Modeling of Cascade Buck Converters

Mummadi VeeracharyDepartment of Electrical EngineeringIndian Institute of Technology Delhi

New Delhi - 110 016, India.

ABSTRACT- In this paper signal flow graph non-linearmodeling of cascade buck converters is presented. Uni-fied signal flow graph model of the converter is developedand then deduction of large, small-signal and steady-state models from the unified graph is demonstrated.Converter performance expressions are derived. Large-signal model is developed and programmed in TUTSIMsimulator. Large-signal responses against supply andload disturbances are obtained. Validity of the proposedsignal flow graph modeling is verified through PSIM sim-ulator results.

I. INTRODUCTION

Development of cascade converters with new controlstrategies is coming up to increase the Power Process-ing capability and to improve the reliability of the powerelectronic system. Particularly, aeronautics and telecom-munication appliances require large conversion ratios.These requirements can he fulfilled either with the helpof isolated stepdown/ stepup Pulse width modulated(PWM) dc-dc converters or non-isolated converters. How-ever, the use of stepdown/ step-up converters with trans-formers (isolated) results in large switching -~-snwes thatmay damage the switching devices [l-21. Further, use oftransformer limits the switchins frequency of the con-verter. An alternative option, for realizing larger dcconversion ratios, is cascading of the converters. Thisscheme mainly uses multi-stage approach that consistsof n-basic converters connected in cascade

State-space averaging is the well-known analysis methodfor dc-dc converters [3]. However, this method is tediouswhen the converter circuit contains a large number of el-ements. Furthermore, the linearised models, obtainedfrom state-space averaging, do not predict the large-signal stability information, and are only sufficient topredict small-signal stability. To overcome this proh-lem, a signal flow graph (SFG) modeling method hasbeen developed for PWM converters [4-51. This model-ing method translates a given switching converter intoits dynamic model directly and provides a visual under-standing of the switching converter system with a possi-bility of incorporating the cause and effect relationship ofthe dynamics. The important advantage of this modelingmethod is that, it converts the two or multi-stage switch-ing converter into a unified dynamic model from which itis possible to derive large, small-signal and steady-statemodels with minimum mathematical manipulations.

This paper presents the SFG modeling of dc-dc cascadebuck converters. For illustration, a 2-cell cascade buckconverter is considered in this paper. SFG's for large-signal and steady-state cases are drawn. Further, large-signal dynamic behavior of the converter system is pre-sented for supply voltage, load disturbances. To validatethe SFG modeling method the results, obtained fromSFG method, are compared PSIM simulator results.

11. DEVELOPMENT OF SIGNAL FLOW GRAPH FOR THEDC-DC CASCADE CONVERTER

This section describes SFG development of a dc-dc cas-cade converter (consisting two identical/ non-identicalbuck cells connected in cascade) as shown in ~i~. 1.The analysis of the system is carried out under the fol-lowing assumptions:

(i) Switching elements of the basic converter cellsare assumed to be ideal.

(ii) The individual cells of the cascade convertersystem operate in the continuous inductor cur-rent mode.

(iii) The switches SI, Sz, , S, operate in synchrenism fashion.

(iv) The ESR of the capacitance and stray capaci-tances are neglected.

(v) Passive components (R, L, and C) are assumedto he linear time-invariant.

The assumed switching sequence results in two modesof operation. During the time 0 < t 5 TON the switches4,S,; and during tON < t 5 T the diodes DD1,DD1

are respectively conducting and thus generating two dif-ferent sub-circuits. The converter switches between these

Fig: 1. 2-cell cascade buck converter.

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Table 1. Steady-state performance expressions

(1+sRC.)

Fig. 2. Unified SFG of the cascade buck converter

two subcircuits, which are linear and a linear systemtheory can be extended. Considering the switch Sl operation as reference, signal flow graphs GON,GOFF aregenerated for ON, OFF subcircuits respectively sharingcommon nodes and part of the branches. The two signalflow graphs GON,GOFF are combined to form a sim-plified signal flow graph. While merging the two signalflow graphs (CO,, GOFF) into a single graph G, someof the branches exist in the two graphs and some maynot. Branches that exist in GON but not in GOFF arereplaced by K1 branches, and the branches that existin GOFF hut not in GON are replaced by KZ branches.The resulting graph topology, shown in Fig. 2, can bemathematically written as

G = KIGON + K~GOFF (1)

where K1,Itz are the switching functions, whose valuesdepend on the switching times, defined by the followingexpressions.

for 0 < t < toNfor toN <t<Tfor 0 < t < tON

for toN <t <T.II(2)

(3)

Assuming filter corner frequency is much smaller [4] thanthe switching frequency, the effective signals carried atthe outputs of K1, Kz branches having an average valuesdl(t), &(t) respectively are

y(t)=x(t)d1(t),

!At) = +)dz(t).

(4)

(5)

Incorporation of these large-signal models for the switch-ing functions in the simplified SFG, results in a large-signal flow graph model of the converter. From thelarge signal switching branch models the steady-stateswitching branch models can easily be derived. In thesteady-state, K1 branch will have a transmittance ofml(t) = 01 and Kz branch will have a transmittanceof mz(t) = D2. Simplifying the large-signal flow graphwith the above steady-state switching branch models andsetting complex frequency s +» 0, a steady-state model isobtained. From this switching flow graph various steady-state relations can be derived by employing the wellknown Mason's gain formula [6]. Using this steady-stateSFG model various performance characteristics expres-sions are derived using the Mason's gain formula and

(MIL

Rfc?

0o<J±io(s)

M')

Table

[*•<•

2.

+[*

+E»

+['

+1'

Small-signal

Vsfci[fi+(Jtj -j

transfer

hr3)|ClS(Z

t-r3)+*J](]

*+rJ)](l +

functions

.«Ci)+flCI^1.+r1)+Rl• is+ri)+l]

sflC2)+/JCis(Ll5+r1) + fl|

l+3fiC2)+fiClSj

Ap = Cis(LyS + n)(L2s + r2)(l + sRC2)\ n, ^2 are theseries resistances of the inductors.

tabulated in Table 1. A small'signal SFG of the con-verter can be obtained from the unified SFG model, byreplacing the switching branches with their small-signalequivalents. On the assumption of neglecting second-order perturbations, the small-signal switching equationsfor K1, KZ branches, respectively are

(6)

(7)g(t) = Dze(t) - Xd(t).

Upon substitution of the above small-signal models forswitching branches in the simplified SFG, a small-signalflow graph is generated. Using this small-signal SFGmodel various small-signal transfer functions are derivedusing the Mason's gain formula and tabulated in Table 2.These expressions are in agreement with those obtainedfrom state-space averaging method.

111. RESULTS AND DISCUSSIONS

Comprehensive simulation studies were made to inves-tigate the signal flow graph modelling of dc-dc cascadeconverters as shown in Fig. 1. To verify the theoreti-cal analysis and signal flow graph modelling equations,developed in the previous sections, the following designexample was considered. The buck converter parameterschosen are: n = 0.01 mn, rZ = 0.01 mR, R = 1.0 0,Li = 200 pH, LZ = 200 pH, C1 = 200 P F , C2 = 200 pF,

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SFC method

PSIM simulation

- Vg = IOV -

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0,

Time (s)

Fig. 3. Load voltage response (supply disturbance)

SFG method

PSIM simulation

R = 1 R = l n

0 0.01 0.02 0.03 004 0,05 0.06 0.07 0.08

Time (s)

Fig. 5. Load voltage response (load disturbance).

SFG method

VE = IOV - = i6V •

0 0.01 0.02 0.03 0.04 0,05 0.06 0.07 0.08

Time (s)

Fig. 4. Load current response (supply disturbance).

/ = 10 kHz. For illustration, large-signal response simu-lations are obtained for a 2-cell dc-dc cascade buck con-verter. Largesignal SFG is programmed in the TUT-SIM simulator to determine the large-signal responses.These responses were obtained for two different cases:(i) supply voltage is changed from 10 V to 16 V, (ii)load resistance is changed from 1 R to 0.5 R and thenback to 1 R. For different values of the duty ratios, thestep responses of the source current and load voltage ofthe cascade converter are obtained. For illustration, fewsample results are presented here for the duty ratio of0.5 and they are shown in Figs. 3 to 6. To validate thelarge-signal response characteristics obtained from thesignal flow graph modeling method, PSIM simulator re-sults are provided and they are also given in the Figs. 3and 6. These results closely match with those obtainedfrom the signal flow graph modeling. Slight differences

SFG method

PSIM ~imulation

k^ ,

R=O,5ni--

0 0.01 O.M 0.03 O.M 0.05 O.M o w 0 8

Time (6)

Fig. 6. Load current response (load disturbance).

in these results are attributed to the following factors:(i) in the signal flow graph modeling, non-idealities ofthe converter elements such as forward voltage drops,on-state resistances of the switching devices and otherparasitics are not taken into account, (ii) use of in-builtdevice models available in the PSIM simulator, (iii) useof built-in integration methods etc.

Iv. CONCLUSIONS

In this paper, the SFG approach was extended to modelthe dc-dc cascade buck converter operating in continu-ous current mode. Large, small signal and steady-statemodels lead to simple graphical circuits that are verymuch suitable for analysis and simulation. To confirm

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the modelling method theoretical results, obtained from [3] R. D. Middiebrook, Siobodan Cuk, "A ~~~~~~l uni-SFG analysis, were compared with PSIM simulations. fied Approach to Modeling Switching Converter Power

Stages," IEEE Power Electronics Specialist Conference,Vo1.4, pp. 18-34, 1976.

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