Organization of the Thesis

First of all, various SWPWM are studied in Chapter 2 and Chapter 3 and several SWPWM types will be recommended. Then, the comparison between two back-EMF detecting circuits will be finished in Chapter 4 and it follows that BEDC-II will be used in the following sensorless control. Afterward, twelve-step BDCM sensorless control will be presented in Chapter 5.

CHAPTER 2

SIX-STEP SQUARE-WAVE PWM (SWPWM)

2.1 Introduction

This chapter studies all the possible six-step square-wave PWM (SWPWM) schemes and classifies them according to the gate power loss and thermal distribution. Then, we select best ones of all 16 types for

120

^° SWPWM schemes and 64 types for

180

^° SWPWM schemes as the recommended ones implemented and integrated in SOC. The provided experimental results also confirm the selections.

The simplified schematic of three-phase six-switch voltage source inverter (VSI) is shown in Fig. 2.1 where three-phase balanced resistances R are connected to it. The supply voltage magnitude of individual gate driving circuit is

V . The input capacitances

_GS

C of power

_iss switches are also plotted in order to represent the power loss due to the transition of gate voltage. The neutral voltage

V of balanced three-phase impedances can be expressed as an

_n average value of the three terminal voltages

V ,

_a

V , and

_b

V .

_c

There are two common six-step square-wave modulations (SWM) for six-switch VSI. One is

120

^° modulation and the other is

180

^° modulation. The simplest way of adjusting output voltage magnitude is to change the dc-link voltage

V (i.e. using pulse amplitude

_d modulation PAM). However, in most cases, the dc-link voltage

V is not controllable.

_d Therefore, in the applications of uncontrolled dc-link-voltage, SWM should be work together with pulse width modulation (PWM) to become square-wave pulse-width-modulation (SWPWM) in order to generate the controllable output voltage. It follows that the switching signals G_T1 ~

G may keep discontinuous “on” with duty ratio D between any two

_T6 commutation instants.

With the PWM frequency

f

_PWM and the desired fundamental frequency f₁, the frequency modulation ratio

m is defined as

_f

f1

m_f

=

f^PWM (2.4)

Due to the input capacitance

C of power switch, any transition (0-to-

_iss

V as shown in

_GS Fig. 2.1) in the switch gate voltage would consume some power even when the power switch

Vd

Fig. 2.1. Simplified schematic of three-phase six-switch VSI.

does not conduct any current. Depending on the various gate signal generating schemes, the

output transition rate can be slower than the PWM frequency

f

_PWM . To better represent this behavior, the effective PWM transition rate

α

_{PWM ,x} experienced per fundamental cycle will be introduced. Then, the average power loss due to input capacitance

C becomes

_iss

x

In addition, the power switch loss

P

_S,Tx can be represented as the average value of the multiplication product of switch current

i and voltage

_Tx

v

_Tx

= ∫

¹

With the thermal consideration, the induced switching loss occurring in the upper switch and the lower switch equally is better than the switching loss is poured to either upper switch or the lower switch. In order to represent the loss distribution, the thermal distribution factor

TD

F

G_, of gate driving power loss and the thermal distribution factor

F

_S,TD of power switch loss are defined in the followings,

4 six commutations in each periodic cycle. The available six conduction patterns are tabulated in Table 2.1. We can find that at any instant, only two switches conducts. Therefore, a

60

^°

dead-time exists between the switching signals of upper switch and the lower switch of the same leg, which avoids the occurrence of short of dc voltage.

Table 2.1. Conduction pattern in common

120

^° SWPWM Pattern Duration Conducting devices

1 60^° T1 T₂

From Table 2.1, it is clear that

120

^° conduction scheme in each switch is composed of two neighbor conduction patterns. There are total four conduction patterns involved in the two switching signals of each leg. In order to apply SWPWM to adjust the output voltage, the conduction state in each conduction pattern is not always continuous “on”. It may be either continuous “on” or discontinuous “on” (i.e. PWM “on”). Consequently, it follows that there are

2

⁴

= 16

possible types in

120

^° SWPWM. The definition of

120

^° SWPWM scheme is switching signals are always continuous “on”. For

120

^o SWPWM type-01_01, all switching signals, the output voltage and current waveforms are plotted in Fig. 2.3 [5]. We can find that

each switching signal holds the voltage level at the first

60

^° period and then alternates for triple harmonics (3, 9, 15 ...) in v¹²⁰_ab ^° are zeros. By calculation, the fundamental component

°

Obviously, the fundamental output voltage in

120

^o SWPWM type 01_01 can be linearly controlled from near zero to its maximum value 3

V

_d /

π

by adjusting the duty ratio D.

However, not all

120

^° SWPWM types possess the same linear equation in (2.12). Fig. 2.4

plots the output voltage waveforms of all

120

^° SWPWM types. Obviously, in types 00_00, 00_01, 00_10, 01_00, 01_10, 10_00, and 10_01, their fundamental output voltages are not always linear with duty ratio D. Additionally, the other types 00_11, 01_01, 01_11, 10_10, 10_11, 11_00, 11_01, 11_10 and 11_11 possess the same controllable output voltage as (2.12) and they are named linear

120

^° SWPWM schemes.

According to (2.11), the hth-order harmonic voltage can be expressed as:

h DV V

V_abh h _d ^ab

°

= =

°

120 1 , 120,

3 1

π

^h

^{= r 6 ± 1}

^;integer

^r

^{≥1 (2.13)} Equation (2.13) shows that nonzero hth-order harmonic component

V

_ab¹²⁰_,h^° increases with PWM duty ratio D and the fundamental voltage

V

_ab¹²⁰_,1^°. However, (2.13) holds only when its

harmonic order h is significantly lower than the frequency modulation ratio

m . As listed in

_f Table 2.2, the other harmonics centered around the PWM frequency

f

_PWM and its multiples do not increase with duty ratio D. Fig. 2.5 shows the output voltage harmonic spectrum with duty ratioD

= 0.5

.

.

Fig. 2.3. Illustrated waveforms for

120

^o SWPWM type-01_01 [5].

Fig. 2.4. The output voltage v¹²⁰_ab^° of all

120

^° SWPWM schemes.

Table 2.2. Harmonic components in linear

120

^° SWPWM and linear

180

^° SWPWM 7 0.0142 0.0285 0.0428 0.0571 0.0714 0.0857 0.1 0.1142 0.1285 0.1428 11 0.0091 0.0182 0.0273 0.0364 0.0455 0.0545 0.0636 0.0727 0.0818 0.0909 13 0.0077 0.0154 0.0231 0.0308 0.0385 0.0462 0.0538 0.0615 0.0692 0.0769

±1

From Fig. 2.5, we can clearly find that the harmonics due to PWM appear as sidebands centered around the PWM frequency

f

_pwm =

m

_f

f

₁ and its multiples. The main difference between the voltage spectrums of SWPWM and the common sinusoidal pulse width modulation (SPWM) is the nonzero subharmonics in SWPWM.

1

Moreover, due to the input capacitance

C of the power switch, the more time the

_iss switching signals take to change for PWM, the more driving loss the circuit can yield. The effective PWM transition rate of upper switching signals G_T1,

G ,

_T3

G and lower

_T5 represented by

)

α

PWM . In fact, these four types are often found in the literatures [4-9]. But the other types with higher value

α

¹²⁰_PWM^° had not yet found in any literatures.

Additionally, it is noted that based on the thermal consideration, the distribution of power switch loss in upper switch and the lower switch is another important issue. From the voltages and the currents of the switches as shown in Fig. 2.3, we can find that in type-10_10, the

switch power loss

P

_S,T1 and

P

_S,T4 are equal to each other.

On the other hand, from the illustrated waveforms of type-11_00 plotted in Fig. 2.6, all the switching loss occurs only in the upper switch. Therefore, not all the linear

120

^° SWPWM types possess good thermal distribution. The thermal distribution factors

F

_G,TD

and

F

_S,TD defined in (2.7) and (2.8) of all linear

120

^° SWPWM types can be found in Table 2.3. Zero thermal distribution factors means that the thermal distribution due to power loss is even. It follows that no hot spot can be found in the main power circuit of small-sized motor drivers by using the linear

120 °

SWPWM with zero thermal distribution factors,

Therefore, it follows that type-01_01 and type-10_10 are the best types of

120

^° SWPWM.

Table 2.3. Loss and thermal characteristics of various linear

120 °

SWPWM

type

120o

α

PWM

F

_G¹²⁰_,TD^o vT1

×

iT1 v_T4

×

i_T4

F

_S¹²⁰_,TD^o

01_01 1 0 0

10_10 1 0 0

00_11 1 1 1

11_00 1 1 1

01_11 1.5 0.333 0.5

11_01 1.5 0.333 0.5

10_11 1.5 0.333 0.5

11_10 1.5 0.333 0.5

11_11 2 0 0

1

GT

4

GT

π

π 2

ωt

ωt

1

iT 1

vT

4

vT

4

iT

ωt

ωt

ωt

ωt ωt

1

1 T

T i

v ×

4

4 T

T i

v ×

ωt Vd

Vd

R V_d 2

R V_d 2

Fig. 2.6. Illustrated waveforms for

120

^o SWPWM type-11_00.

2.3 180 SWPWM

^°

In the common

180

^° SWM, each switch conducts for

180

^°. Like

120

^° SWPWM, there are also six commutations per periodic cycle in

180

^° SWPWM and thus, there are six conduction patterns tabulated in Table 2.4. From Table 2.4, the conduction of the upper switch closely follows the conduction of the lower switch, and vice versa. Therefore, in practice, some blanking time must be included in their switching signals to avoid the short circuits of the dc bus due to the simultaneous conduction of the lower and upper switches.

Table 2.4. Conduction pattern in common

180

^° SWPWM Pattern Duration Conducting devices

1 60^° T1 T₂

T

3

From Table 2.4, it is clear that

180

^° conduction in each switch is composed of three neighbor conduction patterns. There are total six conduction patterns involved in each leg. In order to apply SWPWM to adjust the output voltage, the conduction state in each conduction pattern is not always continuous “on”. It may be either continuous “on” or discontinuous “on”

(i.e. PWM “on”). Consequently, there are

2

⁶

= 64

possible types in

180

^° SWPWM. Each

180

° SWPWM is represented as type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃ where the definitions of

U

_m and

L

m can be found in (2.9)-(2.10). Fig. 2.7 shows the Illustrated waveforms for

180

^o SWPWM type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃. By following the above definition, the common

180

^° SWM can be seen as

180

^° SWPWM type-000_000 whose switching signals are always continuous “on”. All switching signals for

180

^° SWPWM type-010_010 and the output voltages are plotted in Fig. 2.8.

1

1

Fig. 2.8. Illustrated waveforms for

180

^o SWPWM type-010_010.

We can find that in Fig. 2.8, each switch conducts continuously for

60

^°, then conducts discontinuously for the following

60

^° and conducts continuously for the remaining

60

^°. Because that the output voltage v¹⁸⁰_ab ^° can be expressed in Fourier series as:

⎟⎠

Obviously, the fundamental output voltage

V

_ab¹⁸⁰_,1^° can be easily controlled from near zero to maximum output voltage

2 3

V_d

/

π by adjusting the duty ratio D. According to (2.18), the hth-order harmonic voltage can be expressed as:

h However, not all

180

^° SWPWM types possess the controllable fundamental voltage as

(2.19). Table 2.6 tabulates the classification of various output voltage waveforms for all

180

^° SWPWM types. Obviously, according to Table 2.5, only 25 types possess the controllable fundamental voltage, i.e. linear modulation.

Equation (2.20) shows that nonzero hth-order harmonic component increases with PWM duty ratio D and the fundamental voltage

V

_ab¹⁸⁰_,1^° and, however, it holds only when the harmonic order h is significantly lower than the frequency modulation ratio

m

_f . Since the voltage harmonics

V

_ab¹⁸⁰_,h^° in linear

180

^° SWPWM types possess the same proportionality as (2.13) in linear

120

^° SWPWM schemes. Therefore, the voltage harmonics of linear

180

^° SWPWM types can also be found from the same Table 2.3 and Fig. 2.5 as for linear

120

^° SWPWM schemes.

Table 2.5. Classification of output voltage v¹⁸⁰_ab ^° for all

1 8 0

^° SWPWM types

Types Output voltage waveform

000_000

000_001, 000_100, 001_000, 100_000

000_010, 000_101, 001_010, 100_010, 101_000, 101_010, 010_000, 010_001, 010_100, 010_101

000_011, 000_110, 001_101, 001_011, 001_110, 100_110, 101_100, 100_101, 101_110, 011_000, 011_001, 011_100, 011_101, 110_000, 110_001, 110_100, 110_101, 100_011, 101_001, 101_011

001_001, 100_100

001_100, 100_001

000_111, 001_111, 010_010, 010_011, 010_110, 010_111, 011_010, 011_011, 011_110, 011_111, 100_111, 101_101, 101_111, 110_010, 110_011, 110_110, 110_111, 111_000, 111_001, 111_010, 111_011, 111_100, 111_101, 111_110, 111_111 [10]

By following (2.17), the effective PWM transition rate of

180

^o SWPWM type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃ can be represented by

) 2 (

1

³

1 3

1

180

= ∑ + ∑

=

=

°

i i i i

PWM U L

α (2.21)

Then the total driving loss P_G^180° of

180

^o SWPWM type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃ is

°

=

°

°

= ∑

⁶

=

^{2 180}

1 180,

180 iss GS PWM PWM

x GTx

G P C V f

P α (2.22)

The higher value α¹⁸⁰_PWM^° of

180

^o SWPWM type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃ is, the higher driving

loss of type-

U

₁

U

₂

U

₃_

L

₁

L

₂

L

₃ is. According to effective PWM transition rate α¹⁸⁰_PWM^° , all 25 linear

180

^° SWPWM types can be classified into Table 2.6. From Table 2.6, we can find that only type- 010_010 has minimum value α¹⁸⁰_PWM^°

= 1

.

However, based on the thermal consideration, the distribution of power switch loss in upper switch and the lower switch is another important issue. From the voltages and the currents of the switches as shown in Fig. 2.8, we can find that in type-010_010, the switch power loss

P

_S,T1 and

P

_S,T4 are equal to each other.

On the other hand, from the illustrated waveforms of type-111_000 plotted in Fig. 2.9, all the switching loss occurs only in the upper switch. Therefore, not all the linear

180

^° SWPWM types possess good thermal distribution. The thermal distribution factors

F

_G,TD

and

F

_S,TD defined in (2.7) and (2.8) of all linear

180

^° SWPWM types can be found in Table 2.6. It follows that by using the linear

180 °

SWPWM with zero thermal distribution factors, no hot spot due to small-sized motor driver circuit can be expected in the main power circuit of small-sized motor drivers.

By considering the linear modulation in Table 2.5 and thermal distribution in Table 2.6, we can find that only type-010_010 is the best selection of

180

^° SWPWM.

Table 2.6. Loss and thermal characteristics of various linear

1 8 0 °

SWPWM

type

0o 8 1PWM

α F

_G¹⁸_,TD^0o vT1

×

iT1 v_T4

×

i_T4

F

_s¹_,TD^80o

010_010 1 0 0

000_111 1.5 1 1

010_011 1.5 0.333 0

010_110 1.5 0.333 0

011_010 1.5 0.333 0

101_111 1.5 0.333 0.5

110_010 1.5 0.333 0

111_000 1.5 1 1

001_111 2 0.5 1

010_111 2 0.5 0.5

011_011 2 0 0

011_110 2 0 0

100_111 2 0.5 1

101_101 2 0 0

110_011 2 0 0

110_110 2 0 0

111_001 2 0.5 1

111_010 2 0.5 0.5

111_100 2 0.5 1

011_111 2.5 0.25 0.5

110_111 2.5 0.25 0.5

111_011 2.5 0.25 0.5

111_101 2.5 0.25 0.5

111_110 2.5 0.25 0.5

111_111 3 0 0

1

GT

4

GT

π 2π

ωt

ωt

Pattern 6 1 2 3 4 5 6 1 2

1

v

T

1

i

T

R V_d 3 2

4

vT

Vd

4

iT

ωt

ωt

ωt ωt ωt

1

1 T

T i

v ×

3 4 5

ωt

4

4 T

T i

v ×

R V_d 3 2 Vd

R V_d 3

R V_d 3

Fig. 2.9. Illustrated waveforms for type-111_000.

2.4 Experimental Result

As shown in Fig. 2.1, a MCU-based SWPWM inverter is implemented. The DC link voltage of the voltage source inverter is

V

_DC =150

V

and the used balanced load resistances are

100 Ω

. All the six-step SWPWM switching signals G_T1~

G

_T6 with blanking time are implemented in MCU.

2.4.1 Experimental Result for 120

^o

SWPWM

The experimental waveforms for

120 °

SWPWM type-01_01 and type-11_00 are plotted in Fig. 2.10(a) and Fig. 2.10(b), respectively. Compared Fig. 2.10 with Fig. 2.3 and Fig. 2.6, we can make sure that the illustrated waveforms are very close to the experimental results.

Thus, the classification of effective PWM transition α¹²⁰_PWM^° and thermal distribution of gate

driving loss

F

_G¹²⁰_,TD^° in Table 2.3 are correct.

Due to the used six-switch power module, it is hard to measure the switch currents i_T1 and i_T4 directly. However, based on (2.2)-(2.3), the currents i_T1 and i_T4 can be synthesized from the inverter output current

i

_a. Therefore, the illustrated waveforms of products v_T1

×

i_T1 and v_T4

×

i_T4 in Fig. 2.3 and Fig. 2.6 and the following thermal distribution factor

F

_S¹²⁰_,TD^° in Table III can be trusted. It follows that type-01_01 and type-10_10 with minimum effective PWM transition rate and even thermal distribution are the best selection of

120 °

SWPWM.

1

GT

4

GT

vab

ia

ms 2 V

200

A 1

1

vT

4

vT

(a)

4

GT

vab

ia

ms 2 V

200

A 1

1

vT

4

vT 1

GT

(b)

Fig. 2.10. Experimental waveforms: (a) for type-01_01; (b) for type-11_00.

2.4.2 Experimental Result for 180

^o

SWPWM

The experimental waveforms for

180 °

SWPWM type-010_010 and type-111_000 are plotted in Fig. 2.11(a) and Fig. 2.11(b), respectively. By Comparing Fig. 2.11 with Fig. 2.8 and Fig. 2.9, we can also find that the illustrated waveforms are very close to the experimental results. Thus, the classification of effective PWM transition α¹⁸⁰_PWM^° and thermal distribution

of gate driving loss

F

_G¹⁸⁰_,TD^° in Table 2.6 are correct.

The illustrated waveforms of products v_T1

×

i_T1 and v_T4

×

i_T4 in Fig. 2.8, Fig. 2.9 and the thermal distribution of switching power loss

F

_S¹⁸⁰_,TD^° in Table VII also can be trusted. It demonstrates that type-010_010 is the best selection of

180 °

SWPWM.

In order to integrate the small-sized motor drivers into SOC design, we should minimize the power consumption due to gate driving circuit, reduce temperature rise and eliminate possible hot spot in the PWM motor driver. With the common six-switch circuit topology, best selection of SWPWM scheme is an effective way to overcome the SOC challenges – power consumption and thermal balance. After the classification of linear modulation, effective PWM transition rate

α

_PWM , thermal distribution factor

F

_G,TD of gate driving power loss, thermal distribution factor

F

_S,TD of power switch loss and experimental demonstration, we recommends the following best SWPWM schemes for SOC design -- type

01_01 and 10_10 for

120

^° SWPWM and type 010_010 for

180

^° SWPWM. The results are very important for small-sized motor drivers in some portable and electric equipment powered by batteries.

1

GT

4

GT

vab

ia

2ms V

200

A 1

1

vT

4

vT

(a)

4

GT

vab

ia

ms 2 V

200

A 1

1

vT

4

vT 1

GT

(b)

Fig. 2.11. Experimental waveforms: (a) for type-010_010; (b) for type-111_000.

CHAPTER 3

TWELVE-STEP SQUARE-WAVE PWM (SWPWM)

3.1 Introduction

The definition of six-step square-wave PWM has been studied in Chapter 2. At any instant, only two switches conduct in the

120

^° SWPWM but three switches conducts in the

180

^° SWPWM. In

150

^° SWPWM, the sequences of conducting devices are tabulated in Table 3.1.

There are twelve switching patterns Each switch conducts

150

^° and a

30

^° dead-time exists between the switching signals of switch pair which avoids the occurrence of short of dc voltage. Therefore,

150

^° SWPWM is also called twelve-step SWPWM.

Table 3.1. Conduction pattern in common

150

^° SWPWM Pattern Duration Conducting devices

1

30

^o T1 T₂

T

3

2

30

^o T2

T

3

3

30

^o T2

T

₃ T₄

4

30

^o

T

³ T₄

5

30

^o

T

³ _T4

T

₅

6

30

^o T4

T

5

7

30

^o T4

T

₅

T

₆

8

30

^o

T

⁵

T

₆

9

30

^o

T

⁵

T

_{6 T1}

10

30

^o

T

⁶ _T1

11

30

^o

T

6 T₁ T₂

12

30

^o T1 T₂

It is clear that

150

^° conduction scheme in each switch is composed of five neighbor conduction patterns. There are total ten conduction states involved in one switch pair. In order to apply SWPWM to adjust the output voltage, the conduction state in each conduction pattern is not always continuous “on”. It may be either continuous “on” or discontinuous “on”

(i.e. PWM “on”). Consequently, it follows that there are

2

¹⁰

= 1024

possible types in

150

^° SWPWM. In the sense definition in (2.9)-(2.10), the definition of

150

^° SWPWM scheme is defined as type-

U

₁

U

₂

U

₃

U

₄

U

₅_

L

₁

L

₂

L

₃

L

₄

L

₅. Fig. 3.1 shows the Illustrated waveforms for

180

o SWPWM type-

U

₁

U

₂

U

₃

U

₄

U

₅_

L

₁

L

₂

L

₃

L

₄

L

₅

1

G

T 2

G

T 3

G

T 4

G

T 5

G

T 6

G

T

ω

t

ω

t

ω

t

ω

t

ω

t

ω

t

3

π

/

π

π

2

U1 U₂U₃ U₄ U₅

U1 U₂ U₃ U₄ U₅

U1 U₂U₃U₄ U₅

U1 U₂U₃U₄ U₅

U1 U₂ U₃U₄ U₅

U1 U₂U₃

U5 U₄

L1 L₂ L₃ L₄ L₅

L1 L₂ L₃ L₄ L₅

L1 L₂ L₃ L₄ L₅

L1 L₂ L₃ L₄ L₅ L1 L₂ L₃ L₄ L₅

L1 L₂ L3 L₄ L₅

Fig. 3.1. Illustrated waveforms for

150

^° SWPWM type-

U

₁

U

₂

U

₃

U

₄

U

₅_

L

₁

L

₂

L

₃

L

₄

L

₅.

3.2 150 SWPWM

^°

In the common

150

^° SWPWM, for type-00110_00110, switching signals G_T1 and G_T4, the output voltage and current waveforms are plotted in Fig. 3.1. We can find that each switching signal holds the “on” voltage level at the first

120

^° period, alternates with a fixed

duty ratio D for the next

120

^° period and then return to keep the “on” voltage level at the last

By calculation, the fundamental component

V

_ab¹⁵⁰_,1^°of output voltage

v

¹⁵⁰_ab ^ocan be expressed as:

d

According to (3.1), the hth-order harmonic voltage can be expressed as:

cos12

nonzero hth-order harmonic component V_ab,h^150° increases with PWM duty ratio D. However, (3.3) holds only when its harmonic order h is significantly lower than the frequency modulation ratio

m

_f. As listed in Table 3.2, the other harmonics centered around the PWM frequency

f

_PWM and its multiples do not increase with duty ratio D. Fig. 3.3 shows the output voltage harmonic spectrum with duty ratioD

= 0.5

.

1

Fig. 3.2. Illustrated waveforms for

150

^° SWPWM type-00110_00110.

The higher value α¹⁵⁰_PWM^° of

150

^o SWPWM type-

U

₁

U

₂

U

₃

U

₄

U

₅_

L

₁

L

₂

L

₃

L

₄

L

₅ is, the higher driving loss of type-

U

₁

U

₂

U

₃

U

₄

U

₅_

L

₁

L

₂

L

₃

L

₄

L

₅ is. By calculating, there are four linear

150

^o SWPWM schemes type-00100_01110, type-01110_00100, type-00110_00110 and type-01100_01100 with the minimum value α¹⁵⁰_PWM^°

= 1

.

According to the above cases of

120

^o SWPWM and

180

^o SWPWM, we can find that due to the symmetry of the switching signals, some linear SWPWM schemes including type-00110_00110 and type-01100_01100 tabulated at the diagonal line of Table 3.4 always possess zero thermal distribution factor

F

_G,TD =0. It follows that type-00110_00110 and type-01100_01100 are the best selection of

150

^° SWPWM. Fig. 3.4 plots the illustrated waveform for type-01100_01100.

Table 3.2. Harmonic components in

150

^o SWPWM

D h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

5 0.0061 0.0122 0.0182 0.0243 0.0304 0.0365 0.0426 0.0486 0.0547 0.0608 7 0.0044 0.0087 0.0130 0.0174 0.0217 0.0261 0.0304 0.0348 0.0391 0.0435 11 0.0103 0.0206 0.0309 0.0412 0.0516 0.0619 0.0722 0.0825 0.0928 0.1031 13 0.0087 0.0175 0.0262 0.0349 0.0436 0.0523 0.0611 0.0698 0.0785 0.0872

±1

RMS 0.2266 0.3208 0.3925 0.4532 0.5067 0.5551 0.5996 0.6412 0.6802 0.7171 THD(%) 304.46 203.61 155.68 125.19 102.66 84.37 68.37 53.37 37.74 16.87 WTHD(%) 20.71 16.97 14.40 12.34 10.57 9.00 7.75 6.29 5.27 4.77

Note: The value tabulated in，

V

_abh

V

_d

Table 3.3. Harmonic components in modified

150

^o SWPWM

D h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 0.0928 0.1857 0.2785 0.3713 0.4641 0.5570 0.6498 0.7426 0.8354 0.9283

5 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0

11 0.0248 0.0330 0.0412 0.0495 0.0577 0.0660 0.0743 0.0825 0.0248 0.0330 13 0.0210 0.0279 0.0349 0.0419 0.0488 0.0559 0.0628 0.0698 0.0210 0.0279

±1 RMS 0.2181 0.3080 0.3770 0.4359 0.4871 0.5334 0.5764 0.6157 0.6530 0.6885 THD(%) 305.29 212.18 163.25 132.53 109.69 91.33 75.74 61.23 47.12 31.65 WTHD(%) 20.78 17.08 14.55 12.49 10.73 9.15 7.71 6.43 5.24 4.37

Note: The value tabulated in，

V

_ab^M_h

V

_d

1

Table 3.4. Linear

150 °

SWPWM U₁U₂U₃U₄U₅

5 4 3 2 1LLLL

L 00000 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111

00000

00001

00010

00011

00100

00101

00110

00111

01000

01001

01010

01011

01100

01101

01110

01111

10000

10001

10010

10011

10100

10101

10110

10111

11000

11001

11010

11011

11100

11101

11110

11111

Linear

150

^o SWPWM types.

1

Fig. 3.4. Illustrated waveforms for type 01100_01100.

3.3 Modified 150 SWPWM

^°

In all

120

^o,

150

^°,

180

^o SWPWM, the SWPWM duty ratios are fixed when the switching signals are discontinous ”on”. However, from Fig. 3.3 and Fig. 3.4, we can find that in

150

^° SWPWM, the output voltage harmonics can be closely to those of pure sinusoidal voltages if the SWPWM duty ratios of odd-number patterns shows in Fig. 3.2 and Fig. 3.4 decrease adequately. By calculation, keeping duty ratio D at even-number patterns and reducing duty ratio to ( 3 2)

D

at odd-number patterns would result in the desired harmonics reduction. This modification of

150

^° SWPWM is called modified

150

^° SWPWM[16].

Fig. 3.5 and Fig. 3.6 plot the illustrated waveforms for modified

150

^° SWPWM type-00110_00110 and modified

150

^° SWPWM type-01100_01100, respectively. In order to represent the reduction effect of duty ratio at odd-number patterns, the plotted output voltage is changed to

( 3 2 )

V_d without the change of the average value in individual

switching period. Its output voltage v_ab^M150° as shown in Fig. 3.5 can be expressed in Fourier series as: expressed as:

d

voltage can be expressed as:

Equation (3.8) shows that nonzero hth-order harmonic component

V

_ab^M_,¹⁵⁰_h ^° increases

proportionly with PWM duty ratio D and the fundamental voltage

V

_ab^M_,¹⁵⁰₁ ^°. Compared with

150

° SWPWM, no

12

r

± 5

harmonics can be found in modified

150

^° SWPWM. As listed in Table 3.3, the other harmonics centered around the PWM frequency

f

_PWM and its multiples have maximum values at D=0.5. Fig. 3.3(b) shows the output voltage harmonic spectrum with modified

150

^° SWPWM.

1

GT 2

GT 3

GT 4

GT 5

GT 6

GT

vab

vbc

vca

ωt ωt ωt ωt ωt ωt

ωt

ωt

ωt 2

3V_d

Vd

Vd

3 π/

π

2π

ωt ia

R V_d 6

3 V_d 2R 3V_d 3R

Vd

Fig. 3.5. Illustrated waveforms for modified

150

^° SWPWM type-00110_00110.

1

GT 2

GT 3

GT 4

GT 5

GT 6

GT

vab

vbc

vca

ωt ωt ωt ωt ωt ωt

ωt

ωt

ωt Pattern 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Vd

Vd

Vd

3 π/

π

π 2

ωt ia

R Vd 6

3 Vd 2R 3Vd 3R

2 3Vd

Fig. 3.6. Illustrated waveforms for modified

150

^° SWPWM type-01100_01100.

3.4 Experimental Result

As shown in Fig. 2.1, a MCU-based SWPWM inverter is implemented. The DC link voltage of the voltage source inverter is

V

_DC =150

V

and the used balanced load resistances are

100 Ω

. All the twelve-step SWPWM switching signals G_T1~

G with blanking time are

_T6 implemented in MCU.

3.4.1 Experimental Result for 15 0

^o

SWPWM

The experimental waveforms for

1 5 0 °

SWPWM type-00110_00110 and type-01100_01100 are plotted in Fig. 3.2 and Fig. 3.4., respectively. By Comparing Fig. 3.7 with Fig. 3.2 and Fig. 3.4, we can also find that the illustrated waveforms are very close to the experimental results.

The illustrated waveforms of products v_T1

×

i_T1 and v_T4

×

i_T4 in Fig. 3.2, Fig. 3.4 can be trusted. It demonstrates that type-00110_00110 and type-01100_01100 are both the best

The illustrated waveforms of products v_T1

×

i_T1 and v_T4

×

i_T4 in Fig. 3.2, Fig. 3.4 can be trusted. It demonstrates that type-00110_00110 and type-01100_01100 are both the best

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