Controlling Transformer Magnetizing Offset Current in Isolated Phase-Shift Full-Bridge Converters Using a Luenberger Observer

%is paper proposes a flexible digital control scheme for isolated phase-shift full-bridge (PSFB) converters. %e required transformer suffers from inevitable imbalance of magnetic flux resulting in an increased magnetizing DC-offset current that threatens system reliability due to saturation effects.%e paper addresses two major issues of the occurrence of a magnetizing DCoffset current. First, caused by the change of duty cycle due to output power regulation and second caused by initial manufacturer tolerances of devices. In contrast to common methods the novel control scheme uses a Luenberger observer to estimate the magnetizing current requiring only simple measurement of transformer voltages without additional and lossy auxiliary networks. %e observer model, in combination with a PI-controller, directly interventions the duty cycle and removes any DC-offset current resulting from both issues. A detailed deviation of the state-space model of the transformer and a subsequently design of the observer are presented. Simulation and experimental results on a PSFB prototype verify the principal functionality of the proposed control scheme to prevent transformer saturation.


Introduction
Regarding the worldwide increasing climate problematic the portion of clean and renewable energy generation has increased over three-fold since the last ten years [1]. Unfortunately, renewable energies like photovoltaic (PV) or wind power reduce the reliability of power generation with its dependency on weather conditions [2,3]. To utilize the maximum capacity of fluctuating renewable energies different storage technologies have been developed. Although lithium-ion battery systems are not the only solution to store energy, this technology has prevailed in various mobile DC-DC applications and also becomes more and more important for the high and medium power DC-DC charging sector. Typically, high power DC-DC systems use a liquid cooling system due to large heat development while charging batteries, but especially non-industrial DC-DC applications with medium power have to manage cooling without liquid cooling system and therefore a high efficiency is very important [4]. e high efficiency and power density of an isolated phase-shift full-bridge (PSFB) converter in Figure 1 make this topology very attractive for DC-DC medium and high power applications such as DC-storage battery systems for PV or battery chargers for electric vehicles [5][6][7]. Its phaseshift control scheme allows for zero-voltage-switching (ZVS) operation with negligible switching losses and reaches higher efficiency compared to conventional hard-switched topologies [8].
Two full-bridges on primary H A and secondary H B side make the topology suitable for bidirectional operation. e transformer TR, with ratio r � N p /N s between primary N p and secondary N s winding, compensates a high voltage difference between input voltage V A and output voltage V B � V A /r d eff to avoid low duty cycles d eff with decreased efficiency. Additionally, its naturally galvanic isolation satisfies the normative requirements for e.g. battery chargers in electric vehicles for safety reasons [9]. Nevertheless, a transformer isolated converter suffers from issues such as an imbalance of magnetic flux resulting in an increasing DCoffset current I h,DC that drives the transformer into saturation [10]. To prevent damage due to high saturation currents and guarantee system reliability I h,DC must be eliminated.
Common topologies connect a blocking capacitor in series to the transformer in the main power path to suppress any DC-offset I h,DC . However, placed in series with the transformer the blocking capacitor must be dimensioned for the maximum current of the converter at full power and additionally increases conduction losses due to its series resistance. As consequence the blocking capacitor is not preferred for applications with high power density and efficiency [11]. Considering bidirectional operation, a blocking capacitor must be placed in series to primary and secondary side of transformer. is method needs several additional power devices compared to controller-based methods and therefore results in increased size of power stage and overall system costs [12,13].
Without additional passive devices in the main power path a controller needs to eliminate I h,DC [14]. Designing of a stable feedback control loop requires the measurement of I h,DC as process variable. However, considering the transformer as a four terminal device the acquisition of I h,DC is not directly possible as it overlays the transformer primary current I p [15]. e current-mode control (CMC) is a very popular analog controller-based method to prevent the transformer from saturation. Its fast response due to analog comparators allows for stable operating conditions even at high switching frequencies f sw . Unfortunately, in certain operating conditions instabilities, known as sub-harmonic oscillations, occur [16,17]. e slope compensation technique normally is used to compensate the oscillation but overcompensation may result in a response delay [18,19]. Additionally, the CMC method is not suitable for flexible digital control schemes as it needs analog comparators [20].
Numerous research works concern the adjustment of PWM signals of power switches to balance the transformer magnetizing current in isolated DC-DC converters. However, the method of gathering the magnetizing current as process variable for a controller differs. [12] extends currentmode control with a hybrid peak and valley current control and does not need for slope compensation. e valley current detection requires AC-current sensing on the primary side of transformer with a suitable bandwidth for high switching frequencies. Considering bidirectional operation, the study in [21] uses primary and secondary transformer current to estimate magnetizing current from the previous switching period. A similar method in [22] only uses the averaged primary current to estimate the DC-offset from the previous period assuming the inductor current to be in steady state and average secondary current to be zero. [23] addresses the problem of magnetizing DC-offset causing DC-offsets in primary and secondary current. Here, two control loops need to scene also both, primary and secondary current and calculate the magnetizing current accordingly. In [24] a sensor directly measures the magnetic flux in the transformer core. e sensor requires modification of transformer core to mount the sensing coil through hole. An extended Kalman-Filter-approach to estimate the primary transformer current is presented in [25]. However, the study only concerns the discrimination of the transformer inrush current from internal faults to trigger a protective relay.
is article presents a novel digital feedback control loop scheme for eliminating DC-offset I h,DC without additional devices in the main power path. In contrast to related work a Luenberger observer is used that only requires the measurements of the transformer voltages to estimate the magnetizing current as process variable for a PI-controller. Main purpose of the study is to develop a digital controller that allows for simple application or flexible modification of existing controller structures only by adding the voltage measurement feature and to improve system reliability. At Transformer TR partially compensates high voltage difference between V A and V B and provides galvanic isolation.
2 International Transactions on Electrical Energy Systems the beginning Section 2 explains the inevitable occurrence of DC-offset I h,DC in isolated PSFB converters in detail. In Section 3 conventional methods, like a blocking capacitor or CMC, used to overcome this problem are introduced. Section 4 explains the structure of the proposed digital control scheme. Section 5 describes an approach of acquiring the not measurable DC-offset I h,DC using a Luenberger observer with a detailed derivation of the state space model of the transformer and the subsequently design of the observer. In Section 6 the principal functionality of the proposed control scheme is evaluated by simulation and experimental results with an isolated PSFB converter prototype operating with a power P � 1kW at a switching frequency f sw � 100kHz. Finally, Section 7 gives a conclusion of the presented results.

Transformer DC-Offset Current
e magnetizing current I h creates the magnetic field of the transformer required for power transfer from primary to secondary side and flows through the magnetizing inductor L h of the transformer. I h adds either to primary I p or secondary current I s of transformer depending on power transfer direction. e transformer is not capable to transfer any DC component of I h [26]. erefore, the magnetizing current I h needs to be detected on primary and secondary side when operating in bidirectional direction. e following considerations refer exemplary to a power transfer direction from primary side A with I p to secondary side B with I s � (I p − I h )r. Figure 2 shows the typical schematic characteristics of applied voltage V p in Figure 2(a), primary current I p in Figure 2(b) and magnetization current I h in Figure 2(c) for transformer of the PSFB converter shown in Figure 1. e power transfer cycles PC 1 and PC 2 are determined according to the effective duty cycle d eff � 2t on /T applied to the transformer TR. For desired duty cycle d eff the phaseshift between lagging S 1 + S 3 and leading leg S 2 + S 4 of H A varies and applies either the positive or the negative input voltage V A to the magnetizing inductor L h of the transformer TR according to Figure 2(a). For positive voltages V p > 0 the magnetizing current I h increases and for negative voltages V p < 0, I h decreases respectively. During the interval t off no voltage is applied to L h resulting in a constant I h . e alternating positive and negative voltage-second products applied to the transformer within one period T(n) International Transactions on Electrical Energy Systems 3 determines the peak value I h,peak,1 and I h,peak,2 , assuming to be constant during T(n), in Figure 2(c) according to: erefore, the DC-offset I h,DC (n) can approximately be defined as the average value of magnetizing current I h (n) during one period T(n) with: (2) Figure 2(b) explains the sloped magnetizing current I h as an additional comparable small portion to the high main primary current I p,n with I h � I p − I p,n [15]. As consequence I h cannot be measured directly considering the transformer as a four terminal device. Assuming negligible variations of I p,n and d eff during one switching period T(n) the DC-offset I h,DC (n) can be calculated from the difference between the peak values of I p during one complete period T(n) according to: For a present asymmetrical voltage-second product V 1 ≠ V 2 , the absolute value of I h,DC (n) increases each period T(n) with I h,DC (n + 1) � I h,DC (n) + ΔI h,DC (4) and the resulting DC-offset I h,DC ≠ 0 drives the transformer into saturation. Ideally, the symmetrical structure of the PSFB topology ensures no DC-offset I h,DC as positive and negative current slope of I h balance each other. Unfortunately, various inevitable effects can cause a small asymmetrical voltage-second product on primary side of transformer [14]. One transient reason is the unbalanced duty cycle between power cycle PC 1 for the lagging leg and PC 2 for the leading leg due to ZVS-operation or the change of d eff for regulating the output current I B of the PSFB [12]. Even in steady state operation manufacturing tolerances of devices as well as circuit board layout (PCB) cause different voltage drops and apply different voltages to the magnetizing inductor L h and increase I h,DC [10]. Beside of the transformer the resulting high saturation current threatens the switching devices H A and H B and general system reliability. Additionally, the conduction loss unnecessarily increases with an unbalanced magnetizing current I h as it does not contribute to power transfer from primary to secondary side.
erefore, system efficiency η is directly related to minimize I h,DC .

Conventional Methods
Several methods have been proposed either to suppress or to control DC-offset I h,DC in [11,12,16,17,27,28].

Blocking Capacitor.
Connecting a blocking capacitor in series to the transformer suppresses any DC component in the transformer current I p and allows for symmetrical operation [11]. erefore, this method is suitable for acquiring only secondary side output current and apply proportional integral current control [27]. Additionally, in phase-shift control scheme the blocking capacitor method reduces the circulating currents during the free-wheel interval of the lagging leg and minimizes conduction loss. However, the timing of the leading leg becomes more complicated [12]. On the other hand, the blocking capacitor method simultaneously increases the conduction loss as the capacitor is located in the main power path with maximum current of the PSFB [28]. For bidirectional operation the magnetizing current I h occurs on either side, primary and secondary, according to power direction. erefore, this method needs two blocking capacitors and significantly increases volume and overall costs.

Current-Mode Control.
e current-mode control (CMC) method is well known for non-isolated converters such as buck or boost converter topologies [16][17][18]. For nonisolated converters the inductor current I L serves as process variable for the control loop. Fast response analog comparators generate PWM signals and directly trigger the power switches when inductor current I L reaches a given threshold value and provide output control. Reference [18] distinguishes CMC between peak detect, valley detect and emulated control mode. However, either method suffers from sub-harmonic oscillation when inductor current I L does not return to its initial value by the start of next switching cycle in continuous current-mode (CCM). In peak detect CMC the sub-harmonic oscillations occur at duty cycles d eff > 0.5 and for valley detect at duty cycles d eff < 0.5 respectively. A slope compensation circuit needs to correct the ripple current ΔI L and results in a response delay for the controller [19]. In isolated converters the CMC method can additionally provide the elimination of DC-offset I h,DC with small modifications. Any DC component of I h cannot pass the transformer. erefore, the input current I A must serve as process variable for I h,DC control with power direction from A to B side. Figure 3 explains the compensation principle of CMC with the schematic characteristic curves of primary voltage V p in Figure 3(a), input current I A in Figure 3(b) and magnetizing current I h in Figure 3(c) of transformer with a present flux imbalance. Neglecting parasitic inductors, the magnetizing current I h slopes up with m 1 � V 1 /L h during power cycle PC 1 and with m 2 � V 2 /L h during power cycle PC 2 respectively. Parasitic effects reduce the applied voltage V 1 for power cycle PC 1 in Figure 3(a). Assuming a symmetrical duty cycle d 1 � d 2 � d eff for both power cycles the peak of input current I A,peak,1 does not reach the threshold value I th of the analog comparator and the negative DCoffset I h,DC < 0 rises [12]. e current-mode controller directly influences the duty cycle d 1,CMC � 2t on,CMC /T and extends t on,CMC > t on until I A,peak,1 reaches the threshold current I th . erefore, the CMC ensures flux balance of the transformer with I h,DC � 0 and provides a symmetrical voltage-second product V 1 t on,CMC � V 2 t on [12,18].
Common control schemes prefer an analog controller with comparators compared to digital controller as the erratic increasing of I A needs high bandwidth to detect the peak current I A,peak exactly. In consequence, the CMC is hard to combine with a flexible digital control scheme [20]. An additional challenge for the measured process variable I A is the accuracy to properly detect the small portion of I h as Figure 2(b) explains I h as a comparable small additional sloped offset to the high input current I A or rather I p,n .

Proposed Control Scheme ctrIh
e proposed control scheme ctr Ih uses flexible digital design. It requires neither lossy and expensive additional passive devices in the main power path of the PSFB nor analog comparator for triggering power switches. Similar to CMC, the proposed digital control scheme ctr Ih directly interventions the duty cycle d 2 of the leading leg to equalize the peak values of primary current |I p,peak,1 | � |I p,peak,2 | and eliminates any magnetizing DC-offset I h,DC within a few switching cycles. Figure 4 depicts the control loop of the PSFB converter with ctr IL for output current regulation and ctr Ih for DCoffset I h,DC control. e output controller ctr IL is built up with a PI-controller PI IL (z) and feedback control for the output voltage V B ′ � V B + ΔV B for better adjustment of output operating point. e transfer function of the PSFB G duty determines the effective duty cycle d eff � d 1 � V B ′ /V A r for desired output current I B,set . e proposed magnetization controller ctr Ih adds or subtracts a small offset Δ d between duty cycle d 1 for the lagging and duty cycle d 2 � d 1 + Δ d for the leading leg until I h,DC � 0. As I p,n and d 1 vary only a little during one switching period and also the impact of parasitic effects contribute only to a small amount, the ctr Ih limits the magnitude of Δ d to a few percentage of d 1 . erefore, the influence of ctr Ih on the output controller ctr IL is negligible [14]. According to d 1 and d 2 the phase-shift calculator provides eight separate PWM signals for power switches S 1 -S 8 of PSFB power electronics. In the feedback path the main control unit (MCU) performs the acquisition of process variables V A , V B , I L and I h with analog digital converters (ADC). For output control the rippled inductor current I L averages to the desired output current I B .
erefore, the moving average finite impulse response (FIR) International Transactions on Electrical Energy Systems 5 filter with G IL (z) averages I L over N number of periods with: Accordingly, the DC-offset I h,DC is defined as the average of rippled magnetization current I h over several periods.
Again, a moving average FIR filter with G Ih (z) provides the process variable I h,DC of ctr Ih . As V A and V B are already DC values there is no need for averaging with a FIR filter.
For proper design of ctr Ih the relation G Ih,DC (z) � I h,DC (z)/Δ d(z) must be obtained. As mentioned before several effects, such as an unbalanced duty cycle either due to ZVS operation or output regulation as well as parasitic effects influence the occurrence of Figure 4: Control loop scheme of PSFB for I h,DC elimination. Output controller ctr IL with PI IL (z) and G duty determines duty cycle d 1 for desired I B,set . e ctr Ih adds a small offset Δ d to d 1 of lagging leg until I h,DC � 0. e phase-shift calculator translates the duty cycle information into eight phase shifted PWM signals for bidirectional operation of PSFB power electronics. Figure 5: Influence of Δ d on ΔI h,DC,duty over two periods T(n) and T(n + 1). According to phase shifted PWM signals for S 1 -S 4 in (a) the voltage V p applies to transformer shown in (b). In (c) the different duty cycles d 1 (n + 1) > d 1 (n) cause a small difference ΔI h,DC,duty between I h,DC (n + 1) and I h,DC (n).

6
International Transactions on Electrical Energy Systems unbalanced flux leading to I h,DC ≠ 0. erefore, these effects have to be considered in the transfer function G Ih,DC (z). Figure 5 illustrates the influence of Δ d on the DC-offset ΔI h,DC ,duty for analyzing the impact of unbalanced duty cycle due to output regulation ctr IL over two consecutive switching periods T(n) and T(n + 1). According to phase shifted PWM signals for power switches S 1 -S 4 in Figure 5(a) the voltage V p in Figure 5(b) applies to the transformer. Figure 5(c) shows the time course of magnetizing current I h resulting from unbalanced duty cycles d 1 (n) < d 1 (n + 1). e PWM signals of power switches S 1 -S 4 obtain the duty cycle difference Δ d to: Assuming equal voltages V 1 � V 2 � V A for power cycles PC 1 (n) and PC 2 (n) the magnetizing current I h slopes with the same gradient In period T(n + 1), the output regulator ctr IL increases d 1 (n + 1) � d 1 (n) + Δ d. Due to the increased duty cycle d 1 (n + 1) the magnetizing current I h surpasses the peak value I h,peak (n) before the end of power cycle PC 1 (n + 1), thus, leading to an increased DC-offset in the following period T(n + 1) according to: is behavior expresses also for small differences between duty cycle d 1 (n) and d 2 (n) in the same period T(n).
Furthermore, I h,DC (n) defines as the average of magnetizing current I h (n) with a linear slope V A /L h . erefore, ΔI h,DC,duty between two consecutive switching periods due to unbalanced duty cycle approximately calculates as follows: As long as there is a duty cycle imbalance Δ d present, the magnetizing current I h,DC needs to diverge until the PSFB gets damaged if not compensated.
Another reason causing I h,DC are different parasitic resistances such as switch resistances or general PCB-design resulting in unequal voltages V 1 ≠ V 2 . [14] derives the influence of parasitic elements ΔR and can approximately be calculated as follows: As the impact of resistance is comparable low during transition times, (9) only considers the dominant time intervals of free-wheeling and power cycle. Regarding (9) parasitic resistances ΔR contribute proportionally with K res to magnitude of I h,DC . erefore, the impact of ΔI h,DC,res represents a steady-state fault of ctr Ih [14].
During the transition between power PC x and freewheeling FC x cycles mainly the switching characteristics of the power switches and dead time control influence the DC-offset I h,DC . e dead time control determines different duty cycle d 1 for the lagging and d 2 for the leading leg according to ZVS-conditions. In [14], the detailed derivation results in: again, with propoportional gain K trans . For a unified transfer function G Ih,DC (z) the three main effects influencing I h,DC must be considered according to [14] with:  [14].
Equation (12) calculates the transfer function of Δ d and I h,DC in z domain as follows: According to (12) the magnetization controller ctr Ih requires an integrator for steady state stability. For the experimental results the ctr Ih is implemented as PI-controller PI Ih (z).

Derivation of Observer Model
In the proposed digital control scheme of ctr Ih , I h,DC � I h represents the process variable. Considering TR as a four terminal device with I p and V p as inputs and I s and V s as outputs, I h,DC cannot be measured at clamps directly. erefore, I h,DC needs to be calculated from a related value depending on physically measurable in-and outputs.
In technical systems, a state observer is able to estimate an internal state value, such as I h,DC from measurable in-and outputs of the real system [29,30]. Figure 6 simplifies [30]. For the design of a Luenberger observer, in Section 5.1 first a stable linear state space model of the real transformer must be developed. Although the observer allows for controlling a time continuous system, a discretization is necessary for realistic applications with a digital control as described in Section 5.2. Section 5.3 examines the observability criteria according to Kalman of derived transformer model and Section 5.4 explains the design of the Luenberger gain parameters for observer model.

Transformer State Space Model.
e Luenberger observer requires a linear state space model of transformer in form of: In (14a) and (14b) x represents the state variables with input current i p , output current i s and magnetization current i h . e primary voltage V p represents the system input u and the secondary voltage V s serves as measurable system output y according to: y � V s .
(15c) Figure 7 shows the equivalent circuit diagram (ECD) of a real transformer. Parasitic resistances R p and R s,u and leakage inductors L p and L s,u depict the loss component of primary and secondary windings while R Fe simulates the magnetic resistance of the transformer core. e core magnetization is regarded with magnetizing inductor L h .
Nonlinear behavior of magnetizing inductor L h is neglected as the controller ctr Ih prevents saturation of transformer. e ratio r of primary winding N p to secondary winding N s represents the ideal transformer TR with: In this model the parasitic capacitors are neglected as their time constants are very small compared to switching frequency and time constant of inductors. Z 0 represents the impedance of the load for gathering secondary current i s . e reference-arrow system in Figure 7 defines the sign of variables for further derivations. Secondary side values with indexes s, u are referred to primary side with respect to transformer ratio r according to: e Kirchhoff current (KCL) and voltage (KVL) laws obtain the main equations of the transformer model according to: Reshaping (18a)-(18d) delivers the time continuous state space vector _ x to: Figure 7: Equivalent circuit diagram of a real transformer TR regarding parasitic elements for loss and magnetization effects. Values with indexes s, u are referred to primary side with transformer ratio r.

International Transactions on Electrical Energy Systems
Referred to (14a) the system A and input B matrices can be obtained to: e terms of voltage drop (L p /r)i . p and (L s,u /r 2 )i . s,u in (18c) depend on the dynamic change of input and output current due to leakage inductors. However, the small time constant of transient process of leakage inductors in μ H-range is not detected due to FIR filtering of signal with G Ih . Neglecting the dynamic terms in (18c) provides the equation for system output V s to: with output matrix C and direct feed-through matrix D from (14b) according to:

Analysis and Discretization of the Transformer
Model. e transformer model includes three state variables i p , i s and i h building a third order system with rank(A) � 3.
From the derived transformer model in the previous Section 5.1 the eigenvalues or poles of the system matrix A describe the dynamic behavior of the system. e values in Table 1 refer to the measured values from PSFB prototype or from the corresponding data from manufacturer. Analyzing the system with transformer parameter according to Table 1 delivers the poles for all three state variables i p , i s and i h shown in Table 2. As all poles are in the left-hand plane (LHP), the system provides stability [31]. e pole of i p according to Table 2 is too fast for realistic sampling time t s � (T/5) � 2μs required for discretization of observer model. As the desired value of the observer only includes the state variable i h , the state variable i p with the fastest pole can be ignored. e reduction of system order eliminates the first state variable i p and reconfigures the matrices A red , B red and C red with rank(A red ) � 2 and poles according to Table 3.
Again, all poles of the system with reduced order are located in LHP with negative values. erefore, also the reduced system is stable.
For realistic application of the observer, the reduced system matrices must be available in a discrete time form with sampling time t s . e transfer of the reduced transformer model into discrete form affects only the system matrix A red and the input matrix B red [32]. One approach of discretization presented in [20] according to: delivers the discrete system matrix A d,red without approximation. e MATLAB order of matrix exponential function exp m(A red t s ) can directly perform the discretization and is suitable for the experimental approach. With reduced discrete system matrix A d,red the reduced discrete input matrix B d,red calculates as follows: e reduced discrete output matrix C d,red � C red and reduced discrete direct feed-through matrix D d,red � D red are not affected and remain the same. e analysis of the reduced discrete transformer model in Z-domain delivers the poles of A d,red according to Table 4 with one stable pole at |z| < 1 and one critically stable pole at |z| � 1. erefore, a suitable design of the observer gain factors must guarantee the stability of the system.

Observability of the Transformer Model.
To use the reduced discrete transformer model and realizing a Luenberger observer the reduced system must be observable. For a completely observable system the initial state x(t � 0) � x 0 must be reconstructible form known input u and output value y within a finite time interval [31]. According to Kalman criteria of observability, the observability is proved if the observability matrix S B d,red in (25) According to (25) the rank of observability matrix rank(S B d,red ) � rank(A d,red ) � 2 equals the rank of reduced discrete system matrix and so the system is completely observable.

Design of Luenberger Observer.
In the structure of a Luenberger observer the derived transformer model works in parallel to real transformer. e observer compares the measured output y with the calculated value of the transformer model y and feeds back the estimation error Δy error � L(y − y) with feedback matrix L to the model [30].
From observer model the estimated state vector _ x calculates as follows: Suitable values of feedback matrix L adjust the dynamical behavior of the model to match the real transformer and react to disturbances or transient variations of duty cycle d 1 .
e design of feedback matrix L with pole placement according to Ackermann [31,33] is used and the known poles of A d,red from Table 4 deliver the first approach according to:

Results
e proposed digital control scheme regulates the magnetizing current I h of a PSFB to prevent transformer saturation. First approaches concern the simulation of electronics to prove principal functionality of ctr Ih , supported through experimental tests on a PSFB prototype with P � 1kW. Table 5 lists the specifications of analyzed operating point of PSFB prototype electronics, also used for simulation.
To validate the simulated results in Section 6.1 the following Sections 6.2 and 6.3 explain the experimental measuring setup up and show the achieved results in operating point according to Table 5.

Simulation of Observer.
For the evaluation of proposed ctr Ih in Section 4 and derived observer model in Section 5,    Table 1 and operating point listed in Table 5. e control scheme for ctr Ih and observer model are implemented in MATLAB/SIMULINK. e simulation results in Figure 8 demonstrate the behavior of the PSFB magnetizing current I h at operating point in Table 5 for the different disturbances mentioned in Section 2, dynamic change of output current I B through duty cycle d 1 and initial static differences due to manufacturing process of devices. Fluctuations of d 2 due to ctr Ih thereby influence the magnetization current I h in a similar way as d 1 .
e simulation starts at t 0 with a desired output current I B � 9.3A and a corresponding duty cycle d 1 (t 0 ) � 0.76. e series resistance R S2 � R S1 + ΔR � 0.2Ω of switch S 2 is doubled compared to other switches with R S1,3,4 � 0.1Ω, simulating an initial voltage-second product imbalance. Figure 8(a) shows the simulated average values for magnetizing current I h and estimated observer current I h,obs . Without ctr Ih , the magnetizing current I h,0 steadily raises due to the unbalanced voltage-second product till saturation of transformer. In Figure 8(a) the observer current I h,obs reflects the dynamic behavior of simulated current I h with the model error ΔI h,error � I h − I h,obs in Figure 8(b) fluctuating around zero with a DC-offset ΔI h,error,DC � 11mA. e influence of ctr Ih on controlled variable Δ d � |d 1 − d 2 | ≈ 0.07 is shown in Figure 8(c). e increased resistance R S2 forces ctr Ih to increase d 2 > d 1 and compensates the voltage-second product imbalance. After initial transition process the ctr Ih is able to correct the error resulting from static difference due to R S2 within t 1 ≈ 7.7ms. Either change due to d 1 step at t 3 or d 2 control at t 2 and t 4 causes fluctuations in I h . e estimated observer current I h,obs follows the dynamics of I h with a small delay Δt delay � 0.5ms mainly resulting from computation and FIR filter. According to I h,obs the ctr Ih adjusts d 2 for regulating I h,DC � 0A. Figure 9 explains the dynamic changes exemplary on d 1 in detail. At simulation time t 3 � 50ms the desired output current raises to I B � 14.1A resulting in a dynamic change of duty cycle d 1 (t 3 ) � 0.8 . For t < t 3 the magnetizing current I h,0 approximately slopes up with (d/dt)I h � 3.2As − 1 . For time t 3 ≤ t < t 3 + 50μs the magnetizing current I h,0 steps up according to equation (8) and (9), raising I h,DC . e step expresses as slope with (d/dt)I h,0 � 4.7mA(50μs − 1 ) due to FIR filter G Ih in MCU. With increased magnetizing current I h,0 the rising slope of magnetizing current (d/dt)I h � 6.6As − 1 due to R S2 enlarges after t 3 + 50μs encouraging saturation of transformer. e controlled estimated observer current I h,obs matches the dynamics of magnetizing current I h and the change can be controlled out by ctr Ih with I h,DC ≈ 0A. According to applied to voltage V p to L h and moving average filter, the magnitude value results to I h,pp ≈ I h,obs,pp � 0.2A.
For analyzing the resistance of ctr Ih , I h,obs,fail exemplary represents the observer current with a model failure in magnetizing inductance L h,fail � 1.1L h . e model failure produces a time delay Δt fail � 1ms for I h,fail as the bigger magnetizing inductance reduces the slope of magnetizing current and enlarges the initial time t 1 for regulating the initial differences. Nevertheless, the ctr Ih can handle the change in d 1 as well as in d 2 . A similar result is achieved with an examined model failure in primary resistance R p and load impedance Z 0 .

Measurement Application.
In the proposed control scheme the ctr Ih only needs the average values of voltage V p and V s as measuring inputs to control the magnetization DC-offset I h,DC . Typically, voltage measurements only need a network of low-cost resistors and operational amplifiers with naturally high bandwidth and low self-consumption. erefore, voltage measurements reach higher efficiency than current measurements with high loss due to shunt resistors or hall sensors with limited bandwidth. Figure 10 shows the measuring application for acquiring the input V p and output V s for the observer model. A simple voltage divider sets the transformer voltages to required 0 . . . 5V level of the main control unit MCU with V p′ and V s′ . Depending on the switching state of full-bridge H A the potential of the transformer voltage V p is floating. As in single ended mode the MCU refers its input signals to GND A , an operational amplifier OP p' performs the differential measurement V p′ � V p′1 − V p′2 and provides stable and resilient signals for further processing in MCU.  Additionally, the differential measurement does not require a floating supply voltage V CC for amplifiers. To maintain isolation of the PSFB the voltage measurement on secondary side V s′ needs to be decoupled with an isolated amplifier OP s' after differential measurement. e observer model only needs the average of voltage signals V p and V s . erefore, the low pass filters LP p' and LP s' smooth the signals V p′ and V s′ and minimize the measuring noise due to rising edges with high dV/dt. In the MCU, the digital FIR filters G Vp′ (z) and G Vs′ (z) calculate the moving average over 1000 samples for averaging the signals. According to the observer model the MCU estimates I h,DC for ctr Ih . Figure 11 shows the PSFB prototype used for the validation of the proposed control scheme. e modular structure splits the PSFB into 5 separate parts, full-bridges H A and H B in the front, transformer TR and filter inductor L in the back and a third circuit board equipped with the measuring devices and controller on the second floor to the right. erefore, separate components can be substituted easily and allows for testing different power devices and combinations of the PSFB for optimization.

Experimental Results.
In the experimental test set up a National Instrument (NI) Data Acquisition (DAQ) system acquires the values from the measuring circuit board with 500MSs − 1 and performs the calculations of the MCU. erefore, the NI system includes the observer model, ctr Ih control scheme with PI-controller and phase-shift calculator. e resulting eight PWM signals are generated on the counter outputs of NI system and directly connected to the drivers for fullbridge switches S 1 -S 8 .
For validating the observer model and simulation the effects of initial resistance differences and variable operating points are observed. Due to layout and manufacturing tolerances of devices, initial differences are not avoidable and LPs' ctr Ih V s Figure 10: Measuring principle for in-and outputs of observer model. Voltage divider sets V p and V s to MCU voltage level. OP p' and OP s' perform differential measurement for V p′ and V s′ smoothened by LP p' and LP s' . MCU averages values with G Vp′ (z) and G Vs′ (z) and estimates I h,DC according to observer model for ctr Ih .  present in prototype electronics. As the effect of resistance ΔR in real prototype is much smaller than in simulation, the rising slope of magnetizing current is smaller in experiment and allows for reacting before the prototype gets damaged. First, Figure 12 compares the measured primary currents of the PSFB prototype in a steady state operating point according to Table 5. Figure 12(a) shows the primary current I p,0 without any control of magnetization current while Figure 12(b) illustrates I p,obs controlled by ctr Ih with observer model. Figure 12(c) explains the according courses of I p,0 and I p,obs at higher temporal resolution. e initial differences cause an unbalanced voltagesecond product and lead to an asymmetrical course of I p,0 with a negative average offset value of I p,0 � − 848mA. Without compensation this imbalance drives the transformer into saturation for continuous operating time. e ctr Ih detects the initial imbalance and is able to regulate the average offset to a constant value of I p,obs � − 42mA. us, the ctr Ih removes the offset difference ΔI � I p,0 − I p,obs � − 806mA and improves the system reliability and efficiency. Figure 13 illustrates the results of the PSFB prototype controlled with ctr Ih switching between different operating points. erefore, during the experiment duty cycle d 1 varies each time t 0 − t 5 resulting in changed output current I B shown in Figure 13(d).
In Figure 13 (13). Although the magnitude of observer current I h,obs is scaled due to feedback matrix L, the dynamical behavior reflects the magnetizing current I h and keeps the model error ΔI h,error � I h − I h,obs in Figure 13(b) around zero. Equal to theory and simulation in Figure 13(c) the ctr Ih adjusts d 2 with a constant offset Δ d � 0.1 due to initial resistance differences and controls variations of d 1 . e exact measurement of the initial differences in the full-bridges of the prototype is very difficult and also vary due to temperature effects. erefore, there is a difference between simulated duty offset Δ d � 0.07 and prototype offset. However, the difference does not affect the functionality of the ctr Ih . e magnetizing current in Figure 13(a) oscillates around its DC-offset I h,DC � − 30mA while the ctr Ih shows a better performance on I h,obs,DC � 1mA. is difference results from measuring voltage offset due to operational amplifier and is the main drawback of the control scheme. erefore, the measuring offset must be minimized by using offset compensated operational amplifiers. Compared to simulation the magnitude of magnetizing current is reduced. In real prototype there are also parasitic resistances in the PCB-Layout and in the supply lines from the source. As consequence the given input voltage V A drops till the transformer with V p < V A and decreases the slope of magnetizing current I h . e magnitude of observer I h,obs and magnetizing current I h vary each time t 0 − t 5 the operating point changes, as the controlling parameter for PI-controller PI IL (z) depend on I B and were not modified during the experiment. Nevertheless, the effect on average values of I h or observer current I h,obs is negligible and saturation can be prevented. erefore, the experimental results confirm the results of simulation and prove the functionality of ctr Ih for observed disturbances.

Conclusion
is paper presents a digital control scheme ctr Ih for preventing saturation of the transformer due to flux In (c) ctr Ih adjusts d 2 due to variation of d 1 and initial resistance differences. Changes of output current I B due to d 1 are shown in (d).
imbalances in isolated PSFB converters. As the magnetizing current I h of a transformer, which is responsible for the saturation problem, cannot be measured directly, the proposed ctr Ih uses a Luenberger observer for estimating the magnetizing current. Conventional methods overcome this problem either with increased costs and system size due to additional passive devices or need for lossy and bandwidth limited current measuring. As measuring inputs, the proposed ctr Ih only needs the average values of transformer voltages V p and V s that can be measured with simple operational amplifiers with low self-consumption and naturally high bandwidth. e digital design of ctr Ih allows for flexible implementation in the main control unit MCU of PSFB converters. erefore, the MCU includes the ctr Ih with Luenberger observer model and PI-controller, the output regulator ctr IL as well as a phase-shift calculator and directly interventions the PWM signals for all power switches S 1 -S 8 via driver. e paper identifies two inevitable effects as main reasons for increased magnetizing DC-offset current I h,DC . First, initial resistance differences due to manufacturing tolerances of devices and second dynamic changes of duty cycle for regulating the output power. It is shown that without any control both effects would lead to transformer saturation and threaten system reliability. e proposed ctr Ih with Luenberger observer model is evaluated by simulation and experimental results on a PSFB prototype with power P � 1kW for the identified effects. e ctr Ih is able to compensate initial resistance differences within 7.7ms by independently controlling the duty cycles of leading and lagging leg of the full-bridge. Also dynamic changes in duty cycle can be controlled out and guarantee a stable continuous operation with an magnetizing DC-offset |I h,DC | � 30mA at V A � 200V. e differences between simulation and experiment can be explained through the additional voltage drop in the supply lines and PCB-layout that result in a smaller magnitude of I h for the real prototype. e main drawback of the proposed scheme can be eliminated by using offset compensated operational amplifiers.
Data Availability e datasets supporting the research are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare no conflicts of interest.