Transfer Function Based Input Impedance Determination of Triple Active Bridge Converter

The concept of multiport dc-dc converter was proposed to reduce the conversion stages of dc microgrid on more electric aircraft (MEA). The structure of multiport dc-dc converter is basically developed from the dual active bridge (DAB) converter because of its galvanic isolation and bidirectional power flow. A power electronics converter as a key element of the electrical power distribution system may cause stability issues. To address these challenges, the impedance characteristic of the multiport converter will be analyzed. In this paper, a transfer function based small signal model is developed and validated with a switching model, to figure out the characteristic of input impedance of triple active bridge (TAB) converter. Preliminary experimental results are presented to be as a support.


I. INTRODUCTION
Nowadays, the electrification of transportation becomes an irresistible trend to cope with the lack of conventional energies and environmental deterioration. In the aspect of aerospace, the concept of all electric aircraft (AEA) which emphasizes using onboard electrical devices to power main engine and propulsion engine at startup, was considered by military aircraft engineers since World War II [1]. However, this thorough change of onboard devices will increase the unknown stability issues in aircraft while the gains cannot satisfy the conservative aerospace industry [2]. As a step-wise approach, the concept of more electric aircraft (MEA) which proposes to achieve electrification of subsystems on aircraft, is more preferred by aerospace industry and researchers. Compared to conventional aircraft, MEA has advantages of lower maintenance cost, lower weight and positive environmental impacts [3]. During the last several decades, significant progress has been achieved to move toward MEA. Many on-board devices which use hydraulic, mechanical and pneumatic power have been replaced by electrical devices [4].
As on-board electrical devices increase, the potential risks in power distribution are increasing as well. The aircraft can be regarded as an isolated system connected to generators and loads, and the distribution system can be regarded as an on-board microgrid [5]. To standardize different requirements according to the number of installed power electronics and the characteristics of the actuator, several different voltage standards containing both ac and dc levels for electrical system on large civilian aircraft are stated in [6]. Compared with ac microgrid, dc microgrid (±270 V dc bus and 28 V dc bus) is preferred by more researchers because of its lower current rating and fewer conversion stages, indicating higher efficiency [5], [7]. The structure of dc microgrid is shown in Fig. 1. A conventional dc microgrid consists of multiple dc-dc converters to achieve different dc voltage levels for individual use. Although this pattern allows the power to be transferred simultaneously, it has many conversion stages and communication devices between converters, resulting in complex structure and high cost. Based on this fact, multiport dc-dc converter was proposed in [8] to reduce the conversion stages. Moreover, droop control based hierarchical control proposed in [9] is deemed effective to cut down the communication devices in microgrid.
Considering the safety purpose, the dc-dc converter used for MEA must have the characteristic of galvanic isolation [10]. The most popular topology is dual active bridge (DAB) converter which is composed of two H-bridges and a high frequency (HF) transformer. With such a structure, bidirectional power flow can be easily achieved by means of control. Based on this topology, a triple active bridge (TAB) converter was proposed and verified in [11] and a quadruple active bridge (QAB) converter was proposed and verified in [12]. Although these multiport dc-dc converters are thought highly to replace general two-port dc-dc converters in the future, the stability of them needs to be assessed when they are interfaced to sources and loads in microgrid of MEA.
In this paper, the characteristic of the input impedance of TAB converter is investigated and validated. Section II presents the structure and operating principle of TAB converter. Section III shows the transfer function based small signal model of TAB converter. Section IV gives the specifications for TAB converter. Section V investigates and validates the characteristic of input impedance of TAB converter by using both Matlab and PLECS. Section VI shows preliminary experimental results. Section VII draws the conclusion.

II. TRIPLE ACTIVE BRIDGE CONVERTER
Generally, the TAB converter can be used to interface multiple buses or as a storage management system. In this paper, the TAB converter is supposed to connect dc bus and resistive loads to facilitate the simulation and validation of model. As Fig. 2 shows, Port 1 is connected to dc bus, Port 2 and Port 3 are connected to resistive loads. By controlling the phase shift among the switching signals of all three ports, the power flow among three ports can be enabled. Also, lead and lag of switching signals will decide the direction of power flow. The individual power transferred between any two ports is given as

A. The structure
Where Vi is the voltage of Port i, Vj is the voltage of Port j, nij is the turn ratio between Port i and Port j, Lij is the leakage inductance between Port i and Port j, dij is the phase shift ratio normalizing the phase shift angle to π. Hence, to obtain the real value of the leakage inductance in Equation (2.1), the proper transformer model needs to be figured out. Fig. 3 shows the transformer models of TAB converter, containing both star model and delta model. In Fig. 3(b) the star model, the inductance and voltage of Port 2 and Port 3 referred to Port 1 are calculated as:

2)
To be consistent with the inductance used in Equation (2.1), the delta model is adopted by calculating its leakage inductance. The approach was introduced in [11]. Assuming the magnetizing inductance Lm is super large that can be regarded as open circuit, the inductance on each branch in Fig. 3(c) can be derived as

B. Operating principle
The operating principle of TAB converter is similar to DAB converter. The switching signals and leakage inductance current are represented in Fig. 4. Considering the conservation of power, the port power P1, P2 and P3 designated in Since Port 2 and Port 3 are supposed to connect the resistive loads, P2 and P3 cannot be negative. This can be concluded to the individual phase shift that two conditions will never happen in nearly linear operating region: (a) d31 < d23 and (b) d21 < d32. When d31 = d23 or d21 = d32, it means resistive load at Port 2 or Port 3 is disconnected.

III. TRANSFER FUNCTION BASED SMALL SIGNAL MODEL
State-space averaging and circuit averaging are two conventional techniques to obtain the small signal model of converter [13]. However, it becomes difficult to use these two techniques to model the converter including more components, such as DAB converter. In this section, an alternative technique is proposed, which uses the small signal equations derived from power equation, to acquire the small signal model of input impedance of TAB converter. Assume Port 1 as the driving point port, according to Equation (2.1) and (2.5), the current of each port can be obtained as Then the small signal equations can be obtained by deriving Equation (3.1) -(3.3) partially to voltage and phase shift ratio. In the calculation of the small signal equations, d23 is supposed to be substituted by d21 -d31, since this substitution will simplify the equations and undermentioned transfer function block scheme largely. The small signal equations are derived and represented as Matrix (3.4) at the bottom side of this page. Based on this, a transfer function block scheme can be drawn as Fig. 5.
In Fig. 5, there are totally twelve gain blocks 1-12 corresponding to the gains in Matrix (3.4) multiplied with a firstorder transfer function which presents the time delay in both switches and controllers. The specific equations of these twelve blocks are depicted as where τ is one switching period of TAB converter.
By simplifying the block scheme, the input admittance of TAB converter can be derived as (3.6) The coefficients are as follows (3.7) where Ctrl2, Ctrl3, Load2 and Load3 are the transfer functions of controllers and loads in Fig. 5. By considering the input capacitance C1, the input impedance of TAB converter at Port 1 is calculated as (3.8)

IV. SPECIFICATIONS
As the structure of TAB converter has been already aforementioned in Section II and according to the requirement of MEA, the feasible parameters are listed in Table 1 By employing the parameters above, the structure of TAB converter is designed to be symmetrical and it will facilitate the calculation and observation of power flow inside the converter. Besides, since the converter will be unstable if the phase shift ratio goes out the nearly linear operating region of power (-0.5, 0.5), d21 and d31 are limited in (0.2, 0.2), which not only guarantees d21 and d31 are nearly linear with power, but also indirectly makes d23 settle in (-0.4, 0.4), which is still in the nearly linear operating region of power.

V. THE CHARACTERISTIC OF INPUT IMPEDANCE OF TAB CONVERTER
In this section, Matlab is used to plot the characteristic of input impedance of TAB converter and PLECS is used to validate the characteristic. In addition, both symmetrical and asymmetrical power flow mode of TAB converter are investigated to see the influence on the input impedance of TAB converter.

A. Matlab plot
The characteristic of input impedance with both symmetrical and asymmetrical power flow mode is presented in Fig. 6 below: As shown in Fig. 6, in both symmetrical and asymmetrical power flow modes, the input impedance of TAB converter behaves as constant power load (CPL) at low frequencies and behaves as capacitor at high frequencies, which is same as the characteristic of input impedance of DAB converter. Moreover, it is obvious that the port power P1 which is transferred to Port 2 and Port 3 will influence the input impedance at low frequencies.
To be more specific, the lower the P1, the higher the input impedance at low frequencies.

B. PLECS validation
A switching model and a transfer function model are established in PLECS to validate the bode diagram shown in Fig.  6. The methodology is to inject sinusoidal ac current with different frequencies at the input port and observe the variations in input voltage. To make the sinusoidal waveforms more visible, the source resistance Rs is increased to 1 Ω. Although this change will cause a drop of input voltage, it does not influence the input impedance of TAB converter. Since it will be a huge work if validating all cases shown by Fig. 6, two of them are chosen to validate the input impedance at 1 Hz and 100 Hz in which power flows symmetrically and asymmetrically: Case 1: d21 = d31 = 0.1 and Case 2: d21 = 0.1, d31 = 0.05.   Fig. 7, the input impedance in both symmetrical and asymmetrical power flow modes are tested. In symmetrical power flow mode, the resistance of two resistive loads at Port 2 and Port 3 are set to be same to ensure the same power flow. As for asymmetrical power flow mode, the resistive load at Port 3 is disconnected. The magnitude of input impedance at specific frequency can be calculated as By observing the waveforms of Fig. 7, the input impedance of (a) has magnitude of approximately 30.4 dB with -180° phase, the input impedance of (c) has magnitude of approximately 32 dB with -180° phase, the input impedance of (b) and (d) has lower magnitude of approximately 12.9 dB with -90° phase. Hence, all input impedance values calculated are almost consistent with the curves in Fig. 6. It proves that the transfer function based small signal model is correct.

VI. EXPERIMENTAL RESULTS
This section gives the experimental setup and preliminary experimental results. The experimental prototype of TAB converter is shown in Fig. 8, which is actually a QAB converter with one port disconnected.
During the experiment, both symmetrical and asymmetrical cases are implemented. As for experimental specifications, the input voltage and output voltage are chosen to be 60 V, the switching frequency is chosen to be 10 kHz, the control parameter Ki is chosen to be 10, other parameters are kept being same as simulation. In symmetrical case, the resistance of both loads is set to be 44 Ω to achieve the same power flow from source port to load ports. In asymmetrical case, one load is disconnected. By injecting 5 V ac voltage with different frequencies at source port, the input current will fluctuate with different amplitude. The preliminary experimental measurements of input impedance are presented in Table 2.  The captures of key waveforms at 2 Hz and 100 Hz in symmetrical case and asymmetrical case are shown below: In Fig. 9, Channel 1 is the input voltage, Channel 4 is the input current, Channel 2 and Channel 3 are the output voltage.
By observing measured data and captured waveforms, it can be concluded that the input impedance is almost complying with the curve in Fig. 6. Furthermore, the amplitude of input impedance in the range from 2 Hz to 10 Hz in asymmetrical case is much larger than it in symmetrical case.

VII. CONCLUSION
A transfer function based small signal model is proposed to determine the input impedance of TAB converter with both symmetrical and asymmetrical power flow modes in this paper. The delta model of transformer is adopted to calculate the real leakage inductance between any two ports in TAB converter. By considering the equations of both individual power and total power, a set of small signal equations can be obtained to link the plants and controllers, hence a block scheme of small signal model is established. By using Matlab, the bode diagram of input impedance of TAB converter is plotted, which shows that the input impedance behaves as CPL at low frequencies and behaves as capacitor at high frequencies. Moreover, the output power from source will influence the input impedance, the lower the output power of source, the higher the magnitude of input impedance at low frequencies. To validate the bode diagram of input impedance, both simulation and preliminary experiments are carried out. As for simulation, a transfer function model is constructed to compare with a switching model in PLECS. By injecting sinusoidal ac current with different frequencies at input port, the input voltage of both models fluctuates equally. Similarily, in experiment the input