Power Electronics
AC-AC Converters
Dr. Firas Obeidat
1
Table of contents
1
• The Three Phase AC Voltage Controller – Y
Connected Resistive Load
2
• The Three Phase AC Voltage Controller – Δ
Connected Resistive Load
2
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
S1
 The power delivered to the load in
three phase AC voltage controller
with Y-connected resistive load is
controlled by the delay angle α on
each thyristor. The six thyristors
are turned on in the sequence 1-23-4-5-6, at 60 intervals. Gate
signals
are
maintained
throughout
the
possible
conduction angle.
-
+
S4
S3
A
B
-
+
S6
b
R
a
R
n
N
R
S5
C
-
+
c
S2
 The instantaneous voltage across each phase of the load is determined by
which thyristors are conducting. At any instant, three thyristors , two
thyristors , or no thyristors are ON.
 The instantaneous load voltages are either a line-to-neutral voltage (three
thyristors ON), one-half of a line-to-line voltage (two thyristors ON), or
zero (none on).
 Which thyristors are conducting depends on the delay angle α and on the
source voltages at a particular instant. The ranges of α that produce
particular types of load voltages are 0< α<60o, 60o< α<90o and 90o< α<150o.
3
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
 To study the output voltage Van, Vbn and Vcn, triggering angle α must be
determined first.
 Suppose that the triggering angle α equal to 30o.
 The output voltage Van will be studies when α= 30o, and the same procedure
can be applied to get the output voltages Vbn and Vcn at any triggering angle
from 0 to 60o.
VAN
VBN
VCN
Let
𝑉𝐴𝑁 = 𝑉𝑚 sin𝜔t
𝑉𝐵𝑁 = 𝑉𝑚 sin(𝜔t-2π/3)
𝑉𝐶𝑁 = 𝑉𝑚 sin(𝜔t-4π/3)
30o 60o
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
𝑉𝐴𝐵 = 3𝑉𝑚 sin(𝜔t+π/6)
𝑉𝐵𝐶 = 3𝑉𝑚 sin(𝜔t-π/2)
𝑉𝐶𝐴 = 3𝑉𝑚 sin(𝜔t-7π/6)
4
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
1. Determine the triggering angle α (for example α= 30o).
2. Determine the conducting period for each thyristor.
• As seen from the input phase voltages VAN, VBN and VCN, every 30o there is
a change that affect on the conduction of each thyristor.
VBN
VAN
VAN=VCN
VAN=VBN
VCN=0
30o 60o
VCN
VBN=0
VBN=VCN
VAN=0
VCN=0
VBN=0
VAN=0
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
VBN=VCN
VAN=VCN
VAN=VBN
5
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
VAN
• Because α=30o, S1 will be
triggered at 30o, S3 will be
triggered at 150o (120o+α), S5
will be triggered at 270o
(240o+α).
• S4 will be triggered after S1
by 180o, S6 will be triggered
after S3 by 180o S2 will be
triggered after S5 by 180o.
• The six SCRs are turned on
in the sequence 1-2-3-4-5-6,
at 60o intervals.
30o 60o
VBN
VCN
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
iG1
iG2
S1
iG3
-
+
S4
S3
iG4
A
B
-
+
S6
b
R
a
R
n
iG5
N
R
S5
C
-
+
c
iG6
S2
Dr. Firas Obeidat
Faculty of Engineering
6
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
3. Study the conduction period for each thyristor.
• Each thyristor will be studied at each angle that affect on the conduction of
each thyristor to determine if the thyristor will conduct or not.
• The conduction period will be studied for S1.
• At 30o, VAN=VCN, VAN and VCN are positive, while VBN is negative. So, the
current will flow from phase A and C to phase B. S1 will be on from 30o to
60o.
S1
VAN
+
VBN
VCN
-
+
S4
S3
A
b
-
+
B
S6
30o
a
R
60o
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
R
n
N
R
S5
+
C
-
+
S2
c
iG1
7
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
• At 60o, VCN=0, VAN is positive, while VBN is negative. So, the current will
flow from phase A to phase B. S1 will be ON from 60o to 90o.
S1
+
VAN
-
+
VBN
VCN
S4
S3
A
B
b
-
+
S6
30o 60o
a
R
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
R
n
N
R
S5
0
C
-
+
c
iG1
S2
8
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
• At 90o, VBN=VCN, VAN is positive, while VBN and VCN are negative. So, the
current will flow from phase A to phase B and phase C. S1 will be ON from
90o to 120o.
+
VAN
-
+
VBN
VCN
S4
S3
A
B
b
-
+
S6
30o 60o
a
R
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
R
n
N
R
S5
C
-
+
c
iG1
S2
9
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
• At 120o, VBN=0, VAN is positive, while VCN is negative. So, the current will
flow from phase A to phase C. S1 will be ON from 120o to 150o.
S1
VAN
+
VBN
VCN
-
+
S4
S3
0
A
b
-
+
B
S6
30o 60o
a
R
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
R
n
N
R
S5
C
-
+
c
iG1
S2
10
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
• At 150o, VAN=VBN, VAN and VBN are positive, VCN is negative. So, the current
will flow from phase A and B to phase C. S1 will be ON from 150o to 180o.
VAN
S1
VBN
VCN
+
-
+
S4
S3
+
A
B
30o 60o
b
-
+
S6
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
a
R
R
n
N
R
S5
C
-
+
S2
c
iG1
11
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
• At 180o, VAN=0, VBN is positive, VCN is negative. So, the current will flow
from phase B to phase C and the current will not flow in phase A. S1 will be
OFF from 180o to 210o.
• At 210o, VAN= VCN, VBN is positive, VAN and VCN are negative. So, the
current will flow from phase B to phase A and C. S1 will be OFF from 210o
to 240o.
• At 240o, VCN= 0, VBN is positive, VAN is negative. So, the current will flow
from phase B to phase A. S1 will be OFF from 240o to 270o.
• At 270o, VBN=VCN, VBN and VCN are positive, VAN is negative. So, the current
will flow from phase B and C to phase A. S1 will be OFF from 270o to 300o.
• At 300o, VBN=0, VCN is positive, VAN is negative. So, the current will flow
from phase C to phase A. S1 will be OFF from 300o to 330o.
• At 330o, VAN=VBN, VCN is positive, VAN and VBN are negative. So, the current
will flow from phase C to phase A and B. S1 will be OFF from 330o to 360o.
12
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
VAN
VBN
VCN
•
The same procedure can be
done for other thyristors to
determine the conduction
period for each thyristor.
4. Determine the thyristors
that will be conducted for
each period. For example,
when 0<ωt<30o, S5 and S6
will be conducted, when
30o<ωt<60o, S5, S6 and S1
will be conducted, ect…
30o 60o
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
iG1
iG2
iG3
iG4
iG5
iG6
5,6 5,6,1 6,1 6,1,2 1,2 1,2,3 2,3 2,3,4 3,4 3,4,5 4,5 4,5,6
13
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤
S1
α<60o
0
-
+
S4
5. Determine the output voltages (Van,
Vbn and Vcn) at each period.
• The output voltage Van can be
found by applying the thyristors
that will be conducted at each
period.
• when 0<ωt<30o, S5 and S6 will be
conducted which means that the
current will flow from phase C to
phase B, and there is no current
will flow in phase A. In this period
Van=0.
• when 30o<ωt<60o, S5, S6 and S1 will
be conducted which means that the
current will flow from phase A and
C to phase B. In this period
Van=VAN.
S3
A
S6
Philadelphia University
a
R
R
n
N
R
S5
+
-
+
C
c
S2
0<ωt<30o
S1
+
-
+
S4
S3
A
B
b
-
+
S6
a
R
R
n
N
R
S5
+
30o<ωt<60o
Faculty of Engineering
+
B
C
Dr. Firas Obeidat
b
-
-
+
S2
c
14
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
•
•
VAN
when 60o<ωt<90o, S6 and S1 will be
conducted which means that the
current will flow from phase A to
phase B, and there is no current will
flow in phase C. In this period
Van=VAB/2.
After completing the output voltage
Van for the whole period, the
waveform of Van can be found from
the waveforms VAN, VAB/2, VAC/2 and
zero.
S1
30o 60o
VBN
VCN
90o 120o 150o 180o 210o 240o 270o 300o 330o 360o
iG1
iG2
+
-
+
iG3
S4
iG4
S3
A
B
b
-
+
S6
a
R
R
n
N
R
iG5
iG6
S5
60o<ωt<90o
0
C
-
+
c
S2
Dr. Firas Obeidat
Faculty of Engineering
Van
5,6 5,6,1 6,1 6,1,2 1,2 1,2,3 2,3 2,3,4 3,4 3,4,5 4,5 4,5,6
0
VAN VAB/2 VAN VAC/2 VAN
0
VAN VAB/2 VAN VAC/2 VAN
15
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
VAN
30o
 The figure below shows the
final shape for the output
voltage Van. The rms voltage
for this waveform will be
less than the rms voltage for
the input voltages VAN, VBN
and VCN.
60o
VBN
VCN
90o 120o 150o 180o 210o 240o 270o 300o 330o
360o
iG1
iG2
iG3
iG4
iG5
iG6
Van
Van
5,6
0
5,6,1
6,1
6,1,2
1,2
1,2,3
VAN VAB/2 VAN VAC/2 VAN
2,3
0
2,3,4
3,4
3,4,5
4,5
4,5,6
VAN VAB/2 VAN VAC/2 VAN
Van
VAN
VAC/2
VAB/2
30o
60o
90o
120o 150o
180o
o
210o 240
270o
300o
330o 360o
16
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 0 ≤ α<60o
 The other output waveforms (Vbn
and Vcn) can be found by the
same way, and the resulting
waveform of Vbn will be the same
as the waveform of Van shifted by
120o, and the waveform of Vbn
will be the same as the waveform
of Van shifted by 240o.
 At any instant, three thyristors
or two thyristors are ON. The
instantaneous load voltages are
either a line-to-neutral voltage
(three thyristors ON), one-half of
a line-to-line voltage (two
thyristors ON), or zero (none
ON)
 Because the load is resistive load,
the shape of the output current is
similar to the shape of the output
voltage.
17
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 60o ≤ α<90o
 Only two thyristors conduct at any one time when the delay angle is
between 60o and 90o.
 For α=75o, prior to 75o, S5 and S6 are conducting, and Van=0. When S1 is
turned on at 75o, S6 continues to conduct, but S5 must turn off because VCN
is negative. Voltage Van is then VAB/2. When S2 is turned on at 135 , S6 is
forced off, and Van= VAC/2. The next thyristor to turn on is S3, which forces
S1 off, and Van=0. One thyristor is always forced off when thyristor is
turned on for α in this range. Load voltages are one-half line-to-line
voltages or zero.
18
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
When 90o ≤ α<150o
 Only two thyristors conduct
at any one time when the
delay angle is between 90o and
150o.
 For α=120o, prior to 120o, no
thyristors are on, and Van=0.
At α=120o, S1 is turned on at
120o, S6 still has a gate signal
applied. Since VAB is positive,
both S1 and S6 are forwardbiased and begin to conduct,
and Van=VAB/2. Both S1 and S6
turn off when VAB becomes
negative. When a gate signal
is applied to S2, it turns on,
and S1 turns on again.
For α>150o, there is no time interval when thyristor is forward-biased
while a gate signal is applied. Output voltage is zero for this condition.
19
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
The rms output voltage for a Y– connection loads are found to be:
 For 0o≤ α<60o
𝑽𝒐,𝒓𝒎𝒔 =
𝑽𝒐,𝒓𝒎𝒔 =
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝝅
𝝅 𝟑
𝜶
𝟏
𝝅
𝟏
𝟐𝝅
𝝅 𝟑
𝜶
𝟐𝝅
𝟎
𝑽𝒂𝒏 𝟐 𝒅𝝎𝒕
𝝅 𝟑+𝜶
𝟐
𝑽𝑨𝑵 𝒅𝝎𝒕 +
(𝑽𝒎 sin𝜔t )𝟐 𝒅𝝎𝒕 +
𝝅 𝟑+𝜶
(
𝝅 𝟑
𝑽𝒐,𝒓𝒎𝒔 = 𝟑𝑽𝒎
𝝅 𝟑
𝑽𝑨𝑩 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝟑𝑽𝒎 sin(𝜔t+π/6) 𝟐
) 𝒅𝝎𝒕 +
𝟐
𝟐𝝅 𝟑
𝟐
𝑽𝑨𝑵 𝒅𝝎𝒕 +
𝝅 𝟑+𝜶
𝟐𝝅 𝟑
𝟐𝝅 𝟑+𝜶
(𝑽𝒎 sin𝜔t )𝟐 𝒅𝝎𝒕 +
𝝅 𝟑+𝜶
𝟐𝝅 𝟑
𝟐𝝅 𝟑+𝜶
(
𝟐𝝅 𝟑
𝑽𝑨𝑪 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝝅
𝑽𝑨𝑵 𝟐 𝒅𝝎𝒕
𝟐𝝅 𝟑+𝜶
𝟑𝑽𝒎 sin(𝜔t−π/6) 𝟐
) 𝒅𝝎𝒕 +
𝟐
𝝅
(𝑽𝒎 sin𝜔t )𝟐 𝒅𝝎𝒕
𝟐𝝅 𝟑+𝜶
𝟏 𝝅 𝜶 𝒔𝒊𝒏𝟐𝜶
− +
𝝅 𝟔 𝟒
𝟖
20
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
The rms output voltage for a Y– connection loads are found to be:
 For 60o≤ α<90o
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝟐𝝅
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝝅
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝝅
𝟐𝝅
𝑽𝒂𝒏 𝟐 𝒅𝝎𝒕
𝟎
𝝅 𝟑+𝜶
𝑽𝑨𝑩 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝜶
𝟐𝝅 𝟑+𝜶
𝟐𝝅 𝟑
𝑽𝑨𝑪 𝟐
(
) 𝒅𝝎𝒕
𝟐
𝝅 𝟑+𝜶
𝜶
𝑉𝑜,𝑟𝑚𝑠 = 3𝑉𝑚
𝟑𝑽𝒎 sin(𝜔t+π/6) 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝟐𝝅 𝟑+𝜶
(
𝟐𝝅 𝟑
𝟑𝑽𝒎 sin(𝜔t−π/6) 𝟐
) 𝒅𝝎𝒕
𝟐
1 𝜋 3𝑠𝑖𝑛2𝛼
3𝑐𝑜𝑠2𝛼
+
+
𝜋 12
16
16
21
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
The rms output voltage for a Y– connection loads are found to be:
 For 90o≤ α<150o
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝟐𝝅
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝝅
𝑽𝒐,𝒓𝒎𝒔 =
𝟏
𝝅
𝟐𝝅
𝑽𝒂𝒏 𝟐 𝒅𝝎𝒕
𝟎
𝟓𝝅 𝟔
𝑽𝑨𝑩 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝜶
𝟓𝝅 𝟔
𝜶
𝑽𝒐,𝒓𝒎𝒔 = 𝟑𝑽𝒎
𝟕𝝅 𝟔
𝝅
𝑽𝑨𝑪 𝟐
(
) 𝒅𝝎𝒕
𝟐
𝟑+𝜶
𝟑𝑽𝒎 sin(𝜔t+π/6) 𝟐
(
) 𝒅𝝎𝒕 +
𝟐
𝟕𝝅 𝟔
(
𝝅 𝟑+𝜶
𝟑𝑽𝒎 sin(𝜔t−π/6) 𝟐
) 𝒅𝝎𝒕
𝟐
𝟏 𝟓𝝅 𝜶 𝒔𝒊𝒏𝟐𝜶
𝟑𝒄𝒐𝒔𝟐𝜶
− +
−
𝝅 𝟐𝟒 𝟒
𝟏𝟔
𝟏𝟔
22
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Y Connected Resistive Load
Example: a three phase bidirectional AC voltage controller Y-connected
connected to resistive load (R=10Ω). The supply voltage VL-L=208V, f=60Hz. If
α=π/6, find the output voltage rms value and the input power factor.
𝑉𝑜,𝑟𝑚𝑠 = 3𝑉𝑚
𝐼𝑜,𝑟𝑚𝑠
1 𝜋 𝛼 𝑠𝑖𝑛2𝛼
1 𝜋 π/6 𝑠𝑖𝑛2π/6
− +
= 208
−
+
= 83𝑉
𝜋 6 4
8
𝜋 6
4
8
𝑉𝑜,𝑟𝑚𝑠 83
=
=
= 8.3𝐴
𝑅
10
𝑃𝑜,𝑎𝑐 = 3𝐼𝑜,𝑟𝑚𝑠 2 𝑅 = 3 × 8.32 × 10 = 2066.7𝑊
𝑉𝑖𝑛𝑝𝑢𝑡,𝑟𝑚𝑠 =
208
2 3
= 84.9𝑉
𝑆 = 3𝐼𝑜,𝑟𝑚𝑠 𝑉𝑖𝑛𝑝𝑢𝑡,𝑟𝑚𝑠 = 3 × 8.3 × 84.9 = 2114.4𝑉𝐴
𝑝𝑓 =
𝑃𝑜,𝑎𝑐 2066.7
=
= 0.977
𝑆
2114.4
23
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Δ Connected Resistive Load
 The voltage across a load resistor is the corresponding line-to-line voltage
when a thyristor in the phase is on. The delay angle is referenced to the
zero crossing of the line-to-line voltage. thyristors are turned on in the
sequence 1-2-3-4-5-6.
The line current in each
phase is the sum of two of
the delta currents:
24
Dr. Firas Obeidat
Faculty of Engineering
Philadelphia University
The Three Phase AC Voltage Controller – Δ Connected Resistive Load
The relationship between rms line
and delta currents depends on the
conduction angle of the thyristors.
For small conduction angles (large α),
the delta currents do not overlap, and
the rms line currents are
For large conduction angles (small α),
the delta currents overlap, and the
rms line current is larger than √2IΔ.
In the limit when γ (α=0), the delta
currents and line currents are
sinusoids. The rms line current is
determined from ordinary threephase analysis.
Current waveforms for α=130o
The range of rms line current is
therefore
depending on α
Dr. Firas Obeidat
Faculty of Engineering
Current waveforms for α=90o
Philadelphia University
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