Sneak Circuit Analysis in n-Stage Resonant Switched Capacitor Converters

Wenjuan Tu, Dongyuan Qiu*, Bo Zhang, Jianyuan Li

School of Electric Power, South China University of Technology, Guangzhou, 510641

*Email: epdyqiu@scut.edu.cn

ABSTRACT: Based on graph theory, this paper proposed a general method to analyze sneak circuit paths in n-stage

resonant switched capacitor (RSC) converters. Firstly, the n-stage step-up RSC converter was simplified into the

directed graph. Then all of the possible paths in the converter could be obtained by calculating the connection matrix.

Based on the operating principle of RSC converter, the useless paths had been gotten rid of, and the effective ones were

left. Comparing the effective paths with the normal operating condition, sneak circuit paths could be found out. At

last, the proposed sneak circuit analysis method has been verified in a 3-stage step-up RSC converter.

1. INTRODUCTION

The concept of sneak circuit was firstly put forward at the end of 1960s when NASA carried out the Apollo

moon-landing program. Sneak circuit is an unexpected path or logic flow within a system that may, under certain

conditions, lead to an unwanted or unintended action or inhibit a desired action. The reasons of sneak circuits are not

component or hardware failures, but the latent conditions that have been inadvertently designed into the system.

Therefore, sneak circuit analysis (SCA) is very important for the complex systems [1, 2].

As more and more new topologies of power electronic converters have been proposed in recent years, system

security and reliability become more and more important. Correspondingly, some methods of fault diagnosis,

topology identification and analysis have been developed, but the effect of sneak circuit is left out of account. Sneak

circuit phenomena had been found in some basic resonant switched capacitor (RSC) converters, the sneak circuit path

appeared under certain circuit operating conditions and affected the converter performance [3, 4]. The discovery of

sneak circuits in RSC converters shows that circuit disturbance will bring about undesirable action. Therefore, it is

strongly suggested to search for the sneak circuits during the topology design period.

The sneak circuit paths in the basic RSC converters could be pointed out easily by the fundamental electric circuit

theory [3, 4]. However, for the n-stage RSC converters, especially when n>2, it is difficult to find out the sneak circuit

paths directly [5, 6]. Therefore, it is necessary to propose a universal method that can analyze the sneak circuit paths

in all kinds of RSC converters. In this paper, graph theory is used to realize sneak circuits analysis. Firstly, an

n-stage step-up RSC converter is simplified into the directed graph, all of the operating paths will be figured out by

connection matrix calculation. After eliminating the false paths, the sneak circuit paths can be identified by converter

operating principle. Finally, the theoretical analysis has been verified by the experimental results.

2. OPERATING PRINCIPLE OF N-STAGE STEP-UP RSC CONVERTER

A n-stage step-up RSC converter is shown in Fig.1, including two switching components S1, S2, resonant inductor

Lr and n-1 basic switched capacitor step-up cells, where o iV nV= in the normal operation. The basic switched

capacitor step-up cell is composed of Dam, Dbm, Com and Crm, where 1,2, 1m n∈ − . Fig.2 illustrates the equivalent

circuits in normal operation. When S2 is turned on, Crm is charged by Co(m-1) except that Cr1 is charged by the input

source Vi. When S1 is turned on, the energy accumulated in Crm releases to Com [5].

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(a) Topology (b) Basic switched capacitor step-up cell

Fig.1 n-stage step-up RSC converters

(a) S2 ON (b) S1 ON

Fig.2 Equivalent circuits of n-stage step-up RSC converter in normal conditions

3. SNEAK CIRCUIT ANALYSIS BASED ON GRAPH THEORY

3.1 Definitions of The Directed Graph and Connection Matrix

In order to draw the directed graph of the n-stage step-up RSC converter, every node and edge should be named [7].

As shown in Fig.3(a), node 1 represents anode of power supply while node (2n+2) represents ground. eS1, eS2, eLr, er1,

eo1, ea1, eb1 denotes the edges with S1, S2, Lr, Cr1, Co1, Da1, Db1 respectively. The arrowhead stands for the likely current

flowing direction, which is determined by the characteristic of components in the edge. For example, the inductor

current is bidirectional, but not the diode current. Thus, the directed graph of the n-stage step-up RSC converter is

demonstrated in Fig.3(b).

Based on the connection matrix algorithm, the element of connection matrix cij is defined by

1 when

0 when , and there is no edge between nodes and

when , and there are edges between nodes and

ij

k

i j

c i j i j

e i j i j

== ≠

≠ (1)

then the connection matrix C of Fig.3 is expressed by

(a) Circuit diagram (b) Directed graph

Fig.3 Directed graph of the n-stage step-up RSC converter.

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1 1

1 2

( 1)

( 1)

( 1)

( 1)

( 1)

2 ( 1) ( 1)

1 0 0 0 0 0 0

1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0 0 0 0 1

s a

s s

a m m

b mn

o m

b n

o n

s om o m o n

e e

e e

e e

eC

e

e

e

e e e e

+

+

+

−

−

+ −

= (2)

3.2 Path Calculation

Based on graph theory, the components in the determinant of ij∆ , which is the cofactor of C, are the paths from

node j to node i ( ji ≠ ) [7]. As some false paths are included in the calculation result, path eliminating principles

should be set up to obtain the effect paths. Based on the operating theory of n-stage step-up RSC converter, the

principles are listed as following:

(1) S1 and S2 cannot be turned on at the same time, then the path containing es1 and es2 should be removed.

(2) It is impossible that Dam and Dbm or Db(m-1) and Dam are conducted together, so the paths containing two nearby

diodes should be gotten rid of.

(3) The resonant capacitors Cr1, Cr2, , Cr(n-1) cannot be connected in series, thus the path with er(m-1)erm is the

false one.

(4) The paths with quadratic terms are false paths, because they do not represent the practical circuit paths.

Therefore, effect paths from power supply to ground are

(2 2)1 1 1 2 1 1 1 1 1 1 ( 1) ( 1) ( 1) n n a r Lr s s Lr r b o s Lr rm bm om s Lr r n b n o nP e e e e e e e e e e e e e e e e e e e− + − − −= + + + + + (3)

Paths from ground to power supply are

1(2 2) 1 2 2 1 ( 1) ( 1) 1 ( 2) ( 1) ( 1) 1n n o a r Lr s om a m r m Lr s o n a n r n Lr sP e e e e e e e e e e e e e e e− + + + − − −= + + + + (4)

Cycles in the directed graph include

1 2 2 2 ( 1) ( 1) 2 ( 2) ( 1) ( 1) 2

1 1 1 2 2 ( 1) ( 1) ( 1) 2 1 1 1

C o a r Lr s om a m r m Lr s o n a n r n Lr s

Lr r b o s Lr rm bm om s Lr r n b n o n s Lr s a r

P e e e e e e e e e e e e e e e

e e e e e e e e e e e e e e e e e e e

+ + − − −

− − −

= + + + +

+ + + + + + (5)

3.3 Sneak Circuit Path Identification

Comparing with the equivalent circuits of n-stage step-up RSC converter in Fig.2, it is obviously that ea1er1eLres2,

eo1ea2er2eLres2, , eo(n-2)ea(n-1)er(n-1)eLres2 and es1eLrer1eb1eo1, , es1eLrer(n-1)eb(n-1)eo(n-1) are normal operating paths.

However, eLrer1eb1eo1es2, , eLrer(n-1)eb(n-1)eo(n-1)es2 and eLres1ea1er1, eo1ea2er2eLres1, , eo(n-2)ea(n-1)er(n-1)eLres1 maybe the

sneak circuit paths.

The equivalent circuits of the sneak circuit paths are shown in Fig.4. It is found that the inductor current doesn’t

stop after half-cycle resonance when S2 or DS2 is ON, but keeps running in the reverse direction though n-1 paths: Cr1,

Db1, Co1, Ds2, Lr; Cr2, Db2, Co2, Ds2, Lr; ; Cr(n-2), Db(n-2), Co(n-2), Ds2, Lr; Cr(n-1), Db(n-1), Co(n-2), Ds2, Lr. Similarly, when

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S1 or DS1 is ON, there are n-1 paths for the reverse inductor current, which are Cr1, Da1, Ds1, Lr; Co1, Da2, Cr2, Lr, Ds1, Vi;

; Co(n-2), Da(n-1), Cr(n-1), Lr, Ds1, Vi.

Vi

Cr1

Lr

S1

Ds1

S2

Ds2Co1

Cr2

Co(n-1) RL

Co3

Cr3 Cr(n-1)

Da1 Da2 Da3Db2 Da(n-1) Db(n-1)

Vo

iL

Db1

(a) S2 or Ds2 ON (b) S1 or Ds1 ON

(eLrer1eb1eo1es2, , eLrer(n-1)eb(n-1)eo(n-1)es2) (eLres1ea1er1, eo1ea2er2eLres1, , eo(n-2)ea(n-1)er(n-1)eLres1, eo(n-1))

Fig.4 Equivalent sneak circuits of n-stage step-up RSC converter with sneak circuit

4. EXPERIMENTAL RESULTS

In this paper, a 3-stage step-up RSC converter is used as an example. As shown in Fig.5, the connection matrix

of 3-stage step-up RSC converter is expressed in equation (6).

1

2

3

4

5 6 7

8

es2

ea1

eLr

eb1 ea2 eb2

er1

eo1

er2

eo2

es1

Fig.5 3-stage step-up RSC converter and its directed graph

1 1

1 2

1 1

1 2

3

2 1

2 2

2

2 1 2

1 0 0 0 0 0

1 0 0 0 0

0 0 1 0 0 0

0 1 0 0 0

0 0 0 0 1 0

0 0 0 0 1 0

0 0 0 0 0 0 1

0 0 0 0 1

s a

s Lr s

r b

Lr r r

a o

r b

o

s o o

e e

e e e

e e

e e eC

e e

e e

e

e e e

= (6)

Based on (3)~(6), all of the effective paths in 3-stage step-up RSC converter are

3 1 1 2 1 2 2 2 1 1 1 1 1 2 2 2 1 1 1 2 2 2 2 2

1 1 1 1 2 2 1

a r Lr s o a r Lr s s Lr r b o s Lr r b o Lr r b o s Lr r b o s

Lr s a r o a r Lr s

P e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

e e e e e e e e e

= + + + + ++ +

(7)

The normal paths and sneak ones can be identified by the converter’s operating principle, which are listed in Tab.1.

Tab.1 Normal and sneak paths in 3-stage step-up RSC converters

Normal paths Sneak paths

ea1er1eLres2, eo1ea2er2eLres2 eLrer1eb1eo1es2, eLrer2eb2eo2es2

es1eLrer1eb1eo1, es1eLrer2eb2eo2 eLres1ea1er1, eo1ea2er2eLres1

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In the 3-stage step-up RSC prototype, the switching devices and diodes are power MOSFET IRFZ44N and

Schottky diode S30SC4M, the specification are Lr=320nH, Cr1=Cr2=2µF, Co1=Co2=330µF, fr=140kHz, fs=42kHz, Vi=2V.

The waveforms in normal operating condition are shown in Fig.6(a). By reducing the load resistance, the sneak circuit

phenomena show up in Fig.6(b). Therefore, the theoretic analysis is confirmed by the experimental results.

(a) 15LR = Ω (b) 8.9LR = Ω

Fig. 6 Experimental waveform at different load in 3-stage step-up RSC prototype

5. CONCLUSION

Under certain conditions, sneak paths will join in circuit operation in RSC converters and affect the output

characteristics. Because of the complexity of the high n-stage switched capacitor converters, it is difficult to find out

the sneak circuit paths directly. This paper proposed a sneak circuit analysis method based on graph theory, which can

be summarized as following: (1) Determine the edge direction and simplify the circuit information into directed graph;

(2) Write out the connection matrix for the directed graph, figure out all of the paths and cycles, and then eliminate the

false paths; (3) Screen effective paths according to the converter operating principle, judge which one is normal

operating path and which one is sneak circuit path. The validity of this method has been confirmed by the

experimental results, and can be applied in other sorts of circuits.

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[4] D. Qiu, B. Zhang, “Analysis of Step-down Resonant Switched Capacitor Converter with Sneak Circuit State”, in

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