2008 Lecture 6 - Uncontrolled rectifier circuits II 6... · 2009-03-31 · Elec4614 Power...
Transcript of 2008 Lecture 6 - Uncontrolled rectifier circuits II 6... · 2009-03-31 · Elec4614 Power...
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Lecture 6 - Diode 6-1 F. Rahman rectifier circuits II
Lecture 6 - Uncontrolled rectifier circuits II
Single-phase and center-tapped rectifier circuits
Used for full-wave rectification at low voltage.
i2 D 2
D 1i1
V m ax sinω tvs
ip R
L
iL
vo
Figure 6.1
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Lecture 6 - Diode 6-2 F. Rahman rectifier circuits II
Figure 6.2
Output voltage harmonics
The output voltage waveform can be expressed in a Fourier series
( )1 2 32
ov a cos n t b sin n to n nn , , ,...
aω ω
∞= + +∑
= 6.3
In this case,
( ) ( )o maxd max
0
a 2V1V V sin t d t2
π
ω ωπ π
= = =∫ 6.4
( ) ( ) ( )n max0
2b V sin t sin n t d t 0π
ω ω ωπ
= =∫ 6.5
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Lecture 6 - Diode 6-3 F. Rahman rectifier circuits II
( ) ( ) ( )n max0
max
n 2,4 ,6 ,...
2a V sin t cos n t d t
4V 1( n 1)( n 1)
πω ω ω
π
π
∞
=
=
−=
+ −
∫∑
6.6
Hence,the output voltage waveform can be expressed in a Fourier series as
max maxo
max max
2V 4Vv cos 2 t
34V 4V
cos 4 t cos6 t ...15 35
ωπ π
ω ωπ π
= −
− − − 6.7
The first term on the RHS is the DC value. The second term is the dominant output ripple, which in this case is at twice the supply frequency, the third and other terms are the higher order ripples. The amplitudes of these ripples reduce as the harmonic number increases. Presence of all ripple components in the output voltage is undesirable.
Figure 6.3
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Lecture 6 - Diode 6-4 F. Rahman rectifier circuits II
Input Current Harmonics
If ripple-free load current in the steady-state is assumed, the input current waveform of the above rectifier may then be indicated as in figure 6.2 for an ideal input transformer. The diodes D1 and D2 carry each half cycle of the load current and input current pi is a squarewave ac waveform. The actual input current waveform includes the transient behaviour of the load in each half cycle (see figure 6.1) and its harmonics are the quite difficult to obtain analytically. With the assumption of perfectly smooth and ripple free load current, which implies an infinitely large load inductance, it is quite straightforward to obtain an analytical expression for the input current harmonics.
Figure 6.4
For this waveform, which is an odd function because f(t) = -f(-t),
( ) ( )n d0
2b I sin n t d tπ
ω ωπ
= ∫ ; and an = 0
π 2π0
Id
-Id
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Lecture 6 - Diode 6-5 F. Rahman rectifier circuits II
= d4I
nπ for n = 1, 3, 5, 7, ….. 6.8
pi∴ = ( )d
n 1,3,5
4I / N sin n tn
ωπ
∞
=
= ∑
( )
( )
d dp
d
4I / N 4I / Ni sin t sin 3 t
34I / N
sin 5 t ......5
ω ωπ π
ωπ
= +
+ + 6.9
where N is the transformer turns ratio between primary and secondary windings. The transformer magnetizing current has been neglected in this analysis.
Note that in the above circuit, the input current waveform ip has zero DC value. The first term, for n = 1, is called the fundamental and the higher order terms are the harmonics, which are unwanted. The harmonic amplitudes reduce as the harmonic number n increases.
Figure 6.5
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Lecture 6 - Diode 6-6 F. Rahman rectifier circuits II
Single-phase bridge rectifier (p = 2)
Bridge rectifiers (see figure 6.3) do not suffer from the problem of DC magnetization (present in center-tap rectifiers which will be described shortly) and low device and transformer utilisation. They also offer higher DC output voltage for a given AC supply voltage. This is at the cost of lower efficiency, because there are now two diode drops between the load voltage and the AC supply voltage.
maxV sin tω
Figure 6.6 Diode-bridge rectifier
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Lecture 6 - Diode 6-7 F. Rahman rectifier circuits II
maxV sin tω
Figure 6.7
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Lecture 6 - Diode 6-8 F. Rahman rectifier circuits II
Figure 6.8 Diode current waveforms
Figure 6.9. FFT of out voltage and input current
waveforms of figure 6.2
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Lecture 6 - Diode 6-9 F. Rahman rectifier circuits II
Output DC voltage s maxv V sin tω=
( ) ( ) maxd max0
2V1V V sin t d tπ
ω ωπ π
= =∫ 6.10 where Vmax is the peak of the input AC voltage to the rectifier. Note that the PRV of each diode is Vmax, not 2Vmax, as in the case of the center-tapped rectifier. Output voltage harmonics The rectifier output voltage contains only even order harmonics. The Fourier coefficients of the output voltage waveform are
2
n max0
max
n 2,4,6 ,...
2a V cos t cos n td( t )
4V 1( n 1)( n 1)
πω ω ω
π
π
∞
=
=
−=
+ −
∫∑ 6.11
2 42
34 4
4 615 35
max maxo
max max
V Vv cos t
V Vcos t cos t
ωπ π
ω ωπ π
= −
− − − ⋅ ⋅ ⋅ ⋅ 6.12
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Lecture 6 - Diode 6-10 F. Rahman rectifier circuits II
Figure 6.11 Output DC and ripple voltage magnitudes
Input current harmonics
Figure 6.12 Input current waveform for ripple-free load
current
As before, assuming ripple-free load current, Fourier analysis of the square-wave input current waveform is given by,
1 3 5
4 dp
n , , .....
I / Ni sin n t
nω
π
∞
=
= ∑ 6.13
Vd V2 V4 V6 V8 V10
Id/N
− Id/N
π 2π 3π
ip
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Lecture 6 - Diode 6-11 F. Rahman rectifier circuits II
d
DrmsII2
= ; d
DdcII2
= 6.14
d
prmsIIN
= ;
d
1rms4I / NI
2 π= ; 6.15
d
3rms4I / NI
2 3π= and so on.
Figure 6.13 RMS input current harmonics
I1 I3 I5 I9 I11
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Lecture 6 - Diode 6-12 F. Rahman rectifier circuits II
3-phase center-tap diode rectifier (p = 3)
The three-phase center-tap rectifier uses the neutral connection of the supply as the return path for the load.
Figure 6.14
D2
D1
D3
c
n
vbn
vcn
a
b L
Rvo
iLvan
ic
ib
ia iR
iY
iB
R
C
B
ia/N
ic/N
iB/N
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Lecture 6 - Diode 6-13 F. Rahman rectifier circuits II
Figure 6.15 Waveforms and diode reverse blocking
voltage of 3-phase CT rectifier. Note that i1 is the input current waveform of a delta-connected primary input
transformer.
iR
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Lecture 6 - Diode 6-14 F. Rahman rectifier circuits II
For this circuit, it can be shown that,
5max
d max6
3 3V1V V sin td( t )2 / 3 2
ππ ω ω
π π= =∫ 6.16
where Vmax is the peak line-neutral voltage of the supply. The peak reverse voltage (PRV) across each diode
max3V=
an max
bn max
cn max
v V sin t2v V sin t3
4v V sin t3
ωπω
πω
=
⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠
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Lecture 6 - Diode 6-15 F. Rahman rectifier circuits II
Figure 6.16
If we assume that the load is highly inductive, the load current can be taken to be smooth and ripple free. In that case, in the steady-state, the diode and the secondary current waveforms (ia − ic) can be approximated as flat-topped waveforms of 120° of conduction followed by 240° of non conduction, as indicated in the above traces. Note that the secondary windings of the supply transformer carry unidirectional currents, which leads to DC magnetization of the transformer core. This implies that the transformer cores have DC flux, so that for a given AC voltage and flux swing, it must have larger core size than is necessary. This problem of DC magnetization is avoided in the hexa-phase rectifier circuit of figure 6.16. The output voltage waveform of this rectifier has six positive voltage pulses per AC cycle (a 6-pulse rectifier).
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Lecture 6 - Diode 6-16 F. Rahman rectifier circuits II
CT rectifiers with higher pulse number (Hexa-phase rectifier
Load
iL
vo
n v'an van
vbn
v'cn
v'bn
vcn
R
Y
B Vd
Three Phase
Ac Supply
Van
VRY VYB
VBR Vbn
Vcn
V'an
V'bn
V'cn
Figure 6.17 Hexa-phase diode rectifier with delta-connected primary
The output DC voltage Vd is given by
/ 6
maxd max
6
3V1V V cos td( t )/ 3
π
π ω ωπ π−
= =∫ 6.17
where Vmax is the peak line-neutral voltage of the supply to the rectifier.
Note that the PRV of the diodes is 2Vmax.
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Lecture 6 - Diode 6-17 F. Rahman rectifier circuits II
Figure 6.18 Waveforms in a hexa-phase rectifier
Figure 6.18 Waveforms in a hexa-phase rectifier
vanv'bn vcnvbnv'cn v'an
ωt
ia
iRY
iR
vo
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Lecture 6 - Diode 6-18 F. Rahman rectifier circuits II
Hexa-phase rectifier with inter-phase reactor
The hexa-phase rectifier does not utilize the input transformer or the switches well. However, the conduction period for each winding and diode is only 60° per cycle. This is avoided in the rectifier of figure 6.19 in which two CT rectifiers operate independently and their output voltages add across an inter-phase reactor (inductor) which carries half of the load current and supports the potential difference between the two rectifiers.
nv'an van
ib
v'cn
v'bn
ic
Inter-phase reactor
Load
Id/2
Id/2
Id
Vd
Van
Vcn
VbnV'cn
V'bn
V'an
ia
vbn
vcn
i'a
i'c
i'b
vRY
vYB
vBR
iY
iB
iR
B
Y
R
VRY
VYB VBR
Figure 6.19 Hexa-phase rectifier with inter-phase reactor The output voltage waveform is a 6-pulse waveform (i.e., six voltage pulses per cycle of the input ac waveform), the dominant ripple being at six times the supply frequency. The output DC voltage is given by,
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Lecture 6 - Diode 6-19 F. Rahman rectifier circuits II
max
d3VVπ
= 6.18
Note that each diode and each transformer secondary winding now conducts for 120°. The inter-phase reactor has bi-directional current, hence it also does not suffer from any dc magnetization.
Note that '
RY a ai i i= − 6.19
,
BR c ci i i= − 6.20
and R RY BRi i i= − 6.21
Note that the voltage across the reactor is AC, a roughly triangular waveform of amplitude which is 0.5Vmax.
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Lecture 6 - Diode 6-20 F. Rahman rectifier circuits II
Figure 6.20 Waveforms in the hexa-phase rectifier of Figure 6.19.
v a vcvbv' b v' c v'a
vipr
vo
vo
ia
ib
ic
i'b
i'c
i'a
iRY
iR
iYB
0.5Vmax
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Lecture 6 - Diode 6-21 F. Rahman rectifier circuits II
Center-tap rectifiers with 12 and higher pulse numbers
12- and 24-pulse rectifiers can be formed by connecting six-pulse rectifier circuits, as shown in figure 6.21, in series or parallel. The circuit of figure 6.21 shows two parallel connected hexa-phase (6-pulse) rectifiers forming a 12-pulse rectifier.
Load
n
Id
Vd
Id/2
Id/4
Id/4
Id/4 Id/2
Id/4 Y B
R
n
iR
iR2
iR1
2
1
Figure 6.21 Connection of two hexa-phase rectifier to
form a 12-pulse rectifier One of the input voltage waveforms, van, and the output voltage vo are indicated in the figure 6.22(a). The input primary currents iR1, iR2 for converter groups 1 and 2 respectively, and the total primary input current waveform iR to the transformer are also indicated in figure 6.22(b).
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Lecture 6 - Diode 6-22 F. Rahman rectifier circuits II
Figure 6.22(a) Output and input voltage waveforms of a 12-pulse CT rectifier.
Figure 6.22(b) Input current waveforms of the 12-pulse, center-tapped diode rectifier.
The above waveforms for the 12-pulse rectifier show that the DC output voltage waveform now has much lower ripple and that the input current waveform iR is now more closer to a sinusoid.
vO
van
vo
van
sec
iR1
iR2
iR