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**Contents:**

Three Phase Magic Sinewaves

By Don Lancaster and Synergetics

http://www.tinaja.com

The Issue...

Magic Sinewaves offer maximized efficiency with minimized low harmonics for emerging power electronics applications. Per these tutorials. But ordinary magic sinewaves are not three phase compatible because they would need extra drivers, equipment rewiring, and have other restrictions. Fortunately, a special class of Delta Friendly magic sinewaves can instead be generated that can end up fully three phase compatible.

Delta Friendly Features...

Fully three-phase compatible. Lengths of n=12, 28, 44, 60, 76,... available. Zero out the first (3n/4) + 1 harmonics. Table lookup storage only one-half Analysis and design is faster.

of usual.

Why Three Phase Power?

Power flow is continuous. Motors start and reverse easier. Less noise and vibration. Smaller wiring sizes. Better use of copper and iron.

Delta Friendly Switching...

x +dc y z a c b

Eight Allowable Switch States...

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

Produce These Current Patterns...

c=

a + 240

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

o

b=

a + 120

o

a+0

a= o

0 0 ccw ccw cw cw 0 0

0 cw 0 cw ccw 0 ccw 0

0 ccw cw 0 0 ccw cw 0

To ALWAYS force a zero average!

c=

a + 240

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

o

b=

a + 120

o

a+0

a= o

sum

zero! zero! zero! zero! zero! zero! zero! zero!

0 0 ccw ccw cw cw 0 0

0 cw 0 cw ccw 0 ccw 0

0 ccw cw 0 0 ccw cw 0

The Key Delta Friendly Rule...

Because of the permissible switching combinations...

All triad samples MUST sum to zero !

Which leads to this strict rule...

NO TRIAD HARMONICS !

Thus, delta friendly magic sinewaves must have precisely zero 3rd, 9th, 15th, 21st, ... harmonics.

This Works...

fundamental

0o

tracking pulses COMBINE for fundamental but CANCEL for triad harmonics

90o

third harmonic

So Does This...

fundamental

0o

mirrored pulses COMBINE for fundamental but CANCEL for triad harmonics

90o

third harmonic

Leading to our Delta Design rules...

If there is ZERO energy in a narrow

interval x in the 60 to 90 degree region of the first quadrant, then there must also be ZERO energy in intervals x-60 and 120-x.

If there is ONE energy in a narrow interval

x in the 60 to 90 degree region of the first quadrant, then there must also be ONE energy in EITHER interval x-60 OR in the interval 120-x. But not both.

Delta Friendly Synthesis Starts...

... by picking a number k of whole and bounded pulses placed in the 60 to 90 degree quadrant interval...

value of k 1 2 3 4 5 k pulses per quadrant 3 7 11 15 19 ( 4k - 1 ) edges per quadrant 6 14 22 30 38 2 ( 4k - 1 ) pulses per cycle n 12 28 44 60 76 4 ( 4k - 1 ) harmonics zeroed 10 22 34 46 58 ( 12k - 2 ) = ( 3n / 4 ) + 1

With a Goal of...

... creating one equation for each available first quadrant pulse edge. Specifically...

One half of the pulse edges get used as edge

tracking for zero triad harmonics.

One pulse edge will set the amplitude. Remaining pulse edges zero 5, 7, 11, 13, 17,

19, ... non-triad odd harmonics.

Ferinstance...

On a delta friendly n=28 Magic Sinewave, there are fourteen first quadrant pulse edges. One edge sets the amplitude. Seven edges zero out all triad harmonics 3, 9, 15, 21, 27... and will guarantee three phase compatibility through edge tracking. Six edges zero harmonics 5, 7, 11, 13, 17, & 19. Harmonics 23 and 25 will end up fairly strong.

Continue Synthesis with Wrap Map...

Arrange your pulses into a wrap map of (2k-1), k, and k pulses. The map must obey the delta rules...

60o 60o 0o

6s

6e 7s

7e

90o 30o

5e

5s

4e

4s

1s

1e 2s

2e

3s

3e

30o

Wrap Map Guidelines...

Vertical positions MUST have ZERO or TWO pulses. The 60 to 30 interval INCREASES to the LEFT. Initially center k pulses in 60 to 90 interval. Left 60 to 30 pulse aligns LEFT. Rest CENTER. Left 0 to 30 pulse aligns RIGHT. Rest BY PAIRS. PAIRS of pulse edges must be perfectly aligned.

Write the Tracking Equations...

Pairs of pulse edges found from the wrap map must be locked together for tracking in order to eliminate all of the triad harmonics...

p1s = 60 - p5s p1e = p6e - 60 p2s = p7s - 60 p2e = 60 - p4e p3s = 60 - p4s p3e = p7e - 60 p5e = 120 - p6s

Then Write the Full Equations...

Shown for a "n=28" delta friendly Magic Sinewave. Because of edge locking, there will be only seven independent equations in seven unknowns. Only one-half of the normal storage is needed.

cos ( 1*p1s ) - cos (1* p1e ) + ... + cos ( 1*p7s ) - cos ( 1*p7e ) = ampl * pi/4 cos ( 5*p1s ) - cos ( 5*p1e ) + ... + cos ( 5*p7s ) - cos ( 5*p7e ) = 0 cos ( 7*p1s ) - cos ( 7*p1e ) + ... + cos ( 7*p7s ) - cos ( 3*p7e ) = 0 cos (11*p1s) - cos (11*p1e) + ... + cos (11*p7s) - cos (11*p7e) = 0 cos (13*p1s) - cos (13*p1e) + ... + cos (13*p7s) - cos (13*p7e) = 0 cos (17*p1s) - cos (17*p1e) + ... + cos (17*p7s) - cos (17*p7e) = 0 cos (19*p1s) - cos (19*p1e) + ... + cos (19*p7s) - cos (19*p7e) = 0

Equation Solution...

As before, the equations are elegantly solved by using Newton's Method, aka "shake the box". A guess is made to get you near the solution. This time, pairs of edges are then moved slightly to see if the distortion gets better or worse. Final pulse edge locations are related to three phase port patterns by using this Tutorial Resource and these Waveform Plots.

For Additional Help...

Magic Sinewave calculators and tutorial... http://www.tinaja.com/magsn01.asp Magic Sinewave development proposal... http://www.tinaja.com/glib/msinprop.pdf Magic Sinewave seminars and consulting... http://www.tinaja.com/info01.asp

This has been...

... a presentation by Don Lancaster and Synergetics, 3860 West First Street, Box 809, Thatcher, Arizona, 85552. (928) 428-4073. mailto:don@tinaja.com

Copyright c 2003 and earlier by Don Lancaster and Synergetics. Linking usually welcome. All media, web, and ALL other rights fully reserved. Mirroring or reposting is expressly forbidden.

By Don Lancaster and Synergetics

http://www.tinaja.com

The Issue...

Magic Sinewaves offer maximized efficiency with minimized low harmonics for emerging power electronics applications. Per these tutorials. But ordinary magic sinewaves are not three phase compatible because they would need extra drivers, equipment rewiring, and have other restrictions. Fortunately, a special class of Delta Friendly magic sinewaves can instead be generated that can end up fully three phase compatible.

Delta Friendly Features...

Fully three-phase compatible. Lengths of n=12, 28, 44, 60, 76,... available. Zero out the first (3n/4) + 1 harmonics. Table lookup storage only one-half Analysis and design is faster.

of usual.

Why Three Phase Power?

Power flow is continuous. Motors start and reverse easier. Less noise and vibration. Smaller wiring sizes. Better use of copper and iron.

Delta Friendly Switching...

x +dc y z a c b

Eight Allowable Switch States...

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

Produce These Current Patterns...

c=

a + 240

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

o

b=

a + 120

o

a+0

a= o

0 0 ccw ccw cw cw 0 0

0 cw 0 cw ccw 0 ccw 0

0 ccw cw 0 0 ccw cw 0

To ALWAYS force a zero average!

c=

a + 240

z y x

0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

o

b=

a + 120

o

a+0

a= o

sum

zero! zero! zero! zero! zero! zero! zero! zero!

0 0 ccw ccw cw cw 0 0

0 cw 0 cw ccw 0 ccw 0

0 ccw cw 0 0 ccw cw 0

The Key Delta Friendly Rule...

Because of the permissible switching combinations...

All triad samples MUST sum to zero !

Which leads to this strict rule...

NO TRIAD HARMONICS !

Thus, delta friendly magic sinewaves must have precisely zero 3rd, 9th, 15th, 21st, ... harmonics.

This Works...

fundamental

0o

tracking pulses COMBINE for fundamental but CANCEL for triad harmonics

90o

third harmonic

So Does This...

fundamental

0o

mirrored pulses COMBINE for fundamental but CANCEL for triad harmonics

90o

third harmonic

Leading to our Delta Design rules...

If there is ZERO energy in a narrow

interval x in the 60 to 90 degree region of the first quadrant, then there must also be ZERO energy in intervals x-60 and 120-x.

If there is ONE energy in a narrow interval

x in the 60 to 90 degree region of the first quadrant, then there must also be ONE energy in EITHER interval x-60 OR in the interval 120-x. But not both.

Delta Friendly Synthesis Starts...

... by picking a number k of whole and bounded pulses placed in the 60 to 90 degree quadrant interval...

value of k 1 2 3 4 5 k pulses per quadrant 3 7 11 15 19 ( 4k - 1 ) edges per quadrant 6 14 22 30 38 2 ( 4k - 1 ) pulses per cycle n 12 28 44 60 76 4 ( 4k - 1 ) harmonics zeroed 10 22 34 46 58 ( 12k - 2 ) = ( 3n / 4 ) + 1

With a Goal of...

... creating one equation for each available first quadrant pulse edge. Specifically...

One half of the pulse edges get used as edge

tracking for zero triad harmonics.

One pulse edge will set the amplitude. Remaining pulse edges zero 5, 7, 11, 13, 17,

19, ... non-triad odd harmonics.

Ferinstance...

On a delta friendly n=28 Magic Sinewave, there are fourteen first quadrant pulse edges. One edge sets the amplitude. Seven edges zero out all triad harmonics 3, 9, 15, 21, 27... and will guarantee three phase compatibility through edge tracking. Six edges zero harmonics 5, 7, 11, 13, 17, & 19. Harmonics 23 and 25 will end up fairly strong.

Continue Synthesis with Wrap Map...

Arrange your pulses into a wrap map of (2k-1), k, and k pulses. The map must obey the delta rules...

60o 60o 0o

6s

6e 7s

7e

90o 30o

5e

5s

4e

4s

1s

1e 2s

2e

3s

3e

30o

Wrap Map Guidelines...

Vertical positions MUST have ZERO or TWO pulses. The 60 to 30 interval INCREASES to the LEFT. Initially center k pulses in 60 to 90 interval. Left 60 to 30 pulse aligns LEFT. Rest CENTER. Left 0 to 30 pulse aligns RIGHT. Rest BY PAIRS. PAIRS of pulse edges must be perfectly aligned.

Write the Tracking Equations...

Pairs of pulse edges found from the wrap map must be locked together for tracking in order to eliminate all of the triad harmonics...

p1s = 60 - p5s p1e = p6e - 60 p2s = p7s - 60 p2e = 60 - p4e p3s = 60 - p4s p3e = p7e - 60 p5e = 120 - p6s

Then Write the Full Equations...

Shown for a "n=28" delta friendly Magic Sinewave. Because of edge locking, there will be only seven independent equations in seven unknowns. Only one-half of the normal storage is needed.

cos ( 1*p1s ) - cos (1* p1e ) + ... + cos ( 1*p7s ) - cos ( 1*p7e ) = ampl * pi/4 cos ( 5*p1s ) - cos ( 5*p1e ) + ... + cos ( 5*p7s ) - cos ( 5*p7e ) = 0 cos ( 7*p1s ) - cos ( 7*p1e ) + ... + cos ( 7*p7s ) - cos ( 3*p7e ) = 0 cos (11*p1s) - cos (11*p1e) + ... + cos (11*p7s) - cos (11*p7e) = 0 cos (13*p1s) - cos (13*p1e) + ... + cos (13*p7s) - cos (13*p7e) = 0 cos (17*p1s) - cos (17*p1e) + ... + cos (17*p7s) - cos (17*p7e) = 0 cos (19*p1s) - cos (19*p1e) + ... + cos (19*p7s) - cos (19*p7e) = 0

Equation Solution...

As before, the equations are elegantly solved by using Newton's Method, aka "shake the box". A guess is made to get you near the solution. This time, pairs of edges are then moved slightly to see if the distortion gets better or worse. Final pulse edge locations are related to three phase port patterns by using this Tutorial Resource and these Waveform Plots.

For Additional Help...

Magic Sinewave calculators and tutorial... http://www.tinaja.com/magsn01.asp Magic Sinewave development proposal... http://www.tinaja.com/glib/msinprop.pdf Magic Sinewave seminars and consulting... http://www.tinaja.com/info01.asp

This has been...

... a presentation by Don Lancaster and Synergetics, 3860 West First Street, Box 809, Thatcher, Arizona, 85552. (928) 428-4073. mailto:don@tinaja.com

Copyright c 2003 and earlier by Don Lancaster and Synergetics. Linking usually welcome. All media, web, and ALL other rights fully reserved. Mirroring or reposting is expressly forbidden.