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~
2. 7
Reacti ve power consi derati ons
Due to the phase- angl e cont r ol w th del ay angl e i n t he case
of 1i ne- commutated conver tors the di spl acement to be seen i n
Fi g. 23 of t he phase cur rent s r esul t i ng f r om t he val ve currents
w t h r espect t o t he phase vol t age i n t he suppl y syst em gi ves r i se
to cont rol react i ve power .
I n addi t i on, as descr i bed i n 2. 3,
the reactances i n the cornmutat i ngci r cui t cause over l aps between
the consecut i ve cur rent bl ocks of the i ndi vi dual phases whereby
the f undamental osci l l at i ons of the phase cur rents are f urt her
r etarded. The react i ve power caused by thi s i s the cornmutat i ng
reacti ve power .
The cont r ol r eact i ve power i s consi der ed f i r st .
2. 7. 1 Contr ol react i ve power
I n Fi g. 23 t he vol t age and cur rent of one phase are separat el y
pl ot ted i n accordance w th t he r epresent at i on i n Fi g. 12 f or two
del ay angl es a = 0
and a =
~5.
I t was assumed f or t hi s t hat t he
commutat i ng ci rcui t contai ns no react ances ( L
K
= O) , t he di r ect
cur r ent I d i s compl etel y smoothed and the phase vol t age curve i s
si nusoi dal .
wt
Fi g. 23 Cur ves of phase cur r ent i R and phase vol t age u
R
pl ot ted agai nst t i me for a si x-pul se conver tor when
f ree- f i r i ng = 0
and w th phase shi f t cont r ol
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I n t he f r ee- f i r i ng condi t i on t he f undament al osci l l at i on i
1R
of t he
cur r ent i R i s i n phase w t h t he si nusoi dal phase vol t age ~ due t o
i t s ~h order symnet r y. Wt h phase shi f t cont r ol t he cur r ent bl ocks
ar e di spl aced backwar ds t hr ough
t hus di spl aci ng i t s f undament al
osci l l at i on by ~1. Because of t he ~h order syr nmet r y of t he bl ock
wavef orm resul t i ng f rom t he assumed omssi ons,
= ~ as can be
1
seen f r om Fi g. 23b. Hence
cos ~1 = cos a
2. 30
The r eal power Pl i n t he t hree- phase syst em i s t hen
U
rms
I
1rms
3
U
rms
I
1rms
cos
a
2. 31
The r . m s. val ue of t he f undamental osci l l at i on of t he squar e- wave
cur r ent i s obt ai ned by Four i er anal ysi s:
I
1rms
~
I d
30
J
72
cos
2
.f3
J
. [ 2 .
I d
Usi ng equat i on 2. 3 and i nsert i ng val ues f or U
rms
and I
1rms
:
3 J
2
Udi o cos
2. 33
I gnor i ng t he conver t or l osses, t he r eal power of t he f undament al
osci l l at i on on t he a. c. si de i s i dent i cal t o t he power on t he d. c.
si de.
The appar ent power of t he f undament al osci l l at i on i s gi ven by
The t ot al apparent power S becomes
S
=
3 Urms
I
rms 2. 35
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- ~6 -
Thus t he
power
f actor
A .
i s
def i ned by
Pl
3
U
I
1rms
cos
rms
A .
S
3
U
1
ms
rms
Taki ng 9
I
1rms
rms
as t he f undament al osci l l at i on cont ent ,
9 cos
For t he cur r ent wavef orm i l l ust r at ed i n Fi g. 23 ( e. g. i n t he
t hree- phase br i dge connect i on) , accordi ng to equat i on 2. 32
I
1rms
2
.f3
I d
and moreover
= ; i
. J 2
2
I d
2. 37
rms
.f3
whence
9
0 955
From equat i on~ 2. 30 and 2. 36 the r el at i onshi p between a and cos ~1
i s obt ai ned f r om
A . =g cos = 9 cos
1
I t can be seen t hat , due t o t he cur rent wavef orm bei ng burdened
w t h harmoni cs, t he power f act or A i s l ess t han 1 even i n
t he f r ee- f i r i ng condi t i on ( cos = cos
fJ J
1
= 1) .
The react i ve power Q1 of t he f undament al osci l l at i on i s gi ven by
Ql
S
2
P
1
2
2
3
U I
1rms
cos
rms
3 U I 1rms
.
si n
2. 39rms
Ql
U
di o
I d
si n
2.~
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I f t he f undament al osci l l at i on r eact i vepower i s r ef erred t o t he
f undament al osci l l at i on apparent power Si = U
di
I d equat i ons
2. 39 and 2. ~1 produce t he expressi on
1 ( cos e x . _ d ) 2
x
Wth t he rel at i ve di rect vol t age
cos e x . d
x
t he equat i on f or an el l i pse i s obt ai ned, whi ch t ur ns i nt o a
ci r cl e when bot h r adi i ar e made t he same.
d e x .
2
U
di
1
2. ~~
pos i t i ve, i s r epresented by a semci rc l e.
The capaci t i ve r ange
I n Fi g. 25 t he i nducti ve r ange, i n whi ch t he react i ve power i s
w t h negat i ve r eact i ve power cannot be obtai ned w th t he
l i ne- commutat ed conver t or s di scussed here w t hout addi t i onal sel f com
mutat i ng devi ces. Wt h const ant rel at i ve i nduct i ve di r ect vol t age
drop d
x
i . e. constant suppl y syst em shor t - ci rcui t vol t age u
k
t he del ay angl e e x . i s t he paramet er of t hi s ci r cl e. The si ze of
t he angl e depends upon t he magni t ude of d
x
I n Fi g. 25 t he react i ve
power consumpt i on at e x = O and e x = 160
0
i s pl ot t ed f or
5
val ues of
d
x
i n accordance w t h t he rel at i onshi p uo
=
ar c cos ( 1 2 dx) .
_
-
_ . _
Wth posi t i ve r el at i ve di r ect vol t age oper at i on i s i n t he r ect i f i er
mode and w th negat i ve r el at i ve di r ect vol t age i n the i nver ter mode.
Fi g. 25 Var i at i on of r eact i ve power i n mai ns w t h
di r ect vol t age at constant di r ect cur r ent output
Paramet er: Overl ap angl e u
Margi n of commut at i on 20