singer1992 - canonical modeling of power

7/28/2019 Singer1992 - Canonical Modeling of Power

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38 IE E E TRANSACTIONS ON POWER ELECTRONICS. V O L . 7 . NO. I . JANUARY 1992

process, in which it is desired that the rectifier present alinear resistive load to the ac system. The power pro-cessed by this effective load is transferred to a dc output.Such a network is called a loss-free resistor [ 5 ] . InSection V, the basic equations of loss-free resistor net-works are described. It is shown that a number of com-monly known converters operate naturally as loss-free re-sistors, including the discontinuous conduction modebuck-boost and Cuk converters, without additional feed-back. Alternatively, feedforward and feedback can beused to convert a different type of POPI network, such asa time-variable transformer, into a loss-free resistor [ 5 ] .In Section VI, another type of natural POPI device, theconstant power network, is described, and a current pro-grammed discontinuous buck boost example is given.

11. THE POPI CONCEPTThe POPI network is a black box power processingunit that controls the inputioutput voltages and currentssuch that the instantaneous output power and input powerare equal. For a two-port network, denote the input volt-age and current quantities as u 1 an d i , , and the output

quantities as u2 an d i2 . As with any two-port device, onemay denote two of these quantities as independent vari-ables and solve for the remaining dependent quantities. Ingeneral form, the POPI equations can be writtenu , ( r ) i 1 ( r )= - uZ(r) i2(r) (1)

x d f ) = f(xi(t), u( r ) ) (2 )where x , ( r ) an d xi(f)are vectors containing the dependentand independent quantities, respectively, and u(r) is acontrol parameter. In the case where the POPI network islinear and time invariant, it is well known that only theideal transformer and gyrator satisfy ( 1 ) an d ( 2) . The pa-rameter u(r) then has the physical interpretation of trans-former turns ratio and gyration constant, respectively.

111. T H ET I M E - V A R I A B L ER A N S F O R M E RThe transformer model is well known and has been ap-plied for the description of dc-dc as well as dc-ac con-verters [11 , [6 ] . Because in most power processing appli-cations the input and/or output may vary, a controlledtime-variable transformer ( TV T) is required. The definingequations of the time variable transformer are

where M ( t ) is the conversion ratio. The first-order powerconversion properties of many types of converter circuitssatisfy ( 3 ) , including the continuous conduction modePWM b uck, boost, buck-boo st, Cuk and other converters.A notable property of the transformer is that it preservesthe nature of the source input: if a voltage source is con-nected to the transformer input, then the transformer out-put also exhibits voltage source characteristics. This

property is not shared by other types of POPI networks,and whether it is desirable depends on the intended powerprocessing application.I V . T H EG Y R A T O R

The gyrator is a lossless two-port network defined bythe following input-output relationship:

Gyrators have the property that they reflect networks astheir duals with respect to the gyration conductance g [7].The gyrator converts a capacitor into an inductor, resistorinto admittance, voltage source into current source, andvice versa. For example, the gyrator concept has been ap-plied for control of switching networks that convert a ca-pacitor into an equivalent inductor [SI. Hence it is suit-able for the modeling of POPI circuits, which are pow-ered by a voltage source and which have current sourceoutput characteristics. The transformer model is not suit-able for the description of such circuits as it cannot con-vert the voltage source (w hich powers the circuit ) into acurrent source.Even more importantly, the transformer is not a suit-able element for the coupling of an ideal voltage sourceto a load that has a voltage source characteristic becausethis kind of coupling would result in a K irchhoffs voltagelaw (KV L) violation 121. Examp les of such loads are stor-age batteries, gas discharge devices, voltage-stabilizedloads, and distribution networks. Coupling such loads bymeans of a gyrator is very convenient because the gyratorconverts the voltage source i.nput into a current source asviewed by the load, implying a stable operating point [2].An example of a converter that naturally operates as agyrator is the series resonant converter operated in th e k= 2 discontinuous conduction m ode. Fig . 2 , in which thetank rings for exactly two complete half cycles during eachhalf switching period [9], [ I O ] . The fact that this circuitexhibits current-source output characteristics suggestsmodeling it as a gyrator. Indeed, it can be shown that theaverage switching cell input and output currents are givenby

I ] = 822 ( 5 )i 2 = (6 )

where

1f -~- 2rJLc


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SI N G ER A N D ER I C K SO N : C A N O N I C A L M O D ELI N G OF PO WER - PR O C ESSI N G C I R C U I TS-39

il =

_ _ _ - - - - - _ _ _ _ _ _ _ -tcontrolFig. 3. Steady-state equivalent circuit m odel for the circuit of Fig. 2 . Theconverter behaves as a gyrator. .

(b )Fig. 2. The series resonant converter. (a) Schematic. (b) Tank inductorcurrent waveform, k = 2 discontinuous conduction mode.

L an d C are the tank inductance and cap acitance, and f, isthe switching frequency. Equations (5) and (6) are of theform of a gyrator, with gyration conductance g control-lable by the switching frequency fs. An equivalent circuitis given in Fig. 3 , which is valid provided that the con-verter indeed operates in k =2 discontinuous conductionmode. The effect of the single high-frequency tank capac-itor state can also be included as a further refinement tothe model.The form of a POPI network can be altered by the useof feed forward and feedback. In [3], a converter withgyrator characteristics was constructed by controlling thevoltage conversion ratio M of a PW M converter (whichwould otherwise exhibit transformer characteristics) ac-cording to the law M = - i l / g v l . Substitution of this lawinto (3 ) yields the gyrator characteristics of (4). Any net-work that obeys (4) can be regarded as a g yrator, regard-less of whether it was obtaine d naturally o r through feed-back techniques.Consider a converter with gyrator characteristics thatcouples a power input zll of voltage sourc e characteristicsto a load (such as a storage battery) that consists of a volt-age source E in series with a small resistance r as in Fig.4.Let us calcula te the input current by transferring theload to the input terminals. Because the gyrator trans-forms a series into a parallel graph, with each of the com-ponents chan ged into its dual, the transferred load is com-posed of equivalent current souce gE in parallel withequivalent admittance g 2 r . The input current is , there-fore,

il = gE +g 2 r v l . (7)Note that, unlike the transformer case, the system cur-rents are well controlled, even when the value of r tendsto zero. Coupling v l o E by means of a transformer vi-olates the KVL when r is zero.Thus, the gyrator is useful as a power processing ele-

(C)Fig. 4. Alteration of Characteristi cs of a POPI network via feedback. (a)Schematic. (b) Equivalent circuit model based on the gyrator. (c) Calcu-lation of input current by transferring load to input side of gyrator.

ment. T he rules for manipulating circui ts containing gy-rators are well known and easy to use. T he gyrator is well-suited for interconnecting stiff voltage sou rces. Con vert-ers exist today with natural gyrator characteristics.V . THE LOSS-FREE ESISTOR

The loss-free resistor (LFR) is a two-port POPI devicewhose input port obeys Oh m's law [5]:v 1 = i , R , (8 )

The pow er consumed by the input port(9)

is transmitted to the output port. Hence, the output portbehaves as a source of constant power which obeys thelawP = v2i2. (10)


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40 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL 7. NO . I . JANUARY 1992

Equations (8)-( 10)are the defining relations of a loss-freeresistor. The effective resistance R , is generally someLoss-free resistor characteristics are desirable in a num-ber of applications, including the stabilization of gas-dis-charge tubes, lossless dam ping [ lo ], and high-power fac-tor rectification. These characteristics are usually obtainedby nonlinear feedback and feedfonvard around a PWM

converter operating in continuous conduction mode; in ef-fect, this converts the time-variable transformer modelinto a loss-free resistor. Some converter topologies, how-ever, behave naturally as loss-free resistors without theneed for external feedback. Examples include the buck-boost, flyback, and Cuk converters operated in discontin- DTS Tsuous conduction m ode.Consider the buck-boost converter operated in discon-tinuous conduction mode, as illustrated in Fig. 5. It canbe shown that the average cell input and output currentsare given by

function of a control input. "1d'-i"m. R4- _ _ _ _ __ - - - - - - - - - - - - - -;(;r k,;.( a )I D i l ( 0- - - - - - - - - - - - - - - - - 595 - - - - -

DTs (D+4?)Ts Ts( 1 1) ( b )2) II , =-Re Fig. 5 . Discontinuous conduction m ode buck-boost converter. (a) Sche-rnatic. (b) Input and output current waveforms.

zji z u 2=- PRe (1 2 )whereR , =2 L / D 2 T s . (13)

L is the inductance, D is the switch duty cycle, and T, isthe switching period. An equivalent circuit utilizing theLFR is shown in Fig. 6.The output source is neither con-stant voltage nor constant current but rather supplies con-stant power P according to (14). This model is valid bothfor dc and for large ac variations, provided that the con-verter is operated correctly in discontinuous conductionmode.Fig. 6predicts that DCM buck-boost (and flyback) con-verters, operated at constant duty cycle in a rectifier ap-plication, should produce no ac line harmonics other thanswitching harmonics (which can be reduced to low valueby other means). The con verter input appears as a linearresistive load to the ac system. Hence, if the input voltageis

(14)1 = Vp k sin (u t ) /

tD I \U?

Fig. 6 . An averaged equivalent circuit model of the D C M buck-boost con-verter. Th e converter inherently behaves as a loss-free resisto r.

If the output capacitor is large so that v2 has negligibleripple, then the equilibrium load resistor current is equalto the average value of iz:

then the input current is Elimination of R , using (13) yields

and the instantaneous power is where K = 2 L / R T , . Hence, the output voltage can beregulated as usual by simple control of D , provided thatD does not change so fast that harmonics are introducedinto the input ac line current.Another converter that naturally behaves as a loss-freeresistor is the Cuk converter (Fig. 7(a)) whose internal

P , ( 1 7 ) capacitor voltage state is operated i n the discontinuous21 2 R , 12 conduction mode [ 1 2 ] . t can be shown that the average

(16)v 2 . 2P = u I i l = uz i2 = - s i n (ut).R ,The output current is

iz =- = ~ sin' (ut).


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42 I EFE TR A N SA C TI O N S O N P O WE R FLFC TR O N I C S. VO I 7 NO I . JANUARY I Y Y ?

1, 12

i Llock& paratorcontrolinput

( b )Fig. 9. The current-programmed discontinuous conduction mode buck-boost converter. (a) Schematic. (b ) Input and output current wave-forms.

VII. C O N C L U S I O N SThe generalized POPI network is an effective way tomodel the first-order power conversion properties of

switching converters. A special case of POPI networks-the ideal transformer-is well known as a suitable meansfor modeling most continuous conduction mode PWMdc-dc converters. However, many oth er types of con-verters do not possess transformer properties and natu-rally process power in other ways. Examples discussedhere include a k = 2 discontinuous conduction mode(DCM ) series resonant conve rter, which operates a s a gy-rator, a PWM-DCM buck-boost con verter, which oper-ates as a loss-free resistor, and a current-programmedDCM buck-boost converter, which operates as a constantpower network.Furthermore, in many applications, a power conversioncharacteristic othe r than that of the transformer is needed.The insight gained by use of these POPI models may in-spire new and simpler system realizations. For example,understanding that some types of converters operate nat-urally as loss-free resistors can lead one to the possibilityof constructing a simp ler unity power facto r ac-dc con-

buck

1 ontrolP=;Lip$fs

I I I 1

buck- boost

1 - 1

P.+LIPkZfSFig. I O . Averaged equivalent circuit models lor the current-programmeddircontinuous conduction mode buck, boost. and buck-boost converters.The buck-boost inherently beh aves as a constant power network.

verter. Another example is the use of the gyrator to inter-connect stiff voltage sources.It is possible to obtain these other properties by addi-tion of (possibly nonlinear) feedback to a converter withtransformer characteristics. Nonetheless, use of a topol-ogy that naturally processes power in the desired fashionmay lead to a simpler solution. Regardless of the con-verter realization, a proper choic e of POPI network m odelcan lead to a better understanding of the system powerprocessing me chanisms, and to development of more ef-fective solutions to meet power processing requirements.

R E F E R E N C E S[ I ] R . D. Middlebrook and S. uk . " A gencral unified approach to mod-elling switching-converter power stages." / E E Power Elarrronics

Spccirr/isr.\ C o n f : . 1976 Record, pp. 18-34. June 1976.121 S . Singer, "Gyrators application in powe r processing circuits." l E E ETrun.5. I ndusrr iu l E lmron . . v o l . IE-34. no. 3 . pp. 313-318. August1987.

[ 3 ] S . Singer. "Loss free gyrator realization," / E Trmns. CircuitsS w . . vol . CAS-35. no. I . pp. 26-34, Jan. 1988.[4] E . S . Kuh and R . A. Rohner . 7hror\ofLirirrrrAcrir'r Nrrnorks. Ne wYork: Holden Day, 196 7, pp. 96-103.


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SI N G ER A N D ER I C K SO N : C A N O N I C A L M O D ELI N G OF PO WER - PR O C ESSI N G C I R C U I TS 43

S . Singer, Realization of loss-free resistive elements, IEEE Trans.C i rcu it s Sys t . , vol . 37, no. I , pp. 54-60, Jan 1990.S . Singer, Power conversion and control with zero ac current har-monics by means of a time-variable transformer, Proc. Inst . Elec.E ng. , vol. 131, pt. G , no. 4, pp. 147-150, Aug . 1984.R. W Newco mb, T he sem istate description of nonlinear time-variable circuits, IEEE Trans. Circuits Syst . , vol . CAS-28, no. 2 ,pp. 62-71, Feb 1981.D. M Divan, Non-dissipative switching networks for high powerapplications, Electron. Lett , vol. 20, no. I , pp 277-279, March29, 1984.V . Vorperian and S Cuk, A complete dc analysis of the series res-onant converter, IEEE Power Electronics Special is ts Con&, 1982Record, pp. 85-100, June 1982.A F Witulski and R. W. Erickson, Steady-state analysis of th eseries resonant converter, IEEE Trans. Aerospace Electronic Syst . ,vol AES-21, no 6 , pp. 791-799, Nov. 1985.S . Singer, T he application of loss-free resistors in power process-ing circuits, IEEE Power Electronics Special is ts Conf. , 1989 Rec-o r d , , ~ ~43-846, June 1989.S . Cuk, General topological properties of switching structures,IEEE Power Electronics Special is ts Conf . , 1979 Record, pp 109-130, June 1979.S . Freeland, I . A unified analysis of converters with resonantswitches, 11. Input current shaping for single phase ac-dc pha se con-verters, Ph.D . dissertation, California Institute of Technology, Pas-adena , CA, October 1987.

Darlington Award by

monic rectif ication.

Sigm unt S inger received the B.Sc. and D.Sc. de-grees from the Technion, Haifa, Israel, in 1967and 1973, respectively.In 1978 he joined the Faculty of Engineering,Tel-Aviv University. His fields of research arepower electronics, power processing, switchingnetworks, time-variable coupling networks, en-ergy conversion, photovoltaic systems, modeling,and simulation of circuits operated at low lossesconditions. His paper Realization of loss-freeresistive elements has been granted the 1990the CAS Transactions Best Paper Award Co mmittee.

Robert W . Erickson (SBILM83) was born inSanta Monica, CA, on August 3 , 1956. He re -ceived the B . S . , M.S. , and Ph.D. degrees f romthe California Institute of Technolog y, Pasadena,CA, in 1978, 1980, and 1983, respectively.In 1982, he joined the D epartment of Electricaland Computer Engineering at the University ofColorado, Boulder, where he is currently an As-sociate Professor. Hi s research interests inculderesonant power conversion, converter modeling,high-frequency power compon ents, and low har-

singer1992 - canonical modeling of power

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