A Nonlinear Resonant Switch.pdf

7/27/2019 A Nonlinear Resonant Switch.pdf

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A Nonlinear Resonant Switch

R. W. Erickson, A. F. Hernandez, A. F. Witulski, R.Xu

Departmentof Electrical and Computer Engineering

University of Colorado at Boulder, 80309-0425

ABSTRACTA new resonant switch is introduced which employs nonlinear

tank elements. Zero current switching is obtained. yet the peaktransistor voltage and current stresses can approach those of anequivalent ideal PW M converter. Reduced switching loss without asubstantial increase in conductb n oss is therefore possib le.

An approximate analysis isoutlined,and transistorpeak voltageand current stresses are shown to be much lower than those of linearresonant switch technologies. Single-transistor mplementations of thebuck, boost. and bu ck-boost nonlinear resonant switch converters aregiven. The validiry of the nonlinear resonant switch concept, aswell asof the approximate analy sis, isproven e.xperimentalty.

I . IntroductionThe objective of high frequency operation in dc-dc converters is

the reduction of transformer, ilter inductor. and filter capacitor size andcost. As in any power application, high efficiency is essential, andhence increase of the switching frequency to SOOkHzormore can beproblematic because of the direct dependence of transistor switchingloss on frequency. The use of resonant switching techniques is anattempt to substantially reduce transistor switchingloss. and hence attainhigh efficiency at increased frequency. Other sources such astransformer core loss then become the limitation on maximum switchingfrequency.

A class of resonant converters which have received much recentattention is the resonant switch, or quasi-resonant, converters [1-51.Resonant tank elements are connected to the semiconductor switchcomponents of a PWM converter such that the switch waveform isinitially zero, rings posit ive, and then again reaches zero. The transistorturn-on and turn-off transitions naturally occur at zero voltage [2,3,4] orzcro current [1,3.51, and hence switching loss is reduced substantiallyor eliminated. Since resonant switch versions of any previously knownPWM converter are possible, this is an extremely general concept

applicable to a wide variety of applications.Although switching losses are nearly eliminated in quasi-resonant converters, the peak transistor stresses, and hence conductionlosses, are substantially increased. Typical ideal full-load transistorcurrent waveforms of conventional pulse-width-modulated (PWM) andzero-current resonant switch(ZCS) 11 converters are compared in Fig.lab and thecorrespondingcircuitsare given in Wg. 2ab. Because of thequasi-sinusoidalnature of theZCS waveform, its peak and RMS valuesare significantly higher than those of a PWM converter for the sameoutput power, and hence higher conductionlossesoccur. In the case ofthe zero-voltage resonant switch (ZVS) 2], the transistor blockingvoltage is substantially increased. This necessitatesuse of high voltageMOSFETs with significantly increased on resistance, and hence alsoincreased conductionloss. The multiresonant switch ( M R S ) [4] allowsa further reduction in switching loss and a compromise between theZCS and ZV S n peak switch voltage and current stresses, but itsconduction loss is nonetheless significantly higher than in the PWMcase.

Afurther problem arises when a wide range of load currentsand/or input voltages must be tolerated. By skillful choice of the tankcharacteristic impedance, it is possible to minimize peak transistor

Stresses at a single operating point. However , these stresses canincrease significantly at other operating points. In the case of the zero

This work wa s supported in part by the Delco-RemyDivision of the General Motors Corporation, Anderson IN.

iL

L

Fig. 1 . Comparison of ransistor current waveform: (a) conventionalPW M switch, (b ) zero-current linear resonant switch (ZCS).(c) zero-current nonlinear resonant switch (NRS).

voltage switch, a 1O:l load current variation can result in a similarvariation of transistor peak voltage; extremely high switch blockingvoltage ratings are then necessary. In the case of a zero current switchbuck converter, input voltage variations directly affect the peaktransistor current. In addition, he peak switch current does not dependstrongly on load current, eading to poor efficiency at light load.

In consequence of these problems, the overall efficiency of anyof the presently known linear resonant switch converters, operating atany switching fr uency,must alwaysbe significantly lower than that

of an eq uh le nt %W z PWM converter. The significant reductions inswitching loss obtained through resonant switching are offset byincreases in conduction loss. It has been estimated that the increasedconductionloss of the zero current switch is approximately equal to theswitching loss of an equivalent 5OOkHz PWM switch [6]; hence, theoverall efficiency of the ZCS s lower than the PWM switch atfrequenciesbelow approximately SoOkHz. A similar result holds for theother types of resonant switches. Therefore the goal of increasing theswitching kequency without reducing efficiency has not been achicved.

It is apparent that the zero-current or zero-voltage switchingproperty must be obtained in a way which does not significantlyincrease transistor peak current and voltage stress. To accomplish his,a new nonlinear resonant switch[7] as been developed which exhibitszero current switching. Its peak transistor voltage is identical to, and the

peak current is typically 20%greater than, that of an equivalent PWMconverter. The efficiency of a nonlinear resonant switch ( N R S )

converter can therefore approach that of a low-frequency PWMconverter. Furthermore, the nonlinearity can be controlled such that thepeak transistor current is closely related to the load current; improvedefficiency at light load is then obtained.

These advantages are realized through the use of nonlinear tankelements whose natural response is quasi-trapezoidal as in Fig. IC,rather than quasi-sinusoidal. As described in Section 11. a biasedsaturating nductor isused as the tankelement in a zero-current resonantswitch. A buck conver ter implemented with a half-wave nonlinearresonant switch is shown in Fig. 2c. The tank inductorL is normallybiased into saturation by the dc current IT in a secondary winding; thetank inductance is then similar in value to that of a linear zero-currentresonant switch. When the tank current iL becomes sufficiently large,

43

CH2721-9/89/0000-0043181.00' 989 IEEE



Fig. 2 . Buck conver ter, implemented using (a ) conventional PW M

switch; (b ) linear zero current resonant switch; and (c)nonlinear resonant switch. tank inductance

winding

9ZL

L(iL1

the tank inductor desaturates, which causes its inductance to increaseand hence reduces the peak value of the switch current. Near the end ofthe interval, the tank inductor resaturates and the current rings back tozero.

As in the case of linear resonant switches, the nonlinear resonantswitch cell can be inserted into most of the common convertertopologies. The properties of such converters can be found byanalyzing the nonlinear resonant switch cell to find i ts average terminalwaveforms, then inserting the results into the converter equations. Suchan analysis is performed in Section 111 using a simple piecewise-constant model for the tank inductor. The averaged switch waveformsare found for the half-wave and full-wave cases, and the results aretabulated for the buck, boost, and buck-boost converters. The additionof a third winding to the nonlinear tank inductor is also considered; thisallows another degree of control of the switch characteristics. The range

of switching frequency variations can then be reduced, and constantfrequency operation is feasible if the range of operating points issufficiently limited.

The peak switch stresses for a lOOW, 50 V input dc-dc converterutilizing PWM and various resonant switches are compared in SectionIV . A uadeoff between N R S peak switch current, transistor switchingtimes, and minimum tank inductance is also discussed. The transistorpeak current can approach the ideal PWM value, but this requires aninstantaneous switching time and a zero minimum tank inductance.Nonetheless, full-load peak switch currents which exceed the PWMvalue by only 20% are feasible.

The analysis and discussion results are applied in Section V tothe design of a 50 W full-wave buck converter. Experimentalverification of the nonlinear resonant switch concept and the analysis ofSection I11 is presented in Section VI. The results are summarized inSection VII.

bias winding

> , 4 -

>dc currentd->-

11. Introduction of nonlinearity into zero current switchThe objective of reducing and controlling the peak switch current

of the zero-current resonant switch can be attained through the use ofnonlinear tank elements. By proper variation of the tank inductancecharacteristic, a quasi-trapezoidal tank current natural response can beobtained. In particular, the incremental inductance must increase withcurrent, thereby yielding the electrical analog of a hard spring. Apractical implementation of this which employs a biased saturatinginductor is described here.

In the zero-current resonant switch with linear tank capacitance,the tank capacitor must charge to twice the applied input voltage duringevery ringing interval. This requires that the capacitor current i, bepositive during the fust portion of the ringing interval. where

Hence, one cannot expect to limit the peak iL,or peak switch current, toless than IT, the current conducted by an equivalentPWM switch.

However, it is possible to limit the peak iL to a value slightly

higher than IT. This is done by increasing the incremental tankinductance L(iL)when iL exceeds IT, such that iL does not change veryrapidly but the capacitor continues to charge. Hence, it is desirable thatthe incremental inductance vary as shown in Fig. 3a.

(1),= i,- IT

(a )

actual saturation curve

Ll

where I = (N - 1)IT+ I,. Since the converter characteristics dependstrongly on the value of L , , and to a lesser extent on L, , it may bedesirable to accurately control these values by introducing gapped

44



" itFig. 4 . Nonlinear resonant switch cell. Switch SI is implemented

using a transistor Q l and diode DI, onnected in series (hay-wave) or anti-parallel (full-wav e).

inductors in series (L,) and parallel(Ldwith the saturating inductor.Thus, peak transistor current is limited by causing the tank

inductance to increase at high tank current. The nonlinear tankinductance characteristic s realized by biasing a saturating inductor withdc currents in oneor more secondary windings.

111. Ana lys i sThe principal steps in the analysis of the nonlinear resonant

switch cell of Fie. 4 with Diecewise constant tank inductance areoutlincd here. It i's assumed'that the converter imposes an essentiallyconstant current IT on switch terminal B as shown; this generally occursbecause a converter low-frequency filter inductor is connected there.Likewise, the converter imposes an essentially constant voltage VTbetween terminalsA andC through hc connectionof the converter dcinput voltage or a low frequency filter capacitor.

It is convenient to normalize all variables with respect to thefollowing base quantities:

Vbsre = vT

Ibase= V T / R o

1oo=-m

The normalized tank waveforms

(3 )

jL = iL/lbase (4)

mc= vc / Vbarc

are plotted vs normalized time in Fig. 5, and in the phase plane in Fig.6.

Interval I : 0 -< w0t ISwitch S1 and diode D2 conduct during this interval, and the

tank inductance is saturated: L=Ll. The tank current increases linearlyfrom 0 to IT:

Vidt) = t

e1 ( 5 )

( 6 )

This can beexpressed in normalized form as

The interval ends when iL = IT and oot= a. Hence,

jdoo t) =mot

JT=IT/Ib.se = a (7)

Dz

Iankmductorsute: s s U s s s(S = aaumed,

U =UISallxaud)

Fig. 5 . Normalized tank inductor urrent and capacitor voltage: ( a)hagwave waveforms; (b ) ull wave waveforms.

Interval2: a I ot 5 a + f iSwitchS1 conducts during this interval. The tank elements are

excited by the essentially constant quantities V, and IT and ringsinusoidally. The normalized solution is:

j L = J T + sin(wot- a)

mC =1 - cos(w,,t - a)

Interval2 ends when the tank inductor comes out of saturation.This occurs when the tank current iL reaches the critical value I + IT,corresponding to the bottom of the core B-H loop. The length of this

interval is found by substitution of coot=a+pintoEq . (8). resulting in

(8)

(9 )

p = sin-' (J,,~J

J a i l = IcritIIbm

where

The capacitor voltage at the end of the interval is

(10)

m c ( a + p) = 1 - cos(p)= 1 -41 - J* nt (11)

Interval3: a ii I ,t 5 a ii iThe tank inductor is unsaturated during this interval, and hence

the effective ank inductance is L2, a quantity much largerthanL1 in awell-designed converter. The normalized state equations for this

45



J T

0

........................

3

full wave half wave

7

Ma = I + ? / l - J :

Fig. 6. No ml iz ed phase plane trajectory j L V S.p.

interval are:

1 dmc-1

(12)d j , = k ( l - m , )1 dt W 1 & kGL-JT)

where0 -- 1 k=?!=&<<l1-m WO

Al l quantities are again normalized with respect to the base values of Eq .(3). The solution is an ellipse in the phase plane, of the form

where

mc = l-Acoscp+Bsincp

j L = JT+kAsincp+Bcoscpk

c p = 0'1- k(a + B )

A = cosp =

B = sin fi = Jcfi,

Note that the solution canbemanipulated into the following form:

k2(m,- 1)2+( i~-JT) z=(kA )2+Bz (14)

This describes an ellipse centered at m , = 1, j L = JTwhich is

symmetrical about its horizontal an d vertical axes. The vertical

(semiminor) axis is of lengthm,nd hence the peaknormalized tank current is

J,, = J+

The interval ends when the tank inductance resaturates, i.e.,whenjL= J&l + JTagain. Because of the symmetry of the ellipse, thisoccurs at a normalized capacitor voltage of

Substitution of this relation into the waveform expression and solution

for the interval length 6 yields

(16),(a+ p + 6)= 1+

2 1 k A

k B= - l a n - (-1

Interval4: a + p + 6 I o,r I a + p + 6 + 5

Thc tank inductance is again saturated, and the circuit is the sameas during interval 2. The solutions are again sinusoidal, and lead to acircular arc in the phase plane (Fig. 6) centered at rn , = 1, L = JT withradius 1. This interval ends at the first (half-wave) or second (full-wave) zero crossing of the tank inductor current.

This interval can be solved by simple geometry. As show on thephase plane diagram, the final value of tank capacitor voltage is-

1 +4 1- ; half-wave (18)m c ( a + p + 6+ 5 )= Mcl=

subject to JTI

arc, orThe length of this interval is given by the angle subtendcd by the

p + sin-' JT half wave

full wave (19)= { p + n: - sin-' jT

with 0 S sin-' JT S ?t/2

Interval5: a+ p+6+ { I WJ I a + P + S + { + C

During this interval, all transistors and diodes are in the off state.The tank capacitor is discharged linearly by the filter inductor current I ,and the tank inductor current is zero. The expression for the normalizedcapacitor voltage is

This interval ends when the tank capacitor voltage reaches zeroSubstitution of r n , = 0 into the abovend diode D2 turns on.

expression allows one to solve for the length of this interval:

Interval6: a + p + 6 + { + C I OJ I a + p + 6 + { + [ + o

switching period. The tank states remain at zero.

Circuit AveragingThe static and low-frequency dynamic characteristics of

converters containing a nonlinear resonant switch can be found by'avcraging the switch terminal waveforms (tank inductor current iL andcapacitor voltage v,) over one switching period. This can be doneusing the solutions derived above. The result for the averagenormalized tank voltage is:

During this interval, diode D2 conducts for the remainder of h e

and F = fJfo = normalized switching frequency. To the extent thatlosses can be neglected, the resonant switch input and output powers areequal since the switch does not contain dynamic states. In normalizedform, this means that

Equations (22) and (24), and the use of circuit averaging techniques[9,10] are sufficient to model converters incorporating nonlinearresonant switches. For example, in the case of the buck converter,

and the overall dc conversion ratio is

The averaged function P(IT,J& is plotted in Fig. 7 for the half-wave nonlinear rcsonant switch wi h JB=O. The switch conversion ratio

46



P

P

0.0 0.2 0.4 0.6 0.8 1.0

JT

Fig. 7 . Switch conversion factor P(JT )for the half-wave nonlinearresonant sw itch with JB=O and N > I .

p=FP is dependent on JT and load current in this case. A goodapproximation when N>1 and Jn=O s

p - N

(27)

N-1) A JT

This is valid fork << (N-1) JT. and iswithin 10%of the exact value forany allowed load current when N = 1.2 and k=O. The dc conversionratio M for a half-wave buck converter is plotted in Fig. 8 for variousvalues o f 0=R R*.

M

1o

0.8

0.6

0.4

0.2

0.0

0.0 0.1 0.2 0.3 0.4 0. 5

F

F i g . 8 . Voltage conversion ratio M=VIV, for the half-wave buckconverter, fo r various values of load resistance R . NoteQ=RIR,.

The averaged function P(JT,Jd) is plotted in Fig. 9 for the full-wave switch with N=l. The switch conversion ratio p=FP is thenessentially independent of JT and load current. The converter outputvoltage depends on switching frequency and bias current. A goodapproximation for P in this case is

, -

with J&t= (N-1) JT+ J8. When N= l and k << Jg, this further reducesto

When the inequality k <<JB is not satisfied, P(JB) is reduced in valuebut is nonetheless independent of load current and JT. Hence thevoltage convers ion ratio V/V of the buck converter in th i s case is alinear function of switching brequency, a nonlinear function of biascurrent, and independent of load current.

0.4 0.6 0.8 1.0.0 0.2

J B

F i g . 9. Switch conversion factor P(JB ) o r the full-wave nonlinearresonant switch with N = l, or various values of k.

Full-wave nonlinear resonant switch versions of the buck,boost, and buck-boost converters are shown in Fig. 10. For each case,the relation between VT and the average tank capacitor voltage <vc> isgiven by Eq. 22). However, VT does not in general coincide with theinput voltage: VT = V for the boost converter, and VT =V +V forthe buck-boost. Expressions for M, JT, and JB are given in Taille 1.

TABLE1. STEADY-STATE CONVERTER QUATIONS

Converter M = VIV, JT J B

Buck ITRo/v, IBRolv,

with p = F P ( Jv JB)

Q = R/R,J T = M/ Q = ITR,/VT

As for the linear zero-currcnt resonant switch, there are two

basic restrictions for operation in the mode described here. First, thenormalized output filter inductor current JT must not exceed 1;otherwise, the phase plane trajectory during the fourth interval cannotreach the jL = 0 axis because the radius of the circle is too small (see

Fig. 6). Attemptingto operate at currents greater thanthisvalue resultsin high transistor turn-off stress. The curves of Fig. 8 end at M=Q forQ<1 because arger output voltages would violate the J+l condition.

Second, the length of the sixth interval must be positive, and t h i slimits the maximum switching frequency. This limit can be expressedas

I V . DiscussionAS in the linear zero current switch, the energy stored in the

MOSFET drain-to-source capacitance is lost during the transistorswitching transitions, and results in switching loss [6]. The recoverytime and stored charge of diode D1 can also contribute to switchingloss. These mechanisms can limit the maximum switching frequency of

the linear zero current switch to roughly 1 MHz with current componenttechnology. In the nonlinear resonant switch, this frequency limit issimilar but somewhat higher because he reduced conduction loss allowsthe use of smaller MOSFET devices with higher on resistance and loweroutput capacitance.

The nonlinear resonant switch is also somewhat less tolerant oflong transistor switching times and large transformer leakageinductance. However, a reasonable tradeoff between these quantities,

47



II I I I

1T

IT

at I I

Fig. 10. Nonlinear resonant switch implementations of three basic

switching frequency, and peak transistor current can be obtained.Consider a full-wave buck converter example. I n the linear zero currentswitch, the voltage conversion ratio M = V / V, and the normalizedswitching frequency F = f, /f a are directly related: M= F. Also, thetransistor peak current I always exceeds the output filter inductorCurrent 1 by the amount e',/ R,; hence, (Ipk- ,) is equal to V, / R,.

For the nonlinear resonant switch, the peak transistor current canbe varied continuously from the PWM switch value (Ipk =, IT) 10 thelinear resonant switch value (Ipk = I, + V IR ), depending on the

design of the tank inductor and the choice ohd. Maintaining givenconverter operating conditions requires that the switching frequencyvary also. Equations (15), (22), and (26) can be used to determine thenormalized switching frequency of an equivalent nonlinear zero currentswitch for various values of (1,k - T). The results are plotted in Fig. 11. It can be seen that a design in which the peak transistor CUITent Ipkexceeds the output current IT by 0.2 V P , results in F= 0.5M, orP =2. For the same voltage conversion ratio M and switching frequency f,, his nonlinear design requires a tank frequency f, twice as high as inthe linear case. Hence, the tank inductance and capacitance must besmaller by a factor of two. and the transistor switching times must behalf as long. This is a reasonable price to pay for the greatly reducedconduction loss of the nonlinear resonant switch.

Estimated peak transistor current Ipkand peak transistor voltageV of various switch technologies are compared in Table 2 for thefoflowing nonisolated buck convener application:

input voltage V = 50volts f 20%output voltage .$ = 25 voltsoutput power = 1OW- OOW

converters: (a)buck, (b)boost, (c) buck-boost .

The zcro-current nonlinear resonant switch (MS ) was designed to

operate at a maximumJT

of 0.8 and a full load JB of 0.125. The linearzero current switch (ZCS) and linear zero voltage switch (ZVS) weredesigned with similar margins. The linear multiresonant switch (MRS)values were derived from results reported in [4,8].

It can be secn that the NR S case attains zero current switching

PWM zcs

1 o

0.8

0.6

0.4

0.2

0.0

0.0 0.2M 0.4 M 0.6 M 0.8M 1.0 M

F

Fig. 11 . The relation between normalized w itching frequ ency F andpeak transistor current Ipk

TABLE2.PEAK TRANSISTOR STRESSES USDJG VARIOUS

SWITCH TECHNOLOGIES

Switch typePWMNRSzcszvsMRS

Current, A4.0

4.8

11.54.06.4

Voltage, V6060

60810200

without substantially increasing the peak transistor current or voltageabove the PW M values. The linear ZCS results in significantlyincreased conduction loss because of its high peak currents. The linearZV S and MR S require high voltage MOSFET switches, withcorrespondingly high onresistance and high conduction loss.

V . Steady State Des ign Example

following specifications:

The desired dc conversion ratio is M = 30/50 = 0.6, at a switchingfrequency off, = 500kHz. The following identities hold for the buckconverter:

As discussed previously, the normalized current JT must not exceedunity. Let us choose JT = 0.75 at full load. Hence,

To obtain a control characteristic which is independent of load currentvariations, i.e., in which variations in IT do not require an adjustment ofswitching frequency or bias current, choose the full-wave mode ofoperation with N=l and with a bias winding current IB . Then,

(32)

(33)

Consider the design of a 50 watt buck converter to meet the

input V =5Ov0ltsoutput 4 = 30volts at I = 1.67 amperes

VT = v, = 5ovoltsI, = I = 1.67 amperes

R , = J VT/IT = (0.75)(50V)/(l.67A) 22.5 q 3 1 )

IC*t = (N-1) IT + I B = I B

Let us design for a full load peak transistor current which is

20% greater than the output current I:I,, = 1.21 = 2amperes

J p k = I , ~ R , / V T= 0.9

Equation (15) has no solution for positive J ~ tnless the quantity k isless than J - , = 0.15. The choice k = 0.1 is therefore acceptable.Equation ('is) can be used to solve for the normalized bias current JB =

48



This value of bias winding current can be reduced by increasing the biaswinding turns ratio.

The converter solution is given by Eq. 26) in conjunction withEqs. (22). (10). (17). (18) and (19); evaluation of these expressionsyields

P = 2.837F = M I P = 0.211 (35)

The required tank resonant frequency is

This leads to the ollowing tank component values:(36)

(37)

f, = f , / F = 2.36MHz

C = 1/2xf,R, = 3 n FL , = R,/2xf0 = 1.5pH

L, = L l / k 2 = 150pH

VI. Experimental VerificationA prototype nonlinear resonant switch buck converteras in Fig.

10a was constructedusing the following(different)values:input voltage V =24VtransistorQ1: IkF530

tank capacitor C=9.1nFdiode D ,, 4: MBR 1045

Initially, these components were connected in a full-wave configuration.The nonlinear tank inductor was constructed using a Magnetics, Inc.80523-1/8-D-MA tape wound core (1/8 mil Permalloy 80). Theprimary and secondary windings each contained 12 turns. A third

winding, driven by the dc bias currentIB,

consisted of36 turns.

Themeasured primary inductance was 1WpH with no bias, and was within20% of 0.3pH for total bias currents between 130mAand 1.2A referredto the bias winding. The tank characteristic mpedanceis therefore

R, =& 5.7R

and the base current is

VIbase= -E = 4.15A

R O (39)

Full wave results: comtant bias current, variable requency control

The theoretical curves and measured data points for the full waveconverter using a constant bias current I, of 0.54A are shown in Fig. 12. It can be seen that the converter behaves nearly as a voltage source..with output voltage directly dependent on switching frequency andnearly independent of load current. Good agreement between the modeland experiment is obtained, except at light load currents where the coredoes not fully saturate. Prediction of the observed behavior at light 1 0 4requires a more accuratecore saturationmodel.

Full wav e res&: constnnt freq uen cy, variable bias current control

Next, the switching frequency was held constant at 500kHz andthe bias current I, was varied. The esults are given in Rg. 13. The

1, = 0.54A

4

3

I (AI

2

1

0

0 5 10 15 20

Fig . 1 2 . Output characteristic theoretical curves (solid lines) andexperimental data pointsfor he fi ll wave constant bias currentvariable switchinR frequency controlled buck converter.

v VI

output voltage now depends directlyon IB and is nearly independent ofload current, except at light load or low bias current where incompletesaturation of the core again causes the converter characteristicso deviatefrom the model. Hence, constant frequency operation is possible, andis desirableprovided that the range of voltage conversion ratio variationsis not too large. Constant frequency regulation for a wide range ofvoltage variationsrequires large bias currents, and consequently leads tolarge peak tank currents and lower efficiency. Nonetheless, it isapparently feasible to control voltage variations of a factor of two withthisdesign.

f, = 500kHz5

l b = .M A .54A.42A .3A

0 5 10 IS 20

v VI

Fig. 13 . Output characierisiic theoretical curves (solid lines) andexperimental data points for the full wave consiant switchingfrequen cy variable bias current controlled buck converter.

Ha ywa ve results: variable requen cy control,no bias current

The same components were connected in the half waveconfiguration (as in Fig. 2c). and the nonlinear tank inductor wasredesigned. The same 80523-lfi-D-MA tape wound core was used,but with a 12 turn primary, a 15 turn secondary, and no additional biaswinding. Experiment and theory are compared in Fig. 14, and goodagreement is obtained. Aspredicted, he output voltage is strongly load-dependent in this case. No data is shown for load currents under 1

ampere because the tank inductor saturated during the third (6 ) interval;additional tums are required to support the applied volt-seconds in thiscaw.

0 5 10 15 20 25

v (V)

Fig. 1 4 . Output characteristic theoretical curves (solid lines) andexperimental data points for the hawwave variable switchingfrequen cy buck converier with no additional bias winding .

Results with errite tank inductor: full wave case

The core loss of the tape-wound core was fairly high at 500kI-I~.It is possible to obtain high efficiency operation and low peak tankcurrents by use of a ferrite core material. However, the convertercharacteristics are more difficult to predict because of the gradualsaturation characteristic of ferrite.

The saturating inductor of the full-wave converter describedabove was rewound as follows: two TDKH7C4 T4-8-2 cores werestacked and wound with primary and secondary windings of 5 NmS and

49



an additional bias winding of 15 turns. The measured primaryinductance with zero bias was 44pH, and varied from 10pH to 2.8pHas the bias was increased from lOOmA to 2A. Note that even at 2A biascurrent, the core relative permeability was over 100.

Because of the significant variation of the “saturated”inductance, the piecewise constant inductance model of section 111doesnot directly apply. Nonetheless, the converter worked well: zerocurrent switching occurred for load currents up to approximately 3A atV, = 24V, the maximum switching frequency was over 5OOkHz, and

the peak transistor current was limited. A typical transistor currentwaveform is shown in Fig. 15for a load current of 1.95 amperes, a biascurrent of 245mA. an output voltage of 13.5V, an input of 24V at 1.26amperes, and a switching frequency of 330kHz. With a 24V input andbias current of 0.25A. the peak transistor current was approximately 0.6amperes greater than the load current.r---7-vT-- 7- 7 , 1 ,

I

i

1 ’ ;I

Fig. I S . Experimentally measured tank current waveform fo r the fullwave nonlinear switch with ferrite core material. Vertical scale:IAld iv., horizontal scale: 500nsecldiv.

Results with ferrite tank inductor : ha lfwa ve case

Measured tank waveforms for the half wave case are shown inFig. 16. The primary was wound with five turns. the secondarycontained six turns, and there was no additional bias winding. Twostacked TDK H7C4 T4-8-2 cores were again used. The measuredoperating conditions were: 24V input at 1.88A. 19V output at 2.15A,peak transistor current: 2.4A. switching frequency: 5OOkHz. It can beseen that zero current switching occurs and that the peak transistorcurrent is limited to 0.25A geatcr than the load current. The maximumload current for thisconverter was again approximately 3 ampercs.

~ .Fig. 16 . Experimentally measured tank voltage and current waveforms

for the half wav e nonlinear switch with ferrite core material.Upper trace: capacitor voltage, ZOVldiv., ower trace: inductorcurrent, IAld iv., horizontal scale: lpse cld iv.

V I I . Summary an d Conclusions

Linear resonant switches exhibit greatly reduced switchingloss,and can be employed in a wide variety of converter topologies.However, with these advantages come the problems of poor switchutilization and substantially increased conduction losses. High

efficiency is essential if the converter power density is to be increased,and poor switch utilization limits the number of useful applications ofthe resonant switch.

A resonant switch employing nonlinear tank elements has beendeveloped which overcomes these disadvantages. The nonlinearity

forces the tank natural response to exhibit a reduced peak value whilemaintaining the zero current switching property. Peak switch stressesapproaching those of ideal PWM converters can be obtained, yieldinglow conduction losses and high efficiency.

The nonlinear inductance characteristic is realized using asaturating core, biased by dc currents in one or more secondarywindings. By forcing a current proportional to the load through one ofthe secondary windings, the peak transistor current can be made to scaledirectly with load current. Constant frequency operation can also be

achieved by control of the secondary bias current.A piecewise constant inductance model can be employed to solvefor approximate switch terminal characteristics; this works well forsquare loop core materials, but needs further refinement fo r predictionof the characteristics obtained using ferrite core material. The resonantswitch voltage conversion ratio is equal to the normalized switchingfrequency F times a function P which depends on the load and biascurrents. The dependence of P on load current is essentially removed inthe full wave case with N=l; the switch voltage conversion ratio thenexhibits an inverse dependence on bias current I,. The equilibriumcharacteristics of buck, boost, and buck-boost converters utilizingnonlinear resonant switches have been determined.

This analysis can also be used to examine the tradeoff betweenpeak switch current, saturated tank inductance, and transistor switchingtime. By proper design, any value of peak switch current between thatof a linear resonant switch and an ideal PWM switch can be obtained;however, low peak currents require a small tank inductance and fasttransistor switching times. Nonetheless, peak currents which exceedthe ideal PWM value by only 20% are feasible, and require reduction oftransistor switching times by approximately50%.

Extensive experimental verification of the nonlinear resonant

switch concept has been performed. Buck converter experimental datais presented here, including results for full wave switches withfrequency and bias current control, and half wave switches withfrequency control. The nonlinear tank inductor has been implementedusing both ferrite and square loop tape wound core material. The resultsshow that use of the nonlinear resonant switch technique is an effectiveway to simultaneously obtain low conduction loss, low switching loss,and good switch utilization in high frequency converters.

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