An Analysis-Supported Design of a Single Active Bridge (SAB) Converter

Abstract

Currently, due to its various applications, the high-performance isolated dc-dc converter is in demand. In applications where unidirectional power transfer is required, the single active bridge (SAB) is the most suitable one due to its simplicity and ease of control. The general schematic of the SAB converter consists of an active bridge and a passive bridge, which are connected through a high-frequency transformer thus isolated. The paper summarizes the behavior of this converter in its three operation modes, namely the continuous, discontinuous, and boundary modes. Later, the features of this converter, such as its input-to-output and external characteristics are discussed. Input-to-output characteristics include the variation of converter output power, voltage, and current with an input control variable i.e., phase-shift angle, whereas the external characteristic is the variation of the output voltage as a function of output current. In this discussion, the behavior of this converter in its extreme operating conditions is also examined. The features of the characteristics are elucidated with the help of suitable plots obtained in the MATLAB environment. Afterward, the specifications of a SAB converter are given and, based on the results of the analysis, a detailed design of its electrical elements is carried out. To validate the features and the design procedures presented in this paper, a prototype is developed. An element-wise loss estimation is also carried out and the efficiency of the converter has been found to be approximately equal to 93%. Lastly, the test was executed on this prototype, confirming the theoretical findings concerning this converter.

1. Introduction

In todays’ applications, isolated dc-dc converters are becoming more prevalent. Their main advantages include a compact structure, galvanic isolation, and excellent efficiency [1,2,3,4,5,6]. A variety of converters exist; of these the dual active bridge (DAB) [7,8,9], single active bridge (SAB) [10,11], semi-DAB [12,13], and phase shift full bridge (PSFB) [14,15,16] are the most commonly used isolated dc-dc converters. Resonant dc-dc converters such as LLC [17], CLLC [18,19], and LC [20,21] are also popular due to their virtues of soft switching and better efficiency. However, the resonant converters are out of the scope of this paper and are not discussed hereafter.

Figure 1 shows the general schematic of an isolated dc-dc converter. Bridge 1 is usually of the active kind and is utilized in all types of dc-dc converters. In contrast, bridge 2 is of different types; for the DAB converters, it is of the active type and the bridge sides are composed of controlled switches; for the SAB converter, it is a passive type and the bridge sides are composed of diodes; for the semi DAB converters, the bridge sides are a combination of controlled switches and diodes [13]; for the PSFB, for unidirectional application, it is a diode rectifier [14,15] and, for bidirectional application, it is of the active type with a clamping diode in bridge 1 [16]. Moreover, the PSFB converter contains an inductor at the output of bridge 2. The schematic in Figure 1 includes, besides the two bridges, the following elements: (i) the inductor L, which accounts for the total leakage inductance of the isolation transformer, (ii) the capacitor C_o, which filters the output voltage V_o, and (iii) the resistor R_l, which represents the load of resistive nature.

For applications where unidirectional power transfer is required, PSFB and SAB are suitable. As mentioned earlier, the PSFB converter constitutes a reactor at the output. This converter can realize a unity voltage conversion ratio, as the reactor is bulky, and can operate in a continuous current mode. Furthermore, due to the surge voltage generated by the reactor and the parasitic capacitor of the diode rectifier, high voltage diodes must be used to realize the diode rectifier [22]. However, refs. [23,24] have suppressed the voltage surge with the use of an auxiliary circuit. As an alternative, moving the reactor to the primary side of the converter can reduce the inductance by a factor of the square of the transformer turn-ratio. Furthermore, this external inductor can also be realized by the leakage inductance of the transformer; thus, no extra inductor is needed. This topology is referred to as the SAB.

Since bridge 2 of the SAB converter consists of diodes only, this is the most convenient converter among the unidirectional converters in terms of simplicity. This bridge 2 configuration not only decreases the cost and volume of a dc-dc converter but may also be utilized in a series connection to obtain a dc-dc converter with a medium-voltage rating. Furthermore, it simplifies SAB converter control, making the SAB a very appealing alternative for unidirectional power flow applications. The possibility of its use for power transmission from an offshore wind farm to the onshore network via a high voltage DC system has also been demonstrated in [25].

Existing literature on the SAB converters is mainly focused on topologies with half-bridge (HB) or full-bridge (FB) input, and deals with either their working principles or soft-switching capabilities of the transistors or other specific aspects [10,11,26,27,28,29,30,31]. Specifically, in [10], the dynamics and basic working principle of an HB SAB converter are presented; in [11], the switching losses of the transistors of an FB SAB converter were mitigated by employing a dual-current pulse control; in [26], the snubber capacitor is optimized to improve the FB SAB converter efficiency; in [27], a partial-resonant FB SAB converter is arranged for the reduction of the transistor conduction losses; in [28], an open-circuit fault detection method and the tolerant control strategy for N parallel-connected FB SAB converters are reported; in [29], the impact of transformer turns ratio and leakage inductance on the operating behavior of the FB SAB converter with a voltage doubler is studied, where one leg of the bridge 2 diode rectifier is replaced with two capacitors; in [30,31], respectively, the design characteristics and soft-switching capabilities of an FB SAB converter are compared to those of the DAB converter.

There are numerous papers available that deal with the functioning, design, and control of dc-dc converters. In [32], the design and performance analysis of a 6 kW DAB converter utilized to charge a Li-ion battery-based storage system are discussed; in [33], operational principles and design guidelines are exposed for a three-phase SAB converter; in [34], the design and control of a 10 kW DAB converter are presented, preceded by a study of its behavior in different modes of operation; in [13], a 1 kW semi-DAB converter is analyzed and its performance is documented, including its ZVS capabilities; in [35], a 10 kW three-phase DAB is designed and developed; in [36] a 1400 W PSFB converter for server applications is developed; and in [14], a PSFB converter of 350 W is designed and implemented. Despite the fact that there is substantial literature on dc-dc converters, there are no articles that address the characteristics of the SAB converter with two full bridges and how to design its elements. In this regard, the purpose of this paper is to encompass such issues. The objectives of this paper and the methodology used are as follows. The paper starts by summarizing the working principle of the SAB converter at a steady state in its three operation modes, namely continuous, discontinuous, and boundary modes. It should be noted that a similar study is also presented in [30] but, to discuss the SAB converter characteristics and its design procedure, which is the novel part of this paper, it is necessary to formulate some basic relationship, to be included in the paper. Since there is only one control parameter in this converter, called the phase-shift angle, it is a matter of interest to observe how the output parameters such as output power, voltage, and current vary with respect to the phase-shift angle. The relationship of the output quantities with the phase-shift angle, as mentioned earlier, is termed here as the input-to-output characteristics of the converter. Moreover, another important observation where the output voltage is plotted against output current is also included here and termed as the converter external characteristic. The external characteristic is very useful to locate the converter operating point and subsequently detect the corresponding operation mode. The same characteristic is also useful to examine converter behavior in extreme operating conditions. These characteristics are first formulated and later plotted in the MATLAB environment and then elaborated to provide design hints.

From the design perspective, the specifications of this converter are specified. It should be noted that the design specifications do not target any particular application, but are incorporated here to validate the simulation findings. However, a similar approach can be used to design this converter for any power ratings. Later, the element-wise design is included in the paper followed by the losses associated with the individual elements and calculation of the overall converter efficiency. Based on the specifications and design approach, a SAB prototype is developed and put to the test. Finally, the experimental results are also encompassed in the paper. Figure 2 includes the used methodology for the paper.

The organization of this paper is as follows. Section 2 describes the basic circuit of the SAB. Section 3 studies its three operational modes. Section 4 and Section 5 include the characteristics and element-wise design, respectively. Section 6 includes the experimental results and Section 7 concludes the paper.

2. SAB Converter Description

As shown in Figure 3, the circuit diagram of a SAB converter is composed of two bridges coupled through a transformer. Bridge 1, also known as the input bridge, is an active-type bridge that is powered by the DC voltage V_i. It functions as an inverter and generates the AC high-frequency voltage v₁ at its output. The active bridge’s switches T_i are made up of two elements (transistor Q_i and freewheeling diode D_i) as well as a snubber circuit C_s,i. Transistor Q_i might be a MOSFET or an IGBT for medium- and high-power applications. The snubber circuit is used to soft-switch Q_i, while the diode D_i permits negative current to flow through T_i. The output bridge of the SAB converter is a passive bridge that is powered by voltage v₂ at the isolation transformer’s secondary. The bridge functions as a pure rectifier and, in conjunction with the filtering effect of C_o, produces the dc voltage V_o at the output of the SAB converter.

3. SAB Converter Analysis

Depending on the current in the inductor L, three operation modes can occur for the SAB converter, yielding peculiar sets of equations relating the output quantities to the input ones. Therefore, each mode is analyzed separately. The equations are formulated under the following hypotheses: (i) the turns ratio n of the isolation transformer is equal to 1, (ii) its magnetization inductance is large enough to make the magnetizing current negligible, and (iii) C_o is large enough to make the ripple of the output voltage V_o negligible.

Waveforms of voltages and currents for the three operation modes are plotted in Figure 4 as a function of the angular quantity $\theta$, which is proportional to the time according to $\theta ={\omega }_{s}t$; quantity ${\omega }_s$ is the working angular frequency in rad/sec of the SAB converter and is given by 2π/T_s, where T_s is the working period. The phase-shift angle between the driving of the two transistors on the same side is designated with β and ranges from 0 to π. The two sets of waveforms at the top of Figure 4 show the voltage v₁ at the output of the active bridge and the voltage v₂ at the input of the passive bridge. The two sets at the bottom of Figure 4 show the current i_L in the inductor L and the current i_out at the output of the passive bridge.

Looking at the waveforms, it emerges that they exhibit the odd symmetrical property with respect to π.

3.1. Continuous Conduction Mode

The circuit analysis is simplified by replacing the transformer with a leakage inductance L. A pair of output bridge diodes conduct at any moment in the Continuous Conduction Mode (CCM). Then, the voltage v₂ can be either V_o or −V_o depending on the sign of the current i_L. The waveforms of voltages and currents in CCM are presented in Figure 4a, showing that the voltage v₁ has a quasi-square waveform, whereas the voltage v₂ has a pure square waveform with the same zero crossings as the current i_L; the delay angle of v₂ to v₁ is indicated with φ. The current i_L, in turn, has a linear piecewise waveform, which is forced by the voltage $v_L$ applied to the terminals of the inductor L. Such voltage is given by ($v_1-v_2)$ and has a piecewise constant waveform, made up of the three intervals [0 ÷ φ], [φ ÷ β] and [β ÷ π] within the half-period [0 ÷ π]. The equations of $v_l$ and i_L in the three intervals are as follows:

3.1.1. Interval 1

Span of Interval 1 is [0 ÷ φ]. Figure 5a depicts the currents paths and devices conduction in this interval. The voltage v_L is equal to (V_i + V_o), and the current i_L grows linearly from i_L(0) until it vanishes at $\theta =\phi$; the expression of the current at φ is

$$i_L\left(\phi \right)=i_L\left(0\right)+\frac{V_i+V_o}{{\omega }_{s}L}\phi \equiv 0$$

(1)

3.1.2. Interval 2

Span of Interval 2 is [φ ÷ β]. Figure 5b depicts the currents paths and devices conduction in this interval. The voltage v_L is equal to (V_i − V_o), and the current i_L grows linearly from 0 to its peak value, which is reached at $\theta =\beta$; the expression of the peak current is

$$i_L\left(\beta \right)=i_L\left(\phi \right)+\frac{V_i-V_o}{{\omega }_{s}L}\left(\beta -\phi \right)$$

(2)

3.1.3. Interval 3

Span of Interval 3 is [β ÷ π]. Figure 5c depicts the currents paths and devices conduction in this interval. The dc voltage source V_i does not feed the inductor L, and the voltage v_L is equal to −V_o; then, the current i_L drops linearly and its expression at $\theta =\pi$ is

$$i_L\left(\pi \right)=i_L\left(\beta \right)-\frac{V_o}{{\omega }_{s}L}\left(\pi -\beta \right)$$

(3)

By the odd symmetrical property, the current $i_L\left(\pi \right)$ is equal to $-i_L\left(0\right)$.

The average value of v_L in the interval [$\phi$÷ π +$\phi$] is zero since the current i_L takes the same value, equal to zero, at the two extremes of the interval. It follows that

$$\frac{V_o}{V_i}=\frac{\beta -2\phi }{\pi }$$

(4)

As V_o is a positive value, Equation (4) reveals some basic properties of the SAB converter operation in CCM. Firstly, $2\phi$ is lower than $\beta$. Secondly, V_o is lower than V_i and this indicates that the SAB converter works as a step-down converter. Thirdly, it is

$$\frac{V_o}{V_i}<\frac{\beta }{\pi }$$

(5)

The Equation (5) is the distinctive condition for the SAB converter to operate in CCM.

By obtaining φ from Equation (4) and substituting it in Equations (1) and (3), the expressions of the current $i_L$ at 0 and $\beta$ can be written as a function of the input and output voltage, V_i and V_o, and the phase-shift angle β. It is

$$i_L\left(0\right) =-\frac{V_i}{2{\omega }_{s}L}\left(1+\frac{V_o}{V_i}\right)\left(\beta -\frac{V_o}{V_i}\pi \right)$$

(6)

$$i_L\left(\beta \right) =\frac{V_i}{2{\omega }_{s}L}\left(1-\frac{V_o}{V_i}\right)\left(\beta +\frac{V_o}{V_i}\pi \right)$$

(7)

3.2. Boundary Conduction Mode

The Boundary Conduction Mode (BCM) separates CCM from the Discontinuous Conduction Mode (DCM). It stands out because the current i_L becomes zero exactly when $\theta =$0, and then when $\theta =\pi$. The waveforms of voltages and currents in DCM are plotted in Figure 4b and show that, in the half-period [0 ÷ π], the voltage v_L is equal to V_i − V_o from 0 to β and to −V_o from β to π. If the average value of v_L, in this half-period, is equal to zero, we obtain

$$\frac{V_o}{V_i}=\frac{\beta }{\pi }$$

(8)

Equation (8) is the distinctive condition for the SAB converter to operate in BCM. It discloses that, for an equal value of the input voltage V_i and phase-shift angle β, the voltage V_o in BCM is higher than in CCM.

The peak value of the current i_L is given by

$$i_L\left(\beta \right)=\frac{V_i}{{\omega }_{s}L}\left(1-\frac{\beta }{\pi }\right)\beta$$

(9)

3.3. Discontinuous Conduction Mode

In the DCM, the current i_L does not flow within an interval in the half-period [0 ÷ π]. The waveforms of voltages and currents in DCM are plotted in Figure 4c and show that the current i_L is zero from [π − α] to π. Within this interval, also the voltage v₂ is zero because all four diodes of the passive bridge remain reverse biased.

Moreover, the voltage v_L is equal to V_i − V_o from 0 to β and to −V_o from β to [π − α]. If the average value of v_L in the half-period is equal to zero, then

$$V_o=V_i\frac{\beta }{\pi -\alpha }$$

(10)

Since [$\pi -\alpha$] is greater than $\beta$, Equation (10) reveals that, like in CCM, V_o is lower than V_i, and the SAB converter works again as a step-down converter. From Equation (10) it follows that

$$\frac{V_o}{V_i}>\frac{\beta }{\pi }$$

(11)

Equation (11) is the distinctive condition for the SAB converter to operate in DCM. The expression of the peak of the current i_L is still given by Equation (9).

4. SAB Converter Characteristics

The input-to-output and external characteristics of the SAB converter can be formulated using the preceding results. The phase-shift angle β, in addition to the voltage V_i, is the input quantity. The voltage V_o, the average value I_o of the current i_out, and the power P_o, given by V_oI_o, are the output quantities. They represent the voltage across, current into, and power drawn by the load resistor R_l, and are referred to as output voltage, current, and power hereafter.

At steady state, the input-to-output characteristics clearly provide the control characteristics of the SAB converter. The external characteristic is a relationship between the output voltage and the output current; in other words, it describes the SAB converter’s electrical behavior as observed from its output terminals.

The current I_o is calculated as the integral of i_out over the half-period [0 ÷ π], which takes advantage of the odd symmetrical property of the current i_out. I_o is given as

$$I_o=\frac{1}{\pi }{{\displaystyle \int }}_0^{\pi }{i}_{out}\left(\theta \right)d{\theta }_{}$$

(12)

To set forth expressions of general validity, the equations of the characteristics are formulated in terms of “per unit (pu)” quantities. They are worked out in the next subsections and collected in Table 1.

4.1. Per Unit Quantities

Let us define the voltage, current and angle bases as

$$V_b=V_i;Z_b={\omega }_{s}L;{\theta }_b=\pi$$

(13)

By Equation (13), the power and current bases are derived as

$$P_b\equiv \frac{V_b^2}{Z_b}=\frac{V_i^2}{{\omega }_{s}L};I_b\equiv \frac{V_b}{Z_b}=\frac{V_i}{{\omega }_{s}L}$$

(14)

4.2. CCM

Looking at the graph at the bottom of Figure 4a, the current $I_o$ in CCM can be calculated as the sum of the areas of three simple geometrical Figures, namely two triangles and one trapezoid, that extend respectively in the intervals [0 ÷ φ], [φ ÷ β], and [β ÷ π]. Further to the above observation and Equation (4), the expression of $I_o$ in pu is formulated as in Equation (15). It follows the expression of P_o,pu in Equation (16)The expression of V_o,pu is found by equating I_o,pu to V_o,pu/R_l,pu and extracting V_o,pu, thus yielding Equation (17). The latter Equation outlines that, in pu quantities, the output voltage V_o depends only on the phase-shift angle β and the load resistor R_l.

It is of interest to formulate the expression of the load resistor in CCM. By taking the ratio of V_o,pu to I_o,pu, one can obtain it as Equation (18).

4.3. BCM

Looking at the graph at the bottom of Figure 4c, the current $I_o$ in BCM can be calculated as the area of the triangle that extends in the half-interval [0 ÷ π]. In light of the preceding observation and Equation (8), I_o,pu is expressed as Equation (19). It follows the expression of P_o,pu in Equation (20) and that of V_o,pu in Equation (21). Note that they all depend only on β.

It would be interesting to define the load resistance expression in BCM as well. By taking the ratio of V_o,pu to I_o,pu, one can obtain it as Equation (22).

Table 1. Equations for Output Voltage, Current and Power in the Three Operation Modes.

	CCM	BCM	DCM
$I_{o,pu}$	$\frac{\pi }{4}\left[2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2\right] \left(15\right)$	$\frac{\pi }{2}\left(1-{\beta }_{pu}\right){\beta }_{pu} \left(19\right)$	$\frac{\pi }{2}\left(1-V_{o,pu}\right)\frac{{\beta }_{pu}^2}{V_{o,pu}} \left(23\right)$
$P_{o,pu}$	$\frac{\pi }{4}{V}_{o,pu}\left[2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2\right] \left(16\right)$	$\frac{\pi }{2}\left(1-{\beta }_{pu}\right){\beta }_{pu}^2 \left(20\right)$	$\frac{\pi }{2}\left(1-V_{o,pu}\right){\beta }_{pu}^2 \left(24\right)$
$V_{o,pu}$	$\frac{2}{\pi }\left[-\frac{1}{R_{l,pu}}+\sqrt{{\left(\frac{1}{R_{l,pu}}\right)}^2+{\left(\frac{\pi }{2}\right)}^2\left(2-{\beta }_{pu}\right){\beta }_{pu} }\right] \left(17\right)$	${\beta }_{pu} \left(21\right)$	$\frac{\pi }{4}{R}_{l,pu}{\beta }_{pu}^2\left(-1+\sqrt{1+\frac{8}{R_{l,pu}{\beta }_{pu}^2\pi }}\right) \left(25\right)$
$R_{l,pu}$	$\frac{4}{\pi }\frac{V_{o,pu}}{2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2}\left(18\right)$	$\frac{2}{\pi }\frac{1}{1-{\beta }_{pu}}\left(22\right)$	$\frac{2}{\pi }\frac{V_{o,pu}{}^2}{\left(1-V_{o,pu}\right){\beta }_{pu}{}^2}\left(26\right)$

Table 1. Equations for Output Voltage, Current and Power in the Three Operation Modes.

	CCM	BCM	DCM
$I_{o,pu}$	$\frac{\pi }{4}\left[2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2\right] \left(15\right)$	$\frac{\pi }{2}\left(1-{\beta }_{pu}\right){\beta }_{pu} \left(19\right)$	$\frac{\pi }{2}\left(1-V_{o,pu}\right)\frac{{\beta }_{pu}^2}{V_{o,pu}} \left(23\right)$
$P_{o,pu}$	$\frac{\pi }{4}{V}_{o,pu}\left[2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2\right] \left(16\right)$	$\frac{\pi }{2}\left(1-{\beta }_{pu}\right){\beta }_{pu}^2 \left(20\right)$	$\frac{\pi }{2}\left(1-V_{o,pu}\right){\beta }_{pu}^2 \left(24\right)$
$V_{o,pu}$	$\frac{2}{\pi }\left[-\frac{1}{R_{l,pu}}+\sqrt{{\left(\frac{1}{R_{l,pu}}\right)}^2+{\left(\frac{\pi }{2}\right)}^2\left(2-{\beta }_{pu}\right){\beta }_{pu} }\right] \left(17\right)$	${\beta }_{pu} \left(21\right)$	$\frac{\pi }{4}{R}_{l,pu}{\beta }_{pu}^2\left(-1+\sqrt{1+\frac{8}{R_{l,pu}{\beta }_{pu}^2\pi }}\right) \left(25\right)$
$R_{l,pu}$	$\frac{4}{\pi }\frac{V_{o,pu}}{2{\beta }_{pu}-V_{o,pu}^2-{\beta }_{pu}^2}\left(18\right)$	$\frac{2}{\pi }\frac{1}{1-{\beta }_{pu}}\left(22\right)$	$\frac{2}{\pi }\frac{V_{o,pu}{}^2}{\left(1-V_{o,pu}\right){\beta }_{pu}{}^2}\left(26\right)$

It can readily be demonstrated that Equation (22) is greater than Equation (18); this shows that the SAB converter operates in CCM for load resistances lower than Equation (22). Therefore, by Equation (22), when β is close to π, the resistance $R_{l.BCM,pu}$ becomes very high and the SAB converters operate in CCM for any practical load.

4.4. DCM

Looking at the graph at the bottom of Figure 4b, the current $I_o$ in DCM can be calculated as the area of the triangle that extends in the half-interval [0 ÷ π − a]. Further to the above observation and Equation (10), the expression of $I_{o,pu}$ is formulated as in Equation (23). It follows the expression of P_o,pu in Equation (24) and that of V_o,pu in Equation (24). As before, Equation (25) outlines that, in pu quantities, the output voltage V_o depends only on the angle β and the resistor R_l.

It is of interest to formulate the value of the load in DCM. By taking the ratio of V_o,pu to I_o,pu, one can obtain it as Equation (26).

It can readily be demonstrated that Equation (26) is greater than Equation (22); this shows that the SAB converter operates in DCM for load resistances greater than Equation (22). It is worth noting that Equations (18), (22) and (26) on the load resistance are alternatives to Equations (5), (8) and (11) on the output voltage in discriminating between the SAB converter operation modes.

4.5. Characteristic Features

The features of the SAB converter characteristics are elucidated by resorting to some graphs. Let us start by drawing the characteristic that relates the output power to the phase-shift angle in Figure 6, as per Equations (16), (20) and (24). The Figure highlights that the characteristic is composed of a family of curves, each of them distinguished by a different value of the output voltage.

The curves demonstrate that, for a fixed value of $V_{o,pu}$, the output power increases monotonously with the phase-shift angle so that the maximum output power is reached at β_pu = 1, i.e., when the input bridge generates a square-wave voltage, and the SAB converter operates in CCM. What may be surprising is that the highest value of the output power is not achieved when $V_{o,pu}$ is maximum, but when it is equal to $1/\sqrt{3}\cong 0.58$, being the related power equal to 0.3. In BCM, the highest output power is delivered at β_pu ≡ V_o,pu = 2/3 and is equal to about 0.23, which is 2.5 times less than that in CCM. All these values can be directly computed from Equations (16) and (20). Given that the DCM region extends below the BCM curve, the output power in DCM does not exceed 0.23.

Figure 7 illustrates the characteristic relating the output voltage to the phase-shift angle. It is composed of a family of curves, each of them distinguished by a different value of the load resistor R_l,pu, as per Equations (17), (21), and (25). The main outcome of Figure 7 is that the SAB converter does not behave like a voltage source, since the output voltage changes with the load. Other outcomes of interest are: (i) the output voltage increases with R_l,pu, while still remaining less than 1, and (ii) the output voltage rises nearly proportional to the phase-shift angle for low values of β_pu, and with a saturated behavior for β_pu approaching 1, (iii) the BCM plot is a straight line, and (iv) the lighter the load, the wider the range of β_pu, where the SAB converter operates in DCM.

Figure 8 illustrates the characteristic relating the output current to the phase-shift angle. It is composed of a family of curves, each of them distinguished by a different value of the output voltage, as per Equations (15), (19), and (23). Similarly to Figure 6, Figure 8 shows that, for a fixed V_o,pu, the curves of the output current have an increasing monotonic behavior so that they reach their maximum at β_pu = 1, i.e., when the input bridge generates a square-wave voltage and the SAB converter operates in CCM. Differently from Figure 6, the peak value of the output current is reached for V_o,pu = 0, whilst the maxima of the output current decrease as V_o,pu increases. The BCM curve is a parabola, as can be realized from Equation (19).

Figure 9 illustrates the external characteristic of the SAB converter; it is composed of a family of curves relating the output voltage to the output current for different values of the phase-shift angle. The equations of the curves are obtained by (i) substituting Equations (17) and (25) in place of Equations (15) and (23), (ii) tracing I_o,pu as a function of V_o,pu, and, lastly, (iii) inverting the graph. The Equation of the BCM curve is found by substituting V_o,pu to β_pu in Equation (19), obtaining

$$I_{o,pu}=\frac{\pi }{2}\left(1-V_{o,pu}\right)V_{o,pu}$$

(27)

Equation (27) outlines that, in BCM, I_o,pu is a parabolic function of V_o,pu. The curves of Figure 9 highlight that the SAB converter operates in DCM when the working point lies in between the V_o,pu axis and the BCM curve, whereas it operates in CCM when I_o,pu is greater than Equation (27), calculated for the actual value of $V_{o,pu}$. Once again, it can be seen that the SAB converter operates in DCM over a wide range of the output current when it is lightly loaded.

The curves of Figure 9 also reveal two remarkable features of the SAB converter. The first one is that the SAB converter gets the maximum output voltage at no-load; it has a finite value that is equal to 1, irrespective of β_pu. The second feature is that the SAB converter gets the maximum output current at short-circuit, with the input bridge generating a square-wave voltage. By replacing β_pu = 1 and V_o,pu = 0 in Equation (15), the maximum value of $I_o$ turns out to be equal to

$$I_{o,pu,MAX}=\frac{\pi }{4}\cong 0.79$$

(28)

Therefore, the SAB converter is built to withstand the extreme working conditions of open or short-circuited output terminals.

5. SAB Design

This section is devoted to the design of the SAB converter by sizing and selecting its circuit elements. Supplementary Equations, which are needed to support the design, are first formulated.

5.1. Design-Supporting Equations

Active Bridge: Besides the voltage solicitation, the sizing of the active bridge switches requires the valuation of the peak and rms values of current flowing through each of them. Let us consider first the value of the peak current.

From the current waveforms reported in Figure 4a,c for the CCM and DCM modes, it emerges that, regardless of the operating mode, the peak of the current circulating in the switches is reached at β and is equal to i_L(β). The relationship between i_L(β) and the average output current I_o of the SAB converter is again obtained with the aid of the p.u. quantities. Equation (7), conveniently scaled by the base current, and Equation (15) are used for this purpose in the case of the CCM operation mode. A similar approach is used for the DCM. Because the resulting expressions are extremely complex, it was thought that exposing them in graphical form would be more beneficial. This is done in Figure 10, where the relationship is plotted for different output voltage values. The Figure emphasizes that the peak current circulating in the switches has an increasing monotonic trend with the current I_o. Moreover, from the Figure, it emerges that: (i) as V_o,pu increases, the peak current delivered by the active converter reduces significantly, and (ii) in DCM, the curve piece belonging to V_o,pu = 0.1 coincides with that belonging to V_o,pu = 0.9, and the same is true when V_o,pu is equal to 0.7 and 0.3. The curves also confirm that, as the current I_o increases, the operation of the SAB converter swaps from DCM (dashed lines) to CCM (continuous lines).

To complete the analysis of the current flowing in the switches of the active bridge, the rms values are calculated for the CCM and DCM modes and reported in the four upper rows of

Table 2. rms Values of Currents.

	CCM	DCM
$I_{Q1, rms}$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right)\left(\pi -\phi \right)+i_L^2\left(0\right)\left(\pi -\beta \right)+i_L\left(\beta \right)i_L\left(0\right)\left(\beta -\pi \right)\right]} \left(29\right)$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right)\left(\pi -\alpha \right)\right]} \left(34\right)$
$I_{D1, rms}$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(0\right) \phi \right]} \left(30\right)$	0 (35)
$I_{Q3, rms}$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right)\left(\beta -\phi \right)\right]} \left(31\right)$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right) \beta \right]} \left(36\right)$
$I_{D3, rms}$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right)\left(\pi -\beta \right)+i_L^2\left(0\right)\left(\pi -\beta +\phi \right)+i_L\left(\beta \right)i_L\left(0\right)\left(\beta -\pi \right)\right]}\left(32\right)$	$\sqrt{\frac{1}{6\pi }\left[i_L^2\left(\beta \right)\left(\pi -\alpha -\beta \right)\right]}\left(37\right)$
$I_{out,rms}$	$\sqrt{\frac{1}{3\pi }\left[i_L^2\left(\beta \right)\left(\pi -\phi \right)+i_L^2\left(0\right)\left(\pi -\beta +\phi \right)+i_L\left(\beta \right)i_L\left(0\right)\left(\beta -\pi \right)\right]} \left(33\right)$	$\sqrt{\frac{1}{3\pi }\left[i_L^2\left(\beta \right)\left(\pi -\alpha \right)\right]} \left(38\right)$

. To avoid burdening the notation, the subscript “pu” has been omitted from the equations. In any case, the way the equations are written allows the currents to be treated as either dimensional quantities or per-unit quantities. Table 2 shows only the equations for the transistors Q₁, Q_3, and for the diodes D₁, D₃. It is worth noting that the same equations hold for the transistors Q₂, Q_4, and for the diodes D₂, D₄ since their waveforms are just shifted by an angle θ equal to π with respect to the waveforms of the same components in the same leg. The plot of these equations yields curves similar to those of Figure 10; therefore, the conclusion is drawn that the switches must be sized by taking into account the working point of the SAB converter where I_o is maximum.

In addition to the rms currents in the active bridge switches, Table 2 also reports the rms value of the current i_out. The rms value of the converter output current is equivalent to the rms value of the current i_L. Equations (33) and (38) come into play in the selection of the capacitor C_o. Indeed, the rms value of the current i_Co can be found with

$$I_{Co,rms}=\sqrt{I_{out,rms}^2-I_o^2}$$

(39)

Output Capacitor: The task of the capacitor $C_o$ is to filter the output voltage of the passive bridge in order to mitigate the ripple of the voltage V_o. This ripple can be calculated using the following formula:

$$\Delta V=\frac{\Delta Q}{C_o}$$

(40)

where ∆Q is the variation of the charge stored in the capacitor that produces the maximum excursion $\Delta V$ of the voltage V_o. The charge ∆Q is graphically represented by the two colored areas in Figure 11. At steady state, the two areas are equal and the charge can be calculated for any of them. In the case of Figure 11, in which I_o < i_L(π), the calculation is eased by referring to the triangle lying in the interval [θ₂÷ (π + θ₁)]. As a result, ∆V is given by

$$\Delta V=\frac{1}{{\omega }_s{C}_o}\frac{I_o\left(\pi +{\theta }_1-{\theta }_2\right)}{2}$$

(41)

In turn, the angles θ₁ and θ₂ can be calculated by using a straight-line Equation approach. In the case of Figure 11, they are

$${\theta }_1=\phi +\frac{I_o}{i_L\left(\beta \right)}\left(\beta -\phi \right)$$

(42)

$${\theta }_2=\pi +\phi +\frac{I_o}{i_L\left(0\right)}\phi$$

(43)

Putting together Equations (41) with (42) and (43), the value of the capacitor C_o for an assigned ripple magnitude can be readily found. If I_o is greater than the current i_L(π), the equations above are no longer valid, but the process for obtaining the value of C_o remains the same.

5.2. Elements Sizing and Selection

The specifications for the design of the SAB converter are listed in

Table 3. SAB Converter Specifications.

Parameter	Symbol	Nominal Data
Input voltage	Vi	130 V
Output voltage	Vo	48 V
Output power	Po	200 W
Switching Frequency	fs	20 kHz

in terms of nominal data for the input and output voltages, the output power, and the active bridge switching frequency. From them, it follows that the values of the nominal output current and resistance are given by

$$I_{o,nom}=\frac{P_{o,nom}}{V_{o,nom}}$$

(44)

$$R_{l,nom}=\frac{V_{o,nom}}{I_{o,nom}}$$

(45)

From Table 3, I_o,nom and R_l,nom are calculated in 4.16 A and 11.54 Ω, respectively. The sizing of the circuit elements starts from the transformer.

Transformer Sizing: The key parameter that is crucial for the SAB operation is the inductance L, which usually coincides with the transformer leakage inductance, appositely designed. One of the criteria for the selection of L is to make the converter operate at the maximum power transfer capability. Referring to Figure 6, this point is obtained with β_pu = 1 and V_o,pu ≈ 0.58, and the corresponding per unit power is P_o,pu ≈ 0.3. By equating P_o in Table 3 with P_o,pu multiplied by P_base given by the first part of Equation (14) and solving for L, the leakage inductance is found to be around 202 μH. However, designing the converter to settle on this point is not practically advisable, mainly for two reasons: (i) if the measured leakage inductance obtained after the transformer construction is slightly higher than 202 μH, there is no way to make the converter work at the nominal power; (ii) if, for a small amount of time, the converter needs to operate at higher power, this cannot be achieved due to the fact that β_pu is already at its maximum. For this reason, with a safety margin, the choice of designing the converter for P_o,pu = 0.25 is made. In this case, the leakage inductance L is 168 μH obtained with the same procedure explained before. With this value of L, the converter operation point in Figure 6 is forced to be in the horizontal line P_o,pu = 0.25. The exact point in this line is obtained by setting the value of β_pu or equivalently by setting V_o,pu. To take into account the transformer turns ratio n, the analysis carried out in Section 4 is easily exploited by considering all the output quantities referred to the primary side.

Thus, the quantity V_o,pu depends on the transformer turn ratio according to V_o,pu = nV_o/V_i. A possible way to select n is to choose the value that allows the converter to work with the minimum circulating currents. Figure 12 shows the transformer turns ratio versus the rms value for the current I_L,rms in the primary side and the secondary side; since the rms of the current that flows in the leakage inductance is the same as the rms of the output current I_out,rms, both curves are derived from Equations (33) and (38). All points of the curves are calculated considering a constant value of the output current I_o given by I_o,nom. When the curves drop to the horizontal axis (i.e., I_L,rms becomes 0), it means that the SAB converter is no longer able to guarantee such current for the relevant values of n. From Figure 12, it is easy to see that higher values for the turns ratio bring benefits in terms of circulating currents, which means higher efficiency for the converter. In light of this, the final selection has fallen on a value of n equal to 2. After the construction of the transformer, the leakage inductance L has been measured equal to 170 µH. With the allocated values of the turns ratio and the leakage inductance, the shift angle, as well as the operation mode of the SAB converter at the nominal working point, can be established. Numeric coordinates of this point are calculated by entering V_o,pu = 0.738 and I_o,pu = 0.342 in Equations (15) and (23). Equation (15) offers two solutions for β_pu, namely 0.85 and 1.15, but the latter one must be disregarded as it is outside the range; Equation (23) does not comply with the given pair V_o,pu, I_o,pu. Therefore, the prototypal SAB converter, in nominal conditions, operates in CCM with β_pu = 0.85. This agrees with the fact that V_o,pu is less, albeit slightly, than β_pu, which is confirmed by Figure 13, where Figure 6 is redrawn, and the nominal working point P is highlighted. The design process of the transformer is also shown in the flow chart given in Figure 14.

Output Capacitor: The selection of the output capacitor has been dictated mostly by the rms of the current that has to flow into it. Considering the transformer parameters obtained previously, Equation (39) provides a value of 2.3 A for the current. The chosen capacitor was the Panasonic EEU-EE2C331, which has a capacitance of 330 ± 20% μF and is rated for 160 V, being able to withstand an rms current of 2.6 A. The capacitance is well beyond the value useful for the ripple limitation. In fact, Equation (41) yields a ripple of 0.075 V, corresponding to 0.16% of the output voltage. The design process is also included in the flow chart given in Figure 14.

Active Bridge: The active bridge for the SAB converter must sustain the highest peak current through the switches. From the previous section, this quantity is given by i_L(β) and occurs when the SAB works at the nominal working point. Using Equation (7), the value of i_L(β) is 4 A. For the prototype, the selected devices have been four STF16NF25 N-Channel MOSFETs manufactured by STMicroelectronics in the TO-220 package using the STripFET process.

The rated values for the component are a continuous drain current capability at 25 °C equal to 14 A, and 250 V maximum reverse blocking voltage. The MOSFETs intrinsically have an antiparallel diode that allows them to conduct the current in both directions. In this case, with the aid of the equations of Table 2, the rms values for the current through the MOSFETs (useful for assessing the conduction losses and thus for the thermal calculations) can be easily found with the equation $\sqrt{I_{Q,i}^2+I_{D,i}^2}$, where the index i refers to any switches of the H-bridge. The rms current is equivalent for all the MOSFETs and was found to be 1.7 A. The switches rating process can also be seen in the flow chart given in Figure 14.

To switch the MOSFETs on or off, two (one per leg) gate drivers are used. The selected gate drivers are the IR2110 produced by Infineon. In the prototype, the gate drivers accept the incoming logical values of 5 V and are supplied with 15 V in order to properly turn on the MOSFETs. They implement the bootstrap technique to turn on the upper switches.

Passive Bridge: For the specification of the SAB converter in Table 3, the average output current that the diode rectifier has to sustain is 4.16 A, while the peak value is 8 A, obtained from Equation (7). Four MBR10100G Schottky diodes produced by On Semiconductor are used for the passive bridge. Each diode can conduct an average forward current of 10 A, withstands an inverse voltage of 100 V, and has a forward voltage drop of about 0.7 V. The switches rating process is also included in the flow chart given in Figure 14.

SAB Converter Control: The algorithms for the generation of the proper gate signals to the MOSFET drivers have been implemented in the ARM CORTEX-M4 STM32F401RE microcontroller mounted in the Nucleo-F401RE development board produced by STMicroelectronics. The SAB converter has been operated in the open-loop setting without considering any feedback signal.

5.3. Loss Estimation and Efficiency Calculation

To incorporate the efficiency of the converter, the converter is operated at the nominal operating point which is highlighted as point p in Figure 13. Furthermore, losses are calculated for each element, and the efficiency is evaluated.

Passive Bridge: Diode turn-on loss is generally neglected whilst its turn-off loss depends on the reverse recovery charge Q_c and the cathode-anode voltage, and is given as

$$E_{off,D}=\frac{Q_C}{4}V$$

(46)

Substituting Q_c = 53 nC, obtained from the datasheet of MBR10100G, and V = 48 V, gives E_off,D = 0.63 µJ and the corresponding power loss is 0.01 W. For four diodes, the total turn-off losses will be 0.04 W. The conduction loss of the diode is proportional to the forward voltage drop V_F and is given as

$$P_{con,D}=V_F{I}_{Dk,avg}$$

(47)

where I_Dk,avg is the average current flowing in the diode D_k. Using some basic calculations, the average current through D₅, namely I_D5,_avg, is obtained as 2.07 A. It should be noted that all the diodes of the passive bridge have the same average current, as can be observed from Figure 4a. Furthermore, substituting V_F = 0.7 V, obtained from the datasheet of MBR10100G, and I_D5,avg = 2.07 A into Equation (47), conduction loss is found to be equal to 1.45 W. Considering all four diodes in the rectifier, the total conduction losses are 5.8 W.

As a result, the total losses in the passive bridge are 5.84 W. With the nominal power of 200 W, the input power is 205.84 W and the estimated efficiency is 97%.

Transformer: Transformer losses are divided into the copper loss and the core loss. Copper loss can be evaluated by measuring the winding resistance and the rms current flowing through it. Based on the measurement, the primary and the secondary winding resistances are found to be 0.1 Ω and 0.05 Ω, respectively. Rms currents through the primary and the secondary sides are calculated to be equal to 2.38 A and 4.76 A, respectively. Furthermore, the total copper loss is calculated as 1.7 W. The core loss of the transformer is calculated following the rigorous mathematical steps suggested in [37] and was found to be equal to 1.2 W. As a result, the total power losses in the transformer are 2.9 W. With a nominal output power of 205.84 W, its input power is 208.74 W, giving a transformer efficiency value equal to 98.6%.

Active Bridge: A close examination of Figure 4a reveals that the turn-on of the switches occurs when the corresponding intrinsic diodes are already conducting i.e., zero voltage switching (ZVS) occurs during their turn-on. At this node, it can be stated that the turn-on losses of the switches are approximately equal to zero. However, the turning off of the switches is associated with losses due to their hard switching. Referring to the datasheet of STF16NF25 provided by the manufacturer, turn-off losses of the switches Q₁ and Q₄ are given as 0.12 W and 0.26 W, respectively. Here, two points are notable: (i) since commutation of Q₁ and Q₄ occurs at different current levels, i.e., for Q₁ it is i_L(π) and for Q₄ it is i_L(β), the turn-off losses are different for them; (ii) as can be seen from Figure 4a, turn-off losses for Q₂ and Q₃ will be similar to Q₁ and Q₄, respectively. As a result, the total turn-off losses of the switches Q_i in a complete cycle are given as 0.8 W.

Conduction loss of the switch Q_i can be approximated by again referring to the datasheet and finding the conduction resistance, namely R_DS(on) which is given as $I_{Q,i}^2{R}_{DS\left(on\right)}$, where I_Q,i is the rms current flowing through the switch Q_i. Exploiting Equations (29) and (31) and substituting R_DS(on) = 0.18 Ω, the conduction losses for Q₁ and Q₃ are calculated as 0.5 W and 0.38 W, respectively. Again, the difference in these two losses can be better appreciated by referring to Figure 4a, where the conduction duration of Q₁ is more than that of Q₃. It should be noted that the switches Q₂ and Q₄ have similar conduction losses to Q₁ and Q₃, respectively. As a result, the total conduction losses of the switches are given as 1.76 W.

Considering the intrinsic diode of the switch, it is desirable to calculate its switching loss as well as the conduction loss. Furthermore, the turn-on loss of the diode is negligible and the turn-off loss can be evaluated by taking into account the Q_c of the intrinsic diode. From the datasheet, Q_c is obtained as 895 nC. Substituting this value of Q_c and V = 130 V into Equation (46) gives turn-off loss of the diode as 0.58 W. The conduction loss of the diode can be evaluated using Equation (47), where V_F is 1.6 V. The average currents for diodes D₁ and D₃ are calculated as 0.02 A and 0.22 A, respectively, and their corresponding conduction losses are found to be 0.04 W and 0.35 W, respectively. Again, conduction losses of D₁ and D₃ are equal to D₂ and D₄, respectively. Total losses in the intrinsic diodes are calculated as 3.1 W.

As a result, total losses in the active bridge are obtained as 5.66 W. For nominal power to be equal to 208.74 W, the input power will be 214.4 W and the corresponding efficiency is 97%.

The overall efficiency of the SAB converter operating at its nominal point is found to be approximately equal to 93%.

6. Experimental Results

An experimental prototype of the SAB converter has been arranged according to the findings of Section 5. Experimental voltage and current waveforms in the CCM and DCM modes are shown in Figure 15 and Figure 16, respectively. Since only a two-channel oscilloscope was available in the laboratory, two Figures are captured for each conduction mode and, to make the waveforms easy to correlate, v₂ is kept common to both the plots. Waveforms in Figure 15 correspond to the converter nominal operating point, and can be compared with the theoretical ones shown in Figure 4a. Since in this Figure, point β is close to π, it is very difficult to appreciate the four different slopes for the current in each half-period of 25 μs. However, a meticulous check of the relevant values obtained from Figure 15 reveals that the theoretical analysis and the experimental outcomes are in good agreement. The waveforms of Figure 16, instead, match very well with the waveforms drawn in Figure 4c. They are obtained starting from the nominal condition, reducing β and at the same time increasing the load resistance in order to maintain a constant output voltage equal to 48 V.

The different characteristic plots of the SAB converter presented in Section 4 are compared here with the experimental results and the outcome is shown in Figure 17. The Figure shows that the star points obtained from measurements in the prototype are very close to the curves that arise from the theoretical analysis. To understand the steps to be followed to obtain the experimental measurements, let us take the case of Figure 17a which plots P_o,pu as a function of β for various V_o,pu. The steps are outlined as follows: (i) the prototype is operated for a particular angle β, (ii) the load resistor is varied to attain the desired load voltage, for instance V_o = 33.6 V, which corresponds for the V_o,pu = 0.7, (iii) once the desired V_o is achieved at any load resistance R_l, P_o is calculated using $P_o=V_o^2/R_l$, (iii) steps (i) to (iii) are repeated for other values of β. A similar approach is also used to obtain the results for Figure 17b–d.

7. Discussion and Conclusions

7.1. Discussion

The preceding analysis has discussed the behavior of this converter. However, the important observations can be highlighted as follows:

• The output power depends not only on the phase-shift angle but also on the load voltage. The highest power that can be drawn from this converter is obtained when the phase-shift angle becomes equal to π and the load voltage is maintained at 0.58 V_nom, where the converter operates in CCM mode (see Figure 6).
• Load voltage increases with the phase shift angle for any value of R_L and the region of DCM increases with the increase in R_L value (see Figure 7).
• The load current as a function of alpha increases monotonically and is at its maximum when Vo = 0 V i.e., when the load does not draw any current from the converter (see Figure 8).
• When the V_o is zero, i.e., output is shorted, the load current is finite and is given as Equation (27). Again when I_o = 0 i.e., output is open circuit, circuit output voltage is V_o,nom. This confirms the suitability of this converter in adverse operating conditions (see Figure 9).
• The transformer turns ratio can be utilized to reduce the inductor current, which, consequently, will improve efficiency. However, it should be noted that the ability of the converter to provide the nominal load current is not possible for all values of the turns ratio (see Figure 12).

7.2. Limitations, Future Scope, and Applications

7.2.1. Limitations: Limitations of this Converter Are Outlined as Follows

• This converter suffers from large switching losses. In CCM, ZVS occurs during the turning on of the active bridge switch thus enabling soft switching, whereas turning off is hard switching. In DCM, ZCS occurs during the turn-off of the switch, whereas ZVS during switch turn-on is lost [11].
• When the transformer ratio is equal to unity and the voltage conversion ratio is kept at unity, the power transfer is not possible because the passive bridge diode remains reverse biased. When the voltage conversion ratio is kept below unity, the total power factor of the high-frequency transformer is low [38].
• Since primary voltage and the peak current are high in this converter, the high-voltage and high-current switching devices and the high-voltage transformer are then required.

7.2.2. Future Scope

As mentioned earlier, the converter suffers from large switching losses, which could be important to the scope of the research. The authors are also planning to continue the work on the soft switching of this converter. Another scope of the research could be the dynamic modeling of the converter [4,5,6].

7.2.3. Application

Application of this converter is generally in the area of EV charging and wind farms, where unidirectional power transfer is required.

7.3. Conclusions

The paper has presented the analysis and the step-by-step design of the SAB converter. The basic principle of the SAB converter has been reviewed, and the procedure followed in designing the converter has been thoroughly described. A systematic analysis of the converter’s characteristics has also been carried out. Afterward, a prototypal SAB converter has been set up according to the design results and tested in different operation modes. Several measurements obtained from the prototype have been reported that fully confirm the soundness of both the design procedure and the characteristics of the SAB converter.