Solid-State Tuning Restorer for Second-Harmonic LC Filter in Single-Phase Converters

Second-harmonic ($2\omega$) filter is a critical constituent in the single-phase power conversion system and continues to trigger significant research interest. Due to the design and performance trade-offs between passive and active filters, hybrid filters have emerged as an attractive solution that combines the desired features. Moreover, in an existing system, re-tuning, retrofitting, or replacement of the passive filters becomes inevitable due to degraded filter characteristics with parameter drifts. Considering these requirements, an easily integrable active circuit, named solid state tuning restorer (SSTR), is proposed in this work to enhance the performance of an existing LC filter. Along with SSTR, the second-harmonic tuned passive LC filter forms a hybrid filter with adjustable characteristics capable of maintaining the tuned state over a range of operating frequencies and LC parameters. The low volt-ampere (VA) rating of SSTR, being less than 2% of the main converter, facilitates easy integration with the existing dc bus LC filter. Even in case of SSTR failure, the configuration offers a graceful degradation in the filter characteristics without disrupting the main converter. Thus, it improves the performance of an existing second-harmonic LC filter while not compromising its reliability. The operation, design constraints, and control methodology of the proposed SSTR are discussed, and the performance is validated through experiments on a hardware prototype.

Solid-State Tuning Restorer for Second-Harmonic LC Filter in Single-Phase Converters Anwesha Mukhopadhyay , Member, IEEE, and Vinod John Abstract-Second-harmonic (2ω) filter is a critical constituent in the single-phase power conversion system and continues to trigger significant research interest.Due to the design and performance trade-offs between passive and active filters, hybrid filters have emerged as an attractive solution that combines the desired features.Moreover, in an existing system, re-tuning, retrofitting, or replacement of the passive filters becomes inevitable due to degraded filter characteristics with parameter drifts.Considering these requirements, an easily integrable active circuit, named solid state tuning restorer (SSTR), is proposed in this work to enhance the performance of an existing LC filter.Along with SSTR, the second-harmonic tuned passive LC filter forms a hybrid filter with adjustable characteristics capable of maintaining the tuned state over a range of operating frequencies and LC parameters.The low volt-ampere (VA) rating of SSTR, being less than 2% of the main converter, facilitates easy integration with the existing dc bus LC filter.Even in case of SSTR failure, the configuration offers a graceful degradation in the filter characteristics without disrupting the main converter.Thus, it improves the performance of an existing second-harmonic LC filter while not compromising its reliability.The operation, design constraints, and control methodology of the proposed SSTR are discussed, and the performance is validated through experiments on a hardware prototype.Modulation index of main converter and SSTR.S m , S SST R VA rating of main converter and SSTR.

χ, ψ
Ratio of S m to S SST R and ratio of χ to I s /V dc .x, X, X Amplitude, RMS, and phasor of any ac signal/variable x. x Small-signal perturbations considered on any variable x.

I. INTRODUCTION
S INGLE-PHASE power converters find wide applications in different power conversion systems, ranging from power levels of a few watts to several megawatts.Starting from a home appliance like UPS, micro-inverter for roof-top solar photovoltaic (PV) systems [1], battery charger for electric vehicles to megawatt level front-end converter in railway traction systems [2], single-phase voltage source converters (VSC) are adopted at large.Despite its widespread applications, standardised design and control practices [3], proficient handling of second-harmonic ripple remains a challenge in single-phase VSCs, especially in applications demanding high power density, reliable operation, and prolonged life [4].Conventional filter design practices use huge electrolytic capacitors that are vulnerable to premature failure under operating electrical and thermal stresses [5].Continuous assessments of the capacitor's health and residual life are advised [6], [7], [8] to prevent catastrophic failure.The potential failure risk of electrolytic capacitors has led to research on the feasibility of using polypropylene (PP) film capacitors or multilayer ceramic capacitors (MLCCs) in dc-link design [9].However, the cost and volume per unit capacitance, voltage de-rating, and mechanical failures [10] often make such designs impracticable.Alternative dc bus filter structure, consisting of a combination of an inductor (L) and capacitor (C), tuned at second-harmonic (2ω) frequency [11], reduces the capacitance requirement, enhancing the likelihood of deployment of film capacitors [12], [13].
Fig. 1(a) depict single-phase VSCs with dc-bus having (i) capacitive (C) and (ii) tuned LC filters.In Fig. 1(b), the impedance characteristics of C-and LC-filter indicate that the capacitive filters require much higher capacitance to offer a second-harmonic impedance as small as that of a perfectly tuned LC filter.Therefore, applications with stringent ripple constraints prefer a tuned-LC configuration over a capacitive dc bus.In solar PV inverter [12], as a higher ripple in dc voltage affects the maximum power point tracking performance [14], a tuned LC filter is an attractive option [12] for second-harmonic ripple filtering.In railway rolling stock architectures, LC tuned filter is used [2], [15] to mitigate dc bus voltage ripple, which, otherwise, generates beat frequencies [16].
Even for very high power density single-phase VSC designs [17], an LC-tuned filter has been used for second-harmonic ripple filtering by Zhao et al. [18].It is shown that with a high flux magnetic core inductor, LC filter can be designed compact enough to be compared to popular active filter topologies [19].Moreover, better efficiency and reliability are achieved with the adoption of a passive filter.
So far as the filter characteristics are concerned, despite offering excellent attenuation at the resonance frequency, LC filters can not ensure consistent filtering performance under frequency and parameter variations [16], [20].While a high-Q LC filter provides excellent second-harmonic ripple attenuation at its resonance frequency, its efficacy deteriorates drastically at off-resonance frequencies.On the other hand, a low-Q LC filter ensures moderate filtering performance over a wider frequency range around the resonance frequency.Therefore, a tunable LC filter with adjustable filter characteristics is desirable to achieve consistent filter performance over a range of frequencies and parameter variations.A smooth adjustment in the inductance and capacitance values is difficult to accomplish through a switched capacitor or tapped inductor circuit.An active inductor in series with a passive capacitor is emulated in [21] to achieve the above flexibility without introducing a bulk inductor.However, the filter and the main converter fail to operate if the active inductor fails.Adjustable filter characteristics with dc-side shunt- [14], [19], [22] and series-active filters [23], [24] have become popular for their ability to operate independent of the main converter topology, operation, and control.Filtering techniques incorporating an L or LC branch with an additional switching leg [25], [26] or shared main converter leg [27] to reroute and store the 2ω ripple power have been studied in the literature at length.However, the participation of one of the H-bridge legs of the main converter in the filtering makes these solutions specific to the main converter topology and modulation.A common concern for all these topologies is the device or driver failures in the active filter, which leads to a complete shutdown of the main converter.Thus, their adaptability to critical applications gets limited.
Contrary to such approaches, the present work provides a solution which retains the basic LC passive filter and enhances its characteristics with a solid-state tuning restorer (SSTR).As per the tuning requirement of the LC filter, the low-power active circuit of SSTR acts as an electronic inductor or capacitor.Also, it ensures a graceful degradation in the filter characteristics even when it fails.The series LC filter, depicted in Fig. 1(a)(ii), is considered the basic passive filter unit, the performance of which is enhanced by augmenting the proposed SSTR.The preliminary study on the proposed filter configuration and its operation is discussed in [28].The derivation of the proposed filter configuration with SSTR, detailed design considerations, modelling, and closed-loop control scheme discussed in the present article provide a complete guideline for implementing SSTR augmented LC filter.The major contributions of this work are as follows.
1) A low-power rated solid-state tuning restorer (SSTR) is proposed for improving the performance of a secondharmonic tuned LC filter.2) Filter characteristic that is resilient to filter parameters and operating frequency variation over a given range is synthesized using the proposed SSTR.
3) The operation of SSTR as a variable inductance or capacitance is established.The closed-form expressions of the synthetic inductance and capacitance are obtained.4) The proposed SSTR rating is only 2% of the main converter VA for mitigating parameter and frequency variations up to ±10%.
5) The proposed filter is compared for efficiency, volume, cost, reliability, and ease of adoption, with state-of-the-art active second harmonic filters.The results highlight the advantages of the proposed configuration.6) Modelling and closed-loop control design are elaborated with relevant experimental results to validate the filter performance.A brief review of the existing passive and active filters is provided in the next section.Section III proposes and compares different possible restorer connections.The operation of the most reliable filter configuration with a solid-state tuning restorer (SSTR) is discussed in Section IV.The rating of the proposed SSTR is compared with that of the main converter in Section V.The key design criteria and parameters are discussed in Section VI, based on which the proposed LC+SSTR filter is compared with popular active 2ω filters in Section VII.The modelling and closed-loop control are elaborated in Sections VIII and IX.The validation of SSTR operation through experimental results is presented in Section X.

A. Ripple Power in Single-Phase Converters
In a single-phase converter, a mismatch exists between the instantaneous input and output power, resulting in a secondharmonic component in the dc-link current.If the instantaneous ac voltage and current are v ac (t) = vac cos ωt and i ac(t) = îac cos(ωt + φ), respectively, the instantaneous power is given by, where, P dc and p r (t) are the dc and second-harmonic ripple power, respectively.Assuming a stiff dc bus voltage V dc , the second-harmonic ripple current in the dc-link due to p r (t) is calculated in (1), where modulation index,

B. Capacitive DC Bus
The assumption of a stiff dc bus voltage is valid if the secondharmonic ripple current i r (t) is filtered with a capacitor (C dc ), large enough to ensure a small voltage ripple across it.To limit the peak-to-peak voltage ripple to 2.5% of the average dc-bus voltage V dc , the required capacitance [29] is Thus, for a 1 kVA rated converter having a 400 V dc bus, the capacitance requirement is calculated as 795 μF when the operating frequency (f = ω 2π ) is 50 Hz.With increased power levels in the range of 100s of kW to a few MW, as in railway traction or industrial variable frequency drives, the capacitance

C. DC Bus with Tuned LC-Filter
Capacitance requirement is brought down in LC-tuned filter using an inductor, which can be designed as a more reliable component [5] than electrolytic capacitors.Though the realisation of the second-harmonic tuned LC filter is possible with different combinations of L and C values, a cost-optimised selection approach is suggested in [13], where two different combinations are found to be cost-optimised for a 2 kVA converter design.Table I contains the two combinations which will be considered subsequently.The equivalent series resistance (ESR) R r is the combined ESR of the laminated steel-core inductor and film capacitor, measured at the second-harmonic of the nominal operating frequency (f = 50 Hz).
Thus, the capacitance requirement (2 × 795 μF) for a 2 kVA converter is reduced by 5 to 7 times while reducing the ripple voltage at perfectly tuned conditions.In Fig. 1(b), the comparison of the filter impedance characteristics depicts that a capacitive filter requires ten times higher capacitance than an LC filter to achieve second-harmonic impedance of similar order.However, the sensitivity of the LC filter impedance with drift in parameters and operating frequency results in a substantial ripple across the dc bus under detuned conditions, especially for a high-Q LC combination.

D. Active Filters
Adaptable impedance over a range of frequencies and parameter variations is an excellent feature of active filters (AF), apart from the improved power density.The effective use of secondharmonic active filters in Google Little Box Challenge [17] produced significant research on generalized topology synthesis [30], optimal design [31], modelling, and control [32] of active second-harmonic filters.Out of different variants, the distinct active filter (AF) topologies that became popular afterwards were (a) buck converter-based shunt AF [19], (b) series AF [24], and (c) series capacitor stacked buffer (SSB)-based AF [21], depicted in Fig. 2.However, the added devices and drivers increase the system complexities, escalating the possibility of outages due to component failures.In addition, switching losses in the added devices deteriorate the efficiency of most active filters compared to their passive counterparts [18].The proposed SSTR augmented LC-tuned filter combines passive and active filters to retain the merits of the individuals that offer a reliable and efficient filter structure with tunable impedance characteristics.The evolution of the proposed filter configuration out of different possibilities is discussed in Section III.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

III. TUNING RESTORER CONFIGURATIONS
Considering an H-bridge converter as the basic building block for SSTR, three possible filter configurations are compared in Fig. 3.

A. Restorer in Series
Fig. 3(a) depicts a configuration where SSTR is in series with the series LC filter.In the case of detuning, SSTR injects a small voltage that makes the second-harmonic ripple voltage across the filter-branch zero.As connected in series, the SSTR circuit always carries the entire second-harmonic component (i r ) of the dc-link current, with no regard to the degree of detuning.The reliability of the overall system suffers due to the insertion of a series element [33].

B. Restorer across Capacitor
In Figs.3(b), the tuning restorer is connected across the capacitor of the LC branch.While in Fig. 3(b)(i), the ac terminals of SSTR are connected across C r , the dc terminals are connected across C r for Fig. 3(b)(ii).In both the configurations, across terminals 'a' and 'b', SSTR maintains a second-harmonic voltage, superimposed on average dc voltage V dc .Thus, the voltage rating of the SSTR becomes greater than V dc .However, these configurations allow LC filters to operate independently even in the event of failure or non-availability of the restorer.While in operation, only a fraction of the dc-link ripple current (i r ) flows into the restorer, depending on the degree of detuning.

C. Proposed Configuration
In order to synthesise a restorer using low voltage and low current devices, the configuration of Fig. 3(c) is proposed in this work.The passive LC filter, together with active SSTR acts as a hybrid filter unit (LC+SSTR).The restorer is connected across the inductor L r of the LC filter so that an ac voltage appears across its terminals 'a' and 'b'.The peak of this ac voltage decides SSTR voltage stress which can be optimised with respect to the capacitance of C r by the selection of L r and C r combination.Thus, the proposed configuration is realised with lower voltage devices compared to the configurations of Fig. 3(b).The current into SSTR depends on the degree of drift in the resonance frequency from the targeted filtering frequency.Therefore, unlike in the configuration of Fig. 3(a), it does not need to carry the entire second-harmonic component of the dc-link current.Hence, this topology is selected over the other configurations.Fig. 4 depicts a single-phase VSC with SSTR augmented LC filter, which operates as described in the next section.

A. Impedance Emulation
The operation of SSTR is explained with the help of a phasor diagram, shown in Fig. 5 [28], where the phasor of any variable u is represented as U.
1) Perfectly Tuned LC-Branch: At perfectly tuned condition, the voltage across terminals 'a'-'b' is exactly equal and opposite to the second-harmonic voltage drop across C r (v cr,2ω ), making the net second-harmonic voltage (ΔV dc ) across the dc-link zero, as depicted in Fig. 5(a).However, in a practical circuit, due to the equivalent series resistance of L r -C r combination, a small second-harmonic voltage ripple is expected to appear on top of the average dc bus voltage V dc .
2) Detuned LC-Branch: Detuned conditions of the LC-filter are depicted in Fig. 5(b) and (c).For below nominal operating  frequency or negative drift in the L r or C r , the resonance frequency of the LC branch exceeds the second-harmonic of the operating frequency, making the branch capacitive, as indicated by the ΔV dc phasor lagging the 2ω current phasor I 2 by 90 • in Fig. 5(b).Similarly, for positive drift in frequency (ω), L r , or C r from their respective nominal values, the resonance frequency becomes higher than the second-harmonic of the operating frequency.As a result, the LC branch becomes inductive, causing the ΔV dc phasor to lead the 2ω current phasor I 2 by 90 • , as depicted in Fig. 5(c).
3) SSTR as Capacitor: In the case of detuned condition caused by below nominal parameter or frequency drifts, SSTR is operated to act as a synthetic capacitor which appears in parallel to L r and compensates ΔV dc by increasing the effective inductance of SSTR augmented LC-branch.Fig. 5(d) depicts the phasors pertaining to this.

4) SSTR as Inductor:
For detuning caused by parameters or frequency drifts above nominal values, SSTR is controlled to mimic an inductor.The emulated inductor effectively reduces the inductance of the LC branch and restores the tuned condition.Fig. 5(e) depicts the phasors at this condition with SSTR enabled.
To obtain the closed-form expressions of the emulated capacitance and inductance values, a ±10% variation is considered in the operating frequency, capacitance, and inductance ω, C r , and L r , respectively.The emulated capacitance and inductance are expressed as per unit of the nominal capacitance (C rn ) and inductance (L rn ) of the LC branch, respectively.Table II provides the expression of emulated impedance Z e , and its corresponding L e or C e for ±10% variation in the nominal frequency ω and parameters (L r , C r ).

B. Emulation Strategy
The emulation of required value of inductance and capacitance by SSTR is achieved by regulating its terminal voltage v ab or the current drawn i 2A .In v ab control, the SSTR tracks the second-harmonic voltage drop (v cr,2ω ) across C r but with the opposite phase.This ensures Δv dc = 0 across the main dc bus.
As the amplitudes of both v cr,2ω and i r are decided by the main converter power level, the volt-ampere (VA) rating of the SSTR is directly related to the main converter VA.In addition, the selection of C r and L r combination decides SSTR voltage and current ratings for the parameter and frequency variations within the range of interest.

V. RATING OF SSTR
The volt-ampere (VA) rating is an important figure of merit for active filters, indicative of their silicon area and loss.A lower VA rating allows the filter to be designed with higher efficiency [21], [24].The SSTR VA rating is decided by the emulated capacitance and inductance values and the voltage v ab across Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
it.Thus, for an emulated impedance Z e , the VA of SSTR is where, V ab is the RMS value of the voltage (v ab ) across L r .

A. Voltage Rating
The voltage rating of the SSTR is decided by the voltage across its terminals 'a' and 'b', which is also same as the voltage across L r .As the second-harmonic ripple filtering demands v ab = −v cr,2ω , using (1), V ab can be calculated for the rated power operation, For linear modulation, the dc terminal voltage of SSTR across capacitor C A should be At the rated operating condition, considering the worst case of parameters and frequency drifts (i.e.C r = 0.9 C rn and ω = 0.9 ω n ), the maximum value of V ab is obtained.

B. Current Rating
In the proposed configuration, SSTR appears in parallel to the inductor L r .Therefore, the second-harmonic component (i 2 ) drawn from the dc-link current gets divided between the SSTR and L r .The current through SSTR is decided by the emulated impedance Z e , as given in Table II for different cases and degrees of detuning.Therefore, the maximum current through SSTR is calculated considering the worst-case deviation in frequency and mismatch in parameters.Depending on the emulation of inductance L e or capacitance C e , SSTR draws current either in phase or in phase opposition to i 2 , as depicted in Fig. 5(d) and (e).Referring to Fig. 5, applying KCL at node 'a' in Fig. 4 for the capacitance emulation (C e ) case, Thus, for capacitance emulation, the current i 2A,C e in SSTR is maximum when L r draws the maximum current.Similarly, for L e emulation current drawn by SSTR, Thus, for inductance emulation, the current i 2A,L e in SSTR is maximum when L r draws the minimum current.For ±10% deviation, the maximum currents in SSTR during capacitive and inductive emulation are calculated as 0.524 î2 and 0.3174 î2 , respectively.Taking the maximum amplitude of current (0.524 î2 ) into account, devices and the filters L A , C A , and C 3 are designed.
The steps of derivations are given in the Appendix.

C. VA Rating
From the RMS values of v ab and i 2A , the ratio of SSTR VA to the main converter VA is derived as where, I s and V dc are the average values of i dc and v dc , respectively.The expression in (8) indicates that I s and hence, χ increase with increased load.Let, χ per unit of the fraction The variation in ψ with L r and C r is graphically depicted in Fig. 6 for nominal operating frequency ω n , and its ±10% variations.It is noted that for ω = 0.9 ω n (Fig. 6(a)), SSTR always acts as capacitor, making ψ negative.For ω = 1.1 ω n , SSTR mostly acts as an inductor (Fig. 6(c)).However, at the nominal frequency ω = ω n (Fig. 6(b)), SSTR needs to emulate both inductance and capacitance, depending on the sense of deviation in L rn and C rn .The higher absolute value of ψ (1.515), which corresponds to capacitive emulation (Fig. 6(a)), decides the VA rating of the SSTR.Thus, referring to Fig. 6, for a 2 kW rated main converter with 400 V dc bus voltage, the SSTR should be rated for × 2000 = 37.9 VA (10) which is less than 2% of the main converter VA rating.The derivation is elaborated on in the Appendix.The design of SSTR is discussed briefly in the next Section to facilitate a consistent comparison of VA rating, efficiency, volume and cost with existing second-harmonic filters.

VI. DESIGN PARAMETERS
The design of the proposed LC+SSTR filter is divided into (a) passive LC filter selection and (b) active SSTR design for mitigating a specified degree of detuning.However, the passive filter parameters also decide the active SSTR design.
The proposed filter is designed for a 2 kVA rated main converter with 400 V dc bus voltage and 230 V, 50 Hz ac voltage.Table III contains the design details of the LC filter and SSTR.

A. Design of L r and C r
For the selection of L rn and C rn , out of the two combinations listed in Table I, the second set of parameter values are chosen due to (a) lower off-resonant impedance and (b) lower VA requirement for the SSTR.The detailed considerations for withstand voltage and ripple current rating of C r are discussed in [13].For inductor design, two different magnetic materials are considered: (a) Si-steel laminations [34] and (b) Kool-Mu HF [35].Given the L r , C r parameters, the design of the SSTR is described next.

B. SSTR Design
r Choice of L A : Inductor L A is designed to limit the switch- ing ripple to less than 40% of the steady-state amplitude of î2A , maximum value of which is î2A,C e = 0.524 × î2 = 2.6A.Therefore, the peak-to-peak switching ripple is limited to less than 1 A. As the switching ripple gets further filtered by C 3 and is not injected into the grid, allowing 40% ripple in i 2A is acceptable considering the trade-off between inductor size and ripple magnitude.However, the grid-side filter (L ac ) of the main converter is designed [32] to keep the switching ripple small enough to satisfy the standard for harmonic limits [36].
r Choice of C 3 : As C 3 appears in parallel with L r , the with- stand voltage rating (V u ) of C 3 depends on the maximum amplitude of the voltage vab that appears across L r .The voltage across L r is given by For 2 kVA rated main converter design, considering the maximum parameter variations of ±10% for both C r and L r from their nominal values, the worst-case amplitude of For inductive emulation with L r = 1.1L rn and ω = 1.1ω n , the maximum possible value of v ab is 24.5 V. Thus, the capacitor C 3 must be rated for a withstand voltage greater than 30 V for the present design case.
r Choice of V A and C A : The average value of v A is chosen so that ( 5) is satisfied.The assumption is true as long as the capacitance C A is large enough to ensure the ripple in v A is negligible, i.e., v A = V A .In practice, a higher ripple in It is to be noted that the ripple across C A appears at 4ω frequency since SSTR draws second-harmonic current i 2A .
If the modulation index of SSTR is M A , at steady-state, the peak to peak 4ω ripple in v A is The average voltage across C A is maintained at 40 V (≥ vab,max = 30 V).To limit the ripple Δv A to less than 10% of V A , the capacitance for C A is chosen as 680 μF.Similarly, the selections of the main converter filter parameters (L ac , C f ) are governed by switching ripple criteria, following the steps suggested in the design in [32].

VII. COMPARISON WITH EXISTING FILTER TOPOLOGIES
In literature, the LC-hybrid filters and their applications are discussed in the context of ac power line harmonics mitigation [37].As opposed to such hybrid filters, the proposed filter is designed to filter out the second-order harmonic component from the dc-link current of the single-phase converters.Hence, its comparison with the ac-side hybrid filter topologies is not accordant.Therefore, dc bus second-harmonic filters are considered for a fair comparison to assess the proposed configuration in terms of reliability, cost, volume, and efficiency.The secondharmonic active filter topologies that stand out for efficiency and power density in the Google Little Box Challenge [38] are chosen here for comparison and are depicted in Fig. 2. The following attributes are compared, viz., 1) Ease of retrofitting: Conventionally, single-phase converters are found with a capacitive or LC-tuned dc-link in different applications [2], [12], [15], [39].Adoption of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IV COMPARISON OF FAILURE IMPACT OF DIFFERENT ACTIVE SECOND-HARMONIC FILTERS IN SINGLE-PHASE CONVERTERS TABLE V COMPARISON OF LC+SSTR WITH STATE-OF-THE-ART SECOND-HARMONIC ACTIVE FILTERS FOR 2 KW-RATED MAIN CONVERTER
the shunt, series or SSB-based active filters to an existing system requires access to the dc-links and involves substantial modification.However, the SSTR, rated for less than 2% of the main converter rating, can be easily retrofitted across the existing LC filter inductor, requiring minimal intervention in the dc links.2) Impact on reliability: A major concern with active filter (AF) is its reliability.Failure in the AF components can damage the dc source and loads by injection of secondharmonic current.Table IV compares the failure impact of different AF circuits on the main converter operation.The comparison indicates that the impact of active filter fault and subsequent isolation causes increased voltage stress on switches.However, even with SSTR inoperative in the proposed configuration, the LC-filter still provides a path for the second-harmonic ripple current with gracefully degraded impedance characteristics.3) VA rating: Referring to Figs. 2 and 4, the volt-ampere (VA) rating of the active circuits in the filter are calculated below.
A comparison of different attributes in Table V indicates that the SSTR in the proposed filter configuration has the lowest VA rating.
The above three aspects of different filters can be compared qualitatively from their configurations.However, in the state-of-the-art topologies, the literature reports the use of varied technologies for devices, capacitors, and inductors, making the comparison of efficiency, volume, and cost challenging.Moreover, most of the references of Google Little Box Challenge provide the combined efficiency, volume and cost of the main converter and second-harmonic active filter.Therefore, to ensure a fair and consistent comparison, our best efforts have been put into designing and estimating the efficiency, volume and cost of the existing topologies using devices and passives of similar technologies.For devices: Infineon Optimos (100 V for Series, SSB, and SSTR and 650 V for Shunt AF) [40], for capacitors: polypropylene film capacitors and for switching ripple filter inductor design: Kool Mu HF cores have been chosen.4) Efficiency: As the filter principally consists of reactive elements (capacitor C r , inductor L r , and emulated inductor or capacitor by SSTR), the efficiency figure of merit (η F OM ) is defined as below.
where P LC+SST R is the total loss in the filter and V dc I 2 is the input VA to the filter.The input volt-ampere to the SSTR augmented LC filter is calculated as the product of V dc and RMS value of i 2 .Since the capacitive emulation results in higher current flow in both SSTR and L r , losses are slightly higher for capacitive emulation.As it is equally probable for SSTR to emulate capacitive and inductive characteristics, to estimate the worst-case efficiency of the proposed filter, the capacitive emulation case is considered here.In Fig. 7, the distribution of losses between SSTR and LC filter is shown for both the designs of L r , indicated in Table III.It can be noted that the core loss is negligible for the Kool Mu High Flux core.However, its lower permeability leads to a higher number of turns in the inductor.Consequently, copper loss contributes a significant part of the total loss in this case.On the other hand, the core loss has the major contribution to losses in the Si-steel laminated core inductor.To calculate the efficiency of the filter, consisting of LC and SSTR, different loadings on the main converter are considered.Fig. 8 depicts the efficiency FOM of SSTR for 400 V main dc bus.It is seen that, at lower power levels, the efficiency of the Kool Mu HF core is higher as the copper loss is small at light loads.For the Si-steel core, the dominance of core loss over copper loss results in better efficiency as the operating power level increases.
The efficiencies of the other AF topologies are calculated using similar FOM, based on their individual VA ratings calculated using (14).Table V compares the efficiencies, which indicate that SSTR augmented LC filter has an efficiency closest to SSB, which is reportedly the most efficient amongst the existing active filters [21].5) Volume: The volume of the SSTR augmented LC filter is calculated for both the Kool Mu HF core and Si-steel core based L r designs.Fig. 9, depicting the volumes of different components, indicate that the overall volumes in both designs are nearly equal (0.56 ltr).The volume for the proposed and the existing filters are calculated based on individual component volume approach [21].6) Cost: The cost of the SSTR augmented LC filter is calculated for both Kool Mu HF and Si-steel core inductorbased design in Fig. 10.The cost is calculated based on individual component prices from Digikey catalogue [41] and quotations by local manufacturers.In hardware proto-type design, Si-steel core is used in manufacturing the inductor L r due to its lower cost and ready availability.

VIII. DYNAMIC MODELLING
The switching cycle averaged equivalent circuit of SSTR, along with the LC filter, main converter, and dc bus components, are depicted in Fig. 11(a) where the main converter is represented as a current source [32].Fig. 11(b) depicts the small-signal     A1 in the Appendix.

A. Interpretation of Load Characteristics
The load characteristics in Fig. 12(a) indicate the following: 1) The effective loading (Z L ) on SSTR changes from inductive to capacitive beyond the frequency 2) For the present design, ω rl is always greater than 2ω considering the possible variations in the operating frequency.Hence, the load characteristics remain predominantly inductive.3) Since Z 2 << Z 3 < Z 1 for 2ω < ω rl , ĩ2A changes the current distribution in L r only.Thus, the current in C r does not exceed the rated value of −i r , but i 2L does for C e emulation, as discussed in Section V.

B. Plant Model for Voltage and Current-Mode Control
Considering ṽab and ĩ2A as change in outputs due to control input dA for the voltage-mode and current-mode control, respectively, the transfer functions for the plant models are found as Table VI contains the expressions for G pv (s) and G pi (s) in terms of Z L (s).The frequency responses are depicted in Fig. 12(b).It can be noted that the effect of change in the loading characteristics is more prominent in the current-controlled plant than in the voltage-controlled plant.Referring to the expressions of G pv (s) and G pi (s) in Table VI, the dominance of Z L over the filter impedance (sL A + R A ) justifies the visible impact of Z L (s) characteristics on G pi (s).In a voltage-controlled plant, owing to the presence of Z L (s) in both numerator and denominator (Table VI), the change in the characteristics of Z L (s) is not captured in the plant model.Based on the derived plant models, the design of the control loop for SSTR is described in Section IX.

IX. CLOSED-LOOP CONTROL
The control of SSTR can be performed either by current mode control, which ensures i 2A = i 2 − i 2L or by voltage mode control which makes v ab = −v cr,2ω .SSTR is operated in closedloop control with unipolar modulation [42].The control process consists of two steps: (i) synthesis of reference and (ii) tracking of reference.
A. Synthesis of Reference 1) Current Reference: Detuning, irrespective of the causes, is reflected in the current i 2 , drawn by SSTR augmented LC filter.At perfectly tuned condition and nominal operating frequency, i 2L = i 2 = −i r which makes i 2A = 0. Otherwise, i 2L = i 2 < −i r , as explained by the phasor diagram in Section IV.Thus, the difference between −i r and i 2L gives the The dc-link current i dc is either measured [21] or estimated [29] to obtain its second-harmonic component i r .Both methods have certain limitations, as discussed in [29].Therefore, in the present work voltage-mode control scheme is implemented for SSTR.
2) Voltage Reference: In voltage mode control, ideally, the overall second-harmonic ripple across the main dc bus is made zero.This indicates that the second-harmonic ripple i r flows through a zero impedance path, consisting of an LC filter and SSTR.In order to achieve this, the instantaneous voltage across L r is made to track the negative of the second-harmonic voltage component (−v cr,2ω ) across C r .Fig. 13 shows the closedloop control schematic for the voltage-mode control where the second-harmonic voltage across C r is sensed and filtered through a second-harmonic bandpass filter (2ω BPF).The output of the BPF, after phase inversion, gives the reference voltage Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.v * ab,q for the voltage mode control.If SSTR is controlled to apply v * ab,q across L r , it ensures zero second-harmonic ripple across the main dc link, irrespective of the operating frequency and the tuning status of L r and C r .However, SSTR can apply the desired voltage across L r as long as v A > v ab .This condition can get violated due to the losses in SSTR switches and passives.
The loss is compensated by a small amount of active power drawn from the main converter at second-harmonic voltage and current, as suggested in [21].The control loop to maintain v A is shown in Fig. 13, which adds v * ab,d to v * ab,q to synthesize the resultant reference voltage for v * ab .The design of the voltage controller to track the reference is explained next.

B. Design of Controller
In the voltage-mode controller design, the lightly damped resonant pole in the plant model (G pv (s)) makes the design of a high-bandwidth controller challenging.However, considering variations in operating frequency, high gain at 100 Hz (±10%) is essential.With these considerations, a resonant controller H v (s) is designed as below, following the methods suggested in [32].
The controller resonant pole at 2ω is changed adaptively with operating frequency ω.The steady-state and dynamic performance of the controller is verified in experimental hardware.

X. EXPERIMENTAL VERIFICATION
The performance of the proposed filter configuration is verified in a scaled prototype of a single-phase VSC operated in inverter mode.The design parameters of the filter components are as given in Table III.With a nominal ac voltage of 110 V, 50 Hz, the main converter is tested up to 500 W, and its average dc-link voltage is regulated at 200 V. Fig. 14 depicts the major components of the experimental prototype.The steady-state and dynamic performances are captured to validate the operation of the filter configuration.
1) Verification of Frequency Variation Effect on LC Filter Performance: Main converter ac voltage v ac , current i ac , dc voltage ripple Δv dc , and second-harmonic current i 2 , drawn by LC filter are depicted in Fig. 15 for operating frequency equals to the nominal frequency ω n , below ω n , and above ω n .At ω = ω n , depicted in Fig. 15(b), dc bus ripple is negligibly small.However, it is not zero since the capacitor C r and L r are not perfectly tuned at f n = 50 Hz.Also, the resistive drop across the LC branch causes a small voltage drop across the dc bus.It can be noted that the ripple in the dc bus voltage is almost in-phase with the current i 2 .For ω < ω n (Fig. 15 2) Effect of SSTR on Ripple: Fig. 16(a) and (b) depict the main converter source current i s and ripple in the dc bus voltage Δv dc without and with SSTR, respectively at an operating frequency of 45 Hz.It can be noted that without SSTR, the second-harmonic ripple is substantial in both i s and v dc .On enabling SSTR, the ripple gets reduced to nearly zero.Fig. 16(c) and (d) depict a similar impact of SSTR on the ripple for 55 Hz operating frequency.
3) Dynamic Change in DC Voltage Ripple on Enabling SSTR: The reduction in the second-harmonic ripple in the dc bus voltage on enabling SSTR is depicted in Fig. 17 for operating frequency f = 45 Hz (90%f n ) and f = 55 Hz (110%f n ), respectively.The dynamic performance indicates that it takes 80 ms to reach the steady state.
4) Dynamic Change in Operating Frequency: The operating frequency is dynamically changed from the nominal frequency by ±10%, while the main converter runs without and with SSTR enabled.In Fig. 18    ms, without enabling SSTR.It can be noted that the steady-state voltage ripple (Δv dc ) increases from 2.5 V to 6 V in both cases.Next, the frequency is changed, keeping SSTR enabled, as depicted in Fig. 18(c) and (d).It is seen that even at the nominal frequency of 50 Hz, the dc bus ripple is reduced to 0.7 V as against 2.5 V in the previous case.This is due to LC parameter mismatch in the practical implementation, which causes the resonance frequency to deviate from 100 Hz (2 × 50 Hz).The effect of parameter mismatch on Δv dc at the nominal operating frequency is mitigated by enabling SSTR.When the frequency changes to 45 Hz and 55 Hz in Fig. 18(c) and (d), respectively, the steady-state dc bus ripple ΔV dc does not increase irrespective of the frequency change.The SSTR operation ensures a small ripple (0.7 V-0.8 V) throughout.During transient, a higher Δv dc is observed due to the low-bandwidth control dynamics; however, the transient ripple remains less than the ripple without SSTR.
5) SSTR Current: Fig. 19 shows the current drawn by SSTR at ±10% off-nominal frequencies.The ripple voltage across C r is measured by configuring the voltage probe in ac coupled mode.The figure shows the inverted ripple as -v cr,2ω .
For f = 45 Hz (Fig. 19(a)), before enabling SSTR -v cr,2ω > v Lr and current drawn by SSTR i 2A = 0. SSTR is enabled, the voltages gradually change to make -v cr,2ω = v Lr .The current i 2A is nearly in phase opposition with respect to i 2L , indicating capacitance emulation by SSTR.Also, the inductor current i 2L increases as predicted in Section V. Similarly, for f = 55 Hz (Fig. 19

Fig. 1 .
Fig. 1.Single-phase H-bridge converter (a) with passive (i) capacitive and (ii) LC tuned filter and (b) frequency responses of the impedance characteristics of capacitive and LC filters for different parameter values.

Fig. 3 .
Fig. 3. Circuit configurations of LC filter where solid-state tuning restorer (SSTR) is connected (a) in series with LC, (b) across C, and (c) across L.

Fig. 6 .
Fig. 6.Variation in the ratio of SSTR to main converter VA, ψ expressed per unit of the effective dc admittance, with respect to variation in C r and L r for (a) ω = 0.9 ω n , (b) ω = ω n , and (c) ω = 1.1 ω n .

Fig. 7 .
Fig. 7. Loss breakup in LC+SSTR filter, designed for 2 kW rated main converter having 400 V dc bus: Filter inductor L r designed with (a) Si-steel laminated core and (b) Kool Mu HF core.

Fig. 8 .Fig. 9 .
Fig. 8. Efficiency figure of merit for SSTR augmented LC filter at different main converter loading.

Fig. 10 .
Fig. 10.Comparison of the cost of LC filter and SSTR for two different considerations of the core material, viz.Si steel laminations and Kool Mu HF.

Fig. 14 .
Fig. 14.Hardware set-up depicting main converter, LC filter, and solid-state tuning restorer (SSTR) together with necessary sensing, control, protection, and measurement circuits and instruments.

Fig. 16 .
Fig. 16.DC link current (i dc ) and ripple in dc bus voltage (Δv dc ) without and with SSTR enabled, respectively, at an operating frequency of (a) and (b) 45 Hz; (c) and (d) 55 Hz.
(a) and (b), the operating frequency is changed from 50 Hz to 45 Hz and 50 Hz to 55 Hz, respectively, over 120

Fig. 17 .
Fig. 17.Main converter ac voltage, current, and dc voltage ripple without and with SSTR enabled for operating frequency (a) 45 Hz and (b) 55 Hz.

Fig. 19 .
Fig. 19.Voltage across C r (captured in inverted ac coupled mode), across L r , current (i 2L ) though L r , and current (i 2A ) drawn by SSTR for operating frequency (a) 45 Hz and (b) 55 Hz.
(b)), SSTR emulates an inductive behaviour.XI.CONCLUSION This work proposes a hybrid filter configuration with a solidstate tuning restorer for enhancing the performance of a passive LC-tuned filter.The low VA rating of the SSTR (2% S m ) makes it suitable for use in high-power applications.Positive aspects of reliability, retrofit ability and efficiency are assessed and compared with the state of art filters.The impact of frequency and filter parameter variations on the filter performance is studied through phasor analysis.Dynamic modelling and closed-loop control scheme are elaborated for voltage-mode control of SSTR and validated with detailed experiments.The results confirm the flexible filter characteristics, resilient to the resonance and operating frequency deviation.

hybrid filter, solid-state tuning restorer, online tuning restoration, graceful performance degradation.
Main converter dc input voltage and current.V dc , I s Average values of v dc , i s .p, P dc , p r Instantaneous, dc, and second-harmonic ripple power.i dc , i r , i sw Total dc-link current, its second-harmonic and switching ripple component.C f , C dc Dc bus capacitance for filtering switching ripple and second-harmonic ripple.L r , C r Inductance, capacitance of second-harmonic tuned LC filter.L rn , C rn Nominal values of L r , and C r .i 2 , i 2L , i 2A Second-harmonic current through C r , L r , and SSTR.v ab , v A Ac and dc terminal voltages of SSTR.V A , d A Average of v A and duty cycle of SSTR.C 3 , C A Ac and dc port capacitance of SSTR.Δv dc , Δv A Peak-to-peak ripple in v dc and v A .M m , M A

TABLE II EXPRESSIONS
OF EMULATED L e OR C e FOR ±10% VARIATION IN L r , C r , AND ω FROM THEIR RESPECTIVE NOMINAL VALUES

TABLE III DETAILS
OF COMPONENT AND PARAMETERS OF LC+SSTR FILTER FOR 2 KVA RATED MAIN CONVERTER

TABLE VI IMPEDANCE
OF EQUIVALENT LOADS AND TRANSFER FUNCTIONS OF PLANT MODELS