A Generalized Input Impedance Model of Multiple Active Bridge Converter

The electrical power distribution system (EPDS) of the more electric aircraft (MEA) is a fundamental component that needs to be efficient and resilient. The commonly considered architectures feature separate buses to achieve separation between different subsections of the EPDS. Although effective, this implies an over design, since all subsections are sized for the local worst case scenarios. In the MEA concept, multiport converters could connect the whole EPDS while guaranteeing the galvanic isolation between buses. Since multiport converters would give rise to a completely different EPDS topology, dominated by power electronics interfaces, the stability of such a system must be assessed. This article investigates the input impedance of multiple active bridge (MAB) converters when interfaced with a single dc bus and multiple resistive loads. A transfer function-based input impedance model of the MAB converter is proposed. To validate the proposed input impedance model, the verification of input impedances of a triple active bridge (TAB) converter and a quadruple active bridge (QAB) converter is carried out using both simulation and experimental results.


I. INTRODUCTION
N OWADAYS, the concept of the more electric aircraft (MEA) has attracted more and more consideration by researchers, which is aimed at electrifying the subsystems on aircraft. In recent years, several projects have been started to achieve the electrification of subsystems on aircraft, such as Totally Integrated More Electric Systems (TIMESs) sponsored by the Department of Trade and Industry (DTI) under the Civil Aircraft Research and technology Demonstration (CARAD) program [1] and More Electric Initiative developed by the U.S. Air Force Research Laboratory [2]. In an MEA, most of the onboard pneumatic, hydraulic, and mechanical devices are supposed to be replaced by electrical devices, benefiting the aircraft with lower weight, lower maintenance cost, and lower environmental impact [3]. So far, significant progress has been made in the electrification of subsystems on aircraft, for example, the main engine generator is directly coupled to the jet engine via a gearbox without the integrated drive generator (IDG) in the latest and the most advanced commercial aircraft, including the Boeing 787 and the Airbus A380 [4].
As onboard electrical devices increase, the demands of electrical power increase, resulting in an onboard electrical power system (EPS) with larger size and heavier weight. Meanwhile, the architecture and individual subsystems of the EPS become more complicated, increasing the potential risks of instability. In addition, the large-scale redundancy of the electrical power distribution system (EPDS) is supposed to be minimized to diminish the overall installed power in an aircraft, which increases the unpredictability of power requested [5]. To meet the different requirements of power according to the number of installed power electronic converters and the characteristics of actuators, a number of voltage standards including both ac and dc levels on commercial aircraft exist [6]. Since the aircraft can be regarded as an isolated system with generators and loads, the EPDS can be regarded as an onboard microgrid [7]. Compared to the ac microgrid, the dc microgrid has attracted interest from researchers due to the advantages of fewer conversion stages and lower current ratings, simplifying the architecture and improving the efficiency [8]. Fig. 1 shows a simplified structure of dc microgrid for an MEA. It can be observed that the dc microgrid may contain several ac-dc converters to convert ac waveforms from the generator to dc and many dc-dc converters to achieve different dc voltage levels for the various loads. Whereas the dc microgrid contains many individual converters and communication devices among converters due to the multiple conversion stages and load characteristic, the structure is still complex and the cost is high. To cope with these challenges, the concept of multiport dc-dc converters was proposed in [9], to reduce the number of conversion stages by combining individual dc-dc converters in an integrated converter with multiple inputs and outputs.  Considering the safety requirement of the MEA, the installed dc-dc power converters must have galvanic isolation [5], [7]. Among a variety of dc-dc converters, the dual active bridge (DAB) converter is one of the most popular topologies because of its bidirectional power flow, high efficiency, high power density, low device and component stresses, and low switching losses [10]. Benefiting from the DAB converter, it becomes a consensus to most researchers that it is better if the feature of multiport dc-dc converters can imitate the DAB converter, i.e., multiple active bridge (MAB) converters which contain multiple H-bridges and a multiwinding highfrequency (HF) transformer. Some researches about multiport dc-dc converters have been published, including investigations of the triple active bridge (TAB) converter and the quadruple active bridge (QAB) converter [11], [12].
Although multiport dc-dc converters are expected to replace the single-input-single-output (SISO) dc-dc converters in microgrids, the effects of them on system stability needs to be assessed. There are several methods to analyze stability. Among these methods, the impedance-based method is preferred by the majority of researchers because the system stability can be readily evaluated by modeling the impedance of system. Changes of the system structure and the parameters of sources or loads only influence the impedance characteristics [13]. Fig. 2 shows the equivalent circuit of a dc system consisting of two individual stable subsystems. To assess the stability of dc systems based on impedance, the concept of minor loop gain (MLG) is introduced, which is the ratio of the output impedance Z O of source subsystem and the input impedance Z I of load subsystem. The MLG is also the term responsible for stability in the input-to-output transfer function of the whole system [14]. Using the Nyquist criterion, the system will be stable if the MLG does not encircle (−1, 0) in the Nyquist contour. Based on this, several stability criteria have been proposed, the most conservative is the Middlebrook criterion [15]. The stability condition of the Middlebrook criterion can be expressed as where GM is the expected gain margin. Although the Middlebrook criterion can ensure the stability of a system, it could sacrifice the size of the system by implementing large passive components in the input filter design [14], [16]. To avoid this overdesign, the gain margin phase margin (GMPM) criterion was proposed in [17], which is more moderate by giving conditions in both gain margin and phase margin to system design. The stability condition of the GMPM criterion can be given as where GM is the expected gain margin and PM is the expected phase margin. It is obvious that the GMPM criterion gives a less restrictive condition by considering the phase margin when the gain condition is not satisfied. However, the GMPM criterion requires the information of the amplitude and phase of subsystems, and a proper design for gain margin and phase margin [14]. So far, some works related to the impedancebased stability analysis of systems have been done. It is shown in [18] that the stability issues caused by dc-link voltage control of grid-connected voltage-source converters can be investigated and addressed by analyzing the impedance of the grid and the converter. It is also shown in [19] that the stability of a single dc-bus, multigenerator EPS supplying a constant power load (CPL) can be analyzed in terms of the impedance of the source and load subsystems, with the cases of a different number of generators and power-sharing ratio. Therefore, it is credible that the impedance-based method can be used to assess the stability of large systems by modeling the impedance of subsystems inside. As the dc-dc converters are supposed to connect to the dc buses in EPDS, to avoid the instability issues, it is necessary to find the input impedance and output impedance of dc-dc converters. In this article, a generalized input impedance model of the MAB converter is proposed and validated with both simulation and experimental results. Section II presents the power transmission characteristics of the MAB converter. Section III gives the deduction of the generalized input impedance model of the MAB converter based on the power equations of any two ports. Section IV provides reasonable specifications for the MAB converter applied for MEA. Section V validates the generalized input impedance model in terms of simulating the switching model of the TAB converter and the QAB converter. Section VI shows the experimental results. Section VII summarizes the work and draws the conclusions.

II. MULTIPLE ACTIVE BRIDGE CONVERTER
In general, the MAB converters can be interfaced with buses and a variety of loads to achieve simultaneous power flow between any two ports. In this article, the MAB converter is supposed to connect to a single bus and several resistive loads, and the generalized input impedance model of it is investigated and derived. Fig. 3 shows the basic structure of the MAB converters. Port 1 is connected to a dc bus while other ports are connected to different resistive loads. By enabling the single-phase shift modulation of all H-bridges, the multidirectional power flow among all ports can be achieved. Assuming that the MAB converter has n ports in total, the individual power transmitted between any two ports can be given as where V j and V m are the port voltage of Port j and Port m, respectively, N j,m is the turn ratio between Port j and Port m, L j,m is the leakage inductance between Port j and Port m, and d j,m is the phase shift ratio normalized to π. Consider the conservation of power, the total power in Fig. 3 satisfies the equation Since the power of each port can be regarded as the combination of individual power between any two ports, the equation of power of port j can be written as To obtain the real leakage inductance in (2.1), the appropriate transformer model needs to be worked out. Fig. 4 shows different kinds of transformer models, containing both the star model and the delta model. In Fig. 4(b), which is the star model, the leakage inductance and the voltage of each port Since the star model can only represent the leakage inductance and the voltage of each port referring to the specific port, it cannot be used to obtain the leakage inductance of the MAB converter in which the leakage inductance is coupled. To model the leakage inductance of the MAB converter properly, the delta model is adopted. The methodology of the delta model for a TAB converter was introduced in [20]. Based on this and assuming the magnetizing inductance is so large that it can be regarded as an open circuit, the inductance between two ports of the MAB converter can be deduced as , (j, m = 1, 2,. . . , n and m = j ).
(2.6) III. GENERALIZED SMALL-SIGNAL MODEL OF MAB CONVERTER State-space averaging and circuit averaging are two conventional techniques to obtain the small-signal model of converters [21]. Both techniques can show the system dynamics and the coupling among system state variables. The statespace averaging technique allows deriving the small-signal model with the known system state equations, while the circuit averaging technique allows manipulating on the circuit diagram directly if the waveforms of switch terminal over a switching period are known. However, both techniques need detailed information about the system to achieve high accuracy of modeling. When it comes to the converter with more components, it becomes difficult to obtain the small-signal model by using these two techniques. An input impedance model of the DAB converter based on improved state-space averaging technique was proposed in [22], where the ac components are also considered for state variables and the computation procedure is complex. By contrast, a transfer function block scheme based on the small-signal equations derived from the power equation of the DAB converter was proposed in [23], which is apparently much easier. Based on this, an input impedance model of the TAB converter has been worked out in [24]. In this section, a further generalized model of the input impedance of the MAB converter is developed. According to (2.1) and (2.3), the average current of port j can be written as  The small-signal equation of port current is worked out by deriving (3.1) partially to voltages and phase shift ratios, which is represented in (3.2). Note that the d j,m is substituted by d j ,1 − d m,1 , since this substitution will simplify the structure of the transfer function block scheme by using fewer phase shift ratios. To get the input impedance of the MAB converter in a more organized way, the gains are numerically sorted and labeled. Based on (3.2), there are totally 2n 2 -2n smallsignal gains and they are defined in Table I, where T D is a first-order delay function with time constant of one switching period. 5 shows the transfer function block scheme. It can be seen that the small signals on each branch also have effects on other branches and this coupling between branches makes the derivation of input impedance more difficult. Hence, it is necessary to perform a decoupling technique to separate each branch. The small signals in Table I can be represented as where G Branch, j and G Branch,m are the final gains of "Branch j " and "Branch m," respectively, shown in Fig. 5; G Ctrl,m and G Ctrl, j are the transfer function of "Controller m" and "Controller j ," respectively; G Load, j is the transfer function of "Load j "; and G 2 j −2 and G 2m−2 are the numerically labeled small-signal gains shown in Table I. Combining Table I As (3.4) shows,V 1 is influenced by each branch. This can be rearranged to be a matrix shown in (3.5), at the bottom of the previous page. Since it is aforementioned that the branches are also coupled together, to cancel the coupling effects, (3.5) is transformed as where B is the matrix of branches and A −1 is the inverse of Matrix A. By performing this transformation, the branches are decoupled and can be represented byV 1 only. By summing the elements in each row of A −1 , the coefficients vector ofV 1 can be obtained. Thus, (3.6) can be written as Considering the direction of current I 1 and input capacitor C 1 in Fig. 3, the input impedance of the MAB converter is calculated as

IV. SPECIFICATIONS OF MAB CONVERTER IN MEA
With the structure of the MAB converter shown in Fig. 3, to facilitate the calculation and observation of power flow inside the MAB converter, all H-bridges are designed to be the same, leading to a symmetrical structure of the converter. Considering the requirements and standards of MEA, a set of feasible parameters for simulation is listed in Table II. Since the converter will be unstable if the phase shift ratio goes out of the nearly linear operating region (−0.5, 0.5), the phase shift ratios d 2,1 , d 3,1 , . . . , d n,1 are limited in (−0.2, 0.2).  This limitation indirectly ensures the potential phase shift ratios such as d 2,3 , which is the difference between d 2,1 and d 3,1 , to be in (−0.4, 0.4). Hence, with this limitation, all possible phase shift ratios will be in the nearly linear operating region. In addition, it is worth noticing that the power of load ports is larger than or equal to zero because of the resistive load. Therefore, based on (2.3), it exists

V. INPUT IMPEDANCE VALIDATION OF MAB CONVERTER
In this section, the TAB converter and the QAB converter are chosen to be validated with the generalized model, while MATLAB is used to plot the input impedance of converter and an ideal switching model is built in PLECS to measure and verify the input impedance. A preliminary validation for the input impedance of the TAB converter has been already done and shown in [24].

A. Simulation Verification of TAB Converter
Considering the TAB converter has three ports, totally 12 small signal gains can be obtained for the transfer function block scheme. The gains are given in Table IV. According to (3.5), Matrix A can be written as With the known Matrix A in (5.1), the input impedance of the TAB converter can be obtained following the equations from (3.6) to (3.9). Fig. 6 shows the bode plot of the input impedance of the TAB converter. It can be observed that the input impedance of the TAB converter behaves as a CPL at low frequencies and behaves as a capacitor at high frequencies in both symmetrical (i.e., same load power) and asymmetrical (i.e., different load power) power flow modes. Furthermore, the input power influences the input impedance at low frequencies because of the negative incremental impedance characteristic. The lower the input power, the higher the input impedance and vice versa.
To verify the input impedance of the TAB converter, a switching model is created in PLECS. The methodology is injecting sinusoidal ac current with different frequencies into input current while observing the variations in input voltage. In the following verification, two cases in Fig. 6 are chosen in which power flows symmetrically and asymmetrically: Case 1: As Figs. 7 and 8 show, the simulation measurements are consistent with the bode plot. This proves that the input impedance model is correct for both symmetrical and asymmetrical power flow modes.

B. Simulation Verification of QAB Converter
Since the QAB converter has four ports, there are totally 24 gains. They are given in Table V. According to (3.5), Matrix A can be written as (5.2), as shown at the bottom of the next page.
Again, following the equations from ((3.6) to (3.9), the input impedance bode diagram of the QAB converter is worked out and shown in Fig. 9, which is similar to Fig. 6. The input impedance of QAB converter also behaves as a CPL at low frequencies and behaves as a capacitor at high frequencies.
In addition, the input power influences the input impedance at low frequencies. The higher the input power, the lower the input impedance and vice versa. To verify the input impedance of the QAB converter, the methodology is same as the verification approach for the TAB converter. Two cases in Fig. 9

VI. EXPERIMENTAL RESULTS
In this section, the experimental setup and experimental results are given. As for the experimental setup, a QAB converter experimental prototype with a symmetrical structure is built and shown in Fig. 12. The experimental setup of the TAB converter can be realized by disconnecting an H-Bridge from the transformer. As for experimental results, the input impedance of the TAB converter and the QAB converter is measured and compared with the proposed input impedance model, and with the PLECS switching model in which the value of magnetizing inductance and winding resistance of transformer, ON-state resistance of MOSFET, and dead time of switching are added in accordance with the experimental setup.
Considering the specifications of the experimental setup, a different set of parameters is chosen for the experimental verification; also, the nonideal characteristics, which were not considered before, are measured from this setup directly. They are listed in Table III. During the experiments, both symmetrical power flow case and asymmetrical power flow case of the TAB converter and the QAB converter are carried out.

A. Experimental Verification of TAB Converter
To measure the input impedance of the TAB converter, Port 1 is connected to the power supply, Port 2 and Port 3 are connected to loads, and the H-Bridge of Port 4 is disconnected from the transformer.   Fig. 13.
2) Asymmetrical Power Flow Case: In asymmetrical power flow case of the TAB converter, Port 2 is connected to a 16.7-resistive load and Port 3 to a 10-resistive load. The simulation results and experimental results are shown in Fig. 14.

B. Experimental Verification of QAB Converter
As for the input impedance measurements of the QAB converter, Port 1 is connected to the power supply and Port 2, Port 3, and Port 4 are connected to resistive loads.

1) Symmetrical Power Flow Case:
In symmetrical power flow case of the QAB converter, Port 2, Port 3, and Port 4 are connected to the same 8.3-resistive load. The simulation results and experimental results are shown in Fig. 15.   of the PLECS switching model, it is noted that the mismatch between the proposed model and measurements can be caused by the nonideal characteristics of the converter. These nonideal characteristics can be regarded as damping, making the input impedance more resistive. As a result, the amplitude of input impedance is a bit lower and the phase is a bit higher in reality.

VII. CONCLUSION
This article has thoroughly investigated the impedance modeling for multiport power converters in symmetrical and asymmetrical power transfer conditions. The theoretical model is validated against switching simulations and experimental results, showing a good matching. It can be concluded that the proposed model can be used to represent the multiport system in the near actual condition. The multiport converter exhibits a CPL behavior at lower frequencies, whereas at higher frequencies, the input capacitor dominates the input impedance. Once the input impedance is known, further stability studies based on the impedance criterion can be carried out to evaluate the system behavior in a complex network configuration. These results can be also used for system design and optimization, allowing to find the minimum input capacitance value that guarantees the system stability.

APPENDIX
The small-signal gains of the TAB converter are given in Table IV.
The small-signal gains of the QAB converter are given in Table V.