Analysis and Design of the Resonant Converter for Low Output Current Ripple

2772 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 7, JULY 2012

Analysis and Design of the LmC ResonantConverter for Low Output Current Ripple

B. C. Hyeon, Student Member, IEEE, and B. H. Cho, Fellow, IEEE

Abstract—In this paper, an LmC resonant converter is pro-posed for low output current ripple, and its steady-state analysisand design guidelines are investigated. In the LmC converter,the current stress of the output capacitors is reduced due to theinductive output filter, and the series resonant inductor is removedwhile maintaining the benefits of an LLC resonant converter suchas zero-voltage switching and narrow switching frequency range.The mode analysis is described, and the steady-state characteris-tics are explored using the extended describing function. Based onthe result, the design guidelines are suggested. A 240-W 24-V/10-Ahardware prototype is built and tested to verify the analysisresults and to evaluate the performance of the LmC resonantconverter.

Index Terms—Low current ripple, resonant magnetizing induc-tance, resonant power conversion, zero-voltage switching (ZVS).

I. INTRODUCTION

IN RECENT years, the LLC resonant converter has beenwidely used for an isolated dc/dc converter due to its attrac-

tive features such as the following: 1) zero-voltage switching(ZVS) of MOSFETs and zero-current switching (ZCS) of rec-tifier diodes; 2) narrow switching frequency range accordingto the load current; 3) simple circuitry and low cost; and4) utilization of parasitic elements for soft switching [1]–[9].

However, since an LLC resonant tank has a current-sourcecharacteristic and it is connected to a capacitive output filter,the ripple current of the output capacitor is large. Generally,an electrolytic capacitor has a large equivalent series resistanceand a low current rating. Hence, several parallel capacitors areneeded to satisfy the current rating and to reduce the voltageripple and power loss. To overcome the problem, a multiphaseLLC resonant converter is suggested in [10] and [11]. Althoughthe N -phase converter reduces the ripple current of the outputcapacitor to 1/N , active and passive elements of the converterare increased by N times. Furthermore, an additional control isrequired for the current sharing [12]–[16].

To reduce the ripple current without additional elements,a half-bridge magnetizing-inductance (LmC) resonant dc/dc

Manuscript received March 24, 2010; revised September 16, 2010 andFebruary 20, 2011; accepted May 23, 2011. Date of publication December 9,2011; date of current version February 17, 2012.

B. C. Hyeon was with the School of Electrical Engineering, SeoulNational University, Seoul 151-744, Korea. He is now with the AdvancedResearch Group, Visual Display Division, Samsung Electronics, Suwon-si 443-742, Korea (e-mail: [email protected]).

B. H. Cho is with the School of Electrical Engineering, Seoul NationalUniversity, Seoul 151-744, Korea (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2011.2179269

Fig. 1. Circuit diagram of the half-bridge LmC resonant converter.

converter was proposed as a topology challenge [17]. Thecircuit diagram of the LmC resonant converter is shown inFig. 1. While it is similar to the LLC half-bridge resonantconverter, there are two differences. In the LmC resonantconverter, the series resonant inductor is eliminated in theprimary side of the transformer (T1), and the main resonantinductor, which transfers the power to the load, is changed to themagnetizing inductance (Lm). The output filter inductor (Lo)is used to reduce the ripple current of the output capacitor (Co),maintaining the advantages of the LLC resonant converter suchas soft switching and narrow switching frequency range. Thus,the converter is suitable for a high-current application with alow cost and a high power density.

In this paper, the steady-state characteristics of the half-bridge LmC resonant converter, such as voltage gain andpeak value of the resonant elements, are analyzed. For this,the extended describing function (EDF) method with high-order terms is used. Based on the result, the practical designconsiderations are proposed for the efficient power stage. Toshow the validity of the analysis results and the performanceof the converter such as ZVS and low output ripple current,the simulation and experiments with a 24-V/10-A hardwareprototype are carried out for 320- to 400-V input range.

II. MODE ANALYSIS OF THE PROPOSED CONVERTER

A. Circuit Descriptions

A circuit diagram of the LmC resonant converter is shownin Fig. 1. It is composed of a half-bridge switch network, anLmC resonant tank, a transformer, a center-tapped rectifier, andan output filter. For comparison among the previous resonantconverter topologies, the equivalent resonant circuits during theenergy-transferring period are shown in Fig. 2. In the case of theLLC resonant converter, the main resonant tank is composed ofLs and Cs. Lm did not resonate because its voltage is clampedto the reflected output voltage. Due to the relatively large Ls, anadditional inductor is required, or a sectional winding techniqueof the transformer is applied [4], [6].

0278-0046/$26.00 © 2011 IEEE

HYEON AND CHO: ANALYSIS AND DESIGN OF LmC RESONANT CONVERTER FOR LOW OUTPUT CURRENT RIPPLE 2773

Fig. 2. Comparison of the resonant circuit configurations. (a) LLC resonantconverter. (b) LmC resonant converter. (c) Series parallel resonant converter.(d) Parallel resonant converter.

However, the sectional winding technique makes the designof the transformer difficult. Since the resonant tank of the LLCconverter has a current-source characteristic and a capacitiveoutput filter, the ripple current of the output capacitor andrectifier is large.

On the other hand, the main resonant tank of the LmCresonant converter is configured with Lm and Cs. Thus, theLmC resonant converter shows the simplest circuit structure inthe primary side of the transformer among the various resonanttank configurations, and it provides the following advantages.

First, the peak and rms currents of the resonant tank arereduced by the large resonant inductance Lm. Second, theripple current of the output capacitor is decreased due to theinductive output filter. As a result, a number of output capacitorsare replaced by a small inductor Lo. Therefore, the volume andcost of the converter are decreased.

B. Mode Analysis of the LmC Resonant Converter

Before the explanation of the modes of operation, the fol-lowing assumptions are made: 1) SH and SL are driven by apush–pull-type gate driving voltage, and both switches have50% on-time with respect to the switching period TS ; 2) thecurrent of Lo is constant, and its average value is the same as theload current; and 3) the junction capacitances of the MOSFETsand diodes are neglected.

Before t0, SH and DH are turned on. Lm and Cs resonatewith an input voltage source vi, and then, power is transferredto the load from vi. At t0, SH is turned off, and mode 1 starts.The circuit operations according to each mode are shown inFig. 3, and the waveforms are shown in Fig. 4.

Mode 1 (t0−t1): After SH is turned off, the voltage acrossSL is discharged by the current of Lm. When this voltagebecomes zero, the body diode of SL conducts, and ZVS isachieved. SL is turned on during this mode. The negativevoltage across Cs decreases the current of Cs and DH . Sincethe current of Lo is almost constant, DL starts conducting, andthe currents of DH and DL are commutated. This commutationmeans that the terminal voltage of T1 is zero and the current

of Lm is constant. The absolute voltage across Cs reaches itsminimum value, and it can be approximated to be constantduring mode 1 because the time period of this mode is muchshorter than TS . At t1, the current of DH becomes zero, andthe voltage across DH starts to increase.

Mode 2 (t1−t2): The main resonance of Lm and Cs oc-curred, and the power is transferred from Cs to the load. Theleakage inductance of T1 can be neglected because Lm is large.

Mode 3 (t2−t3): At t2, SL is turned off. The negative currentof Cs conducts through the body diode of SH . The currentcommutation is similar to that in mode 1.

Mode 4 (t3−t4): At t3, SH is turned on. The voltage acrossthe input terminal of the resonant tank is equal to vi. The circuitoperation of this mode is analogous to that of mode 2 except thepolarity of the current and voltage of the resonant elements.

In the waveforms, the dc value of the capacitor voltage isequal to half of the input voltage due to the 50% duty ratio of theswitches. It means that the transformer is reset by the resonantcapacitor voltage naturally. Due to the full-wave rectification,the reflected current of Lo is a square wave. Thus, the currentshape of Cs is triangular because it is the sum of the resonantcurrent of Lm and the reflected current of Lo.

III. STEADY-STATE CHARACTERISTICS

OF THE CONVERTER

Various methods have been studied to analyze the resonantconverter [18]–[23]. To find the steady-state characteristics ofthe LmC resonant converter, EDF is applied because it providesthe steady-state solutions of all states regardless of the resonantscheme [19], [21].

In the LmC resonant converter, the output voltage is theaveraged value of the voltage across Lm (vLm). However, vLm

is unusable in a state equation because it is not a state of Lm.Therefore, a dummy element (Cd) which is not a physicaldevice is inserted. Then, the state variable vCd is usable in thestate equation as a voltage across Lm because vCd is equal tovLm. In the final result, Cd will be zero to cancel its effect.

To establish the state equations, nonlinear waveforms areexpressed using the functions {f1, f2, f3, f4}. Thus, the equiv-alent circuit for the state equations is configured and shown inFig. 5.

As observed in Fig. 4, the waveforms of the states{iLm, vCs, vCd} are quite similar to a sinusoidal waveformbecause the switching frequency is near the resonant frequencyωo defined in

ωo(2πfo) =1√

LmCs

. (1)

Therefore, the states of the resonant elements can beapproximated to the combination of sine and cosine termslike

vCs(t) ∼= vCss(t) sin ωst + vCsc(t) cos ωst

vCd(t) ∼= vCds(t) sin ωst + vCdc(t) cos ωst

iLm(t) ∼= iLms(t) sin ωst + iLmc(t) cos ωst. (2)


Fig. 3. Modes of operation of the LmC resonant converter. (a) Mode 1 (t0−t1). (b) Mode 2 (t1−t2). (c) Mode 3 (t2−t3). (d) Mode 4 (t3−t4).

The nonlinear terms such as va and ip in (3) are substitutedto the combination of sine and cosine terms with the functioncoefficients {f1, f2, f3, f4} as in (4)

vo = (iLo + io)rc +(

1 − rc

RL

)vCo

va = vCs + vCd

CsdvCs

dt− Cd

dvCd

dt= iLm + ip

ip = sgn(vCd) ·iLo

n(1 +

RCo

Ro

)Co

dvCo

dt+

vCo

RL= iLo + io

LmdiLm

dt= vCd

LodiLo

dt=

|vCd|n

− vo, where rc = (RCo//RL). (3)

To find out the describing functions, Fourier series expan-sion is applied. Thus, the coefficients in (4) are obtainedas (5)

va(t) ≈ f1(vi) sin ωst

|vCd| ≈ f4(vCds, vCdc)

sgn(vCd) ≈ f2(vCds, vCdc, iLo) sin ωst

+ f3(vCds, vCdc, iLo) cos ωst (4)

f1 =2vi

πf2 =

4iLovCds

nπApf3 =

4iLovCdc

nπAp

f4 =2Ap

nπ, where Ap =

√v2

Cds + v2Cdc. (5)


Fig. 4. Operating waveforms of the LmC resonant converter.

Fig. 5. Equivalent circuit for the EDF analysis.

By substituting (2), (4), and (5) into (3) and by equating thecoefficients of the dc, sine, and cosine terms, respectively, theequations in (6) are obtained

f1 = vCss + vCds

0 = vCsc + vCdc

Cs

(dvCss

dt− ωsvCsc

)−Cd

(dvCds

dt− ωsvCdc

)= iLms+f2

Cs

(dvCsc

dt+ ωsvCss

)−Cd

(dvCdc

dt+ ωsvCds

)= iLmc+f3

Lm

(diLms

dt− ωsiLmc

)= vCds

Lm

(diLmc

dt+ ωsiLms

)= vCdc

(1 +

RCo

RL

)Co

dvCo

dt+

vCo

RL= iLo + io

LodiLo

dt= f4 − vo

vo = (iLo + io)rc +(

1 − rc

RL

)vCo. (6)

Fig. 6. Voltage gain curves according to fs/fo.

Since the coefficients (or envelope) of the states in (2) have aslow time-varying characteristic, they are treated as a constantwith respect to the switching period. Thus, the steady-statesolutions of (6) are obtained as in

VCdc =2ReβVi

π (1+(α−β)2R2e)

VCsc =−VCdc

VCds =2(α−β)R2

eβVi

π (1+(α−β)2R2e)

VCss =2Vi

π−VCds

ILmc = − VCds

βILms =−VCdc

βVo =

2Ap

nπ,

where α=Csωs; β=1

Lmωs; Re =

n2π2

8RL.

(7)

The dc voltage gain curves according to the load resistanceand normalized switching frequency are shown in Fig. 6. Asthe normalized switching frequency increases, two interestingfeatures are observed. First, the voltage gains of the curvesare converged to 0.5 due to the half-bridge inverter with 50%duty ratio. In that region, pulse frequency modulation (PFM)method is not applicable because the slope of the voltagegain becomes zero. Since this flat gain means that the inputimpedance characteristic of the resonant tank is resistive, theZVS is not achievable. The voltage gain variations according tothe Q factor are small when the normalized switching frequencyrange is on 2–3. This means that the switching frequency rangeaccording to the load variation is narrow like the conventionalLLC resonant converter. The Q factor is defined in

Q =1

Re

√Lm

Cs(8)

where Re is defined in (7). The normalized peak current ofiLm is shown in Fig. 7, and two interesting facts are observed.First, the peak points are located below (about half) the resonantfrequency. In Fig. 6, the voltage gain is decreased as thenormalized frequency decreases in region 2. However, in thisregion, the normalized peak current of iLm is increased. Thesecond interesting fact is that the normalized peak current isdecreased as the Q factor is decreased. This means that the


Fig. 7. Normalized peak magnetizing current.

relative increment of the peak current becomes small along withan increased load current. Thus, the converter is suitable for ahigh-current low-voltage application, and it is advantageous fortransformer design due to the low normalized peak current.

IV. DESIGN CONSIDERATIONS

Based on the analysis result, the design considerations forhigher efficiency are as follows. First, the operation of the LmCresonant converter in region 1 is recommended from the resultsin Fig. 6 and previous results of the analysis. Even thoughregion 2 offers ZCS-off of MOSFET and smaller fs range forthe load current variation than region 1, the peak current ismuch higher for the same output power. At points A and B inFig. 6, the two operating points provide the same voltage gainwhen Q = 0.8 (the same load resistance). On the other hand,the peak current of point B is higher than that of point A inFig. 7. This means that the circulating current is large. Thus,the conversion efficiency is deteriorated at point B.

To achieve ZVS at minimum-load condition, it is recom-mended that the maximum fs/fo is less than 2.5 becausethe gain slope becomes flat. Enough voltage gain range isprovided at the same load condition for a wide input and outputapplication.

Based on the aforementioned considerations, the pro-posed converter is designed as follows. The input range is320–400 Vdc, the output voltage is 24 V, and the output rangeis 1–10 A. fsmax is limited to 150 kHz at minimum loaddue to electromagnetic interference in the application. For theregulation of the output voltage and lower switching loss at no-load condition, the burst-mode operation is used as in the LLCresonant converter [24]–[26]. The required maximum gain(Mmax) is 24/320 = 0.075, and the minimum gain (Mmin)is 24/400 = 0.06. Thus, Mmax/Mmin = 1.25. Considering a10% gain margin, the required gain range is 1.375. Using thedetailed gain curve according to Q in Fig. 8, the Q value atminimum load and maximum input can be selected. To satisfythe two conditions (fs/fo < 2.5 and Mmax/Mmin = 1.375) forfixed Q, Qmin = 0.05, and fs/fo = 2.02 is selected (point A).Since fsmax is 150 kHz, fo is 74.26 kHz. fsmin is obtainedfrom point B in Fig. 8 as 96.5 kHz.

Fig. 8. Detailed voltage gain according to Q where n = 1.

Fig. 9. Operating region of the designed converter.

Since Mmin is 0.06 and the normalized voltage gain atpoint A is 0.6, the turns ratio (n) is selected to be ten. Thevalues of the resonant inductor and capacitor are obtained from

Lm =Qmax × Remax

ωo= 317 μH

Cs =1

Lm × ω2o

= 14.37 nF. (9)

Since the copper loss of Lo significantly affects the con-version efficiency, particularly for low-voltage high-currentapplication, a small winding resistance is desirable. A smallinductance value was enough to reduce the ripple current effec-tively (one-sixth) in [17]. The relation between fsmin and Lo isgiven in

fsmin =k

2π√

(n2Lo//Lm)Cs

. (10)

It shows the relative magnitude of Lo for a negligible effecton the LmC resonance. The value of k is recommended to be1.1–1.3 by the empirical data through simulation.

In the prototype hardware, two standard 33-nF capacitorsare paralleled for Cs (16.5 nF), and Lm is 300.4 μH. Theoperating region of the designed converter is shown in Fig. 9.The minimum switching frequency is higher than 100 kHz, andthe maximum frequency when the load is 1 A is lower than


TABLE IEXPERIMENTAL CONVERTER PARAMETERS

150 kHz as designed. In the case of no-load condition, theconverter is entered to burst-mode operation due to the higherswitching frequency (> 150 kHz).

From (10), the value of Lo is selected to be 3.5 μH with abar-type core of l = 35 mm and ϕ = 6 mm.

V. SIMULATION AND EXPERIMENTAL VERIFICATION

The designed circuit parameters and devices of the LmCresonant converter for the hardware experiment are listed inTable I. For comparison, an LLC resonant converter is built.

Using a simulation tool, the validity of the EDF analysiswith high-order harmonics is verified. The steady-state modelfrom the EDF solutions of (7) is built, and it is comparedto the switching circuit model in Fig. 10. The input voltageis 400 V, Ro = 2.4 Ω, and fs = 140 kHz. From Fig. 10(a)and (b), the error between the steady-state values of the EDFmodel and the switching circuit is decreased when the first andthird harmonics are included in the EDF model. The high-orderharmonics of the states are obtained using the relation in

xn(t) =n∑

k=1

xs|k · sin kωst + xc|k · cos kωst. (11)

The high-order harmonics of the describing function in (5)are obtained as

f1|k=1,3,5,... =2vi

kπf2|k=1,3,5,... =

1k

4iLovCds

nπAp

f3|k=1,3,5,... =1k

4iLovCdc

nπApf4|k=1,3,5,... =

2Ap

nkπ. (12)

The voltage gain according to switching frequency is shownin Fig. 11. The theoretical gain curve with fundamental compo-nent shows similar shape to the experimental result while thereis offset error. It is observed that the error becomes smaller

Fig. 10. Simulation results of the EDF model and the switching circuit model.(a) Simulation results with the first harmonic. (b) Simulation results with thefirst and third harmonics.

Fig. 11. Comparison of voltage gains when Ro = 2.4 Ω.

as the high-order harmonics are included. Fig. 12 shows theZVS of the MOSFET at no-load [Fig. 12(a)] and full-load[Fig. 12(b)] conditions. The current shape of the resonant tank(iCs) shows good agreement with the analyzed waveform inFig. 4. For both load conditions, the switching periods are


Fig. 12. Steady-state waveforms and ZVS achievement of the MOSFET.(a) 400-V input voltage and no-load current (0 A). (b) 400-V input voltageand full-load current (10 A).

almost the same because the gain characteristic according tothe load is quite similar to that of the LLC resonant converter,while the resonant tanks are different. The measured ripplecurrent of the output capacitors of the LLC resonant convertershows 20 App in Fig. 13(a). Therefore, several capacitors arerequired due to the current ratings. On the other hand, theripple current of the LmC resonant converter is reduced to6.3 App as in Fig. 13(b). Therefore, the current stress and powerlosses in the output capacitors and secondary windings of thetransformer are reduced.

In the primary side of the transformer, the rms and peakvalues of iCs are reduced due to the increased characteristicimpedance of the resonant tank. To meet the input voltage rangeand narrow switching frequency, the LLC resonant converter isdesigned Lm/Ls = 4. Since the LLC resonant converter usesa sectional winding method, the magnitudes of Ls and Lm aresmaller than Lm of the LmC resonant converter for the sameferrite core.

Thus, it is expected that the conduction loss and turnoff lossof the MOSFET are decreased. Furthermore, the circulationcurrent also decreased; the light load efficiency is increased.

The measured switching frequency ranges according to theload and input voltage are shown in Fig. 14. When vi = 400 V,the switching frequency range from 0 to 10 A is 138–168 kHz(Δfs = 30 kHz). The burst-mode operation in Fig. 15 is

Fig. 13. Comparison of the ripple currents of the output capacitor. (a) Oper-ating waveforms of the LLC resonant converter (400 V; 10 A). (b) Operatingwaveforms of the LmC resonant converter (400 V; 10 A).

Fig. 14. Switching frequency range of the LmC resonant converter.

helpful to limit the maximum switching frequency under theextremely light load (Δfs = 5 kHz).

In the case of the LLC resonant converter, the switching fre-quency range from 0 to 10 A when vi = 400 V is 112–138 kHz(Δfs = 26 kHz). Therefore, the switching frequency range ofthe LmC resonant converter is quite similar to that of the LLCresonant converter.


Fig. 15. Operating waveforms (burst) at no-load condition.

Fig. 16. Conversion efficiency according to the load current (vi = 400 V).

When vi = 320 V and load current range is from 1 to 10 A,the frequency range of the LmC resonant converter is102.7–114.23 kHz. On the other hand, in the LLC case, therange is 89–96.4 kHz.

From the experimental result, the LmC converter shows anarrow frequency range which is almost the same as that ofthe LLC resonant converter for a small input range. However,as the input range is increased, the switching frequency range isincreased more in the LmC resonant converter than in the LLCresonant converter.

The conversion efficiencies of the LmC and LLC resonantconverters are shown in Fig. 16. Compared to the efficienciesof the LLC resonant converter, the efficiencies of the LmCresonant converter at light- and middle-load conditions areincreased because the turnoff loss and conduction loss in theprimary side of the MOSFET are decreased as the peak and rmsvalues of the resonant current are decreased. Since the dominantloss term is the conduction loss of the rectifier diode as theload increased like (13), the efficiencies of the two convertersbecome the same

Pdiode = vf × Idiode.avg = vf × ILoad (13)

where vf is the forward voltage drop of the rectifier diode andIdiode.avg is the average current of the rectifier diode. In the

experiments of the two converters, the same switches, rectifierdiodes, and transformer core are used except the number ofoutput capacitors and small output inductor. Thus, the reducedvolume in the LmC resonant converter is affected by thedecreased number of capacitors, and it is 9462 mm3.

VI. CONCLUSION

This paper has presented the steady-state analysis and de-sign considerations of the half-bridge LmC resonant converter.Furthermore, the performance evaluations are shown comparedto the LLC resonant converter. The large-signal characteristicssuch as voltage gain, peak stress, and frequency ranges aredetermined by the steady-state solutions using the EDF method.The accuracy increment of the analysis result according to thehigh-order terms is verified. Based on the result, the require-ments for a practical circuit design under the given specificationare suggested. Simulations and experiments using a 24-V/10-A hardware prototype for laser printer applications arecarried out to show the validity of the analysis results andcompared to the conventional LLC resonant converter. Eventhough the obtained EDF model shows a small amount of error,it is applicable to the initial design procedure of the converter.The LmC resonant converter shows a narrow switching fre-quency range for the input and load conditions like the LLCresonant converter. Moreover, it provides a higher efficiencythan the LLC converter at light and middle loads. Furthermore,the number of output capacitors is much reduced due to thedecreased capacitor current.

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B. C. Hyeon (S’07) received the B.S. degree fromKyungpook National University, Daegu, Korea, in2004, and the M.S. and Ph.D. degrees in electricalengineering from Seoul National University, Seoul,Korea, in 2006 and 2011, respectively.

Since 2011, he has been with the Advanced Re-search Group, Visual Display Division, SamsungElectronics. His research interests are dc–dc convert-ers, power factor correction, digital power conver-sion, and contactless charging systems.

B. H. Cho (M’89–SM’95–F’11) received the B.S.and M.E. degrees in electrical engineering from theCalifornia Institute of Technology, Pasadena, and thePh.D. degree in electrical engineering from VirginiaPolytechnic Institute and State University (VirginiaTech), Blacksburg.

Prior to his research at Virginia Tech, he workedfor two years as a Technical Staff Member in thePower Conversion Electronics Department, TRWDefense and Space System Group. From 1982 to1995, he was a Professor with the Department of

Electrical Engineering, Virginia Tech. In 1995, he joined the School of Electri-cal Engineering, Seoul National University, Seoul, Korea, where he is currentlya Professor. His main research interests include power electronics, modeling,analysis, and control of spacecraft power processing equipment, power systemsfor space stations and space platforms, and distributed power systems.

Prof. Cho is a member of Tau Beta Pi. He was the recipient of a 1989Presidential Young Investigator Award from the National Science Foundation.

Analysis and Design of the Resonant Converter for Low Output Current Ripple

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