14.2.1 Fundamental Topologies

Buck converter is a step-down converter, which is shown in Fig. 14.2a, the equivalent circuits during switch-on and -off periods are shown in Figs. 14.2b and c. Its output voltage and output current are

(14.1)

and

(14.2)

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

This converter may work in discontinuous mode if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high.

Boost converter is a step-up converter, which is shown in Fig. 14.3a, the equivalent circuits during switch-on and -off periods are shown in Figs. 14.3b and c. Its output voltage and current are

(14.3)

and

(14.4)

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

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

Buck–boost converter is a step–down/up converter, which is shown in Fig. 14.4a, the equivalent circuits during switch-on and -off periods are shown in Figs. 14.4b and c. Its output voltage and current are

(14.5)

and

(14.6)

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

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

14.2.2 Developed Topologies

For convenient applications, all developed converters have output voltage and current as

(14.7)

and

(14.8)

Positive output (P/O) Luo-converter is a step-down/up converter, and is shown in Fig. 14.5. This converter may work in discontinuous mode if the frequency f is small, k is small, and inductance L is small.

FIGURE 14.5 Positive output Luo-converter.

Negative output (N/O) Luo-converter is shown in Fig. 14.6. This converter may work in discontinuous mode if the frequency f is small, k is small, inductance L is small, and load current is high.

FIGURE 14.6 Negative output Luo-converter.

Double output Luo-converter is a double output step-down/up converter, which is derived from P/O Luo-converter and N/O Luo-converter. It has two conversion paths and two output voltages V_o+ and V_o− It is shown in Fig. 14.7. If the components are carefully selected the output voltages and currents (concentrate the absolute value) obtained are

(14.9)

and

(14.10)

FIGURE 14.7 Double output Luo-converter.

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 is small, and load current is high.

CÛk-converter is a negative output step-down/up converter, which is derived from boost and buck converters. It is shown in Fig. 14.8.

FIGURE 14.8 Cúk converter.

Single-ended primary inductance converter is a positive output step-down/up converter, which is derived from boost converters. It is shown in Fig. 14.9.

FIGURE 14.9 SEPIC.

14.2.3 Transformer-type Topologies

All transformer-type converters have transformer(s) to isolate the input and output voltages. Therefore, it is easy to obtain the high or low output voltage by changing the turns ratio N, the positive or negative polarity by changing the winding direction, and multiple output voltages by setting multiple secondary windings.

Forward converter is a step-up/down converter, which is shown in Fig. 14.10. The transformer turns ratio is N (usually N > 1). If the transformer has never been saturated during operation, it works as a buck converter. The output voltage and current are

(14.11)

and

(14.12)

FIGURE 14.10 Forward converter.

This converter may work in discontinuous mode if the frequency 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 converters, a tertiary winding is applied. The corresponding circuit diagram is shown in Fig. 14.11.

FIGURE 14.11 Forward converter with tertiary winding.

To obtain multiple output voltages we can set multiple secondary windings. The corresponding circuit diagram is shown in Fig. 14.12.

FIGURE 14.12 Forward converter with multiple secondary windings.

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

(14.13)

and

(14.14)

FIGURE 14.13 Push-pull converter.

This converter may work in discontinuous mode if the frequency f is small, conduction duty k is small, inductance L is 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 in fly-back operation to obtain high surge voltage induced, then get high output voltage. It works likely in buck–boost operation as a buck–boost converter. Its output voltage and current are

(14.15)

and

(14.16)

FIGURE 14.14 Fly-back converter.

Half-bridge converter is a step-up converter, which is shown in Fig. 14.15. There are two switches and one double secondary coils transformer required. The transformer turns ratio is N. It works as a half-bridge rectifier (half of V₁ 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 voltage and current are

(14.17)

and

(14.18)

FIGURE 14.15 Half-bridge converter.

This converter may work in discontinuous mode if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high.

Bridge converter is a step-up converter, which is shown in Fig. 14.16. There are four switches and one double secondary coils transformer required. The transformer turns ratio is N. It works as a full-bridge rectifier (full V₁ 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 voltage and current are

(14.19)

and

(14.20)

FIGURE 14.16 Bridge converter.

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 (L₁) plus a buck–boost converter plus a low-pass filter (L₂ –C₂). Its output voltage and current are

(14.21)

and

(14.22)

FIGURE 14.17 ZETA (zeta) converter.

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

Because of the effect of the parasitic elements, the voltage conversion gain is limited. Especially, when the conduction duty k is towards unity, the output voltage is sharply reduced. Voltage-lift technique is a popular method used in electronic circuit design. Applying this technique can effectively overcome the effect of the parasitic elements, and largely increase the voltage transfer gain. In this section, we introduce seven self-lift converters which are working in continuous mode.

All self-lift converters (except enhanced self-lift circuit) have the output voltage and current to be

(14.23)

and

(14.24)

The voltage transfer gain in continuous mode is

(14.25)

P/O self-lift Luo-converter is shown in Fig. 14.18. The variation ratio of the output voltage v_O in continuous conduction mode (CCM) is

(14.26)

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

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

(14.27)

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

N/O self-lift Luo-converter is shown in Fig. 14.20. The Variation ratio of the output voltage v_o in CCM is

(14.28a)

FIGURE 14.20 N/O self-lift Luo-converter.

Reverse N/O self-lift Luo-converter is shown in Fig. 14.21. The variation ratio of the output voltage v_O in CCM is

(14.28b)

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

Self-lift Cúk-converter is shown in Fig. 14.22. The variation ratio of the output voltage v_O in CCM is

(14.28c)

FIGURE 14.22 Self-lift Cúk-converter.

Self-lift SEPIC is shown in Fig. 14.23. The variation ratio of the output voltage v_O in CCM is

(14.28d)

FIGURE 14.23 Self-lift SEPIC.

Enhanced self-lift Luo-converter is shown in Fig. 14.24. Its output voltage and current are

(14.29)

and

(14.30)

FIGURE 14.24 Enhanced self-lift Luo-converter.

The voltage transfer gain in continuous mode is

(14.31)

The variation ratio of the output voltage v_o in CCM is as in Eq. (14.26)

14.2.5 Tapped Inductor (Watkins–Johnson) Converters

Tapped inductor (Watkins-Johnson) converters have been derived from fundamental converters, which circuit diagrams are shown in Table 14.1. The voltage transfer gains are shown in Table 14.2. Here the tapped inductor ratio is n = n1/(n1 + n2).

TABLE 14.1 The circuit diagrams of the tapped inductor fundamental converters

TABLE 14.2 The voltage transfer gains of the tapped inductor fundamental converters

14.3.1 Positive Output Luo-converters

Positive output (P/O) Luo-converters perform the voltage conversion from positive to positive voltages using the voltage lift technique. They work in the first-quadrant with large voltage amplification. Their voltage transfer gains are high. Five circuits are introduced in the literature. They are:

Further lift circuits can be derived from the above circuits. In all P/O Luo-Converters, we define normalized inductance L = L₁ L₂ /L₁ + L₂ ) and normalized impedance z_N = R/fL.

P/O Luo-converter elementary circuit is shown in Fig. 14.25a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.25b and c. Its output voltage and current are

and

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

(14.32)

The variation ratio of the output voltage v_o in CCM is

(14.33)

This converter may work in discontinuous conduction mode if the frequency f is small, conduction duty k is small, inductance L is small, and load current is high. The condition for discontinuous conduction mode (DCM) is

(14.34)

The output voltage in DCM is

(14.35)

P/O Luo-converter self-lift circuit is shown in Fig. 14.26a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.26b and c. Its output voltage and current are

and

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

The voltage transfer gain in continuous mode is

(14.36)

The variation ratio of the output voltage v_O in CCM is

(14.37)

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

(14.38)

The output voltage in DCM is

(14.39)

P/O Luo-converter re-lift circuit is shown in Fig. 14.27a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.27b and c. Its output voltage and current are

and

FIGURE 14.27 P/O Luo-converter re-lift circuit: (a) circuit diagram; (b) switch on; and (c) switch off.

The voltage transfer gain in CCM is

(14.40)

The variation ratio of the output voltage v_O in CCM is

(14.41)

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

(14.42)

The output voltage in DCM is

(14.43)

P/O Luo-converter triple-lift circuit is shown in Fig. 14.28a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.28b and c. Its output voltage and current are

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

and

The voltage transfer gain in CCM is

(14.44)

The variation ratio of the output voltage v_O in CCM is

(14.45)

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

(14.46)

The output voltage in DCM is

(14.47)

P/O Luo-converter quadruple-lift circuit is shown in Fig. 14.29a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.29b and c. Its output voltage and current are

and

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

The voltage transfer gain in CCM is

(14.48)

The variation ratio of the output voltage v_O in CCM is

(14.49)

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

(14.50)

The output voltage in DCM is

(14.51)

Summary for all P/O Luo-converters:

To write common formulas for all circuits parameters, we define that subscript j = 0 for the elementary circuit, j = 1 for the self-lift circuit, j = 2 for the re-lift circuit, j = 3 for the triple-lift circuit, j = 4 for the quadruple-lift circuit, and so on. The voltage transfer gain is

(14.52)

The variation ratio of the output voltage is

(14.53)

The condition for discontinuous conduction mode is

(14.54)

The output voltage in discontinuous conduction mode is

(14.55)

where

(14.56)

is the Hong function.

14.3.2 Simplified Positive Output (S P/O) Luo-converters

Carefully check P/O Luo-converters, we can see that there are two switches required from re-lift circuit. In order to use only one switch in all P/O Luo-converters, we modify the circuits. In this section we introduce following four circuits:

Further lift circuits can be derived from the above circuits. In all S P/O Luo-converters, we define normalized impedance z_N = R/fL.

S P/O Luo-converter self-lift circuit 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

and

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

The voltage transfer gain in CCM is

(14.57)

The variation ratio of the output voltage v_O in CCM is

(14.58)

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

(14.59)

The output voltage in DCM is

(14.60)

S P/O Luo-converter re-lift circuit is shown in Fig. 14.31a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.31b and c. Its output voltage and current are

and

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

The voltage transfer gain in CCM is

(14.61)

The variation ratio of the output voltage v_O in CCM is

(14.62)

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

(14.63)

The output voltage in DCM is

(14.64)

S P/O Luo triple-lift circuit is shown in Fig. 14.32a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.32b and c. Its output voltage and current are

and

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

The voltage transfer gain in CCM is

(14.65)

The variation ratio of the output voltage v_O in CCM is

(14.66)

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

(14.67)

The output voltage in DCM is

(14.68)

S P/O Luo quadruple-lift circuit is shown in Fig. 14.33a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.33b and c. Its output voltage and current are

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

and

The voltage transfer gain in CCM is

(14.69)

The variation ratio of the output voltage v_O in CCM is

(14.70)

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

(14.71)

The output voltage in DCM is

(14.72)

Summary for all S P/O Luo-converters:

To write common formulas for all circuits parameters, we define that subscript j = 1 for the self-lift circuit, j = 2 for the re-lift circuit, j = 3 for the triple-lift circuit, j = 4 for the quadruple-lift circuit, and so on. The voltage transfer gain is

(14.73)

The variation ratio of the output voltage is

(14.74)

The condition for discontinuous mode is

(14.75)

The output voltage in discontinuous mode is

(14.76)

14.3.3 Negative Output Luo-converters

Negative output (N/O) Luo-converters perform the voltage conversion from positive to negative voltages using the voltage-lift technique. They work in the third-quadrant with large voltage amplification. Their voltage transfer gains are high. Five circuits are introduced in the literature. They are:

Further lift circuits can be derived from above circuits. In all N/O Luo-converters, we define normalized impedance z_N = R/fL.

N/O Luo-converter elementary circuit is shown in Fig. 14.34a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.34b and c. Its output voltage and current (the absolute value) are

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

and

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

The voltage transfer gain in CCM is

(14.77)

The variation ratio of the output voltage v_O in CCM is

(14.78)

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

(14.79)

The output voltage in DCM is

(14.80)

N/O Luo-converter self-lift circuit is shown in Fig. 14.35a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.35b and c. Its output voltage and current (the absolute value) are

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

and

The voltage transfer gain in CCM is

(14.81)

The variation ratio of the output voltage v_O in CCM is

(14.82)

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

(14.83)

The output voltage in DCM is

(14.84)

N/O Luo-converter re-lift circuit is shown in Fig. 14.36a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.36b and c. Its output voltage and current (the absolute value) are

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

and

The voltage transfer gain in CCM is

(14.85)

The variation ratio of the output voltage v_O in CCM is

(14.86)

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

(14.87)

The output voltage in DCM is

(14.88)

N/O Luo-converter triple-lift circuit is shown in Fig. 14.37a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.37b and c. Its output voltage and current (the absolute value) are

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

and

The voltage transfer gain in CCM is

(14.89)

The variation ratio of the output voltage v_O in CCM is

(14.90)

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

(14.91)

The output voltage in DCM is

(14.92)

N/O Luo-converter quadruple-lift circuit is shown in Fig. 14.38a. The equivalent circuits during switch-on and -off periods are shown in Figs. 14.38b and c. Its output voltage and current (the absolute value) are

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

and

The voltage transfer gain in CCM is

(14.93)

The variation ratio of the output voltage v_O in CCM is

(14.94)

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

(14.95)

The output voltage in DCM is

(14.96)

Summary for all N/O Luo-converters:

To write common formulas for all circuits parameters, we define that subscript j = 0 for the elementary circuit, j = 1 for the self-lift circuit, j = 2 for the re-lift circuit, j = 3 for the triple-lift circuit, j = 4 for the quadruple-lift circuit, and so on. The voltage transfer gain is

(14.97)

The variation ratio of the output voltage is

(14.98)

The condition for discontinuous conduction mode is

(14.99)

The output voltage in discontinuous conduction mode is

(14.100)

where

is the Hong function.

14.5.1 P/O Super-lift Luo-converters

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

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

P/O SL Luo-converter elementary circuit is shown in Fig. 14.43a. The equivalent circuits during switch on and switch off are shown in Figs. 14.43b and c. Its output voltage and current are

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

and

The voltage transfer gain is

(14.131)

The variation ratio of the output voltage v_O is

(14.132)

P/O SL Luo-converter re-lift circuit is shown in Fig. 14.44a. The equivalent circuits during switch on and switch off are shown in Figs. 14.44b and c. Its output voltage and current are

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

and

The voltage transfer gain is

(14.133)

The variation ratio of the output voltage v_O is

(14.134)

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

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

and

The voltage transfer gain is

(14.135)

The variation ratio of the output voltage v_O is

(14.136)

P/O SL Luo-converter additional circuit is shown in Fig. 14.46a. The equivalent circuits during switch on and switch off are shown in Figs. 14.46b and c. Its output voltage and current are

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

and

The voltage transfer gain is

(14.137)

The variation ratio of the output voltage v_O is

(14.138)

P/O SL Luo-converter additional re-lift circuit is shown in Fig. 14.47a. The equivalent circuits during switch on and switch off are shown in Figs. 14.47b and c. Its output voltage and current are

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

and

The voltage transfer gain is

(14.139)

The variation ratio of the output voltage v_O is

(14.140)

P/O SL Luo-converter additional triple-lift circuit is shown in Fig. 14.48a. The equivalent circuits during switch on and switch off are shown in Figs. 14.48b and c. Its output voltage and current are

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

and

The voltage transfer gain is

(14.141)

The variation ratio of the output voltage v_O is

(14.142)

14.5.2 N/O Super-lift Luo-converters

There are several subseries of N/O Super-lift Luo-converters:

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

N/O SL Luo-converter elementary circuit is shown in Fig. 14.49. Its output voltage and current are

FIGURE 14.49 N/O SL Luo-converter elementary circuit.

and

The voltage transfer gain is

(14.143)

The variation ratio of the output voltage v_O is

(14.144)

N/O SL Luo-converter re-lift circuit is shown in Fig. 14.50. Its output voltage and current are

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

and

The voltage transfer gain is

(14.145)

The variation ratio of the output voltage v_O is

(14.146)

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

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

and

The voltage transfer gain is

(14.147)

The variation ratio of the output voltage v_O is

(14.148)

N/O SL Luo-converter additional circuit is shown in Fig. 14.52. Its output voltage and current are

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

and

The voltage transfer gain is

(14.149)

The variation ratio of the output voltage v_O is

(14.150)

NO SL Luo-converter additional re-lift circuit is shown in Fig. 14.53. Its output voltage and current are

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

and

The voltage transfer gain is

(14.151)

The variation ratio of the output voltage v_O is

(14.152)

N/O SL Luo-converter additional triple-lift circuit is shown in Fig. 14.54. Its output voltage and current are

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

and

The voltage transfer gain is

(14.153)

The variation ratio of the output voltage v_O is

(14.154)

14.5.3 P/O Cascade Boost-converters

There are several subseries of P/O cascade boost-converters (CBC):

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

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

FIGURE 14.55 P/O CBC elementary circuit.

and

The voltage transfer gain is

(14.155)

The variation ratio of the output voltage v_O is

(14.156)

P/O CBC two-stage circuit is shown in Fig. 14.56. Its output voltage and current are

FIGURE 14.56 P/O CBC two-stage circuit.

and

The voltage transfer gain is

(14.157)

The variation ratio of the output voltage v_O is

(14.158)

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

FIGURE 14.57 P/O CBC three-stage circuit.

and

The voltage transfer gain is

(14.159)

The variation ratio of the output voltage v_O is

(14.160)

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

FIGURE 14.58 P/O CBC additional circuit.

and

The voltage transfer gain is

(14.161)

The variation ratio of the output voltage v_O is

(14.162)

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

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

and

The voltage transfer gain is

(14.163)

The variation ratio of the output voltage v_O is

(14.164)

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

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

and

The voltage transfer gain is

(14.165)

The variation ratio of the output voltage v_O is

(14.166)

14.5.4 N/O Cascade Boost-converters

There are several subseries of N/O CBC:

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

N/O CBC elementary circuit is shown in Fig. 14.61. Its output voltage and current are

FIGURE 14.61 N/O CBC elementary circuit.

and

The voltage transfer gain is

(14.167)

The variation ratio of the output voltage v_O is

(14.168)

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

FIGURE 14.62 N/O CBC two-stage circuit.

and

The voltage transfer gain is

(14.169)

The variation ratio of the output voltage v_O is

(14.170)

N/O CBC three-stage circuit is shown in Fig. 14.63. Its output voltage and current are

FIGURE 14.63 N/O CBC three-stage circuit.

and

The voltage transfer gain is

(14.171)

The variation ratio of the output voltage v_O is

(14.172)

N/O CBC additional circuit is shown in Fig. 14.64. Its output voltage and current are

FIGURE 14.64 N/O CBC additional circuit.

and

The voltage transfer gain is

(14.173)

The variation ratio of the output voltage v_O is

(14.174)

N/O CBC additional two-stage circuit is shown in Fig. 14.65. Its output voltage and current are

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

and

The voltage transfer gain is

(14.175)

The variation ratio of the output voltage v_O is

(14.176)

N/O CBC additional three-stage circuit is shown in Fig. 14.66. Its output voltage and current are

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

and

The voltage transfer gain is

(14.177)

The variation ratio of the output voltage v_O is

(14.178)

14.6.1 Continuous Conduction Mode

Referring to Figs. 14.67b and c, we have got the current i_L1 increases with the slope +V₁/L₁ during switch on, and decreases with the slope –V₁/L₁ during switch off. In the steady state, the current increment is equal to the decrement in a whole period T. The relation below is obtained

(14.180)

Thus,

(14.181)

The current i_L2 increases with the slope +(V_i –V₁)/L₂ during switch on, and decreases with the slope –(V₁ –V_O)/L₂ during switch off. In the steady state, the current increment is equal to the decrement in a whole period T. We obtain the relation below

(14.182)

(14.183)

(14.184)

The voltage transfer gain is

(14.185)

From Eq. (14.185) we can see that the voltage transfer gain of UL Luo-converter is very high which is the product of those of VL Luo-converter and SL Luo-converter. We list the transfer gains of various converters in Table 14.3 for reference.

TABLE 14.3 Comparison of various converters gains

The variation of inductor current i_L1 is

(14.186)

and its variation ratio is

(14.187)

The variation of inductor current i_L1 is

(14.188)

and its variation ratio is

(14.189)

The variation of capacitor voltage v_C1 is

(14.190)

and its variation ratio is

(14.191)

The variation of capacitor voltage v_C2 is

(14.192)

and its variation ratio is

(14.193)

From the analysis and calculations, we can see that all variations are very small. A design example is that V_i = 10V, L₁ = L₂ = 1 mH, C₁ = C₂ = 1, μ F, R = 3000 ω, f = 50 kHz, and conduction duty cycle k varies from 0.1 to 0.9. We then obtain the output voltage variation ratio ε, which is less than 0.003. The output voltage is very smooth DC voltage nearly no ripple.

14.6.2 Discontinuous Conduction Mode

Referring to Fig. 14.67d, we have got the current i_L1 decreases to zero before t = T, i.e. the current becomes zero before next time the switch turns on. The DCM operation condition is defined as

or

(14.194)

The normalized impedance z_N is,

(14.195)

We define the filling factor m to describe the current exists time. For DCM operation, 0 < m = 1,

(14.196)

Thus,

(14.197)

We finally obtain the relation below

(14.198)

(14.199)

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

(14.200)

14.7.1 Forward Two-quadrant DC/DC Luo-converter

Forward Two-quadrant (F 2Q) Luo-converter is shown in Fig. 14.68. The source voltage (V₁) and load voltage (V₂) are usually considered as constant voltages. The load can be a battery or motor back electromotive force (EMF). For example, the source voltage is 42 V and load voltage is + 14V. There are two modes of operation:

FIGURE 14.68 Forward two-quadrant operating Luo-converter.

Mode A: The equivalent circuits during switch-on and -off periods are shown in Figs. 14.69a and b. The typical output voltage and current waveforms are shown in Fig. 14.69c. We have the output current I₂ as

(14.201)

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

and

(14.202)

The minimum conduction duty k corresponding to I₂ = 0 is

(14.203)

The power transfer efficiency is

(14.204)

The variation ratio of capacitor voltage v_C is

(14.205)

The variation ratio of inductor current i_L1 is

(14.206)

The variation ratio of inductor current i_L2 is

(14.207)

The variation ratio of diode current i_D2 is

(14.208)

If the diode current becomes zero before S₁ switch on again, the converter works in discontinuous region. The condition is

(14.209)

Mode B: The equivalent circuits during switch-on and -off 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 I₁ as

(14.210)

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

and

(14.211)

The minimum conduction duty k corresponding to I₁ = 0 is

(14.212)

The power transfer efficiency

(14.213)

The variation ratio of capacitor voltage v_C is

(14.214)

The variation ratio of inductor current i_L1 is

(14.215)

The variation ratio of inductor current i_L2 is

(14.216)

The variation ratio of diode current i_D1 is

(14.217)

If the diode current becomes zero before S₂ switch on again, the converter works in discontinuous region. The condition is

(14.218)

14.7.2 Two-quadrant DC/DC Luo-converter in Reverse Operation

Reverse two-quadrant operating (R 2Q) Luo-converter is shown in Fig. 14.71, and it consists of two switches with two passive diodes, two inductors and one capacitor. The source voltage (V₁) and load voltage (V₂) are usually considered as constant voltages. The load can be a battery or motor back EME For example, the source voltage is 42 V and load voltage is − 14 V. There are two modes of operation:

FIGURE 14.71 Reverse two-quadrant operating Luo-converter.

Mode C: The equivalent circuits during switch-on and -off periods are shown in Figs. 14.72a and b. The typical output voltage and current waveforms are shown in Fig. 14.72c. We have the output current I₂ as

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

(14.219)

and

(14.220)

The minimum conduction duty k corresponding to I₂ = 0 is

(14.221)

The power transfer efficiency is

(14.222)

The variation ratio of capacitor voltage v_C is

(14.223)

The variation ratio of inductor current i_L1 is

(14.224)

The variation ratio of inductor current i_D2 is

(14.225)

The variation ratio of inductor current i_L2 is

(14.226)

If the diode current becomes zero before S₁ switch on again, the converter works in discontinuous region. The condition is

(14.227)

Mode D: The equivalent circuits during switch-on and -off 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 I₁ as

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

(14.228)

and

(14.229)

The minimum conduction duty k corresponding to I₁ = 0 is

(14.230)

The power transfer efficiency is

(14.231)

The variation ratio of capacitor voltage v_C is

(14.232)

The variation ratio of inductor current i_L1 is

(14.233)

And the variation ratio of inductor current i_D1 is

(14.234)

The variation ratio of inductor current i_L2 is

(14.235)

If the diode current becomes zero before S₂ switch on again, the converter works in discontinuous region. The condition is

(14.236)

14.7.3 Four-quadrant DC/DC Luo-converter

Four-quadrant DC/DC Luo-converter is shown in Fig. 14.74, which consists of two switches with two passive diodes, two inductors, and one capacitor. The source voltage (V₁ ) and load voltage (V₂) are usually considered as constant voltages. The load can be a battery or motor back EMF. For example, the source voltage is 42 V and load voltage is ± 14 V. There are four modes of operation:

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

Each mode has two states: “on” and “off.” Usually, each state is operating in different conduction duty k. The switches are the power MOSFET devices. The circuit 1 in Fig. 14.74 implements Modes A and B, and the circuit 2 in Fig. 14.74 implements Modes C and D. Circuits 1 and 2 can changeover by auxiliary switches (not in the figure).

Mode A: During state-on switch S₁ is closed, switch S₂ and diodes D₁ and D₂ are not conducted. In this case inductor currents i_L1 and i_L2 increase, and i₁ = i_L1 + i_L2. During state-off switches S₁, S₂, and diode D₁ are off and diode D₂ is conducted. In this case current i_L1 flows via diode D₂ to charge capacitor C, in the meantime current i_L2 is kept to flow through load battery V₂. The free-wheeling diode current i_D1 = i_L1 + i_L2. Mode A implements the characteristics of the buck–boost conversion.

Mode B: During state-on switches S₂ is closed, switch S₁ and diodes D₁ and D₂ are not conducted. In this case inductor current i_L2 increases by biased V₂, inductor current i_L1 increases by biased V_C Therefore capacitor voltage V_C reduces. During state-off switches S₁, S₂, and diode D₂ are not on, and only diode D₁ is on. In this case source current i₁ = i_L1 + i_L2 which is a negative value to perform the regenerative operation. Inductor current i_L2 flows through capacitor C, it is charged by current i_L2. After capacitor C, i_L2 then flows through the source V₁. Inductor current i_L1 flows through the source V₁ as well via diode D₁. Mode B implements the characteristics of the boost conversion.

Mode C: During state-on switch S₁ is closed, switch S₂ and diodes D₁ and D₂ are not conducted. In this case inductor currents i_L1 and i_L2 increase, and i₁ = i_L1 During state-off switches S₁, S₂, and diode D₁ are off and diode D₂ is conducted. In this case current i_L1 flows via diode D₂ to charge capacitor C and the load battery V₂ via inductor L₂. The free-wheeling diode current i_D2 = i_L1 = i_C + i₂ - Mode C implements the characteristics of the buck–boost conversion.

Mode D: During state-on switches S2 is closed, switch S₁ and diodes D₁ and D₂ are not conducted. In this case inductor current i_L1 increases by biased V_2, inductor current i_L2 decreases by biased (V₂ –V_C). Therefore capacitor voltage V_C reduces. Current i_L1 = i_C-on + i₂. During state-off switches S₁, S₂, and diode D₂ are not on, and only diode D₁ is on. In this case source current i₁ = i_L1 which is a negative value to perform the regenerative operation. Inductor current i₂ flows through capacitor C that is charged by current i₂, i.e. i_C-off = i₂ Mode D implements the characteristics of the boost conversion.

Summary: The switch status is shown in Table 14.4.

TABLE 14.4 Switch's status (the blank status means OFF)

The operation of all modes A, B, C, and D is same to the description in Sections 14.7.1 and 14.7.2.

14.8.1 Two-quadrant Switched-capacitor DC/DC Luo-converter

This converter is shown in Fig. 14.75. It consists of nine switches, seven diodes, and three capacitors. The high source voltage V_H and low load voltage V_L are usually considered as constant voltages, e.g. the source voltage is 48 V and load voltage is 14 V There are two modes of operation: