Upcoming SlideShare
×

# Ripple theorem

361
-1

Published on

0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
361
On Slideshare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
13
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Ripple theorem

1. 1. t is well known that the switching ripple informa-tion is lost in a linearized average model of a PWMconverter. In this work it is shown that the switch-ing ripple component of any state in a PWM dc-to-dcconverter operating in continuous conduction mode canbe recovered with remarkable accuracy from the lin-earized, average, small-signal model of the converter byreplacing the duty-ratio control signal, d , with the accomponent of the switching PWM drive signal, s(t) -D, in the control-to-state transfer function. The theoremis very useful for estimating the peak-to-peak ripple inany state of a converter and is used here to determine,for the first time, the actual ripple in the zero-ripple,coupled Cuk converter as a function of the load.A RippleTheoremffoorr DDcc--ttoo--DDcc CCoonnvveerrtteerrssOOppeerraattiinngg iinnCCoonnttiinnuuoouuss CCoonndduuccttiioonn MMooddeeby Dr. Vatché VorpérianJet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena, CaliforniaI1©Copyright 2005 Switching Power Magazine
2. 2. The determination of the steady-state ripple current in aninductor in a dc-to-dc converter is a fairly simple task that iseasily accomplished by balancing the flux in the inductorwhile ignoring at first the ripple voltage across thecapacitor(s). Later, the ripple voltage across a capacitoris determined assuming the entire ripple current isabsorbed by the capacitor.When this simplistic approach is applied to converters withcoupled inductors, such as the Cuk [1] converter or any of itsderivatives, the ripple current in the inductor is shown to van-ish when the coupling coefficient of the two inductors is setequal to their turns ratio. Such converters have been referredto as zero-ripple converters even though, when ideal compo-nents are used, the ripple current does not vanish.The determination of the ripple current in such convertersunder zero-ripple condition has not been presented before andhas sometimes been mistakenly attributed to non-ideal com-ponents. In this paper, it is observed that there is an interest-ing relationship between the ripple component of a voltage orcurrent in a PWM converter, operating in continuous conduc-tion mode, and the small-signal transfer function relating theduty-ratio control signal, d, to that voltage or current.Because of its generality, this observation is formulated interms of a theorem and two corollaries. A simple proof, usingthe model of the PWM switch, is given. The theorem is par-ticularly useful for analytically determining the peak-to-peakvalue of a ripple voltage or current and is easy to implementin circuit simulation programs.AA RRIIPPPPLLEE TTHHEEOORREEMMSome definitions are reviewed before presenting the theorem.The steady-state (non-modulated) switching signal, s(t),isthe normalized drive signal applied to the gate of the activeswitch and is shown in Fig. 1a. The steady-state duty-ratio,D, is the dc component of s(t), and p(t) is the ac compo-nent of s(t), which is shown in Fig. 1b and defined as:(1)When the switching drive signal is modulated, a low-frequen-cy sideband accompanies the duty-ratio, D, which, in the lit-erature of small-signal analysis, is commonly known as thecontrol signal, d .Although any one of the well-known analytical methods fordetermining the small-signal characteristics of a dc-to-dcconverter can be used to prove the theorem, we shall usethe technique of the PWM switch model [2] becauseof its simplicity.Proof: In all PWM dc-to-dc converters with two switchedstates, the voltage across the inductor and the ripple currentthrough it have the shape shown in Fig. 2 in which the volt-ages V1 and V2 are related to the input and output voltages ofthe converter and satisfy the condition:(2)The effective circuit that generates the voltage and current inFig. 2 can be configured using the PWM switch as shown inFig. 3. In this figure we see that the dc voltage between theactive and passive terminals is given by:(3)To obtain the small-signal equivalent circuit of the circuit inFig. 3, we set V1 = V2 = 0 and replace the PWM switch withits small-signal equivalent circuit as shown in Fig. 4.Now, if we replace in this figure d with p(t) , the voltageacross the inductor will be given by:in which we have made use of Eqs. (3)and (4). Hence the small-signal modelin Fig. 4, with d replaced by p(t) ,generates the correct switching voltagewaveform across the inductor andhence the correct ripple currentthrough it. This concludes the proof.Theorem: The steady-state ripple current in the switched inductor(s)in a PWM dc-to-dc converter operating in continuous conduction mode canbe obtained from the average, small-signal, control-to-inductor transferfunction by replacing the duty-ratio control signal, d , with the ac compo-nent of the steady-state switching signal, p(t) = s(t) - D .Figure 1: a) The switching signal and b) its ac component.(4)2©Copyright 2005 Switching Power MagazineRipple Theorem
3. 3. Figure 2: The voltage across a switched inductor and thecurrent through it.Figure 3: The effective circuit of a switched inductor using thePWM switch.Figure 4: The small-signal equivalent circuit of the circuit in Fig.3 using the small-signal model of the PWM.3©Copyright 2005 Switching Power MagazineThe remaining ripple components in a converter are derivedeither directly from the inductor ripple current (as in thecase of the ripple voltage of the capacitor in the output fil-ter of a buck converter), or indirectly from the ac compo-nent of the active terminal current (as in the case of the rip-ple voltage of the capacitor in the input filter of the buckconverter), or from the ac component of the passive termi-nal current (as in the ripple voltage of the capacitor in theoutput filter of a boost or a buck boost converter). Hence,we compare next the active terminal current in the small-signal model in Fig. 4 and the actual circuit model in Fig.3 in order to assess the accuracy of the switchingcurrents obtained form the small-signal model.From the small-signal model in Fig. 4, we see that Ia(t) isgiven by:(5a,b)in which we have used Ic = IL. The actual Ia(t) from theswitching circuit in Fig. 3 is given by:(6a,b)These two currents are sketched in Figs. 5a, b and c forthree different cases of the conduction parameter, K, whichcorrespond to heavy, critical load and no load conditions.Figure 5: Comparison of the active terminal current from theactual and small-signal models in a) heavy load b) critical loadand c) no load.Ripple Theorem
4. 4. The conduction parameter is defined as:(7)When synchronous rectification is not used, the critical valueof K = 1 corresponds to boundary between continuous anddiscontinuous conduction modes.It can be easily deduced from Figs. 5a, b and c that when K>> 1, the actual and small models predict nearly the sameswitching current waveforms so that the ripple voltageinduced on the capacitors which absorb these switching cur-rents must be nearly identical in both models. When K<1however, we do not expect the shape of the induced ripplevoltages in the two models to be the same, but we expect thatthe peak-to-peak ripple in both models to be very closebecause the average value of the active and passive terminalcurrents in each sub-interval is the same in both models. Wecan now state the following two corollaries.Often, it is the peak-to-peak ripple and not its exact shapethat is of interest to a designer. Hence, regardless of the valueof K, the peak-to-peak ripple can be obtained to a very goodapproximation from the magnitude response of Gxd (s) to thefundamental component of p(t) which is given by:(10)RRIIPPPPLLEE AANNAALLYYSSIISS OOFF TTHHEE ZZEERROO--RRIIPPPPLLEE,,CCOOUUPPLLEEDD CCUUKK CCOONNVVEERRTTEERRThe ripple theorem is now applied to the coupled Cuk con-verter [2], with a synchronous rectifier, shown in Fig. 6. Thecurrent ripple steering properties of this converter from theoutput inductor to the input inductor and vice-versa as a func-tion of the coupling coefficient are well known. For the con-verter in Fig. 6, the triangular ripple current in the outputinductor can be steered to the input inductor by letting n = kwhich is known as the zero-ripple condition. Although incomparison to the uncoupled case the ripple current in theoutput inductor under zero-ripple conditions is far less, it isnot zero. We shall now determine, for the first time since theintroduction of this converter, the actual output ripple currentand ripple voltage when the zero-ripple condition is satisfied.The small-signal model of the Cuk converter using the modelof the PWM switch is shown in Fig. 7 in which the parame-ters of the PWM switch are defined as:Before analyzing the small-signal circuit using the ripple the-orem, a simulation is performed of the circuits in Figs. 6 and7 as shown in Figs. 8a-c. The actual converter is simulated inFig. 8a in which:(13)Corollary 1: For values of the conductionparameter greater than unity, the ripple compo-nent of any state in a PWM converter can becalculated from the small-signal control-to-statetransfer function, Gxd (s), according to:(8)inwhich:(9a,b)Corollary 2: The peak-to-peak value of the rip-ple component in a state can be obtained, to avery good approximation, from the magnituderesponse of the control-to-state transfer functionevaluated at the switching frequency, Fs ,according to:(11)4©Copyright 2005 Switching Power Magazine(12 a-g)Figure 6: The coupled Cuk converter.Ripple Theorem
5. 5. Design WorkshopPower SupplyGain a lifetimeof design experience. . . in four days.FFoorr ddaatteess aanndd rreesseerrvvaattiioonnss,, vviissiitt wwwwww..rriiddlleeyyeennggiinneeeerriinngg..ccoommMorning Theory● Converter Topologies● Inductor Design● Transformer Design● Leakage Inductance● Design with Power 4-5-6Afternoon Lab● Design and BuildFlyback Transformer● Design and BuildForward Transformer● Design and BuildForward Inductor● Magnetics Characterization● Snubber Design● Flyback and ForwardCircuit TestingDay 1 Day 2Morning Theory● Small Signal Analysisof Power Stages● CCM and DCM Operation● Converter Characteristics● Voltage-Mode Control● Closed-Loop Designwith Power 4-5-6Afternoon Lab● Measuring Power StageTransfer Functions● Compensation Design● Loop Gain Measurement● Closed Loop PerformanceWorkshop AgendaMorning Theory● Current-Mode Control● Circuit Implementation● Modeling of Current Mode● Problems with Current Mode● Closed-Loop Design forCurrent Mode w/Power 4-5-6Afternoon Lab● Closing the Current Loop● New Power StageTransfer Functions● Closing the VoltageCompensation Loop● Loop Gain Designand MeasurementMorning Theory● Multiple Output Converters● Magnetics Proximity Loss● Magnetics Winding Layout● Second Stage Filter DesignAfternoon Lab● Design and Build MultipleOutput Flyback Transformers● Testing of Cross Regulationfor Different Transformers● Second Stage Filter Designand Measurement● Loop Gain withMultiple Outputs andSecond Stage FiltersDay 3 Day 4Only 24 reservations are accepted. \$2495 tuition includes POWER 4-5-6 Full Version, lab manuals,breakfast and lunch daily. Payment is due 30 days prior to workshop to maintain reservation.Ridley Engineeringwww.ridleyengineering.com770 640 9024885 Woodstock Rd.Suite 430-382Roswell, GA 30075 USA
6. 6. The gates of M1 and M2 are driven with a 5V, PWM wave-form which result in an on-resistance of 150m⍀ for each ofthe MOSFETs:(14)The small-signal equivalent circuit in Fig. 7 is simulated inFig. 8b in which the sub-circuit U1=VMSSCCM is the small-signal model of the PWM switch in continuous conductionmode available from the ORCAD/Pspice library calledswit_rav. The numerical values of the small-signal parametersof the PWM switch in Eqs. (12) which are supplied toVMSSCCM are shown in Fig. 8b whereRSW = rt = RdsON and RD = rd = RdsON .The voltage applied to terminal Vc is converted to the duty-ratio control signal, d , by the Ramp parameter according to:(15)The default value of the Ramp is 1V. The PWM switchingfunction, s(t), and the gate-drive voltage generating circuit isshown in Fig. 8c. The ac component of the PWM switchingfunction s(t) - D, is obtained at the output of the high-passfilter formed by Cx and Rx in Fig. 8b.The results of the simulation for heavy load (R_load = 5⍀)are shown in Figs. 9 and 10. The agreement between thesmall-signal model and the actual circuit is seen to be excel-lent for the input and output ripple currents as well as for theripple voltages across both capacitors. It can also be seen thatthe output ripple current and the output ripple voltage areboth non-zero even though the zero-ripple condition is satis-fied. The results of the simulation for light load (R_load =500⍀) are shown in Figs. 11 and 12.The agreement between the small-signal model and the actu-al circuit is seen to be fairly good especially if one is inter-ested in determining the value of the peak-to-peak ripple.The cause of discrepancy in the small-signal result for theripple voltage across Cc owes to the inability of the small-signal model to capture accurately the active and passive ter-minal currents when K < 1 as explained earlier in connec-tion with Fig. 5. The excellent agreement in Fig. 11 in theinput ripple current predicted by the small-signal and theactual models owes to the fact that the input ripple current,under zero-ripple conditions, mostly consists of the com-mon-terminal current which is captured very accurately bythe small-signal model. Not withstanding these dis-crepancies, the usefulness of the small-signal resultscan clearly be appreciated.Figure 7: The small-signal equivalent circuit model of the Cukconverter using the model of the PWM switch5©Copyright 2005 Switching Power MagazineFigure 8: (a) The coupled Cuk converter, (b) the small-signalequivalent circuit model using the model of the PWM switch(VMSSCCM from the Pspice library swit_rav) and (c) the switch-ing functions, p(t) and 1 - p(t) , and gate-drive signal generatorsE_D and E_Dp.Ripple Theorem
7. 7. Figure 9: Comparison of the input ripple current (a) and the out-put ripple current (b) obtained by the theorem and the actual cir-cuit in the coupled Cuk converter in Fig. 8 under heavy load condi-tions (Rload = 5⍀⍀) and under zero-ripple conditions.Figure 10: Comparison of the ripple voltage across Cc (a) and theoutput ripple voltage across C1 (b) obtained by the theorem and theactual circuit in the coupled Cuk converter in Fig. under heavyload conditions (Rload = 5⍀⍀) and under zero-ripple conditions.6©Copyright 2005 Switching Power MagazineFigure 11: Comparison of the input ripple current (a) and the out-put ripple current (b) obtained by the theorem and the actual cir-cuit in the coupled Cuk converter in Figs. 6 and 8a under verylight load (Rload = 500⍀⍀) and zero-ripple conditions.Figure 12: Comparison of the ripple voltage across Cc (a) and theoutput ripple voltage across C1 (b) obtained by the theorem andthe actual circuit in the coupled Cuk converter in Figs. 6 and 8aunder very light load (Rload = 500⍀⍀) and zero-ripple conditions.Ripple Theorem
8. 8. Only Need One Topology?Buy a module at a time . . .Buck Converter \$295Boost Converter \$295Buck-Boost Converter \$295Flyback Converter \$595Isolated Forward, Half Bridge, \$595Full-Bridge, Push-PullModulesAABBCCDDEEBundlesPOWER 4-5-6 Features●● Power-Stage Designer●● Magnetics Designer with core library●● Control Loop Designer●● Current-Mode and Voltage-Mode Designer andanalysis with most advanced & accurate models●● Nine power topologies for all power ranges●● True transient response for step loads●● CCM and DCM operation simulated exactly●● Stress and loss analysis for all power components●● Fifth-order input filter analysis of stability interaction●● Proprietary high-speed simulation outperforms anyother approach●● Second-stage LC Filter Designer●● Snubber Designer●● Magnetics Proximity Loss Designer●● Semiconductor Switching Loss Designer●● Micrometals Toroid Designer●● Design Process Interface accelerated and enhancedBundle A-B-C \$595All Modules A-B-C-D-E \$1295Ridley Engineeringwww.ridleyengineering.com770 640 9024885 Woodstock Rd.Suite 430-382Roswell, GA 30075 USA