An Energy-Based Model of Four-Switch Buck–Boost Converters

The four-switch buck–boost (FSBB) topology is often used in combination with other isolated converters to extend the voltage range capability of the overall structure. In such applications, the duty-cycles of the two legs of the FSBB are independently controlled, and a phase shift is typically used to shape the inductor current ripple and, thus, achieve zero voltage switching. This article proposes a nonlinear average model, and the corresponding linearized small-signal model for the FSBB operating in the described way. Notably, the proposed approach is based on the modeling of the inductor energy, which allows the correct description of the dynamics related to the phase shift, in addition to those related to the duty-cycles. The derived models are shown to be in excellent agreement with simulation results and are also validated by measurements on an experimental prototype.


I. INTRODUCTION
T HE four-switch buck-boost (FSBB) topology, shown in Fig. 1, is a well-known noninverting step-up/down topology, which is used in combination with other isolated converters for higher input/output voltage ranges.In [1], an integrated buck-boost LLC converter is proposed, cascading an FSBB stage and an LLC stage with passive rectification sharing a switching leg to widen the input voltage range.The LLC stage works in zero-current switching operation throughout the full-operating range.Here, the two stages are driven independently with different switching frequencies for the two legs.The same converter is used in [2] as intermediate bus converter in distributed telecom power systems.The legs of the FSBB work at the same switching frequency and the shared leg operates at 50% duty-cycle.The output voltage is regulated by the input-leg duty-cycle.The Ezio Gallo, Davide Biadene, and Tommaso Caldognetto are with the Department of Management and Engineering, University of Padova, 36100 Vicenza, Italy (e-mail: ezio.gallo@phd.unipd.it;davide.biadene@unipd.it;tommaso.caldognetto@unipd.it).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TPEL.2023.3349327.
Digital Object Identifier 10.1109/TPEL.2023.3349327phase shift is modulated to reduce the current ripple in the buck-boost inductor.Similarly, in [3] and [4], the converter is used with push-pull rectifier and full-bridge rectifier, respectively, with the bipolar symmetric phase-shift modulation.In these applications, the shared leg duty-cycle is fixed at 50% and the duty-cycle of the other leg and the phase shift are modulated to regulate the output voltage and modify the inductor current.In [5], a parallel structure using two interleaved FSBB stages working in counter-phase is also proposed.The FSBB is used with a dual-active-bridge (DAB) converter to match the high-and low-voltage side reducing, in this way, the current stress and increasing the soft-switching range [6].Other articles emphasize the fundamental role of the FSBB as front-end stage in unidirectional dc-dc converters [7], [8], [9], [10], [11] or as an additional stage in power factor correction or in bidirectional converters, widening the converter voltage range and eventually improving the efficiency of the other stage in a wide range of operation [12], [13].
Despite the simple structure of the FSBB, models covering the dynamics of the converter during the variation of both the two duty-cycles and the phase shift are not presented.Typical FSBB average models account for buck or boost operating modes, where only one of the legs is modulated to reach the desired output voltage, or operating in buck-boost mode where S 1 and S 3 of Fig. 1 are turned-ON and -OFF simultaneously.When these operating modes are considered, as shown in [14], [15], and [16], the canonical method in [17] is generally used neglecting the dynamics of the converter in the transition between the buck and boost mode of operation in the so called dead zone.In [18], a quadrangle current control for the FSBB is presented and, in [19], a model of the converter under that control is developed, considering the variation of the duty-cycles and the phase shift.With such a control, since the minimum current to reach zero voltage switching (ZVS) operation is fixed, one of the three variables (i.e., the two duty-cycles or the phase shift) becomes dependent to the others, limiting the validity of the model.
The transition between the buck and boost modes is considered in several papers.In [20], control strategies utilizing buck-boost operation are employed for a smooth transition between the modes.The strategy proposed in [20] is used also in [21] and [22], changing the overlap between the carrier of the switches S 1 and S 2 to gently reduce one duty-cycle and increase the other, effectively changing the phase shift.However, a single small-signal model for the FSBB in all the three operation modes (i.e., buck, boost, and buck-boost modes) is derived and the effect of the phase shift, present in the buck-boost mode, is neglected.On the other hand, in the literature, the phase-shift dynamic is considered in other converter topologies, such as the dual half-bridge or DAB [23], [24], [25], [26] in which, however, the phase shift is used to change the power transferred at the output.
The aim of this article is to develop a nonlinear average model for the converter in Fig. 1 in the most general case where both input and output legs operate with independent duty-cycles, and considering the effect of the phase shift between the two legs.
From the nonlinear average model, a simple linearization technique is used to derive the desired small-signal linear model.
This article develops on the preliminary results reported in [27].Herein, the theoretical treatment is enhanced by performing the analysis based on a general energy modeling of the converter.Moreover, additional results are reported, including phase-shift perturbations, comparisons with average state-space models, and validations comprising all the switching patterns.
The proposed modeling approach, besides encompassing effects of variables like the input and output duty-cycles, akin to other average modeling approaches in the literature, it also allows the modeling of the converter behavior in response to phase-shift variations, which is typically not included in common models, as also elaborated in other research papers (see, e.g., [19]).The rest of this article is organized as follows.Section II, the converter modes of operation are reviewed.Section III presents the energy-based approach and shows the derivation of the average and small-signal models of the converter.In Section IV, the modeling approach is verified by simulations in MATLAB/Simulink also providing a control application example.Section V reports the experimental validation.Finally, Section VI concludes this article.

II. REVIEW OF THE FSBB OPERATING MODES
The main waveforms of the FSBB converter are displayed in Fig. 2. D g and D o are defined as the duty-cycles of the top switch of the input and output legs, respectively, denoted as S 1 and S 2 in Fig. 1.The phase shift β is defined as the relative time between the center instants of the voltage pulses of v 1 and v 2 and it is positive when v 2 leads v 1 .
From the volt-second balance of the inductor, the voltage gain is derived as a function of the two duty-cycles.Thus, the phase shift β can be used to shape the piece-wise linear current waveform i L to achieve ZVS for the four switches as well as to minimize its rms value.Depending on the phase shift β, four different modes are possible for each of the step-up or step-down cases.In these operating regions, the switching period can be divided into four different subperiods Δt k bounded by the leg commutation instants t k , as shown in Fig. 2. In (1), the subperiods Δt k and their normalized duration δ k are described as a function of the two duty-cycles, D g and D o , and the phase shift The boundaries of the operating modes are dependent on the defined quantities D g , D o , and β.Each boundary satisfies the particular condition δ k = 0 defining a hyperplane in the (D g , D o , 2β)-space.The resulting space-partition defines the different converter modes.The expressions of the intervals δ k are shown in Table I for the regions R 4-8 (i.e., R 4 and R 8).
A graphical representation of the operating regions is reported in Fig. 3, where they are evaluated for D g equal to 0.4 and 0.6.This allows us to project the space-partition on the (D o , 2β)plane.
Given the intervals definition in (1), the corresponding inductor current at the instant t k [i.e., I k = i L (t k )] can be described by where V Lk is the voltage across the inductor L during the k-th subperiod.Iterating (2) for all the intervals Δt k , all the switching current values I k are found with respect to I 0 .In this article, the analysis is focused on regions R 4-8 and R 2-6 since they allow ZVS turn-ON transitions for all four switches by modulating the phase shift.Operation in regions R 1-5 precludes complete ZVS and is, thus, typically avoided.
In the following, the use of the overlap interval δ 2 as a control variable is preferred over β.

A. Limitation of Average Quantities on a Switching Period
Let us first consider an example to highlight a limitation of considering a model based on the average quantities computed over a generic switching period.To this end, the modeling of the effect of a phase-shift variation is analyzed in the following.(i.e., δ 2 ) causes a variation in the average inductor current i L , where the function x is the moving average of the signal x Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The variation observed in i L is expected and should correspond to an associated variation in the modeled average inductor voltage.However, commonly used average models of FSBBs approximate v L with the average value of the inductor voltage computed on a switching period, which does not allow us to catch such a variation due to the phase-shift perturbation.This is illustrated in Fig. 4 by reporting with circles the values of the average quantities computed on a switching period; remarkably, the average inductor voltage computed on a switching period does not capture the variations observed in the moving average v L (e.g., the negative values of v L ), which actually have an effect on the inductor current as shown in the figure.
To overcome this limitation, a model based on the energy balance is investigated herein.

B. Energy-Based Average Model
The proposed model exploits the fact that the difference between the energy at the FSBB input and output ports must equal the variation in the inductor energy, which is related to the average inductor current on a switching period.Beginning with the input port, the energy w g entering the port during a switching period is with integration limited over the interval where the switch S 1 is conducting.In general, the current integral can be split into a contribution related to the initial current I 0 plus a remaining piecewise linear contribution Considering the output port, the energy w o leaving the port during a switching period is with integration limited over the interval where the switch S 2 is conducting.In this case, the current integral is split into a contribution related to the final value I 4 plus a piecewise linear contribution The use of the initial and final current values I 0 and I 4 allows tracking the variation of the average inductor current i L across a switching period, as shown hereafter.From ( 5) and ( 7), it emerges that the average input and output currents are functions of the ports voltages V g and V o , the duty-cycles D g and D o , the voltage-overlap interval δ 2 and the considered current values I 0 and I 4 .
The difference between the energy entering at the input port w g and the energy leaving the output port w o corresponds to a variation of the energy stored in the inductor Δw L in the switching cycle.From ( 5) and (7), such variation results

Δw
Using ( 2), the final value I 4 can be expressed in terms of the initial value I 0 as Substituting ( 9) in ( 8), it yields Equation (10) shows two key components: the average voltage across the inductor in a switching period and the mean between the initial and the final inductor currents of the switching period, respectively: It is worth noting that, in (11), īe is not equal to the average current īL and does not take into account the phase shift β.The current īe may be associated with the lower or higher envelope (in the sense of lower or higher bounding function) of i L .In the proposed modeling, īe is considered as a new state variable for the inductor.

C. Input and Output Currents
The average input and output currents in a switching period can be derived from the energy exchanged at the respective ports.The input current īg is given using (5) as while the output current īout is derived from (7) as From ( 9) and ( 11), the terms I 0 and I 4 can be expressed as a function of the new variable īe , which relates these variables to the energy variation Substituting ( 14) in ( 12) and ( 13), the input and output current equations for region R 4-8 can be derived R 4-8 : Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The same procedure can be used for the other operating region R 2-6 to find the input and output average currents considering the initial current in the period I 0 as in Fig. 2(a), yielding R 2-6 :

TABLE II SMALL-SIGNAL PARAMETERS OF THE INPUT AND THE OUTPUT CURRENTS IN THE CONSIDERED REGIONS
As expected, the input and output currents are functions of the overlap δ 2 and the current īe .The input and the output currents can be represented as the sum of the current īe multiplied by D g and D o , respectively, and the additional currents īx and īy as follows: where these additional terms depend on the operating region R 4-8 : 2f sw L .
The resulting equivalent nonlinear average model of the converter is shown in Fig. 5. Based on the relationships existing between ve , īe , īg , and īout , an ideal transformer is used to couple the input and output ports and the inductor L, as often done in the literature.The linear small-signal model is derived by perturbing and linearizing the average quantities in (11), (15), and (16).As commonly done (see, e.g., [17]), each variable is decomposed as x = X + x, where X represents the steady-state component while x represents small-signal variations.Retaining only the first-order terms, the model reported in Fig. 6 yields.All the coefficients are a function of the considered operating point.The average and small-signal models, reported in Figs. 5 and  6, have the same structure in the other operating regions.Table II reports the small-signal coefficients related to Fig. 6 for the considered region.

IV. SIMULATION RESULTS
The proposed large-signal and small-signal models are verified with simulations in MATLAB/Simulink using ideal components (i.e., ideal inductors, capacitors, and switches).The parameters used for the average and small-signal models are listed in Table III.If needed, parasitic elements can be included in simulations, as done, for example, considering the results reported in the next Section V. State-of-the-art symmetric-ontime single-update digital pulsewidth modulators (DPWM) with triangular carriers are used.With such a modulation, the input control variables are D g , D o , and β.The derivations described in the previous Section II can be applied based on (1), considering the following equivalence:

A. Large-Signal Model Validation
To verify the average model in (17), a circuit simulation using MATLAB/Simulink is built comparing the responses of the proposed large-signal model in Fig. 5 and an instantaneous switching model developed in simulation.In the first simulation, a step excitation of Δd = 5% is applied to the control variable D o , and the response of the output voltage V o and the input and output currents īg and īout are compared.To excite the derived large-signal model, it is necessary to simultaneously impose also  a step change of δ 2 , with an amplitude of Δd/2, according to (18).In the second simulation, a step variation of Δδ = 0.2 is applied to δ 2 to show the accuracy of the model in describing the dynamics of the input and output quantities after a change in the phase shift.The initial circuit parameters are shown in Table III.Figs. 7 and 8 illustrate the low-frequency dynamics of the input and output currents and the output voltage responding to the D o and δ 2 step, respectively, compared with the switching model.
The presented results show that the predicted low-frequency response of the proposed model agrees with the results provided by the instantaneous switching model.

B. Small-Signal Model Validation
This section aims to verify the proposed small-signal model of the FSBB, comparing the transfer functions predicted by the model with those obtained from the frequency response measurements performed in the simulation.In particular, the analyzed transfer functions are i) the output duty-cycle do to output-voltage vo and ii) the phase shift β to output-voltage vo .The transfer function G do (s) = vo (s)/ do (s) is obtained setting dg = vg = 0 and δ2 = do /2, which stems from ( 18) and β = 0; the resulting expression is Simulation and experimental measurements also include the delays associated with the DPWM, which should be integrated with the theoretical G do (s).Let us call G mod (s) the transfer function between the output voltage and the perturbed modulation signal.The final overall transfer function including DPWM delays [28] is In simulations, the perturbation of the duty-cycle is done by superimposing a sine wave to the steady-state value of D o at different frequencies between 1 and 50 kHz.In the same way, the transfer function G δo (s) = vo (s)/ δ2 (s) can be calculated considering dg = vg = do = 0, yielding Again, to compare the model with the simulation, the considered transfer function is G δo (s) • G mod (s).The perturbation of the overlap δ 2 is obtained by adding a sine wave to the steady-state reference of δ 2 in the same frequency range of the first simulation (i.e., 1 and 50 kHz).From (18), δ 2 is a function of D o , D g and β, then the phase shift needs to be perturbed during the subinterval δ 4 .Then, the perturbation of δ 2 corresponds only to the perturbation of the phase shift β, avoiding unintended variation of the duty-cycles.Consequently, the update of δ 2 reference is provided by sampling the phase shift at the end of the switching period (i.e., end of δ 4 ).Due to this forced sampling at the switching frequency f sw , the sinusoidal characteristic of the perturbation is lost approaching the Nyquist frequency, leading to a consequent error in the frequency response.
Comparisons between simulation results and analytical predictions are shown in Figs. 9 and 10.The model accurately predicts the response across the whole considered frequency range.
The gain error between the analytical and simulated G do (s) transfer function is reported in Fig. 11, for all the regions.The considered frequency range is f r /5 to 5f r where f r is the resonant frequency of the FSBB (i.e., D o /2π √ LC o ).Electrical quantities (i.e., V g , f sw , R L , L, C o ) are given by Table III.Since δ 2 does not represent the overlap between the two voltage pulses in every region, the operating point is given in terms of D g , D o , and β.The models accurately predict the small-signal gain in the considered frequency range for all the regions.The maximum gain error is lower than 0.6 dB, thus validating  the proposed modeling approach.Additionally, a comparison of the G do (s) transfer function between the proposed model and the standard model of the FSBB from [17] are reported in Fig. 12.The electrical parameters are the same as before, with D g = 0.5, D o = 0.9, and δ 2 = 0.45.The proposed model can follow the phase response, unlike the standard model in [17].

C. Control Application Example
A cascade control of FSBB, composed of an outer voltage control loop with an inner current control loop, is designed in this section using the proposed energy model.The plant transfer functions, namely G di (s) = îe (s)/ do (s) and G io (s) =   vo (s)/ îe (s), are derived from Fig. 6 and reported in ( 22) This control is tested by imposing a step voltage variation of 5% of the output voltage reference.The current i L is sampled at the beginning of each switching cycle, in order to acquire the desired current value īe .Remarkably, the approach does not require the information on the true average current.Parameters are listed in Table III.A lower value of the output capacitor C o = 15 μF is used to consider a more critical control problem.The target phase margin is Φ M = 70 • , while crossover frequencies are f i cr = 15 kHz and f v cr = f i cr /6 (i.e., f v cr = 2.5k Hz) for current and voltage loops, respectively.The voltage reference step response is reported in Fig. 14.The figure highlights an accurate matching between the expected response of the controller designed using the proposed models and the actual response when  the same designed controller is interfaced with the instantaneous simulation model of the converter.

V. EXPERIMENTAL VALIDATION
To corroborate the validation performed by means of numerical simulations, additional results from experimental measurements are reported in the following.To this end, an FSBB prototype using SiC devices UJ4SC075009K4S for the input leg and UF3SC065007K4S devices for the output leg has been implemented.The inductor is based on a PQ32-N97 core with 9 turns, using 800 × 71 μm litz wire, and an air-gap of 1.7 mm. Circuit parameters are listed in Table IV.A Texas Instruments microcontroller TMS320F28379D is used to implement the DPWM.
Fig. 15 displays the experimentally obtained time-domain waveforms of the input and output switching node voltage v 1 and v 2 , and inductor current i L of the tested FSBB in the region R 8; the considered operating point is V g = 50 V, f sw = 100 kHz, D g = 0.5, D o = 0.6, δ 2 = 0.3.Frequency response measurements were performed to experimentally validate the transfer function from the output duty-cycle to the output voltage.To this end, the Bode functionality of the Rohde&Schwartz RTA4004 oscilloscope was used, which computes the Bode plot considering the signals applied at two input channels of the oscilloscope as input and output variables.The waveform generator signal of the RTA4004 is used to stimulate the system to be characterized by a series of tones at specified frequencies.In this case, the stimulus signal is given as input to the microcontroller TMS320F28379D and, after proper filtering and scaling, used to impress the sinusoidal perturbations on the modulation variables (e.g., the duty-cycle of the output leg).This signal is considered as input for the generation of the Bode plot, while the output voltage V o of the FSBB is considered as output.The Bode Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
plot measurement procedure is performed through the following steps.
1) Calibration: By connecting the two probes together, the measurement offset, which will later be subtracted from the main results, is measured.2) Measurement: Using the Bode application, the frequency response of the target transfer function is measured.3) Postprocessing: The results from the previous step are postprocessed, taking into consideration the microcontroller operation and subtracting the offset calculated from the calibration step.These steps are applied for the experimental measurement of the transfer function among the perturbed modulation signal and the output voltage.
The experimental measurements are compared in the following with results obtained analytically and in simulation.Parasitic elements present in the experimental setup are taken into account in the models used in this section, which include inductor resistance R ind , MOSFETs on-resistance R dson , capacitor ESR R C , and deadtimes t d .In the simulation model, parasitic resistances are added in series with the corresponding elements (e.g., inductors, switches, etc), while the deadtime effect is included by delaying the rising edges of all of the MOSFETs' gate signals.The proposed analytical model is also enhanced to include the aforementioned nonidealities.For example, the circuit model in Fig. 6 is modified by adding the resistances related to the inductor, the MOSFETs, and the capacitor losses in series with the respective circuit elements.On this basis, the impedances of inductance and capacitance become Z L = sL + R ind + R dson1 + R dson2 and Z out = 1/sC o + R C , respectively, where R dson1 and R dson2 are the on-resistances of the first and second leg.This yields the following output duty-cycle to output voltage transfer function: The deadtime is taken into account as a reduction of the effective duty-cycles D g and D o .
Comparison results between the proposed small-signal model taking into account converter nonidealities and the previously explained experimental and simulated frequency response measurements, are provided in Fig. 16 for the step-up and step-down case.Simulation results match the analytical predictions in the whole considered frequency range.Around the resonant frequency, a small mismatch is observed in the experimental results.This is assumed to be related to the nonlinear effect of the deadtimes, as reported in [29], of the core [30], and of the switching losses [31].

VI. CONCLUSION
This article has presented a nonlinear average model and the corresponding small-signal model of the FSBB converter under both duty-cycle and phase-shift modulation.This article focused on two specific regions of operation that allow zero voltage switching of all four switches, the same model procedure can be used for the other operating region.In the presented modeling approach, the average inductor current is not one of the state variables, and instead, a new state variable based on the inductor energy variations is introduced.The model allows the implementation of current control based on the sampling of the initial current at the beginning of the switching period.This avoids postprocessing computation or low-pass filters to sense the average current.The simulation and experimental results have verified the nonlinear large-signal and linearized small-signal models of the FSBB converter for both duty-cycle and phase-shift variations.

Fig. 2 .
Fig. 2. Example of converter waveforms for different phase-shift values.(a) Region R 2 in step-up case (i.e., D o < D g ).(b) Region R 8 in step-down case (i.e., D o > D g ).

Fig. 4
displays the main waveforms of an FSBB converter when a phase-shift variation is applied while keeping constant the duty-cycles D g and D o .The change of the v 1 −v 2 overlap

Fig. 4 .
Fig.4.Instantaneous and moving average waveforms in R 4-8.A variation in the overlap δ 2 generates a step response on the voltage v L that causes a variation on the inductor current i L and, thus, in its moving average i L .Circles • mark the values of the average quantities computed on a switching period.

Fig. 7 .
Fig. 7. Response of the switching and the proposed model to a step increment of output duty-cycle D o of 5%.In blue the currents and voltage of the simulation, in dashed orange lines the model quantities.

Fig. 8 .
Fig. 8. Response of the switching and the proposed model to a step increment of phase shift δ 2 of 0.2.In blue the currents and voltage of the simulation, in dashed orange lines the model quantities.

Fig. 9 .
Fig. 9. Simulation validation of the output duty-cycle to output voltage transfer function.Comparison between frequency response obtained from simulations and the model G do (s) • G mod (s) in region R 4. The results are given for the following operating point: D g = 0.4, D o = 0.6, δ 2 = 0.2.

Fig. 10 .
Fig. 10.Simulation validation of the phase shift to output voltage transfer function.Comparison between frequency response obtained from simulations and the model G δ 2 (s) • G mod (s) in region R 4. The results are given for the following operating point: D g = 0.4, D o = 0.6, δ 2 = 0.2.

Fig. 12 .
Fig. 12. Bode plots comparison of G do = vo / do from proposed model, average state-space model, and simulation for D g = 0.5, D o = 0.9 and δ 2 = 0.45.

Fig. 14 .
Fig. 14.Voltage reference step response of the controlled FSBB as shown in Fig. 13.

Fig. 16 .
Fig. 16.Verification of the output duty-cycle to output voltage transfer function.Comparison between frequency response obtained from simulations, experimental results, and model G par do (s) • G mod (s).The results are given for the operating point: D g = 0.5, D o = 0.4, δ 2 = 0.23.

TABLE I NORMALIZED
TIME INTERVAL EQUATIONS FOR THE REGION R 4-8

TABLE III LIST
OF PARAMETERS USED IN THE SIMULATION

TABLE IV PARAMETER
VALUES OF THE CIRCUIT