Efficacy of STC in Magnetically Coupled High Frequency Power Electronics Systems: Full-Bridge DC-DC Converter

— Due to its simple gain selection and implementation procedures high-gain sliding mode control idea was introduced. Apart from achieving robustness features, the aim was to reduce product-design cycle time. To overcome its basic chattering issues, it was replaced by complex super twisting control (STC). Subsequently, the research in this field has been extensive, intense and persistent. The idea has been validated in several applications generating great hope in minds of industry experts. Its real success, however, would depend on its adoption and diffusion into industry domain, where an analysis of validating STC based products is due. In STC, selection of gains is based on worst-case values of disturbance and/or its derivative. Can such procedure ensure controller’s reliable functioning? This article proposes that such simplistic procedure of gain selection may not work for power electronics controllers where, particularly, the actuation or controlled power transfer to load is through magnetically-coupled system. Using practical approach, this article elaborates that optimally designed transformer, driven by high-gain super twisting controller, invites problem of core saturation leading to higher switching losses, poor operating duty cycle of the system and there could be problem of reliability. It further details an alternate high-gain controller that generates superior efficacy.


I. INTRODUCTION
RADUALLY, for effective and efficient use, major part of energy demand would be handled through use of compatible power electronics systems [1]- [4]. Full-bridge DC-DC converter FBC (see Fig. 1) is one popular configuration being used for control of power flow to wide range high-power applications [4]- [8]. High-power FBC consists of several functional blocks, such as, 1. Power converter and topologies [8] used, therein 2. Transformer for isolation and voltage translation [9] 3. Sensors and actuators [10] 4. Embedded system [11] 5. Control concepts [11]- [22], etc.
Ideally, for effective utility of each block, they should be functionally compatible to each other; should not pose as any constraint to other. Then, say, for control of secondary side outputs (e.g., current I a or voltage V L ), the system in Fig. 1 could be represented as secondary side controller (see Fig. 2a). In industrial domains, the utility of PI control [11]- [15] is unmatched. Since its inception its application potential was recognized and the concept was soon integrated into industrial control system applications. Due to integral action, it could achieve zero steady state error in presence of uncertainty [13]. PI control is conservative because its two gains are chosen based on the desired response characteristics involving the complete dynamical system that includes the power controller. The gain selection is flexible to desired response characteristics.
The super twisting control (STC) [16]- [22] is an improved version of chattering-prone discontinuous sliding mode control function [23], the control is continuous. Though, it is perceived, at least, in theoretical domain, as a superior concept, its utility in industrial application is still to be demonstrated [24], [25]. From industry perspective, it is important to analyze the issue in details purely in practical domain. Like PI control, STC also needs setting of two control loop gains [21], [22]. But, the approach of gain selection is different, it is based on the magnitude of matched disturbances present in process or load. Neither the primary side of transformer TR (see Fig. 1) nor the inverter switching frequency f s play any role in gain selection. Compared to PI, value of each gain is large. It makes the duty cycle d pwm of PWM pulses sensitive to error. For magnetically coupled controlled actuation, fast rate of change in d pwm , caused by disturbance or ripple content in STC function, could make adverse impact on nonlinear magnetic circuit. The prospect of DC bias in core would be more [26]- [28]. Presence of DC flux in nonlinear core would result increase in power loss in inverter of multiple origin. The worst could lead to core saturation where one pair of devices draw large magnetizing current and incur large power loss. There could be pre-matured thermal tripping of the controller and or failure of power devices. The article analyzes with detailed practical demonstration that high gain fractional order STC function may be less effective for control application where power transformer is part of controlled energy transfer to the load. The article is structured as follows: Section II briefly discusses a typical FBC. It further elaborates the results of PI concept for control of current for a particular load profile. Section III develops one STC based SOSM controller for the same load profile, and further validates the concept with superior control response. Section IV elaborates the next step of innovation process i.e., complete product validation. It details that STC fails to clear a magnetically coupled product on several aspects such as on power loss, poor availability and reliability issues. Finally, Section V, details, with practical validation, an alternate high-gain controller for robust control of secondary side variables with better efficacy.

II. FB DC-DC CONVERTER AND PI CONTROL
The power circuit of FBC of Fig. 1 could be used for control of either current or voltage of load. Though, load characteristics compatible several soft-switched topologies exist. for clarity, hard-switched version would be discussed here. It is compatible to any transient load conditions. It consists of power inverter, transformer TR, secondary side fast rectifier and an inductor L1. Due to small conduction loss, metallized polypropylene capacitor is used in DC link for C DC . Each IGBT turns-on at half the supply voltage V DC , the turn-on loss of inverter is small. For ideal TR with zero leakage inductance l TR , IGBTs turn-off at peak magnetizing current I m , not dependent on load current. Turn-off loss of each IGBT and recovery and conduction losses of diodes D1-D4 would as well be small. The value of switching frequency f s could be increased if SiC Mosfets are used in place of Q1-Q4, or, SiC diodes are used in place of D1-D4. Here, as representative load, constant current (CC) arc welding process would be considered. From control perspective of secondary variables, FBC could be represented as shown in Fig. 2a. When either pair of (Q1, Q4 or Q2, Q3) or equivalently Q5 turns on the dynamics of current I a is expressed as, When Q1-Q4 or Q5 is turned off, D5/D6 or equivalently D8 conducts, corresponding current dynamics is, V oc is controlled input voltage to arc circuit, V L is arc voltage. L1, of significant value, is pre-requisite to contain ripple for wide range arc loads and also for ensuring arc stability. Furthermore, arc is extinguished when short-circuiting takes place, but the energy stored in it assists re-ignition of arc. During switching on and off of Q5 the desired current I a in both cases are shown in Fig. 2b. For a particular application, with fixed ripple current Δ , the value of L1 depends on f s , because, At high f s , small value of L1 would ensure continuity of I a for arc stability. To reduce associated high-frequency power loss, load-compatible soft-switching topologies [29], [30] are mostly adopted.

A. PI Control for Constant Current Applications
For control of I a , the control inputs V oc in power domain and u 1 in control circuit are expressed as, where, = 1 (max) 1 = 2 1 (5) V oc (max) is maximum value of V oc , k 1 and k 2 are constant. V DC is input voltage, n is turns ratio of TR, d pwm is PWM duty cycle of inverter. PI control is mostly used in power electronics controllers. The control function for power actuation is, K P and K I are constant and e is current error. The control u 1 in PI i.e., u PI could be expressed as, The closed loop transfer function G CL (s) could be written as, = ⁄ represents the equivalent resistive load in V L . The denominator d(s) could be written as ( ) = 2 + 2 + 2 where L1 and K I decide the natural frequency ω n and damping ratio ξ depends on controller gain K P . In PI control, ω n is related to frequency of rectified voltage V oc i.e., 2f s , like [14], Where τ c is time constant of the system and τ S is settling time.
To improve response time τ S , f s is increased.
Considering the value of ξ as 0.8, the value of ω n for V oc at 40 kHz is 10 kHz. The value of K P and K I are [12], Considering load R L of arc at 0.09Ω and with L1 at 75 µH, to generate controlled V oc , the value K P is 1.2 and K I is 7500.

B. Practical Implementation of PI Control
For practical validation, one 20 kHz, 400A FBC was developed. For sensing of I a , a Hall effect-based transducer (output: 1A =10mV) was used. The value of k 1 and k 2 a) b) Fig. 2: a) For secondary side control, ideal representation of full-bridge converter, and, b) Typical secondary side DC waveforms respectively was 8 and 0.1. For practical implementation, in control circuit domain i.e., for u PI the value of k p and k i were chosen as 0.2 and 1000 respectively. At nominal value of V DC at 580V the value of V oc (max) was 80V.
Waveforms of different dynamic parameters for making two welding joints under different welding conditions are shown in Fig. 3. In arc welding occasional shorting of electrode with weld pool is considered as disturbance. Here, arc or load disappears, but current continues to flow through the electrode. Arc needs to be regained when shorting gets removed. It is clear from Fig. 3a, that while welding at 100A, in spite of overshoot and undershoot during the disturbances, there was no problem in making a good joint. Similarly, as shown in Fig. 3b, there was no problem of arc stability when welding was performed at higher current level 200A.

III. STC BASED DESIGN AND CONCEPT VALIDATION
Gains in STC are decided based on the process disturbance, it needs the process or load to be clearly modelled. The arc is nonlinear. In stable welding arc zone, arc voltage V L written is, E (V/m) is field intensity of arc, V 0 is anode and cathode drop together, h a is arc length and R a is arc resistance.
Considering error e (6) as sliding surface, the objectives of SOSMC are to meet following conditions, The derivative of sliding surface e for control of CC arc is, Compared to PI control, the approach for gain selection in STC is different. It is ignorant about the parameters of the inverter e.g., frequency f s , etc. For calculation of gains ρ 0 and ρ 1 the information on maximum value of matched disturbances and ̇ are needed, they depend on process behavior. For CC process, at rated current of, say, 400A, the value of is 36 V. Consider, conservatively the current error at 1.0 % of rated current i.e., at 4 A. and the value of R a at 0.025Ω. The bounded values of and ̇ depend on the dynamics of the process, their values for different value of 1̇ are listed in Table I. For a dynamical system where ̇ is dominant, compared to the approach given in [22], [21] would yield smaller gains. To cater different transient conditions, respective set of gains are listed in Table I. It is clear that for design of controller for complete arc welding process [7], [17] the value of each gain would be large. To study the efficacy of STC, simple CC arc is considered where, even, ̇ is ignored where gains are least (see Table I).

A. SOSMC Implementation
Here, to validate the STC function, simplest dynamical situation of arc load was considered, the gains were selected based on minimum disturbance. The controller was analyzed for CC arc welding process where the disturbance ̇ was zero. As shown in Table I, for V oc (14), the calculated gains were ρ 0 (=40) and ρ 1 (> 25000). To generate u STC (15), gains were changed to ρ 01 (=5.0) and ρ 11 (= 3125). To validate high gain STC practically, hardware-based implementation was preferred for prompt corrective action where input (15) was simplified as, Where, Fig. 3: There was no problem in welding, but, overshoot and undershoot were noticed in PI control (k p = 0.  Table II. The hardware circuit for u STC [19] was based on analog switch (see Fig. 5).

B. Experimental Results
The STC controller needs to meet the desired performance in presence of wide range disturbances with multiple origin, like, 1. Sharp change in load or process disturbance 2. Sharp change in reference or process requirement 3. Increase in ripple content in error, particularly, at large current, caused by drooping nature of L1, and or, 4. Ripple content at source V DCdue to availability of metallized polypropylene capacitors with high RMS current handling capability the value of DC link capacitance C DC (µF/A) is drastically reduced.
Multiple welding experiments were conducted using STC functions. For concept validation, values of C DC and L1 were varied. Initially, their values (C DC =55µF and L1=50µH) were kept same like PI control of Fig. 3. Fig. 5 shows waveforms of electrical variables for two different welding conditions. Control of current while welding manually at 100A (see Fig.  5a) during sudden load change was robust, there was no overshoot or undershoot. The control performance was repeated when the load current was 200A (see Fig. 5b). When compared with PI control (see Fig. 3), performance of STC was definitely superior, but the ripple content in u STC was more than u PI . Control response was robust (see Fig. 6a & b) and ripple content in u STC was reduced when L1 was increased to 75 µH. It was also noticed that the ripple content in u STC further reduced when the value of C DC was increased to 110 µF. Increase in value of C DC and L1 would increase cost and reduce power density, might not be acceptable in price sensitive competitive world.

IV. PRODUCT VALIDATION OF STC BASED FBC
Often, even practical validation of a concept might not be a good indicator for its adoption and diffusion into (certain) industry or commercial domains; at least, for SOSMC, it did not happen that way. Apart from demonstrating superior control performance against process disturbances, efficacy of STC, as a superior control function, could be judged if the concept mandatorily meets or does not violate following criteria of a product deemed to be commercialized: 1. High degree of availability of the controller 2. Long service life i.e., the reliability estimate 3. Efficiency or power loss characteristics, and, 4. Cost competitiveness or 5. Favorable performance vs cost parameterization While testing the controller for continuous duty applications in simple CC process, when compared with PI controller's performance, following observations were recorded: 1. STC controller's operating duty cycle (continuous run time in each ten-minute cycle) was much less. Due to frequent thermal shut down, its availability was poor. 2. The power losses in Q1-Q4 and D1-D4 were more 3. PWM controller entered cycle-by-cycle current mode shut-down frequently, and, 4. Prospect of IGBT failure increased, it would make impact on product reliability To understand the origin of frequent thermal shut-down, separate heatsinks i.e., HS1 for Q1-Q4 and HS2 for D5-D6 were used. It was noticed that the thermal tripping was from HS1. It The turn-on power loss of Q1-Q4 depends on current I a reflected at primary of TR whereas their turn-off loss depends on the value of current at turn-off moment i.e., the peak magnetizing current I m . The recovery loss of D1-D4 also depends on I m , plus presence of l TR adds more losses to them. It is important to study the impact of STC on functioning of magnetically coupled inverter.

A. Impact of Control Function on Transformer
For design of TR, N87 Grade EE8020 cores were used. The primary to secondary turns ratio (n p :n s :n s ) was 15:2:2. At V DC of 580V, the value of peak operating flux density B m for maximum value of d pwm (=0.8) was 0.197 T. At 100 0 C, the value of saturation flux density B sat was 0.39 T. Among soft ferrites, linearity of B-H (H: magnetomotive force) curve of N87 Grade was found to be superior. To reduce the prospect of volt-second unbalance in core [28], each IGBT pair (Q1, Q4) or (Q2, Q3) was driven by single pulse transformer. For variable d pwm , employing DC blocking capacitor in inverter would not be effective. Though the inverter was protected by cycle-bycycle current limit, it was effective only at large current. The expression of peak magnetizing current I m is, B m is peak flux density, n p is number of primary turns of TR, l mag is mean length of core and μ r is its relative permeability. B m is directly linked to the control input, like, A C is core area and n s is number of secondary turns. For a particular application, constant value of u STC (its slope m=0 in a cycle) is desirable because it maintains B m at constant value. The switch-on duration of (Q1, Q4) or (Q2, Q3) is same at, say, T1 (see Fig. 7) to ensure average flux in core at zero value. If m is other than zero in a cycle, d pwm would be different in negative (say, T2) and positive (T3) half cycles (see Fig. 7, bottom). The flux variation ΔΦ over one complete cycle is [26], The rate of change of flux density Δ v caused by non-zero m is, C is slope of sawtooth waveform with peak at V st (see Fig. 7) and m is slope of u PI or u STC . The time t sat taken for the core to attain the core saturation is sat = sat ∆ v ⁄ . In this application, sat ≈ 2 m . For reliable operation of the system the condition sat < s (9) needs to be satisfied. If, by design, m is kept comparable to C, swing in B m in a cycle would be large where non-linearity in B-H loop could play adverse impact. Actually, m is considered small so that the value Δ v in one complete cycle is negligible. By design, the operating point is kept in linear zone and I m solely depends on d pwm . Its effective swing depends on the operating point in the B-H loop. For persistent non-zero value of m the value of d pwm keeps increasing (see Fig. 8a) to make the operating B-H loop shift towards one particular quadrant (see Fig. 8b) where the B-H curve for a particular half-cycle would be increasingly non-linear, leading to DC bias in core. The value of µ r (21) would be less in that half cycle, could lead to different value of I m in positive and negative half-cycles. Fig.  8b shows that with positive m, DC bias in core is introduced where H needed to maintain the flux swing is large i.e., the value of I m is large. I m does not contribute to power transfer but causes turn-off loss in Q1-Q4 and recovery losses in D1-D4, its minimum value is desired for reliable performance.

B. Influence of PI Control on Primary Current
In FBC the current at secondary DC side is controlled. But, the actuation for control execution takes place at primary of TR. Ideally, the controller should be designed so that the u PI does not disturb TR in its functioning for voltage translation in isolated manner. It is possible if the average flux in a PWM cycle is negligible. It is decided by the slope m PI of u PI , like, , k hs is constant for current to voltage conversion in hall sensor (10mV/A). Consider transient disturbance at 100 kA/s i.e., ⁄ of 1000 V/s, here, with k p at 1.0 and k i at 1000, the value of m PI is 1000. Fig. 9a and 9b show waveforms of I pri for two different applications. Unform waveform of I pri suggested that the secondary control did not disturb the magnetic circuit of TR. The average flux inside the a) b) Fig. 8: a) Primary voltage of TR with increasing PWM duty cycle, and, b) it shifts the operating hysteresis loop and creates of DC bias in core Fig. 7: Relation between slope of control function and PWM duty cycle core was zero. The value of I m was negligible because the peak value of I pri measured during turn-off at welding current 100A was 14A, and for welding at 200A, it was 27A. Current I pri and I a were relatable and following relation was established: With V st of 10V, for 20 kHz inverter the value of C would be 400000, and the value of Δ v would be at 0.25% of B m .

C. Influence of STC on I pri when C DC = 55 µF and L1=50µH
The slope m STC of u STC is complex and quite different from m PI , it is difficult to define its trajectory. It is expressed as, For same value of ⁄ of 1000 V/s, the maximum value of m STC is 28125. The corrective measure is 28-time faster than PI. It is sensitive near zero error; even ripple in I a could modulate u STC at fast rate at frequency 2f s . The prospect of core saturation is large because the value of Δ v is large, at 7.6% of B m . Fig. 10 shows two sets of waveforms with same value of C DC and L1 as used for PI control in Fig. 9. Even in steady state condition, due to large ripple in V DC and, particularly, in I a resulted large ripple in u STC (see Figs. 10a and 10b). High ripple in u STC resulted DC bias in core and eqn. (26) was violated.

D. Influence of STC on I pri when L1 or CDC was increased
In order to reduce the ripple in I a as well as in u STC , the value of L1 and CDC were increased, one at a time. It is clear from Fig. 11 that, in steady state condition, the ripple content in I a and u STC were only marginally reduced. Still, as shown in two welding conditions, I pri in each half cycle in both cases were quite different (see Figs. 11a and b). The current in one pair of IGBTs was different from the other and (26) was not met.

E. Influence of STC on I pri during Disturbance
Here, to reduce ripple in V DC , C DC was increased to 110 µF. Two experimental waveforms in Fig. 12 demonstrate the response of STC in presence of sharp transient disturbance for welding at 220A. Here, to create anti-sticking force during short-circuiting of electrode with weld pool (V L ≈ 0), transient reference equivalent to ⁄ at 3000V/s was added. STC reacted sharply and changed the value of d pwm almost cycle-bycycle basis (see Fig. 12a). Large peak value of I pri in one pair of IGBT at 60A (220 % of PI) signified that current limit was operational. It resulted increase in conduction and highfrequency losses in inverter. Fig. 12b shows the behavior of I pri when u STC was gradually stabilizing from transient disturbance. The DC bias in core persisted because the operating point was already in nonlinear zone of B-H loop.

V. MODIFIED CONTROL FUNCTION TO BOOST EFFICACY
It was clear that 20 kHz PI controller was reliable but it had overshoot and undershoot. To improve the control response the value of f s is increased (9). Calculated values of different control parameters (see Table III) get better. Soft-switching topologies [29], [30] are employed to eliminate associated highfrequency losses. Power density of inverter also improves  STC could resolve the control related issues. But, even small disturbance or, ripple in I a changed u STC instantly resulting increase in power loss in magnetically coupled controller. There were concerns on reliability and availability features. Ideally, to cater wide range need of the process the theoretical value of two gains would be extremely large (see parameters in red ink in Table I). To avoid DC bias in core, respective value of m STC (27) would need unrealistically large value of C and f s where PI performance would be great (see Table III).
It was noticed that the primary reasons for creation of DC bias in core and associated excess power loss in inverter were: 1. Extremely high gain near zero error (see Table II), and, 2. Existence of ripple in output, or ripple in error.
To make high-gain function reliable, L1 is increased to ensure ripple in e (12) is under acceptable limit, and for soft corrective measure around zero error, following alternate control u SMP [7], [15] of u STC (15) is proposed, It has been practically validated in several process applications [7], [19], [31] that (28) strongly meets conditions of (12). Here, like m PI , slope m SMP of u SMP depends on ⁄ as, SMP = ( 01 + 11 ) sgn( ) = ( , sgn( )) For same value of ⁄ of 1000 V/s, the maximum value of m SMP is around 8000. For C at 400000, compared to STC, the value of Δ v was small at 2.0% of B m . Fig. 13 shows control of current using new function (28) in similar applications (see Fig. 3, 5, 6), control responses were robust. Fig. 14 shows respective primary current I pri . Though, same gains (ρ 10 : 5.0, ρ 11 : 3125) of u STC were used, but DC bias in core was negligible and conditions of (26) were satisfied.

VI. CONCLUSION
This article, for the first time, addressed the critical issues pertaining to the extremely slow pace of adoption and diffusion of widely researched STC concept. It detailed that efficacy of a new concept lied on their effective and efficient utilization to yield definite benefits. It further addressed that, apart from concept validation, commercialization of a product based on a novel idea needed to meet availability and reliability criteria. When the role of control was to directly modulate the duty cycle of PWM based converter, the control function that caused minimum stress to high-frequency power switching devices of a product would be preferred. It elaborated that the processdisturbance based gain selection procedure in STC might not be suitable for vast majority of products that used magnetically coupled power transfer mechanism. The high-gain STC could vary the duty cycle of PWM switches in each half-cycle, it was also highly sensitive to output ripple. Such fast change in duty cycle could inject DC bias in magnetic circuit of transformer that caused excess power loss in inverter. On the other hand, the choice of gains in PI control was more collective. It, not only involved the process dynamics, parameters of power controller were part of the design; smoothing inductance and switching frequency of inverter were part of gain selection.