Harmonic Analysis of Single-Phase Full Bridge Rectifiers Based on ...

IEEE ISIE 2006, July 9-12, 2006, Montreal, Quebec, Canada
<strong>Harmonic</strong> <strong>Analysis</strong> <strong>of</strong> <strong>Single</strong>-<strong>Phase</strong> <strong>Full</strong> <strong>Bridge</strong>
<strong>Rectifiers</strong> <strong>Based</strong> on Fast Time Domain Method
K. L. Lian (University <strong>of</strong> Toronto), and P. W. Lehn (University <strong>of</strong> Toronto)
Abstract- <strong>Single</strong>-phase full bridge rectifiers are heavily used in
low voltage distribution system. It is therefore essential to predict
the harmonic levels these converters produce. In this paper, a fast
time domain method is presented to solve for the current and
voltage harmonics injected by single-phase diode bridge rectifiers.
The method is highly accurate and it does not suffer from
harmonic truncation errors or aliasing effects that exist in
harmonic domain analyses or brute force time domain simulation
methods.
I. INTRODUCTION
SINGLE-phase full bridge rectifiers are widely used in low
voltage distribution systems. They constitute the input
conversion stage for power supplies <strong>of</strong> many electronic
appliances such as TV sets, computers, light dimmers, and
battery chargers. The main drawback <strong>of</strong> these rectifiers is that
they generate significant harmonic distortion. In some cases,
the total harmonic current distortion can be as high as 100%.
The harmonics injected into the power network can have an
adverse impact on the power systems. It is therefore essential
for engineers to be able to predict these harmonics accurately
and to determine how these harmonics may interact with other
system components.
Brute force time domain simulation, in general, can provide
accurate results. However, they require significant computation
time to reach steady state at the required accuracy. <strong>Harmonic</strong>
domain analysis methods [1], [2] are far more computationally
efficient, although they suffer from truncation errors. In
particular, truncation <strong>of</strong> the harmonic series that represent
square-wave or sawtooth waveforms introduces Gibbs
oscillations at the point <strong>of</strong> discontinuity [3]. In case where these
artificial oscillations occur near a zero crossing <strong>of</strong> a diode
current, incorrect diode turn-on or turn-<strong>of</strong>f times result -
leading erroneous harmonics.
In this paper, an efficient time domain method is presented to
solve for the current and voltage harmonics injected by
single-phase full bridge rectifiers. The computation time is very
fast compared to brute force time domain simulators such as
PSCAD/EMTDC because the proposed method directly
pinpoints the steady state solution without stepping through
numerical transients. In addition, the results are highly accurate
because all harmonics are calculated analytically. Therefore,
the results do not suffer from harmonic truncation errors or
1-4244-0497-5/06/$20.00 © 2006 IEEE 2608
aliasing effects.
The paper is organized as follows: Section II introduces the
model <strong>of</strong> a single phase diode bridge rectifier. Section III and
IV show how to use the proposed method to analyze continuous
and discontinuous conduction modes [4]. Section V presents
numerical results, and compares computed results with those
obtained by PSCAD/EMTDC simulation to demonstrate the
validity <strong>of</strong> the method.
II. MODEL OF A SINGLE-PHASE FULL BRIDGE RECTIFIER
Fig. 1 shows the single-phase capacitor-filtered diode bridge
rectifier [4], [5], [6], [7] where the dc load is modeled as an
equivalent resistance, Req,
VS
R L idi
LD1I
LD4
LD3
DD2
Fig. I <strong>Single</strong>-phase capacitor-filtered diode bridge rectifier
C Req
If one models the diode as a variable resistor, one can
convert Fig. 1 to Fig. 2, where Rd= Ron when the diode turns on,
and Rd- Rff when the diode turns <strong>of</strong>f. Note that the diode
voltage drop is neglected to simplify the formulation.
Vs
Fig.2 Equivalent circuit model <strong>of</strong> Fig. I
Req
The state space form describing Fig. 2 can be written as in (1).
di = Ax
dt
or
dFX1 FAx N1Fx
dtLzL ° QiLzi
(1)

-(R + Rd)
where x = [id Vo IT Ax = L
I
0
N=0 N L L ~~... L
ii
L
-1I
ReqC
z =uxi uy1 ux2 uy2 |u uyn and
Q= blkdiag {K? 0 0}[2? 0 0'[n j}
Note that:
1. A is a time varying matrix because Rd changes its values
depending on the diodes' conducting or non-conducting
states. To distinguish these two cases, A,,. and A0ff will be
used to denote Rd= Ron and Rd= Ro,^ respectively.
2. The source voltage, v, is assumed to take the following
n
general form: vs = E Ak sin(kt + bk) .
k=I
3. uxk and uyk represents a pair <strong>of</strong> kth harmonic oscillator states,
where the source voltage may be expressed in terms <strong>of</strong> these
n
states according to: vS = E uyk, provided that the states, uxk
k=I
and uyk are initialized at Akcos(o), and Aksin(o), respectively.
4. A more involved modeling approach, based on an ideal
switch model, may also be developed based on the concept
<strong>of</strong> injection and projection matrices [8], [9].
III. CONTINUOUS CONDUCTION MODE
During the positive half period <strong>of</strong> the fundamental, the diode
D1 and D2 can be characterized by either a single conduction
interval or by multiple conduction intervals.
G. Carpinelli et. al., name these as "continuous" and
"discontinuous" conduction modes respectively [4]. These
differing conduction conditions result from differing harmonic
contents present in the supply system voltage [4].
When the ac source is free <strong>of</strong> harmonics, typically only the
continuous conduction mode exists. Fig. 3 shows the voltage
and current waveform in the case <strong>of</strong> continuous conduction
mode. In the figure, y represents the time from voltage zero
crossing until the diode starts to conduct, and t represents the
conduction interval duration.
In order to fully describe the behavior <strong>of</strong> the rectifier in this
mode, one needs to solve for 7, and t.
A. Diode Constraint Equations
To solve for 7, and t, one needs to
diode constraint equations:
Diode turn-on occurs when
formulate appropriate
MD=vo (d ) - Vt(o ) = vo(w) - n uly
Diode turn-<strong>of</strong>f occurs when
(2)
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M2= id(y+,u)=Cle onX(y (3)
where C1=[1 0 0 0].
In order to solve for (2) and (3), one needs to find an
expression linking v0 and y. This is characterized by the steady
state constraint equations.
a1)
>)
-c
o0- 7
Time
-v
Fig. 3 Voltage and current waveforms in the case <strong>of</strong> continuous conduction
B. Steady State Constraint Equations
Since the current waveform exhibit half-wave symmetry, only
half period needs to be considered.
The states at the end points <strong>of</strong> the interval are linked by (4)
x(T /2+ y) =eA<strong>of</strong>f (T 2-u) eAOnYx(y) =4D(Q) (4)
In addition, based on half period mapping [10],
x(T /2 +y)=I2x) (5)
where 12= diag([-1,1,-1,-1]);
Combing (2) and (3), one gets
(12 - D)iX(y) =0 (6)
Partitioning the above equation [10] gives:
K f x(y)0Lo (7)
x(y) = -E- Fz(y) = -E- Fe 2z(0) (8)
Since (2), (3), and (8) are transcendental equations, numerical
iteration is required to obtain solutions, which is described in
section C.
C. Numerical Iteration For Finding Unknowns
Fig. 4 shows the overall flow diagram <strong>of</strong> the proposed
method. First, 4=[y g]t iS initialized, allowing determination <strong>of</strong>
x(y) based on (8). Then, x(y) is substituted into the diode
constraint mismatch equations (2) and (3). The mismatch
equations then set the stage for a Newton-type iterative method,
which generates the sequence, { }n 0 by (9)
.(k+l) = .(k) j-IM
(9)

4+vl 4+vl Second diode turn-<strong>of</strong>f occurs when
where M M 11,and J=
LM2 j
a'
aM2
aJ
/'
aM2
a,J
M4 i(y+a+o)=c e Aon/
The iteration terminates when the difference between i(k+1)
and 4(k) reaches a required tolerance, F_ egin
D. Analytic <strong>Harmonic</strong> <strong>Analysis</strong> Initialize Y(k) P(k)
Once y, t, and x(y) are found, one can proceed to solve for (with k= 0)
the current and voltage harmonics injected by the single-phase Set ;(k) = !yk) A(k)]
full bridge rectifier.
<strong>Based</strong> on [11], the system harmonic is a two stage process. +(k)
First, the system matrix is augmented with one additional Calculate steady state
equation for each harmonic <strong>of</strong> interest, leading to the form: solutions, x(y) based
x Ax N O x on (8)
dt z = O Q O] z (10) x(y)
-Y P
where y=[Idh V0] T
Q_ -Y
and dh and VI represents the hth ac
Mll
current, and lth dc voltage harmonics, respectively, and based on (2), and (3)
r2e jh(T/2+y) 0 M
P=T
°+ ~2-jl(T/2+y)
Ae
e A
Calculate M= kk
k kJ 1M whereJ Lam
0 2ejlT
Q Ljh 01.
L0 11w] No
Second, the analytical solution for the augmented system is l (k
found at time, t= T/2+ y by solution <strong>of</strong> the augmented state
equations. The solution for the augmented states then yields the Yes
harmonic components <strong>of</strong> the interest. j
IV. DISCONTINUOUS CONDUCTION MODE Fig. 4 Flow diagram <strong>of</strong> the proposed method
The discontinuous conduction mode takes place when the
source voltage is characterized by the presence <strong>of</strong> harmonics
superimposed on the fundamental. In such a case, the diodes
can conduct more than one time interval during half period. To
simplify the analysis, only two conduction intervals (Fig. 5) are dv
considered here. vs
A. Diode Constraint Equations
<strong>Based</strong> on Fig. 5, the diode constraint equations can be m X
formulated as follows: ,
First diode turn-on occurs when
n ........ -------------_._._-------- .----f --
M1= vo (y)-vs(y)= vo (Y)- Uyk (Y)= (11)
k=l
First diode turn-<strong>of</strong>f occurs when
M?= id (y+) =Cl eAOng(y) O (12)
Second diode turn-on occurs when 0* T/ Yc 2 T/2 T
A.ffaA u I fn Time
M3=: C2 eA f e on, x(y)-E Uyk (r+,U +aQ) = o (13) Tm
k= ( Fig. 5 Voltage and current waveforms in the case <strong>of</strong> discontinuous conduction
where C21[0 1 0 0].
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U1)
a)
>0
B. Steady State Constraint Equations
The steady state equation in this case is also characterized by
(8).
C. Numerical Iteration To Find Unknowns
With slight modifications, the numerical analysis described in
III. C. may again be applied. In the flow chart <strong>of</strong> Fig. 4, 4 is
redefined as [y g oc H- , and the mismatch matrix, M is
redefined as [Ml I2 I3 M4]T.
D. Analytic <strong>Harmonic</strong> <strong>Analysis</strong>
The harmonics <strong>of</strong> interest can be easily found by evaluating
(1O) at t=T/2+ ,which has the form <strong>of</strong> (15).
x(T12+ Fx(y)
z(T12+ Y) =e e on, eAOff eAonA z()
-y(T1/2+ y) A(Y(Y)T)
(15)
where A is equal to the state matrix <strong>of</strong> (10), with subscript,
"on", and "<strong>of</strong>f' representing the cases when Rd- Ron and Rd-
R<strong>of</strong>f respectively.
V. EXAMPLES
In order to demonstrate the validity <strong>of</strong> the proposed method,
two sets <strong>of</strong> parameters are chosen to result in continuous and
discontinuous conduction modes. Solutions are compared with
those <strong>of</strong> PSCAD/EMTDC.
A. Continuous Conduction Mode
System parameters are as follows: R= 0.005Q, L= 0.5mH,
C= lOOOgF, Req=IOQ, Ron= 0.005Q, R<strong>of</strong>f-106Q and vs
= X x 1 12 sin(t), continuous conduction mode results. Fig. 6
shows the ac current waveform, id, dc capacitor voltage, vo,
together with the source voltage, vs, predicted by
PSCAD/EMTDC.
250 0
0o
-50
-100
-150
d
-v
0 Y- < > 0.0083 0.0167
Time (secs)
Fig. 6 Voltage and current waveforms in the case <strong>of</strong> continuous conduction
V
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Table I shows the diode conduction instant, y, conduction
interval, t and steady state capacitor voltage, v0 at t= y,
predicted by both PSCAD/EMTDC, and the proposed fast time
domain method. As can be noted, excellent agreement exists.
TABLE I
RESULTS FROM PSCAD/EMTDC AND THE FAST TIME DOMAIN METHOD IN
CONTINUOUS CONDUCTION MODE
PSCAD/EMTDC Fast Time Domain
0.0021s 0.0021s
RL 0.0032s 0.0031s
vo(y) 113.5273V 113.7460V
Fig. 7 and 8 compare ac current and dc voltage harmonics
obtained from PSCAD/EMTDC, and from the fast time domain
method.
Ac Current (Q) Spectrum
30
- PSCAD/EMTDC
Fast Time Domain
25
< 20
E 15
.a
I 10
1 3 5 7
<strong>Harmonic</strong> number
Fig. 7 Current harmonics for the case <strong>of</strong> continuous conduction
150
> 100
E c
.g
co
I 50-
u
0
Dc Voltage (VO) Spectrum
2 4
<strong>Harmonic</strong> number
- PSCAD/EMTDC
= Fast Time Domain
El . F-.
6
Fig. 8 Voltage harmonics for the case <strong>of</strong> continuous conduction
B. Discontinuous Conduction Mode
To analyze discontinuous conduction mode the following
system parameters are used: R= 0.005Q, L= 0.5mH, C=500gF,
Req= 1OQ, Ron= 0.005Q, R<strong>of</strong>0 106Q and v5
= Xx 112 sin(S) + 2x 11.2 sin(3t) .
Fig.
9 shows the ac
current waveform, id, dc capacitor voltage, vo, together with the
source voltage, vs, predicted by PSCAD/EMTDC.
9

2 142.
: 75.,
8
c 0 6
0 -50
-100
-150
0 -Y- P -a- 0 X 0.0083
Time (secs)
0.0167
Fig. 9 Voltage and current waveforms in the case <strong>of</strong> discontinuous conduction
Table II shows the values <strong>of</strong> y, t, cc, j, vo(y), and vo(y+±+oc)
predicted by both PSCAD/EMTDC, and the proposed fast time
domain method. As can be clearly noted, the values are in
perfect agreement.
TABLE II
THE RESULTS FROM PSCAD AND FAST TIME DOMAIN METHOD IN
DISCONTINUOUS CONDUCTION MODE
PSCAD/EMTDC Fast Time Domain
0.001 s 0.001 ls
0.0025s 0.0025s
cc 9.1520x 10-4s 9.3397x10-4s
0.0021s 0.0021s
vO(Y) 75.8683V 75.8982V
vI(y±+±L+o) 142.4148V 142.4172V
Furthermore, as shown in Fig. 10 and 11, good agreement
exists between PSCAD/EMTDC, and the fast time domain
method for predicting ac current and dc voltage harmonics.
VI. CONCLUSIONS
A method for the computation <strong>of</strong> the ac current and dc
voltage harmonic generated by a capacitor filtered
single-phase rectifier is presented. The approach employs
numerical iteration to determine the diode conduction and
non-conduction period, and steady state solutions. <strong>Based</strong> on
the determined conduction intervals, harmonics <strong>of</strong> interest
are solved analytically through a state augmentation method.
The results have been validated with those <strong>of</strong>
PSCAD/EMTDC to show the accuracy <strong>of</strong> the method.
One potential application <strong>of</strong> the proposed model would be
to incorporate it into a harmonic power flow program to
improve the accuracy <strong>of</strong> the existing harmonic domain
methods.
ACKNOWLEDGMENT
The authors would like to thank Dr. Brian Perkins for his
valuable input to this paper.
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25r
20
c 15
E
10 E
Ac Current (Q) Spectrum
<strong>Harmonic</strong> number
- PSCAD/EMTDC
Fast Time Domain
Fig. 9 Current harmonics for the case <strong>of</strong> discontinuous conduction
140
120
100 _
w' 80
E
I
-
60 -
40
20-
0
Dc Voltage (VO) Spectrum
I1 I
- PSCAD/EMTDC
Fast Time Domain
0 2 4 6
<strong>Harmonic</strong> number
Fig. 10 Voltage harmonics for the case <strong>of</strong> discontinuous conduction
REFERENCES
[1] M. Chen, Z. Qian, and X. Yuan "Frequency-domain analysis <strong>of</strong>
uncontrolled rectifiers," Applied Power Electronics Conference and
Exposition, vol. 2, pp. 804-809, 2004
[2] A. D. Graham "Frequency-domain analysis for multiple controlled
rectifiers," Fifth European Conference on Power Electronics and
Applications, vol.8, pp.205-210, 1993.
[3] A. V. Oppenheim, and A. S. Willsky, and H. Nawab, Signal and Systems,
Prentice Hall, 2nd edition, 1996.
[4] G. Carpinelli, F. lacovone, P. Varilone, and P. Verde, "<strong>Single</strong> phase
voltage source converters: analytical modeling for harmonic analysis in
continuous and discontinuous current conditions," International Journal
<strong>of</strong>Power and Energy Systems, vol. 23, No. 1, pp. 37-48, 2003.
[5] A. Mansoor, W. M. Grady, R. S. Thallam, M. T. Doyle, S. D. Krein, and
M. J. Samotyj, "Effect <strong>of</strong> supply voltage harmonics on the input current <strong>of</strong>
single -phase diode bridge rectifier loads," IEEE Transactions on Power
Delivery, vol. 10, No. 3, pp. 1416-1421, July 1995.
[6] A. Mansoor, W. M. Grady, A. H. Chowdhury, and M. J. Samotyj, "An
investigation <strong>of</strong> harmonic attenuation and diversity among distributed
single -phase power electronic loads," IEEE Transactions on Power
Delivery, vol. 10, No. 3, pp. 1416-1421, July 1995.
[7] N. Mohan, T. Undeland, and W. Robins, Power Electronics: Converters,
Applications, and Design, John Wiley & Sons, 2nd edition 1995.
[8] I. Dobson, and S. G. Jalali, "Suprising simplification <strong>of</strong> the Jacobian <strong>of</strong> a
diode switching circuit," IEEE International Symposium on Circuits and
Systems, pp. 2652-2655, May 1993.
[9] B. K. Perkins, and M. R. Iravani, "Novel calculation <strong>of</strong> HVDC converter
harmonics by linearization in the time-domain," IEEE Transaction on
Power Delivery, vol. 12, No. 2, pp. 867-873.

[10] P. W. Lehn, "Exact modeling <strong>of</strong> the voltage source converter," IEEE
Transactions on Power Delivery, vol. 17, No. 1, pp. 217-222, January
2002.
[11] P. W. Lehn, "Direct harmonic analysis <strong>of</strong> voltage source converter," IEEE
Transactions on Power Delivery, vol. 18, No. 3, pp. 1034-1042, July
2003.
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