Applications of DC–DC the push–pull converter. There is no danger of transformer sat-

13.10 Applications of DC–DC the push–pull converter. There is no danger of transformer sat-

uration in the half-bridge converter. It requires, however, two

Converters

additional input capacitors to split in half the input dc source. The full-bridge converter is used at high (several kilowatts)

Step-down choppers ﬁnd most of their applications in high- power and voltage levels. The voltage stress on power switches performance dc drive systems, e.g. electric traction, electric is limited to the input voltage source value. A disadvantage of vehicles, and machine tools. The dc motors with their winding the full-bridge converter is a high number of semiconductor inductances and mechanical inertia act as ﬁlters resulting in devices. high-quality armature currents. The average output voltage

The dc–dc converters are building blocks of distributed of step-down choppers is a linear function of the switch power supply systems in which a common dc bus voltage duty ratio. Step-up choppers are used primarily in radar is converted to various other voltage according to require- and ignition systems. The dc choppers can be modiﬁed for ments of particular loads. Such distributed dc systems are two-quadrant and four-quadrant operation. Two-quadrant common in space stations, ships and airplanes, as well as in choppers may be a part of autonomous power supply system computer and telecommunication equipment. It is expected that contain battery packs and such renewable dc sources as that modern portable wireless communication and signal photovoltaic arrays, fuel cells, or wind turbines. Four-quadrant processing systems will use variable supply voltages to min- choppers are applied in drives in which regenerative breaking imize power consumption and extend battery life. Low output of dc motors is desired, e.g. transportation systems with fre- voltage converters in these applications utilize the synchronous quent stops. The dc choppers with inductive outputs serve as rectiﬁcation arrangement. inputs to current-driven inverters.

Another big area of dc–dc converter applications is related An addition of ﬁltering reactive components to dc choppers to the utility ac grid. For critical loads, if the utility grid fails, results in PWM dc–dc converters. The dc–dc converters can there must be a backup source of energy, e.g. a battery pack.

be viewed as dc transformers that deliver to the load as This need for continuous power delivery gave rise to various

13 DC–DC Converters 263 types of uninterruptible power supplies (UPSs). The dc–dc regulated output voltage and for one or more non-critical

converters are used in UPSs to adjust the level of a rectiﬁed grid other output voltage levels. voltage to that of the backup source. Since during normal operation, the energy ﬂows from the grid to the backup source and

during emergency conditions the backup source must supply Further Reading

the load, bidirectional dc–dc converters are often used. The dc–dc converters are also used in dedicated battery chargers.

1. R. P. Severns and G. Bloom, Modern DC-to-DC Switchmode Power Power electronic loads, especially those with front-end rec-

Converter Circuits, New York: Van Nostrand Reinhold Company, 1985. tiﬁers, pollute the ac grid with odd harmonics. The dc–dc

2. D. W. Hart, Introduction to Power Electronics, Englewood Cliffs, NJ: converters are used as intermediate stages, just after a rectiﬁer

Prentice Hall, 1997.

and before the load-supplying dc–dc converter, for shaping 3. P. T. Krein, Elements of Power Electronics, New York: Oxford the input ac current to improve power factor and decrease the

University Press, 1998.

harmonic content. The boost converter is especially popular in 4. A. I. Pressman, Switching Power Supply Design, 2nd Ed., New York: such power factor correction (PFC) applications. Another util-

McGraw-Hill, 1998.

ity grid related application of dc–dc converters is in interfaces 5. A. M. Trzynadlowski, Introduction to Modern Power Electronics, between ac networks and dc renewable energy sources such as

New York: Wiley Interscience, 1998.

fuel cells and photovoltaic arrays. 6. R. Erickson and D. Maksimovic, Fundamentals of Power Electronics, In isolated dc–dc converters, multiple outputs are possi- 2nd Ed., Norwell, MA: Kluwer Academic, 2001. 7. M. H. Rashid, Power Electronics Circuits, Devices, and Applications

ble with additional secondary windings of transformers. Only 3rd Ed., Upper Saddle River, NJ: Pearson Prentice Hall, 2003. one output is regulated with a feedback loop. Other out-

8. N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics: puts depend on the duty ratio of the regulated one and on

Converters, Applications and Design, 3rd Ed., New York: John Wiley & their loads. A multiple-output dc–dc converter is a convenient

Sons, 2003.

solution in application where there is a need for one closely

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266 F. L. Luo and H. Ye

14.1 Introduction

drawn much attention from the research workers and man- ufacturers. However, most of these converters in the literature

DC/DC converters are widely used in industrial applications perform single-quadrant operation. Some of them work in and computer hardware circuits. DC/DC conversion tech- the push–pull status. In addition, their control circuit and nique has been developed very quickly. Since 1920s there topologies are very complex, especially, for the large difference have been more than 500 DC/DC converters’ topologies between input and output voltages. developed. Professor Luo and Dr. Ye have systematically

Switched-inductor (SI) DC/DC converters are made of only sorted them in six generations in 2001. They are the first- inductor, and have been derived from four-quadrant choppers. generation (classical) converters, second-generation (multi- They usually perform multi-quadrant operation with very sim- quadrant) converters, third-generation (switched-component) ple structure. The significant advantage of these converters is converters, fourth-generation (soft-switching) converters, its simplicity and high power density. No matter how large fifth-generation (synchronous-rectifier) converters and sixth- the difference between the input and output voltages, only one generation (multi-element resonant power) converters.

inductor is required for each SI DC/DC converter. Therefore, The ﬁrst-generation converters perform in a single quad- they are widely required for industrial applications. rant mode with low power range (up to around 100 W), such

The fourth-generation converters are called soft-switching as buck converter, boost converter and buck–boost converter. converters. Soft-switching technique involves many meth- Because of the effects of parasitic elements, the output volt- ods implementing resonance characteristics. Popular method age and power transfer efﬁciency of all these converters are is resonant-switching. There are three main groups: zero- restricted.

current-switching (ZCS), zero-voltage-switching (ZVS), and The voltage-lift (VL) technique is a popular method that zero-transition (ZT) converters. They usually perform in sin- is widely applied in electronic circuit design. Applying this gle quadrant operation in the literature. We have developed technique effectively overcomes the effects of parasitic ele- this technique in two- and four-quadrant operation with high ments and greatly increases the output voltage. Therefore, these output power range (say thousands watts). DC/DC converters can convert the source voltage into a higher

Multi-quadrant ZCS/ZVS/ZT converters implement ZCS/ output voltage with high power efﬁciency, high power density, ZVS technique in four-quadrant operation. Since switches turn and a simple structure.

on and off at the moment that the current/voltage is equal to The VL converters have high voltage transfer gains, which zero, the power losses during switching on and off become increase in arithmetical series stage-by-stage. Super-lift (SL) zero. Consequently, these converters have high power den- technique is more powerful to increase the converters voltage sity and transfer efﬁciency. Usually, the repeating frequency transfer gains in geometric series stage-by-stage. Even higher, is not very high and the converters work in a mono-resonance ultra-lift (UL) technique is most powerful to increase the frequency, the components of higher order harmonics is very converters voltage transfer gain.

low. Using fast fourier transform (FFT) analysis, we obtain that The second-generation converters perform in two- or the total harmonic distortion (THD) is very small. Therefore, four-quadrant operation with medium output power range the electromagnetic interference (EMI) is weaker, electro- (say hundreds watts or higher). Because of high power con- magnetic sensitivity (EMS) and electromagnetic compatibility version, these converters are usually applied in industrial (EMC) are reasonable. applications with high power transmission. For example, DC

The ﬁfth-generation converters are called synchronous rec- motor drives with multi-quadrant operation. Since most of tiﬁer (SR) DC/DC Converters. Corresponding to the develop- second-generation converters are still made of capacitors and ment of the microelectronics and computer science, the power inductors, they are large.

supplies with low output voltage (5 V, 3.3 V, and 1.8 ∼ 1.5 V) The third-generation converters are called switched- and strong output current (30 A, 50 A, 100 A up to 200 A) component DC/DC converters, and made of either inductor are widely required in industrial applications and computer or capacitors, which are so-called switched-inductor and peripheral equipment. Traditional diode bridge rectiﬁers are switched-capacitors. They usually perform in two- or four- not available for this requirement. Many prototypes of SR quadrant operation with high output power range (say DC/DC converters with soft-switching technique have been thousands watts). Since they are made of only inductor or developed. The SR DC/DC converters possess the technical capacitors, they are small.

feathers with very low voltage and strong current and high Switched-capacitor (SC) DC/DC converters are made of power transfer efﬁciency η (90%, 92% up to 95%) and high

only switched-capacitors. Since switched-capacitors can be power density (22–25 W/in 3 ).

integrated into power semiconductor integrated circuits (IC) The sixth-generation converters are called multi-element chips, they have limited size and work in high switching resonant power converters (RPCs). There are eight topolo- frequency. They have been successfully employed in the induc- gies of 2-E RPC, 38 topologies of 3-E RPC, and 98 topologies torless DC/DC converters and opened the way to build the of 4-E RPC. The RPCs have very high current transfer gain, converters with high power density. Therefore, they have purely harmonic waveform, low power losses and EMI since

14 DC/DC Conversion Technique and 12 Series Luo-converters 267 they are working in resonant operation. Usually, the sixth- current is ζ. Voltage transfer gain is M and power transfer

generation RPCs used in large power industrial applications efﬁciency is η. with high output power range (say thousands watts).

The DC/DC converter family tree is shown in Fig. 14.1. Professor F. L. Luo and Dr. H. Ye have devoted in the sub-

14.2 Fundamental, Developed,

ject area of DC/DC conversion technique for a long time and harvested outstanding achievements. They have created

Transformer-type, and Self-lift

twelve (12) series converters namely Luo-converters and more

Converters

knowledge which are listed below: Positive output Luo-converters;

The ﬁrst-generation converters are called classical converters Negative output Luo-converters;

which perform in a single-quadrant mode and in low. Histori- cally, the development of the ﬁrst generation converters covers

Double output Luo-converters; very long time. Many prototypes of these converters have been Positive/Negative output super-lift Luo-converters;

created. We can sort them in six categories: Ultra-lift Luo-converter;

Multiple-quadrant Luo-converters; • Fundamental topologies: buck converter, boost converter, Switched capacitor multi-quadrant Luo-converters;

and buck–boost converter.

Multiple-lift push-pull switched-capacitor Luo-converters; • Developed topologies: positive output Luo-converter,

negative output Luo-converter, double output Luo- Switched-inductor multi-quadrant Luo-converters; converter, Cúk-converter, and single-ended primary Multi-quadrant ZCS quasi-resonant Luo-converters;

inductance converter (SEPIC).

Multi-quadrant ZVS quasi-resonant Luo-converters; • Transformer-type topologies: forward converter, push– Synchronous-rectiﬁer DC/DC Luo-converters;

pull converter, ﬂy-back converter, half-bridge converter, Multi-element resonant power converters;

bridge converter, and ZETA.

Energy factor and mathematical modeling for power DC/DC • Voltage-lift topologies: self-lift converters, positive output converters.

Luo-converters, negative output Luo-converters, double output Luo-converters.

All of their research achievements have been published • Super-lift topologies: positive/negative output super-lift in the international top-journals and conferences. Many

Luo-converters, positive/negative output cascade boost- experts, including Prof. Rashid of West Florida University,

converters.

Prof. Kassakian of MIT, and Prof. Rahman of Memorial • Ultra-lift topologies: ultra-lift Luo-converter. University of Newfoundland are very interested in their work,

and acknowledged their outstanding achievements. In this handbook, we only show the circuit diagram and list

14.2.1 Fundamental Topologies

a few parameters of each converter for readers, such as the Buck converter is a step-down converter, which is shown in output voltage and current, voltage transfer gain and output Fig. 14.2a, the equivalent circuits during switch-on and -off voltage variation ratio, and the discontinuous condition and periods are shown in Figs. 14.2b and c. Its output voltage and output voltage.

output current are

After a well discussion of steady-state operation, we prepare one section to investigate the dynamic transient process of

V 2 = kV 1 (14.1) DC/DC converters. Energy storage in DC/DC converters have been paid attention long time ago, but it was not well inves- and tigated and deﬁned. Professor Fang Lin Luo and Dr. Hong Ye

have theoretically deﬁned it and introduced new parameters:

I 2 = I 1 (14.2) energy factor (EF) and other variables. They have also funda-

mentally established the mathematical modeling and discussed This converter may work in discontinuous mode if the fre- the characteristics of all power DC/DC converters. They have quency f is small, conduction duty k is small, inductance L is

successfully solved the traditional problems.

small, and load current is high.

Boost converter is a step-up converter, which is shown in age is V O or V 2 . Pulse width modulated (PWM) pulse train Fig. 14.3a, the equivalent circuits during switch-on and -off has repeating frequency f, the repeating period is T = 1/f . periods are shown in Figs. 14.3b and c. Its output voltage and

In this chapter, the input voltage is V I or V 1 and load volt-

Conduction duty is k, the switching-on period is kT, and current are switching-off period is (1 − k)T. All average values are in cap-

V 1 (14.3) v 1 (t) or v 1 . The variation ratio of the free-wheeling diode’s

ital letter, and instantaneous values in small letter, e.g. V 1 and

1 −k

268 F. L. Luo and H. Ye

Buck Converter

Fundamental Circuits

Boost Converter Buck-Boost Converter

Positive Output Luo-Converter Negative Output Luo-Converter

Developed

Double Output Luo-Converter 1G

Classical

Forward Converter

Fly-Back Converter

Tapped-Inductor Converters

Push-Pull Converter

Transformer

Half-Bridge Converter

7 Self-Lift Converter

Bridge Converter

Positive Output Luo-Converter

ZETA Converter

Negative Output Luo-Converter Modified P/O Luo-Converter

Voltage Lift

Double Output Luo-Converter

Positive Output Super-Lift Luo-Converter

Super-Lift

Negative Output Super-Lift Luo-Converter Positive Output Cascade Boost Converter

Negative Output Cascade Boost Converter Ultra-Lift Luo-Converter

2G Multi-Quadrant

Transformer-type Converters

Converters

Developed

Multi-Quadrant Luo-Converter

Two Quadrants SC Luo-Converter

Switched-Capacitor Converter

Four Quadrants SC Luo-Converter

P/O Multi-Lift Push-Pull DC/DC Converters

Switched- Luo-Converter Component

Multi-Lift

N/O Multi-Lift Push-Pull Converters

Luo-Converter

Transformer-type Converters Switched-Inductor Converter Four Quadrants SI Luo-Converter

4G ZCS-QRC ----- Four Quadrants Zero-Current Switching Luo-Converter Soft-Switching

ZVS-QRC ----- Four Quadrants Zero-Voltage Switching Luo-Converter Converters

ZTC ----- Four Quadrants Zero-Transition Luo-Converter Flat-Transformer Synchronous Rectifier Converter

5G Synchronous

Synchronous Rectifier Converter with Active Clamp Circuit

Rectifier

Double Current Synchronous Rectifier Converter

Converters

ZCS Synchronous Rectifier Converter ZVS Synchronous Rectifier Converter

6G 2-Elements

Multi-Elements Resonant Power

3-Elements

P-CLL Current Source Resonant Inverter

Converters

4-Elements

Double Gamma-CL Current Source Resonant Inverter Reverse Double Gamma-CL Resonant Power Converter

FIGURE 14.1 DC/DC converter family tree.

14 DC/DC Conversion Technique and 12 Series Luo-converters 269

FIGURE 14.2 Buck converter: (a) circuit diagram; (b) switch-on equivalent circuit; and (c) switch-off equivalent circuit.

FIGURE 14.3 Boost converter: (a) circuit diagram; (b) switch-on equivalent circuit; and (c) switch-off equivalent circuit.

and

voltage and current are

I 2 = (1 − k)I 1 (14.4)

V 1 (14.5)

1 −k

The output voltage is higher than the input voltage. This and converter may work in discontinuous mode if the frequency f is small, conduction duty k is small, inductance L is small, and

1 −k

load current is high.

I 1 (14.6)

Buck–boost converter is a step–down/up converter, which is shown in Fig. 14.4a, the equivalent circuits during switch-on

When k is greater than 0.5, the output voltage can be and -off periods are shown in Figs. 14.4b and c. Its output higher than the input voltage. This converter may work in

FIGURE 14.4 Buck-boost converter: (a) circuit diagram; (b) switch-on equivalent circuit; and (c) switch-off equivalent circuit.

discontinuous mode if the frequency f is small, conduction i I i LO i 2 duty k is small, inductance L is small, and load current is high.

14.2.2 Developed Topologies

v 2 For convenient applications, all developed converters have

output voltage and current as

FIGURE 14.6 = Negative output Luo-converter.

V 2 V 1 (14.7)

1 −k

and Negative output (N/O) Luo -converter is shown in Fig. 14.6. This converter may work in discontinuous mode if the fre-

1 −k

quency f is small, k is small, inductance L is small, and load

I 1 (14.8)

current is high.

Double output Luo -converter is a double output step- Positive output (P/O) Luo-converter is a step-down/up down/up converter, which is derived from P/O Luo-converter converter, and is shown in Fig. 14.5. This converter may work and N/O Luo-converter. It has two conversion paths and two in discontinuous mode if the frequency f is small, k is small, output voltages V O + and V O − . It is shown in Fig. 14.7. If and inductance L is small.

the components are carefully selected the output voltages and currents (concentrate the absolute value) obtained are

I 2 − 1 1 (14.10) V 1

When k is greater than 0.5, the output voltage can be higher than the input voltage. This converter may work in discontinuous mode if the frequency f is small, k is small, inductance L

FIGURE 14.5 Positive output Luo-converter.

is small, and load current is high.

14 DC/DC Conversion Technique and 12 Series Luo-converters 271

FIGURE 14.7 Double output Luo-converter.

FIGURE 14.8 Cúk converter. FIGURE 14.10 Forward converter.

Cúk-converter is a negative output step-down/up converter, positive or negative polarity by changing the winding direction, which is derived from boost and buck converters. It is shown and multiple output voltages by setting multiple secondary in Fig. 14.8.

windings.

Single-ended primary inductance converter is a positive Forward converter is a step-up/down converter, which is output step-down/up converter, which is derived from boost shown in Fig. 14.10. The transformer turns ratio is N (usually converters. It is shown in Fig. 14.9.

N > 1). If the transformer has never been saturated during operation, it works as a buck converter. The output voltage and current are

V O = kNV I (14.11) +

and

I I (14.12) −

FIGURE 14.9 SEPIC. This converter may work in discontinuous mode if the fre-

quency f is small, conduction duty k is small, inductance L is small, and load current is high.

To avoid the saturation of transformer applied in forward

14.2.3 Transformer-type Topologies converters, a tertiary winding is applied. The corresponding

circuit diagram is shown in Fig. 14.11. All transformer-type converters have transformer(s) to isolate

To obtain multiple output voltages we can set multiple sec- the input and output voltages. Therefore, it is easy to obtain the ondary windings. The corresponding circuit diagram is shown high or low output voltage by changing the turns ratio N, the in Fig. 14.12.

272 F. L. Luo and H. Ye 1:1:N

FIGURE 14.14 Fly-back converter. FIGURE 14.11 Forward converter with tertiary winding.

1:1:N 1 in ﬂy-back operation to obtain high surge voltage induced, +

then get high output voltage. It works likely in buck–boost

O/P 1

operation as a buck–boost converter. Its output voltage and N 2 current are

V I (14.15) −

1 −k

O/P 3 and

FIGURE 14.12 Forward converter with multiple secondary windings.

1 −k

I I (14.16)

1:N

Half-bridge converter is a step-up converter, which is shown +

in Fig. 14.15. There are two switches and one double secondary V I

T 1 V' C R

coils transformer required. The transformer turns ratio is N. −

It works as a half-bridge rectiﬁer (half of V 1 inputs to primary winding) plus a buck converter circuit in secondary side. The T 2 conduction duty cycle k is set in 0.1 < k < 0.5. Its output

D 2 voltage and current are

FIGURE 14.13 Push–pull converter.

V O = 2kN

= kNV I (14.17)

Push–pull converter is a step-up/down converter, which and is shown in Fig. 14.13. It is not necessary to set the tertiary winding. The transformer turns ratio is N (usually N > 1).

I I (14.18)

If the transformer has never been saturated during operation,

it works as a buck converter with the conduction duty cycle k < 0.5. The output voltage and current are

This converter may work in discontinuous mode if the frequency f is small, conduction duty k is small, inductance L is

V O = 2kNV I (14.13) small, and load current is high.

and

1:N

I I (14.14)

C 3 R V O This converter may work in discontinuous mode if the fre-

2kN

− quency f is small, conduction duty k is small, inductance L is in

small, and load current is high. Fly-back converter is a high step-up converter, which is

shown in Fig. 14.14. The transformer turns ratio is N (usually N > 1). It effectively uses the transformer leakage inductance

FIGURE 14.15 Half-bridge converter.

14 DC/DC Conversion Technique and 12 Series Luo-converters 273

Voltage-lift technique is a popular method used in electronic T 1 T

1:N

circuit design. Applying this technique can effectively over- +

C 1 come the effect of the parasitic elements, and largely increase R V O the voltage transfer gain. In this section, we introduce seven

V in

C self-lift converters which are working in continuous mode. • Positive output (P/O) self-lift Luo-converter;

− T 3 T 4 D 2 • Reverse P/O self-lift Luo-converter; • Negative output (N/O) self-lift Luo-converter;

FIGURE 14.16 Bridge converter. • Reverse N/O self-lift Luo-converter; • Self-lift Cúk-converter; • Self-lift SEPIC;

Bridge converter is a step-up converter, which is shown in • Enhanced self-lift Luo-converter. Fig. 14.16. There are four switches and one double secondary

All self-lift converters (except enhanced self-lift circuit) have coils transformer required. The transformer turns ratio is N. the output voltage and current to be

It works as a full-bridge rectiﬁer (full V 1 inputs to primary

winding) plus a buck-converter circuit in secondary side. The

conduction duty cycle k is set in 0.1 < k < 0.5. Its output

V I (14.23) voltage and current are

1 −k

V O = 2kNV I (14.19) and

and

I O = (1 − k)I I (14.24)

I I (14.20)

2kN

The voltage transfer gain in continuous mode is ZETA (zeta) converter is a step-up converter, which is shown in Fig. 14.17. The transformer turns ratio is N. The

transformer functions as a inductor (L 1 ) plus a buck–boost

V I = I O = 1 −k

converter plus a low-pass ﬁlter (L 2 –C 2 ). Its output voltage

and current are P/O self-lift Luo-converter is shown in Fig. 14.18. The vari-

ation ratio of the output voltage v O in continuous conduction

NV I (14.21)

1 −k

mode (CCM) is

I I (14.22)

8M S f 2 C O L 2

Reverse P/O self-lift Luo-converter is shown in Fig. 14.19.

The variation ratio of the output voltage v O in CCM is

FIGURE 14.17 ZETA (zeta) converter.

14.2.4 Seven (7) Self-lift DC/DC Converters

C 1 − Because of the effect of the parasitic elements, the voltage

v C1 C O

conversion gain is limited. Especially, when the conduction duty k is towards unity, the output voltage is sharply reduced.

FIGURE 14.18 P/O self-lift Luo-converter.

FIGURE 14.22 Self-lift Cúk-converter. FIGURE 14.19 Reverse P/O self-lift Luo-converter.

FIGURE 14.23 Self-lift SEPIC.

FIGURE 14.20 N/O self-lift Luo-converter. Self-lift SEPIC is shown in Fig. 14.23. The variation ratio

of the output voltage v O in CCM is

N/O self-lift Luo-converter is shown in Fig. 14.20. The

v O /2

3 (14.28d) variation ratio of the output voltage v O in CCM is

128 f L O C 1 C O R Enhanced self-lift Luo-converter is shown in Fig. 14.24. Its

v O /2

(14.28a) output voltage and current are

128 f 3 L O C 1 C O R

2 −k

V I (14.29) Reverse N/O self-lift Luo -converter is shown in Fig. 14.21.

1 −k

The variation ratio of the output voltage v O in CCM is

and

v −k O /2 k 1 I O = I I (14.30) ε =

The voltage transfer gain in continuous mode is Self-lift Cúk -converter is shown in Fig. 14.22. The variation

I I 1 2 −k ratio of the output voltage v O in CCM is

FIGURE 14.21 Reverse N/O self-lift Luo-converter. FIGURE 14.24 Enhanced self-lift Luo-converter.

14 DC/DC Conversion Technique and 12 Series Luo-converters 275 TABLE 14.1 The circuit diagrams of the tapped inductor fundamental converters

Rail to tap Buck

Standard converter

Switch tap

Diode to tap

Buck–Boost

The variation ratio of the output voltage v O in CCM is as number of up-to-date converters. There are three series of in Eq. (14.26)

Luo-converters introduced in this section: v O /2

1 • Positive output Luo-converters;

• Simpliﬁed positive output Luo-converters;

V O 8M S f 2 C O L 2 • Negative output Luo-converters.

14.2.5 Tapped Inductor (Watkins–Johnson)

14.3.1 Positive Output Luo-converters Converters

Positive output (P/O) Luo-converters perform the voltage Tapped inductor (Watkins–Johnson) converters have been conversion from positive to positive voltages using the volt-

derived from fundamental converters, which circuit diagrams age lift technique. They work in the ﬁrst-quadrant with large are shown in Table 14.1. The voltage transfer gains are shown voltage ampliﬁcation. Their voltage transfer gains are high. in Table 14.2. Here the tapped inductor ratio is n = n1/ Five circuits are introduced in the literature. They are: (n1 + n2).

• Elementary circuit; • Self-lift circuit;

14.3 Voltage-lift Luo-converters

• Re-lift circuit; • Triple-lift circuit;

Voltage-lift (VL) technique is very popular for electronic circuit

• Quadruple-lift circuit.

design. Professor Luo and Dr. Ye have successfully applied this Further lift circuits can be derived from the above circuits. technique in the design of DC/DC converters, and created a In all P/O Luo-Converters, we deﬁne normalized inductance L =L 1 L 2 /(L 1 +L 2 ) and normalized impedance z N = R/fL. P/O Luo-converter elementary circuit is shown in

TABLE 14.2 The voltage transfer gains of the tapped inductor Fig. 14.25a. The equivalent circuits during switch-on and -off fundamental converters periods are shown in Figs. 14.25b and c. Its output voltage and

Converter No tap Switched to tap

Diode to tap

Rail to tap

current are

k −n

Buck k

Boost 1 n + k(1 − n)

Buck–Boost

1 −k

1 −k n(1 − k)

1 −k

FIGURE 14.25 P/O Luo-converter elementary circuit; (a) circuit diagram; (b) switch on; and (c) switch off.

The voltage transfer gain in continuous mode is The voltage transfer gain in continuous mode is

1 −k The variation ratio of the output voltage v O in CCM is

The variation ratio of the output voltage v O in CCM is v O /2

C O L 2 This converter may work in discontinuous conduction mode

16M S f 2

if the frequency f is small, conduction duty k is small, induc- This converter may work in discontinuous conduction mode tance L is small, and load current is high. The condition for if the frequency f is small, conduction duty k is small, induc-

discontinuous conduction mode (DCM) is tance L is small, and load current is high. The condition for DCM is

E ≤k

(14.38) The output voltage in DCM is

1 The output voltage in DCM is

V O = k(1 − k)

V I with

V I with k ≥

1 −k The equivalent circuits during switch-on and -off periods are

P/O Luo-converter self-lift circuit is shown in Fig. 14.26a.

2fL

(14.39) shown in Figs. 14.26b and c. Its output voltage and current are

1 P/O Luo-converter re-lift circuit is shown in Fig. 14.27a.

V I The equivalent circuits during switch-on and -off periods are

1 −k

shown in Figs. 14.27b and c. Its output voltage and current are and

I O = (1 − k)I I

1 −k

14 DC/DC Conversion Technique and 12 Series Luo-converters 277

FIGURE 14.26 P/O Luo-converter self-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

i C + V L2 −

V s c IN 2 i L2

D i D i D1

V IN

i L1

L 1 C 1 i C1 L 3 C 2 i L2 C O

+ S 1 V S1

− (a)

L 2 i L2 i IN

i L2

+ i L1

3 V O V L1

V O L1

− L IN

V V V IN

V IN

2 L3

(b)

(c)

278 F. L. Luo and H. Ye and

P/O Luo-converter triple-lift circuit is shown in Fig. 14.28a. The equivalent circuits during switch-on and -off periods

I O −k = I I are shown in Figs. 14.28b and c. Its output voltage and

2 current are

The voltage transfer gain in CCM is

The variation ratio of the output voltage v O in CCM is

V O 16M R f 2 C O L 2 The voltage transfer gain in CCM is This converter may work in discontinuous conduction mode

I I 3 if the frequency f is small, conduction duty k is small, induc-

1 tance L is small, and load current is high. The condition for

−k DCM is

The variation ratio of the output voltage v O in CCM is M R ≤ kz N

v O /2

(14.45) The output voltage in DCM is

16M T f 2 C O L 2

This converter may work in discontinuous conduction mode

V O = 2 +k (1 − k)

if the frequency f is small, conduction duty k is small, induc- 2fL

V I with

tance L is small, and load current is high. The condition for (14.43) DCM is

FIGURE 14.28 P/O Luo-converter triple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

14 DC/DC Conversion Technique and 12 Series Luo-converters 279

The variation ratio of the output voltage v O in CCM is M T ≤

(14.49) The output voltage in DCM is

16M Q f 2 C O L 2

3 This converter may work in discontinuous conduction mode

V O = 3 +k 2 (1 − k)

V I with

2fL ≥ 1 −k

2fL

if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for

(14.47) DCM is

P/O Luo-converter quadruple-lift circuit is shown in M Q ≤ 2kz N (14.50) Fig. 14.29a. The equivalent circuits during switch-on and -off

periods are shown in Figs. 14.29b and c. Its output voltage and

The output voltage in DCM is

current are

V I with k

Summary for all P/O Luo -converters:

V O The voltage transfer gain in CCM is

L 1 +L 2 fL

M Q = To write common formulas for all circuits parameters, we = = (14.48)

1 −k

deﬁne that subscript j = 0 for the elementary circuit, j = 1

FIGURE 14.29 P/O Luo-converter quadruple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

280 F. L. Luo and H. Ye for the self-lift circuit, j = 2 for the re-lift circuit, j = 3 for

14.3.2 Simpliﬁed Positive Output (S P/O)

the triple-lift circuit, j = 4 for the quadruple-lift circuit, and

Luo-converters

so on. The voltage transfer gain is Carefully check P/O Luo-converters, we can see that there are

k h(j) [j + h(j)]

two switches required from re-lift circuit. In order to use only

M j = (14.52) one switch in all P/O Luo-converters, we modify the circuits.

In this section we introduce following four circuits: The variation ratio of the output voltage is

1 −k

• Simpliﬁed self-lift circuit; • Simpliﬁed re-lift circuit;

v O /2

• Simpliﬁed triple-lift circuit;

V O 16M j f C O L 2 • Simpliﬁed quadruple-lift circuit. The condition for discontinuous conduction mode is

Further lift circuits can be derived from the above circuits. In all S P/O Luo-converters, we deﬁne normalized impedance k [1+h(j)] j + h(j)

z N = R/fL.

M S P/O Luo-converter self-lift circuit

2 z N ≥1

is shown in Fig. 14.30a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.30b and c. Its output voltage and current are

The output voltage in discontinuous conduction mode is %

[2−h(j)] 1 −k

V O −j = j +k

z N V I (14.55)

The voltage transfer gain in CCM is

1 if j =0

= (14.57) is the Hong function.

FIGURE 14.30 S P/O Luo-converter self-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

14 DC/DC Conversion Technique and 12 Series Luo-converters 281 The variation ratio of the output voltage v O in CCM is

The voltage transfer gain in CCM is v O /2

1 −k This converter may work in discontinuous conduction mode

128 f L O C 1 C O R

The variation ratio of the output voltage v O in CCM is if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for

v O /2

(14.62) DCM is

128 f 3 L O C 1 C O R

This converter may work in discontinuous conduction mode

2 if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for

The output voltage in DCM is

V I with

The output voltage in DCM is

S P/O Luo-converter re-lift circuit is shown in Fig. 14.31a.

2 The equivalent circuits during switch-on and -off periods

V I with k ≥ are shown in Figs. 14.31b and c. Its output voltage and

V O = 2 +k (1 − k)

1 −k current are

V I S P/O Luo triple-lift circuit is shown in Fig. 14.32a. The

equivalent circuits during switch-on and -off periods are and

1 −k

shown in Figs. 14.32b and c. Its output voltage and current are

FIGURE 14.31 S P/O Luo-converter re-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

FIGURE 14.32 S P/O Luo-converter triple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

and S P/O Luo quadruple-lift circuit is shown in Fig. 14.33a. The equivalent circuits during switch-on and -off periods are

I O −k = I I shown in Figs. 14.33b and c. Its output voltage and current are

The voltage transfer gain in CCM is

The variation ratio of the output voltage v O in CCM is

v O /2

The voltage transfer gain in CCM is

128 f 3 L O C 1 C O R

I I 4 This converter may work in discontinuous conduction mode

1 −k if the frequency f is small, conduction duty k is small, induc-

tance L is small, and load current is high. The condition for The variation ratio of the output voltage v O in CCM is DCM is

v O /2

3 (14.70) M T ≤

This converter may work in discontinuous conduction mode The output voltage in DCM is if the frequency f is small, conduction duty k is small, induc-

tance L is small, and load current is high. The condition for

V O = 3 +k (1 − k)

V I with

M Q ≤ 2kz N (14.71)

14 DC/DC Conversion Technique and 12 Series Luo-converters 283

FIGURE 14.33 S P/O Luo-converter quadruple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

The output voltage in DCM is The output voltage in discontinuous mode is

2 # R √ 2R 4 z N

V O = 4 +k (1 − k)

V I (14.76) 2fL

V I with

14.3.3 Negative Output Luo-converters

Summary for all S P/O Luo -converters: Negative output (N/O) Luo-converters perform the voltage

conversion from positive to negative voltages using the voltage- M =

lift technique. They work in the third-quadrant with large voltage ampliﬁcation. Their voltage transfer gains are high.

To write common formulas for all circuits parameters, we Five circuits are introduced in the literature. They are: deﬁne that subscript j = 1 for the self-lift circuit, j = 2 for

• Elementary circuit;

the re-lift circuit, j = 3 for the triple-lift circuit, j = 4 for the

• Self-lift circuit;

quadruple-lift circuit, and so on. The voltage transfer gain is

• Re-lift circuit; • Triple-lift circuit;

• Quadruple-lift circuit.

1 −k

Further lift circuits can be derived from above circuits. The variation ratio of the output voltage is

In all N/O Luo-converters, we deﬁne normalized impedance z N = R/fL.

1 N/O Luo-converter elementary circuit is shown in ε j =

Fig. 14.34a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.34b and c. Its output voltage and

The condition for discontinuous mode is

current (the absolute value) are

jkz N

1 −k

FIGURE 14.34 N/O Luo-converter elementary circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

and shown in Figs. 14.35b and c. Its output voltage and current (the absolute value) are

1 −k

−k

When k is greater than 0.5, the output voltage can be higher

than the input voltage.

and

The voltage transfer gain in CCM is

I O = (1 − k)I I

The voltage transfer gain in CCM is

1 −k

I I 1 The variation ratio of the output voltage v O in CCM is

1 −k v O /2

The variation ratio of the output voltage v O = in CCM is

V O = 128 f 3 CC O L O R

1 This converter may work in discontinuous conduction mode ε =

v O /2

128 f 3 CC O L O R if the frequency f is small, conduction duty k is small, induc-

tance L is small, and load current is high. The condition for This converter may work in discontinuous conduction mode DCM is

if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for

E ≤k

(14.79) DCM is

The output voltage in DCM is

1 The output voltage in DCM is

V O = k(1 − k)

V I with

V I with k ≥ N/O Luo-converter self-lift circuit is shown in Fig. 14.35a.

V O = 1 +k (1 − k)

1 −k The equivalent circuits during switch-on and -off periods are

2fL

14 DC/DC Conversion Technique and 12 Series Luo-converters 285

FIGURE 14.35 N/O Luo-converter self-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

N/O Luo-converter re-lift circuit is shown in Fig. 14.36a.

The output voltage in DCM is

The equivalent circuits during switch-on and -off periods are shown in Figs. 14.36b and c. Its output voltage and current

2 (the absolute value) are

V O = 2 +k (1 − k)

V I with k ≥

N/O Luo-converter triple-lift circuit is shown in Fig. 14.37a. The equivalent circuits during switch-on and -off

and periods are shown in Figs. 14.37b and c. Its output voltage and current (the absolute value) are

I O −k = I I

1 −k

The voltage transfer gain in CCM is

The variation ratio of the output voltage v O in CCM is The voltage transfer gain in CCM is v O /2

1 −k The variation ratio of the output voltage v O in CCM is

This converter may work in discontinuous conduction mode if the frequency f is small, conduction duty k is small, induc-

1 tance L is small, and load current is high. The condition for

v O /2

3 V (14.90) O 128 f CC O L O R DCM is

This converter may work in discontinuous conduction M R ≤ kz N

(14.87) mode if the frequency f is small, conduction duty k is small,

FIGURE 14.36 N/O Luo-converter re-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

FIGURE 14.37 N/O Luo-converter triple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

14 DC/DC Conversion Technique and 12 Series Luo-converters 287 inductance L is small, and load current is high. The condition

The voltage transfer gain in CCM is for DCM is

= (14.93) M T ≤

The output voltage in DCM is The variation ratio of the output voltage v O in CCM is

V I with

V O = 128 f 3 2fL (14.94) 2fL 1 −k CC O L O R

This converter may work in discontinuous conduction mode N/O Luo-converter quadruple-lift circuit is shown in if the frequency f is small, conduction duty k is small, induc-

Fig. 14.38a. The equivalent circuits during switch-on and -off tance L is small, and load current is high. The condition for periods are shown in Figs. 14.38b and c. Its output voltage and DCM is current (the absolute value) are

M Q ≤ 2kz N (14.95)

1 −k

The output voltage in DCM is

V I with k

FIGURE 14.38 N/O Luo-converter quadruple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

288 F. L. Luo and H. Ye Summary for all N/O Luo -converters:

Further lift circuits can be derived from above circuits. In all D/O Luo-converters, each circuit has two conversion

paths – positive conversion path and negative conversion path. M =

The positive path likes P/O Luo-converters, and the negative path likes N/O Luo-converters. We deﬁne normalized impedance z N

To write common formulas for all circuits parameters, we + = R/fL for positive path, and normalized

1 /fL 11 . We usually purposely select R deﬁne that subscript j 1 − =R =R

impedance z N

for the self-lift circuit, j =z N + =z N .

= 0 for the elementary circuit, j = 1 and L =L 11 , so that we have z N

= 2 for the re-lift circuit, j = 3 for

D/O Luo-converter elementary circuit is shown in the triple-lift circuit, j = 4 for the quadruple-lift circuit, and Fig. 14.7. Its output voltages and currents (absolute values) so on. The voltage transfer gain is

are

k h(j) [j + h(j)]

The variation ratio of the output voltage is

I I − The condition for discontinuous conduction mode is

k [1+h(j)] j + h(j) When k is greater than 0.5, the output voltage can be higher

2 z N ≥1

(14.99) than the input voltage.

2 The voltage transfer gain in CCM is The output voltage in discontinuous conduction mode is

V O −j = j +k [2−h(j)]

1 −k

2 The variation ratio of the output voltage v O + in CCM is where

z N V I (14.100)

h(j) = The variation ratio of the output voltage v O

1 if j =0

− in CCM is

1 is the Hong function.

3 V (14.103) O

128 f C 11 C 10 L 12 R 1

This converter may work in discontinuous conduction mode

14.4 Double Output Luo-converters

if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for

Double output (D/O) Luo-converters perform the voltage DCM is conversion from positive to positive and negative voltages simultaneously using the voltage-lift technique. They work in

M E ≤k

the ﬁrst- and third-quadrants with high voltage transfer gain.

There are ﬁve circuits introduced in this section: • D/O Luo-converter elementary circuit;

The output voltages in DCM are

• D/O Luo-converter self-lift circuit; • D/O Luo-converter re-lift circuit;

• D/O Luo-converter triple-lift circuit; =V O + = |V O − | = k(1 − k) V I with

2 2 1 −k • D/O Luo-converter quadruple-lift circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 289

FIGURE 14.39 Double output Luo-converter self-lift circuit.

D/O Luo-converter self-lift circuit is shown in Fig. 14.39.

The output voltages in DCM are

Its output voltages and currents (absolute values) are

2 1 −k D/O Luo-converter re-lift circuit is shown in Fig. 14.40. Its

and output voltages and currents (absolute values) are

I O − = (1 − k)I I −

V O + = |V O − |=

1 −k The voltage transfer gain in CCM is

The variation ratio of the output voltage v O + in CCM is

2 − v O + /2

1 The voltage transfer gain in CCM is ε + =

3 V (14.107) O

V I V I 1 −k The variation ratio of the output voltage v O − in CCM is

The variation ratio of the output voltage v O + in CCM is v O − /2

128 f 3 C 11 C 10 L 12 R 1 v O + /2

128 f 3 + (14.112) L 2 C O C 2 R This converter may work in discontinuous conduction mode

The variation ratio of the output voltage v O − in CCM is if the frequency f is small, conduction duty k is small, induc-

tance L is small, and load current is high. The condition for

1 DCM is

v O − /2

3 V (14.113) O

128 f C 11 C 10 L 12 R 1

This converter may work in discontinuous conduction

2 mode if the frequency f is small, conduction duty k is small,

L 12 I O − FIGURE 14.40 D/O Luo-converter re-lift circuit.

inductance L is small, and load current is high. The condition D/O Luo-converter triple-lift circuit is shown in Fig. 14.41. for DCM is

Its output voltages and currents (absolute values) are

1 −k The output voltages in DCM are

with kz N ≥

FIGURE 14.41 D/O Luo-converter triple-lift circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 291

FIGURE 14.42 D/O Luo-converter quadruple-lift circuit.

The voltage transfer gain in CCM is D/O Luo-converter quadruple-lift circuit is shown in Fig. 14.42. Its output voltages (absolute values) are

1 −k The variation ratio of the output voltage v O + in CCM is

I −k

v O /2

3 V (14.117) O + 128 f L 2 C O C 2 R

and

The variation ratio of the output voltage v O − in CCM is

1 −k

4 − v O /2

3 V (14.118) O

128 f C 11 C 10 L 12 R 1 The voltage transfer gain in CCM is This converter may work in discontinuous conduction mode

4 if the frequency f is small, conduction duty k is small, induc- Q =

|V O − |

V I V I 1 −k tance L is small, and load current is high. The condition for DCM is

The variation ratio of the output voltage v O + in CCM is

3kz N

128 f 3 L 2 C O C 2 R The output voltages in DCM are

The variation ratio of the output voltage v O − in CCM is

This converter may work in discontinuous conduction

2 1 −k

mode if the frequency f is small, conduction duty k is small,

292 F. L. Luo and H. Ye inductance L is small, and load current is high. The condition

14.5 Super-lift Luo-converters

for DCM is Voltage-lift (VL) technique has been successfully applied in

M Q ≤ 2kz N (14.124) DC/DC converter’s design. However, the output voltage of all VL converters increases in arithmetic progression stage-

The output voltages in DCM are by-stage. Super-lift (SL) technique is more powerful than VL

2 z N V # technique. The output voltage of all SL converters increases in

O =V O + = |V O − |= 4 +k (1 − k)

2 geometric progression stage-by-stage. All super-lift converters

4 are outstanding contributions in DC/DC conversion technol- with

2kz N ≥ (14.125) ogy, and invented by Professor Luo and Dr. Ye in 2000–2003.

1 −k

There are four series SL Converters introduced in this section: Summary for all D/O Luo -converters:

1. Positive output (P/O) super-lift Luo-converters;

V O |V O | L 1 L 2 2. Negative output (N/O) super-lift Luo-converters; M

; L =L 11 ; R =R 1 ;

V I V I L 1 +L 2 3. Positive output (P/O) cascade boost-converter;

4. Negative output (N/O) cascade boost-converter;

− = fL 11

14.5.1 P/O Super-lift Luo-converters

so that There are several sub-series of P/O super-lift Luo-converters:

z N =z N + =z N −

• Main series;

To write common formulas for all circuits parameters, we

• Additional series;

deﬁne that subscript j = 0 for the elementary circuit, j = 1

• Enhanced series;

for the self-lift circuit, j = 2 for the re-lift circuit, j = 3 for

• Re-enhanced series;

the triple-lift circuit, j = 4 for the quadruple-lift circuit, and

• Multi-enhanced series.

so on. The voltage transfer gain is We only introduce three circuits of main series and addi-

h(j)

[j + h(j)]

tional series.

P/O SL Luo-converter elementary circuit is shown in

Fig. 14.43a. The equivalent circuits during switch on and The variation ratio of the output voltage v O + in CCM is

1 −k

switch off are shown in Figs. 14.43b and c. Its output voltage and current are

The variation ratio of the output voltage v O − in CCM is v O − /2

I I The condition for DCM is

2 −k

k [1+h(j)] j + h(j)

2 z N ≥1

The voltage transfer gain is

2 −k

The output voltage in DCM is

V I 1 −k

O −j = j +k [2−h(j)] 1 V −k z N V I (14.130)

2 The variation ratio of the output voltage v O is where

v O /2

2RfC 2 h(j)

P/O SL Luo-converter re-lift circuit is shown in Fig. 14.44a. is the Hong function.

The equivalent circuits during switch on and switch off are

14 DC/DC Conversion Technique and 12 Series Luo-converters 293

FIGURE 14.43 P/O SL Luo-converter elementary circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

FIGURE 14.44 P/O SL Luo-converter re-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

shown in Figs. 14.44b and c. Its output voltage and current are

The voltage transfer gain is

The variation ratio of the output voltage v O is

−k 2

v O /2

2 −k

2RfC 4

FIGURE 14.45 P/O SL Luo-converter triple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

P/O SL Luo-converter triple-lift circuit is shown in and Fig. 14.45a. The equivalent circuits during switch on and switch off are shown in Figs. 14.45b and c. Its output voltage

1 −k

I I and current are

3 −k

−k

3 The voltage transfer gain is

3 The variation ratio of the output voltage v O is

2RfC 12 The voltage transfer gain is

P/O SL Luo-converter additional re-lift circuit is shown

in Fig. 14.47a. The equivalent circuits during switch on and M T =

switch off are shown in Figs. 14.47b and c. Its output voltage and current are

The variation ratio of the output voltage v O is

2RfC 6 and

I −k 1 O −k = I I Fig. 14.46a. The equivalent circuits during switch on and

P/O SL Luo-converter additional circuit is shown in

2 −k 3 −k switch off are shown in Figs. 14.46b and c. Its output voltage and current are

The voltage transfer gain is

3 −k

2 −k 3 −k

V I M AR =

1 −k

V I 1 −k 1 −k

14 DC/DC Conversion Technique and 12 Series Luo-converters 295

FIGURE 14.46 P/O SL Luo-converter additional circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

FIGURE 14.47 P/O SL Luo-converter additional re-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

The variation ratio of the output voltage v O is

and current are

P/O SL Luo-converter additional triple-lift circuit is shown

in Fig. 14.48a. The equivalent circuits during switch on and

−k

1 −k

I I switch off are shown in Figs. 14.48b and c. Its output voltage

2 −k

3 −k

− (c) FIGURE 14.48 P/O SL Luo-converter additional triple-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

The voltage transfer gain is We only introduce three circuits of main series and additional series.

N/O SL Luo-converter elementary circuit is shown in M AT =

−k

3 −k

Fig. 14.49. Its output voltage and current are The variation ratio of the output voltage v O is

14.5.2 N/O Super-lift Luo-converters

There are several subseries of N/O Super-lift Luo-converters: V L 1 C 1 in V C1 − −

• Main series;

R V O • Additional series;

C V C2 • + Enhanced series; 2 + • Re-enhanced series;

• Multi-enhanced series. FIGURE 14.49 N/O SL Luo-converter elementary circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 297 and

I in

I O = (1 − k)I I S

The voltage transfer gain is V in L L 1 C 1 3 V C1 L 2 C 3 V C3 C 5 V C5 _

V I 1 −k

+ The variation ratio of the output voltage v 6

C 2 V C2 C 4 V C4 C V C6

O is

FIGURE 14.51 N/O SL Luo-converter triple-lift circuit. ε =

N/O SL Luo-converter re-lift circuit is shown in Fig. 14.50.

The voltage transfer gain is

Its output voltage and current are

The variation ratio of the output voltage v O is and

N/O SL Luo-converter additional circuit is shown in The voltage transfer gain is

Fig. 14.52. Its output voltage and current are

1 −k The variation ratio of the output voltage v O is

N/O SL Luo-converter triple-lift circuit is shown in Fig. 14.51. Its output voltage and current are

The voltage transfer gain is

The variation ratio of the output voltage v O is

FIGURE 14.50 N/O SL Luo-converter re-lift circuit. FIGURE 14.52 N/O SL Luo-converter additional circuit.

FIGURE 14.53 N/O SL Luo-converter additional re-lift circuit.

NO SL Luo-converter additional re-lift circuit is shown in

and

Fig. 14.53. Its output voltage and current are

−k 2 3 −k

−1 V I

1 −k 1 −k The voltage transfer gain is

1 −k The variation ratio of the output voltage v O is

V I 1 −k

The voltage transfer gain is

= V = −1 (14.151) O 2RfC 12

The variation ratio of the output voltage v O is

14.5.3 P/O Cascade Boost-converters

There are several subseries of P/O cascade boost-converters

2RfC 12 • Main series;

N/O SL Luo-converter additional triple-lift circuit is

• Additional series;

shown in Fig. 14.54. Its output voltage and current are

• Double series; • Triple series;

2 • Multiple series.

We only introduce three circuits of main series and additional series.

FIGURE 14.54 N/O SL Luo-converter additional triple-lift circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 299 i IN

P/O CBC two-stage circuit

is shown in Fig. 14.56. Its output

D 1 voltage and current are

FIGURE 14.55 P/O CBC elementary circuit.

I 2 O = (1 − k) I I

The voltage transfer gain is

P/O CBC elementary circuit is shown in Fig. 14.55. Its output voltage and current are

The variation ratio of the output voltage v O is and

The voltage transfer gain is P/O CBC three-stage circuit is shown in Fig. 14.57. Its output voltage and current are

The variation ratio of the output voltage v O is

2RfC 1 I O = (1 − k) I I

FIGURE 14.56 P/O CBC two-stage circuit.

FIGURE 14.57 P/O CBC three-stage circuit.

300 F. L. Luo and H. Ye i IN

P/O CBC additional two-stage circuit is shown in L 1 D 1 D 11 D 12 Fig. 14.59. Its output voltage and current are

FIGURE 14.58 P/O CBC additional circuit.

− k) 2

2 The voltage transfer gain is

The voltage transfer gain is

The variation ratio of the output voltage v O is The variation ratio of the output voltage v O is

V O = 2RfC 3

P/O CBC additional three-stage circuit is shown in P/O CBC additional circuit is shown in Fig. 14.58. Its Fig. 14.60. Its output voltage and current are output voltage and current are

2 The voltage transfer gain is

The voltage transfer gain is

V I 1 −k The variation ratio of the output voltage v O is

V I 1 −k

The variation ratio of the output voltage v O is

2RfC 12 V O

FIGURE 14.59 P/O CBC additional two-stage circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 301

FIGURE 14.60 P/O CBC additional three-stage circuit.

14.5.4 N/O Cascade Boost-converters

+ There are several subseries of N/O CBC:

C C − 1 − i 2 IN • Main series;

D 1 L 2 • Additional series;

D 3 i O • Double series; • D Triple series; V

V O • Multiple series.

+ We only introduce three circuits of main series and addi-

tional series. FIGURE 14.62 N/O CBC two-stage circuit. N/O CBC elementary circuit is shown in Fig. 14.61. Its

output voltage and current are N/O CBC two-stage circuit is shown in Fig. 14.62. Its output

voltage and current are

The voltage transfer gain is

The variation ratio of the output voltage v O is

v O /2

(14.170) i IN

2RfC 2

N/O CBC three-stage circuit is shown in Fig. 14.63. Its

V IN output voltage and current are L

−1 V I FIGURE 14.61 N/O CBC elementary circuit.

1 −k

FIGURE 14.63 N/O CBC three-stage circuit.

and

The voltage transfer gain is

1 +k

V I −1 1 −k The variation ratio of the output voltage v O is

The voltage transfer gain is

N/O CBC additional two-stage circuit is shown in Fig. 14.65. Its output voltage and current are The variation ratio of the output voltage v O is

N/O CBC additional circuit is shown in Fig. 14.64. Its

output voltage and current are

The voltage transfer gain is

1 The variation ratio of the output voltage v O −k is

2RfC 12 N/O CBC additional three-stage circuit is shown in

Fig. 14.66. Its output voltage and current are

3 FIGURE 14.64 N/O CBC additional circuit.

14 DC/DC Conversion Technique and 12 Series Luo-converters 303

FIGURE 14.65 N/O CBC additional two-stage circuit.

FIGURE 14.66 N/O CBC additional three-stage circuit.

The voltage transfer gain is

voltage transfer gain

k M A3 =

V I 1 −k The variation ratio of the output voltage v O is

The voltage transfer gain of P/O SL Luo-converters is

2RfC 12 V I 1 −k where n is the stage number, j is the multiple-enhanced num-

ber. n = 1 and j = 0 for the elementary circuit with gain (as

14.6 Ultra-lift Luo-converters

in Eq. (14.131))

Ultra-lift (UL) Luo-converter performs very high voltage

2 −k transfer gain conversion. Its voltage transfer gain is the product

V I 1 −k of those of VL Luo-converter and SL Luo-converter.

We know that the gain of P/O VL Luo-converters (as in The circuit diagram of UL Luo-converter is shown in Eq. (14.52)) is

Fig. 14.67a, which consists of one switch S, two inductors L 1 and L 2 , two capacitors C 1 and C 2 , three diodes, and the

= load R. Its switch-on equivalent circuit is shown in Fig. 14.67b.

k h(n)

[n + h(n)]

V I = 1 −k

Its switch-off equivalent circuit for the continuous conduction mode is shown in Fig. 14.67c and switch-off equivalent

where n is the stage number, h(n) (as in Eq. (14.56)) is the circuit for the discontinuous conduction mode is shown in Hong function.

Fig. 14.67d.

14.6.1 Continuous Conduction Mode

Referring to Figs. 14.67b and c, we have got the current i L1 (from Eq. (14.32)) n = 0 for the elementary circuit with the increases with the slope +V I /L 1 during switch on, and

FIGURE 14.67 Ultra-lift (UL) Luo-converter: (a) circuit diagram; (b) switch on; (c) switch off in CCM; and (d) switch off in DCM.

TABLE 14.3 Comparison of various converters gains steady state, the current increment is equal to the decrement

decreases with the slope −V 1 /L 1 during switch off. In the

0.2 0.33 0.5 0.67 0.8 in a whole period T. The relation below is obtained 0.9

0.25 0.5 1 2 4 = (1 − k)T 9

Buck–Boost

L 1 L 1 VL Luo-converter

SL Luo-converter

Thus,

UL Luo-converter

V C1 =V 1 =

V I (14.181)

1 −k The current i L2 increases with the slope +(V I −V 1 )/L 2 dur-

ing switch on, and decreases with the slope −(V 1 −V O )/L 2

of VL Luo-converter and SL Luo-converter. We list the transfer

gains of various converters in Table 14.3 for reference. during switch off. In the steady state, the current increment The variation of inductor current i L1 is is equal to the decrement in a whole period T. We obtain the

relation below

and its variation ratio is

2L 1 I 2 2L 1 M 2(2 − k)fL 1 (14.187)

− k) 2

I I (14.184)

The variation of inductor current i L2 is The voltage transfer gain is

(14.185) and its variation ratio is

From Eq. (14.185) we can see that the voltage transfer gain

= (14.189) of UL Luo-converter is very high which is the product of those

I L2

2L 2 I 2 2L 2 M

2(2 − k)fL 2

14 DC/DC Conversion Technique and 12 Series Luo-converters 305

The variation of capacitor voltage v C1 is

V I (14.197)

(1 − k)m and its variation ratio is

C 1 C 1 (1 − k)C 1

We ﬁnally obtain the relation below v C1 /2

− k)V (14.198) − k) fC 1 R = (1 − k)T L 2 L 2

2 kT

The variation of capacitor voltage v C2 is

V I (14.199)

C 2 C 2 The voltage transfer gain in DCM is higher than that in CCM.

and its variation ratio is

M CCM v C2 /2

k(2 − k)

with m < 1 ε =σ 2 =

(14.200) From the analysis and calculations, we can see that all vari-

V C2 2V O C 2 2fC 2 R

ations are very small. A design example is that V I = 10 V,

14.7 Multiple-quadrant Operating

L 1 =L 2 = 1 mH, C 1 =C 2 and conduction duty cycle k varies from 0.1 to 0.9. We then

Luo-converters

obtain the output voltage variation ratio ε, which is less than 0.003. The output voltage is very smooth DC voltage nearly no Multiple-quadrant operating converters are the second- ripple.

generation converters. These converters usually perform between two voltage sources: V 1 and V 2 . Voltage source V 1 is proposed positive voltage and voltage V 2 is the load voltage.

In the investigation both voltages are proposed constant volt- Referring to Fig. 14.67d, we have got the current i L1 decreases age. Since V 1 and V 2 are constant values, voltage transfer gain to zero before t

14.6.2 Discontinuous Conduction Mode

= T , i.e. the current becomes zero before next is constant. Our interesting research will concentrate the work- time the switch turns on. The DCM operation condition is ing current, minimum conduction duty k min , and the power deﬁned as

transfer efﬁciency η.

Multiple-quadrant operating Luo-converters are the second-

≥1 generation converters and they have three modes: • Two-quadrant DC/DC Luo-converter in forward opera-

tion;

4 • Two-quadrant DC/DC Luo-converter in reverse opera- − k)

1 2(2 − k)fL 1 • Four-quadrant DC/DC Luo-converter. The normalized impedance Z N is,

The two-quadrant DC/DC Luo-converter in forward operation has been derived from the positive output Luo-converter.

It performs in the ﬁrst-quadrant Q I and the second-quadrant

fL 1 Q II corresponding to the DC motor forward operation in motoring and regenerative braking states.

We deﬁne the ﬁlling factor m to describe the current exists The two-quadrant DC/DC Luo-converter in reverse opera- time. For DCM operation, 0 < m ≤ 1,

tion has been derived from the N/O Luo-converter. It performs in the third-quadrant Q III and the fourth-quadrant Q IV corre-

sponding to the DC motor reverse operation in motoring and m =

1 2L 1 G 2(2 − k)

ξ = 1 k(1 − k) 2 TR = (1 − k) 4 Z (14.196) N regenerative braking states.

The four-quadrant DC/DC Luo-converter has been derived

= (1 − k)mT

from the double output Luo-converter. It performs four- L 1 L 1 quadrant operation corresponding to the DC motor forward

306 F. L. Luo and H. Ye and reverse operation in motoring and regenerative braking

The minimum conduction duty k corresponding to I 2 = 0 is states. In the following analysis the input source and output load

k min =

V 1 +V 2 −V S −V D S 1 and S 2 in this diagram are power metal oxide semiconductor ﬁeld effect transistor (MOSFET) devices, and they are

are usually constant voltages as shown, V 1 and V 2 . Switches

The power transfer efﬁciency is

driven by a PWM switching signal with repeating frequency

f and conduction duty k. In this paper the switch repeating

period is T = 1/f , so that the switch-on period is kT and switch-off period is (1

− k)T . The equivalent resistance is R 1

for each inductor. During switch-on the voltage drop across ( 1 + S +V D )/V 2 2 /V 2 ) '1+((1−k)/k) 2

the switches and diodes are V S and V D respectively.

(14.204) The variation ratio of capacitor voltage v C is

14.7.1 Forward Two-quadrant DC/DC

Luo-converter 2 ρ − k)I

V C 2fC(V 1 − RI 2 (1/(1 − k))) Forward Two-quadrant (F 2Q) Luo -converter is shown in

The variation ratio of inductor current i L1 is usually considered as constant voltages. The load can be a

Fig. 14.68. The source voltage (V 1 ) and load voltage (V 2 ) are

V 1 −V S − RI 1 battery or motor back electromotive force (EMF). For example,

i L1 /2

(14.206) the source voltage is 42 V and load voltage is

2fL 1 I +14 V. There are 1 two modes of operation:

I L1

The variation ratio of inductor current i L2 is

1. Mode A (Quadrant I): electrical energy is transferred

from source side V 1 to load side V 2 ;

2. Mode B (Quadrant II): electrical energy is transferred

2fL 2 I 2

from load side V 2 to source side V 1 .

The variation ratio of diode current i D2 is Mode A: The equivalent circuits during switch-on and -off

periods are shown in Figs. 14.69a and b. The typical output

=k voltage and current waveforms are shown in Fig. 14.69c. We

2fL(I 1 +I 2 )

2fLI 1

have the output current I 2 as

1 If the diode current becomes zero before S 1 −k switch on again,

I 1 (14.201) the converter works in discontinuous region. The condition is

2 2fLI 1 and

ζ D2 = 1, i.e. k =

V 1 −V S − RI 1

V 1 −V S −V D −V 2 ((1 − k)/k)

Mode B: The equivalent circuits during switch-on and -off

R periods are shown in Figs. 14.70a and b. The typical output voltage and current waveforms are shown in Fig. 14.70c. We

have the output current I 1 as

V 2 − (V 1 +V S +V D )((1

− k)/k)

R The minimum conduction duty k corresponding to I 1 = 0 is

+V S

V 1 +V D

(14.212) FIGURE 14.68 Forward two-quadrant operating Luo-converter.

min =

V 1 +V 2 +V S +V D

14 DC/DC Conversion Technique and 12 Series Luo-converters 307

FIGURE 14.69 Mode A: (a) switch on; (b) switch off; and (c) waveforms.

FIGURE 14.70 Mode B: (a) switch on; (b) switch off; and (c) waveforms.

The power transfer efﬁciency The variation ratio of diode current i D1 is P O

2fL(I 1 +I 2 )

2fLI 2

1 + ((V S +V D )/V 1 ) + (RI 1 /V 1 ) [1 + ((1 − k)/k) 2 ] (14.213)

If the diode current becomes zero before S 2 switch on again, the converter works in discontinuous region. The condition is

The variation ratio of capacitor voltage v C is

2 ζ D1 − k)) − V (14.218) 1 − RI 1 (k/(1 − k) ) ] = 1, i.e. k =

V 2 −V S − RI 2

14.7.2 Two-quadrant DC/DC Luo-converter in

The variation ratio of inductor current i L1 is

Reverse Operation

i L1 /2

V 2 −V S − RI 2

-converter is shown in Fig. 14.71, and it consists of two switches with two The variation ratio of inductor current i L2 is

ξ 1 = =k

(14.215) Reverse two-quadrant operating (R 2Q) Luo

I L1

2fL 1 I 1

passive diodes, two inductors and one capacitor. The source voltage (V 1 ) and load voltage (V 2 ) are usually considered as

2 = =k (14.216) constant voltages. The load can be a battery or motor back

i L2 /2

−V S − RI 2

I L2 2fL 2 I 2 EMF. For example, the source voltage is 42 V and load voltage

308 F. L. Luo and H. Ye

The power transfer efﬁciency is S 1 S 2 L 2 R

i L2

1 + ((V S +V D )/V 2 )(k/(1 − k)) + (RI 2 /V 2 ) [1 + (1/(1 − k)) 2 ] R

(14.222) The variation ratio of capacitor voltage v C is

FIGURE 14.71 Reverse two-quadrant operating Luo-converter.

v C /2

kI 2

V ( C 2fC '(k/(1 − k))V 1 − ((RI 2 )/(1 2 − k) ) (14.223)

is −14 V. There are two modes of operation:

1. Mode C (Quadrant III): electrical energy is transferred The variation ratio of inductor current i L1 is

from source side V 1 to load side −V 2 ;

2. Mode D (Quadrant IV): electrical energy is transferred

2fL 1 I 1 Mode C: The equivalent circuits during switch-on and -off

from load side −V 2 to source side V 1 .

I L1

periods are shown in Figs. 14.72a and b. The typical output The variation ratio of inductor current i D2 is voltage and current waveforms are shown in Fig. 14.72c. We

have the output current I 2 as

The variation ratio of inductor current i L2 is and

i L2 /2

V 1 −V S −V D −V 2 ((1 − k)/k)

I 2 16f 2 CL 2

R [(1/(k(1 − k))) + ((1 − k)/k)] If the diode current becomes zero before S 1 switch on again,

The minimum conduction duty k corresponding to I 2 = 0 is the converter works in discontinuous region. The condition is

2fL 1 I 1 k min =

ζ D2 = 1, i.e. k =

FIGURE 14.72 Mode C: (a) switch on; (b) switch off; and (c) waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 309

FIGURE 14.73 Mode D: (a) switch on; (b) switch off; and (c) waveforms.

Mode D: The equivalent circuits during switch-on and -off And the variation ratio of inductor current i D1 is periods are shown in Figs. 14.73a and b. The typical output voltage and current waveforms are shown in Fig. 14.73c. We

have the output current I 1 as

I L1

2fL 1 I 2

I 1 −k = I 2 (14.228)

1 The variation ratio of inductor current i L2 is

V 2 − (V 1 +V S +V D )((1 − k)/k)

If the diode current becomes zero before S 2 switch on again,

R [(1/(k(1 − k))) + (k/(1 − k))] the converter works in discontinuous region. The condition is

The minimum conduction duty k corresponding to I 1 = 0 is

2fL 1 I 2

ζ D1 = 1, i.e. k =

14.7.3 Four-quadrant DC/DC Luo-converter

The power transfer efﬁciency is Four-quadrant DC/DC Luo-converter is shown in Fig. 14.74,

1 which consists of two switches with two passive diodes, two

V 2 I 2 inductors, and one capacitor. The source voltage (V 1 ) and load

1 voltage (V 2 ) are usually considered as constant voltages. The

load can be a battery or motor back EMF. For example, the (14.231) source voltage is 42 V and load voltage is ±14 V. There are four

1 + ((V S +V D )/V 1 ) + (RI 1 /V 1 ) [(1/(1 − k) 2 ) + (k/(1 − k)) 2 ]

modes of operation:

1. Mode A (Quadrant I): electrical energy is transferred from source side V 1 to load side V 2 ; v C /2

The variation ratio of capacitor voltage v C is

ρ = 2. Mode B (Quadrant II): electrical energy is transferred =

kI 1

V C 2fC [((1 − k)/k)V 1 + ((RI 1 )/(k(1 − k)))]

from load side V 2 to source side V 1 ;

3. Mode C (Quadrant III): electrical energy is transferred from source side V 1 to load side −V 2 ;

The variation ratio of inductor current i L1 is

4. Mode D (Quadrant IV): electrical energy is transferred from load side −V 2 to source side V 1 .

Each mode has two states: “on” and “off.” Usually, each

I L1 2fL 1 I 1 state is operating in different conduction duty k. The switches

FIGURE 14.74 Four-quadrant operating Luo-converter: (a) circuit 1 and (b) circuit 2.

are the power MOSFET devices. The circuit 1 in Fig. 14.74 currents i L1 and i L2 increase, and i 1 =i L1 . During state-off implements Modes A and B, and the circuit 2 in Fig. 14.74 switches S 1 ,S 2 , and diode D 1 are off and diode D 2 is con- implements Modes C and D. Circuits 1 and 2 can changeover ducted. In this case current i L1 ﬂows via diode D 2 to charge by auxiliary switches (not in the ﬁgure).

capacitor C and the load battery V 2 via inductor L 2 . The Mode A: During state-on switch S 1 is closed, switch S 2 and free-wheeling diode current i D2 =i L1 =i C +i 2 . Mode C

diodes D 1 and D 2 are not conducted. In this case inductor implements the characteristics of the buck–boost conversion.

currents i L1 and i L2 increase, and i 1 =i L1 +i L2 . During state-

Mode D: During state-on switches S 2 is closed, switch S 1 off switches S 1 ,S 2 , and diode D 1 are off and diode D 2 is and diodes D 1 and D 2 are not conducted. In this case induc- conducted. In this case current i L1 ﬂows via diode D 2 to charge tor current i L1 increases by biased V 2 , inductor current i L2 capacitor C, in the meantime current i L2 is kept to ﬂow through decreases by biased (V 2 −V C ). Therefore capacitor voltage V C load battery V 2 . The free-wheeling diode current i D2 =i L1 + reduces. Current i L1 =i C −on +i 2 . During state-off switches

i L2 . Mode A implements the characteristics of the buck–boost S 1 ,S 2 , and diode D 2 are not on, and only diode D 1 is on. In conversion.

this case source current i 1 =i L1 which is a negative value Mode B: During state-on switches S 2 is closed, switch S 1 and to perform the regenerative operation. Inductor current i 2 diodes D 1 and D 2 are not conducted. In this case inductor cur- ﬂows through capacitor C that is charged by current i 2 , i.e. rent i L2 increases by biased V 2 , inductor current i L1 increases i C −off =i 2 . Mode D implements the characteristics of the

by biased V C . Therefore capacitor voltage V C reduces. During boost conversion. state-off switches S 1 ,S 2 , and diode D 2 are not on, and only

Summary: The switch status is shown in Table 14.4. diode D 1 is on. In this case source current i 1 =i L1 +i L2 which The operation of all modes A, B, C, and D is same to the is a negative value to perform the regenerative operation. description in Sections 14.7.1 and 14.7.2. Inductor current i L2 ﬂows through capacitor C, it is charged by current i L2 . After capacitor C, i L2 then ﬂows through the

14.8 Switched-capacitor Multi-quadrant

source V 1 . Inductor current i L1 ﬂows through the source V 1 as

well via diode D 1 . Mode B implements the characteristics of

Luo-converters

the boost conversion. Mode C: During state-on switch S 1 is closed, switch S 2 and Switched-component converters are the third-generation diodes D 1 and D 2 are not conducted. In this case inductor converters. These converters are made of only inductor

TABLE 14.4 Switch’s status (the blank status means OFF) Switch or diode

Mode A (QI)

Mode B (QII)

Mode C (QIII)

Mode D (QIV)

State-off Circuit

State-on

State-off

State-on

State-off

State-on

State-off

State-on

Circuit 1

Circuit 2

S 1 ON

D 1 ON

S 2 ON

D 2 ON

14 DC/DC Conversion Technique and 12 Series Luo-converters 311

FIGURE 14.75 Two-quadrant switched-capacitor DC/DC Luo-converter.

or capacitors. They usually perform in the systems between polarity. It performs four-quadrant operation corresponding two voltage sources: V 1 and V 2 . Voltage source V 1 is proposed to the DC motor forward and reverse operation in motoring

positive voltage and voltage V 2 is the load voltage that can and regenerative braking states.

be positive or negative. In the investigation both voltages are From the analysis and calculation, the conduction duty k proposed constant voltage. Since V 1 and V 2 are constant val- does not affect the power transfer efﬁciency. It affects the input ues, so that voltage transfer gain is constant. Our interesting and output power in a small region. The maximum output research will concentrate on the working current and the power corresponds at k = 0.5. power transfer efﬁciency η. The resistance R of the capacitors and inductor has to be considered for the power transfer

14.8.1 Two-quadrant Switched-capacitor

efﬁciency η calculation.

DC/DC Luo-converter

Reviewing the papers in the literature, we can ﬁnd that almost of the papers investigating the switched-component This converter is shown in Fig. 14.75. It consists of nine converters are working in single-quadrant operation. Profes- switches, seven diodes, and three capacitors. The high source

sor Luo and colleagues have developed this technique into voltage V H and low load voltage V L are usually considered multi-quadrant operation. We describe these in this and next as constant voltages, e.g. the source voltage is 48 V and load sections.

voltage is 14 V. There are two modes of operation: Switched-capacitor multi-quadrant Luo-converters are the

• Mode A (Quadrant I): electrical energy is transferred third-generation converters, and they are made of only capac-

from V H side to V L side;

itors. Because these converters implement voltage-lift and • Mode B (Quadrant II): electrical energy is transferred current-ampliﬁcation techniques, they have the advantages of

from V L side to V H side.

high power density, high power transfer efﬁciency, and low EMI. They have two modes:

Each mode has two states: “on” and “off.” Usually, each state is operating in different conduction duty k. The switching

• Two-quadrant switched-capacitor DC/DC Luo-converter;

• Four-quadrant switched-capacitor DC/DC Luo-converter. = 1/f , where f is the switching frequency.

period is T where T

The switches are the power MOSFET devices. The parasitic

The two-quadrant switched-capacitor DC/DC Luo-converter resistance of all switches is r S . The equivalent resistance of all

in forward operation has been derived for the energy transmis- capacitors is r C and the equivalent voltage drop of all diodes sion of a dual-voltage system in two-quadrant operation. The is V D . Usually we select the three capacitors having same capac- both, source and load voltages are positive polarity. It performs itance C =C 1 =C 2 =C 3 . Some reference data are useful:

C D = 0.5 V, f = 5 kHz, and sponding to the DC motor forward operation in motoring and C = 5000 µF. The switch’s status is shown in Table 14.5. regenerative braking states.

in the ﬁrst-quadrant Q I and the second-quadrant Q II corre- r S

For Mode A, state-on is shown in Fig. 14.76a: switches The four-quadrant switched-capacitor DC/DC Luo- S 1 and S 10 are closed and diodes D 5 and D 5 are con- converter has been derived for the energy transmission of a ducted. Other switches and diodes are open. In this case dual-voltage system in four-quadrant operation. The source capacitors C 1 ,C 2 , and C 3 are charged via the circuit V H – voltage is positive and load voltage can be positive or negative S 1 –C 1 –D 5 –C 2 –D 6 –C 3 –S 10 , and the voltage across capacitors

312 F. L. Luo and H. Ye TABLE 14.5 Switch’s status (the blank status means OFF)

The variation of the voltage across capacitor C 1 is:

Switch or diode

Mode A

Mode B

2.4k(1 − k)(V H − 3V L − 5V D )

(2.4 + 0.6k)fCR AN

After calculation,

k(V H − 2V D ) + 2.4(1 − k)(V L +V D )

The average output current is

3 V C1 −V L −V D

C 1 ,C 2 , and C 3 is increasing. The equivalent circuit resistance

2V D = 1 V. State-off is shown in Fig. 14.76b: switches S 2 ,S 3 , and S 4 are closed and diodes D 8 ,D 9 , and D 10 are conducted.

The average input current is

Other switches and diodes are open. In this case capacitor

C 1 (C 2 and C 3 ) is discharged via the circuit S 2 (S 3 and S 4 )–

V kT

1 V H − 3V C1 − 2V D capacitor C 1 (C 2 and C 3 ) is decreasing. Mode A implements

L –D 8 (D 9 and D 10 )–C 1 (C 2 and C 3 ), and the voltage across

i C1 (t )dt ≈k

R AN the current-ampliﬁcation technique. The voltage and cur-

rent waveforms are shown in Fig. 14.76c. All three capacitors are charged in series during state-on. The input current ﬂows

Therefore, we have 3I H =I L .

through three capacitors and the charges accumulated on the

Output power is

three capacitors should be the same. These three capacitors are discharged in parallel during state-off. Therefore, the output

V C1 L P D

−V −V

(14.241) current is ampliﬁed by three times.

FIGURE 14.76 Mode A operation: (a) state-on; (b) state-off; and (c) voltage and current waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 313 Input power

After calculation

V H − 3V C1 −V D 1 −k

P I =V H I H = kV H (14.242)

V C1 = k(V L −V D ) +

(V H −V L +V D ) (14.245)

R AN

The transfer efﬁciency is

The average input current is

For Mode B, state-on is shown in Fig. 14.77a: switches S 8 ,

(14.246) ducted. Other switches and diodes are off. In this case all three

S 9 , and S 10 are closed and diodes D 2 ,D 3 , and D 4 are con-

capacitors are charged via each circuit V L –D 2 (and D 3 ,D 4 )–

The average output current is

C 1 (and C 2 ,C 3 )–S 8 (and S 9 ,S 10 ), and the voltage across three

capacitors are increasing. The equivalent circuit resistance is

1 3V C1 +V L −V H −V D

R BN =r S +r C and the voltage deduction is V D in each circuit.

i C1 (t )dt ≈ (1 − k)

R BF

State-off is shown in Fig. 14.77b: switches S 5 ,S 6 , and S 7 are

closed and diode D 1 is on. Other switches and diodes are open. (14.247)

In this case all capacitors is discharged via the circuit V L –S 7 –

From this formula, we have 4I H =I L . is decreasing. Mode B implements the voltage-lift technique.

C 3 –S 6 –C 2 –S 5 –C 1 –D 1 –V H , and the voltage across all capacitors

Input power is

The voltage and current waveforms are shown in Fig. 14.77c.

P I =V L I L

All three capacitors are charged in parallel during state-on. The input voltage is applied to the three capacitors symmetri-

V L −V C −V D 3V C +V L −V H −V D cally, so that the voltages across these three capacitors should be

=V L 3k

+(1−k)

R BF same. They are discharged in series during state-off. Therefore,

R BN

(14.248) the output voltage is lifted by three times.

The variation of the voltage across capacitor C is:

Output power is

3V C +V L −V H = D −V

k(1 − k)[4(V L

−V −V H v ] C1 (14.244)

P O =V H I H =V H (1 − k)

fCR BN

(14.249) R BF

314 F. L. Luo and H. Ye The efﬁciency is

TABLE 14.6 Switch’s status (mentioned switches are not open)

V H Quadrant No. Condition

State

Source side Load side

and mode

QI, Mode A

V 1 > V 2 S 1,4,6,8

S 2,4,6,8 V 1 + V 2 +

Forw. Mot.

V 1 < V 2 S 1,4,6,8

S 2,4,7 I 1 + I 2 +

14.8.2 Four-quadrant Switched-capacitor

QII, Mode B

V 1 > V 2 S 2,4,6,8

S 1,4,7 V 1 + V 2 +

DC/DC Luo-converter

Forw. Reg.

V 1 < V 2 S 2,4,6,8

S 1,4,6,8 I 1 − I 2 −

QIII, Mode C

V 1 > |V 2 |S 1,4,6,8

S 3,5,6,8 V 1 + V 2 −

Four-quadrant switched-capacitor DC/DC Luo-converter is

Rev. Mot.

V 1 < |V 2 |S 1,4,6,8

S 3,5,7 I 1 + I 2 −

shown in Fig. 14.78. Since it performs the voltage-lift technique,

QIV Mode D

V 1 > |V 2 |S 3,5,6,8

S 1,4,7 V 1 + V 2 −

it has a simple structure with four-quadrant operation. This

Rev. Reg.

V 1 < |V 2 |S 3,5,6,8

S 1,4,6,8 I 1 − I 2 +

converter consists of eight switches and two capacitors. The

source voltage V 1 and load voltage V 2 (e.g. a battery or DC

motor back EMF) are usually constant voltages. In this paper

they are supposed to be ±21 V and ±14 V. Capacitors C 1 and

C 2 are same and C 1 2 =C “off.” Usually, each state is operating in various conduction = 2000 µF. The circuit equivalent resistance R

duty k for different currents. As usual, the efﬁciency of all SC operation for this converter:

DC/DC converters is independent from the conduction duty cycle k. The switching period is T where T = 1/f . The switch

1. Mode A: energy is converted from source to positive status is shown in Table 14.6.

voltage load; the ﬁrst-quadrant operation, Q I ;

As usual, the transfer efﬁciency only relies on the ratio of

2. Mode B: energy is converted from positive voltage load the source and load voltages, and it is independent on R, C, f,

to source; the second-quadrant operation, Q II ;

and k. We select k = 0.5 for our description. Other values for

3. Mode C: energy is converted from source to negative the reference are f = 5 kHz, V 1 = 21 V, V 2 = 14 V, and total voltage load; the third-quadrant operation, Q III ;

4. Mode D: energy is converted from negative voltage load For Mode A1, condition V 1 > V 2 is shown in Fig. 14.78a.

to source; the fourth-quadrant operation, Q IV .

Since V 1 > V 2 , two capacitors C 1 and C 2 are connected in The ﬁrst-quadrant (Mode A) is so called the forward motor- parallel. During switch-on state, switches S 1 ,S 4 ,S 6 , and S 8

ing (Forw. Mot.) operation. V 1 and V 2 are positive, and are closed and other switches are open. In this case, capaci-

I 1 and I 2 are positive as well. The second-quadrant (Mode tors C 1 //C 2 are charged via the circuit V 1 –S 1 –C 1 //C 2 –S 4 , and

B) is so called the forward regenerative (Forw. Reg.) braking the voltage across capacitors C 1 and C 2 is increasing. During operation. V 1 and V 2 are positive, and I 1 and I 2 are negative. switch-off state, switch S 2 ,S 4 ,S 6 , and S 8 are closed and other The third-quadrant (Mode C) is so-called the reverse motor- switches are open. In this case capacitors C 1 //C 2 are discharged ing (Rev. Mot.) operation. V 1 and I 1 are positive, and V 2 and via the circuit S 2 –V 2 –S 4 –C 1 //C 2 , and the voltage across capac-

I 2 are negative. The fourth-quadrant (Mode D) is so-called the itors C 1 and C 2 is decreasing. Capacitors C 1 and C 2 transfer reverse regenerative (Rev. Reg.) braking operation. V 1 and I 2 the energy from the source to the load.

are positive, and I 1 and V 2 are negative.

The average capacitor voltage

Each mode has two conditions: V 1 > V 2 and V 1 < V 2 (or

V C = kV 1 + (1 − k)V 2 |V (14.251) | for Q

2 III and Q IV ). Each condition has two states: “on” and

V C1 C C1 1 S 7 C 2

FIGURE 14.78 Four-quadrant sc DC/DC Luo-converter.

14 DC/DC Conversion Technique and 12 Series Luo-converters 315

(iii) FIGURE 14.78a Mode A1 (QI): forward motoring with V 1 > V 2 : (i) switch on: S 1 ,S 4 ,S 6 , and S 8 on; (ii) switch off: S 2 ,S 4 ,S 6 , and S 8 , on; and

(iii) waveforms.

The average current is

V 1 –S 1 –C 1 //C 2 –S 4 , and the voltage across capacitors C 1 and C 2 is increasing. During switch-off state, switches S 2 ,S 4 , and S 7

1 V C −V 2

are closed and other switches are open. In this case, capacitors

i C (t )dt ≈ (1 − k) (14.252) C 1 and C 2 are discharged via the circuit S 2 –V 2 –S 4 –C 1 –S 7 –C 2 , T

and the voltage across capacitor C 1 and C 2 is decreasing. Capacitors C 1 and C 2 transfer the energy from the source to and

the load.

The average capacitor voltage is

0.5V 1 +V 2

The average current is

The transfer efﬁciency is

For Mode A2, condition V 1 < V 2 is shown in Fig. 14.78b. Since V 1 < V 2 , two capacitors C 1 and C 2 are connected in and

parallel during switch on and in series during switch off. This is so-called the voltage-lift technique. During switch-on state, kT

1 V 1 −V C

switches S 1 ,S 4 ,S 6 , and S 8 are closed and other switches are

i C (t )dt ≈k

open. In this case, capacitors C 1 //C 2 are charged via the circuit

(iii) FIGURE 14.78b Mode A2 (QI): forward motoring with V 1 < V 2 : (i) switch on: S 1 ,S 4 ,S 6 , and S 8 , on; (ii) switch off: S 2 ,S 4 , and S 7 , on; and

(iii) waveforms.

(iii) FIGURE 14.78c Mode B1 (QII): forward regenerative braking with V 1 > V 2 : (i) switch on: S 2 ,S 4 ,S 6 , and S 8 , on; (ii) switch off; S 1 ,S 4 (S 5 ), and S 7

on; and (iii) waveforms.

The transfer efﬁciency is

The average current is

For Mode B1, condition V 1 > V 2 is shown in Fig. 14.78c. and Since V 1 > V 2 , two capacitors C 1 and C 2 are connected in

parallel during switch on and in series during switch off.

The voltage-lift technique is applied. During switch-on state,

R tors C

switches S 2 ,S 4 ,S 6 , and S 8 are closed. In this case, capaci-

1 //C 2 are charged via the circuit V 2 –S 2 –C 1 //C 2 –S 4 , and

the voltage across capacitors C 1 and C 2 is increasing. During The transfer efﬁciency is switch-off state, switches S 1 ,S 4 , and S 7 are closed. In this case,

capacitors C 1 and C 2 are discharged via the circuit S 1 –V 1 –

1 −k V 1 2V C −V 1 V 1

S 4 –C 2 –S 7 –C 1 , and the voltage across capacitor C 1 and C 2 is

η B1 =

(14.262) k V 2 V 2 C = 2V 2

−V

decreasing. Capacitors C 1 and C 2 transfer the energy from the

For Mode B2, condition V 1 < V 2 is shown in Fig. 14.78d. The average capacitor voltage is

load to the source. Therefore, we have I 2 = 2I 1 .

Since V 1 < V 2 , two capacitors C 1 and C 2 are connected in parallel. During switch-on state, switches S 2 ,S 4 ,S 6 , and S 8 are

V C closed. In this case, capacitors C 1 //C 2 = are charged via the circuit = 11.2 (14.259)

0.5V 2 +V 1

2.5 V 2 –S 2 –C 1 //C 2 –S 4 , and the voltage across capacitors C 1 and C 2

(iii) FIGURE 14.78d Mode B2 (QII): forward regenerative braking with V 1 < V 2 : (i) switch on: S 2 ,S 4 ,S 6 , and S 8 , on; (ii) switch off: S 1 ,S 4 (S 5 ), S 6 , and

S 8 on; and (iii) waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 317

is increasing. During switch-off state, switches S 1 ,S 4 ,S 6 , and

The average capacitor voltage is

S 8 are closed. In this case capacitors C 1 //C 2 is discharged via the

circuit S 1 –V 1 –S 4 –C 1 //C 2 , and the voltage across capacitors C 1 V C = kV 1 + (1 − k)|V 2 | (14.267)

and C 2 is decreasing. Capacitors C 1 and C 2 transfer the energy

The average current (absolute value) is The average capacitor voltage is

from the load to the source. Therefore, we have I 2 =I 1 .

V C = kV 2 + (1 − k)V 1 (14.263)

I − |V 2 2 | = i C (t )dt ≈ (1 − k) (14.268)

R The average current is kT

and the average input current is

The transfer efﬁciency is

V 1 V 1 −V C V 1 The transfer efﬁciency is

For Mode C2, condition V 1 < |V 2 | is shown in Fig. 14.78f. Since V 1 < |V 2 |, two capacitors C 1 and C 2 are connected in P O

1 −k V 1 V C −V 1 V 1 parallel during switch on and in series during switch off, apply- η B2 =

P I k V 2 V 2 −V C V 2 ing the voltage-lift technique. During switch-on state, switches S 1 ,S 4 ,S 6 , and S 8 , are closed. Capacitors C 1 and C 2 are charged

For Mode C1, condition V 1 > |V 2 | is shown in Fig. 14.78e. via the circuit V 1 –S 1 –C 1 //C 2 –S 4 , and the voltage across capac- Since V 1 > |V 2 |, two capacitors C 1 and C 2 are connected in itors C 1 and C 2 is increasing. During switch-off state, switches parallel. During switch-on state, switches S 1 ,S 4 ,S 6 , and S 8 S 3 ,S 5 , and S 7 are closed. Capacitors C 1 and C 2 is discharged via are closed. In this case, capacitors C 1 //C 2 are charged via the the circuit S 3 –V 2 –S 5 –C 1 –S 7 –C 2 , and the voltage across capaci- circuit V 1 –S 1 –C 1 //C 2 –S 4 , and the voltage across capacitors C 1 tor C 1 and C 2 is decreasing. Capacitors C 1 and C 2 transfer the and C 2 is increasing. During switch-off state, switches S 3 ,S 5 , energy from the source to the load. We have I 1 = 2I 2 .

S 6 , and S 8 are closed. Capacitors C 1 and C 2 are discharged via

The average capacitor voltage is

the circuit S 3 –V 2 –S 5 –C 1 //C 2 , and the voltage across capacitors

C 1 and C 2 is decreasing. Capacitors C 1 and C 2 transfer the

0.5V 1 + |V 2 |

energy from the source to the load. We have I 1 =I 2 .

(iii) FIGURE 14.78e Mode C1 (QIII): reverse motoring with V 1 > |V 2 |: (i) switch on: S 1 ,S 4 ,S 6 , and S 8 on; (ii) switch off: S 3 ,S 5 ,S 6 , and S 8 on; and

(iii) waveforms.

(iii) FIGURE 14.78f Mode C2 (QIII): reverse motoring with V 1 < |V 2 |: (i) switch on: S 1 ,S 4 ,S 6 , and S 8 , on; (ii) switch off: S 3 ,S 5 , and S 7 , on; and

(iii) waveforms.

The average currents are S 3 ,S 5 ,S 6 , and S 8 are closed. In this case, capacitors C 1 //C 2 are charged via the circuit V 2 –S 3 –C 1 //C 2 –S 5 , and the voltage

across capacitors C 1 and C 2 is increasing. During switch-off

(14.272) state, switches S 1 ,S 4 , and S 7 are closed. Capacitors C 1 and T

C 2 are discharged via the circuit S 1 –V 1 –S 4 –C 2 –S 7 –C 1 , and the voltage across capacitor C 1 and C 2 is decreasing. Capacitors

and

C 1 and C 2 transfer the energy from the load to the source. We

have I 2 = 2I 1 .

The average capacitor voltage is

2.5 = 11.2 The transfer efﬁciency is

The average currents are

(14.276) P I k

For Mode D1, condition V 1 > |V 2 | is shown in Fig. 14.78g. and

Since V 1 > |V 2 |, two capacitors C 1 and C 2 are connected in

parallel during switch on and in series during switch off, apply-

I |V 2 |−V = i C (t )dt ≈k (14.277) ing the voltage-lift technique. During switch-on state, switches

(iii) FIGURE 14.78g Mode D1 (QIV): reverse regenerative braking with V 1 > |V 2 |: (i) switch on: S 3 ,S 4 ,S 6 , and S 8 , on; (ii) switch off: S 1 ,S 4 , and S 7 on;

and (iii) waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 319

(iii) FIGURE 14.78h Mode D2 (QIV): reverse regenerative braking with V 1 < |V 2 |: (i) switch on: S 3 ,S 5 ,S 6 , and S 8 , on; (ii) switch off: S 1 ,S 4 ,S 6 , and S 8

on; and (iii) waveforms.

The transfer efﬁciency is

14.9 Multiple-lift Push–Pull Switched-capacitor Luo-converters

Micro-power-consumption technique requires high power density DC/DC converters and power supply source. Voltage-

For Mode D2, condition V 1 < |V 2 | is shown in Fig. 14.78h. lift (VL) technique is a popular method to apply in electronic Since V 1 < |V 2 |, two capacitors C 1 and C 2 are connected in circuit design. Since switched-capacitor can be integrated parallel. During switch-on state, switches S 3 ,S 5 ,S 6 , and S 8 into power integrated circuit (IC) chip, its size is small. are closed. In this case, capacitors C 1 //C 2 are charged via the Combining switched-capacitor and VL techniques the DC/DC circuit V 2 –S 3 –C 1 //C 2 –S 5 , and the voltage across capacitors C 1 converters with small size, high power density, high voltage and C 2 is increasing. During switch-off state, switches S 1 ,S 4 , transfer gain, high power efﬁciency, and low EMI can be S 6 , and S 8 are closed. Capacitors C 1 and C 2 are discharged via constructed. This section introduces a new series DC/DC con- the circuit S 1 –V 1 –S 4 –C 1 //C 2 , and the voltage across capacitors verters – multiple-lift push–pull switched-capacitor DC/DC

C 1 and C 2 is decreasing. Capacitors C 1 and C 2 transfer the Luo-converters. There are two subseries:

• P/O multiple-lift (ML) push–pull (PP) switched- The average capacitor voltage is

energy from the load to the source. We have I 2 =I 1 .

capacitor (SC) DC/DC Luo-converter; • N/O multiple-lift push–pull switched-capacitor DC/DC

V C = k|V 2 | + (1 − k)V 1 (14.279)

Luo-converter.

The average currents are

14.9.1 P/O Multiple-lift Push–Pull

Switched-capacitor DC/DC

1 V C −V 1 Luo-converter

i C (t )dt ≈ (1 − k)

P/O ML-PP SC DC/DC Luo-converters have several subseries:

• Main series;

and

• Additional series; • Enhanced series;

• Re-enhanced series;

• Multiple-enhanced series.

0 We only introduce three circuits of main series and additional series in this section.

The transfer efﬁciency is P/O ML-PP SC Luo-converter elementary circuit is shown in Fig. 14.79a. Its output voltage and current are

1 −k V 1 V C −V 1 V 1

η D2 = =

P I k |V 2 | |V 2 |−V C |V 2 |

V O = 2V I

and

The voltage transfer gain is

M T =8

P/O ML-PP SC Luo-converter additional circuit is shown The voltage transfer gain is

in Fig. 14.80a. Its output voltage and current are

P/O ML-PP SC Luo-converter re-lift circuit is shown in and Fig. 14.79b. Its output voltage and current are

V O = 4V I The voltage transfer gain is

and

P/O ML-PP SC Luo-converter additional re-lift circuit is The voltage transfer gain is

shown in Fig. 14.80b. Its output voltage and current are

P/O ML-PP SC Luo-converter triple-lift circuit is shown in and Fig. 14.79c. Its output voltage and current are

= 8V

and

The voltage transfer gain is

M AR =

14 DC/DC Conversion Technique and 12 Series Luo-converters 321 I in

P/O ML-PP SC Luo-converter additional triple-lift circuit and is shown in Fig. 14.80c. Its output voltage and current are

I O =I I

V O = 12V I The voltage transfer gain is

and

= V I =1 (14.289)

= 12

The voltage transfer gain is N/O ML-PP SC Luo-converter re-lift circuit is shown in Fig. 14.81b. Its output voltage and current are

14.9.2 N/O Multiple-lift Push–Pull

and

Switched-capacitor DC/DC

Luo-converter

N/O ML-PP SC DC/DC Luo-converters have several subseries:

The voltage transfer gain is

• Main series; • Additional series;

(14.290) • Enhanced series;

M R =3

• Re-enhanced series; N/O ML-PP SC Luo-converter triple-lift circuit is shown • Multiple-enhanced series.

in Fig. 14.81c. Its output voltage and current are We only introduce three circuits of main series and addi-

V O = 7V I

tional series in this section. N/O ML-PP SC Luo-converter elementary circuit is shown and in Fig. 14.81a. Its output voltage and current are

V O =V I

The voltage transfer gain is

M T =7

(14.293) N/O ML-PP SC Luo-converter additional circuit I is shown

M AR =

V =5

in Fig. 14.82a. Its output voltage and current are

N/O ML-PP SC Luo-converter additional triple-lift circuit

is shown in Fig. 14.82c. Its output voltage and current are

The voltage transfer gain is

N/O ML-PP SC Luo-converter additional re-lift circuit

shown in Fig. 14.82b. Its output voltage and current are

V O = 5V I 14.10 Switched-inductor Multi-quadrant

Operation Luo-converters

and

I O = I I Switched-capacitor converters usually have many switches and

5 capacitors, especially for the system with high ratio between

14 DC/DC Conversion Technique and 12 Series Luo-converters 323 I in

source and load voltages. Switched-inductor converter usually in the third-quadrant Q III and the fourth-quadrant Q IV has only one inductor even if it works in single-, two-, and/or corresponding to the DC motor reverse operation in motoring four-quadrant operation. Simplicity is the main advantage of and regenerative braking states. all switched inductor converters.

The four-quadrant switched-inductor DC/DC Luo-converter

Switched-inductor multi-quadrant Luo-converters are the has been derived for the energy transmission of a dual-voltage third-generation converters, and they are made of only induc- system. The source voltage is positive and load voltage can be tor. These converters have been derived from chopper circuits. positive or negative polarity. It performs four-quadrant oper- They have three modes:

ation corresponding to the DC motor forward and reverse operation in motoring and regenerative braking states.

• Two-quadrant switched-inductor DC/DC Luo-converter in forward operation; • Two-quadrant switched-inductor DC/DC Luo-converter in reverse operation; • Four-quadrant switched-inductor DC/DC Luo-converter.

14.10.1 Two-quadrant Switched-inductor DC/DC Luo-converter in

The two-quadrant switched-inductor DC/DC Luo-converter

Forward Operation

in forward operation has been derived for the energy trans- Forward operation (F) 2Q SI Luo

-converter is shown in mission of a dual-voltage system. The both, source and Fig. 14.83, and it consists of two switches with two passive load voltages are positive polarity. It performs in the ﬁrst- diodes, two inductors, and one capacitor. The source voltage

quadrant Q I and the second-quadrant Q II corresponding to

(V 1 ) and load voltage (V 2 ) are usually considered as constant the DC motor forward operation in motoring and regenerative voltages. The load can be a battery or motor back EMF. For braking states. example, the source voltage is 42 V and load voltage is The two-quadrant switched-inductor DC/DC Luo-converter

+14 V.

There are two modes of operation:

in reverse operation has been derived for the energy transmission of a dual-voltage system. The source voltage is

1. Mode A (QI): electrical energy is transferred from positive and load voltage is negative polarity. It performs

source side V 1 to load side V 2 ;

324 F. L. Luo and H. Ye

The power transfer efﬁciency is

P I kV 1 + V high

The boundary between continuous and discontinuous −

+ V low

I low regions is deﬁned as

high

ζ ≥ 1 i.e.

V 2 R FIGURE 14.83 Switched-inductor QI and II DC/DC Luo-converter.

k(1 − k)V 1 R

≥ 1 or k ≤

+ k(1 − k) (14.298)

V 1 2fL Average inductor current I L in discontinuous region is

1 −V 2 2fL

2. Mode B (QII): electrical energy is transferred from load

V 1 V 1 −V 2 − RI L 2

k (14.299)

side V 2 to source side V 1 .

V 2 + RI L

2fL

Mode A: The equivalent circuits during switch-on and -off

The power transfer efﬁciency is

periods are shown in Figs. 14.84a and b. The typical output voltage and current waveforms are shown in Fig. 14.84c.

V 2 V 2 R We have the average inductor current I L as

η A −dis =

with k ≤ + k(1 − k)

P I V 2 + RI L

V 1 2fL (14.300)

kV 1 −V 2

Mode B: The equivalent circuits during switch-on and -off periods are shown in Figs. 14.85a and b. The typical output

The variation ratio of the inductor current i L is voltage and current waveforms are shown in Fig. 14.85c.

The average inductor current I L is

V 2 = 1 = (14.296) − (1 − k)V

V high

V low

− i − V high D2 i 2 − low o i 1 i 2

kT T t (a)

(b)

(c) FIGURE 14.84 Mode A of F 2Q SI Luo-converter: (a) state-on: S 1 on; (b) state-off: D 2 on, S 1 , off; and (c) input and output current waveforms.

i 1 2 high

− low

− i 2 − high

kT T t (a)

(b)

(c) FIGURE 14.85 Mode B of F 2Q SI Luo-converter: (a) state-on: S 2 on; (b) state-off: D 1 on, S 2 off; and (c) input and output current waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 325 The variation ratio of the inductor current i L is

− (1 − k)V 2 2fL V + L high 1 − V low

The power transfer efﬁciency

P I V 2 The boundary between continuous and discontinuous FIGURE 14.86 Switched-inductor QIII and IV DC/DC Luo-converter.

regions is deﬁned as ζ ≥ 1, i.e.

voltages. The load can be a battery or motor back EMF. For k(1 −k)V 1 R

example, the source voltage is 42 V and load voltage is −14 V.

≥1 or k ≤ 1 −

+k(1−k)

V 2 −(1−k)V 1 2fL

V 1 2fL

There are two modes of operation:

1. Mode C (QIII): electrical energy is transferred from source side V 1 to load side −V 2 ;

Average inductor current I L in discontinuous region is

2. Mode D (QIV): electrical energy is transferred from

V 1 V 2 − RI L

load side −V 2 to source side V 1 .

Mode C: The equivalent circuits during switch-on and -off periods are shown in Figs. 14.87a and b. The typical output

The power transfer efﬁciency is voltage and current waveforms are shown in Fig. 14.87c.

We have the average inductor current I L as η B −dis =

V 2 − RI L

kV 1 − (1 − k)V 2

(14.307) with k ≤ 1 −

The variation ratio of the inductor current i L is

14.10.2 Two-quadrant Switched-inductor

i L /2

k(1

− k)(V +V

DC/DC Luo-converter in Reverse

kV 1 − (1 − k)V 2 2fL

Operation

The power transfer efﬁciency is

Reverse operation (R) 2Q SI Luo -converter is shown in Fig. 14.86, and it consists of two switches with two passive diodes, two inductors, and one capacitor. The source voltage

(V 1 ) and load voltage (V 2 ) are usually considered as constant

P I kV 1

kT T t −

+ V high

V low

V high

(c) FIGURE 14.87 Mode C of F 2Q SI Luo-converter: (a) state-on; S 1 on; (b) state-off: D 2 on, S 1 off; and (c) input and output current waveforms.

326 F. L. Luo and H. Ye The boundary between continuous and discontinuous

The boundary between continuous and discontinuous regions is deﬁned as

regions is deﬁned as

≥1, i.e. ζ ≥1, i.e.

k(1 −k)(V 1 +V 2 ) R

V 1 ≥1 or k ≤ R +k(1−k) −k)(V +V kV 1 2 2fL

k(1

≥1 or k ≤ +k(1−k) −(1−k)V

V 1 2 2fL

kV 2 −(1−k)V 1 2fL

Average inductor current I L in discontinuous region is Average inductor current I L in discontinuous region is

The power transfer efﬁciency is

− RI L

1 − RI

C −dis = =

η D −dis =

P I V 1 V 2 + RI L

P I V 2 V 1 + RI L

with k ≤

+ k(1 − k) (14.318)

V 1 +V 2 2fL Mode D: The equivalent circuits during switch-on and -off

periods are shown in Figs. 14.88a and b. The typical output voltage and current waveforms are shown in Fig. 14.88c.

14.10.3 Four-quadrant Switched-inductor

The average inductor current I L is

DC/DC Luo-converter

Switched-inductor DC/DC converters successfully overcome

I L = (14.313) the disadvantage of switched-capacitor converters. Usually,

kV 2 − (1 − k)V 1

only one inductor is required for each converter with one- or two- or four-quadrant operation, no matter how large the

The variation ratio of the inductor current i L is difference between the input and output voltage is. There- fore, switched-inductor converter has very simple topology

and circuit. Consequently, it has high power density. This ζ

i L /2 k(1

− k)(V 1 +V

− (1 − k)V paper introduces a switched-inductor four-quadrant DC/DC

This converter, shown in Fig. 14.89, consists of three The power transfer efﬁciency is

switches, two diodes, and only one inductor L. The source voltage V 1 and load voltage V 2 (e.g. a battery or DC motor back

D = EMF) are usually constant voltages. R is the equivalent resis- = (14.315) P I kV 2 tance of the circuit, it is usually small. In this paper, V 1 > |V 2 |,

V high

(c) FIGURE 14.88 Mode D of F 2Q SI Luo-converter: (a) state-on: S 2 on; (b) state-off: D 1 on, S 2 off; and (c) input and output waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 327

I I 1 Switch S

FIGURE 14.89 Four-quadrant switched-inductor DC/DC Luo-converter.

they are supposed as +42 V and ±14 V, respectively. Therefore, During switch-off state, diode D 2 is on. In this case current i L there are four-quadrants (modes) of operation:

ﬂows through the load V 2 via the free-wheeling diode D 2 , and

it decreases.

1. Mode A: energy is transferred from source to positive Mode B is shown in Fig. 14.85. During switch-on state,

voltage load; the ﬁrst-quadrant operation, Q I ;

switch S 2 is closed. In this case the load voltage V 2 supplies the

2. Mode B: energy is transferred from positive voltage

load to source; the second-quadrant operation, Q II ;

inductor L, inductor current i L increases. During switch-off state, diode D is on, current i ﬂows through the source V

3. Mode C: energy is transferred from source to negative

2 via the diode D 1 voltage load; the third-quadrant operation, Q , and it decreases.

and load V

III

Mode C is shown in Fig. 14.87. During switch-on state,

4. Mode C: energy is transferred from negative voltage

switch S

1 load to source; the fourth-quadrant operation, Q is closed. The source voltage V 1 supplies the induc-

IV tor L, inductor current i L increases. During switch-off state, The ﬁrst-quadrant is so-called the forward motoring (Forw. diode D 2 is on. Current i L ﬂows through the load V 2 via the

Mot.) operation. V 1 and V 2 are positive, and I 1 and I 2 are free-wheeling diode D 2 , and it decreases. positive as well. The second-quadrant is so-called the forward

Mode D is shown in Fig. 14.88. During switch-on state, regenerative (Forw. Reg.) braking operation. V 1 and V 2 are switch S 2 is closed. The load voltage V 2 supplies the inductor L, positive, and I 1 and I 2 are negative. The third-quadrant is inductor current i L increases. During switch-off state, diode D 1 so-called the reverse motoring (Rev. Mot.) operation. V 1 and I 1 is on. Current i L ﬂows through the source V 1 via the diode D 1 ,

are positive, and V 2 and I 2 are negative. The fourth-quadrant and it decreases.

is so-called the reverse regenerative (Rev. Reg.) braking oper- All description of the Modes A, B, C, and D is same as in

ation. V 1 and I 2 are positive, and I 1 and V 2 are negative. Each Sections 14.10.1 and 14.10.2.

mode has two states: “on” and “off.” Usually, each state is operating in different conduction duty k. The switching period is T, where T = 1/f . The switch status is shown in Table 14.7.

14.11 Multi-quadrant ZCS

Mode A is shown in Fig. 14.84. During switch-on state,

Quasi-resonant Luo-converters

switch S 1 is closed. In this case the source voltage V 1 sup-

plies the load V 2 and inductor L, inductor current i L increases. Soft-switching converters are the fourth-generation converters. These converters are made of only inductor or capacitors. They

TABLE 14.7 Switch’s status (mentioned switches are not off) usually perform in the systems between two voltage sources:

V 1 and V 2 . Voltage source V 1 is proposed positive voltage and

Q no. State S

Source Load

1 1 2 2 3 voltage V

2 is the load voltage that can be positive or negative.

Q I , Mode A ON ON

ON 1/2 V 1 +

In the investigation, both voltages are proposed constant volt-

Forw. Mot. OFF

ON ON 1/2 I 1 +

age. Since V 1 and V 2 are constant value, the voltage transfer

Q II , Mode B ON

ON 1/2 V 1 +

gain is constant. Our interesting research will concentrate on

Forw. Reg. OFF

ON 1/2 I 1 −

the working current and the power transfer efﬁciency η. The

Q III , Mode C ON ON

ON 3/4 V 1 +

resistance R of the inductor has to be considered for the power

Rev. Mot. OFF

ON ON 3/4 I 1 +

transfer efﬁciency η calculation.

Q IV , Mode D ON

ON 3/4 V 1 +

Reviewing the papers in the literature, we can ﬁnd that most of the papers investigating the switched-component converters

Rev. Reg. OFF

ON 3/4 I 1 −

328 F. L. Luo and H. Ye are working in single-quadrant operation. Professor Luo and

TABLE 14.8 Switch’s status (the blank status means off) colleagues have developed this technique into multi-quadrant

operation. We describe these in this section and the next Mode B (QII) sections.

Switch or diode

Mode A (QI)

State-on

State-off State-on State-off

Multi-quadrant ZCS quasi-resonant Luo-converters are the

S 1 ON

fourth-generation converters. Because these converters imple-

D 1 ON

ment ZCS technique, they have the advantages of high

S 2 ON

power density, high power transfer efﬁciency, low EMI, and

D 2 ON

reasonable EMC. They have three modes: • Two-quadrant ZCS quasi-resonant DC/DC Luo-converter

in forward operation; two switches with their auxiliary components. A switch S a is • Two-quadrant ZCS quasi-resonant DC/DC Luo-converter used for two-quadrant operation. Assuming the main induc- in reverse operation;

tance is sufﬁciently large, the current i L is constant. The source • Four-quadrant ZCS quasi-resonant DC/DC Luo- voltage V 1 and load voltage V 2 are usually constant, V 1 = 42 V converter.

and V 2 = 14 V. There are two modes of operation: The two-quadrant ZCS quasi-resonant DC/DC Luo-

1. Mode A (Quadrant I): electrical energy is transferred converter in forward operation is derived for the energy

from V 1 side to V 2 side, switch S a links to D 2 ; transmission of a dual-voltage system. Both, the source and

2. Mode B (Quadrant II): electrical energy is transferred load voltages are positive polarity. It performs in the ﬁrst-

from V 2 side to V 1 side, switch S a links to D 1 .

Each mode has two states: “on” and “off.” The switch status the DC motor forward operation in motoring and regenerative of each state is shown in Table 14.8.

quadrant Q I and the second-quadrant Q II corresponding to

braking states. Mode A is a ZCS buck converter. The equivalent circuit, The two-quadrant ZCS quasi-resonant DC/DC Luo- current, and voltage waveforms are shown in Fig. 14.91. There

converter in reverse operation is derived for the energy are four time regions for the switching on and off period. The transmission of a dual-voltage system. The source voltage is conduction duty cycle is k = (t 1 +t 2 ) when the input current positive and load voltage is negative polarity. It performs in

the third-quadrant Q III and the fourth-quadrant Q IV corre-

sponding to the DC motor reverse operation in motoring and

regenerative braking states. L

i Lr1

The four-quadrant ZCS quasi-resonant DC/DC Luo- converter is derived for the energy transmission of a dual- voltage system. The source voltage is positive, and load voltage

+ can be positive or negative polarity. It performs four-quadrant

operation corresponding to the DC motor forward and reverse

operation in motoring and regenerative braking states.

14.11.1 Two-quadrant ZCS Quasi-resonant (a) Luo-converter in Forward Operation

Since both voltages are low, this converter is designed as

i Lr1

V 1 /Z 1

a ZCS quasi-resonant converter (ZCS-QRC). It is shown in

I Fig. 14.90. This converter consists of one main inductor L and L

FIGURE 14.90 Two-quadrant (QI+QII) DC/DC ZCS quasi-resonant FIGURE 14.91 Mode A operation: (a) equivalent circuit and Luo-converter.

(b) waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 329 ﬂows through the switch S 1 and inductor L. The whole period are four time regions for the switching on and off period. The

is T = (t 1 +t 2 +t 3 +t 4 ). Some formulas are listed below conduction duty cycle is k = (t 1 +t 2 ), but the output current only ﬂows through the source V 1 in the period t 4 . The whole

V 1 period is T = (t 1 +t 2 +t 3 +t 4 ). Some formulas are listed ω 1 = √

1 L r1

; i 1 −peak =I L +

L r1 C r

Z 1 below

1 L r2

; i 2 −peak =I L + (14.325) t 1 =

L Z 2 (14.326) t 2 =

1 =V 2 1 (1 + cos α 1 ) (14.321) = =sin

1 cosα 2 t 3

= I L (14.328)

V 1 (t 1 +t 2 )

V 1 cos α 1

V 2 I L Z 1 π /2 +α 1 V 2 t 4 t 4 t 1 +t 2 +t = 3 = ; t 4 = (14.329)

; T =t 1 +t 2 +t 3 +t 4 ; f =1/T (14.330)

t 1 +t 2 +t 3 +t 4 t 1 +t 2 +t 3 +t 4

Mode B is a ZCS boost converter. The equivalent circuit, current, and voltage waveforms are shown in Fig. 14.92. There

14.11.2 Two-quadrant ZCS Quasi-resonant Luo-converter in Reverse Operation

Two-quadrant ZCS quasi-resonant Luo -converter in reverse operation is shown in Fig. 14.93. It is a new soft-switching technique with two-quadrant operation, which effectively reduces

L r2

the power losses and largely increases the power transfer efﬁ-

V 2 ciency. It consists of one main inductor L and two switches −

i Lr2

with their auxiliary components. A switch S a is used for two-quadrant operation. Assuming the main inductance L is sufﬁciently large, the current i L is constant. The source voltage

= 42 V and V 2 = −28 V. There are two modes of operation:

V 1 and load voltage V 2 are usually constant, e.g. V 1

(a)

1. Mode C (Quadrant III): electrical energy is transferred i Lr2

V 1 /Z 2 from V 1 side to −V 2 side, switch S a links to D 2 ;

2. Mode D (Quadrant IV): electrical energy is transferred I L

from −V side to V

0 2 1 side, switch S a links to D 1 .

FIGURE 14.92 Mode B operation: (a) equivalent circuit and (b) wave- FIGURE 14.93 Two-quadrant (QIII+IV) DC/DC ZCS quasi-resonant forms.

Luo-converter.

330 F. L. Luo and H. Ye Each mode has two states: “on” and “off.” The switch status

I L L r1

(14.332) of each state is shown in Table 14.9.

; α 1 = sin −1

V 1 +V 2 V 1 +V 2 Mode C is a ZCS buck–boost converter. The equivalent cir-

cuit, current, and voltage waveforms are shown in Fig. 14.94.

(π +α 1 ); v CO = (V 1 −V 2 ) +V 1 sin(π/2 +α 1 ) There are four time regions for the switching on and off period.

The conduction duty cycle is kT = (t 1 +t 2 ) when the input

=V 1 (1 + cos α 1 ) −V 2 (14.333)

current ﬂows through the switch S 1 and the main inductor L.

The whole period is T = (t 1 +t 2 +t 3 +t 4 ). Some formulas

(v CO +V 2 )C r

V 1 (1 + cos α 1 )C r

are listed below

L r1 C −peak =I L +

TABLE 14.9 Switch’s status (the blank status means off)

Switch or diode Mode C (QIII)

Mode D (QIV)

Mode D is a cross ZCS buck–boost converter. The equivalent

S 1 ON

circuit, current, and voltage waveforms are shown in Fig. 14.95.

D 1 ON

There are four time regions for the switching on and off period.

S 2 ON

D 2 ON

The conduction duty cycle is kT = (t 1 +t 2 ), but the output current only ﬂows through the source V 1 in the period t 4 . The

FIGURE 14.94 Mode C operation: (a) equivalent circuit and (b) FIGURE 14.95 Mode D operation: (a) equivalent circuit and (b) waveforms.

waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 331 whole period is T = (t 1 +t 2 +t 3 +t 4 ). Some formulas are current i L remains constant. The source and load voltages are

listed below usually constant, e.g. V 1 = 42 V and V 2 = ±28 V [7–9]. There

are four modes of operation:

1. Mode A (Quadrant I): electrical energy is transferred L r2 C r

1 L r2

; i 2 −peak =I L +

Z 2 from V 1 side to V 2 side, switch S a links to D 2 ;

2. Mode B (Quadrant II): electrical energy is transferred from V 2 side to V 1 side, switch S a links to D 1 ;

3. Mode C (Quadrant III): electrical energy is transferred

V 1 +V 2 V 2 +V 2 from V 1 side to −V 2 side, switch S a links to D 2 ;

4. Mode D (Quadrant IV): electrical energy is transferred t 2 =

(π +α 2 ); v CO = (V 1 −V 2 )

ω 2 −V sin(π/2 +α 2 )

−V Each mode has two states: “on” and “off.” The switch status

from

2 side to V 1 side, switch S a links to D 1 .

=V 1 −V 2 (1 + cos α 2 )

of each state is shown in Table 14.10.

The operation of Mode A, B, C, and D is same as in the t 3 =

(V 1 −v CO )C r

V 2 (1 + cos α 2 )C r

previous Sections 14.11.1 and 14.11.2.

t 1 +t 2 V 1 +V 2 cos α 2 t 4

Z 2 π /2 +α 2 T (14.340)

14.12 Multi-quadrant ZVS Quasi-resonant Luo-converters

V 1 I L Z 2 π /2 +α 2 Multi-quadrant ZVS quasi-resonant Luo-converters are the t 1 +t 2 fourth-generation converters. Because these converters imple-

ment ZCS technique, they have the advantages of high t 1 +t 2 +t 3 +t 4 (14.342) power density, high power transfer efﬁciency, low EMI, and reasonable EMC. They have three modes:

; T =t 1 +t 2 +t 3 +t 4 ; f = 1/T

14.11.3 Four-quadrant ZCS Quasi-resonant

• Two-quadrant ZVS quasi-resonant DC/DC Luo-converter

Luo-converter

in forward operation; • Two-quadrant ZVS quasi-resonant DC/DC Luo-converter

Four-quadrant ZCS quasi-resonant Luo -converter is shown

in reverse operation;

in Fig. 14.96. Circuit 1 implements the operation in quadrants I • Four-quadrant ZVS quasi-resonant DC/DC Luo- and II, circuit 2 implements the operation in quadrants III and

converter.

IV. Circuit 1 and 2 can be converted to each other by auxiliary switch. Each circuit consists of one main inductor L and

The two-quadrant ZVS quasi-resonant DC/DC Luo- two switches. A switch S a is used for four-quadrant operation. converter in forward operation is derived for the energy Assuming that the main inductance L is sufﬁciently large, the transmission of a dual-voltage system. Both, the source and load voltages are positive polarity. It performs in the ﬁrst- quadrant Q I and the second-quadrant Q II corresponding to

D 1 2/4 S a 1/3 the DC motor forward operation in motoring and regenerative

L r1

braking states.

S 1 The two-quadrant ZVS quasi-resonant DC/DC Luo-

converter in reverse operation is derived for the energy

D 2 L r2

transmission of a dual-voltage system. The source voltage is

positive and load voltage is negative polarity. It performs in + 2

the third-quadrant Q III and the fourth-quadrant Q IV corre- V 1

sponding to the DC motor reverse operation in motoring and –

regenerative braking states.

The four-quadrant ZVS quasi-resonant DC/DC Luo-

+ – V 2 converter is derived for the energy transmission of a dual- voltage system. The source voltage is positive, and load voltage

can be positive or negative polarity. It performs four-quadrant FIGURE 14.96 Four-quadrant DC/DC ZCS quasi-resonant Luo-

operation corresponding to the DC motor forward and reverse converter.

operation in motoring and regenerative braking states.

332 F. L. Luo and H. Ye TABLE 14.10 Switch’s status (the blank status means off)

Circuit//switch or diode

Mode A (QI)

Mode B (QII)

Mode C (QIII)

Mode D (QIV)

State-off Circuit

14.12.1 Two-quadrant ZVS Quasi-resonant

DC/DC Luo-converter in Forward

Operation

Two-quadrant ZVS quasi-resonant Luo C -converter in forward r1

+ operation is shown in Fig. 14.97. It consists of one main

c1 D 2 V 2 inductor L and two switches with their auxiliary components.

Assuming the main inductance L is sufﬁciently large, the −

current i L is constant. The source voltage V 1 and load voltage

V 2 are usually constant, e.g. V 1 = 42 V and V 2 = 14 V. There

(a)

are two modes of operation:

1. Mode A (Quadrant I): electrical energy is transferred

from V 1 side to V 2 side;

v C1 Z 1 I L

2. Mode B (Quadrant II): electrical energy is transferred

V from V 1

2 side to V 1 side.

t 1 t 2 t 3 t 4 Each mode has two states: “on” and “off.” The switch status of each state is shown in Table 14.11.

Mode A is a ZVS buck converter shown in Fig. 14.98. There I t L ' 3 I L are four time regions for the switching on and off period.

I L i r01

(b)

FIGURE 14.98 Mode A operation: (a) equivalent circuit and (b) +

The conduction duty cycle is kT = (t 3 +t 4 ) when the input current ﬂows through the switch S 1 and the main inductor L.

FIGURE 14.97 Two-quadrant (QI+QII) DC/DC ZVS quasi-resonant

Luo-converter. +t +t 3 +t 4 ). Some relevant

The whole period is T = (t 1 2

formulas are listed below

TABLE 14.11 Switch’s status (the blank status means off)

Switch Mode A (QI)

Mode B (QII)

; v c1 −peak =V 1 +Z 1 I L (14.343)

=−I L cosα 1 (14.345)

14 DC/DC Conversion Technique and 12 Series Luo-converters 333

(π +α 2 ); i rO2 =I L (1 + cos α 2 ) (14.351) t 3 =

1 +t 2 +t 3 +t 4 ; f =1/T (14.348)

t 1 +t 2 +t 3 +t 4 Mode B is a ZVS boost converter shown in Fig. 14.99. There

−1 (t 2 +t 3 ) −t 1 ; (14.353)

are four time regions for the switching on and off period. The

conduction duty cycle is kT = (t 3 +t 4 ), but the output current

only ﬂows through the source V 1 in the period (t 1 t k 3 +t +t 4 2 ). The =

; T =t 1 +t 2 +t 3 +t 4 ; f = 1/T

whole period is T = (t 1 +t 2 +t 3 +t 4 ). Some relevant formulas

t 1 +t 2 +t 3 +t 4

are listed below (14.354)

14.12.2 Two-quadrant ZVS Quasi-resonant

C −peak =V 1 +Z 2 I L L r C r2 r2

; v C2

DC/DC Luo-converter in Reverse

Operation

V 1 Two-quadrant ZVS quasi-resonant Luo -converter in reverse t 1 =

operation is shown in Fig. 14.100. It consists of one main inductor L and two switches with their auxiliary components. Assuming the main inductance L is sufﬁciently large, the cur-

rent i L is constant. The source voltage V 1 and load voltage V 2 D L

are usually constant, e.g. V 1 = +42 V and V 2 = −28 V. There

are two modes of operation:

1. Mode C (Quadrant III): electrical energy is transferred

from V 1 side to −V 2 side;

V 2 2. Mode D (Quadrant IV): electrical energy is transferred −

v c2 C r2

from −V 2 side to V 1 side.

Each mode has two states: “on” and “off.” The switch status

of each state is shown in Table 14.12. Mode C is a ZVS buck–boost converter shown in Fig. 14.101. There are four time regions for the switching on and off v C2 Z I period. The conduction duty cycle is kT = (t 3 +t 4 2 ) when L

(a)

the input current ﬂows through the switch S 1 and the main V 1

inductor L. The whole period is T = (t 1 +t 2 +t 3 +t 4 ).

FIGURE 14.99 Mode B operation: (a) equivalent circuit and (b) wave- FIGURE 14.100 Two-quadrant (QIII+IV) DC/DC ZVS quasi-resonant forms.

Luo-converter.

334 F. L. Luo and H. Ye TABLE 14.12 Switch’s status (the blank status means off)

i r dt ≈ (I L t 4 ) = I L (14.358)

Switch Mode C (QIII)

Mode D (QIV)

; T =t 1 +t 2 +t 3 +t 4 ; f =1/T (14.360)

t 1 +t 2 +t 3 +t 4

D 1 Mode D is a cross ZVS buck–boost converter shown in Fig. 14.102. There are four time regions for the switching on

3 +t 4 ), C r1

D 2 and off period. The conduction duty cycle is kT = (t

1 in the +

but the output current only ﬂows through the source V i

period (t

1 +t 2 ). The whole period is T = (t 1 +t 2 +t 3 +t 4 ).

V 2 Some formulae are listed below

; v C2 −peak =V 1 +V 2 +Z 2 I L

FIGURE 14.101 Mode C operation: (a) equivalent circuit and (b) waveforms.

(a)

Some formulas are listed below

; v c1 −peak =V 1 +V 2 +Z 1 I L

L sin(π/2

FIGURE 14.102 Mode D operation: (a) equivalent circuit and (b)

V 1 +V 2 V 1 +V 2 waveforms.

14 DC/DC Conversion Technique and 12 Series Luo-converters 335

• Mode A (Quadrant I): electrical energy is transferred ω 2

(π +α 2 ); i rO2 =I L [1 + sin(π/2 + α 2 ) ] (14.363)

from V 1 side to V 2 side;

• Mode B (Quadrant II): electrical energy is transferred

i rO2 L r

I L (1 + cos α 2 )L r

from V 2 side to V 1 side;

V 1 +V 2 V 1 +V 2 • Mode C (Quadrant III): electrical energy is transferred

t 3 from V 1 side to −V 2 side;

1 t 1 +t 2 +t 3 • Mode D (Quadrant IV): electrical energy is transferred

i r dt ≈

from −V 2 side to V 1 side.

Each mode has two states: “on” and “off.” The switch status

4 of each state is shown in Table 14.13.

i r dt ≈ (I L t 4 ) = I L ;

The description of Modes A, B, C, and D is same as in the t 3 previous Sections 14.12.1 and 14.12.2.

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