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

Authors: Angelika Neumann; Sönke Meynen; Ahmad Rahmoun; Daniel Ziegler; Thomas Kirchartz
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

This paper proposes a flexible digital control scheme for isolated phase-shift full-bridge (PSFB) converters. The 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. The paper addresses two major issues of the occurrence of a magnetizing DC-offset 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. The 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.


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## Body

## 1. 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].The high efficiency and power density of an isolated phase-shift full-bridge (PSFB) converter in Figure1 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 [5–7]. Its phase-shift control scheme allows for zero-voltage-switching (ZVS) operation with negligible switching losses and reaches higher efficiency compared to conventional hard-switched topologies [8].Figure 1
Phase-shift full-bridge converter topology. Two full-bridgesHA and HB allow for bidirectional operation with power P. Transformer TR partially compensates high voltage difference between VA and VB and provides galvanic isolation.Two full-bridges on primaryHA and secondary HB side make the topology suitable for bidirectional operation. The transformer TR, with ratio r=Np/Ns between primary Np and secondary Ns winding, compensates a high voltage difference between input voltage VA and output voltage VB=VA/rdeff to avoid low duty cycles deff 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 DC-offset current Ih,DC that drives the transformer into saturation [10]. To prevent damage due to high saturation currents and guarantee system reliability Ih,DC must be eliminated.Common topologies connect a blocking capacitor in series to the transformer in the main power path to suppress any DC-offsetIh,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. This 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 eliminateIh,DC [14]. Designing of a stable feedback control loop requires the measurement of Ih,DC as process variable. However, considering the transformer as a four terminal device the acquisition of Ih,DC is not directly possible as it overlays the transformer primary current Ip [15]. The 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 fsw. Unfortunately, in certain operating conditions instabilities, known as sub-harmonic oscillations, occur [16, 17]. The 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 current-mode control with a hybrid peak and valley current control and does not need for slope compensation. The 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. The 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.This article presents a novel digital feedback control loop scheme for eliminating DC-offsetIh,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 the beginning Section 2 explains the inevitable occurrence of DC-offset Ih,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 Ih,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 fsw=100kHz. Finally, Section 7 gives a conclusion of the presented results.


## 2. Transformer DC-Offset Current

The magnetizing currentIh creates the magnetic field of the transformer required for power transfer from primary to secondary side and flows through the magnetizing inductor Lh of the transformer. Ih adds either to primary Ip or secondary current Is of transformer depending on power transfer direction. The transformer is not capable to transfer any DC component of Ih [26]. Therefore, the magnetizing current Ih needs to be detected on primary and secondary side when operating in bidirectional direction. The following considerations refer exemplary to a power transfer direction from primary side A with Ip to secondary side B with Is=Ip−Ihr.Figure2 shows the typical schematic characteristics of applied voltage Vp in Figure 2(a), primary current Ip in Figure 2(b) and magnetization current Ih in Figure 2(c) for transformer of the PSFB converter shown in Figure 1.Figure 2
Schematic characteristics of transformer of PSFB with a present flux imbalance. (a) shows the applied primary voltageVp. (b) explains the according primary current Ip as overlap between magnetizing current Ih and main primary side current Ip,n. (c) shows the course of magnetizing current Ih. The small asymmetrical voltage-second product in (a) causes a DC-offset Ih,DC and its absolute value increases each period Tn till transformer saturation.The power transfer cyclesPC1 and PC2 are determined according to the effective duty cycle deff=2ton/T applied to the transformer TR. For desired duty cycle deff the phase-shift between lagging S1 + S3 and leading leg S2 + S4 of HA varies and applies either the positive or the negative input voltage VA to the magnetizing inductor Lh of the transformer TR according to Figure 2(a). For positive voltages Vp>0 the magnetizing current Ih increases and for negative voltages Vp<0, Ih decreases respectively. During the interval toff no voltage is applied to Lh resulting in a constant Ih. The alternating positive and negative voltage-second products applied to the transformer within one period Tn determines the peak value Ih,peak,1 and Ih,peak,2, assuming to be constant during Tn, in Figure 2(c) according to:(1a)Ih,peak,1n=1LhV1tonn=1LhV1deffnT2,(1b)Ih,peak,2n=1LhV2tonn=1LhV2deffnT2.Therefore, the DC-offsetIh,DCn can approximately be defined as the average value of magnetizing current I¯hn during one period Tn with:(2)Ih,DCn=I¯hn=Ih,peak,1n+Ih,peak,2n2.Figure2(b) explains the sloped magnetizing current Ih as an additional comparable small portion to the high main primary current Ip,n with Ih=Ip−Ip,n [15]. As consequence Ih cannot be measured directly considering the transformer as a four terminal device. Assuming negligible variations of Ip,n and deff during one switching period Tn the DC-offset Ih,DCn can be calculated from the difference between the peak values of Ip during one complete period Tn according to:(3)Ih,DCn=Ip,peak,1n−Ip,peak,2n2.For a present asymmetrical voltage-second productV1≠V2, the absolute value of Ih,DCn increases each period Tn with(4)Ih,DCn+1=Ih,DCn+ΔIh,DCand the resulting DC-offsetIh,DC≠0 drives the transformer into saturation. Ideally, the symmetrical structure of the PSFB topology ensures no DC-offset Ih,DC as positive and negative current slope of Ih 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 PC1 for the lagging leg and PC2 for the leading leg due to ZVS-operation or the change of deff for regulating the output current IB 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 Lh and increase Ih,DC [10]. Beside of the transformer the resulting high saturation current threatens the switching devices HA and HB and general system reliability. Additionally, the conduction loss unnecessarily increases with an unbalanced magnetizing current Ih as it does not contribute to power transfer from primary to secondary side. Therefore, system efficiency η is directly related to minimize Ih,DC.


## 3. Conventional Methods

Several methods have been proposed either to suppress or to control DC-offsetIh,DC in [11, 12, 16, 17, 27, 28].


### 3.1. Blocking Capacitor

Connecting a blocking capacitor in series to the transformer suppresses any DC component in the transformer currentIp and allows for symmetrical operation [11]. Therefore, 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 Ih occurs on either side, primary and secondary, according to power direction. Therefore, this method needs two blocking capacitors and significantly increases volume and overall costs.


### 3.2. Current-Mode Control

The current-mode control (CMC) method is well known for non-isolated converters such as buck or boost converter topologies [16–18]. For non-isolated converters the inductor current IL serves as process variable for the control loop. Fast response analog comparators generate PWM signals and directly trigger the power switches when inductor current IL 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 IL 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 deff>0.5 and for valley detect at duty cycles deff<0.5 respectively. A slope compensation circuit needs to correct the ripple current ΔIL and results in a response delay for the controller [19]. In isolated converters the CMC method can additionally provide the elimination of DC-offset Ih,DC with small modifications. Any DC component of Ih cannot pass the transformer. Therefore, the input current IA must serve as process variable for Ih,DC control with power direction from A to B side.Figure3 explains the compensation principle of CMC with the schematic characteristic curves of primary voltage Vp in Figure 3(a), input current IA in Figure 3(b) and magnetizing current Ih in Figure 3(c) of transformer with a present flux imbalance. Neglecting parasitic inductors, the magnetizing current Ih slopes up with m1=V1/Lh during power cycle PC1 and with m2=V2/Lh during power cycle PC2 respectively. Parasitic effects reduce the applied voltage V1 for power cycle PC1 in Figure 3(a). Assuming a symmetrical duty cycle d1=d2=deff for both power cycles the peak of input current IA,peak,1 does not reach the threshold value Ith of the analog comparator and the negative DC-offset Ih,DC<0 rises [12]. The current-mode controller directly influences the duty cycle d1,CMC=2ton,CMC/T and extends ton,CMC>ton until IA,peak,1 reaches the threshold current Ith. Therefore, the CMC ensures flux balance of the transformer with Ih,DC=0 and provides a symmetrical voltage-second product V1ton,CMC=V2ton [12, 18].Figure 3
CMC compensation principle of PSFB with a present flux imbalance at transformer. (a) shows the primary voltageVp, (b) the input current IA and (c) the magnetizing current Ih of transformer as function of time. Reduced voltage V1<V2 in power cycle PC1 reduces IA,peak,1 and results in an increasing negative DC-offset Ih,DC. CMC extends ton,CMC until IA,peak,1 matches Ith with Ih,DC=0 and provides a symmetrical voltage-second product.Common control schemes prefer an analog controller with comparators compared to digital controller as the erratic increasing ofIA needs high bandwidth to detect the peak current IA,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 IA is the accuracy to properly detect the small portion of Ih as Figure 2(b) explains Ih as a comparable small additional sloped offset to the high input current IA or rather Ip,n.


## 3.1. Blocking Capacitor

Connecting a blocking capacitor in series to the transformer suppresses any DC component in the transformer currentIp and allows for symmetrical operation [11]. Therefore, 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 Ih occurs on either side, primary and secondary, according to power direction. Therefore, this method needs two blocking capacitors and significantly increases volume and overall costs.


## 3.2. Current-Mode Control

The current-mode control (CMC) method is well known for non-isolated converters such as buck or boost converter topologies [16–18]. For non-isolated converters the inductor current IL serves as process variable for the control loop. Fast response analog comparators generate PWM signals and directly trigger the power switches when inductor current IL 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 IL 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 deff>0.5 and for valley detect at duty cycles deff<0.5 respectively. A slope compensation circuit needs to correct the ripple current ΔIL and results in a response delay for the controller [19]. In isolated converters the CMC method can additionally provide the elimination of DC-offset Ih,DC with small modifications. Any DC component of Ih cannot pass the transformer. Therefore, the input current IA must serve as process variable for Ih,DC control with power direction from A to B side.Figure3 explains the compensation principle of CMC with the schematic characteristic curves of primary voltage Vp in Figure 3(a), input current IA in Figure 3(b) and magnetizing current Ih in Figure 3(c) of transformer with a present flux imbalance. Neglecting parasitic inductors, the magnetizing current Ih slopes up with m1=V1/Lh during power cycle PC1 and with m2=V2/Lh during power cycle PC2 respectively. Parasitic effects reduce the applied voltage V1 for power cycle PC1 in Figure 3(a). Assuming a symmetrical duty cycle d1=d2=deff for both power cycles the peak of input current IA,peak,1 does not reach the threshold value Ith of the analog comparator and the negative DC-offset Ih,DC<0 rises [12]. The current-mode controller directly influences the duty cycle d1,CMC=2ton,CMC/T and extends ton,CMC>ton until IA,peak,1 reaches the threshold current Ith. Therefore, the CMC ensures flux balance of the transformer with Ih,DC=0 and provides a symmetrical voltage-second product V1ton,CMC=V2ton [12, 18].Figure 3
CMC compensation principle of PSFB with a present flux imbalance at transformer. (a) shows the primary voltageVp, (b) the input current IA and (c) the magnetizing current Ih of transformer as function of time. Reduced voltage V1<V2 in power cycle PC1 reduces IA,peak,1 and results in an increasing negative DC-offset Ih,DC. CMC extends ton,CMC until IA,peak,1 matches Ith with Ih,DC=0 and provides a symmetrical voltage-second product.Common control schemes prefer an analog controller with comparators compared to digital controller as the erratic increasing ofIA needs high bandwidth to detect the peak current IA,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 IA is the accuracy to properly detect the small portion of Ih as Figure 2(b) explains Ih as a comparable small additional sloped offset to the high input current IA or rather Ip,n.


## 4. Proposed Control Scheme ctrIh

The proposed control scheme ctrIh 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 ctrIh directly interventions the duty cycle d2 of the leading leg to equalize the peak values of primary current Ip,peak,1=Ip,peak,2 and eliminates any magnetizing DC-offset Ih,DC within a few switching cycles.Figure4 depicts the control loop of the PSFB converter with ctrIL for output current regulation and ctrIh for DC-offset Ih,DC control. The output controller ctrIL is built up with a PI-controller PIILz and feedback control for the output voltage VB′=VB+ΔVB for better adjustment of output operating point. The transfer function of the PSFB Gduty determines the effective duty cycle deff=d1=VB′/VAr for desired output current IB,set. The proposed magnetization controller ctrIh adds or subtracts a small offset Δd between duty cycle d1 for the lagging and duty cycle d2=d1+Δd for the leading leg until Ih,DC=0. As Ip,n and d1 vary only a little during one switching period and also the impact of parasitic effects contribute only to a small amount, the ctrIh limits the magnitude of Δd to a few percentage of d1. Therefore, the influence of ctrIh on the output controller ctrIL is negligible [14]. According to d1 and d2 the phase-shift calculator provides eight separate PWM signals for power switches S1–S8 of PSFB power electronics. In the feedback path the main control unit (MCU) performs the acquisition of process variables VA, VB, IL and Ih with analog digital converters (ADC). For output control the rippled inductor current IL averages to the desired output current IB. Therefore, the moving average finite impulse response (FIR) filter with GIL¯z averages IL over N number of periods with:(5)IBn=1N∑i=0N−1ILn−i.Figure 4
Control loop scheme of PSFB forIh,DC elimination. Output controller ctrIL with PIILz and Gduty determines duty cycle d1 for desired IB,set. The ctrIh adds a small offset Δd to d1 of lagging leg until Ih,DC=0. The phase-shift calculator translates the duty cycle information into eight phase shifted PWM signals for bidirectional operation of PSFB power electronics.Accordingly, the DC-offsetIh,DC is defined as the average of rippled magnetization current Ih over several periods. Again, a moving average FIR filter with GI¯hz provides the process variable Ih,DC of ctrIh. As VA and VB are already DC values there is no need for averaging with a FIR filter.For proper design of ctrIh the relation GIh,DCz=Ih,DCz/Δdz 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 unbalanced flux leading to Ih,DC≠0. Therefore, these effects have to be considered in the transfer function GIh,DCz. Figure 5 illustrates the influence of Δd on the DC-offset ΔIh,DC,duty for analyzing the impact of unbalanced duty cycle due to output regulation ctrIL over two consecutive switching periods Tn and Tn+1. According to phase shifted PWM signals for power switches S1–S4 in Figure 5(a) the voltage Vp in Figure 5(b) applies to the transformer. Figure 5(c) shows the time course of magnetizing current Ih resulting from unbalanced duty cycles d1n<d1n+1.Figure 5
Influence ofΔd on ΔIh,DC,duty over two periods Tn and Tn+1. According to phase shifted PWM signals for S1–S4 in (a) the voltage Vp applies to transformer shown in (b). In (c) the different duty cycles d1n+1>d1n cause a small difference ΔIh,DC,duty between Ih,DCn+1 and Ih,DCn.The PWM signals of power switches S1–S4 obtain the duty cycle difference Δd to:(6)Δd=d1n+1−d1n=tonn+1−tonnT/2.Assuming equal voltagesV1=V2=VA for power cycles PC1n and PC2n the magnetizing current Ih slopes with the same gradient V1/Lh=V2/Lh=VA/Lh. In period Tn+1, the output regulator ctrIL increases d1n+1=d1n+Δd. Due to the increased duty cycle d1n+1 the magnetizing current Ih surpasses the peak value Ih,peakn before the end of power cycle PC1n+1, thus, leading to an increased DC-offset in the following period Tn+1 according to:(7)ΔIh,DC,duty=Ih,DCn+1−Ih,DCn.This behavior expresses also for small differences between duty cycled1n and d2n in the same period Tn.Furthermore,Ih,DCn defines as the average of magnetizing current I¯hn with a linear slope VA/Lh. Therefore, ΔIh,DC,duty between two consecutive switching periods due to unbalanced duty cycle approximately calculates as follows:(8)ΔIh,DC,duty=12VALhΔdT2=VAT4LhΔd.As long as there is a duty cycle imbalanceΔd present, the magnetizing current Ih,DC needs to diverge until the PSFB gets damaged if not compensated.Another reason causingIh,DC are different parasitic resistances such as switch resistances or general PCB-design resulting in unequal voltages V1≠V2. [14] derives the influence of parasitic elements ΔR and can approximately be calculated as follows:(9)ΔIh,DC,res=ΔRT2LhIh,DCn=KresIh,DCn.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 Kres to magnitude of Ih,DC. Therefore, the impact of ΔIh,DC,res represents a steady-state fault of ctrIh [14].During the transition between powerPCx and free-wheeling FCx cycles mainly the switching characteristics of the power switches and dead time control influence the DC-offset Ih,DC. The dead time control determines different duty cycle d1 for the lagging and d2 for the leading leg according to ZVS-conditions. In [14], the detailed derivation results in:(10)ΔIh,DC,trans=KtransIh,DCn.again, with propoportional gainKtrans.For a unified transfer functionGIh,DCz the three main effects influencing Ih,DC must be considered according to [14] with:(11)ΔIh,DC=ΔIh,DC,duty−ΔIh,DC,res−ΔIh,DC,trans.The effects of resistanceΔIh,DC,res and switch transition ΔIh,DC,trans both prevent the development of Ih,DC in either direction and therefore have to be subtracted while the transient ΔIh,DC,duty supports the development of Ih,DC [14].Equation (12) calculates the transfer function of Δd and Ih,DC in z domain as follows:(12)GIh,DCz=Ih,DCzΔdz=VAT4Lh1z−Ktrans−Kres.According to (12) the magnetization controller ctrIh requires an integrator for steady state stability. For the experimental results the ctrIh is implemented as PI-controller PIIhz.


## 5. Derivation of Observer Model

In the proposed digital control scheme of ctrIh, Ih,DC=I¯h represents the process variable. Considering TR as a four terminal device with Ip and Vp as inputs and Is and Vs as outputs, Ih,DC cannot be measured at clamps directly. Therefore, Ih,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 asIh,DC from measurable in- and outputs of the real system [29, 30]. Figure 6 simplifies a part of the PSFB converter with four terminal devices of full-bridge HA and transformer TR. The measurable primary voltage Vp and secondary voltage Vs of transformer serve as in- and output for the observer model and allow for estimating the internal value of Ih,DC.Figure 6
Simplified four terminal device ofHA and TR with voltage measuring method M1 for estimating Ih,DC with observer. Current measuring method Mref serves as reference compared to conventional methods.The state observer method is typically computer-implemented and therefore suitable for the proposed digital controller ctrIh.The measuring methodM1 for the observer model requires only the average value of transformer voltages Vp and Vs and can be measured with operational amplifiers at naturally high bandwidth BW>1MHz. Additionally, the voltage measurement only needs small signal currents and therefore it is comparable lossless. The measuring method Mref serves as reference to compare with conventional methods with an inductor Lh,ref connected in parallel to transformer. According to: (13)Ih=Lh,refLhIh,ref,the reference magnetization current Ih,ref reflects the real magnetization current Ih and can be measured with a current clamp.The proposed digital control scheme ctrIh uses the well-known Luenberger observer for estimation of Ih,DC [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.


### 5.1. Transformer State Space Model

The Luenberger observer requires a linear state space model of transformer in form of:(14a)x˙=Ax+Bu,(14b)y=Cx+Du.In (14a) and (14b) x represents the state variables with input current ip, output current is and magnetization current ih. The primary voltage Vp represents the system input u and the secondary voltage Vs serves as measurable system output y according to:(15a)x=ipisihT,(15b)u=Vp,(15c)y=Vs.Figure7 shows the equivalent circuit diagram (ECD) of a real transformer. Parasitic resistances Rp and Rs,u and leakage inductors Lp and Ls,u depict the loss component of primary and secondary windings while RFe simulates the magnetic resistance of the transformer core. The core magnetization is regarded with magnetizing inductor Lh. Nonlinear behavior of magnetizing inductor Lh is neglected as the controller ctrIh prevents saturation of transformer. The ratio r of primary winding Np to secondary winding Ns represents the ideal transformer TR with:(16)r=NpNs.Figure 7
Equivalent circuit diagram of a real transformer TR regarding parasitic elements for loss and magnetization effects. Values with indexess,u are referred to primary side with transformer ratio r.In this model the parasitic capacitors are neglected as their time constants are very small compared to switching frequency and time constant of inductors.Z0 represents the impedance of the load for gathering secondary current is.The reference-arrow system in Figure7 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:(17)Vs,u=rVs,is,u=−1ris,is,u.=−1ris.,Rs,u=Rs+Z0r2,Ls,u=Lsr2.The Kirchhoff current (KCL) and voltage (KVL) laws obtain the main equations of the transformer model according to:(18a)Vp=Rpip+Lpip.+Lhih.,(18b)Vs=−Rs,ur2is−Ls,ur2is.+Lhrih.,(18c)Vs=−Rs,ur2is−Ls,ur2is.−Lprip.−Rprip+1rVp,withVs=Z0is,(18d)ip=+1ris+iFe+ih,withiFe=LhRFeih..Reshaping (18a)–(18d) delivers the time continuous state space vector x˙ to:(19a)ip.=+1LpVp−RFe+RpLpip+RFeLp1ris+RFeLpih,(19b)is.=+RFeLs,urip−Rs,u+RFeLs,uis−RFeLs,urih,(19c)ih.=+RFeLhip−RFeLh1ris−RFeLhih.Referred to (14a) the system A and input B matrices can be obtained to:(20a)A=−RFe+RpLpRFeLp1rRFeLpRFeLs,ur−Rs,u+RFeLs,u−RFeLs,urRFeLh−RFeLh1r−RFeLh,(20b)B=1Lp00T.The terms of voltage dropLp/rip. and Ls,u/r2is,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 GI¯h. Neglecting the dynamic terms in (18c) provides the equation for system output Vs to:(21)Vs=1rVp−Rprip−Rs,ur2is.with output matrixC and direct feed-through matrix D from (14b) according to:(22a)C=−Rpr−Rs,ur20,(22b)D=1r.


### 5.2. Analysis and Discretization of the Transformer Model

The transformer model includes three state variablesip,is and ih building a third order system with rankA=3.From the derived transformer model in the previous Section5.1 the eigenvalues or poles of the system matrix A describe the dynamic behavior of the system.The values in Table1 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 ip, is and ih shown in Table 2. As all poles are in the left-hand plane (LHP), the system provides stability [31].Table 1
Transformer parameter for simulation and PSFB proto type.
ParametervariablevalueSampling timets2μsTransformer ratior2−Magnetization inductorLh5000μHLeakage inductorLp,Ls6.23μHMagnetic resistanceRFe1000ΩParasitic resistance (prim)Rp0.0045ΩParasitic resistance (sec)Rs0.0070ΩReference measuring inductorLh,ref630μHTable 2
Poles of system matrixA for the three state variables ip, is and ih of transformer model.
state variablepolesip−6.4228×108is−2.0918×104ih−3.1444The pole ofip according to Table 2 is too fast for realistic sampling time ts=T/5=2μs required for discretization of observer model. As the desired value of the observer only includes the state variable ih, the state variable ip with the fastest pole can be ignored. The reduction of system order eliminates the first state variable ip and reconfigures the matrices Ared, Bred and Cred with rankAred=2 and poles according to Table 3.Table 3
Poles of reduced system matrixAred for the two state variables is and ih of transformer model.
state variablepolesis−2.0916×104ih−2.4645Again, all poles of the system with reduced order are located in LHP with negative values. Therefore, 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 timets. The transfer of the reduced transformer model into discrete form affects only the system matrix Ared and the input matrix Bred [32]. One approach of discretization presented in [20] according to:(23a)Ad,red=eAredts,(23b)Ad,red=ℒ−1sI−Ared−1,delivers the discrete system matrixAd,red without approximation.The MATLAB order of matrix exponential functionexpmAredts can directly perform the discretization and is suitable for the experimental approach.With reduced discrete system matrixAd,red the reduced discrete input matrix Bd,red calculates as follows:(24a)Bd,red=∫0tseAreddτBred,(24b)Bd,red=Ared−1Ad,red−IBred.The reduced discrete output matrixCd,red=Cred and reduced discrete direct feed-through matrix Dd,red=Dred are not affected and remain the same.The analysis of the reduced discrete transformer model inZ-domain delivers the poles of Ad,red according to Table 4 with one stable pole at z<1 and one critically stable pole at z=1. Therefore, a suitable design of the observer gain factors must guarantee the stability of the system.Table 4
Poles of reduced discrete system matrixAd,red for the two state variables is and ih of transformer model in Z-domain.
state variablepolesis0.9590ih1.0000


### 5.3. 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 statext=0=x0 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 SBd,red in (25) has the same rankSBd,red as the reduced system matrix Ad,red with(25)SBd,red=Cd,redCd,redAd,red=94.21475.110290.35484.9009.According to (25) the rank of observability matrix rankSBd,red=rankAd,red=2 equals the rank of reduced discrete system matrix and so the system is completely observable. Additionally, its determinant detSBd,red=−7.7102×104≠0 is not zero [30, 33].


### 5.4. Design of Luenberger Observer

In the structure of a Luenberger observer the derived transformer model works in parallel to real transformer. The observer compares the measured outputy with the calculated value of the transformer model y^ and feeds back the estimation error Δyerror=Ly−y^ with feedback matrix L to the model [30].From observer model the estimated state vectorx^˙ calculates as follows:(26a)x^˙=Ad,redx^+Bd,redu+Δyerror,(26b)x^˙=Ad,redx^+Bd,redu+Ly−Cd,redx^−Dd,redu,(26c)x^˙=Ad,red−LCd,redx^+Bd,red−LDd,redu+Ly.Suitable values of feedback matrixL adjust the dynamical behavior of the model to match the real transformer and react to disturbances or transient variations of duty cycle d1. The design of feedback matrix L with pole placement according to Ackermann [31, 33] is used and the known poles of Ad,red from Table 4 deliver the first approach according to:(27)L=0.00135−0.02478T.


## 5.1. Transformer State Space Model

The Luenberger observer requires a linear state space model of transformer in form of:(14a)x˙=Ax+Bu,(14b)y=Cx+Du.In (14a) and (14b) x represents the state variables with input current ip, output current is and magnetization current ih. The primary voltage Vp represents the system input u and the secondary voltage Vs serves as measurable system output y according to:(15a)x=ipisihT,(15b)u=Vp,(15c)y=Vs.Figure7 shows the equivalent circuit diagram (ECD) of a real transformer. Parasitic resistances Rp and Rs,u and leakage inductors Lp and Ls,u depict the loss component of primary and secondary windings while RFe simulates the magnetic resistance of the transformer core. The core magnetization is regarded with magnetizing inductor Lh. Nonlinear behavior of magnetizing inductor Lh is neglected as the controller ctrIh prevents saturation of transformer. The ratio r of primary winding Np to secondary winding Ns represents the ideal transformer TR with:(16)r=NpNs.Figure 7
Equivalent circuit diagram of a real transformer TR regarding parasitic elements for loss and magnetization effects. Values with indexess,u are referred to primary side with transformer ratio r.In this model the parasitic capacitors are neglected as their time constants are very small compared to switching frequency and time constant of inductors.Z0 represents the impedance of the load for gathering secondary current is.The reference-arrow system in Figure7 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:(17)Vs,u=rVs,is,u=−1ris,is,u.=−1ris.,Rs,u=Rs+Z0r2,Ls,u=Lsr2.The Kirchhoff current (KCL) and voltage (KVL) laws obtain the main equations of the transformer model according to:(18a)Vp=Rpip+Lpip.+Lhih.,(18b)Vs=−Rs,ur2is−Ls,ur2is.+Lhrih.,(18c)Vs=−Rs,ur2is−Ls,ur2is.−Lprip.−Rprip+1rVp,withVs=Z0is,(18d)ip=+1ris+iFe+ih,withiFe=LhRFeih..Reshaping (18a)–(18d) delivers the time continuous state space vector x˙ to:(19a)ip.=+1LpVp−RFe+RpLpip+RFeLp1ris+RFeLpih,(19b)is.=+RFeLs,urip−Rs,u+RFeLs,uis−RFeLs,urih,(19c)ih.=+RFeLhip−RFeLh1ris−RFeLhih.Referred to (14a) the system A and input B matrices can be obtained to:(20a)A=−RFe+RpLpRFeLp1rRFeLpRFeLs,ur−Rs,u+RFeLs,u−RFeLs,urRFeLh−RFeLh1r−RFeLh,(20b)B=1Lp00T.The terms of voltage dropLp/rip. and Ls,u/r2is,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 GI¯h. Neglecting the dynamic terms in (18c) provides the equation for system output Vs to:(21)Vs=1rVp−Rprip−Rs,ur2is.with output matrixC and direct feed-through matrix D from (14b) according to:(22a)C=−Rpr−Rs,ur20,(22b)D=1r.


## 5.2. Analysis and Discretization of the Transformer Model

The transformer model includes three state variablesip,is and ih building a third order system with rankA=3.From the derived transformer model in the previous Section5.1 the eigenvalues or poles of the system matrix A describe the dynamic behavior of the system.The values in Table1 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 ip, is and ih shown in Table 2. As all poles are in the left-hand plane (LHP), the system provides stability [31].Table 1
Transformer parameter for simulation and PSFB proto type.
ParametervariablevalueSampling timets2μsTransformer ratior2−Magnetization inductorLh5000μHLeakage inductorLp,Ls6.23μHMagnetic resistanceRFe1000ΩParasitic resistance (prim)Rp0.0045ΩParasitic resistance (sec)Rs0.0070ΩReference measuring inductorLh,ref630μHTable 2
Poles of system matrixA for the three state variables ip, is and ih of transformer model.
state variablepolesip−6.4228×108is−2.0918×104ih−3.1444The pole ofip according to Table 2 is too fast for realistic sampling time ts=T/5=2μs required for discretization of observer model. As the desired value of the observer only includes the state variable ih, the state variable ip with the fastest pole can be ignored. The reduction of system order eliminates the first state variable ip and reconfigures the matrices Ared, Bred and Cred with rankAred=2 and poles according to Table 3.Table 3
Poles of reduced system matrixAred for the two state variables is and ih of transformer model.
state variablepolesis−2.0916×104ih−2.4645Again, all poles of the system with reduced order are located in LHP with negative values. Therefore, 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 timets. The transfer of the reduced transformer model into discrete form affects only the system matrix Ared and the input matrix Bred [32]. One approach of discretization presented in [20] according to:(23a)Ad,red=eAredts,(23b)Ad,red=ℒ−1sI−Ared−1,delivers the discrete system matrixAd,red without approximation.The MATLAB order of matrix exponential functionexpmAredts can directly perform the discretization and is suitable for the experimental approach.With reduced discrete system matrixAd,red the reduced discrete input matrix Bd,red calculates as follows:(24a)Bd,red=∫0tseAreddτBred,(24b)Bd,red=Ared−1Ad,red−IBred.The reduced discrete output matrixCd,red=Cred and reduced discrete direct feed-through matrix Dd,red=Dred are not affected and remain the same.The analysis of the reduced discrete transformer model inZ-domain delivers the poles of Ad,red according to Table 4 with one stable pole at z<1 and one critically stable pole at z=1. Therefore, a suitable design of the observer gain factors must guarantee the stability of the system.Table 4
Poles of reduced discrete system matrixAd,red for the two state variables is and ih of transformer model in Z-domain.
state variablepolesis0.9590ih1.0000


## 5.3. 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 statext=0=x0 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 SBd,red in (25) has the same rankSBd,red as the reduced system matrix Ad,red with(25)SBd,red=Cd,redCd,redAd,red=94.21475.110290.35484.9009.According to (25) the rank of observability matrix rankSBd,red=rankAd,red=2 equals the rank of reduced discrete system matrix and so the system is completely observable. Additionally, its determinant detSBd,red=−7.7102×104≠0 is not zero [30, 33].


## 5.4. Design of Luenberger Observer

In the structure of a Luenberger observer the derived transformer model works in parallel to real transformer. The observer compares the measured outputy with the calculated value of the transformer model y^ and feeds back the estimation error Δyerror=Ly−y^ with feedback matrix L to the model [30].From observer model the estimated state vectorx^˙ calculates as follows:(26a)x^˙=Ad,redx^+Bd,redu+Δyerror,(26b)x^˙=Ad,redx^+Bd,redu+Ly−Cd,redx^−Dd,redu,(26c)x^˙=Ad,red−LCd,redx^+Bd,red−LDd,redu+Ly.Suitable values of feedback matrixL adjust the dynamical behavior of the model to match the real transformer and react to disturbances or transient variations of duty cycle d1. The design of feedback matrix L with pole placement according to Ackermann [31, 33] is used and the known poles of Ad,red from Table 4 deliver the first approach according to:(27)L=0.00135−0.02478T.


## 6. Results

The proposed digital control scheme regulates the magnetizing currentIh of a PSFB to prevent transformer saturation. First approaches concern the simulation of electronics to prove principal functionality of ctrIh, 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.Table 5
Operating point for simulation and PSFB prototype at duty cycled1=0.8.
ParametervariablevalueInput voltageVA200VInput currentIA5.8AOutput voltageVB70VCharging currentIB14.1ASwitching frequencyfsw100kHzTo validate the simulated results in Section6.1 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.


### 6.1. Simulation of Observer

For the evaluation of proposed ctrIh in Section 4 and derived observer model in Section 5, the PSFB topology from Figure 1 is built up in a MATLAB/SIMSCAPE simulation model according to parameters in Table 1 and operating point listed in Table 5. The control scheme for ctrIh and observer model are implemented in MATLAB/SIMULINK.The simulation results in Figure8 demonstrate the behavior of the PSFB magnetizing current Ih at operating point in Table 5 for the different disturbances mentioned in Section 2, dynamic change of output current IB through duty cycle d1 and initial static differences due to manufacturing process of devices. Fluctuations of d2 due to ctrIh thereby influence the magnetization current Ih in a similar way as d1. The simulation starts at t0 with a desired output current IB=9.3A and a corresponding duty cycle d1t0=0.76. The series resistance RS2=RS1+ΔR=0.2Ω of switch S2 is doubled compared to other switches with RS1,3,4=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 ctrIh, 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. The influence of ctrIh on controlled variable Δd=d1−d2≈0.07 is shown in Figure 8(c).Figure 8
Evaluation of observer model. (a) shows the simulated average currentI¯h and the estimated observer current I¯h,obs compared to the uncontrolled current I¯h,0. (b) shows the model error ΔI¯h,error between I¯h and modeled I¯h,obs. (c) shows the influence of ctrIh on controlled variable with d1 and d2 with offset Δd≈0.07.The increased resistanceRS2 forces ctrIh to increase d2>d1 and compensates the voltage-second product imbalance. After initial transition process the ctrIh is able to correct the error resulting from static difference due to RS2 within t1≈7.7ms. Either change due to d1 step at t3 or d2 control at t2 and t4 causes fluctuations in Ih. The estimated observer current I¯h,obs follows the dynamics of I¯h with a small delay Δtdelay=0.5ms mainly resulting from computation and FIR filter. According to I¯h,obs the ctrIh adjusts d2 for regulating Ih,DC=0A. Figure 9 explains the dynamic changes exemplary on d1 in detail. At simulation time t3=50ms the desired output current raises to IB=14.1A resulting in a dynamic change of duty cycle d1t3=0.8 . For t<t3 the magnetizing current I¯h,0 approximately slopes up with d/dtI¯h=3.2As−1. For time t3≤t<t3+50μs the magnetizing current I¯h,0 steps up according to equation (8) and (9), raising Ih,DC. The step expresses as slope with d/dtI¯h,0=4.7mA50μs−1 due to FIR filter GI¯h in MCU. With increased magnetizing current I¯h,0 the rising slope of magnetizing current d/dtI¯h=6.6As−1 due to RS2 enlarges after t3+50μs encouraging saturation of transformer. The controlled estimated observer current I¯h,obs matches the dynamics of magnetizing current I¯h and the change can be controlled out by ctrIh with Ih,DC≈0A. According to applied to voltage Vp to Lh and moving average filter, the magnitude value results to I¯h,pp≈I¯h,obs,pp=0.2A.Figure 9
Behavior of magnetizing current in case of dynamic change ofd1 at t3. Uncontrolled I¯h,0 enlarges slope at t3 with increased duty cycle d1 and encourages transformer saturation. The controlled observer current I¯h,obs matches the dynamic change and keeps I¯h around zero. I¯h,obs,fail varies in phase due to model failure in Lh,fail.For analyzing the resistance of ctrIh, Ih,obs,fail exemplary represents the observer current with a model failure in magnetizing inductance Lh,fail=1.1Lh. The model failure produces a time delay Δtfail=1ms for Ih,fail as the bigger magnetizing inductance reduces the slope of magnetizing current and enlarges the initial time t1 for regulating the initial differences. Nevertheless, the ctrIh can handle the change in d1 as well as in d2. A similar result is achieved with an examined model failure in primary resistance Rp and load impedance Z0.


### 6.2. Measurement Application

In the proposed control scheme the ctrIh only needs the average values of voltage Vp and Vs as measuring inputs to control the magnetization DC-offset Ih,DC. Typically, voltage measurements only need a network of low-cost resistors and operational amplifiers with naturally high bandwidth and low self-consumption. Therefore, voltage measurements reach higher efficiency than current measurements with high loss due to shunt resistors or hall sensors with limited bandwidth.Figure10 shows the measuring application for acquiring the input Vp and output Vs for the observer model. A simple voltage divider sets the transformer voltages to required 0…5V level of the main control unit MCU with Vp′ and Vs′. Depending on the switching state of full-bridge HA the potential of the transformer voltage Vp is floating. As in single ended mode the MCU refers its input signals to GNDA, an operational amplifier OPp’ performs the differential measurement Vp′=Vp′1−Vp′2 and provides stable and resilient signals for further processing in MCU. Additionally, the differential measurement does not require a floating supply voltage VCC for amplifiers. To maintain isolation of the PSFB the voltage measurement on secondary side Vs′ needs to be decoupled with an isolated amplifier OPs’ after differential measurement.Figure 10
Measuring principle for in- and outputs of observer model. Voltage divider setsVp and Vs to MCU voltage level. OPp’ and OPs’ perform differential measurement for Vp′ and Vs′ smoothened by LPp’ and LPs’. MCU averages values with GV¯p′z and GV¯s′z and estimates Ih,DC according to observer model for ctrIh.The observer model only needs the average of voltage signalsVp and Vs. Therefore, the low pass filters LPp’ and LPs’ smooth the signals Vp′ and Vs′ and minimize the measuring noise due to rising edges with high dV/dt. In the MCU, the digital FIR filters GV¯p′z and GV¯s′z calculate the moving average over 1000 samples for averaging the signals. According to the observer model the MCU estimates Ih,DC for ctrIh.


### 6.3. Experimental Results

Figure11 shows the PSFB prototype used for the validation of the proposed control scheme. The modular structure splits the PSFB into 5 separate parts, full-bridges HA and HB 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. Therefore, separate components can be substituted easily and allows for testing different power devices and combinations of the PSFB for optimization.Figure 11
PSFB prototype used for experimental validation of proposed control scheme. The modular structure allows for flexible testing of different combination of power device.In the experimental test set up a National Instrument (NI) Data Acquisition (DAQ) system acquires the values from the measuring circuit board with500MSs−1 and performs the calculations of the MCU. Therefore, the NI system includes the observer model, ctrIh control scheme with PI-controller and phase-shift calculator. The resulting eight PWM signals are generated on the counter outputs of NI system and directly connected to the drivers for full-bridge switches S1–S8.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 present in prototype electronics. As the effect of resistanceΔR 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, Figure12 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 Ip,0 without any control of magnetization current while Figure 12(b) illustrates Ip,obs controlled by ctrIh with observer model. Figure 12(c) explains the according courses of Ip,0 and Ip,obs at higher temporal resolution.Figure 12
Experimental results of PSFB prototype in steady state operating point. (a) shows the primary current without ctrIh. The resulting asymmetrical course of Ip,0 leads to a negative offset I¯p,0<0. In (b) ctrIh compensates the offset I¯p,obs≈0 and ensures a symmetrical course of magnetizing current. (c) explains the courses of primary currents in detail and illustrates the offset difference ΔI¯=−806mA.The initial differences cause an unbalanced voltage-second product and lead to an asymmetrical course ofIp,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. The ctrIh detects the initial imbalance and is able to regulate the average offset to a constant value of I¯p,obs=−42mA. Thus, the ctrIh removes the offset difference ΔI¯=I¯p,0−I¯p,obs=−806mA and improves the system reliability and efficiency.Figure13 illustrates the results of the PSFB prototype controlled with ctrIh switching between different operating points. Therefore, during the experiment duty cycle d1 varies each time t0−t5 resulting in changed output current IB shown in Figure 13(d).Figure 13
Experimental results of PSFB prototype in different operating points at timest0−t5. (a) shows the averaged observer current I¯h,obs following the dynamical behavior of real I¯h calculated from measurable reference current I¯h,ref. (b) illustrates the model error ΔI¯h,error. In (c) ctrIh adjusts d2 due to variation of d1 and initial resistance differences. Changes of output current IB due to d1 are shown in (d).In Figure13(a) the magnetizing current Ih=0.126Ih,ref is calculated from measurable reference current Ih,ref with parallel inductor Lh,ref according to (13). Although the magnitude of observer current Ih,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 ctrIh adjusts d2 with a constant offset Δd=0.1 due to initial resistance differences and controls variations of d1. The exact measurement of the initial differences in the full-bridges of the prototype is very difficult and also vary due to temperature effects. Therefore, there is a difference between simulated duty offset Δd=0.07 and prototype offset. However, the difference does not affect the functionality of the ctrIh. The magnetizing current in Figure 13(a) oscillates around its DC-offset Ih,DC=−30mA while the ctrIh shows a better performance on Ih,obs,DC=1mA. This difference results from measuring voltage offset due to operational amplifier and is the main drawback of the control scheme. Therefore, 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 VA drops till the transformer with Vp<VA and decreases the slope of magnetizing current Ih. The magnitude of observer Ih,obs and magnetizing current Ih vary each time t0−t5 the operating point changes, as the controlling parameter for PI-controller PIILz depend on IB 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. Therefore, the experimental results confirm the results of simulation and prove the functionality of ctrIh for observed disturbances.


## 6.1. Simulation of Observer

For the evaluation of proposed ctrIh in Section 4 and derived observer model in Section 5, the PSFB topology from Figure 1 is built up in a MATLAB/SIMSCAPE simulation model according to parameters in Table 1 and operating point listed in Table 5. The control scheme for ctrIh and observer model are implemented in MATLAB/SIMULINK.The simulation results in Figure8 demonstrate the behavior of the PSFB magnetizing current Ih at operating point in Table 5 for the different disturbances mentioned in Section 2, dynamic change of output current IB through duty cycle d1 and initial static differences due to manufacturing process of devices. Fluctuations of d2 due to ctrIh thereby influence the magnetization current Ih in a similar way as d1. The simulation starts at t0 with a desired output current IB=9.3A and a corresponding duty cycle d1t0=0.76. The series resistance RS2=RS1+ΔR=0.2Ω of switch S2 is doubled compared to other switches with RS1,3,4=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 ctrIh, 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. The influence of ctrIh on controlled variable Δd=d1−d2≈0.07 is shown in Figure 8(c).Figure 8
Evaluation of observer model. (a) shows the simulated average currentI¯h and the estimated observer current I¯h,obs compared to the uncontrolled current I¯h,0. (b) shows the model error ΔI¯h,error between I¯h and modeled I¯h,obs. (c) shows the influence of ctrIh on controlled variable with d1 and d2 with offset Δd≈0.07.The increased resistanceRS2 forces ctrIh to increase d2>d1 and compensates the voltage-second product imbalance. After initial transition process the ctrIh is able to correct the error resulting from static difference due to RS2 within t1≈7.7ms. Either change due to d1 step at t3 or d2 control at t2 and t4 causes fluctuations in Ih. The estimated observer current I¯h,obs follows the dynamics of I¯h with a small delay Δtdelay=0.5ms mainly resulting from computation and FIR filter. According to I¯h,obs the ctrIh adjusts d2 for regulating Ih,DC=0A. Figure 9 explains the dynamic changes exemplary on d1 in detail. At simulation time t3=50ms the desired output current raises to IB=14.1A resulting in a dynamic change of duty cycle d1t3=0.8 . For t<t3 the magnetizing current I¯h,0 approximately slopes up with d/dtI¯h=3.2As−1. For time t3≤t<t3+50μs the magnetizing current I¯h,0 steps up according to equation (8) and (9), raising Ih,DC. The step expresses as slope with d/dtI¯h,0=4.7mA50μs−1 due to FIR filter GI¯h in MCU. With increased magnetizing current I¯h,0 the rising slope of magnetizing current d/dtI¯h=6.6As−1 due to RS2 enlarges after t3+50μs encouraging saturation of transformer. The controlled estimated observer current I¯h,obs matches the dynamics of magnetizing current I¯h and the change can be controlled out by ctrIh with Ih,DC≈0A. According to applied to voltage Vp to Lh and moving average filter, the magnitude value results to I¯h,pp≈I¯h,obs,pp=0.2A.Figure 9
Behavior of magnetizing current in case of dynamic change ofd1 at t3. Uncontrolled I¯h,0 enlarges slope at t3 with increased duty cycle d1 and encourages transformer saturation. The controlled observer current I¯h,obs matches the dynamic change and keeps I¯h around zero. I¯h,obs,fail varies in phase due to model failure in Lh,fail.For analyzing the resistance of ctrIh, Ih,obs,fail exemplary represents the observer current with a model failure in magnetizing inductance Lh,fail=1.1Lh. The model failure produces a time delay Δtfail=1ms for Ih,fail as the bigger magnetizing inductance reduces the slope of magnetizing current and enlarges the initial time t1 for regulating the initial differences. Nevertheless, the ctrIh can handle the change in d1 as well as in d2. A similar result is achieved with an examined model failure in primary resistance Rp and load impedance Z0.


## 6.2. Measurement Application

In the proposed control scheme the ctrIh only needs the average values of voltage Vp and Vs as measuring inputs to control the magnetization DC-offset Ih,DC. Typically, voltage measurements only need a network of low-cost resistors and operational amplifiers with naturally high bandwidth and low self-consumption. Therefore, voltage measurements reach higher efficiency than current measurements with high loss due to shunt resistors or hall sensors with limited bandwidth.Figure10 shows the measuring application for acquiring the input Vp and output Vs for the observer model. A simple voltage divider sets the transformer voltages to required 0…5V level of the main control unit MCU with Vp′ and Vs′. Depending on the switching state of full-bridge HA the potential of the transformer voltage Vp is floating. As in single ended mode the MCU refers its input signals to GNDA, an operational amplifier OPp’ performs the differential measurement Vp′=Vp′1−Vp′2 and provides stable and resilient signals for further processing in MCU. Additionally, the differential measurement does not require a floating supply voltage VCC for amplifiers. To maintain isolation of the PSFB the voltage measurement on secondary side Vs′ needs to be decoupled with an isolated amplifier OPs’ after differential measurement.Figure 10
Measuring principle for in- and outputs of observer model. Voltage divider setsVp and Vs to MCU voltage level. OPp’ and OPs’ perform differential measurement for Vp′ and Vs′ smoothened by LPp’ and LPs’. MCU averages values with GV¯p′z and GV¯s′z and estimates Ih,DC according to observer model for ctrIh.The observer model only needs the average of voltage signalsVp and Vs. Therefore, the low pass filters LPp’ and LPs’ smooth the signals Vp′ and Vs′ and minimize the measuring noise due to rising edges with high dV/dt. In the MCU, the digital FIR filters GV¯p′z and GV¯s′z calculate the moving average over 1000 samples for averaging the signals. According to the observer model the MCU estimates Ih,DC for ctrIh.


## 6.3. Experimental Results

Figure11 shows the PSFB prototype used for the validation of the proposed control scheme. The modular structure splits the PSFB into 5 separate parts, full-bridges HA and HB 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. Therefore, separate components can be substituted easily and allows for testing different power devices and combinations of the PSFB for optimization.Figure 11
PSFB prototype used for experimental validation of proposed control scheme. The modular structure allows for flexible testing of different combination of power device.In the experimental test set up a National Instrument (NI) Data Acquisition (DAQ) system acquires the values from the measuring circuit board with500MSs−1 and performs the calculations of the MCU. Therefore, the NI system includes the observer model, ctrIh control scheme with PI-controller and phase-shift calculator. The resulting eight PWM signals are generated on the counter outputs of NI system and directly connected to the drivers for full-bridge switches S1–S8.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 present in prototype electronics. As the effect of resistanceΔR 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, Figure12 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 Ip,0 without any control of magnetization current while Figure 12(b) illustrates Ip,obs controlled by ctrIh with observer model. Figure 12(c) explains the according courses of Ip,0 and Ip,obs at higher temporal resolution.Figure 12
Experimental results of PSFB prototype in steady state operating point. (a) shows the primary current without ctrIh. The resulting asymmetrical course of Ip,0 leads to a negative offset I¯p,0<0. In (b) ctrIh compensates the offset I¯p,obs≈0 and ensures a symmetrical course of magnetizing current. (c) explains the courses of primary currents in detail and illustrates the offset difference ΔI¯=−806mA.The initial differences cause an unbalanced voltage-second product and lead to an asymmetrical course ofIp,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. The ctrIh detects the initial imbalance and is able to regulate the average offset to a constant value of I¯p,obs=−42mA. Thus, the ctrIh removes the offset difference ΔI¯=I¯p,0−I¯p,obs=−806mA and improves the system reliability and efficiency.Figure13 illustrates the results of the PSFB prototype controlled with ctrIh switching between different operating points. Therefore, during the experiment duty cycle d1 varies each time t0−t5 resulting in changed output current IB shown in Figure 13(d).Figure 13
Experimental results of PSFB prototype in different operating points at timest0−t5. (a) shows the averaged observer current I¯h,obs following the dynamical behavior of real I¯h calculated from measurable reference current I¯h,ref. (b) illustrates the model error ΔI¯h,error. In (c) ctrIh adjusts d2 due to variation of d1 and initial resistance differences. Changes of output current IB due to d1 are shown in (d).In Figure13(a) the magnetizing current Ih=0.126Ih,ref is calculated from measurable reference current Ih,ref with parallel inductor Lh,ref according to (13). Although the magnitude of observer current Ih,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 ctrIh adjusts d2 with a constant offset Δd=0.1 due to initial resistance differences and controls variations of d1. The exact measurement of the initial differences in the full-bridges of the prototype is very difficult and also vary due to temperature effects. Therefore, there is a difference between simulated duty offset Δd=0.07 and prototype offset. However, the difference does not affect the functionality of the ctrIh. The magnetizing current in Figure 13(a) oscillates around its DC-offset Ih,DC=−30mA while the ctrIh shows a better performance on Ih,obs,DC=1mA. This difference results from measuring voltage offset due to operational amplifier and is the main drawback of the control scheme. Therefore, 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 VA drops till the transformer with Vp<VA and decreases the slope of magnetizing current Ih. The magnitude of observer Ih,obs and magnetizing current Ih vary each time t0−t5 the operating point changes, as the controlling parameter for PI-controller PIILz depend on IB 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. Therefore, the experimental results confirm the results of simulation and prove the functionality of ctrIh for observed disturbances.


## 7. Conclusion

This paper presents a digital control scheme ctrIh for preventing saturation of the transformer due to flux imbalances in isolated PSFB converters. As the magnetizing current Ih of a transformer, which is responsible for the saturation problem, cannot be measured directly, the proposed ctrIh 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 ctrIh only needs the average values of transformer voltages Vp and Vs that can be measured with simple operational amplifiers with low self-consumption and naturally high bandwidth. The digital design of ctrIh allows for flexible implementation in the main control unit MCU of PSFB converters. Therefore, the MCU includes the ctrIh with Luenberger observer model and PI-controller, the output regulator ctrIL as well as a phase-shift calculator and directly interventions the PWM signals for all power switches S1–S8 via driver.The paper identifies two inevitable effects as main reasons for increased magnetizing DC-offset currentIh,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. The proposed ctrIh with Luenberger observer model is evaluated by simulation and experimental results on a PSFB prototype with power P=1kW for the identified effects. The ctrIh 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 Ih,DC=30mA at VA=200V. The 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 Ih for the real prototype. The main drawback of the proposed scheme can be eliminated by using offset compensated operational amplifiers.



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*Source: 9243429-2022-09-26.xml*