenance, as well as scalability, redundancy, and fault ride-through capability [13–15]. In modular power converters, each unit handles a small portion of the total input power. Accordingly, the selected power switches are of lower voltage and/or current ratings; therefore, higher switching frequency capability, consequently, reduced weight and size [16–18]. Multimodule DC-DC converters can provide a bidirectional power flow through employing submodules that are based on dual active bridge (DAB), dual half bridge (DHB), or series resonant topologies [19,20].

The DAB configuration, shown in Figure 1, consists of two active bridges that are connected via a medium/high-frequency AC transformer. DAB can be constructed using a single-phase bridge or a three-phase bridge depending on the design criteria. The 2L-DAB, shown in Figure 1, usually operates in a square wave mode. The intermediate transformer leakage inductance limits the maximum power flow and is used as the energy transferring element. This topology is capable of bidirectional power flow that can be achieved by controlling the phase shift between the two bridges and the magnitude of the output voltage per bridge. The switches can be switched at zero voltage switching (ZVS) and/or zero current switching (ZCS). Accordingly, switching losses are reduced, and the power e fficiency is increased.

**Figure 1.** DAB converter circuit diagram.

Figure 2 presents a block diagram for a typical EV UFC that involves an AC-DC stage and a DC-DC stage. This paper will focus on the DC-DC stage employed in EV UFC applications. In the literature, many research studies have been introduced the two stages. In [21], to realize medium-voltage EV UFC stations, a multiport power converter has been proposed. In [22], a bidirectional fast charging system control strategy consisting of two cascaded stages has been proposed, where two DABs are connected in parallel at the battery side. However, in [23], an isolated DAB-based single-stage AC-DC converter has been presented. The charger in [23] contains a single stage that includes the PFC and ensures ZVS over the full load range. In [24,25], a frequency modulated CLLC-R-DAB has been proposed. In this topology, the converter operates over a considerable variation of the input voltage while maintaining soft-switching capability. A smaller switching frequency range is used to modulate the CLLC-R-DAB converter when compared to SR-DAB. In [26], a full-bridge phase-shifted DC-DC converter that combines the characteristics of the double inductor rectifier and the conventional hybrid switching converter is introduced for EV fast chargers. In [27], a medium-voltage high-power isolated DC-DC converter for EVs fast chargers is presented. In [28], the AC-DC and DC-DC stages of an EV charger are studied where the DC-DC stage utilizes interleaved DC-DC converters.

**Figure 2.** Block diagram for a typical EV UFC.

Multimodule converters are considered a suitable choice for realizing the high power and high voltage requirements of the UFC charger. However, an increased number of modules with low power would increase system complexity, cost and losses, which reduces the cooling requirements and

consequently the weight, volume, and cost. However, reduced switching losses can be achieved via soft-switching [29–35]. Nonetheless, introducing a low number of modules with high power would reduce the switching frequency capabilities; therefore, reducing the power density, which increases size and weight.

Accordingly, the main contribution of this work is to introduce a hybrid multimodule DC-DC converter-based DAB topology as the DC-DC stage for EV UFC to achieve high e fficiency, high power density, and reduced weight and cost. The hybrid concept is achieved through employing two di fferent groups of multimodule converters. The first group is designed to be in charge of a high fraction of the total required power while operating relatively at a low switching frequency. Nevertheless, the second group is designed for a low fraction of the total power operating relatively at a high switching frequency. The work presented in this paper includes a generalized small-signal model for the presented converter as well as the control strategy required in achieving uniform power-sharing between the employed modules. Besides, a power loss evaluation has been conducted to compare the proposed converter with the other two options.

To verify the presented concept, the number of modules needed to achieve the required ratings is calculated for both; conventional multimodule DC-DC converters and hybrid multimodule DC-DC converters. In addition, the power loss analysis of the hybrid multimodule converter is provided. To support the power converter controller design, a generalized small-signal model for the hybrid multimodule DC-DC converter is studied in detail. Besides, to ensure equal power-sharing among the employed modules, the control scheme for the hybrid multimodule DC-DC converter with the aforementioned specifications is studied. The main contribution of the paper can be summarized as follows:


This paper is structured as follows: Section 2 presents the hybrid input-series input-parallel output-series output-parallel (ISIP-OSOP) multimodule power converter and the generalized small-signal modeling. Section 3 presents a 200 kW hybrid eight-module ISOP converter. In Section 3, the small-signal model of the presented converter is derived using the analysis provided in Section 2. Section 4 presents the number of modules needed to achieve the required ratings for both; conventional multimodule DC-DC converters and hybrid multimodule DC-DC converters. In addition, the power loss analysis of the conventional and hybrid multimodule converters is provided. Section 5 discusses the control strategy for the proposed hybrid ISOP multimodule DC-DC converters. Section 6 discusses the MatLab/Simulink model and the simulation results. Finally, Section 7 summarizes the key findings of this paper.

#### **2. Generalized Small-Signal Analysis for Dual Series**/**Parallel Input-Output (ISIP-OSOP) Hybrid Multimodule Converters**

In this section, the generalized small-signal modeling for dual series/parallel ISIP-OSOP hybrid multimodule DC-DC converter is introduced.

#### *2.1. Hybrid ISIP-OSOP Generic DC-DC Converter Circuit Configuration*

The hybrid ISIP-OSOP generic DC-DC converter configuration, shown in Figure 3, consists of *n* modules that are connected in series and/or parallel at the input side and in series and/or parallel at the output side. These *n* modules consist of two di fferent multimodule groups. The primary group consists of *L* isolated DC-DC converters that are in charge of a high fraction of the total required power operating relatively at a low switching frequency. The secondary group consists of *M* isolated DC-DC converters that are designed for a small fraction of the total power operating relatively at a high switching frequency. Accordingly, it can be said that the summation of *L* and *M* power converters results in a total of *n* DAB units. To di fferentiate between the primary and secondary multimodule DC-DC converter in the small signal analysis, the set of equations representing the primary group is black colored while the set of equations representing the secondary group is blue colored. In addition, the red colored symbols reflect the parameters defined for the input side, while the blue colored symbols reflects the parameters defined for the output side, as presented in the following.

By ensuring input current sharing (ICS) and input voltage sharing (IVS) for the primary group, the input current for each module in the primary group is reduced to *IinL* α*L***1** , and the input voltage for each module in the primary group is reduced to *VinL* β*L***1** . However, by ensuring ICS and IVS for the secondary group, the input current for each module in the secondary group is reduced to *IinM* α*M***1** , and the input voltage for each module in the secondary group is reduced to *VinM* β*M***1** group is reduced to *VinM* β*M***1** , in which, *IinL* and *VinL* are the input current and the input voltage for the primary group that consists of *L* number of modules, respectively. *IinM* and *VinM* are the input current and the input voltage for the secondary group that consists of *M* number of modules, respectively. α *M***1** represents the number of modules connected in parallel in the secondary group at the input side. β*M***1** represents the number of modules connected in series the secondary group at the input side.

Similarly, by ensuring output current sharing (OCS) and output voltage sharing (OVS) for the primary group, the output current per module in the primary group is *IoL aL***1** , and the output voltage per module in the primary module is reduced to *VoL bL***1** . However, by ensuring OCS and OVS for the secondary group, the output current per module is *IoM aM***1** , and the output voltage for each module in the secondary group is reduced to *VoM bM***1** . In which, *IoL* and *VoL* are the output current and the output voltage for the primary group that consists of *L* number of modules, respectively. *IoM* and *VoM* are the output current and the output voltage for the secondary group that consists of *M* number of modules, respectively. *aM***1** represents the number of modules connected in parallel in the secondary group at the output side. *bM***1** represents the number of modules connected in series in the secondary group at the output side.

The *L* isolated modules are responsible for delivering a portion of *KL* of the total required power, while the *M* isolated modules are responsible for delivering a portion of *KM* of the total required power, where *KL* + *KM* = 1 pu. The input voltages, input currents, output currents, and output voltages are represented in terms of the total input voltage, total input current, total output voltage, and total output current would result in Table 1.


**Table 1.** Individual module system parameters representation in terms of the overall system ratings.

#### *2.2. Hybrid ISIP-OSOP DC-DC Converter Small-Signal Modeling*

Using the model in [36], and expanding the study of the multimodule DC-DC converters in [37–41], the small-signal model for the hybrid multimodule ISIP-OSOP converter shown in Figure 4 is derived. Since each group is responsible for delivering a particular portion of the overall required power, where this portion is defined according to the overall system power ratings. Accordingly, it is worth mentioning that the equivalent load resistance seen by each group of multimodule converters is di fferent.

**Figure 3.** Generalized hybrid multimodule DC-DC converter configuration.

**Figure 4.** Small signal model for the generalized hybrid ISIP-OSOP multimodule DC-DC converter.

Since the input current and input voltage for each module in the primary group are α*L***2** *Iin* α*L***1** and β*L***2** *Vin* β*L***1** , respectively, and the output current and output voltage for each module in the primary group are *aL***2***Io aL***1** and *bL***2***Vo bL***1** , respectively. Therefore, the load resistance for each module in the primary group is *aL***1***bL***2** *aL***2***bL***1** *R*. Accordingly, ˆ *dijL*, ˆ *dvL*j which are the effect of changing the filter inductor current and the effect of changing the input voltage on the duty cycle modulation for the primary group and *IeqL* presented in Figure 4 can be defined as:

$$\hat{d}\_{\rm iLj} = -\frac{4\beta\_{\rm l.1}L\_{\rm lkl.}f\_{\rm sL}}{\beta\_{\rm l.2}K\_1V\_{\rm in}}\hat{\imath}\_{\rm l.Lj.}\ \ j = 1,2,\ldots,L\tag{1}$$

Equation (1) can be written as:

$$\hat{d}\_{\rm IL,j} = -\frac{\mathcal{J}\_{\rm L1} \mathcal{K}\_{\rm I} \mathcal{R}\_{\rm dL}}{\mathcal{J}\_{\rm L2} \mathcal{V}\_{\rm in}} \hat{\imath}\_{\rm LL,j}, \ j = 1, 2, \ldots, L \tag{2}$$

where *RdL* = 4*LlkL fsL K*21

$$\hat{d}\_{\rm vLj} = \frac{4a\_{\rm l,2}\psi\_{\rm l,1}\mathfrak{d}\_{\rm l,1}L\_{\rm lkl}f\_{\rm sL}D\_{\rm eff,1}}{a\_{\rm l,1}\psi\_{\rm l,2}\mathfrak{d}\_{\rm l,2}K\_{\rm l}^2RV\_{\rm in}}\mathfrak{d}\_{\rm cdl,j}, \ j = 1,2,\ldots,L. \tag{3}$$

Equation (3) can be written as:

 .

$$\hat{d}\_{\rm vLj} = \frac{a\_{\rm l.2} \upsilon\_{\rm l.1} \mathfrak{p}\_{\rm l.1} \mathfrak{R}\_{\rm dl.} D\_{\rm eff1}}{a\_{\rm l.1} \upsilon\_{\rm l.2} \mathfrak{p}\_{\rm l.2} \operatorname{RV}\_{\rm in}} \mathfrak{d}\_{\rm cdl.j\prime} \ j = 1, 2, \ldots, L \tag{4}$$

$$I\_{eqL} = \frac{a\_{12}b\_{11}\mathcal{B}\_{12}V\_{\rm in}}{a\_{11}b\_{12}\mathcal{B}\_{11}K\_1R} \tag{5}$$

Since the input current and input voltage for each module in the primary group is α*M*2 *Iin* α*M*1 and β*M*2 *Vin* β*M*1 , respectively, and the output current and output voltage for each module in the primary group is *aM*2*Io aM*1 and *bM*2*Vo bM*1 , respectively. Therefore, the load resistance for each module in the primary group is *aM*1*bM*2 *aM*2*bM*1 *R*. Accordingly, ˆ *dijM*, ˆ *dvMj* which are the effect of changing the filter inductor current and the effect of changing the input voltage on the duty cycle modulation for the primary group and *IeqM* presented in Figure 4 can be defined as:

$$\hat{d}\_{\text{lMj}} = -\frac{4\mathfrak{B}\_{\text{l}\text{1}}L\_{\text{l}\text{-}\text{lM}}f\_{\text{s}\text{M}}}{\mathfrak{B}\_{\text{l}\text{2}}K\_{\text{2}}V\_{\text{in}}}\hat{t}\_{\text{L}\text{M}j\text{-}j} = 1,2,...,M\tag{6}$$

Equation (6) can be written as:

.

$$\hat{d}\_{iMj} = -\frac{\oint\_{\text{L1}} K\_2 R\_{iM} \hat{\imath}\_{i\text{L}M\text{\textdegree}}}{\oint\_{\text{L2}} V\_{i\text{m}} \, V\_{i\text{m}} \, \hat{\jmath}\_{i\text{m}} \, j = 1, 2, \dots, M} \tag{7}$$

where; *RdM* = 4*LlkM fsM <sup>K</sup>*2<sup>2</sup>

$$\hat{d}\_{\text{cM}j} = \frac{4a\_{\text{L2}}b\_{\text{L1}}\mathfrak{g}\_{\text{L1}}L\_{\text{R2}M}f\_{\text{sM}}D\_{eff2}}{a\_{\text{L1}}b\_{\text{L2}}\mathfrak{g}\_{\text{L2}}K\_2^2RV\_{\text{in}}}v\_{\text{cM}Mj}, j = 1,2,...,M\tag{8}$$

Equation (8) can be written as:

$$\hat{d}\_{vMj} = \frac{a\_{L2}b\_{L1}\mathfrak{P}\_{L1}\mathcal{R}\_{dM}D\_{eff2}}{a\_{L1}b\_{L2}\mathfrak{P}\_{L2}\mathcal{R}\_{Vin}}v\_{cdMj\prime} \, j = 1, 2, \dots, M \tag{9}$$

$$I\_{\rm eqM} = \frac{a\_{12}b\_{11}\mathfrak{B}\_{12}V\_{in}}{a\_{11}b\_{12}\mathfrak{B}\_{11}K\_2\mathcal{R}}\tag{10}$$

Based on the feature of modularity and to simplify the analysis, it is assumed that all modules in the primary group and all modules in the secondary group have an equal e ffective duty cycle, transformer turns ratio, capacitor, and inductor values. Accordingly, *KL*1 = *KL*2 = ··· = *KLL* = *K*1, *KM*1 = *KM*2 = ··· = *KMM* = *K*2 *CL*1 = *CL*2 = ··· = *CLL* = *CL*, *C M*1 = *C M*2 = ··· = *CMM* = *C M*, *CdL*1 = *CdL*2 = ··· = *CdLL* = *CdL*, *CdM*1 = *CdM*2 = ··· = *CdMM* = *CdM*, *L*L1 = *L*L2 = ··· = *LLL* = *LL* and *LM*1 = *LM*2 = ··· = *LMM* = *LM*. In addition, it is also assumed that all modules in the primary group share the same input voltage and that all modules in the secondary group share the same input voltage. Accordingly, the DC input voltage of each module in the primary group is β*L*2 *Vin* β*L*1 and the DC input voltage of each module in the secondary group is β*M*2 *Vin* β*M*1 . Although each module has a di fferent duty cycle perturbation, it is assumed that all the DAB units have an equal normalized time shift. Besides, the ESR of the output capacitance is considered in this model. However, the ESR can be neglected compared to the load.

The following equations are obtained from Figure 4:

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*Deff* 1 *K*1 *<sup>v</sup>*<sup>ˆ</sup>*cdL*1 + β*L*2 *Vin* β*L*1 *K*1 ˆ *diL*1 + ˆ *dvL*1 + ˆ *dL*1 = *sLL*<sup>ˆ</sup>*iLL*1 + *<sup>v</sup>*<sup>ˆ</sup>*outL*1 *Deff* 1 *K*1 *<sup>v</sup>*<sup>ˆ</sup>*cdL*2 + β*L*2 *Vin* β*L*1 *K*1 ˆ *diL*2 + ˆ *dvL*2 + ˆ *dL*2 = *sLL*<sup>ˆ</sup>*iLL*2 + *<sup>v</sup>*<sup>ˆ</sup>*outL*2 . . . *Deff* 1 *K*1 *<sup>v</sup>*<sup>ˆ</sup>*cdLL* + β*L*2 *Vin* β*L*1 *K*1 ˆ *diLL* + ˆ *dvLL* + ˆ *dLL* = *sLL*<sup>ˆ</sup>*iLLL* + *<sup>v</sup>*<sup>ˆ</sup>*outLL Deff* 2 *K*2 ˆ *vcdM*1 + β*L*2 *Vin* β*L*1 *K*2 ( ˆ *diM*1 + ˆ *dvM*1 + ˆ *dM*1) = *sL M* ˆ *iLM*1 + ˆ *voutM*1 *Deff* 2 *K*2 ˆ *vcdM*2 + β*L*2 *Vin* β*L*1 *K*2 ( ˆ *diM*2 + ˆ *dvM*2 + ˆ *dM*2) = *sL M* ˆ *iLM*2 + ˆ *voutM*2 . . . *Deff* 2 *K*2 ˆ *vcdMM* + β*L*2 *Vin* β*L*1 *K*2 ( ˆ *diMM* + ˆ *dvMM* + ˆ *dMM*) = *sL M* ˆ *iLMM* + ˆ *voutMM* (11) ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ˆ*iLL*11 + ˆ*iLL*21 + ··· + ˆ*iLLaL*1<sup>1</sup> = *sCL sRcLCL*+1 *<sup>v</sup>*<sup>ˆ</sup>*outL*1 + *<sup>v</sup>*<sup>ˆ</sup>*outL R* ˆ*iLL*12 + ˆ*iLL*22 + ··· + ˆ*iLLaL*1<sup>2</sup> = *sCL sRcLCL*+1 *<sup>v</sup>*<sup>ˆ</sup>*outL*2 + *<sup>v</sup>*<sup>ˆ</sup>*outL R* . . . ˆ*iLL*1*bL*1 + ˆ*iLL*2*bL*1 + ··· + ˆ*iLLaL*1*bL*1 = *sCL sRcLCL*+1 *<sup>v</sup>*<sup>ˆ</sup>*outLL* + *<sup>v</sup>*<sup>ˆ</sup>*outL R* ˆ *iLM*11 + ˆ *iLM*21 + ··· + ˆ *iLMa M*11 = *sC M sRcMCM* + 1 ˆ *voutM*1 + ˆ *voutM R* ˆ *iLM*12 + ˆ *iLM*22 + ··· + ˆ *iLMa M*12 = *sC M sRcMCM* + 1 ˆ *voutM*2 + ˆ *voutM R* . . . ˆ *iLM*1*bL*1 + ˆ *iLM*2*bL*1 + ··· + ˆ *iLMa M*1*bL*1 = *sC M sRcMCM* + 1 ˆ *voutMM* + ˆ *voutM R* (12)

Adding equations representing the primary multimodule group in (12):

$$\sum\_{i=1}^{b\_{L1}} \sum\_{j=1}^{b\_{L1}} \hat{\imath}\_{ILij} = \frac{s\mathcal{C}\_L}{s\mathcal{R}\_{cL}\mathcal{C}\_L + 1} \mathfrak{d}\_{outL} + \frac{b\_{L1}\mathfrak{d}\_{outL}}{R} \tag{13}$$

Equation (13) can be written as:

$$\sum\_{i=1}^{h\_{L1}} \sum\_{j=1}^{h\_{L1}} \hat{\imath}\_{LLij} = \vartheta\_{outL} \left( \frac{sRC\_L + s\nu\_{L1}R\_{cL}C\_L + b\_{L1}}{R(1 + sR\_{cL}C\_L)} \right) \tag{14}$$

Adding equations representing the secondary multimodule group in (12):

$$\sum\_{l=1}^{M1} \sum\_{f=1}^{b\_{M1}} \mathbf{f}\_{LMlj} = \boldsymbol{\theta}\_{outM} \left( \frac{\mathrm{sRC}\_{M} + \mathrm{sB}\_{M1}\mathrm{R}\_{cM}\mathrm{C}\_{M} + \mathrm{b}\_{M1}}{R\left(1 + \mathrm{sR}\_{cM}\mathrm{C}\_{M}\right)} \right) \tag{15}$$

Equation (15) can be written as:

$$\sum\_{l=1}^{3M1} \sum\_{f=1}^{bM1} \mathfrak{l}\_{LM1f} = \mathfrak{d}\_{outM} \left( \frac{\mathrm{sRC}\_{\mathcal{M}} + \mathrm{sb}\_{M1} \mathrm{R}\_{\mathcal{M}} \mathrm{C}\_{\mathcal{M}} + \mathrm{b}\_{M1}}{\mathrm{R} \left(1 + \mathrm{s} \mathrm{R}\_{\mathcal{C}M} \mathrm{C}\_{\mathcal{M}}\right)} \right) \tag{16}$$

Defining the summation terms of the module's input and output voltage appearing after summing up equations representing the primary multimodule group in (11):

$$\sum\_{j=1}^{L} \mathfrak{d}\_{\text{cdl.}j} = \chi\_{\perp} \mathfrak{d}\_{\text{in}L} = \chi\_{\perp} \mathfrak{f}\_{L2} \mathfrak{d}\_{\text{in}} \tag{17}$$

where:


$$\sum\_{j=1}^{L} \mathfrak{d}\_{outLj} = c\_L \mathfrak{d}\_{outL} = c\_L b\_{L2} \mathfrak{d}\_{out} \tag{18}$$

where:


Defining the summation terms of the module's input and output voltage appearing after summing up equations representing the secondary multimodule group in (11):

$$\sum\_{j=1}^{M} \boldsymbol{\theta}\_{cdMj} = \boldsymbol{\varkappa}\_{M} \boldsymbol{\uptheta}\_{lmM} = \boldsymbol{\varkappa}\_{M} \boldsymbol{\uptheta}\_{M2} \boldsymbol{\uptheta}\_{ln} \tag{19}$$

where:


$$\sum\_{j=1}^{M} \mathfrak{d}\_{outMj} = \mathfrak{c}\_{\mathbf{M}} \mathfrak{d}\_{outL} = \mathfrak{c}\_{\mathbf{M}} b\_{\mathbf{M}2} \mathfrak{d}\_{out} \tag{20}$$

where:

• *cM* = 1, if all the modules in the secondary group at the output side are connected in series. • *cM* = *aM***1**, if all the modules in the secondary group at the output side are connected in parallel or connected in both series and parallel.

### 2.2.1. Control-to-Output Voltage Transfer Function

The relation between the output voltage and the duty cycle is obtained by performing two steps. The first step is by adding the *L* equations in (11) to obtain the relation between *<sup>v</sup>*<sup>ˆ</sup>*outL* and ˆ *dLj*, assuming *v* ˆ *inL* = 0, and ˆ *dLk* = 0, where *k* = 1, 2, ... , *L* and *k* - *j*, and substituting (2), (4), (14), (17) and (18). However, the second step is by adding the *M* equations in (11) to obtain the relation between ˆ *voutM* and ˆ *dMj*, assuming ˆ *vinM* = 0, and ˆ *dMk* = 0, where *k* = 1, 2, ... , *L* and *k* - *j*, and substituting (7), (9), (16), (19) and (20).

Adding the *L* equations in (11) would result in:

$$\frac{D\_{eff1}}{K\_1} \sum\_{j=1}^{L} \vartheta\_{\text{cill},j} + \frac{\beta\_{\text{l.2}}}{\beta\_{\text{l.1}} K\_1} \left( \sum\_{j=1}^{L} \hat{d}\_{\text{iL}j} + \sum\_{j=1}^{L} \hat{d}\_{\text{vL},j} + \sum\_{j=1}^{L} \hat{d}\_{\text{Lj}} \right) = sL\_L \sum\_{j=1}^{L} \hat{\imath}\_{\text{l.1},j} + \sum\_{j=1}^{L} \vartheta\_{\text{out},j} \tag{21}$$

Substituting (2), (4) and (14) would result in:

$$\begin{split} \frac{\frac{D\_{eff1}}{K\_{\text{L}}}}{K\_{\text{L}}} \sum\_{j=1}^{L} \mathfrak{d}\_{\text{cdL},j} + \frac{\mathfrak{d}\_{\text{fL}}}{\mathfrak{P}\_{\text{L}1}} \frac{V\_{in}}{K\_{\text{L}}} \Bigg( \begin{split} & -\frac{\mathfrak{P}\_{\text{fL}} K\_{\text{L}} \mathfrak{R}\_{\text{dL}}}{\mathfrak{P}\_{\text{L}1}} \hat{\mathfrak{v}}\_{\text{out}} \mathrm{tf}\_{\text{cdL}} \bigg( \frac{\mathrm{sRC}\_{\text{L}} + \mathfrak{s} \mathfrak{r}\_{\text{fL}} \mathrm{R}\_{\text{L}} \mathrm{C}\_{\text{L}} + \mathfrak{h}\_{\text{fL}}}{\mathfrak{R} (1 + \mathfrak{s} \mathfrak{R}\_{\text{L}} \mathrm{C}\_{\text{L}})} \bigg) + \\ & \sum\_{j=1}^{L} \frac{a\_{1,2} b\_{1,1} \bar{\rho}\_{11} R\_{\text{L}1} D\_{\text{eff}} \bar{D}\_{\text{fL}1}}{a\_{1,1} b\_{1,2} \bar{\rho}\_{12} R\_{\text{L}}} \mathfrak{G}\_{\text{cdL},j} + \mathfrak{d}\_{\text{L}1} \bigg) \\ & = s \mathrm{L}\_{\text{L}} \Big( \frac{\mathrm{sRC}\_{\text{L}} + \mathrm{s} \psi\_{11} R\_{\text{L}} \mathrm{C}\_{\text{L}} + \psi\_{11}}{\mathrm{R} (1 + \mathfrak{s} \mathfrak{R}\_{\text{L}} \mathrm{C}\_{\text{L}})} \bigg) \mathfrak{G}\_{\text{outL}} + \sum\_{j=1}^{L} \mathfrak{D}\_{\text{outL},j} \end{split} \tag{22}$$

Substituting (17) and (18) in (22) results in:

$$\begin{split} \frac{^{D\_{eff}}\mathcal{I}}{^{D\_{L}}\!\!\!\!\!/}\mathcal{V}\_{\rm L}\mathfrak{H}\_{\rm inL} + \frac{\mathcal{S}\_{\rm L}}{\mathcal{P}\_{\rm L1}} \frac{\mathcal{V}\_{\rm in}}{\mathcal{K}\_{\rm}} \left( \begin{array}{c} -\frac{\mathcal{P}\_{\rm L1}\mathcal{K}\_{\rm R}}{\mathcal{P}\_{\rm L2}} \mathcal{I}\_{\rm outL} \mathtt{\mathcal{C}}\_{\rm L} \mathtt{\mathcal{C}}\_{\rm L} + \mathtt{\mathcal{V}}\_{\rm L} \\ \frac{\mathcal{A}\_{\rm L2}\mathcal{B}\_{\rm L1}\mathcal{R}\_{\rm L2}\mathcal{R}\_{\rm eff}}{\mathcal{A}\_{\rm L1}\mathcal{V}\mathcal{D}\mathcal{I}\_{\rm L2}\mathcal{R}\mathcal{V}\mathcal{N}\_{\rm in}} \mathcal{V}\_{\rm I}\mathcal{O}\_{\rm inL} + \mathcal{d}\_{\rm L1} \end{array} \right) \\ = \mathrm{s}\mathcal{L}\_{\rm L} \begin{pmatrix} \frac{\mathcal{s}\mathcal{R}\mathcal{C}\_{\rm L} + \mathcal{s}\mathcal{S}\_{\rm L1}\mathcal{R}\_{\rm eff}}{\mathcal{R}\mathcal{S}\mathcal{I} + \mathcal{s}\mathcal{R}\_{\rm cl}\mathcal{C}\_{\rm L}} \mathcal{I}\_{\rm l} + \mathcal{C}\_{\rm L}\mathcal{O}\_{\rm outL} \end{pmatrix} \end{split} \tag{23}$$

Simplifying (23) results in (24):

$$\begin{split} G\_{\text{rdL}} &= \frac{\dot{v}\_{\text{NL}L}}{\dot{d}\_{Lj}}\\ &= \frac{\frac{\rho\_{\parallel,2} - V\_{\text{int}}}{\dot{P}\_{L1} + \dot{K}\_{L}} (1 + \kappa \mathbb{R}\_{c1} \mathbb{C}\_{L})}{s^{2} L\_{L} \mathbb{C}\_{L} \left(1 + \frac{l\_{\parallel,1} \mathbb{R}\_{c1}}{\mathcal{R}}\right) + s \left(\frac{l\_{\parallel,1} \mathbb{L}\_{L}}{\mathcal{R}} + \mathbb{R}\_{c1} \mathbb{C}\_{L} \left(1 + \frac{l\_{\parallel,1} \mathbb{R}\_{c1}}{\mathcal{R}}\right) + c\_{\perp} \mathbb{R}\_{c2} \mathbb{C}\_{L}\right) + \frac{l\_{\parallel,1} \mathbb{R}\_{c1}}{\mathcal{R}} + c\_{L}} \end{split} \tag{24}$$

Performing the second step which is adding the *M* equations in (11) to obtain the relation between ˆ *voutM* and ˆ *dMj*, assuming ˆ *vinM* = 0, and ˆ *dMk* = 0, where *k* = 1, 2, ... , *M*, and *k* - *j*, and substituting (7), (9), (16), (19) and (20) would result in:

Adding the *M* equations in (11) would result in:

$$\frac{D\_{off2}}{K\_2} \sum\_{f=1}^{M} \partial\_{cathf} + \frac{\mathcal{B}\_{M2}}{\mathcal{B}\_{M1} K\_2} \left( \sum\_{f=1}^{M} \partial\_{iMf} + \sum\_{f=1}^{M} \partial\_{vMf} + \sum\_{f=1}^{M} \partial\_{Mf} \right) = sL\_M \sum\_{f=1}^{M} \mathfrak{k}\_{LMf} + \sum\_{f=1}^{M} \partial\_{outMf} \tag{25}$$

Substituting (7), (9) and (16) would result in:

$$\begin{aligned} \frac{D\_{off2}}{K\_2} + \frac{\mathcal{B}\_{M2}}{\mathcal{B}\_{M1}} \frac{V\_{in}}{K\_2} \left( -\frac{\mathcal{B}\_{M1} K\_2 R\_{dM}}{\mathcal{B}\_{M2} V\_{in}} \boldsymbol{\vartheta}\_{outM} \left( \frac{s R C\_M + s \mathbf{b}\_{M1} R\_{cM} C\_M + \mathbf{b}\_{M1}}{R (1 + s R\_{cM} C\_M)} \right) \right) \\ + \sum\_{j=1}^{M} \frac{a\_{M2} b\_{M1} \theta\_{M1} R\_{cM} D\_{off2}}{a\_{M1} b\_{M2} \theta\_{M2} R V\_{in}} \boldsymbol{\vartheta}\_{outM} \left( \boldsymbol{\vartheta}\_{outM} \right) \\ = s L\_M \left( \frac{s R C\_M + s \mathbf{b}\_{M1} R\_{cM} C\_M + \mathbf{b}\_{M1}}{R (1 + s R\_{cM} C\_M)} \right) \boldsymbol{\vartheta}\_{outM} + \sum\_{j=1}^{M} \boldsymbol{\vartheta}\_{outMj} \end{aligned} \tag{26}$$

Substituting (19) and (20) results in:

$$\begin{aligned} \frac{D\_{eff2}}{K\_2} \mathbf{\hat{y}}\_M \boldsymbol{\hat{\upbeta}}\_{lmM} + \frac{\boldsymbol{\mathcal{B}}\_{M1}}{\boldsymbol{\mathcal{B}}\_{M1}} \frac{\mathbf{\hat{I}}\_{in}}{K\_2} & - \frac{\boldsymbol{\mathcal{B}}\_{M1} K\_2 \mathbf{\hat{I}}\_{dM}}{\boldsymbol{\mathcal{B}}\_{M2}} \boldsymbol{\hat{\upbeta}}\_{outM} \left( \frac{sRC\_M + s\boldsymbol{\mathcal{B}}\_{M1} R\_{cM} \mathbf{\hat{C}}\_M + \mathbf{\hat{b}}\_{M1}}{R\left(1 + s\boldsymbol{\mathcal{R}}\_{cM} \mathbf{\hat{C}}\_M\right)} \right) \\ & + \frac{a\_{M2} b\_{M1} \mathbf{\hat{\upbeta}}\_{M1} R\_{cM} D\_{eff2}}{a\_{M1} b\_{M2} \mathbf{\hat{\upbeta}}\_{M2} RV\_{in}} \boldsymbol{\chi}\_M \boldsymbol{\hat{\upbeta}}\_{lmM} + \boldsymbol{\hat{d}}\_{M1} \\ & = sL\_M \left( \frac{sRC\_M + s\boldsymbol{\mathcal{B}}\_{M1} R\_{cM} \mathbf{\hat{C}}\_M + \mathbf{b}\_{M1}}{R\left(1 + s\boldsymbol{\mathcal{R}}\_{cM} \mathbf{\hat{C}}\_M\right)} \right) \boldsymbol{\upbeta}\_{outM} + \boldsymbol{c}\_M \boldsymbol{\hat{\upbeta}}\_{outM} \end{aligned} \tag{27}$$

Simplifying (27) would result in (28).

$$G\_{\rm{total}} = \frac{\partial\_{\rm{outM}}}{\partial\_{Mf}}$$

$$= \frac{\frac{\beta\_{M2}}{R\_{M1}} \frac{V\_{in}}{K\_{2}} \left(1 + sR\_{\rm{CM}}C\_{M}\right)}{s^{2}L\_{M}C\_{M}\left(1 + \frac{b\_{M1}R\_{\rm{CM}}}{R}\right) + s\left(\frac{b\_{M1}L\_{M}}{R} + R\_{\rm{CM}}C\_{M}\left(1 + \frac{b\_{M1}R\_{\rm{CM}}}{R}\right) + c\_{M}R\_{\rm{CM}}C\_{M}\right) + \frac{b\_{M1}R\_{\rm{CM}}}{R} + c\_{M}}\tag{28}$$

By adding *GvdL* and *GvdM* the control-to-output voltage transfer function can be found.

#### 2.2.2. Control-to-Filter Inductor Current Transfer Function

The relation between the filter inductor current and the duty cycle is derived by performing two steps, where the first step considers the *L* modules in (11) while the second step considers the *M* modules in (11). The first step is by substituting *<sup>v</sup>*<sup>ˆ</sup>*outL* in terms of ˆ *iLLj* using (14) in (23) and assuming *v* ˆ *inL* = 0, and ˆ *dLk* = 0, where *k* = 1, 2, ... , *L* and *k* - *j*. However, the second step is by substituting ˆ *voutM* in terms of ˆ *iLMj* using (16) in (27) and assuming ˆ *vinM* = 0, and ˆ *dMk* = 0, where *k* = 1, 2, ... , *M* and*kj*.

Substituting *<sup>v</sup>*<sup>ˆ</sup>*outL* in terms of ˆ *iLLj* using (14) in (23):

$$\begin{split} \frac{\boldsymbol{D}\_{tf\mid I}}{\boldsymbol{K}\_{1}} \boldsymbol{Y}\_{\text{L}} \boldsymbol{\hat{\upmu}}\_{\text{in}L} &+ \frac{\rho\_{t\mid L}}{\rho\_{\text{L}\mid L}} \frac{\boldsymbol{V}\_{\text{in}}}{\boldsymbol{K}\_{1}} \Bigg( \underbrace{-\frac{\rho\_{\text{L}\mid L} \boldsymbol{K}\_{\text{R}} \boldsymbol{R}\_{\text{d}}}{\rho\_{\text{L}\mid L} \sum\_{i=1}^{L} \boldsymbol{\hat{\upmu}}\_{\text{L}\mid L}^{\text{L}}}\_{\text{in}L \mid \text{s}} + \boldsymbol{1}\_{\text{L}} \\ &= \boldsymbol{s} \boldsymbol{L} \sum\_{i=1}^{a\_{\text{L}}} \sum\_{j=1}^{b\_{\text{L}}} \hat{\boldsymbol{t}}\_{\text{L}\mid Lj} + c\_{\text{l}} \left( \frac{\boldsymbol{R} \left( 1 + \boldsymbol{s} \boldsymbol{R}\_{\text{d}} \boldsymbol{C}\_{\text{L}} \right)}{\boldsymbol{s} \boldsymbol{R} \boldsymbol{C}\_{\text{L}} + \boldsymbol{s} \boldsymbol{I}\_{\text{L}} \boldsymbol{R}\_{\text{d}} + \boldsymbol{c}\_{\text{l}}} \right) \end{split} \tag{29}$$
 
$$\frac{\frac{\beta\_{\text{L}}}{\beta\_{\text{L}}} \frac{\boldsymbol{V}\_{\text{in}}}{\boldsymbol{K}\_{\text{L}}} \boldsymbol{\hat{t}}\_{\text{L}\mid}}{\beta\_{\text{L}} \boldsymbol{K}\_{\text{L}}} \hat{\boldsymbol{t}}\_{\text{L}\mid} - \boldsymbol{R}\_{\text{d}} \sum\_{j=1}^{L} \hat{\boldsymbol{t}}\_{\text{L}\mid Lj} = \boldsymbol{s} \boldsymbol{L}\_{\text{L}} \sum\_{j=1}^{L} \hat{\boldsymbol{t}}\_{\text{L}\mid Lj$$

Simplifying (30) would result in (31).

$$\begin{split} G\_{\text{idL}} &= \frac{\sum\_{j=1}^{L} \hat{\imath}\_{\text{LL}j}}{d\iota\_{j}}\\ &= \frac{\frac{\frac{\hat{P}\_{\text{L}}\mathcal{E}}{\hat{P}\_{\text{L}1}\mathcal{E}\_{\text{L}}} \left(s\mathcal{R}\mathcal{C}\_{L} + s\boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}}\mathcal{C}\_{L} + \boldsymbol{\upsilon}\_{\text{L}1}\right)}{R\left(s^{2}\mathcal{L}\_{\text{L}}\mathcal{C}\_{\text{L}}\left(1 + \frac{\boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}}}{R}\right) + s\left(\frac{\boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}}}{R} + \mathcal{R}\_{\text{dL}}\mathcal{L}\_{\text{L}}\left(1 + \frac{\boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}}}{R}\right) + \boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}}\mathcal{C}\_{\text{L}}\right) + \frac{\boldsymbol{\upsilon}\_{\text{L}1}\mathcal{R}\_{\text{L}L}}{R} + \boldsymbol{\upsilon}\_{\text{L}1}} \end{split} \tag{31}$$

ˆ

Performing the second step which is substituting *voutM* in terms of *iLMj* using (16) in (27) and assuming ˆ *vinM* = 0, and ˆ *dMk* = 0, where *k* = 1, 2, ... , *M* and *k* - *j* would result in:

$$\begin{split} \frac{D\_{eff1}}{K\_{2}} \mathbf{Y}\_{H} \boldsymbol{\theta}\_{lmM} + \frac{\boldsymbol{\mathcal{B}}\_{M2}}{\boldsymbol{\mathcal{B}}\_{M1}} \frac{\mathbf{V}\_{in}}{K\_{2}} \begin{pmatrix} -\frac{\boldsymbol{\mathcal{B}}\_{M1} K\_{2} R\_{dM}}{\boldsymbol{\mathcal{B}}\_{M2}} \sum\_{l=1}^{M11} \boldsymbol{\upvarepsilon}\_{lLMl} + \\ -\frac{a\_{M1}}{\boldsymbol{\mathcal{B}}\_{M2}} \frac{\mathbf{V}\_{lm}}{V\_{lm}} \sum\_{l=1}^{M11} \boldsymbol{\upvarepsilon}\_{lLMl} + \\ \frac{a\_{M1}}{a\_{M1} b\_{M2} \boldsymbol{\upbeta}\_{M2} R\_{M}} \boldsymbol{\upvarepsilon}\_{llm} \end{pmatrix} \boldsymbol{\upvarepsilon}\_{M} \boldsymbol{\upmu}\_{lLM} + \boldsymbol{\uphat{\mathcal{U}}\_{M1}} \\ = \boldsymbol{\upvarepsilon}\_{M} \sum\_{l=1}^{M11} \sum\_{j=1}^{M11} \boldsymbol{\upvarepsilon}\_{lLMj} + \boldsymbol{\upvarepsilon}\_{M} \left( \frac{R(1+s \boldsymbol{\mathcal{B}}\_{lCM} \boldsymbol{\upvarepsilon}\_{l})}{s R \boldsymbol{\mathcal{C}}\_{M} + s \boldsymbol{\upbeta}\_{M1} R\_{cM} \boldsymbol{\upvarepsilon}\_{M} + \boldsymbol{\upbeta}\_{M1}} \right) \sum\_{l=1}^{M11} \boldsymbol{\upvarepsilon}\_{lLM(l)} \end{split} \tag{32}$$

ˆ

$$\frac{\mathcal{B}\_{M2}}{\mathcal{B}\_{M1}} \frac{V\_{\text{in}}}{R\_2} \partial\_{M1} - R\_{\text{dM}} \sum\_{f=1}^{M} \mathfrak{k}\_{\text{LM}f} = \text{sL}\_M \sum\_{f=1}^{M} \mathfrak{k}\_{\text{LM}f} + \mathfrak{c}\_M \left( \frac{R\left(1 + \text{s}R\_{\text{cM}}\mathcal{C}\_M\right)}{\text{s}R\mathcal{C}\_M + \text{s}\mathfrak{b}\_{M1}R\_{\text{cM}}\mathcal{C}\_M + \mathfrak{b}\_{M1}} \right) \sum\_{f=1}^{M} \mathfrak{k}\_{\text{LM}f} \tag{33}$$

Simplifying (33) would result in (34). By adding *GidL* and *GidM*, the control-to-filter inductor current transfer function can be found.

$$\begin{split} G\_{LML} &= \frac{\sum\_{f=1}^{M} \mathbb{1}\_{LMf}}{\bar{d}\_{Mf}} \\ &= \frac{\frac{\beta\_{M2}}{R} \frac{V\_{IN}}{K\_{f}} \left( sRC\_{M} + s b\_{M1}R\_{cM}C\_{M} + b\_{M1} \right)}{\frac{1}{R} \left( s^{2}L\_{M}C\_{M} \left( 1 + \frac{b\_{M1}R\_{cM}}{R} + R\_{dM}C\_{M} \left( 1 + \frac{b\_{M1}R\_{cM}}{R} \right) + c\_{M}R\_{cM}C\_{M} \right) + \frac{b\_{M1}R\_{cM}}{R} + c\_{M}} \right) \end{split} \tag{34}$$

By adding *GidL* and *GidM* the control-to-output filter inductor current transfer function can be found.
