I. INTRODUCTION

Achieving high efficiency on power converters is one of the main issues of power electronics [1], [2], [37], [38]. Two main issues related to efficiency are conduction and switching losses. Conduction losses can be resolved by reducing the ON resistance on power MOSFETs, reducing the forward voltage on power diodes or IGBTs and using passive components with low parasitic resistance [22]-[26]. Semiconductor devices are also becoming faster. Power diodes with low reverse recovery current have also been developed to reduce switching losses [3].

In addition to the reduction of switching losses, semiconductor devices with high reverse voltage have also been developed. Although these new devices expand the possibilities for high-voltage and high-frequency use, classical converters, such as buck or boost, still face some limitations in terms of applications of several kilovolts [4]-[8], [27], [28]. Multilevel converters were proposed to deal with the high-voltage problem of classical converters. Multilevel converters manage the high-voltage problem (several kilovolts) with the use of low-voltage components (hundreds of volts). Among the multilevel converters, some, such as the ladder multilevel DC/DC converter, use only capacitive components (switched capacitors (SC)) in their power circuit, reducing size and weight [9]-[15], [29]-[31]. In [16], the classical ladder topology (CLT) and two variations were presented. The double ladder topology (DLT) was shown to produce less output resistance and less output voltage ripple, which makes this topology highly efficient and suitable for implementation.

In addition to efficiency, a control law is required by power converters. In most cases, a mathematical model of the converter is required. These converters are often modeled with the use of the classical averaging technique [32]. However, this technique works under the assumption of slow dynamics on each switching state. If such assumption is unfulfilled, then an inaccurate model can be obtained.

In an SC converter, capacitors with different voltages can be connected in parallel, producing current spikes. Given that the equivalent series resistance (ESR) of the capacitors and the ON resistance of the interconnection switches are usually small, the equivalent time constant τ of the resulting circuit is also small. Therefore, the dynamics are rapid compared with the switching period T (see Fig. 1). With consideration of the rapid dynamics of the current of the capacitors, the assumption for the classical averaging model is unfulfilled. In [17], a new analytical modeling technique that obtains an equivalent continuous model of a switched converter regardless of the dynamics on each switching state is proposed. The technique is called generalized equivalent continuous model (GECM). In this study, this technique is used to analyze the behavior of the converter and optimize its efficiency.

Fig. 1.Capacitor current waveform for different τ values. T represents the switching period.

Fig. 1 shows that, for a small τ (i.e., small ESR and switch ON resistance), the current waveform presents an exponential shape and higher current spikes than for a large τ value (i.e., higher ESR and switch ON resistance). Moreover, given the charge balance, the waveforms should have the same average in each cycle. If the average is the same, then we can conclude that the exponential waveform has a higher root mean square (RM S) value. Therefore, reducing the resistor value R implies an increase in the RM S value. Given that the power losses in resistor R are expressed as , no conclusion on power losses can be easily drawn.

In [1], [18]-[20], [33], and [34], power losses in SC converters are resolved and analyzed. In SC converters, reducing the ON resistance of the switches or ESR does not necessarily imply higher efficiency. In [1] and [18], analytical calculations of the conduction energy losses of an SC converter are presented. These calculations are essentially the same that the GECM conducted under all matrix operations. Therefore, the results of conduction losses only are the same. However, the GECM is easier to compute using a convenient state space form and works for any circuit topology modeling its dynamics. In [33], an analysis method that completely omits parasitic resistances is presented. Therefore, it cannot be computed in the state space form (GECM). References [20] and [34] provide a basic mathematical demonstration on how an ideal SC converter (without parasitic resistances) generates energy losses and how to prevent energy losses. However, no specific calculation of the losses is conducted. In [19], a different approach is presented. Using the example proposed in Section VI of reference [19], an approximation of the output resistance is obtained at the switching frequency of 30 MHz, producing a 5% error. This error can be eliminated using GECM. Notably, none of the previous works considered switching losses.

In this study, the GECM approach is applied to the DLT converter with eight cells for high-voltage application. The components of the converter are selected to optimize its efficiency with consideration of conduction and switching losses. Moreover, the converter is implemented and experimental measurements are taken.

This paper is organized as follows: Section II presents the DLT. Section III highlights the different modeling techniques. Section IV analyzes the switching losses. Finally, Section V demonstrates the efficiency of the DLT with the use of the new modeling technique.

 

II. LADDER MULTILEVEL CONVERTER

Fig. 2 shows the generalized DLT proposed in [16]. This converter has NC basic cells (Fig. 3) and an ideal output voltage Vout = (NC + 1)Vin . This converter works with two control signals, one with 50% duty cycle for switches SW1 and the other complementary for switches SW2 [21], [35]. The DLT presents less output voltage drop and less ripple in comparison with the CLT [16].

Fig. 2.DLT of a DC/DC voltage multilevel elevator with NC cells. SW1 and SW2 are complementary. Average currents are shown during each cycle.

Fig. 3.Basic cell of the ladder multilevel converter.

The basic cell shown in Fig. 3 consists of two capacitive elements (C1 and C2) and four switching components (two SW1 and two SW2). When switches SW1 are ON while SW2 are OFF , the input voltage source Vin charges the capacitor C1. In the other half cycle, with SW2 ON and SW1 OFF, the capacitor C1 charges the capacitor C2, which ideally results in Vout = 2Vin .

The connection between capacitors is affected by the ON resistance of the switches and the ESR of each capacitor, thus producing voltage drops. Furthermore, the time constant of the equivalent circuit, which depends on the capacitance and resistance values, affects the current waveform on the capacitors: a large time constant produces a square waveform, whereas a small time constant produces an exponential waveform, as previously shown in Fig. 1.

As stated previously, when the time constant is small compared with the switching period, the classical averaging model cannot produce an accurate model [17].

 

III. MODELING TECHNIQUES

The model of ladder converters can be written as a switched system. Mode 1 is represented by σ = 1 and occurs when SW1 = ON (SW2 = OFF) . Mode 2 is represented by σ = 2 and occurs when SW1 = OFF(SW2 = ON) . Thus, the dynamics of the converter can be written as follows:

where x and are the voltages of the capacitors and their derivatives, respectively. u represents input variables (input voltage Vin ) and y represents output variables (output voltage Vout). The matrices of each mode A1, B1, F1, G1 and A2, B2, F2, G2 can be computed using classical circuit analysis. The values of these matrices depend on the value of each capacitor, as well as the values of the ESR and the resistances of the switches.

From this switched model, we can deduce two modeling techniques, namely, the classical averaging model and the GECM.

A. Classical Averaging Model

When we assumed that the state space vector x is constant when each mode is active, the classical averaging technique is deduced [32]. With this assumption, the average model from Eqs. (1) and (2) can be computed as follows:

where 〈∗〉 is the average of the argument and d is the duty cycle. However, as shown in Fig. 1, capacitor voltages in ladder topologies can considerably vary during each mode. Then, classical averaging model is not an appropriate method to characterize ladder converters.

B. Generalized Equivalent Continuous Model

A new analytical modeling technique was proposed in [17] to deal with high-speed dynamics. In this method, the assumption of slow variation in the state vector x during each mode is eliminated. This subsection briefly presents the method. The following steps show the procedure to obtain this model:

1) Obtain the State Space Matrices of Each Switching State: The analysis starts with the matrices Aσ, Bσ, Fσ, and Gσ for σ = {1,2}. These matrices are obtained from the model shown in Eqs. (1) and (2).

2) Reorganize the State Space Matrices: Previous matrices are unsuitable for the model. Then, the matrices , , , and are computed as follows:

with σ = {1,2}. Thus, the system is transformed into

where

3) Calculate Φ1(dT) and Φ2((1 − d)T) : The state transition matrices are computed, but have to be evaluated in dT . Thus, Φ1(dT) and Φ2(dT) are expressed as follows:

4) Calculate : Matrix represents the transition matrix in one switching period T . The two modes are presented in this matrix. is computed as follows:

5) Calculate Ω1 and Ω2 : Ω1 and Ω2 are defined as the integrals of the functions Φ1(t) and Φ2(t), respectively. These integrals enable the generation of the average dynamics of each mode, which are obtained as follows:

An analytical solution of Ω1 and Ω2 exists when , σ = {1,2} is nonsingular. In this case, we obtain the following equation:

where d1 = d and d2 = 1 − d. When is singular, it should be numerically computed. Indeed, numerical integration is an approximation of the integral real value.

6) Calculate Γ : Defining the matrix Γ in the function of Ω1 and Ω2 is also necessary. Using the results of the previous step, the value of Γ can be computed as follows:

7) Obtain the Matrix : The proposed continuous model is expressed as follows:

Information on 〈A〉 and 〈B〉 is inside matrix . This matrix describes the average dynamics and can be obtained as follows:

Γ and are previously defined. Indeed, has the following form:

Then, 〈A〉 and 〈B〉 can be derived from .

8) Calculate ΓƑ : Matrix ΓƑ is necessary to compute the output model, as follows:

9) Obtain the Matrix 〈Ƒ〉: The continuous model for 〈x〉 is obtained using Eq. (14). The continuous model for the output y is obtained as follows:

As shown in Step 7, matrix 〈Ƒ〉 exists, which contains 〈F〉 and 〈G〉. This matrix is expressed as follows:

where 〈Ƒ〉 is derived from:

where ΓƑ and Γ are derived from Eqs. (17) and (13), respectively.

10) Obtain the Final State Space Matrices: Finally, the results from Steps 7 and 9 are separated as shown respectively in Eqs. (16) and (19), to obtain the state space matrices 〈A〉, 〈B〉, 〈F〉, and 〈G〉. These matrices describe the average system behavior.

C. Basic Cell Example

In this section, the GECM is applied to the basic ladder multilevel converter shown in Fig. 3. Fig. 4 shows the two possible modes of the basic cell, with the parasitic resistances of each component (ESR of capacitors denoted as RC and ON resistance of the MOSFETs denoted as RON).

Fig. 4.Switching modes with parasitic resistances. (a) SW1 ON; SW2 OFF. (b) SW1 OFF; SW2 ON.

To obtain a model of the system in state space variables, such as in Eqs. (1) and (2), the state space vector x, the input u, and the output y are defined as follows:

For the mode shown in Fig. 4(a), the state space matrices are calculated as follows:

For the mode shown in Fig. 4(b), the state space matrices are calculated as follows:

with:

.

For this numerical example, we considered the following values of the components: RON = 1.8 Ω, C1 = C2 = 2.2 μF, RC = 2.5 mΩ, and Rbad = 50 Ω. The switching frequency for this example is f = 50 kHz, with a duty cycle d = 0.5.

The equivalent state space matrices that describe the system behavior using classical averaging model are computed. The matrices are expressed as follows:

The matrices for the GECM, following the steps previously explained, are expressed as follows:

Fig. 5 shows a comparison of the step response for both models (classical averaging and GECM) and the real behavior of the converter in simulation. We observed that the GECM produces better results than the classical averaging model because it is capable of computing the real average of Vout. For more information on this example, see [36].

Fig. 5.Comparison of the classical averaging model, the GECM, and the switched response.

 

IV. SWITCHING LOSSES

In this section, an analytical method to estimate the switching losses of the converter is presented. This method uses the results of the GECM to analyze the system dynamics.

A. General Case

Let the voltages of the switches (open) and their currents (closed) be organized as follows:

where j is the number of switches.

The matrix from Eq. (10) defines a discrete model of the initial conditions of the mode σ = 1. Therefore, the following model can be defined using Eq. (26):

where:

with z0 being the discrete version of z and x0 being the discrete version of x.

Matrices and represent the relationship between open voltages and closed currents of the switches, the capacitor voltages, and the input (z0).

Fig. 6 shows the switching waveforms of a switch with a trapezoidal approximation of the switching losses. We assumed that the turn-on and turn-off times are small as compared with the system time constant. Under this assumption, the value of the state space vector is the same right before switching and right after switching (i.e., the value of the state space vector is z0(0) in time instants marked as a and b in Fig. 6). Therefore, the voltages of the switches (open) and their currents (closed) can be computed in terms of z0 regardless their state in each subsystem.

Fig. 6.Switching waveforms of Q1 with switch turn-on time ttom and turn-off time toff.

Notably, all voltages and currents in the instants marked as a and b in Fig. 6 are calculated. Computing the same voltages and currents in the next switching instant z1 is also necessary. The state space vector in the next switching instant can be defined as follows:

Using this definition, additional voltages and currents can be computed and added to the system in Eq. (27), expanding matrices and as follows:

Therefore, the new system is defined as follows:

Similar to the case of the GECM, the matrices of Eq. (31) can be decomposed to be used with the original discrete state space vector x0 and the input u, as follows:

To compute the switching losses in steady state, the steady-state value of the outputs is required. The steady-state values of the voltages and currents can be computed as follows:

Finally, the switching losses in steady state can be obtained using the previous results as follows:

where:

and

where the function diag (∗) generates a diagonal matrix with the elements of the argument. Matrix M is scaled by 1/6 because of the trapezoidal resistive switching approximation [37].

B. Basic Cell Example

In this subsection, a numerical example on how to compute the switching losses is applied to the basic ladder multilevel converter shown in Fig. 3. The switches are numbered from bottom to top (Fig. 3). Only the computation of the losses for switches Q1 and Q2 is shown. Therefore, the following vectors can be defined:

1) Compute the Voltages and Currents of the Switches: We compute the matrices and from Eq. (27). For instance, the calculation of the open voltage of Q1 should be computed in the mode σ = 2 (Fig. 4(b)) and Q2 should be computed in the mode σ = 1 (Fig. 4(a)), as follows:

where:

Replacing numerical values, the following results are obtained:

2) Create the Matrices * and *: We expand the matrices and using Eq. (9), as follows:

3) Create Matrix M : The matrix M contains the information on the turn-on and turn-off time of the switches. For this numerical example, tton = 200 ns and ttoff = 150 ns.

4) Decompose the Matrices: We obtain the matrices from Eq. (32) using the results in Eqs. (10), (43), and (44), as follows:

5) Compute the Steady-State Values of the Currents and Voltages: We compute the steady-state values of the voltages and currents using Eq. (33) and the results of the previous step (u = Vin = 340), as follows:

6) Compute the Switching Power Losses:

Using Eqs. (34) and (45) and the results of the previous step, the total switching power losses for Q1 and Q2 can be computed as follows:

With consideration of the switching power losses, the efficiency of the converter can be computed as follows:

For this case, the output voltage of the converter is not significantly affected by switching losses. Therefore, Pout and PCbss can be computed with the GECM. Fig. 7 shows a comparison between the theoretical calculation of efficiency (with and without switching losses) and the respective simulations for different switching frequencies while considering all switches in the basic cell. The estimation obtained using the analytical method reflects correctly the behavior of these losses.

Fig. 7.Comparison of theoretical efficiency with simulation for the basic cell.

 

V. ANALYSIS OF THE CONVERTER

In this section, the DLT is analyzed using the new techniques introduced in Sections III and IV. An in-depth analysis is conducted for this topology to show the effect on the efficiency caused by the switching frequency and the parasitic resistances of the switches. The GECM is validated using the DLT in [36].

Fig. 8 shows an equivalent model of the converter. In this model, N represents the ideal voltage gain and the resistance Rout represents the losses. The resistance Rout is dependent on the switching frequency of the converter and the values of ON resistances of the switches and ESR of the capacitors.

Fig. 8.Equivalent model of an SC converter with a DC/DC transformer, where N is the gain of the converter with no load and Rout is the output resistance.

Given that the new modeling technique is able to model accurately the SC converter, it is used to analyze and optimize the DLT for implementation.

The nominal values of the converter are as follows: input voltage Vin = 350 V; output voltage Vout = 3,000 V; output current Iout = 100 mA; number of cells Nc = 8. For this example, a commercial capacitor C = 2.2 μF , ESR = 2.5 mΩ is used.

Using the GECM, we observed that the duty cycle of maximum gain is independent of the load value and switching frequency and is always d = 0.5. Moreover, the gain Vout/d is insufficient to use d as control signal. Therefore, the duty cycle is set to a constant value d = 0.5. As mentioned previously, no conclusion on efficiency can be easily drawn because of the effect on the current waveform when the time constants are small compared with the switching period. Using the results obtained in Section IV, the efficiency of the converter can be computed analytically.

The following criteria is considered when selecting an adequate value for the switching frequency and RS : The value of RS should be as high as possible without compromising efficiency. In this converter, the lower the RS, the higher the current spikes in the switches, producing overcurrents.

Fig. 9 shows efficiency as a function of f and RS. The values of RS and f are selected as follows: First, the highest possible value of RS that can be classified into the lighter region is selected and restricted to commercial values. A frequency value that can also be classified into this region is then selected. Fig. 10 shows the detailed effect of switching frequency on a given RS. The value that produces the highest efficiency can be easily selected. The values of RS = 1.8 Ω (N-MOS transistor P8NK100Z) and f = 80 kHz are selected, with an efficiency of 92.5%. We observed no increments in efficiency of over 1% for any other possible values of frequency and resistance.

Fig. 9.Efficiency as a function of the switch resistance RS and switching frequency f. The dot marks the selected values RS = 1.8 Ω and f = 80 kHz, with an efficiency of 92.5%.

Fig. 10.Comparison of the efficiency obtained using the theoretical model for three different values of switch ON resistance RS (conduction losses only (CL) and conduction and switching losses (CL+SL)).

Fig. 10 shows the efficiency of three different switch resistances as a function of switching frequency to compare in detail the influence of switching frequency and switch resistance. The results of the GECM (where no switching losses are considered) are compared with the results of the GECM with the addition of switching losses for each switch. When the charging and discharging process of capacitors is completed in one switching period, the losses are independent of the RS (conduction only) value and are attributed to the inherent energy losses caused by voltage differences of the capacitors when they are connected. However, including switching losses, at low frequencies, the lower the RS, the higher the losses.

Although efficiency is slightly higher with RS = 0.18 Ω at different frequencies in comparison with RS = 1.8 Ω at f = 80 kHz (increment < 1% ), the current spikes are dramatically higher with RS = 0.18 Ω (up to 16.6 A ) in comparison with the selected values (up to 3.3 A).

Experimental measurements are conducted to validate the analysis of the converter. Fig. 11 shows a comparison among the analytical, simulated, and experimental efficiencies. The simulated and experimental results present a relative error below 4% compared with the analytical results, with switching losses for all the tested cases. Power loss attributed to switching is 5.5%, with RS = 1.8 Ω and f = 80 kHz.

Fig. 11.Comparison of the efficiency obtained using the theoretical model with and without switching losses, simulation results, and experimental measurements with RS = 1.8 Ω.

 

VI. CONCLUSIONS

In this study, a method for computing switching losses is introduced, and a new modeling technique is used to analyze the efficiency of an SC converter. This technique is helpful when the dynamics of each mode of the switched system are characterized as rapid when compared with the switching period.

The DLT was analyzed with the use of the new model, and the optimal parameter values were obtained. We observed that the highest efficiency is not necessarily achieved with the smallest resistor values and it depends on the switching frequency. Therefore, careful analysis is required to achieve the highest efficiency point.

Experimental measurements and simulations were conducted to validate the analytical results. An efficiency of 92.5% was obtained at 80 kHz, with switching losses of 5.5%.