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Lectures 10-11 Effect of source inductance on phase
controlled AC-DC converters
10.1 Single-phase converters
10.1.1 Overlap in single-phase, CT fully controlled converter
T1
Ls
iL
i1
ip
R
vi
vo
L
V m ax sin ω t
vs
T2
Ls
i2
Figure 10.1 Single-phase C-T converter with source inductance
The presence of source inductance means that commutation of load current from
one thyristor to the next, as they are triggered with a firing angle α, can not be
instantaneous. This source inductance, Ls, is invariably because of the inductance
of the supply lines and the leakage inductance of the input transformer. For this
circuit, the overlap of conduction for the duration µ makes the output voltage zero
(which is the mean of the overlapping input voltages) during this period .
Lectures 10&11 – Overlap in AC-DC Converters
10&11-1
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T1
Ls
iL
i1
ip
R
vi
vo
L
V m a x s in ω t
vs
T2
Ls
i2
Figure 10.2 Waveforms in the converter of figure 10.1, commutation overlap and
commutation notches, for α = 45°.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-2
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T1
Ls
iL
i1
ip
R
vi
vo
L
V max sin ω t
vs
T2
Ls
i2
Figure 10.3 Waveforms in the converter of figure 10.1, commutation overlap and
commutation notches, for α = 130°.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-3
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The part of the input voltage waveform which is missing from the output voltage
is given by
Vmax sinω t = Ls
∫
α +µ
α
di
dt
10.1
Vmax sin (ω t ) d (ω t ) = ω Ls
∫
Id
0
10.2
di
where Vmax is the peak of the input ac voltage and Id is the dc level of the output
(load) current.
Note that in each half cycle of the above waveform, the input current through the
incoming thyristor rises from zero to Id in time µ/ω, starting from the instant of
firing. Thus, integrating equation 10.2,
−Vmax  cos (α + µ ) − cos α  = ω Ls I d
cos (α + µ ) = cosα −
ω Ls
Vmax
Id
10.3
The commutation or overlap angle µ can be found from expression 10.3. It
increases with load current Id at any firing angle. The output dc voltage is then
given by
Vd =
1
π
∫
π +α
α
Vmax sin (ω t ) d (ω t ) −
1
π
∫
α +µ
α
Vmax sin (ω t )d (ω t )
10.4
=
2Vmax
π
cos α −
ω Ls
I
π d
α = 30°
µ°
Vd
α = 30°
Id
α = −120°
(a)
(b)
Id
Figure 10. 4 Variation of Vd and commutation angle µ with Id and firing angle α.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-4
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10.1.2 Overlap in single-phase, fully controlled bridge converter
Without overlap, two thyristors conduct at all times, if turned on. With overlap, all
four thyristors conduct during overlap (or commutation of current from the out
going pair to the incoming pair of thyristors).
vo
ip
i
Ls
V maxsin ωt
T1
T3
Vd
vsi
T2
Id
Load
T4
(a)
vo
iL
vp
vp
ip
ip
v si
v s -v si
(b)
Figure 10.5 (a) Single-phase, F-C, bridge converter with source inductance and
(b) its waveforms, for α = 45°.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-5
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vs
vs
-vs
vo
iL
vp
ip
vp
ip
vsi
vs-vsi
α =145o
Figure 10.5(c) Waveforms in the converter of figure 10.5(a) for α = 145°
The part of the input voltage waveform which is missing from the output during
overlap is given by
Vmax sin ω t = Ls
di
dt
10.5
As before
∫
α +µ
α
Vmax sin (ω t ) d (ω t ) = ω Ls
cos (α + µ ) = cos α −
∫
Id
−Id
di = 2ω Ls I d
2ω Ls
Id
Vmax
Lectures 10&11 – Overlap in AC-DC Converters
10.6
10&11-6
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Note that during overlap, the input current i, which flows through Ls, changes by
2Id. The overlap angle µ can be found from the above expression. The angle µ
increases when larger load current is commutated.
The dc output voltage of the converter with overlap is given by
Vd =
1
π
=
∫
α +π
α
2Vmax
π
Vmax sin (ω t )d (ω t ) −
cos α −
2ω Ls
π
1
π
∫
α +µ
α
Vmax sin (ωt )d (ω t )
Id
α = 30°
10.7
µ°
Vd
α = 30°
α = 125°
Id
Id
Figure 10.6(a) Variation of Vd and commutation angle µ with load current and
firing angle α for the converter of figure 10.5.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-7
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10.1.3 Overlap in single-phase, half-controlled converter
vo
ip
Ls
T1
T3
iL
Vd
Vmaxsinωt
Df
D2
LOAD
D4
Figure 10.7
The commutation of the load current through the thyristors and the diodes occur in
two stages. At first, the load current commutates to the freewheeling diode at the
zero crossings of the input voltage. When the incoming thyristor is triggered, the
freewheeling load current commutates to this thyristor. The part of the input
voltage waveform missing from the output is thus dropped across the source
inductance due to the load current rising to Id during the commutation overlap
angle µ when a thyristor is triggered. Consequently, the overlap angle µ and the dc
output voltage Vd for this converter with source inductance Ls are given by,
1
ω
∫
α +µ
α
Vmax sin ω td( ω t ) = Ls
∴ cos (α + µ ) = cos α −
ω Ls
Vmax
∫
Id
0
di
Id
10.8
The dc output voltage is given by
1
Vd = 
π
∴ Vd =
∫
π
α
Vmax
π
Vmax sin ω td( ω t ) −
[1 + cos α ] −
∫
α +µ
α

Vmax sin ω td( ω t )

ω Ls
Id
π
Lectures 10&11 – Overlap in AC-DC Converters
10&11-8
10.9
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500
vs
-500
vo
iL
iDf
ip
vp
vsi
vs-vsi
For α = 45°
Figure 10.8. Waveforms in the H-C converter with source inductance; for α = 45°.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-9
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vs
vo
iL
iDf
vp
ip
vsi
vs-vsi
0
π
2π
3π
For α = 145°
Figure 10.9. Waveforms in the H-C converter with source inductance; for α =
145°
Lectures 10&11 – Overlap in AC-DC Converters
10&11-10
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10.2 Three-phase converters
10.2.1 Overlap in a three-phase, CT, fully-controlled converter,
Ls
ic
Ls
ia
c
T3
vcn
n
van
a
iL
vabi
vab
vbn
T1
Ls
b
ib
R
T2
vo
L
Figure 10.10
In this circuit, the converter output voltage during overlap is half of the incoming
and outgoing voltages. For instance, when T1 is triggered with angle α, after the
crossover of van and vcn, the output voltage vo is given by
vo =
van + vcn
2
10.10
The voltage pulse missing from the output voltage waveform is bounded by va and
vo. This is given by
voL = van −
van − vcn van − vcn vacl − l
=
=
2
2
2
10.11
3 Vmax sin ω t
where Vmax is the peak of the line-neutral
2
By expressing vol =
voltage.
1
ω
∫
α +µ
3 Vmax
2
α
sin ω td (ω t ) = Ls
Hence cos (α + µ ) = cos α −
2ω L s
3 V max
∫
Id
di
0
2ω L s
Id
V max l − l
I d = cos α −
10.12
from which µ can be found. The dc output voltage is given by
1 

Vd =
2π / 3 

∫
5π
+α
6
π
6
+α
Vmax sin ω td (ω t ) −
Lectures 10&11 – Overlap in AC-DC Converters
3 Vmax
2
10&11-11
∫
α +µ
α

sin ω td (ω t )


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3 3Vmax
3ω Ls
cos α −
Id
2π
2π
10.13
Figure 10.11 Waveforms in the circuit of figure 10.10 for α = 45°
Lectures 10&11 – Overlap in AC-DC Converters
10&11-12
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Figure 10.12 Waveforms in the circuit of figure 10.10 for α = 130°
Lectures 10&11 – Overlap in AC-DC Converters
10&11-13
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10.2.2 Overlap in 3-φ, fully-controlled bridge converter
vo
+VD/2
ia
van
T3
T1
Ls
R
ib
vbn
vcn
Ls
T4
T6
Load
Vd
ic
n
iL
T5
L
T2
−VD/2
Figure 10.13
In this converter, overlap due to source inductance occur every 60°. When a
thyristor connected with the positive voltage is triggered, the positive dc bus
voltage becomes the average of the incoming and the outgoing phase voltages.
The same happens when one of the thyristors connected with the negative dc bus
is triggered.
During the commutation angle µ, current in the outgoing line falls gradually to
zero, while the current in the incoming line rises to Id. The voltage pulse missing
from the output voltage waveform is the difference between the incoming line
voltage minus the average of the incoming and the outgoing line voltages. For
instance, when thyristor T1 is triggered with angle α, this voltage is given by,
voL = van −
van + vcn van − vcn vacl −l
=
=
2
2
2
from α to α + µ on the vac
waveform.
Using a similar analysis as in the previous section, it can be shown that for this
converter,
cos (α + µ ) = cos α −
and
Vd =
3Vmax l − l
π
cos α −
2ω Ls
Id
Vmax l −l
3ω Ls
π
Id
Lectures 10&11 – Overlap in AC-DC Converters
10.14
10.15
10&11-14
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Figure 10.14 Waveforms in the converter of figure 10.13; α = 45°
Lectures 10&11 – Overlap in AC-DC Converters
10&11-15
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Figure 10.15 Waveforms in the converter of figure 10.13; α = 130°
Lectures 10&11 – Overlap in AC-DC Converters
10&11-16
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10.2.3 Overlap in 3-φ, half-controlled bridge converter
vo
+VD/2
ia
van
Ls
T5
ib
vbn
n
T3
T1
Df
ic
vcn
Ls
T4
T6
iL
iDf
R
Load
Vd
L
T2
−VD/2
Figure 10.16
It can be shown that
Vd =
3Vmax l − l
3ω Ls
Id
[1 + cosα ] −
2π
2π
10.16
The above equation assumes that load current transfers to the free-wheeling diode
completely before the next thyristor is triggered, and that the load current transfers
from the free-wheeling diode to the incoming thyristor when it is triggered.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-17
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10.3 Converter voltage regulation due to source inductance
The commutation overlap affects the performance of all converter circuits,
including the rectifiers which rely for their operation on natural commutation. The
overlap phenomenon affects the operation of the converter in many ways, such as
introducing a voltage
regulation characteristic. The voltage regulation
characteristic of the converter is in general given by
Vd =Vdcmax cosα − XLIL
and
for fully controlled converters
Vd = Vdc max ( 1 + cos α ) − X L I L for half controlled converters.
10.17
10.18
Here Vdcmax is the maximum dc output voltage for the converter circuit and XL is
a parameter determined by the input source inductance Ls and the converter
circuit.
V dm ax
α = 0°
α = 30°
Vd
α = 60°
α = 75°
0
α = 90°
α = 110°
α = 130°
α = 160°
− V dm ax
Id
Figure 10.17 Voltage regulation characteristic of the fully-controlled converter
Lectures 10&11 – Overlap in AC-DC Converters
10&11-18
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10.3.1 Other effects
A. Output voltage ripple
The output voltage waveform of the converter is modified by the loss of voltage
pulses due to overlap. Analysis of the output voltage waveforms shows that the
output ripple frequencies remain the same as for the converter without overlap.
Only the output ripple voltage amplitudes are affected. The overlap actually
reduces the amplitudes of the ripples. For a p-pulse converter the output ripple
voltages, assuming continuous conduction, are given by
 nπ 
Vd max cos   n sina × sinn (ωt + µ ) − cos α cos n (ωt + µ )
 p 

vn ( t ) =
2
+ n sin (α + µ ) sinnωt − cos (α + µ ) cos nωt 
n −1


10.19
where α = firing angle
µ = commutation angle
n = order of the output voltage ripple
B Input current harmonics
The input current waveforms of the converter have gentler rise and fall of current,
rather than the abrupt changes in the converter without overlap. It is obvious that
the harmonic amplitudes of the input current are reduced as a result of overlap.
Assuming linear transitions, of duration m, the input current harmonics of a sixpulse converter is given by
 nπ
2 sin 
 3
bn =
nπ

 nµ 
 sin 

×
 2 
nµ
2
10.20
C. Commutation notches
During commutation overlap, simultaneous conduction of two thyristors makes
the line to line voltage of the two input lines under commutation zero. Other line
to line voltages are found by obtaining the difference of their potentials taking into
account the commutation. If the converter input voltage terminals are shared with
other loads, (these voltages are invariably used as signals which control the
triggering of the thyristors), then adequate filter circuits must be used to reduce
the commutation notches to acceptable levels. If these notches are not adequately
filtered out, firing angles vary from thyristor to thyristor, leading to uneven output
voltage ripples and thyristor currents.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-19
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10.4 Converter characteristics with discontinuous load current
In analyzing ac-dc converter circuits we have so far assumed that the load current
was continuous, i.e., positive at all times. This allowed the output voltage to be
described in terms of the firing angle α and the input ac voltage, Vmax. When load
current becomes discontinuous, the output voltage becomes either zero or equal to
the voltage of the active dc source in the load. In general, it results in an increased
output dc voltage (because of the removal of part of the negative voltage of the ac
source in the load).
The dc output voltage of a fully-controlled converter with discontinuous
conduction and input source inductance are thus of the form as in the figure
below.
Consider the figure below in which a single-phase bridge supplies a load with a
back emf of Eb.
vo
ip
T1
T3
Id
is
R
Vmaxsinωt
Vd
L
T2
T4
Eb
Figure 10.18 Single-phase bridge rectifier with discontinuous current.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-20
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For this circuit,
L
di
+ Ri = Vmax sin ω t − Eb
dt
10.21
The solution for i is
V
E
− R/L t
i = Ae ( ) + max sin( ω t − ϕ ) − b
Z
R
10.22
2
2
where Z = R + ( ω L )
and
10.23
 ωL 

 R 
ϕ = tan −1 
10.24
The instantaneous load current i is obtained by noting that i = 0 at the effective
firing angle α which is either the actual firing angle (when Eb < Vmaxsinα) or the
angle sin− 1(Eb/Vmax)
Thus
V
i = max
R
 R
Eb
 sin( ω t − ϕ ) −
Vmax
 Z
R
 Eb
 − ( ω t −α )

R
sin( α − ϕ ) e ω L
−
 + 
Z
Vmax


10.25
Current i falls to zero at angle β which is obtained by equating the above equation
to zero. Thus
Eb
Vmax
eRβ / ω L ×
= e Rα / ω L
E
cos ϕ sin( α − ϕ ) − b
Vmax
cos ϕ sin( β − ϕ ) −
10.26
The extinction angle β is found by solving the above transcendental equation. Vd
can then be calculated.
By solving for β for a firing angle α, the Vd - IL characteristic of the converter for
different load parameter values (R, L and Eb) can be obtained. The boundary
between continuous and discontinuous conduction is a semicircle. The Vd - IL
Lectures 10&11 – Overlap in AC-DC Converters
10&11-21
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characteristics of a half controlled and a fully controlled converter are shown in
figures below. These also include the effect of commutation overlap.
Vd
V dmax
α = 0°
α = 30°
α = 60°
α = 75°
0
α = 90°
α = 110°
α = 130°
α = 160°
− V dmax
Id
Figure 10.19 Converter Vd vs α and regulation characteristic with source
inductance and discontinuous conduction.
Lectures 10&11 – Overlap in AC-DC Converters
10&11-22
F. Rahman