PWM Rectifiers - DidatticaWEB
Transcript of PWM Rectifiers - DidatticaWEB
1/56POWER ELECTRONICS and ELECTRICAL DRIVES – PWM Rectifiers
PWM
Rectifiers
Prof. Stefano Bifaretti
University of Rome Tor Vergata
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AC/DC grid-connected Converters
• unidirectional converters (Single-phase or Three-phase)
• bidirectional converters (Single-phase or Three-phase)
Also denoted as:
PWM Rectifiers
Active Rectifiers
Power Facor Correctors
AC DC
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Diode rectifiers have a fixed output voltage. In order to regulate
the DC output voltage it is necessary to employ controlled power
semiconductor switches.
A single-phase
unidirectional converter
is obtained by the diode
bridge converter
replacing 2 diodes with
2 static switches.
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If every switch is closed with a
firing instant ton comprised in
range 0-T/4 and opened in the
time instant toff =T/2-ton, the
output voltage and supply current
waveforms are shown in the
figure.
The mean output voltage is:
ton
ton
toff
toff
21sin ( ) cos
a
a
mDC m a
VV V t d t
a=ton
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Neglecting the current ripple on the load, the fundamental
harmonic circulating in the transformer is in phase with the grid
voltage (unitary Power Factor).
The Generalized Power Factor l assumes the following shape as a
function the turn-on angle a=ton:
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For low output
voltages
A diode bridge
rectifier has
l= 0.9 the same value
assumed by this
configuration at the
max VDC = 2Vm/
0.8
Vm/ 2Vm/
VDC
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A different topology having the same behavior as the previous
circuit, but using only one switch.
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Modulated waveforms
A modulation
techniques permits to
shift at higher
frequencies the
harmonic content as
well as reduce the
sizing and volume of
the output filter.
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If an output voltage higher than 2Vm/ is required, a boost rectifier
has to be employed. This solution, commonly known as single phase
Power Factor Corrector (PFC), is a two-stage conversion system
composed of a diode rectifier and a DC-DC boost converter.
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The AC/DC bidirectional four quadrants converters are
made by traditional single-phase or three-phase bridges, where
each static device is composed by a fully controlled switch
(MOSFET, IGBT) and an anti-parallel diode.
Single-phase AC/DC bidirectional converter
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The average value of
the DC voltage must
be larger than the
value obtained by
rectifying the supply
voltage with a diode
bridge converter
Single-phase AC/DC bidirectional converter
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1. In the simplest case the AC filter is an inductive filter that
allows the reduction of the current harmonics injected in the
supply network.
2. H-Bridge, implements the AC/DC conversion
3. DC output filter the reduction of harmonics in the output DC
voltage.
From a functional point
of view, the single
phase grid connected
bidirectional converter
is composed by three
cascaded blocks.
Single-phase AC/DC bidirectional converter
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The design of the input filter is done based on an acceptable
harmonic content of the converter input current and compliant
with Grid Standards (e.g IEC 61000-3-2) or application
specifications. In order to reduce the filter sizing other common
solutions are:
Single-phase AC/DC bidirectional converter
Examples of Grid Standards
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As for the output filter we need to observe that the current i2 at
the output of the converter is periodic with a period equal to half
of that of the grid supply (f) and that its first harmonic voltage (at
frequency 2f) is usually quite substantial.
Therefore the resonant filter L1C1 needs to be tuned on the
frequency 2f so to attenuate the output voltage oscillations due to
the first harmonic current. The capacitor C allows to attenuate
higher order voltage harmonics.
Single-phase AC/DC bidirectional converter
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Single-phase AC/DC bidirectional converter
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The analysis of the converter operation will be carried out
supposing that the output filter has been designed in order to
totally neglect the harmonics on DC side.
Under this Hypothesis, if the converter works synchronized with
the grid supply voltage and with a 3-level modulation, the
voltage vx at the input of the converter has the shape illustrated
in the following figure.
Single-phase AC/DC bidirectional converter
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The fundamental of vx has the same frequency of the supply
voltage and has an amplitude Vx1 equal to:
in which k represents the modulation index (<1)
where N is the number of commutations occurring within one
quarter of a period.
Single-phase AC/DC bidirectional converter
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Considering only the components at fundamental frequency and
choosing Ea in phase with the f axis:
1 1sin cos 0a L xE V V
0a L1 x1E -V -V
1 1cos sin 0L xV V
Projection on the f axis
Projection on the q axis
indicates the phase of the
voltage first harmonic vx with
respect to ea;
Is the displacement angle
between the first current
harmonic and ea.
Vx1
Ea
Ia1
f
q
Ia1f
Ia1q
VL1
VL1
Single-phase AC/DC bidirectional converter
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Therefore the amplitudes If e Iq of the two components of the
input current first harmonic, respectively in phase and in
quadrature with respect to the supply voltage, are:
where
111
111
sincos cos
cossin sin
xLf a
x aLq a
VVI I
L L
V EVI I
L L
Single-phase AC/DC bidirectional converter
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If we neglect the converter losses and suppose that the output power Pu
Is equal to the power drawn by the supply (where R is the resistive
equivalent DC load):
We obtain a second expression for If
Single-phase AC/DC bidirectional converter
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Equating the two
expressions of If
We can obtain the following expression for the output voltage
It is also possible to obtain the following expression for the
reactive power relative to the first harmonic:
Single-phase AC/DC bidirectional converter
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The open loop control of this converter can be designed by
selecting the values of and k imposing, in the previous
equations, the desired output voltage average value and Q=0.
However, a closed loop current control is actually largely
preferred; This control is capable of adapting to the variations
of voltage amplitude and frequency of the supply network and
to load operative conditions. The current reference is usually
chosen in phase with the supply voltage, with an amplitude
value so to obtain the desired value of the output DC voltage.
A closed loop control for the output DC voltage is also
necessary to ensure the load voltage regulation.
The two control loops are structured in a cascade configuration.
Single-phase AC/DC bidirectional converter
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Closed-loop Control
Load Converter
PLL
Rectifier operation
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The control loop for the DC output voltage is implemented bymeans of a PI regulator which ensures a zero steady-state error.
The output of the PI regulator represents the amplitudereference value for the converter input current. This value isthen multiplied by a unity sinusoidal template functionsynchronous with the supply voltage (so to obtain unity powerfactor operation). The sinusoidal template is synthesized usingthe estimation of phase and frequency of the supply voltageobtained by the PLL. The reference for the current control loopis therefore:
* * cos( )s aI I t
VOLTAGE CONTROL LOOP
Closed-loop Control
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CURRENT CONTROL LOOP
The current control loop also uses in this case a Proportional Integral
(PI) regulator. The output of this regulator represents the reference
voltage for the PWM modulator.
It is to be noticed that the current reference is, in steady state, a
sinusoidal signal at the same frequency of the supply voltage (50Hz);
for this reason the bandwidth of the current control loop needs to be
large enough to ensure a satisfactory tracking of the reference signal.
In practical applications, however, delays due to the control
implementation and to the PWM delay, together with sampling
effects, need to be taken into account. Too high values for the
controller gains, could also lead to instability. Therefore even a control
design with a large bandwidth, often produces phase shifts and
attenuation that significantly affect reference tracking.
Closed-loop Control
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CURRENT CONTROL LOOP DESIGN
Given the cascade structure of the nested control loops, the inner current loops
needs to be faster than the outer voltage loop. It is good design practice to
ensure that the bandwidth of the current control loop is at least 10 times larger
that that of the voltage control loop.
The design of the current loop can be derived applying the Kirchhoff voltage
law to the input circuit:
ss a x
diL Ri e v
dt
The input current can be regulated, therefore, by controlling the PWM voltage
at the output of the converter.
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DESIGN OF THE CURRENT LOOP
A simplified scheme of the current loop for control design purposes is:
If we neglect delays and non-linearity typical of a power converter, in an ideal
situation the output voltage vx will be equal in steady-state to the reference signal.
Thus, the behaviour of the PWM converter can be approximated by a unity gain
block.
For a more accurate modelling, if the delays introduced by the converter are
estimated to be equal to a certain time Ts, the converter transfer function can be
represented as:
Is* (s)
R2(s)+
-PWM
Vx* (s) Vx
(s) -+ 1
f fR sL
Ea (s)
Is (s)
14
14
S
s
sT
s
Ts
eT
s
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Is* (s)
R2(s)+
-PWM
Vx* (s) Vx
(s) -+ 1
f fR sL
Ea (s)
Is (s)
DESIGN OF THE CURRENT LOOP
A simplified scheme of the current loop for control design purposes is:
In case of LCL filters a second-order TF has to be considered thus introducing a resonance in the control loop that has to be damped.
For LLCL filters the model is still valid as the resonant LC is tuned on the switching frequency.
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DESIGN OF THE CURRENT LOOP
The PI parameters can be calculated by means of classical design methods:
•Trial and error
•Zigler Nichols
•Root Locus
•Frequency response methods (Bode diagram)
•Advanced identification methods (Genetic Algorithms, Neural Networks…)
•Zero-pole cancellation as done for DC-DC converters
The use of a PI regulator it is not the only possible choice. Any type and order
of linear controller can be used, still considering that in this case a zero steady
state error will never be obtained. It will only be possible to reduce the steady
state error by increasing as much as possible (instability and noise issues) the
control loop bandwidth.
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DESIGN OF THE CURRENT LOOP
A simplified scheme of the current loop for control design purposes is:
The source voltage Ea acts as a disturbance in the direct chain.
For PI regulator the following transfer function can be used:
22 2
2
1( ) i
pi
i
sR s k
s
22 2
2
1 1 1( )
1
fipi
i f f
LsG s k
s R s R
Open-loop transfer function G2(s):
Is* (s)
R2(s)+
-PWM
Vx* (s) Vx
(s) -+ 1
f fR sL
Ea (s)
Is (s)
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DESIGN OF THE CURRENT LOOP
If we choose i2 = the open-loop TF becomes:
2 2
1( ) pi
f
G s ksL
2 2
2 2
1( )
1
f
pi
LW s
s k
The current loop can be approximated with a time constant 2
2
11 s
*( )sI s ( )sI s
Closed-loop TF W2(s):
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DESIGN OF THE VOLTAGE LOOP
When determining an equivalent simplified scheme for the design of the voltage
control loop, the dynamics of the inner current control loop, being much faster
than the one of the outer voltage loop, can be neglected. In fact its transients
effects on the voltage signal decay very quickly.
It is therefore a good approximation to suppose that the current amplitude at the
output of the current loop, tracks in steady state the reference Is* at the output of
the voltage loop PI.
To further simplify the scheme, it is also possible to neglect the presence of the
LC output filter on the DC side; In this case the scheme of the voltage control
loop can be approximated by the DC side capacitor only and by the load.
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DESIGN OF THE VOLTAGE LOOP
Using the previous result of current control loop design, it is possible to
consider the following block scheme for the voltage loop design having
considered an equivalent resistive load Rc.
Kpi1 and i1 values should be selected so that the dynamics of the
inner current control loop will be much faster (at least 10 times) than the one of the outer voltage loop, on the basis of
desired gain and phase margins.
11
1 2
1 1( )
1 1
i cpi
i c
s RG s k
s s sCR
R1(s) 1c
c
RsCR
VDC(s)VDC*(s)
2
11 s
*( )sI s ( )DCI s
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
50
100
150
200
250
300
350
400
450
time [s]
Vu [
V]
Voltage transient on the DC side
Closed-loop Control
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0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4
-200
-150
-100
-50
0
50
100
150
200
Time [s]
is [
A]
vs [
V]
ea
isref
is
Waveforms on the AC side can present a significant phase shift
(in the figure it has been intentionally enlarged)
Closed-loop Control
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CURRENT CONROL BY MEANS OF RESONANT
CONTROLLERS
In order to obtain an accurate current control at the fundamental
frequency, with attenuation of all other frequencies and obtaining
at the same time a zero steady state error, it is possible to emulate
the same behaviour an integrator has with DC signals, utilizing a
proportional plus resonant type controller (PR) tuned at the
fundamental frequency.
2 2( ) r
p res p
K sC s K
s
Kp is the proportional gain, Kr is the gain of the resonant term and
ω is the resonant angular frequency to be set at the value of the
supply voltage frequency (ω=2π50 on 50 Hz networks).
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Load Converter
PLL
Resonant Controllers
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Using this structure, the controller gain at the resonance frequency
is infinite and, in closed loop, the steady state error is theoretically
zero for signals at that resonance frequency.
A too high gain can cause control problems, reducing the system
phase margin and its robustness.
However the gain of the resonant term needs to be high enough to
obtain a zero or nearly zero steady state error.
We can then introduce the quality factor:
fH and fL are the higher and lower frequencies at which the
controller gain has dropped to 0.707 of the value at the resonance
frequency.
H L
fQ
f f
Resonant Controllers
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2 2( )
( / )
rp
K sC s K
s Q s
Bode diagram of C(s) in function of Q
Resonant Controllers
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DESIGN OF THE RESONANT CONTROLLER
The design of Q is a compromise choice:
High values of Q determine a higher gain at the resonant frequency and a
narrower bandwidth around this frequency.
• Advantages: Reduced steady state error and larger selectivity of the control.
• Disadvantages: High sensitivity to noise and possibility of unstable behaviour.
Dual considerations can be done for low values of Q
The resonant term presents only a very low gain outside its pass band, therefore
the desired transient response needs to be obtained by a careful design of Kp
The term Kr is instead selected in order to improve the system
stability, placing the system zeros in a way to push the closed loop poles
towards more stable positions.
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0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4
-200
-150
-100
-50
0
50
100
150
200
Time [s]
is [
A]
vs [
V]
ea
isref
is
Waveforms on the AC side
Resonant Controllers
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Considering the converter as a unity gain block, the system transfer
function in closed loop is:
2 2
*
2 2
1
( / )( )
11
( / )
rp
s
s rp
K sK
R sLs Q sIG s
I K sK
R sLs Q s
Resonant Controllers
2 2
( / )
rK s
s Q s
+
-PWM
-+ 1
f fR sL+
+
Kp
Is* (s) Vx
* (s) Vx (s)
Ea (s)
Is (s)
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( )
2 2
* 2 2 2 2
( / )( )
( / ) ( / )
r ps
s r p
K s K s Q sIG s
I K s K s Q s s Q s R sL
( )
2 2
* 2 2 2 2( )
r ps
s r p
K s K sIG s
I K s K s s R sL
*( ) 1s r
rs
I K sG s
K sI
In the ideal case when Q→∞ it is:
If the reference signal is at the resonance frequency, s=-jω:
Working the equation out we can obtain:
In steady state the output will therefore track the reference with zero error.
Resonant Controllers
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How does the resonant controller behave with respect to the disturbance
represented by the supply voltage?
2 2
1
( )1
1( / )
sD
a rp
I R sLG sE K s
KR sLs Q s
For signals at the resonance frequency (s=-jω) and in the ideal case (Q→∞ ):
GD(s)=0
Hence the supply voltage does not affect the control loop with any disturbance.
Zero steady state error and cancellation of disturbance effects for signals at the
resonance frequency can be obtained independently from the design values of Kp
e Kr and from the system parameters R e L, ensuring the theoretical robustness of
the control.
Resonant Controllers
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Very often the supply voltage is distorted and therefore it contains harmonics at
various frequencies different from the resonance frequency. This can reduce the
effectiveness of the control.
A way forward to solve this problem is to add to the converter reference signal a
feed-forward term represented by the supply voltage. In this way its effect on the
current control loop is cancelled.
Resonant Controllers
2 2
( / )
rK s
s Q s
+
-PWM
-+ 1
f fR sL+
+
Kp
Is* (s) Vx
* Vx (s)
Ea (s)
Is (s)+
+
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Three subsystems can be highlighted:
1. AC-side filter
2. Controlled three-phase bridge
3. DC-side filter
3-phase
grid
ea
ib
ic
Lf
Lf
Lf
eb
ec
ia
N
vaRf
Rf
Rf
vb
vc
CDC
S1
S2
S3 S5
S4S6
Passive Load
or
DC-Source
AC-side filter
DC-side filter
Three-phase AC/DC bidirectional converters
Two goals:
• decouple the grid and the converter voltages
• reduce the input currents harmonics in order to be compliant with
Grid Standards (e.g. IEC 61000-3-2) or application requirements.
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Three subsystems can be highlighted:
1. AC-side filter
2. Controlled three-phase bridge
3. DC-side filter
3-phase
grid
ea
ib
ic
Lf
Lf
Lf
eb
ec
ia
N
vaRf
Rf
Rf
vb
vc
CDC
S1
S2
S3 S5
S4S6
Passive Load
or
DC-Source
AC-side filter
DC-side filter
Three-phase AC/DC bidirectional converters
The bridge technology choice depends on target power and/or power density:
• Discrete devices (low power density, low-cost, accurate PCB design needed)
• IGBT/MOSFET power module (high-power, medium power density)
• SiC MOSFET/IGBT power module (medium-power, high power density)
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Three subsystems can be highlighted:
1. AC-side filter
2. Controlled three-phase bridge
3. DC-side filter
3-phase
grid
ea
ib
ic
Lf
Lf
Lf
eb
ec
ia
N
vaRf
Rf
Rf
vb
vc
CDC
S1
S2
S3 S5
S4S6
Passive Load
or
DC-Source
AC-side filter
DC-side filter
• the energy transfer from the three-phase grid to the DC-side is
almost continuous;
• the output current ripple has a frequency equal to 6f and the
amplitude of its first harmonic is quite reduced compared to the single
phase case
resonant output filter is not necessary.
Three-phase AC/DC bidirectional converters
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For three phase systems
Three-phase AC/DC bidirectional converters
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Control in the natural reference frame abc(rectifier mode)
G
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• Power factor corrector circuit (e.g. to compensate reactive power
absorbed by other loads)
• Active filter (to compensate for harmonics produced by non linear
loads)
Examples of Applications
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Electronic transformers (Frequency links)
Examples of Applications
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AC electrical motor drive
This converter finds its more common application as input stage of
a back to back AC/AC converter with intermediate dc-link and
voltage source inverter since it allows to draw power from the
power supply with high power factor and allows obtaining a high
quality and stabilized dc link voltage.
Examples of Applications
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Grid Interface system of a wind power generator
• Variable input frequency;
• Can be used with different energy sources.
Examples of Applications
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DC/AC three phase grid interface solution for PV systems
LCL grid connecting stage are usually used;
DC/DC boost converters are usually present for MPPT algorithm implementation
Control can be the same as in previous examples
Examples of Applications