issues with inverters

7/27/2019 Issues With Inverters

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EE 497D

Spring 2003 Lecture Notes 4

1 Issues with inverters

1.1 Dead or Blanking Time

When attempting to command accurate voltages from an inverter one must take into account the deadtime effect. Typical wave-

forms for the inverter phase are shown in Fig. 1.1. To avoid shorting the bus voltage, there is a specified delay between when

one switch is opened and the other is closed. This is known as the dead time or blanking time,

. The value of

during

this deadtime is dependent on the direction of the current

. Diodes parallel to the transistors carry the current

during this

deadtime, the lower diode if

is positive and the upper diode if

is negative. Hence, during deadtime the output voltage

will be

when

is positive and

when

is negative.

To determine the effect of the deadtime, we separate the output voltage into two parts: a part

" $ '

corresponding to time

when either the upper or lower transistor is on, and a part (

corresponding to the deadtime. The average value of " $ '

is

given by:

1

"$ '3 2 4 5

6

"

7 8

@

"$ ' A

4 5

6

" B

7 D8

E

G

@ H I

"

Q

A R

78

E

G

D8

H I

"

Q

A U

4 W X

6

" `

HI

"

Q

R W

5

X

6

" `

W

HI

"

Q

`

4W

X

5

Q

`

H I

" d

(1)

or the same as the average-value output voltage when deadtime is neglected. As there are two deadtime intervals per switching

1


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period, the average deadtime voltage is therefore given by:

1

( $& ' 2 4

f h p

$

'

Q

6

"

7

G

@ H I

"

Q

A

4

f h p

$

'

HI

"

6

"

(2)

If

$ '

is a sinusoid (as is expected for steady-state induction machine operation), then1

( $ ' 2

will be a square wave

5 s t u

out of phase with

. The total average-value output voltage, including deadtime, is given by:

1

$0 ' 2 4W

X $& '

5

Q

`

HI

"

f h p

$

'

HI

"

6

"

(3)

This deadtime voltage is annoying for two reaons:

x The fundamental component of the square-wave deadtime voltage will alter the magnitude and phase of our commanded

average-value output. If we are trying to command small voltages, the deadtime voltage will completely overwhelm the

desired voltage.

x The square-wave deadtime voltage will add odd harmonics (3,5,...) of the fundamental frequency to the output voltage

waveform.

1.2 Overmodulation

Overmodulation occurs when one attempts to command an average-value output voltage whose magnitude is larger than can

be achieved by the bus voltage. This results in additional harmonics below the switching frequency. The extreme case of

overmodulation occurs when one commands a square-wave output with the desired frequency. In this case the magnitude of the

fundamental is

y

H I

"

, and the harmonics are odd-valued with magnitude

y

H I

"

.

1.3 Frequency Modulation

The inherent assumption in pulse-width modulation is that the switching frequency is substantially higher than the frequency

of the average-value waveforms we wish to generate. An accepted threshold ratio between the switching frequency and the

desired output frequency is 21. If the switching frequency is significantly less than 21 times the desired output frequency,

additional harmonics can occur in the frequency range of the desired output frequency.


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1.4 Example

To illustrate the effects discussed in the previous sections on overmodulation and frequency modulation, we look at an example.

Fig. 1.4 presents the output of a half-bridge inverter with a bus voltage of t

H

, and a commanded output voltage of 126V

peak, which can be achieved with the stated bus voltage without entering overmodulation. Inspection of the figure reveals that

the desired output frequency harmonic is achieved. Fig. 1.4 reveals the output harmonics if we attempt to command 216V peak,

which is beyond the capability of this half-bridge inverter unless we enter overmodulation. Note that the fundamental voltage

is less than the desired value, and that odd harmonics of the fundamental have now entered the waveform. Fig. 1.4 presents

output harmonics if the command output voltage is returned to 126V peak, but the command frequency is increased to

s 5

.

Because of the close proximity to the switching frequency, the harmonic output in the range of the desired frequency is no longer

a single, pure harmonic. Note also that the harmonic at

s 5

does not have a peak value of 126V.

2 Full-bridge inverter

In the last lecture we discussed the half-bridge inverter. Though conceptually easy to understand, the half-bridge inverter is

typically not used in practice. This is mainly due to two reasons, both having to do with the neutral connection between the two

bus capacitors:

x If possible, it is generally desirable to use only one bus capacitor instead of two, in which case the neutral connection

would not exist. Having two identical bus capacitors in series reduces the effective bus capacitance by a factor of two.

Hence to achieve a desired bus capacitance one must purchase two bus capacitors of twice the size, which is economically

unattractive. Having two bus capacitors in series is typically desirable only when it is necessary to achieve the desired

voltage rating of the application.

x Having a neutral connection between the two bus capacitors forces the load current to flow through these capacitors. In this

situation the impedance of the bus capacitors would have to be sized to minimize the resulting AC voltage across them. If

our load requires DC currents, this will become impossible to achieve.

Hence for single-phase applications we often use the full-bridge inverter, shown in Fig. 2.

The full-bridge inverter consists of 2 half-bridge inverters. The output voltage is the voltage between the outputs of each half

bridge. As discussed previously, each half-bridge can be in one of two states. We will now define state 0 of a half-bridge as


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Most medium- and high-power electrical machines are 3-phase. As we will discover later in the class, it is much easier

to analyze and control 2-phase machines. However, the 3-phase machine has distinct advantages over the 2-phase machine in

implementation. In a 2-phase machine, the two currents aren

tu

out of phase in balanced operation, and therefore do not cancel

as they do in three-phase systems. Hence controlling a 2-phase machine would require 2 full-bridge inverters, or a total of 4

half-bridges, one more than the 3-phase machine. As power electronics switches are still quite expensive (though their price is

dropping), this is a significant advantage.


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VBUS

VBUS

2

+

-

VBUS

2

+

-

vout(t)

T-

T++

-R C

CR

D-

D

+

+

-

n

iout(t)

Figure 1: Half-bridge inverter


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Vbus /2

Vbus /2-

vdead

Vbus /2

Vbus /2-

ge-

iout

vout

ge+

v

v

t

t

t

t

t

T

td

Figure 2: Output voltage waveforms


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0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

20

40

60

80

100

120

140

160

180

Frequency (Hz)

HarmonicVoltageMag.(V)

Switching Frequency 15kHz, Command Frequency 415Hz

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

140

160

180

Frequency (Hz)

HarmonicVoltageMag.

(V)


Figure 3: Half-bridge harmonic output,l

4

5

,l

"4

5 o

,H

I

"4

t

H


9/13

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

20

40

60

80

100

120

140

160

180

200

Frequency (Hz)



0 500 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

Frequency (Hz)

HarmonicVoltageMag.

(V)


Figure 4: Over-modulated half-bridge harmonic output,l

4

5

,l

"4

5 o

,H

I

"4

t

H


10/13

0 0.5 1 1.5 2 2.5 3 3.5

x 104

0

20

40

60

80

100

120

140

160

180

Frequency (Hz)



2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 75000

20

40

60

80

100

120

140

160

180

Frequency (Hz)

HarmonicVoltageMag.

(V)


Figure 5: Half-bridge harmonic output,l

4

s 5

,l

"4

5 o

,H

I

"4

t

H


11/13

VBUS

T

+

T-

T

+

T-

vab(t)

R C

CR

n

+

-

a

aD-

D

a

a

b

bD-

+b

b

D+

-+

Figure 6: Full-bridge inverter


12/13


13/13

VBUS

T +

T-

T +

T-

T +

T-

a

R C

CR

n

+

-

a

aD

-

D

a

a

c

cD

-

+

c

c

D+

b

bD

-

+

b

b

D+

b c

Figure 8: 3-phase inverter

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