2008 lecture 6 - uncontrolled rectifier circuits ii 6... · 2009-03-31 · elec4614 power...

Elec4614 Power Electronics

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



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



( ) ( ) ( )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



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



= 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



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



maxV sin tω

Figure 6.7



Figure 6.8 Diode current waveforms

Figure 6.9. FFT of out voltage and input current

waveforms of figure 6.2



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



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



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



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



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



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

=

⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠



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).



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.



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



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,



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.



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



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).



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

2008 lecture 6 - uncontrolled rectifier circuits ii 6... · 2009-03-31 · elec4614 power...

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