PWM Rectifiers - DidatticaWEB

1/56POWER ELECTRONICS and ELECTRICAL DRIVES – PWM Rectifiers

PWM

Rectifiers

Prof. Stefano Bifaretti

University of Rome Tor Vergata


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


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.


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


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:


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


A different topology having the same behavior as the previous

circuit, but using only one switch.


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.


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.


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


The average value of

the DC voltage must

be larger than the

value obtained by

rectifying the supply

voltage with a diode

bridge converter



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.



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:


Examples of Grid Standards


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.





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.



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.



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



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



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



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:



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.



Closed-loop Control

Load Converter

PLL

Rectifier operation


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


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


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.


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


Is* (s)

R2(s)+

-PWM

Vx* (s) Vx

(s) -+ 1

f fR sL

Ea (s)

Is (s)



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.



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.




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)



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):


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.


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


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


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


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).


Load Converter

PLL

Resonant Controllers


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



2 2( )

( / )

rp

K sC s K

s Q s

Bode diagram of C(s) in function of Q



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.


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



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


2 2

( / )

rK s

s Q s

+

-PWM

-+ 1

f fR sL+

+

Kp

Is* (s) Vx

* (s) Vx (s)

Ea (s)

Is (s)


( )

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.



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.



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.


2 2

( / )

rK s

s Q s

+

-PWM

-+ 1

f fR sL+

+

Kp

Is* (s) Vx

* Vx (s)

Ea (s)

Is (s)+

+


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.



1. AC-side filter


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 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)



1. AC-side filter


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.



For three phase systems



Control in the natural reference frame abc(rectifier mode)

G


• 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


Electronic transformers (Frequency links)



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.



Grid Interface system of a wind power generator

• Variable input frequency;

• Can be used with different energy sources.



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


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