434
11.1. INTRODUCTION
A brief analysis of single- and three-phase semi-controlled bridge converters is pre-
sented in this chapter. This type of converter is also commonly referred to as a line-
commutated converter. The objective is to provide a basic background in converter
operation without becoming overly involved. For this reason, only the constant-current
operation is considered. A more detailed analysis of these and other converters can be
found in References 1–4 . Finally, to set the stage for the analysis of dc and ac drive
systems in later chapters, an average-value model of the three-phase semi-controlled
bridge converter is derived. This model can be used to predict the average-value per-
formance during steady-state and transient operating conditions.
11.2. SINGLE-PHASE LOAD COMMUTATED CONVERTER
A single-phase line-commutated full-bridge converter is shown in Figure 11.2-1 . The
ac source voltage and current are denoted e
ga
and i
ga
, respectively. The series inductance
(commutating inductance) is denoted l
c
. The thyristors are numbered T 1 through T 4,
and the associated gating or fi ring signals are denoted e
f

1
through e
f

4
. The converter
output voltage and current are v

d
and i
d
. The following simplifying assumptions are
Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk,
Scott Sudhoff, and Steven Pekarek.
© 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc.
SEMI-CONTROLLED
BRIDGE CONVERTERS
11
SINGLE-PHASE LOAD COMMUTATED CONVERTER 435
Figure 11.2-1. Single-phase full-bridge converter.
made in this analysis: (1) the ac source contains only one frequency, (2) the output
current i
d
is constant, (3) the thyristor is an infi nite impedance device when in the
reverse bias mode (cathode positive) or when the gating signal to allow current fl ow
has not occurred, and (4) when conducting, the voltage drop across the thyristor is
negligibly small.
Operation without Commutating Inductance or Firing Delay
It is convenient to analyze converter operation in steps starting with the simplest case
where the commutating inductance is not present and there is no fi ring delay. In this
case, it can be assumed that the gating signals are always present, whereupon the thyris-
tors will conduct whenever they become forward biased (anode positive) just as if they
were diodes. Converter operation for constant i
d
with l
c
= 0 and without fi ring delay is
depicted in Figure 11.2-2 . The thyristor in the upper part of the converter ( T1 or T3 ) that

conducts is the one with the greatest anode voltage. Similarly, the thyristor that conducts
in the lower part of the converter ( T 2 or T 4) is the one whose cathode voltage is the
most negative. In this case, the converter operates as a full-wave rectifi er.
Let us begin our analysis assuming that the source voltage may be described by

eE
ga g
= 2cos
θ
(11.2-1)
where

θωφ
ggg
t=+
(11.2-2)
In (11.2-2) , ω
g
and ϕ
g
are the radian frequency and phase of the source, respectively.
We wish to compute the steady-state average-value of v
d
, which is defi ned as

Vvd
ddg
=
−
∫

1
2
π
θ
π
π
(11.2-3)
It is noted that the output voltage is made up of two identical π intervals per cycle of
the source voltage. For the interval − π /2 ≤ θ
g
≤ π /2
436 SEMI-CONTROLLED BRIDGE CONVERTERS

ve
dga
=
(11.2-4)
Using symmetry and (11.2-1)–(11.2-4) , the average output voltage may be determined
by fi nding the average of (11.2-3) over the interval − π /2 ≤ θ
g
≤ π /2. Thus, the average-
value of v
d
may be expressed

VV E d
E
dd gg
==
=

−
∫
0
2
2
1
2
22
π
θθ
π
π
π
cos
/
/
(11.2-5)
where E is the rms value of the source voltage. We will use V
d

0
to denote the average
output voltage without commutation inductance and without fi ring delay.
Figure 11.2-2. Single-phase, full-bridge converter operation for constant output current
without l
c
and fi ring delay.
SINGLE-PHASE LOAD COMMUTATED CONVERTER 437
Operation with Commutating Inductance and
without Firing Delay

When l
c
is zero, the process of “current switching” from one thyristor to the other in
either the upper or lower part of the converter ( T 1 to T 3 to T 1 to . . . , etc., and T 2 to
T 4 to T 2 to . . . , etc.) takes place instantaneously. Instantaneous commutation cannot
occur in practice since there is always some inductance between the source and the
converter. The operation of the converter with commutating inductance and without
fi ring delay is shown in Figure 11.2-3 . During commutation, the source is short-
circuited simultaneously through T 1 and T 3 and through T 2 and T 4. Hence, if we
consider the commutation from T 1 to T 3 and T 2 to T 4 and if we assume that the short-
circuit current during commutation is positive through T 3, then

el
di
dt
ga c
=−
3
(11.2-6)
Figure 11.2-3. Single-phase, full-bridge converter operation for constant output current
with l
c
and without fi ring delay.
438 SEMI-CONTROLLED BRIDGE CONVERTERS
where i
3
is the current in thyristor T3. Substituting (11.2-1) into (11.2-6) and solving
for i
3
yields


i
l
Edt
E
l
C
c
g
gc
g
3
1
2
2
=−
=− +
∫
cos
sin
θ
ω
θ
(11.2-7)
At θ
g
= π /2, i
3
= 0, therefore


C
E
l
gc
=
2
ω
(11.2-8)
whereupon

i
E
l
gc
g3
2
1=−
ω
θ
(sin)
(11.2-9)
At the end of commutation θ
g
= π /2 + γ and i
3
= I
d
, therefore

I

E
l
d
gc
=−
2
1
ω
γ
( cos )
(11.2-10)
where γ is the commutation angle (Fig. 11.2-3 ). The uppercase ( I
d
) is used to denote
constant or steady-state quantities. During commutation, the converter output voltage
v
d
is zero. Once commutation is completed, the short-circuit paths are broken, and the
output voltage jumps to the value of the source voltage since i
d
, and hence i
ga
, are
assumed constant after commutation. Since i
ga
is constant, zero voltage is dropped
across the inductance l
c
. It is recalled that V
d


0
given by (11.2-5) is the average converter
output voltage when l
c
is zero. When l
c
is considered, the output voltage is zero during
commutation. Hence, the average output voltage decreases due to commutation. The
average converter output voltage may be determined by

VEd
V
dgg
d
=
=+
−+
∫
1
2
2
1
2
2
0
π
θθ
γ
πγ

π
cos
( cos )
/
/
(11.2-11)
If (11.2-10) is solved for cos γ and the result substituted into (11.2-11) , the average
converter output voltage with commutating inductance but without fi ring delay becomes

VV
l
I
dd
gc
d
=−
0
ω
π
(11.2-12)
SINGLE-PHASE LOAD COMMUTATED CONVERTER 439
It is interesting to note that commutation appears as a voltage drop as if the converter
had an internal resistance of ω
g
l
c
/ π . However, this is not a resistance in the sense that
it does not dissipate energy.
Operation without Commutating Inductance and
with Firing Delay

Thus far, we have considered the thyristor as a diode and hence have only considered
rectifi er operation of the converter. However, the thyristor will conduct only if the anode
voltage is positive and it has received a gating signal. Hence, the conduction of a thy-
ristor may be delayed after the anode has become positive by delaying the gating signal
(fi ring signal). Converter operation with fi ring delay but without commutating induc-
tance is shown in Figure 11.2-4 .
We can determine the average output by

VEd
E
V
dgg
d
=
=
=
−+
+
∫
1
2
22
2
2
0
π
θθ
π
α
α

πα
πα
cos
cos
cos
/
/
(11.2-13)
where α is the fi ring delay angle (Fig. 11.2-4 ). If the current is maintained constant,
the average output voltage will become negative for α greater than π /2. This is referred
to as inverter operation, wherein average power is being transferred from the dc part
of the circuit to the ac part of the circuit.
Operation with Commutating Inductance and Firing Delay
Converter operation with both commutating inductance and fi ring delay is shown
in Figure 11.2-5 . The calculation of i
3
and
V
d
are identical to that given by (11.2-6)–
(11.2-12) , except that the intervals of evaluation are different. In particular, (11.2-7)
applies, but it is at θ
g
= π /2 + α , where i
3
= 0, thus

C
E
l

gc
=
2
ω
α
cos
(11.2-14)
Commutation ends at θ
g
= π /2 + α + γ , whereupon i
3
= I
d
, thus

I
E
l
d
gc
=−+
2
ω
ααγ
[cos cos( )]
(11.2-15)
From (11.2-13)
440 SEMI-CONTROLLED BRIDGE CONVERTERS
Figure 11.2-4. Single-phase, full-bridge converter operation for constant output current
without l

c
and with fi ring delay.

VEd
V
dgg
d
=
=++
−++
+
∫
1
2
2
2
2
0
π
θθ
ααγ
παγ
πα
cos
[cos cos( )]
/
/
(11.2-16)
Solving (11.2-15) for cos ( α + γ ) and substituting the results into (11.2-16) yields the
following expression for the average output voltage with commutating inductance and

fi ring delay.
SINGLE-PHASE LOAD COMMUTATED CONVERTER 441
Figure 11.2-5. Single-phase, full-bridge converter operation for constant output current
with l
c
and fi ring delay.

VV
l
I
dd
gc
d
=−
0
cos
α
ω
π
(11.2-17)
The equivalent circuit suggested by (11.2-17) is shown in Figure 11.2-6 .
The average-value relations and corresponding equivalent circuit depicted in
Figure 11.2-6 were developed based upon the assumptions that (1) the rms amplitude
of the ac source voltage, E , is constant, and (2) the dc load current i
d
is constant and
hence denoted I
d
. This equivalent circuit provides a reasonable approximation of the
442 SEMI-CONTROLLED BRIDGE CONVERTERS

Figure 11.2-6. Average-value equivalent circuit for a single-phase full-bridge converter.
V
d0
cos
a
w
gc
l
p
d
V
d
I
+
+
-
–
Figure 11.2-7. Single-phase, full-bridge converter operation with RL load. (a) α = 0°; (b)
α = 45°; (c) α ≈ 90°, discontinuous operation.
average dc voltage even if E and i
d
vary with respect to time provided that the varia-
tions from one conduction interval to the next are small.
Modes of Operation
Various modes of operation of a single-phase, full-bridge converter are illustrated by
simulation results in Figure 11.2-7 , Figure 11.2-8 , and Figure 11.2-9 . The source voltage
is 280 V (rms) and the commutating inductance in 1.4 mH. In each case, e
ga
, i
ga

, i
1
, i
3
,
v
d
, and i
d
are plotted, where i
1
and i
3
are the currents through thyristors T 1 and T 3,
respectively. In Figure 11.2-7 , the converter is operating with a series RL load con-
nected across the output terminals, where R = 3 Ω and L = 40 mH. In Figure 11.2-7 a,
SINGLE-PHASE LOAD COMMUTATED CONVERTER 443
Figure 11.2-8. Single-phase, full-bridge converter operation with RL and an opposing dc
source connected in series across the converter terminals. (a) α = 0°; (b) α = 60°.
444 SEMI-CONTROLLED BRIDGE CONVERTERS
Figure 11.2-9. Single-phase, full-bridge converter operation with RL and an aiding dc source
connected in series across the converter terminals. (a) α = 108°; (b) α = 126°.
THREE-PHASE LOAD COMMUTATED CONVERTER 445
the converter is operating without fi ring delay. In Figure 11.2-7 b, the fi ring delay angle
is 45°. In Figure 11.2-7 c, the fi ring delay is slightly less than 90°; the current i
d
is
discontinuous. The output current is nearly constant when the converter is operating
without fi ring delay due to the large-load inductance.
In the case shown in Figure 11.2-8 , the combination of a series RL ( R = 3 Ω ,

L = 40 mH) connected in series with a constant 200-V source is connected across the
output terminals of the converter. The dc source is connected so that it opposes a posi-
tive v
d
. In Figure 11.2-8 a, the converter is operating without fi ring delay, while in Figure
11.2-8 b, the fi ring delay angle is 60°. During the zero-current portion of operation, v
d

is equal to 200 V, the magnitude of the series-connected dc source.
Inverter operation is depicted in Figure 11.2-9 . In this case, the combination of the
RL load and dc source is still connected across the output terminals of the converter,
but the polarity of the dc source is reversed. In Figure 11.2-9 a, the fi ring delay angle
is 108°. In Figure 11.2-9 b, the fi ring delay angle is 126°.
Although (11.2-17) was derived for a constant output current, it is quite accurate
for determining the average values of converter voltage and current, especially if the
current is not discontinuous. The reader should take the time to compare the calculated
converter output voltage and current using (11.2-17) with the average-values shown
in Figure 11.2-7 , Figure 11.2-8 , and Figure 11.2-9 , and to qualitatively justify any dif-
ferences that may occur.
11.3. THREE-PHASE LOAD COMMUTATED CONVERTER
A three-phase, line-commutated, full-bridge converter is shown in Figure 11.3-1 . The
voltages of the three-phase, ac source are denoted e
ga
, e
gb
, and e
gc
, and the phase cur-
rents i
ag

, i
bg
, and i
cg
. The ac source voltages may be expressed as

eE
ga g
= 2cos
θ
(11.3-1)

eE
bg g
=−
⎛
⎝
⎜
⎞
⎠
⎟
2
2
3
cos
θ
π
(11.3-2)

eE

cg g
=+
⎛
⎝
⎜
⎞
⎠
⎟
2
2
3
cos
θ
π
(11.3-3)
where E is the rms magnitude of the source voltage, θ
g
is given by (11.2-2) and is the
angular position of the source voltages, and the source frequency is ω
g
= p θ
g
. The ac
side inductance (commutating inductance) is denoted as l
c
. The thyristors are numbered
T 1 through T 6 in the order in which they are turned on and the gating or fi ring signals
for the thyristors are e
f


1
through e
f

6
. The converter output voltage and current are
denoted v
d
and i
d
, respectively. This circuit also includes a dc inductor and resistor, L
dc

and r
dc
, that may represent the armature inductance and resistance of a dc machine or
the inductance and resistance of a fi ltering circuit. Likewise, the voltage e
d
may repre-
sent the back emf of a dc machine or the capacitor voltage in a dc fi lter.
446 SEMI-CONTROLLED BRIDGE CONVERTERS
Modes of Operation
Before analyzing the converter, it is instructive to consider several modes of operation
of a three-phase, full-bridge converter illustrated in Figure 11.3-2 , Figure 11.3-3 , and
Figure 11.3-4 by simulation results. The line-to-line ac source voltage is 208 V (rms)
and the commutating inductance is 45 μ H. In each case, e
ga
, i
ga
= − i

ag
, i
1
, i
3
, v
d
, and i
d

are plotted where the currents i
1
and i
3
are the currents through thyristors T 1 and T 3,
respectively.
In Figure 11.3-2 , the converter is operating with r
dc
= 0.5 Ω , L
dc
= 1.33 mH, and
e
d
= 0. In Figure 11.3-2 a, the converter is operating without fi ring delay. It is interesting
Figure 11.3-1. Three-phase full-bridge converter.
1f
e
c
l
c

l
c
l
ag
i
as
v
bs
v
cs
v
cg
i
bg
i
ga
e
gc
e
gb
e
dc
L
d
e
d
i
dc
r
4f

e
6f
e
2f
e
3f
e
5f
e
+
–
T1
T4
T6
T2
T3
T5
+
-
+
-
+
-
+
-
+
-
+
-
Figure 11.3-2. Three-phase, full-bridge converter operation with RL load. (a) α = 0°;

(b) α = 45°; (c) α = 90°.
THREE-PHASE LOAD COMMUTATED CONVERTER 447
to note that the output current is nearly constant. In this study, there are alternately two
or three thyristors conducting; hence, this will be referred to as 2-3 mode, which is the
normal mode of operation. The fi ring delay angle is 45° in Figure 11.3-1 b (again 2-3
mode) and 90° in Figure 11.3-1 c where the output current i
d
is discontinuous. Note,
when i
d
is zero, v
d
is also zero. In this case, there are alternately 2 and 0 thyristors
conducting; hence, this will be referred to as 2-0 mode.
In the case depicted in Figure 11.3-3 , the combination of a r
dc
= 50 m Ω and
L
dc
= 133 μ H is connected in series with a e
d
= 260 V dc source is connected across
the output terminals of the converter. In Figure 11.3-2 a (2-3 mode), the converter is
Figure 11.3-3. Three-phase, full-bridge converter operation with RL and an opposing dc
source connected in series across the converter terminals. (a) α = 0°; (b) α = 35°.
448 SEMI-CONTROLLED BRIDGE CONVERTERS
operating without fi ring delay. In Figure 11.3-2 b, the fi ring delay angle is 35°, and the
output current is discontinuous (2-0 Mode). Note that when i
d
is zero, v

d
is 260 V.
Inverter operation is illustrated in Figure 11.3-4 . In this case, r
dc
= 50 m Ω ,
L
dc
= 133 μ H, and e
d
= − 260 V. In Figure 11.3-4 a, the fi ring delay angle is 140° (2-3
mode). The fi ring delay angle in Figure 11.3-4 b is 160° where discontinuous output
current occurs (2-0 mode). Clearly, when i
d
is zero, v
d
is − 260 V.
Note that while these studies depict 2-3 and 2-0 modes, other modes exist. In 3-3
mode, which we will consider later, there are always three thyristors conducting. In 3-4
mode, which occurs under heavy rectifi er loads, there are alternately three and four
thyristors conducting. In this case, the dc link becomes periodically shorted as in the
single-phase case.
Figure 11.3-4. Three-phase, full-bridge converter operation with RL and an aiding dc source
connected in series across the converter terminals. (a) α = 140°; (b) α = 160°.
THREE-PHASE LOAD COMMUTATED CONVERTER 449
Analysis and Average-Value Model
Unlike our work in Section 11.2 , herein we use a qd framework for our analysis, and
include the derivation for the ac currents (represented in terms of qd variables). The
explicit consideration of a slowly varying i
d
will yield a dynamic average-value model

of the load-commutated inverter that more accurately predicts the average dc voltage
during transients. The consideration of the average q- and d -axis components of the ac
source currents allows the model to be used in a system context and to calculate the
real, apparent, and/or reactive power supplied by the ac source using expressions devel-
oped in Chapter 3 . The use of qd variables is desirable because by suitable choice of
reference frame, the qd state variables will be constant in the steady-state, which facili-
tates a variety of analyses. The following simplifying assumptions are made herein: (1)
the three-phase source is balanced, (2) the current i
d
is varying slowly relative to the
converter switching frequency, (3) the thyristor is an infi nite impedance device when
reverse biased or when the gating signal to allow current fl ow has not occurred, (4)
when conducting the voltage drop across the thyristor is negligibly small and (5)
operation is in the 2-3 or 3-3 modes.
In order to put our work into a qd framework, let us transform the source
voltages (11.3-1)–(11.3-3) to qd variables using the reference-frame transformation. In
particular,

vK
qd
g
qg
g
dg
g
s
g
utr
ag
bg

cg
v
v
e
e
e
=
⎡
⎣
⎢
⎤
⎦
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(11.3-4)
where the “g” superscript denotes a reference frame wherein θ = θ
g
and “utr” denotes
upper two rows. This yields


vE
qg
g
= 2
(11.3-5)

v
dg
g
= 0
(11.3-6)
A goal of the model herein will be to accept q - and d -voltages in an arbitrary reference
frame and then to fi nd the currents in that same reference frame. For q - and d -voltages
in an arbitrary reference frame (emphasized with a superscript “a” herein), in which
(11.3-5) and (11.3-6) do not hold, we can utilize a frame-to-frame transformation. In
particular, if the q- and d- axis components of the source voltages are given in the
arbitrary reference frame, the transformation into the reference frame wherein (11.3-5)
and (11.3-6) hold may be deduced from the frame-to-frame transformation

fKf
qd
g
ag
qd
a
=
(11.3-7)
where, from Chapter 3 ,

ag

ga ga
ga ga
K =
−
⎡
⎣
⎢
⎤
⎦
⎥
cos sin
sin cos
θθ
θθ
(11.3-8)
450 SEMI-CONTROLLED BRIDGE CONVERTERS
and where f = [ f
q
f
d
]
T
can be a voltage v or current i, and θ
ga
= θ
g
− θ
a
, where θ
a

is the
position of the arbitrary reference frame. Manipulating (11.3-5) through (11.3-8) ,

θ
ga qg
a
dg
a
vjv=− −angle( )
(11.3-9)

E
vv
qg
a
dg
a
=
()
+
()
1
2
22
(11.3-10)
where
v
qg
a
and

v
dg
a
are the q - and d -axis voltages to the left of the ac side inductor l
c
in
Figure 11.3-1 .
The next step is to derive an expression for the average dc voltage. All dc side and
qd variables are periodic in π /3 of θ
g
. Thus, the average-values may be established for
any π /3 interval of θ
g
. It is convenient to consider the π /3 interval that begins when T 3
begins to conduct and ends when T 4 begins to conduct. The average dc voltage over
this interval may be expressed

ˆ
()vvvd
dbscsg
=−
+
+
∫
3
3
2
3
π
θ

π
π
α
α
(11.3-11)
In (11.3-11) , the “ ∧ ” is used to denote the average-value during dynamical conditions
wherein the dc current i
d
and/or the amplitude of the ac voltages E are allowed to vary,
provided that the variation from one switching interval to the next is relatively small.
In other words, the averaging interval in (11.3-11) is assumed to be small relative to
the longer-term dynamics associated with the variations in E and/or i
d
. Thus, (11.3-11)
may be interpreted as the short-term average of v
d
. Likewise, the short-term average of
i
d
(average of i
d
over a π /3 interval) will be denoted as
ˆ
i
d
. The fi ring delay angle α in
(11.3-11) is defi ned such that T 3 fi res when

θ
π

α
g
=+
3
(11.3-12)
The average dc voltage indicated in (11.3-11) may be evaluated by noting from Figure
11.3-1 that

vel
di
dt
as ga c
ag
=+
(11.3-13)

vel
di
dt
bs gb c
bg
=+
(11.3-14)

vel
di
dt
cs gc c
cg
=+

(11.3-15)
Substituting (11.3-13)–(11.3-15) into (11.3-11) yields
THREE-PHASE LOAD COMMUTATED CONVERTER 451

ˆ
() ()veedlii
dgbgcgcgbgcg
=−+−
+
+
+
+
∫
33
3
2
3
3
2
3
π
θ
π
ω
π
α
π
α
π
α

π
α
(11.3-16)
Substituting (11.3-2) and (11.3-3) into (11.3-16) and simplifying yields

ˆ
cos ( )vE lii
dcgbgcg
=+−
+
+
36 3
3
2
3
π
α
π
ω
π
α
π
α
(11.3-17)
Further simplifi cation can be obtained by observing that prior to the instant when T 3
begins to conduct, only T 1 and T 2 are conducting. Therefore,

i
abcg d d
T

g
ii
θ
π
α
=+
=−
⎡
⎣
⎤
⎦
3
0
ˆˆ
(11.3-18)
Similarly, immediately prior to the instant when T 4 begins to conduct, only T 2 and T 3
are on, therefore

i
abcg d d d d
T
g
iiii
θ
π
α
=+
=−− +
⎡
⎣

⎤
⎦
2
3
0
ˆˆˆˆ
ΔΔ
(11.3-19)
In (11.3-19) ,
Δ
ˆ
i
d
represents the change in average dc current over the given conduction
interval due to long-term dynamics. It follows from this defi nition that the derivative
of the dynamic-average rectifi er current may be approximated as

di
dt
i
dd
g
ˆˆ
=
Δ
π
ω
3
(11.3-20)
Substituting (11.3-18)–(11.3-20) into (11.3-17) and simplifying yields


ˆ
cos
ˆ
ˆ
vE lil
di
dt
dcgdc
d
=−−
36 3
2
π
α
π
ω
(11.3-21)
From Figure 11.3-1 ,
ˆ
v
d
can be related to
ˆ
i
d
and e
d
using


ˆ
ˆ
ˆ
vriL
di
dt
e
ddcd dc
d
d
=+ +
(11.3-22)
Combining (11.3-21) and (11.3-22) yields

di
dt
Erlie
Ll
d
dc c g d d
dc c
ˆ
cos
ˆ
=
−+
⎛
⎝
⎜
⎞

⎠
⎟
−
+
36 3
2
π
α
π
ω
(11.3-23)
452 SEMI-CONTROLLED BRIDGE CONVERTERS
To establish the average q - and d -axis components of the ac currents, it is assumed
that the rectifi er current is constant throughout the interval and equal to
ˆ
i
d
. It is conve-
nient to divide the interval into two subintervals; the commutation interval during which
the current is transferred from T 1 to T 3, and the conduction interval during which only
T 2 and T 3 are conducting. During the commutation interval, T 1, T 2, and T 3 are con-
ducting. Therefore, the current into the ac source must be of the form

i
abcg ag d ag d
T
iiii=−−
⎡
⎣
⎤

⎦
ˆˆ
(11.3-24)
and

vv
as bs
==0
(11.3-25)
Algebraically manipulating (11.2-2) , (11.3-1) , (11.3-2) , (11.3-13) , (11.3-14) , (11.3-24) ,
and (11.3-25) , it is possible to show that

di
dt
E
l
ag
cg
g
=
−
⎛
⎝
⎞
⎠
6
2
5
6
ω

θ
π
cos
(11.3-26)
From (11.3-26) and noting that at θ
g
= α + π /3, we have that
ii
ag d
=−
ˆ
, we conclude that

ii
l
E
ag g d
cg
g
θ
ω
αθ
π
()
=− + − −
⎛
⎝
⎜
⎞
⎠

⎟
⎡
⎣
⎢
⎤
⎦
⎥
ˆ
cos( ) cos
6
23
(11.3-27)
The commutation subinterval ends when the current in T 1, which is the a -phase current,
becomes zero. The angle from the time T 3 is turned on and T 1 is turned off is known
as the commutation angle γ . It can be found by setting (11.3-27) equal to zero. In
particular,

γα
α
ω
=− +
−
⎛
⎝
⎜
⎞
⎠
⎟
arccos
cos

2
6
li
E
cgd
ˆ
(11.3-28)
For (11.3-27) to be applicable, several conditions need to be met. First, for (11.3-28)
to be defi ned

cos
α
ω
−
≤
2
6
1
li
E
cgd
ˆ
(11.3-29)
However, this is not the only condition which must be met. Note that T 3 turns on at

θ
π
α
g
=+

3
(11.3-30)
THREE-PHASE LOAD COMMUTATED CONVERTER 453
For T 3 to turn on, the time derivative of i
ag
must be positive, so that the current in T 3
will increase. Substitution of (11.3-30) into (11.3-26) , we conclude

α
≥ 0
(11.3-31)
Similarly, the end of commutation occurs at

θ
π
αγ
g
=++
3
(11.3-32)
For commutation to complete, the time derivative of i
ag
must be positive at the end of
commutation as well. Substitution of (11.3-32) into (11.3-26) and requiring the time
derivative to be positive,

αγ π
+≤
(11.3-33)
This requirement is particularly relevant to inverter operation in which α is large.

Finally, for 2-3 mode, we must have

γ
π
≤
3
(11.3-34)
Because of these restrictions, it is useful to differentiate between the actual fi ring
delay α and the intended (or commanded) fi ring delay angle α
*
. Suppose we take α = α
*

in our analysis (11.3-28)–(11.3-34) . If all constraints are met, then this is indeed the
case. If the constraints are not met, then the converter is either in another mode or is
not operating properly (and is, e.g., experiencing commutation failures). While our
approach is not valid for these other modes, we can extend it to 3-3 mode. In this mode,
the commutation angle is exactly π /3, and the fi ring delay is increased from the intended
value to the value that corresponds to the aforementioned commutation angle. Setting
γ = π /3 in (11.3-28) and solving for the fi ring angle

α
π
ω
=−
⎛
⎝
⎜
⎞
⎠

⎟
3
2
6
arccos
li
E
cgd
ˆ
(11.3-35)
Note that mathematically, there is another solution with a change of sign on the arccos()
term; however, this alternate solution does not make sense physically since such a
solution would yield a fi ring delay that decreases with increasing
ˆ
i
d
. Note that for (11.3-
35) to be valid,

1
2
2
6
1≤≤
li
E
cgd
ω
ˆ
(11.3-36)

454 SEMI-CONTROLLED BRIDGE CONVERTERS
or α will be negative or the argument to the arccos() function will be out of range.
Summarizing, these results, we have

α
α
π
ω
=
−
⎛
⎝
⎜
⎞
⎠
⎟
⎧
⎨
⎪
⎩
⎪
*
valid - mode
arccos valid - mode
23
3
2
6
33
li

E
cgd
ˆ
(11.3-37)
In other words, if we assume α = α
*
and it yields valid 2-3 mode of operation, then
indeed α = α
*
. If this does not yield valid operation, then we can fi nd α from (11.3-35) .
If this yields valid 3-3 mode, then this is the fi ring delay. If neither assumption yields
a valid result, the operation is in another mode (e.g., 3-4 mode, wherein the converter
operates between three and four thyristors on) or the converter is not operating in a
periodic fashion.
The next step in our analysis is to determine an expression for the ac side currents.
The average q - and d -axis components can be established using

ˆ
()iid
qg
g
qg
g
gg
=
+
+
∫
3
3

2
3
π
θθ
π
α
π
α
(11.3-38)

ˆ
()iid
dg
g
dg
g
gg
=
+
+
∫
3
3
2
3
π
θθ
π
α
π

α
(11.3-39)
Since the expressions for the ac currents are different during the conduction interval
than in the commutation interval, it is convenient to break up (11.3-38) and (11.3-39)
into components corresponding to these two intervals. In particular,

ˆˆ ˆ
,
,
ii i
qg
g
qg com
g
qg cond
g
=+
(11.3-40)

ˆˆ ˆ
,,
ii i
dg
g
dg com
g
dg cond
g
=+
(11.3-41)

where

ˆ
()
,
iid
qg
g
qg
g
ggcom
=
+
++
∫
3
3
3
π
θθ
π
α
π
αγ
(11.3-42)

ˆ
()
,
iid

qg
g
qg
g
gg
cond
=
++
+
∫
3
3
2
3
π
θθ
π
αγ
π
α
(11.3-43)
THREE-PHASE LOAD COMMUTATED CONVERTER 455

ˆ
()
,
iid
dg
g
dg

g
gg
com
=
+
++
∫
3
3
3
π
θθ
π
α
π
αγ
(11.3-44)

ˆ
()
,
iid
dg
g
dg
g
gg
cond
=
++

+
∫
3
3
2
3
π
θθ
π
αγ
π
α
(11.3-45)
The commutation component of the current may be found by substituting (11.3-27)
into (11.3-24) , applying the reference-frame transformation (3.3-1) with θ = θ
g
,
and integrating in accordance with (11.3-42) and (11.3-44) . After considerable
manipulation,

ˆˆ
sin sin
,
ii
E
qg com
g
d
=+−
⎛

⎝
⎜
⎞
⎠
⎟
−−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
+
23 5
6
5
6
32
π
γα
π
α
π
π

ll
E
l
cg
cg
ω
αγα α
πω
ααγ
cos( )[cos( ) cos( )]
[cos( ) cos( )
+−
+−+
1
4
32
222]]
(11.3-46)

ˆˆ
cos cos
,
ii
dg com
g
d
=−+−
⎛
⎝
⎜

⎞
⎠
⎟
+−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
+
23 5
6
5
6
32
π
γα
π
α
π
π
EE
l

E
l
cg
cg
ω
αγα α
πω
ααγ
cos( )[sin( ) sin( )]
[sin( ) sin(
+−
+−+
1
4
32
222))]−
321
2
πω
γ
E
l
cg
(11.3-47)
To compute the conduction component of the average currents, note that after
commutation, the a -phase current remains at zero; therefore

i
abcg d d
T

ii=−
⎡
⎣
⎤
⎦
0
ˆˆ
(11.3-48)
Transforming (11.3-48) to the θ = θ
g
reference frame and utilizing (11.3-43) and
(11.3-45) ,

ˆˆ
sin sin
,
ii
qg cond
g
d
=+
⎛
⎝
⎜
⎞
⎠
⎟
−++
⎛
⎝

⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
23 7
6
5
6
π
α
π
αγ
π
(11.3-49)

ˆˆ
cos cos
,
ii
dg cond
g
d
=−+
⎛

⎝
⎜
⎞
⎠
⎟
+++
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
23 7
6
5
6
π
α
π
αγ
π
(11.3-50)
456 SEMI-CONTROLLED BRIDGE CONVERTERS
At this point, the total q - and d -axis current may be found using (11.3-40) and

(11.3-41) . The resulting currents can then be transformed back to the desired reference
frame using

ˆˆ ˆ
iKi
K
i
qd
aga
qd
g
ga
qd
g
==
[]
−1
(11.3-51)
The relationships between the previous equations are conveniently summarized in
the block diagram illustrated in Figure 11.3-5 , which represents an average-value model
of the load commutated converter. The inputs to this model include the commanded
fi ring delay α
*
, the q - and d -axis components of the source voltage in the arbitrary
reference frame, and the dc source voltage e
d
. The outputs of the model include the
dynamic-average of the rectifi er current
ˆ
i

d
and the dynamic-average of the q - and d -axis
components of the ac currents in the arbitrary reference frame.
To illustrate the dynamic response that is established using the average-value
model, it is assumed that the rated line-to-line source voltage is 208 V (rms). The com-
mutating inductance l
c
is 45 μ H. Also, r
dc
= 0.5 Ω , L
dc
= 1.33 mH, and e
d
= 0. In the
following study, the dc and ac currents are initially zero, and rated voltages are suddenly
applied at t = 0 with the fi ring delay angle α set to zero. The dynamic response is shown
in Figure 11.3-6 , wherein the following variables are plotted: i
d
—the dc current,
i
qg
g
—the
q -axis component of the ac current, and
i
dg
g
—the d -axis component of the ac current.
The ac currents are expressed in the reference frame wherein
v

dg
g
= 0
. The variables
indicated with an “ ∧ ” correspond to the average-value model in Figure 11.3-5 , while
those that do not include the “ ∧ ” correspond to the actual response. At the instant of
time indicated in Figure 11.3-6 , the fi ring delay angle is stepped to 45°. As shown, the
average-value model accurately portrays the dynamic-average dynamic response for
the given study. The steady-state waveforms for α = 0 and α = 45 ° are shown in Figure
11.3-2 a,b, respectively.
11.4. CONCLUSIONS AND EXTENSIONS
The focus of this chapter has been the development of average-value and dynamic
average-value models of line-commutated converters connected to an ideal source. A
natural extension of this work is the consideration of the connection of line-commutated
Figure 11.3-5. Average-value model of three-phase full-bridge converter.
ˆ
a
qg
v
ˆ
a
qg
i
ˆ
g
qg
i
E
ˆ
d

i
ˆ
d
i
ˆ
d
i
ˆ
d
i
ˆ
a
dg
v
ˆ
a
dg
i
ˆ
g
dg
i
(11.3-10)
(11.3-9)
(11.3-51)
(11.3-37)
(11.3-46) (11.3-47)
(11.3-49) (11.3-50)
(11.3-40) (11.3-41)
ˆ

d
i
ˆ
d
pi
ˆ
d
e
(11.3-23)
1
p
a
*
(11.3-28)
g
a
a
a
q
ga
CONCLUSIONS AND EXTENSIONS 457
converters to synchronous machines. One approach to doing this is set forth in Refer-
ences 5 and 6 . An extension of the methodology to a six-phase rectifi er connected to a
sic-phase machine is set forth in Reference 7 . In Reference 8 , a method of determining
instantaneous waveforms from an average-value model is considered using a procedure
referred to as waveform reconstruction. Throughout this work, it was assumed that
enough dc link inductance was included so that the dc current could be considered
constant. An analysis of line-commutated converter systems in which no dc-link is
present is set forth in Reference 9 . Finally, it is appropriate to consider methods for
Figure 11.3-6. Comparison of average-value dynamic response with actual response.

-500
-1000
0
0.02 second
-500
500
0
500
0
1000
a changed from 0 to 45 o
i
d
,A
i
d
i
d
i
qg
i
qg
i
dg
i
dg
i
d
,A
i

dg
,A
g
i
dg
,A
g
i
qg
,A
g
i
qg
,A
g
g
g
g
g
,A
ˆ
,A
d
d
i
i
,A
ˆ
,A
g

qg
g
qg
i
i
,A
ˆ
,A
g
dg
g
dg
i
i
ˆ
d
i
d
i
ˆ
g
qg
i
g
qg
i
g
dg
i
ˆ

g
dg
i
a changed from 0 to 45
oo
458 SEMI-CONTROLLED BRIDGE CONVERTERS
the detailed simulation of line-commutated inverters. While the literature is rich in
this subject, a particularly computationally effi cient methodology is set forth in Refer-
ence 10 .
REFERENCES
[1] C. Adamson and N.G. Hingarani , High Voltage Direct Current Power Transmission ,
Garroway Limited , London , 1960 .
[2] E.W. Kimbark , Direct Current Transmission—Vol. 1 , John Wiley and Sons , New York ,
1971 .
[3] S.B. Dewan and A. Straughen , Power Semiconductor Circuits , John Wiley and Sons , New
York , 1975 .
[4] P. Wood , Switching Power Converters , Van Nostrand Reinhold Co. , New York , 1981 .
[5] S.D. Sudhoff and O. Wasynczuk , “ Analysis and Average-Value Modeling of Line-
Commutated Converter—Synchronous Machine Systems ,” IEEE Trans. Energy Conver-
sion , Vol. 8 , No. 1 , March 1993 , pp. 92 – 99 .
[6] S.D. Sudhoff , K.A. Corzine , H.J. Hegner , and D.E. Delisle , “ Transient and Dynamic
Average-Value Modeling of Synchronous Machine Fed Load-Commutated Converters ,”
IEEE Trans. Energy Conversion , Vol. 11 , No. 3 , September 1996 , pp. 508 – 514 .
[7] S.D. Sudhoff , “ Analysis and Average-Value Modeling of Dual Line-Commutated
Converter—6-Phase Synchronous Machine Systems ,” IEEE Trans. Energy Conversion ,
Vol. 8 , No. 3 , September 1993 , pp. 411 – 417 .
[8] S.D. Sudhoff , “ Waveform Reconstruction in the Average-Value Modeling of Line-
Commutated Converter—Synchronous Machine Systems ,” IEEE Trans. Energy Conver-
sion , Vol. 8 , No. 3 , September 1993 , pp. 404 – 410 .
[9] J.T. Alt , S.D. Sudhoff , and B.E. Ladd , “ Analysis and Average-Value Modeling of an Induc-

torless Synchronous Machine Load Commutated Converter System ,” IEEE Trans. Energy
Conversion , Vol. 14 , No. 1 , March 1999 , pp. 37 – 43 .
[10] O. Wasynczuk and S.D. Sudhoff , “ Automated State Model Generation Algorithm for Power
Circuits and Systems ,” IEEE Trans. Power Systems , Vol. 11 , No. 4 , November 1996 ,
pp. 1951 – 1956 .

PROBLEMS
1 . Using the average-value equations derived in Section 11.2 , calculate the average
dc voltage and current for each of the conditions in Figure 11.2-7 . Compare with
the average-values plotted in Figure 11.2-7 .
2 . Using the average-value equations derived in Section 11.3 , calculate the average
dc voltage and current for each of the conditions in Figure 11.3-2 . Compare with
the average-values plotted in Figure 11.3-2 .
3 . Assume that the ac source voltage applied to the three-phase load commutated
converter have an acb phase sequence. Indicate the sequence in which the thyristors
should be fi red.