Negative output Luo converters: voltage lift technique

Negative output Luo converters: voltage lift technique

F.L. Luo

Abstract: The voltage lift technique is a popular method widely applied in electronic circuit design. Since the effect of parasitic elements limits the output voltage and power transfer efficiency of DC-DC converters, the voltage lift technique can lead to improvement of circuit characteristics. After long term research, this technique has been successfully applied for DC-DC converters. As with positive output Luo converters, negative output Luo converters are another series of new DC-DC step-up (boost) converters, which were developed from prototypes using the voltage lift technique. These converters perform positive to negative DC-DC voltage-increasing conversion with high power density, high efficiency and cheap topology in simple structure. They are different from other existing DC-DC step-up converters and possess many advantages, including a high output voltage with small ripples. Therefore, these converters will be widely used in computer peripheral equipment and industrial applications, especially for high output voltage projects.

1 Introduction

DC-DC step-up converters are widely used in computer hardware and industrial applications [l-151, such as computer peripheral power supplies, car auxiliary power supplies, servo-motor drives, and medical equipment. In recent years, the DC-DC conversion technique has been greatly developed. The main objective is to reach a high efficiency, high power density and cheap topology in a simple structure. For example, the Cuk converter [ 16-22] and class-E converter [23-251 are good topologies which have been developed.

Because of the effect of parasitic elements, the output voltage and power transfer efficiency of all DC-DC converters is restricted. The voltage lift technique is a popular method widely applied in electronic circuit design. It can lead to improvement of DC-DC converter characteristics. After long term research, this technique has been successfully applied to DC-DC converters. As for the positive output Luo converters [ 1-10], the negative output Luo converters are another 0 IEE, 1999 IEE Proceedings online no. 19990302 DOL 10.1049/ip-epa:19990302 Paper first received 16th March and in filial revised form 30th October 1998 The author is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798

series of riew DC-DC step-up (boost) converters, which were developed from prototypes using the voltage lift technique. These converters perform positive to negative DC-DC voltage increasing conversion with high power density, high efficiency and cheap topology in simple structure. They are different from any other existing TIC-DC step-up converters and possess many advantages, including a high output voltage with small ripples. Therefore, these converters will be widely used in computer peripheral equipment and industrial applications, especially for high output voltage projects.

The elementary circuit can perform step-down and step-up DC-DC conversion, which is shown in Fig. 1 and introduced in Section 2. The other negative output Luo converters are derived from this elementary circuit; they are the self-lift circuit, re-lift circuit and multiple- lift circuits (e.g. triple-lift and quadruple-lift circuits), and are introduced in Sections 3 , 4 and 5 , respectively. Switch S in these diagrams is a p-channel power MOS- FET device (PMOS). It is driven by a pulse width modulated (PWM) switching signal with repeating frequency J' and conduction duty k. In this paper, the switch repeating period is T = 1% so that the switch-on period is kT and the switch-off period is (1 - k)T. For all circuits, the load is usually resistive, i.e., R = Vo/Zo; the normalised load is zN = R f L . Each converter consists of a pump circuit S-L-D-(C) and a lI-type filter C-Lo-Co, as well as a lift circuit (without the elementary circuit). The pump inductor L absorbs energy from the source during switch-on, and transfers the stored energy to capacitor C during switch-off. The energy on capacitor C is then delivered to the load during switch-on. Therefore, if the voltage V , is high, the output voltage Vo is correspondingly high.

iD +vD- t "Lo- A $=is

t -

I I I I -. - Fig. 1 Elementary circuit

When the switch S is turned off, the current io flows through the freewheeling diode D. This current descends in a whole switching-off period (1 - k)T. If the current io does not reach zero before switch S is turned on again, we define this working state to be a continuous mode. If the current io reaches zero before switch S is turned on again, we define this working state to be a discontinuous mode.

The directions of all voltages and currents are indi- cated in the Figures. All descriptions and calculations in the text are restricted to the absolute values. In this paper, for any component X, its instantaneous current and voltage values are expressed as i, and vx, or idt)

I€€ Pioc -Electr Power Appl Vu1 146 No 2 Wurch 1999 208

and vdt), and its average current and voltage values are expressed as I, and V,. For the general description, the output voltage and current are Vo and Io; the input voltage and current are VI and I,. Assuming the output power equals the input power, Po = PIN, or V O I O = VII,.

The following symbols are used throughout this paper: voltage transfer gain in continuous mode:

A i l 0 1

variation ratio of current iL:

A i L / 2 ( E -

I L variation ratio of current iLo:

<=- A i L O I 2 I L O

& D / 2 (=- I L

(during switching-off, io = iL) variation ratio of current iLj is:

variation ratio of current io:

....................... ..............+................,.... ......................................

110'10 .......................................... ?--A\ 1. ........... ) ................ 4 ..........................

A i ~ j l 2 xj = ~

j = 1 , 2 , 3 , . . . I L j

variation ratio of voltage vc:

variation ratio of voltage vq:

variation ratio of output voltage vo = vco:

Avo12 VO

E = -

2 Elementary circuit

The elementary circuit is shown in Fig. 1. This circuit can be considered as a combination of an electronic pump S-L-D-(C) and a II-type low-pass filter C-Lo- Co. The electronic pump injects a certain energy into the low-pass filter every cycle. Capacitor C in Fig. 1 acts as the primary means of storing and transferring energy from the input source to the output load. Assuming capacitor C to be sufficiently large, the variation of the voltage across capacitor C from its average value V, can be neglected in the steady state, i.e., vdt) ~ i i Vc, even though it stores and transfers energy from the input to the output.

2. I Circuit description Figs. 2 and 3 refer to the elementary circuit. When switch S is on, the equivalent circuit is shown in Fig. 3a. In this case, the source current iI = IL. Induc- tor L absorbs energy from the source, and the current iL increases linearly with the slope VdL. At the same time, the diode D is blocked since it is reversely biased. Inductor Lo keeps the output current Io continuous and transfers energy from capacitor C to the load R, i.e., iC-on = iLo. When switch S is off, the equivalent

circuit is shown in Fig. 3b. In this case the source current ir = 0. Current iL flows through the freewheeling diode D to charge capacitor C, and enhances current iLo. Inductor L transfers its stored energy to capacitor C and load R via inductor Lo, i.e. iL = iCwoff + iLo. Thus, the current iL decreases. In order to analyse the circuit, these waveforms with enlarged variations are shown in Fig. 2.

......................................................... a ............................................

AiL IL ....... ..................................................... ...............................................

"t : I i j

______*_________.____ i ..............................

t

iLot

I ,t 0 kT T

ON OFF Fig. 2 Elementary circuit: waveform with enlarged variations

L a

b

C

Fig. 3 U Switch-on iL = I,; I ~ . ~ ~ = iLo b Switch-off

c Discontinuous mode

Equivalent circuits of elementary circuit . . .

j , = 0. j L = i , C."f+ i L 0

i r . = i 0 = 0 ; j c - - i LO

IEE Proc-Electr. Power Appl., Vol. 146, No. 2, March 1999 209

2.2 Average voltages and currents The output current Io = ILO, because the capacitor CO does not consume any energy in the steady state. The average output current:

I o = ILO = I c P o n The charge on the capacitor C increases during switch- off and decreases during switch-on:

Q+ = (1 - k ) T I ~ - o f f

(1)

Q- = k T I c P o n (2)

IC-,, = - IO IC-of f = -

In a whole repeating period T, Q+ = Q-, and

k 1 - 5

IO 1 - k

k 1 - 5

Therefore, the inductor current IL is

( 3 ) IL = I C F 0 f f + I o = -

Eqns. 1 and 3 are available for all circuits of negative output Luo converters. The source current iI = iL during the switch-on period. Therefore, its average source current II is

k - IO 1 - k

I1 = k x i~ = k i ~ = k l ~

or

(4) 1 - k

k I o = - 11

and the output voltage is

Vo = - VI 1 - k

The voltage transfer gain in continuous mode is

(5)

0 0 0.2 0.4 0.6 0.8 1 .o

k Voltage transfer gain of elementary circuit Fig.4

The curve of ME against k is shown in Fig. 4. The current iL increases and is supplied by VI during switch- on. It decreases and is reversely biased by -Vc during switch-off:

Therefore ~ T V I = (1 - k ) T V c (7)

2.3 Variations of currents and voltages Current iL increases and is supplied by VI during switch-on. Thus, its peak-to-peak variation is AiL = kTVIIL. Considering eqns. 3 and 6, and R = VdIo, the variation ratio of the current iL is

A ~ L 12 - k (1 - ~ ) V I T I L 2 L I o

(z- -

(9)

Considering eqn. 2, the peak-to-peak variation of voltage vc is:

Q- c c

The variation ratio of voltage vc is

(10)

(11)

AVC = - = - T I 0

A v c f 2 - k I o T - k

Since voltage Vo varies very little, the peak-to-peak variation of current iLo is calculated by the area A of a triangle with width TI2 and height Avc12:

1 - P=- Vc 2CVO 2 f C R

Considering eqn. 1, the variation ratio of current iLo is

Since the voltage vc is a triangle waveform, the difference between vc and the output voltage Vo causes a ripple of the current iLo, and the difference between iLo and the output current Io causes a ripple of the output voltage vo. The ripple waveform of current iLo should be a partial parabola in Fig. 2 because of the triangle waveform of Avc. To simplify the calculation, we can treat the ripple waveform of current iLo as a triangle waveform in Fig. 2, because the ripple of the current iLo is very small. Therefore, the peak-to-peak variation of voltage vco is calculated by the area B of a triangle with width TI2 and height AiL012:

B 1T k CO 2 2 1 6 f 2 C C o L o IO = - == --

The variation ratio of current vco is

A V C O P - k IO Vco 128f3CCoLo % E=- -

(15) k 1 - - -

128 f3CCoLoR Assuming that f = SOkHz, L = Lo = lOOpH, C = CO = 5mF, R = 10Q and k = 0.6, we obtain: ME = 1.5, 5 = 0.16, 5 = 0.03, p = 0.12 and E = 0.0015. The output voltage Vo is almost a real DC voltage with very small ripple. Since the load is resistive, the output current io(t) is almost a real DC waveform with very small ripple as well, and it is equal to Io = VdR.

2.4 Instantaneous values of currents and voltages Referring to Fig. 1, the instantaneous values of the currents and voltages are listed below:

210 IEE Proc.-Electr. Power Appl., Vol. 146, No. 2, March 1999

(17) VI + Vo for 0 < t 5 k T

for kT < t 5 T v D = { o

(19) { :(O) + %t for 0 < t 5 k T

for k T < t 5 T a1 = as =

iL (0 ) + %t i ~ ( k T ) - F(t - k T )

for 0 < t 5 k T for k T < t 5 T i L = {

(20)

0 for 0 < t 5 k T i D = { i L ( k T ) - p(t - k T ) for k T < t 5 T

(21)

(22) - I c - ~ ~ for 0 < t 5 k T { I C p 0 f f for k T < t 5 T ic x

where i~(0) = kI1 - k V 1 / 2 f L i ~ ( k T ) = ~ I I + k V 1 / 2 f L

Since the instantaneous currents iLo and ico are partial parabolas with very small ripples, we can treat them as DC currents.

2.5 Discon tin uous mode Referring to Fig. 3c, we can see that the diode current iD becomes zero during switch-off before the next period switch-on. The condition for a discontinuous mode is 5 2 I , i.e.

-- IC2 > 1 M g 2 f L -

or

The graph of the boundary curve against the normalised load zN = RffL is shown in Fig. 5. It can be seen that the boundary curve is a monorising function of the parameter k.

50 r 20 -

,o continuous mode k=0.9 -

5 -

discontinous mode

I

1 2 5 10 20 50 100 200 500 1000 RL/~L

Fig. 5 put voltage against the normalised load z, = &@ (elementary circuit)

Boundary between continuous and discontinuous modes and out-

filling efficiency and is defined as:

m E = - = - 1 M i 5 k2&

Considering eqn. 16, 0 < mE < 1. Since the current io becomes zero at t = kT + (1 - k)mET, for the current iL we have

kTVI = ( 1 - k ) m E T V c or

1 with - >- J R 2 f L - 1 - k

and for the current iLo, we have

k T ( V I + VC - VO) = ( 1 - k ) m E T V o Therefore, the output voltage in the discontinuous mode is:

1 >- (25)

i.e., the output voltage will increase linearly with load resistance R. The output voltage against the normalised load zN = R/fL is shown in Fig. 5. We can see that a larger load resistance R may result in a higher output voltage in the discontinuous mode.

2.6 Experimental results Experimental results corresponding to k = 0.1, 0.3, 0.6 and 0.8 are shown in Figs. 6-9. For each scope-picture, channel 1 shows the voltage waveform across inductor L, and channel 2 shows the output voltage waveform (absolute value). The voltage of the switching pulse is 5V and the current is very weak, because the switch S is a p-channel MOSFET device. The switch’s voltage and current at turn-on/turn-off are described by eqns. 16 and 19. The diode D is a fast switching diode, so that its effect of reverse recovery is negligible for the analysis and calculation. Referring to eqn. 17, its peak inverse voltage (PIV) should be higher than (VI + Vo). In this circuit, PIV = 200V. These experimental results verified our analysis and calculation.

In this case, the current io exists in the period between kT and tl = [k + (1 - k)mE]T, where mE is the

IEE Proc.-Electr. Power Appl., Vol. 146, No. 2, March 1999

Fig.6 k = 0.1 Frequency = 20 kHz; Vo = 1.1 V

Waveforms of elementary negative output Luo converter for

21 1

Fig. 7 k = 0.3 Frequency = 2OkHz; Vo = 4.3V

Waveforms of elementary negative output Luo convei pter for

Fig.8 k = 0.6 Frequency = 20 kHz; Vo = 15 V


Fig.9 k = 0.8 Frequency = 20kHz; Vo = 40V


3 Self-lift circuit

The self-lift circuit shown in Fig. 10 is derived from the elementary circuit. It consists of eight passive compo- nents: one static switch s; two inductors L, Lo; three capacitors C, C1, and CO; and two diodes D, D1. By comparison with Figs. 1 and 10, we can see that only one more capacitor C1 and one more diode D1 have been added into the self-lift circuit. Circuit C1-D1 is the lift circuit. Capacitor C1 functions to lift the capacitor voltage V , by a source voltage V,. Current icl(t) is an exponential function qt). It has a large value at the moment of power on, but it is small in the steady state, because V,, = Vr

+vci - +'D- - vLO+

Fig. 10 self-lift circuit

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

'L .........................

L * t

vc t

I I . t

iLo I

kT T

ON OFF

0

Fig. 11 Self-lift circuit: waveforms with enlarged variations

a

C

Fig. 12 a Switch-on il = iL + icl; ic.on = iLo b Switch-off

Equivalent circuits of self-lift circuit

il = 0; iL = ic.ofl + iLo = i,, c Discontinuous mode iL = iD = i,, = 0; ic = iLo

212 IEE Proc.-Electr. Power Appl.. Vol. 146. No. 2, Murch 1999

3.1 Circuit description Figs. 11 and 12 refer to the self-lift circuit. When switch S is on, the equivalent circuit is shown in Fig. 12a. In this case the source current iI = iL + icl. Inductor L absorbs energy from the source, and current ir increases linearly with slope VIIL. At the same time, the diode D1 is conducted, and capacitor C1 is charged, by the current icl. The inductor Lo keeps the output current Io continuous, and transfers energy from capacitor C to the load R, i.e. iC-on = iLo. When switch S is off, the equivalent circuit is shown in Fig. 126. In this case the source current iI = 0. Current iL flows through the freewheeling diode D to charge capacitor C, and enhances current iLo. Inductor L transfers its stored energy via capacitor C1 to capacitor C and load R (via inductor Lo), i.e. iL = icl-off = ic-off + i40. Thus, current iL decreases. In order to analyse the circuit, these waveforms with enlarged variations are shown in Fig. 11.

3.2 Average voltages and currents The output current Io = ILo, because the capacitor CO does not consume any energy in the steady state. The average output current is

The charge on the capacitor C increases during switch- off and decreases during switch-on:

Q+ = ( 1 - k )TIc -o f f

Q- = (27) In a whole repeating period T, Q+ = Q-. Thus

k k 1 - k 1 - k IC-off = - IC-on = - IO

Therefore, the inductor current ZL is

(28) IO I L = I c -o f f + Io = -

1 - k We know from Fig. 12a that

and 1 - k 1

ICl-on = - k Icl-off = -Io IC In the steady state we can use

vel = VI Current iL increases during switching-on with slope VIIL, and decreases during switching-off with slope -( Vo - VC#L = -( Vo - VI)lL. Therefore

or kVI = ( 1 - k)(VO - VI)

1 1 - k vo = - VI

and

The voltage transfer gain in continuous mode Io = ( 1 - k)II (32)

(33)

The curve of M , against k is shown in Fig. 13.

fore Circuit (C-Lo-Co) is a II-type low-pass filter. There-

(34)

0 0 0.2 0.4 0.6 0.8 1 .o

k Voltage transfer gain of self-lift circuit Fig. 13

3.3 Variations of currents and voltages The current i, increases, and is supplied by VI during switch-on. Thus, its peak-to-peak variation is AiL = kTVrlL. Considering eqn. 28 and R = VdIo, the variation ratio of the current iL is

(35) k ( 1 - k ) R k R -- - - - - 2MsfL M g 2 f L

Considering eqn. 27, the peak-to-peak variation of voltage vc is

The variation ratio of voltage vc is Avc /2 - kIoT - - k

p=--- -- Vc 2CVo 2 f C R The peak-to-peak variation of voltage Y C ~ is

1

The variation ratio of voltage vcl is

Considering eqn. 12:

the variation ratio of current iLo is A i ~ o / 2 - k 1 ____ < = ~ -

ILO 16 f’CLo Considering eqn. 14:

B 1 T IC CO 2 2 16f2CCoLo Avco = - = -- IO

the variation ratio of current vco is

k 1 - -- 128 f 3CCoLoR

(36)

(37)

(39)

213 IEE Proc.-Electr. Power Appl.. Vol. 146, No. 2, March 1999

Assuming that f = SOkHz, L = Lo = loopH, C = CO = 5pF, R = lOQ, and k = 0.6, we obtain: Ms = 2.5, < = 0.096, 5 = 0.03, p = 0.12, and E = 0.0015. The output voltage V, is almost a real DC voltage with a very small ripple. Since the load is resistive, the output current ic l ( t ) is almost a real DC waveform with a very small ripple as well, and it is equal to I , = VOIR.

3.4 Instantaneous values of the currents and voltages Referring to Fig. 12, the instantaneous values of the currents and voltages are listed below:

i ~ l ( 0 ) + d ( t ) + 3t for 0 < t 5 kT for kT < t 5 T

(44) ( 0

a 1 = as =

iL(0) + Ft i ~ ( k T ) - voLvI ( t - kT)

for 0 < t 5 kT for kT < t 5 T ZL = {

(45)

(47) d ( t ) for 0 < t 5 kT 0 f o r k T < t _ < T

2 0 1 =

- IcPon for 0 < t _< kT I ~ - ~ f f for kT < t 5 T (49)

where i ~ ( 0 ) = k l ~ - k V 1 / 2 f L i ~ ( k T ) = ~ I I + kV1/2fL

Since the instantaneous currents i,, and ic0 are partial parabolas with very small ripples, we can treat them as DC currents.

3.5 Discontinuous mode Referring to Fig. 12c, we can see that the diode current io reaches zero during switch-off before the next period switch-on. The condition for discontinuous mode is < 2 1, i.e.

-- " > 1 MZ2fL -

or

214

The graph of the boundary curve against the normalised load zN = NfL is shown in Fig. 14. It can be seen that the boundary curve has a minimum value of 1.5 at k = 113.

30.0 /- k=0.95 continuous mode 4

20.0

10.0 8.0

r 5.0

3.0

2.0

1.5

1 .o 13.5 124.7 62.5 222.0 842.0

Boundary between continuous and discontinuous modes and out- WfL 16.0

Fig. 14 put voltage against normalised load zN = Rlfl (seFl$t circuit)

In this case, the current in exists in the period between kT and t , = [k + (1 - Iz)ms]T, where ms is the filling efficiency and is defined as:

Considering eqn. 50, 0 < ms < 1. Since the current iD reaches zero at t = kT + (1 - k)msT, for the current i, we have

~ T V I = (1 - k)msT(Vc - V I ) or

VI = l + k 2 ( 1 - k ) - VI 1 (1 - k 1 k)ms i 2f " I L vc= 1+

1 with 6 - > - /27L - 1 - k

and for the current iLo we have

Therefore, the output voltage in the discontinuous mode is:

~ T ( V I + Vc - Vo) = (1 - k)msT(Vo - V I )

1

(52) i.e. the output voltage will increase linearly with load resistance R. The output voltage V, against the normalised load zN = RIfL is shown in Fig. 14. A larger load resistance R leads to a higher output voltage in the discontinuous mode.

3.6 Experimental results Experimental results corresponding to k = 0.1, 0.3, 0.6 and 0.8 are shown in Figs. 15-18. For each scope-picture, channel 1 shows the voltage waveform across inductor L, and channel 2 shows the output voltage waveform (absolute value). The voltage of the switch-

IEE Proc-Electr. Power ,4ppl., Vol. 146, No. 2, March 1999

Fig. 15 vo = 11.1v

Waveforms of self-lift negative output Luo converter f o r k = 0.1

. . . . . . I .

i j . .

Fig. 16 V O = 14.3v

Waveforms of self-liji negative output Luo converter f o r k = 0.3

Fig. 17 vo = 25v

1 Waveforms of self-liff negative output Luo converter for k = 0.6

Fig. 18 Vo = 50V

Waveforms of self-@ negative output Luo converter f o r k = 0.8

ing pulse is 5V and the current is very weak, because the switch S is a p-channel MOSFET device. The switch's voltage and current at turn-on/turn-off are described by eqns. 40 and 44. The diode D is a fast switching diode, so that its effect of reverse recovery is negligible. Referring to eqn. 41, its peak inverse voltage (PIV) should be higher than Vo. In this circuit, PIV = 200V. These experimental results verified the analysis and calculation.

4 Re-lift circuit

The re-lift circuit is shown in Fig. 19. It is derived from the self-lift circuit, and consists of one static switch S, three inductors L, L1 and Lo, four capacitors C, C,, C2, and CO, and diodes. From Figs. 1, 10 and 19, it can be seen that one capacitor C2, one inductor L, and two diodes D2, D,, are added into the re-lift circuit. Circuit C1-D1-Dl1-L1-C2-DZ is the lift circuit. Capacitors C1 and C2 perform characteristics to lift the capacitor voit- age Vc by twice the source voltage 2Vp Inductor L1 performs the function of a ladder joint to link the two capacitors C1 and C2, and raise the capacitor voltage V,. Currents icl(t) and ia(t) are exponential functions s,(t) and &(t). They have large values at the moment of power on, but they are small, because vcl = va = VI in the steady state.

kl+ _iD JL0+ - '0 r m -

'LO

== C =- d1 vo iL 1L Dz Jr D,

Fig. 19 Re-lift circuit

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

'L AiL

i

j

...... 4 . i .............................

IEE Proc.-Electr. Power Appl., Vol. 146. No. 2, March 1999 215

4. I Circuit description Figs. 20 and 21 refer to the re-lift circuit. When switch S is on, the equivalent circuit is shown in Fig. 21a. In this case the source current iI = iL + iC1 + ia. Inductor L absorbs energy from the source, and the current iL increases linearly with slope VIIL. At the same time, the diodes D1, D, are conducted, so that capacitors C1 and C2 are charged by the currents icl and ic2. The inductor Lo keeps the output current Io continuous and transfers energy from capacitor C to the load R, i.e. iC-on = iLo. When switch S is off, the equivalent circuit is shown in Fig. 21b. In this case the source current iI = 0. Current iL flows through the freewheeling diode D, capacitors Cl and C2, inductor L1 to charge capacitor C, and enhances current iLo. Inductor L transfers its stored energy to capacitor C and load R via inductor Lo, i.e. iL = icl-of = ic2-0H = = ic-off + iLo. Thus, current iL decreases. In order to analyse the circuit, these waveforms with enlarged variations are shown in Fig. 20.

=:= CO ". 1

C

Fig. 21 a Switch-on ir = i,,, + i,, + i,; i,. ." = iLO b Switch-off -0 . ; - . c Discontinuous mode iL = iD = i , = iLI = i,, = 0; i, = iL0

Equivalent circuits of re-lift circuit

I - , L - ',..JJ + i,0 = i , = iL, = ic,

4.2 Average voltages and currents The output current Zo = ZLo, because the capacitor CO does not consume any energy in the steady state. The average output current

Io = I L O = ICpon The charge on the capacitor C increases during switch- off and decreases during switch-on:

(53)

Q+ = (1 - k ) T I C - , f f Q- = kTIc-,,

In a whole repeating period T, Q+ = Q-. Thus

IC ICpon = - IO IC-off = -

IC 1 - k 1 - k

Therefore, the inductor current IL is

We know from Fig. 21a that 1

1 - k Icl-off = Ic2-off = I L 1 = I L = - Io (55) and

1 - k 1 ICl-on = Ic1-of . f = -10 k (56)

and 1 - k 1

I~2-on = - ICs-off = -10 k (57)

In the steady state we can use

and VCl = VC.2 = VI

Investigate the current iL. It increases during switching-on with slope VI/L, and decreases during switching- off with slope -(Vo - Vcl - V , - VL1-ojj)/L = -[Vo - 2 VI - k VI/( 1 - k)]/L. Therefore

STV, = (1 - k)T vo - 2v1 - - V I ) ( 1 - k or

(58)

(59)

2 1 - k

1 - k 2

vo = ~ VI

and

Io = - I I


(60) vo Ir 2 VI I , 1 - k

M R = - - = -

The curve of M s against k is shown in Fig. 22.

0 0.2 0.4 0.6 0.8 1 .o k

Fig.22 voltage transfer gain of re-lift circuit

Circuit (C-Lo-Co) is a II-type low-pass filter. There-

vc = vo = - VI

fore

(61) 2

1 - k

4.3 Variations of currents and voltages Current iL increases and is supplied by VI during switch-on. Thus, its peak-to-peak variation is AiL = kTVjL. Considering eqn. 54 and R = VdIo, the variation ratio of the current iL is

IEE Proc.-Electr. Power Appl., Vol. 146, No. 2, March I999 216

A i ~ / 2 k(1-k)VIT k(1-k)R - k - - - - < = - -

IL 2LIo 2hfRfL e The peak-to-peak variation of current iLl is

R

5 L1

A i ~ l = -TV;

The variation ratio of current iL1 is

k ( l - k) R (1 - k ) = ~- A L I I ~ - kTVI X I = - -

IL I 2LiIo -

2MR fL1 (63)

The peak-to-peak variation of voltage vc is

The variation ratio of voltage vC is

The peak-to-peak variation of voltage vc1 is

The variation ratio of voltage vcl is

Taking the same operation, the variation ratio of voltage vc2 is

Considering eqn. 12:

k TI0 = ~

1 T k 2 2 2CL0

ai,, = 8 f 2 C L ~ ' o

The variation ratio of current iLo is Ai,50/2 - k 1 -~ c=--- -

ILO 16 f2CLo Considering eqn. 14:

The variation ratio of current vco is

Assuming thatf= SOkHz, L = Lo = lOOpH, C = CO = 5pF, R = l O Q , and k = 0.6, we obtain: MR = 5, 5 = 0.048, 5 = 0.03, p = 0.12, and E = 0.0015. The output voltage V, is almost a real DC voltage with a very small ripple. Since the load is resistive, the output current io(t) is almost a real DC waveform with a very small ripple as well, and it is equal to Io = Vo/R.

4.4 Instantaneous values of the currents and voltages Referring to Fig. 19, the instantaneous current and voltage values are listed below:

0 for 0 < t 5 kT us={ Vo - ( 2 - &) VI for kT < t 5 T

( 6 9 )

(78)

(79)

& ( t ) for 0 < t 5 kT 0 f o r k T < t < T

& ( t ) for 0 < t I kT 2 0 2 = 0 for k T < t < T

2 0 1 =

for 0 < t 5 kT (82) ic={ I c p 0 f f for kT < t I T

IEE Proc.-Electr. Power Appl., Vol. 146, No. 2, March 1999 217

Since the instantaneous currents iLo and i,, are partial parabolas with very small ripples, we can treat them as DC currents.

60

4o

4.5 Discontinuous mode Referring to Fig. 21c, we can see that the diode current iD reaches zero during switch-off before the next period switch-on. The condition for the discontinuous mode is 5 5 1. i.e.

-- R > l M i f L -

-

k=0.95 continuous mode / c ,

or I

( 8 5 ) That is, the output voltage will increase linearly with load resistance R. The output voltage against the normalised load zN = RifL is shown in Fig. 23. A larger load resistance R may lead to a higher output voltage in the discontinuous mode.

The graph of the boundary curve against the normalised load zN = RIfL is shown in Fig. 23. It can be seen that the boundary curve has a minimum value of 3.0 at k = 113.

Fig.24 f = 2OkHz; Vo = 22.2V

Wuvejorms of re-lft negutive output Luo converter for k = 0.1

Fig.23 put voltuge uguinst normalised loud zN = WfL (re-lft circuit)

Boundury between continuous and discontinuous modes and out-

In this case, the current iD exists in the period between k T and t l = [k + ( 1 - k ) m ~ ] T , where m R is the filling efficiency and is defined as:

Considering eqn. 83, 0 < mR < 1. Because the inductor current iLl = 0 at t = t l ,

Since the current iD becomes zero at t = kT + (1 - k)rnRT, for the current iL we have

kTVI = (1 - k)mRT(VC - 2V1 - V ~ l - ~ f f ) or

2 with dZ - > - d" f L - 1 - k

and for the current i,, we have kT (VJ + VC - vo ) = (1 - k ) m R T (ViJ - 2VJ - VL 1 - o f f )

Therefore, the output voltage in the discontinuous mode is

2

Fig.25 j = 20kHz; Vo = 28.6V

Wavejorms ofre-lft negutive output Luo converterfor k = 0.3

Fig.26 f = 20kHz;

Wuvrforms of re-lft negutive output Luo converter for Vo = 50V

k = 0.6

4.6 Experimental results Experimental results corresponding to k = 0.1, 0.3, 0.6 and 0.8 are shown in Figs. 24-27. For each scope-

IEE Ptw-Elecrr . Power Appl.. Vol. 146, No. 2, March I999 21 8

picture, channel 1 shows the voltage waveform across inductor L, and channel 2 shows the output voltage waveform (absolute value). The voltage of the switching pulse is 5V, and the current is very weak because the switch S is a p-channel MOSFET device. The switch’s voltage and current at turn-oniturn-off are described by eqns. 69 and 74. The diode D is a fast switching diode, so that its effect of reverse recovery is negligible. Referring to eqn. 70, its peak inverse voltage (PIV) should be higher than V,. In this circuit, PIV = 400V. These experimental results verified the analysis and calculation.

Fig.27 f = 2OkHz; V , = 99.9V

Waveforms of re-lift negative output Luo converter f o r k = 0.8

‘c1 + L,

I rl i I h

Fig.28 Triple-lift circuit

Fig. 29 Quadruple-lift circuit

5 Multiple-lift circuits

Referring to Fig. 19, it is possible to build a multiple- lift circuit using only the parts (L1-C2-D2-D1 1) repeat- edly. For example, in Fig. 28 the parts (L2-C3-D3-D1?) were added to the triple-lift circuit. According to this principle, the triple-lift circuit and quadruple-lift circuit were built as shown in Figs. 28 and 29. In this paper, the particular analysis and calculations are unneces- sary, although their calculation formulas are shown in this Section.

5.7 Triple-lift circuit The triple-lift circuit is shown in Fig. 28. It consists of one static switch S, four inductors L, L,, L2 and Lo,

IEE Proc-Electr. Power Appl.. Vol. 146, No. 2, March 1999

five capacitors C, Cl, C2, C3 and CO, and diodes. The circuit C1-D~-Ll-C2-D2-D11-L2-C3-D3-D12 is the lift circuit. Capacitors C1, C2 and C3 perform characteristics to lift the capacitor voltage Vc by three times the source voltage V,. L1 and L2 perform the function of ladder joints to link the three capacitors C1, C2 and C3 and raise the capacitor voltage Vc. The currents icl(t), ia(t) and ic3(t) are exponential functions. They have large values at the moment of power-on, but they are small, because vc1 = va = vc3 VI in the steady state.

30 -

-

24 -

18 -

r - 12 -

0 0.2 0.4 0.6 0.8 1 .o k

Voltage transfer gain of iriple-lij circuit Fig. 30

The output voltage and current are


The curve of M , against k is shown in Fig. 30. Other average voltages:

Other average currents: V f = Vo; Vc1 = Vc2 = vc3 = VI

I L = I,, = I L 2 = - 10 1 1 - k I,o = Io;

Current variation ratios:

k(1 - k ) R k(1 - k ) R X z = ____- XI = ~-

~ M T f h 2h1T f L 2 Voltage variation ratios:

The variation ratio of output voltage Vc is k 1

& = - 128 f3CCoLoR

The output voltage ripple is very small.

(89)

219

The boundary between continuous and discontinuous modes is

It can be seen that the boundary curve has a minimum value of MT equal to 4.5, corresponding to k = 113. The boundary curve against the normalised load zN = RtfL is shown in Fig. 31.

9 0 1

// discontinuous mode I I I

R/f L 48

Fig.31 put voltage against normalised load zN = R/fL (tr@le-l$t circuit)


In discontinuous mode, the current io exists in the period between kT and [k + (1 - k)mT]T, where mT is the filling efficiency:

Considering eqn. 90, 0 < mT < 1. Because inductor current iLI = iL2 = 0 at t = t l ,

V L l - 0 f f = vL2-o f f = Vl (1 - k)mT

Since the current io becomes zero at t = kT + (1 - k)mTT, for the current iL we have

~ T V I = (1 - k)mTT(Vc -3v1- VL1-off - VL2-off) or

and for the current iLo we have

kT (VI + V i - VO) = (1 - k)mTT (VO - 2VI - VL1-off - VL2-Off)

Therefore, the output voltage in the discontinuous mode is

vi= 3 + " ] V I = [ 3 + k 2 ( l - k ) - 2.f " 1 L VI [ (1 -k )mT

3

That is, the output voltage will increase linearly with load resistance R. The output voltage against the normalised load zN = RIfL is shown in Fig. 3 1. We can see that the output voltage will increase with load resistance as the load R increases.

5.2 Quadruple-lift circuit The quadruple-lift circuit is shown in Fig. 29. It consists of one static switch S, five inductors L, LI, L2, L3, and Lp, and six capacitors C, C,, C2, C3, C4, and Co. Capacitors C1, C2, C3, and C4 perform characteristics to lift the capacitor voltage Vc by four times the source voltage V,. L,, L2 and L3 perform the function as ladder joints to link the four capacitors C,, C,, C3 and C4 and raise the output capacitor voltage Vc. The currents icl(t), ia(t), ic3(t) and ic4(t) are exponential functions. They have large values at the moment of power-on, but they are small, because vcl = vc2 = va = vc4 E VI in the steady state.

The output voltage and current are

(93) 4

1 - k v, = - VI and

1 - 5 4 Io = - II

The voltage transfer gain in continuous mode is 4 MQ = VO/VI = -

1 - k The curve of MQ against k is shown in Fig. 32.

(94)

(95)

0 0.2 0.4 0.6 0.8 1 .o k

Voltage transfer gain of quadruple-lift circuit Fig. 32

Other average voltages: vc = vo; vc, = vc2 = vc3 = vc4 = VI

Other average currents: 1

1 - k ILO = IO; IL = I L ~ = I L 2 = I L 3 = - IO


k ( l - k ) R k ( l - k ) R x1= ~-

2MQ f L 1 x2=-- 2-MQ f L 2

220 IEE Proc.-Electr. Power Appl. , Vol. 146, No. 2, March 1999

Voltage variation ratios:

*Q 0 1 = -- k 1 2 fCR 2 f C i R

2 fC2R 2 fC3R

2 fC4R

p = --

MQ 0 2 = -- MQ 0 3 ~-

MQ 0 4 = --

The variation ratio of output voltage Vc is k 1

E = - 128 f 3 C C o L o R

The output voltage ripple is very small.

modes is The boundary between continuous and discontinuous

hfQ<fi@=dG (97)

It can be seen that the boundary curve has a minimum value of M Q equal to 6.0, corresponding to k = 113. The boundary curve is shown in Fig. 33.

120 1 k=0.95 continuous mode .

80

-t

541100 250 889 3360 64 RAL

Fig. 33 put voltage against normalised load zN = Hfl (quadruple-lift circuit)


In the discontinuous mode, the current io exists in the period between kT and [k + (1 - k)mQ]T, where mQ is the filling efficiency:

Considering eqn. 97, 0 < mQ < 1 . Because inductor current iL1 = iL2 = iL3 = 0 at t = tl , so that

Since the current io reaches zero at t = kT + (1 - k)mQT, for the current iL we have

kTVI = (1 - k ) m Q T (VC - 4v1 - V L I - o f f

- V L z - o f f - h 3 - o f . f )

or

I

2 R 4 f L - 1 - k

and for current iLo we have

~ T V I = ( 1 - k ) m Q T (vc - 4v1 - v L 1 - - o f f

- V L 2 - of f - vL 3 - of f ) Therefore, the output voltage in the discontinuous mode is

I

2R 4 f L - 1 - k

(99) That is, the output voltage will increase linearly with load resistance R. The output voltage against the normalised load zN = RIfL is shown in Fig. 33. We can see that the output voltage will increase with load resistance as the load R increases.

6 Summary

From the analysis and calculation in Sections 2 - 5, we can obtain the common formulas for all circuits:


( j = 1 , 2 , 3 , . . .) k ( 1 - k ) R ''= 2 M f L j

Voltage variation ratios: k

E = k p=-

2 f C R 1 2 8 f 3 C C o L o R M

2 f C j R uj = ~ ( j = 1,2,3,4 , . . .)

In order to write common formulas for the voltage transfer gain, and boundaries between continuous and discontinuous modes and output voltage for all circuits, the circuits can be numbered. The definition is as follows: subscript 0 for the elementary circuit, subscript 1 for the self-lift circuit, subscript 2 for the self-lift circuit, subscript 3 for the triple-lift circuit, subscript 4 for the quadruple-lift circuit, and so on. Therefore, the voltage transfer gains in the continuous mode for all circuits are:

j = o , 1 , 2 , 3 , 4 ) . . . [ j + h ( j ) ] k h ( j ) M j = 1 - k

The variation of the freewheeling diode current iD is

The boundaries are determined by the condition:

or <j 2 1

Therefore, the boundaries between continuous and discontinuous modes for all circuits are:

j = O , 1 , 2 , 3 , 4 , . . .

IEE Proc-Electr. Power Appl., Vol. 146, No. 2, March 1999 221

The filling efficiency is

z ._ c 6.0 $ 4.5 m v

2 3.0 s!

- 2 1.5-

c

a, m

9

The voltage across the capacitor C in discontinuous mode for all circuits

- -

- j = O , 1 , 2 , 3 , 4 , . . .

The output voltage in discontinuous mode for all circuits

j = 0,1,2,3,4, where

0 i f j 2 1 1 i f j = O

is the Hong function.

120

100

’- 80 P E? c - 0 c

a 3 50 0

30

10

0 0.2 0.4 0.6 0.8 1 .o I I I I I

0 0.2 0.4 0.6 0.8 1 .o conduction duty k

Fig.34 Output voltuges of all negative output Luo converters ( V I = IOV) (i) Quadruple-lift; ( i i ) triple-lift; ( i i i ) re-lift: (iv) self-lift; (v) elementary Input voltage V , = IOV

The voltage transfer gains in the continuous mode for all circuits are shown in Fig. 34. The boundaries between continuous and discontinuous modes of all circuits are shown in Fig. 35. The curves of M against zN show that the continuous mode area increases from M E via M,, MR, MT to MQ. The boundary of the ele-

mentary circuit is a monotonically rising curve, but there. are minimum values of the boundaries Ms, M,, MT and M Q at k = 113.

,o.ol continuous mode M T e

discontinuous mode

0.51 I I , ,

4 4.5 13.5 27 40.5 54 100 200400 normalised load zN=R/fL

Fig.35 negutive oulput Luo converters

Bounduries between continuous und discontinuous modes of U N

Assuming that f = SOkHz, L = Lo = L, = L, = L, = L4 = 100yH, C = Cl = C2 = C3 = C4 = CO = 5yF, and the source voltage VI = lOV, the value of the output voltage Vo for various k is shown in Fig. 34. Typically, some values of the output voltage Vo for k = 0.33, 0.5, 0.75 and 0.9 are listed in Table 1. The ripple of the output voltage is very small, say smaller than 1%. For example, using the above data and R = lOQ, the variation ratio of the output voltage is E = 0.0025 x k = 0.0008, 0.0012, 0.0019 and 0.0023, respectively. These data indicate that the output voltage of all negative output Luo converters is almost a real DC voltage with a very small ripple. For comparison with all DC-DC step-up (boost) converters, they are shown in Fig. 36. It can be seen that all Luo converters, positive and negative output converters, have higher output voltages [l-lo].

7 Discussion

7. I Discontinuous-conduction mode Industrial applications usually require DC-DC converters to work in a continuous mode. However, it is inevitable that a DC-DC converter will sometimes work in a discontinuous mode. The analysis in Sections 2 - 5 shows that during switch-off, if current iD becomes zero before the next period switch-on, the state is called a discontinuous mode. The following fac- tors cause the diode current iD to become discontinu-

(i) The switching frequency f is too low (ii) The conduction duty cycle k is too small

ous:

Table 1: Comparison of five negative output Luo converters

Negative output Luo vo, v i0 VO

k = 0.33 k = 0.5 k = 0.75 k = 0.9 converters

Elementary circuit (1 - k)/k /I k/( 1 - k ) V/ 5 10 30 90

Self-lift circuit ( 1 - k) / / 1/(1 - k)V/ 15 20 40 100

Re-lift circuit (1 - k)/2 I/ 2/( 1 - k) Vi 30 40 80 200 Triple-lift circuit ( 1 - k)/3 I/ 3/( 1 - k) VI 45 60 120 300

Quadruple-lift circuit ( 1 - k)/4 I, 4/(1 - k)V, 60 80 160 400 Input voltage V,= 1OV

222 IEE Pror.-Eler.fr. Pow,rr Appl.. Vol. 146, No. 2, Murdi I999

http://Pror.-Eler.fr

Fig.

output polarity positive to positive

based circuit

positive to negative

: $ ~ ~ v a

v, I L V Ck I 1 0 g ~ l I

elementaty LUO converter circuit

quadruple-lift 3 circuit

I 1 \ I i DC-DC step-up (boost) converters

(iii) The combined inductor L is too small (iv) The load resistance R is too large A discontinuous mode means that io is discontinuous during switch-off. The output current io(t) is still continuous if Lo and CO are sufficiently large.

7.2 Output voltage Vo against conduction duty k The output voltage V , is positive, and is usually greater than the source voltage VI when the conduction duty ratio k > 0.5 for the elementary circuit, and may take any value in the range 0 < k < 1 for self-lift, re-lift, and multiple-lift circuits. Although a small value of k means that the output voltage V , of self-lift and re-lift circuits is greater than VI and 2V1 etc., when k = 0 it results in Vo = 0, because switch S is never turned on.

If k is close to 1, the ideal output voltage V, should be very large. Unfortunately, because of the effect of parasitic elements, the output voltage V, falls very quickly. Finally, k = 1 results in Vo = 0, not infinity, for all circuits. In this case, the accident of iL tending towards infinity will happen. The recommended range of values for the conduction duty k is 0 < k < 0.9.

7.3 Switching frequency f In this paper, the repeating frequency f = 50kHz was selected, although the switching frequency f can be selected in the range between lOkHz and 5OOkHz. Usu- ally, the higher the frequency, the lower the ripples.

7.4 Sorting DC-DC step-up (boost) converters It is now possible to sort every DC-DC step-up (boost) converter in Fig. 36, and the advantages of Luo converters over other converters can be determined.

8 Conclusions

Since the effect of parasitic elements of DC-DC converters limits their output voltage and power transfer efficiency, the conduction duty k should not be higher than 0.9. This paper introduced the successful application of the voltage lift technique in the design of the DC-DC converter. Negative output Luo converters, a series of new DC-DC step-up (boost) conversion circuits, have been developed. All Luo converters which implement the voltage lift technique avoid taking too high a value of the conduction duty k. This overcomes the effects of parasitic elements and greatly increases the output voltage of the DC-DC converters. It also produces the characteristics of high efficiency, high power density, cheap typology in simple structure, and near-zero output voltage and current ripples. These converters can be used in computer peripheral circuits, medical equipment, and industrial applications, especially for applications with high output voltage.

9 References

1 LUO, F.L.: ‘Luo converters - voltage lift technique’. Proceedings of the IEEE Power Electronics special conference IEEE-PESC’98, Fukuoka, Japan, 17-22 May 1998, pp. 1783-1789 LUO, F.L.: ‘Luo converters - voltage lift technique (negative output)’. Proceedings o f the second World Energy System international conference WES’98, Toronto, Canada, 19-22 May 1998, pp. 253-260 LUO, F.L.: ‘Re-lift converter: design, test, simulation and stabil- ity analysis’, IEE Proc. Electr. Poww Appl., 1998, 145, (4), pp. 3 15-325 LUO, F.L., LEE, W.C., and LEE, G.B.: ‘Self-lift circuit: a new DC-DC converter’. Proceedings of the 3rd National Undergradu- ate Research Programme (NURP), Singapore, 1997, pp. 31-36 LUO, F.L., WONG, K.T., and KOH, K.G.: ‘Investigation of the Cuk converter’. Proceedings of the 3rd National Undergraduate Research Programme (NURP), Singapore, 1997, pp. 3742

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3

4

5

223 IEE Proc.-Elrctr. Power Appl., Vol. 146, No. 2, March 1999

LLO, F.L.: ‘Luo converters: new DC-DC step-up converters’. Proceedings of the IEE international conference ISIC-97, Singa- pore, 1997, pp. 227-230 LUO, F.L.: ‘DSP-controlled PWM Luo converter used for PM DC motor drives’. Proceedings of the IEEE international conference SISCTA’97, Singapore, 1997, pp. 98-102 LUO, F.L.: ‘Luo converters: a series of new DC-DC step-up (boost) conversion circuits’. Proceedings of the IEEE international conference PEDS’97, Singapore, 1997, pp. 882-888 LUO, F.L.: ‘Re-lift circuit: a new DC-DC step-up (boost) converter’. Electron. Lett.. 1997. 33. (1). vv. 5-7

10 LUO, ’F.L. and LIU,. Z.Z.:’ ‘Comp;t&sed PWM DC-DC Luo converter used for PM DC motor drives’. EEE Research, Singa- pore, 1997, pp. 61-62

1 1 MOHAN, N., UNDELAND, T.M., and ROBBINS, W.P.: Power electronics: converters, applications and design’ (John

Wiley & Sons, New York, 1995) 12 RASHID, M.H.: ‘Power electronics: circuits, devices and applica-

tions’ (Prentice-Hall, 1993, 2nd edn.) 13 MASSEY, R.P., and SNYDER, E.C.: ‘High voltage single-ended

DC-DC converter’, IEEE Power Electron. Spec. Conf. Rec., 1977, pp. 156-159

14 JOZWIK, J.J., and KAZIMERCZUK, M.K.: ‘Dual sepic PWM switching-mode DC/DC power converter’, IEEE Trans. Ind. Elec- tron., 1989, 36, (l), pp. 64-70

15 MARTINS, D.C.: ‘Application of the Zeta converter in switch- mode nower sunnlies’. Proceedines of TEEE AP

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SMEDLEY, K.M., and CUK, S.: ‘One-cycle control of switching converters’, IEEE Trans. Power Electron., 1995, 10, (6), pp. 625- 633 MAKSIMOVIC, D., and CUK, S.: ‘Switching converters with wide DC conversion range’, IEEE Trans. Power Electron., 1991, 6, (l), pp. 151-157 MAKSIMOVIC, D., and CUK, S.: ‘A general approach to syn- thesis and analysis of quasi-resonant converters’, IEEE Trans. Power Electron., 1991, 6, (l), pp. 127-140 MAKSIMOVIC, D., and CUK, S.: ‘Constant-frequency control of quasi-resonant converters’, IEEE Trans. Power Electron.. 1991, 6, (I), pp. 141-150 SEVERNS, R.P., and BLOOM, E.: ‘Modern DC-to-DC switch- mode power converter circuits’ (Van Nostrand Reinhold, New York, 1985) MIDDLEBROOK, R.D., and CUK, S.: ‘Advances in switched- mode power conversion - vols. 1 and 2’. Tesla Co., Pasadena, CA, 1981 LIU, Y., and SEN, P.C.: ‘New class-E DC-DC converter topologies with constant switching frequency’, IEEE Trans. Ind. Appl., 1996, 32, (4), pp. 961-969 REDL. R., MOLNAR, B., and SOKAL, N.O.: ‘Class-E resonant DC-DC power converters: analysis of operations, and experimental results at 1.5 Mhz’, IEEE Trans. Power Electron., 1986, 1, pp. 11 1-1 19 KAZIMIERCZUK, M.K., and BUI, X.T.: ‘Class-E DC-DC converters with an inductive impedance inverter’, IEEE Trans. Power Electron., 1989, 4, pp. 124-135

224 IEE Proc.-Electr. Power Appl., Vol. 146, No. 2, March 1999

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Negative output Luo converters: voltage lift technique