Continuous conduction mode operation of three-phase diode bridge rectifier with constant load voltage J.A.M. Bleijs Abstract: Three-phase diode bridge rectifiers with a low source reactance cause considerable current harmonics in the AC supply, which may not meet the latest EMI regulations. At higher source reactance the bridge is more likely to operate in continuous conduction mode (CCM), leading to a reduction in harmonic current level. In the paper accurate analytical expressions are derived for the AC current harmonics, the input power and the power factor in CCM, which have been validated through numerical simulations and practical experiments. They can therefore be used with confidence for the design of power supplies with lower harmonic levels. Remedies are proposed to compensate for the drawbacks of CCM operation, using power factor correction capacitors and a single-switch inductor-less boost rectifier. 1 Introduction Most modern variable-speed motor drives are based on pulsewidth modulated voltage source inverters, supplied from a smooth DC link voltage. The latter is often obtained from a 3-phase diode bridge mains rectifier with a large output capacitor to keep the dc link voltage essentially constant. With a capacitive load, the AC currents at the input of the bridge are often discontinuous and contain high levels of harmonics. Harmonic currents from nonlinear loads are increasingly considered to be unacceptable, as they penetrate the supply system and affect the voltage wave- forms at other locations. A similar situation is found where 3-phase AC generators are used to supply a constant voltage load (e.g. a battery), in which case the harmonic currents can lead to significant losses in efficiency. A large number of 3-phase rectifier circuits with reduced harmonic current levels have been proposed in the literature, but most methods add considerable complexity and costs to the system. Despite its widespread use, the analysis of the 3-phase diode bridge rectifier with constant voltage load, schema- tically depicted in Fig. 1, has attracted surprisingly little attention in the literature. While the case of the bridge rectifier with constant current load (i.e. with a large DC inductance) has received an indepth treatment in many textbooks, constant voltage operation is considered in only a few [1, 2], mostly at a superficial level. This is largely due to the complexity of the mathematical relations governing the behaviour of the bridge in its different modes of operation, as described in [3]. The results of numerical computation and piecewise analysis are reported in [4–7] , summarised in tables or graphs. These do, however, not provide an insight in the mechanisms at work. Moreover, the analysis is often limited to relatively low values of the AC source inductance L, when the bridge operates in discontinuous AC current mode (DCM). In this mode the resulting harmonic currents are likely to exceed the latest EMI regulations [8, 9] , which may prohibit use of the unfiltered bridge rectifier in future applications. Higher reactance values, as required for continuous AC mode (continuos conduction mode (CCM)) operation, are normally not considered in view of the bulk and costs of magnetic components and the effect on power factor and output voltage regulation. However, as shown in [4], CCM operation can result in a considerable reduction in harmonic currents. To fully appreciate the behaviour of the bridge rectifier at higher AC inductance values, a study was undertaken to derive accurate analytical expressions for the harmonic content, root mean square (RMS) and instantaneous values of the line currents, and the input power and power factor in this mode. The results have been validated through numerical simulations as well as practical tests. L L L v c V dc I dc i dc D 4 D 6 D 2 C D 1 D 3 D 5 v b v a i a i b i c . . . . . . . v c ' v b ' v a ' + − Fig. 1 Diode bridge rectifier with AC inductances and capacitive load The author is with the Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom r IEE, 2004 IEE Proceedings online no. 20040684 doi:10.1049/ip-epa:20040684 Paper first received 3rd November 2003 and in revised form 30th April 2004. Originally published online: 8th November 2004 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 359
Transcript
Continuous conduction mode operation ofthree-phase diode bridge rectifier with constantload voltage
J.A.M. Bleijs
Abstract: Three-phase diode bridge rectifiers with a low source reactance cause considerablecurrent harmonics in the AC supply, which may not meet the latest EMI regulations. At highersource reactance the bridge is more likely to operate in continuous conduction mode (CCM),leading to a reduction in harmonic current level. In the paper accurate analytical expressions arederived for the AC current harmonics, the input power and the power factor in CCM, which havebeen validated through numerical simulations and practical experiments. They can therefore beused with confidence for the design of power supplies with lower harmonic levels. Remedies areproposed to compensate for the drawbacks of CCM operation, using power factor correctioncapacitors and a single-switch inductor-less boost rectifier.
1 Introduction
Most modern variable-speed motor drives are based onpulsewidth modulated voltage source inverters, suppliedfrom a smooth DC link voltage. The latter is often obtainedfrom a 3-phase diode bridge mains rectifier with a largeoutput capacitor to keep the dc link voltage essentiallyconstant. With a capacitive load, the AC currents at theinput of the bridge are often discontinuous and contain highlevels of harmonics. Harmonic currents from nonlinearloads are increasingly considered to be unacceptable, as theypenetrate the supply system and affect the voltage wave-forms at other locations. A similar situation is found where3-phase AC generators are used to supply a constantvoltage load (e.g. a battery), in which case the harmoniccurrents can lead to significant losses in efficiency. A largenumber of 3-phase rectifier circuits with reduced harmoniccurrent levels have been proposed in the literature, but mostmethods add considerable complexity and costs to thesystem.
Despite its widespread use, the analysis of the 3-phasediode bridge rectifier with constant voltage load, schema-tically depicted in Fig. 1, has attracted surprisingly littleattention in the literature. While the case of the bridgerectifier with constant current load (i.e. with a large DCinductance) has received an indepth treatment in manytextbooks, constant voltage operation is considered in onlya few [1, 2], mostly at a superficial level. This is largely dueto the complexity of the mathematical relations governingthe behaviour of the bridge in its different modes ofoperation, as described in [3]. The results of numericalcomputation and piecewise analysis are reported in [4–7],summarised in tables or graphs. These do, however, not
provide an insight in the mechanisms at work. Moreover,the analysis is often limited to relatively low values of theAC source inductance L, when the bridge operates indiscontinuous AC current mode (DCM). In this mode theresulting harmonic currents are likely to exceed the latestEMI regulations [8, 9], which may prohibit use of theunfiltered bridge rectifier in future applications. Higherreactance values, as required for continuous AC mode(continuos conduction mode (CCM)) operation, arenormally not considered in view of the bulk and costs ofmagnetic components and the effect on power factor andoutput voltage regulation.
However, as shown in [4], CCM operation can result ina considerable reduction in harmonic currents. To fullyappreciate the behaviour of the bridge rectifier at higher ACinductance values, a study was undertaken to deriveaccurate analytical expressions for the harmonic content,root mean square (RMS) and instantaneous values of theline currents, and the input power and power factor in thismode. The results have been validated through numericalsimulations as well as practical tests.
L
L
Lvc
Vdc
Idcidc
D4 D6 D2
C
D1 D3 D5
vb
va ia
ib
ic. .
.. .
. .
vc'
vb'
va'
+
−
Fig. 1 Diode bridge rectifier with AC inductances and capacitiveload
The author is with the Department of Engineering, University of Leicester,University Road, Leicester LE1 7RH, United Kingdom
r IEE, 2004
IEE Proceedings online no. 20040684
doi:10.1049/ip-epa:20040684
Paper first received 3rd November 2003 and in revised form 30th April 2004.Originally published online: 8th November 2004
IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 359
2 Diode bridge rectifier in CCM
One of the earliest detailed analyses of the 3-phase diodebridge rectifier with inductive source impedance andconstant load voltage is presented in [3], where it is shownthat the circuit can operate in a number of modes withdifferent diode conduction patterns depending on the loadcurrent. The analysis presented here concerns the contin-uous current mode, where 3 diodes (2 diodes in the topgroup plus 1 diode in the bottom group or vice versa)conduct simultaneously (Fig. 1). This implies that commu-tation between diodes in the same group takes one-sixth ofthe AC cycle, as can be seen in Fig. 2, which shows a set oftypical voltage and current waveforms in CCM. For theanalysis of the current and voltage waveforms in Fig. 2, wefirst consider commutation interval I of 601 duration, wherethe current ia is positive (i.e. D1 is conducting), and the(negative) return current commutates from ib in D6 to icin D2.
3 Commutation process and voltage waveforms
It is assumed that the bridge rectifier is supplied from abalanced set of 3 (phase) voltages with RMS value Vph and(angular) frequency o (¼ 2pf):
va tð Þ ¼Vph
ffiffiffi2p
sin ot
vb tð Þ ¼Vph
ffiffiffi2p
sin ot � 120�ð Þvc tð Þ ¼Vph
ffiffiffi2p
sin ot þ 120�ð Þð1Þ
and that the DC output voltage Vdc is kept essentiallyconstant by the DC link capacitor C.
In the absence of a neutral connection to the bridgerectifier, the sum of the AC currents must be identical tozero at all times, i.e.
ia þ ib þ ic � 0 ð2ÞThis also applies to the derivative of the sum of thealternating currents:
ddt
ia þ ib þ icð Þ ¼ 0
which can be written as
dia
dtþ dib
dtþ dic
dt¼ 0 ð3Þ
Neglecting the diode voltage drops, the KVE for the loopva, Vdc and vb during interval I is given by
va � vb ¼ Vdc þ Ldiadt� dib
dt
� �ð4Þ
while that for loop va, Vdc and vc equates to
va � vc ¼ Vdc þ Ldiadt� dic
dt
� �ð5Þ
Adding (4) and (5) yields
2va � vb � vc ¼ 2Vdc þ L 2diadt� dib
dt� dic
dt
� �ð6Þ
The sum of the instantaneous voltages defined by (1) isidentical to zero at all times, i.e.
va þ vb þ vc � 0 ð7ÞUsing (3) and (7) relation (6) can be expressed as
va ¼ 23Vdc þ L
dia
dtð8Þ
Hence, the change in ia during interval I is governed by thefollowing relation:
diadt¼
va � 23Vdc
Lð9Þ
The bridge input voltage v0a, referred to the neutral of thesupply source, is equal to
v0a ¼ va � Ldia
dtð10Þ
Hence, it follows from (9) and (10) that
v0a ¼2
3Vdc ð11Þ
i.e. v0a is constant over interval I.Similarly, the bridge input voltages v0b and v0c can be
In the next interval of 601 (period II in Fig. 2) this process isrepeated when the output current idc commutates from ia in
100
50
0
−50
−100
−200
−400
curr
ent,
A
400
200
volta
ge, V
0
0 5 10 15 20time, ms
ic ia
va'va
ib
�
IIIVIV
III IV
Fig. 2 Waveforms for 3-phase diode bridge rectifier in CCM
360 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005
D1 to ib in D3. During that interval the voltages v0a and v0bchange stepwise to 1/3 Vdc, while v 0c becomes –2/3Vdc.
The result of the above analysis, taken over the sixcommutation intervals of a full cycle, is a stepped waveformfor v0a, as shown in Fig. 2. Fourier analysis shows that onlythe fundamental and non-triplen odd harmonic frequencies(5th, 7th, 11th, 13th, etc.) are present in this waveform. Thebridge input voltage v0a can therefore be written as the sumof the fundamental and a series of harmonic voltages:
v0a tð Þ ¼ 2Vdc
psin ot � fð Þ þ
X1n¼1
sin 6n� 1ð Þot � f½ �( )
ð14ÞThe fundamental component of v0a is
v0a1 tð Þ ¼ 2Vdc
psin ot � fð Þ ð15Þ
which has an RMS value of
V 0a1 ¼Vdc
ffiffiffi2p
pð16Þ
and a phase shift f with respect to the source voltage va, falso represents the difference in angle between the zero-crossings of va and v0a, and, hence, those of v0a and ia.
At the start of interval I the currents ia and ib have thesame value but opposite sign, while at the end of the intervalthis applies to ic. As the commutation process is cyclicallyrepeated for each current, it follows that the values of ia atthe start and end of interval I must be identical, i.e.
ia fþ 2p3
� �¼ ia fþ p
3
� �ð17Þ
The change in ia over interval I can be found from theintegration of (9):
ia fþ 2p3
� �� ia fþ p
3
� �¼ 1
L
Zfþ2p3
fþp3
va �2
3Vdc
� �dt ð18Þ
and according to (17) this change must be equal to zero.After substitution of the first equation of (1) the integral canbe solved, which results in the following condition for f:
cos fþ p3
� �� cos fþ 2p
3
� �¼ p
ffiffiffi2p
9
Vdc
Vphð19Þ
As the left-hand side of the expression in (19) equates tocosf the phase shift angle f is obtained as
f ¼ cos�1pffiffiffi2p
9
Vdc
Vph
!ð20Þ
To guarantee operation in CCM, the commutation of thepositive bridge current from ib to ia must commence at thestart of interval VI, i.e. at f. However, diode D1 will notconduct unless it is forward biased. A condition for CCMis, therefore, that va (f)ZVdc/3, i.e.
Vdc � 3ffiffiffi2p
Vph sinf ð21ÞSubstitution of (20) yields, after some manipulation, thefollowing condition for CCM:
Vdc � Vph
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
4p2 þ 9
r¼ 1:828Vph ð22Þ
Backsubstitution of (22) in (20) shows that in CCM theangle fZ25.51.
4 Fundamental of the line current
From the analysis of the commutation process described inSection 3, it follows that the zero-crossings of the linecurrents coincide with those of the input voltages of thebridge. However, as a result of the distortion of the currentwaveform caused by the commutation process, the currentsdo not possess quarter cycle symmetry. Hence, thefundamental component Ia1 of the a-phase line currentdoes not have the same phase shift f as v0aI , but incurs anadditional phase shift Df, i.e. the total phase shift j1
between the supply voltage Va and Ia1 is equal to
j1 ¼ fþ Df ð23ÞIn Fig. 3 the fundamental voltage and current componentsare shown in a phasor diagram, where the fundamental ofthe voltage drop across the inductance L
VL1¼ joLIa1 ð24Þ
leads Ia1 by 901.Applying the cosine rule to the upper left-hand triangle in
Fig. 3 yields
VL1j j2¼ Vph
�� ��2þ V0
a1
�� ��2�2 Vph
�� �� V 0a1�� ��cosf ð25Þ
Substitution of (16) and (20) into (25) yields, after somealgebraic manipulation, an expression for the fundamentalof the line current:
Applying condition (22) to (26) shows that CCM operationof the diode bridge rectifier is restricted to current levelsbetween 44 % and 100% of the short-circuit current, wherethe RMS value of the latter is defined as:
I s=c ¼Vph
oLð27Þ
The difference DV between V 0aI and the horizontalprojection of Va in Fig. 3 can be written with (16) and(20) as:
DV ¼ Vph cosf� V 0a1 ¼ Vdc
ffiffiffi2p p2 � 9
9p
� �ð28Þ
�
�1
∆�
∆�
∆V
Va1
Ia1
Va
�LIa1
Fig. 3 Phasor relation between fundamental components of bridgerectifier voltages and currents
IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 361
Using the geometrical similarity of the upper right hand andbottom triangle in Fig. 3, the additional phase shift Df canthen be found from (28) and (29) as:
From (20) it follows that DfZ0 and therefore DVZ0,which means that the fundamental component Ia1 of theinput current of a bridge rectifier with capacitive load lagsthe fundamental component of bridge input voltage V 0aI .This additional phase shift is due to the commutationoverlap, similar to that experienced in a bridge rectifier withinductive load. At the boundary of CCM Df attains itsmaximum value of 10.51. This affects the reactive (VAr)power that the bridge rectifier draws from the supply source(see Section 15 Discussion).
5 Line current harmonics
It can be seen from the voltage waveforms in Fig. 2 that thevoltage differences between va and v0a at the start and theend of interval I are not equal. This implies that ia is notsinusoidal during this interval. Also, owing to the commu-tation before and after interval I, the leading and trailingparts of ia over each half cycle are not symmetrical, i.e. iadoes not possess quarterwave symmetry. However, thecurrent waveform has halfwave symmetry, and the linecurrents can therefore contain only odd harmonic compo-nents (see below). Owing to the absence of a neutralconnection in the bridge rectifier, no triplen harmonics canbe present in the input current.
As shown in the preceding analysis, the bridge inputvoltage is composed of a fundamental component plusharmonics at non-triplen odd multiples of the supplyfrequency. According to (14), the RMS value of the voltageharmonic V 0ah is equal to
Vah0 ¼ Vdc
ffiffiffi2p
p 6n� 1ð Þ ð31Þ
where n¼ 1,2,3,y. and h¼ 5,7,11,y.The supply source voltages, described by (1), contain only
fundamental components. Hence, in order for harmoniccurrents to exist, the supply source must act as a short-circuit for these currents. The harmonic current componentsare then solely determined by the harmonic voltages of thebridge, as given by (31), and the impedance of the inputinductance L, which at the harmonic frequencies is given by
Zhj j ¼ 6n� 1ð ÞoL ð32ÞThence, the harmonic current components have an RMSvalue of
Ih ¼V 0ah
Zhj j¼ Vdc
ffiffiffi2p
poL 6n� 1ð Þ2ð33Þ
Using (26) and (33), the relative contribution of eachharmonic component of the current can be expressed as a
percentage of the fundamental:
Ih
I1¼ Vdc
ffiffiffi2p
p 6n� 1ð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2
ph � 0:2418V 2dc
q ð34Þ
The total harmonic distortion (THD) of the input current isdefined as the root of the sum of the squares of thepercentage harmonic components, i.e.
Comparison of (26), (33) and (38) shows that, while thefundamental and harmonic line current components aredetermined by the size of the AC inductance L as well as theDC link voltage Vdc, the percentage THD in CCM is onlyaffected by the value of Vdc relative to the RMS value of thesupply voltage. At the maximum value of Vdc for CCM, asgiven by condition (22), the THD reaches a maximum valueof 8.7%.
6 RMS value of the line current
The RMS value of the line current can be found bycombining the fundamental and harmonic components inthe usual way:
I rms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2a1 þ
Evaluation of the root in (40) under condition (22) showsthat in CCM the RMS value of the line current differs lessthan 1% from that of the fundamental current component.
Substitution of expression (26) for Ia1 yields, finally
362 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005
7 Instantaneous values of the line current
For further analysis it is useful to determine the value of theline currents at certain instances, e. g. the value of ia at thestart and end of period I, i.e. at ot¼f+p/3 andot¼f+2p/3. During this interval the (negative) currentat the bridge input commutates from ib to ic. According to(3) the change in ic is driven by the voltage difference acrossthe input inductance in the c-phase:
dic
dt¼ vc � v
0
c
Lð42Þ
As at the start of the commutation ic¼ 0, the value of icduring period I can be obtained from integration of (42):
ic otð Þ ¼ 1
oL
Zot
fþp3
vc � v0
c
� �dot ð43Þ
By integrating over the full length of period I the value of icat ot¼f+2p/3 can be found. Substitution of the lastequation of (1) and (13) for vc and v0c, respectively, andintegration of the resulting expression yields, after sometrigonometric manipulation,
ic fþ 2p3
� �¼ � Vph
ffiffiffi6p
2oLsinf ð44Þ
At the end of this commutation interval ib¼ 0, while ia andic have equal but opposite values. Furthermore, the value ofia at the start of this interval is equal to that at the end of theinterval. Hence
ia fþ p3
� �¼ ia fþ 2p
3
� �¼ Vph
ffiffiffi6p
2oLsinf ð45Þ
The change in ia during interval I is given by (9), thereforethe instantaneous value of ia during this interval can bewritten as
ia tð Þ ¼ ia t1ð Þ þZ t
t1
diadt
dt ð46Þ
Substitution of (9) and changing the variable from time t toangle ot yields
ia otð Þ ¼ 1
oL
Z ot
fþp3
va �2
3Vdc
� �daþ ia fþ p
3
� �ð47Þ
Substitution of the first equation of (1) and (45), andsubsequent integration, leads to the following expressionfor ia:
ia otð Þ ¼ Vph
ffiffiffi2p
oLcos fþ p
3
� �� cosot
h i(
� 2Vdc
3oLot � f� p
3
� �)þ Vph
ffiffiffi6p
2oLsinf ð48Þ
over the range
fþ p3ootofþ 2p
3
The angle a at which ia reaches its maximum value ininterval I can be found by taking the derivative of (48) withrespect to ot and equating this to zero. This yields
aiamax¼ p� arcsin�1
Vdc
ffiffiffi2p
3Vph
!ð49Þ
assuming that the principal value of the sin�1 function ischosen. After substitution of the value of (49) in (48) themaximum value of ia is obtained.
8 Input power and power factor
As the supply voltages are assumed to be sinusoidal, theharmonic components of the input currents do notcontribute to the active power drawn from the source.According to the definition of average active power, theinput power of the bridge rectifier is given by
Pin ¼ 3VphIa1 cosj1 ð50Þwhere cosj1 is equal to the displacement power factor(DPF) and j1 is the phase shift between fundamentalcurrents and (phase)voltages of the supply.
Using Fig. 3 the following relation between cosj1 andcosf is obtained (see the Appendix, Section 19):
Substitution of expressions (16) and (20) for V 0aI and cosf ,respectively, yields after some mathematical manipulationfor the displacement power factor (DPF):
Using the definition in (54) the reactive power Qin1,associated with the fundamental voltages and currents,can be derived from (53) and (55) to yield
Qin1 ¼3
oLV 2
ph � 0:2222V 2dc
� �ð56Þ
As the current is lagging the supply voltage, the diodebridge rectifier always draws an amount of reactive power,equal to (56), from the supply source.
The total apparent power of the rectifier circuit is definedas the sum of the products of the RMS values of thevoltages and current in each phase, i.e.
S ¼ 3VphIrms ð57ÞSubstituting (41) the total apparent power is found as:
Evaluation of the square root expression in (59) shows thatits value in CCM is always between 0.98 and 1. Hence, theoverall power factor varies (almost) linearly with the DCoutput voltage of the bridge and attains a maximum valueof 0.80 (lagging) at the boundary of CCM operation asgiven by condition (22). The power factor can also beexpressed as
PF ¼ DF cosj1 ð60Þwhere DF is called the distortion factor.
Evaluation of (6) under condition (22) shows that thedistortion factor for the diode bridge under CCM deviatesless than 0.5 % from unity. Hence, for practical purposesthe overall power factor in CCM can be taken to be equalto the displacement power factor.
9 Bridge output (DC) power and current
In CCM the bridge output current idc consists of a pedestalDC current Idc0, on which a 6-pulse ripple Didc issuperimposed. The value of idc during interval I is equalto that of ia and is therefore given by expression (47),repeated every p/3 rad. The average value of the DC currentIdc can be obtained from considering the (active) powerbalance on both sides of the bridge rectifier. In thepreceding analysis it has been assumed that the DC outputvoltage vdc is kept essentially constant by the DC linkcapacitor, i.e. vdc¼Vdc. If the losses in the bridge diodes areneglected, the power balance can be written as
VdcIdc ¼ Pdc ¼ Pin ð62ÞEquating (62) to (53), the DC output power can beexpressed in terms of the AC and DC voltages:
Equally, the equivalence of (50) and (63) permits a relationbetween the magnitudes of the AC and DC currents onboth sides of the bridge rectifier to be found. Aftersubstitution of (51) and some manipulation the followingrelation between the fundamental of the alternating currentand the average direct current is obtained:
As the square root expressions in (65) and (66) differ lessthan 2% from unity under CCM condition (22), therelation between AC and DC is virtually independent of thevoltages and can, to a very good approximation, beexpressed, as:
I rms ¼p
3ffiffiffi2p Idc ¼ 0:74Idc ð67Þ
or
Idc ¼3ffiffiffi2p
pIrms ¼ 1:35Irms ð68Þ
The latter expression has the same form as that of the AC/DC voltage relation of a 3-phase diode bridge rectifier withconstant output current (i.e. a very large DC choke).
The maximum direct current that the bridge rectifier canprovide is determined by the AC short-circuit current, givenby (29).Hence:
Idcmax¼ 3
ffiffiffi2p
Vph
poLð69Þ
10 Maximum power transfer
From equation (53) it can be seen that, for a given supplyvoltage and line inductance value, there is a maximumamount of power that can be transferred from the ACsource to the DC load. The maximum value can be foundby taking the partial derivative of (53) with respect to Vdc
and equating this to zero. This yields the followingcondition:
VdcjPmax¼ 9
2pVph ¼ 1:4324Vph ð70Þ
Therefore the maximum power is equal to
Pmax ¼27
2p2oLV 2
ph ¼ 0:4563
oLV 2
ph ð71Þ
The value of the direct current for maximum power transferis given by
IdcjPmax¼ 3
poLVph ð72Þ
Comparison with (69) shows that the power maximumoccurs at 71% of the DC short-circuit current Idcmax.
11 Characteristics of diode bridge rectifier in CCM
The equations derived in the preceding Sections can be usedto calculate the performance of the rectifier circuit for anyspecific application. In order to see the general trends, it isuseful to per-unit-ise the results using suitable base valuessuch as Idcmax and Pmax. Figure 4 depicts the externalVdc(Idc) characteristic of the bridge in CCM.
The calculated power characteristic of the diode bridgecircuit, operating in CCM, is shown in Fig. 5.
In Fig. 6 the true power factor according to (59) isdisplayed as a full line, with the values of the displacementpower factor according to (52) added as full dots; it can beseen that in CCM the difference between these powerfactors is indiscernible. The percentage total harmonic
364 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005
distortion level as well the 5th and 7th harmonics of thebridge input current are shown in Fig. 7, where it can beseen that they fall rapidly at higher current levels.
12 Validation of theory
The correctness of the developed theory has been verified bysimulation as well as experimentation. Using a proprietarySPICE-based software package with built-in measurementfacilities (including FFT) the diode bridge rectifier circuit,with AC reactors and constant-voltage load, was simulatedover the full range of operating conditions for CCM,assuming ideal diodes (i.e. zero voltage drop duringconduction). The results of the simulations are depicted ascrosses in Figs. 4–7. The deviations from the theoreticalvalues were found to be less than 0.5% for all casesconsidered.
To check the theoretical performance against that of apractical circuit, a test set-up was assembled, consisting ofan adjustable 3-phase transformer, three identical iron-cored inductors of 10 mH, a 3-phase diode bridge with anoutput capacitor of 10000 mF and a DC/AC thyristormains inverter with a DC link choke of 60 mH. Standardinstrumentation including a 3-phase power analyser wasused to measure DC, RMS and percentage harmonic valuesof voltages and currents, as well as power flows and powerfactors. Using the measured input and output voltages,together with the prevailing values of the inductance, thetheoretical performance was also calculated for comparison.The calculated values were corrected for the internal voltagedrop across the bridge, as seen by the output current(Fig. 1), which amounts to two diode voltages (i.e.E2V).
Some results of the tests together with the calculatedvalues are plotted in Figs. 8–10.
0.4 0. 5 0. 6 0. 7 0. 8 0. 9 10
0.5
1.0
1.5
2.0
Idc /Idcmax
Vdc
/Vph
Fig. 4 External DC output characteristic of diode bridge in CCM
0.4 0.5 0.6 0.7 0.8 0.9 1.00
0.2
0.4
0.6
0.8
1.0
Idc /Idcmax
Pdc
/Pm
ax
Fig. 5 Power characteristic of diode bridge in CCM
0.4 0.5 0.6 0.7 0.8 0.9 1.00
0.2
0.4
0.6
0.8
1.0
Idc /Idcmax
pow
er fa
ctor
Fig. 6 Power factor of diode bridge rectifier, operating in CCM
0.4 0.5 0.6 0.7 0.8 0.9 1.00
2
4
6
8
10
Iac /Isc
harm
onic
s, %
THD5 th
7 th
Fig. 7 Harmonic distortion levels of AC input current
0
20
40
60
80
140
120
100
6 8 10 12 14 16 18 20V
dc, V
Idc , A
Fig. 8 Measured and calculated external DC characteristic
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
6 8 10 12 14 16 18 20Idc, A
Pin
, kW
Fig. 9 Measured and calculated power levels
IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005 365
It can be seen that for the external DC characteristic(Fig. 8) the measured values (shown as solid dots) aregenerally in good agreement with the calculated values(open circles connected through a smoothed curve),with a maximum deviation of around 1% at higher currentlevels.
The predicted input power characteristic (open circles,connected through a smoothed curve) is depicted in Fig. 9,together with the measured values (solid dots). The lattercan be seen to exceed the calculated values by an increasingamount at higher current levels. This discrepancy is causedby the resistance and core losses of the AC inductors, whichhave not been considered in the theoretical analysis.
In Fig. 10 the measured and predicted levels of the 5th(diamonds) and the 7th (triangles) harmonics and the THD(squares) are compared. Here, the largest deviations werefound at lower current levels, in particular for the THD and5th harmonics. A check on the waveform of the sourcevoltage revealed considerable harmonic voltage distortion(5th: 2–3%, 7th: 1–1.5%), due to the prevalence ofnonlinear 1-phase loads on the local supply. Theseharmonic voltages are in phase with the harmonic voltagescreated by the diode bridge action, resulting in the observedincrease in the input current harmonics.
It should also be noted that the current range for CCMoperation in the tests exceeded that of the theoretical rangeby a considerable margin (cf. Figs. 4–7). This is caused bysaturation effects in the iron-cored inductors, effectivelyreducing the inductance value at higher AC currents andextending the current range to higher values.
13 Capacitive PF compensation
A disadvantage of the use of higher series inductance valuesto obtain CCM operation is the considerable amount ofreactive power, drawn from the supply source. Capacitivecompensation can be applied to counteract this effect. Asshown in Fig. 5 there is a maximum to the amount of powerthat can be converted from AC to DC, therefore forpractical applications the absorbed reactive power is lessthan the theoretical maximum, set by the phase voltage andinductance value:
Qmax ¼3
oLV 2
ph ð73Þ
Using condition (70) and equation (56) the reactive powerconsumption at maximum power transfer is found to beequal to
QjPmax¼ 0:544� 3
oLV 2
ph ð74Þ
i.e. 54.4% of the theoretical limit, while at the boundaryof CCM operation the reactive power consumptionamounts to
Qmin ¼ 0:2574� 3
oLV 2
ph ð75Þ
Reactive power compensation at the level given by (75)would improve the power factor of the rectifier circuit overthe full power range, with the power factor at maximumpower transfer increasing from 0.642 to 0.847 lagging.
In practical applications, the supply source will have aninternal inductance Ls. The power factor compensationcapacitors, placed at the input of the rectifier circuit,together with the series inductances L, will then form a filterfor the harmonic currents drawn by the rectifier, furtherreducing the harmonic currents injected into the supplysystem.
14 Compensation of voltage regulation
Operation in CCM requires a high value of AC reactance,which leads to a significant regulation of the DC outputvoltage compared to that for the bridge rectifier indiscontinuous mode of operation, where regulation typicallyis less than 5% [4]. Inversion of (64) yields the followingrelation between Vdc and Idc in CCM:
At maximum power transfer of the rectifier, the DC voltageamounts to only 58.5% of the no-load DC voltage (c.f.expression (70)).
However, this drawback can be ameliorated by means ofthe boost rectifier circuit shown in Fig. 11, requiring onlytwo additional power components T1 and D7.
In comparison with the standard DC boost converter, theDC choke is omitted and the AC boost rectifier employs theenergy storage capacity of the AC reactors. As will beshown in a future paper, the DC output voltage of the boostrectifier operating in CCM can be obtained from equation(76), corrected for the duty cycle D (¼%on-time) of switchT1 in Fig. 11. This yields:
Fig. 10 Measured and calculated harmonic current levels
.
..
..
..
va
Vdc
vb
C
D7
T1
D5D3D1
D2D6D4
vc L
L
L
+
−
Fig. 11 Single-switch boost rectifier
366 IEE Proc.-Electr. Power Appl., Vol. 152, No. 2, March 2005
15 Discussion
It is interesting to quantify the effect of the additional phaseshift Df of the diode bridge on the active and reactivepower drawn from the supply. It is sometimes assumed (e.g.[11]) that the fundamental components of the bridge inputvoltages and currents are in phase (i.e. Df¼ 01 in Fig. 3),i.e. the diode bridge rectifier with output capacitor act as apurely resistive load. The voltage drop V 0LI across L due toV 0a1 (primed to distinguish from Ia1 in (26) and further)would then be in quadrature with the fundamental bridgeinput voltage V 0a1. As the voltages now form a right-angledtriangle, the magnitude of the fundamental currentcomponent is found as
Comparison with the expressions in (53) and (56) showsthat the additional phase shift Df has the effect of reducingthe active power and increasing the reactive power drawnfrom the supply source, amounting to more than 30 % athigher values of Vdc. The assumption that the fundamentalcomponents of the bridge input voltages and currents are inphase, therefore leads to an incorrect calculation of theactive and reactive power and power factor of the 3-phasebridge rectifier operating in CCM.
16 Conclusions
In this paper a full analysis is presented of the 3-phase diodebridge rectifier with AC reactors, operating in continuousconduction mode (CCM) and supplying a DC load atconstant voltage. Closed-form analytical expressions havebeen derived for the fundamental and harmonics of the ACinput, the active and reactive input power and the powerfactor in this mode, which have been validated throughsimulation, as well as practical test results.
It has been shown that the phase relation between thefundamentals of the bridge input (phase) voltages andcurrent is not purely resistive as is sometimes assumed, butincurs an additional phase shift due to the currentcommutation in the bridge. It has also been establishedthat the AC/DC power transfer exhibits a maximum inCCM. At higher current levels the harmonic contents of thealternating currents are considerably reduced, which maymake this mode of operation more compliant with the latestlegislation on harmonic limits. Methods have been pro-posed to overcome some of the drawbacks of operation in
CCM, such as low power factor and considerable outputvoltage regulation, by using capacitive compensation and asimple voltage boost arrangement.
17 Acknowledgment
The author wishes to thank Mr Min Chen, an HonoraryResearch Fellow in the Electrical & Electronic PowerEngineering Group, for undertaking the test described inSection 12.
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factor for an uncontrolled 3-phase bridge rectifier with a capacitiveload’. IEE Conf. Publ., London, UK, 1984, 234, pp. 111–114
5 Ray W.F., Davis R., M., and Weatherhogg I., D.: ‘The three-phasebridge rectifier with a capacitive load’. IEE Conf. Publ., London, UK,1988, Vol. 291, pp. 153–156
6 Gr.otzbach, M.: ‘Line side behaviour of uncontrolled rectifier bridgeswith capacitive dc smoothing’. EPE Conf. Proc., Aachen, Germany1989, pp. 761–764
7 Sakui, M., and Fujita, H.: ‘An analytical method for calculatingharmonic currents of a three-phase diode-bridge rectifier with dc filter’.IEEE Trans. Power Electron., 1994, 9, pp. 631–637
8 IEC6 1000-3-2: ‘Limits for harmonic current emissions (equipmentinput current16A per phase): limits for professional equipment withinput power>1000W’
9 IEC6 1000-3-4: ‘Limits for harmonic current emissions in lowvoltage power supply systems for equipment with rated current greaterthan 16A’
