Matrix Converter Technology

IECON 2005 Matrix Converter Tutorial

November 2005

Matrix Converter Technology Dr Pat Wheeler and Prof Jon Clare Power Electronics, Machines and Control Group School of Electrical and Electronic Engineering University of Nottingham, UK Tel. +44 115 951 5591

Email. [email protected]

Presentation Outline I Basic Matrix Converter Concepts (Jon Clare) Power Circuit Implementation (Pat Wheeler) • Bi-directional switch implementation and available semiconductor device products • Status of Devices: SiC, Reverse Blocking IGBTs • Current Commutation strategies • Power circuit protection • Practical circuit layout issues

Modulation Algorithms (Jon Clare) • • • •

Mathematical model Basic Modulation problem and solution Voltage ratio limitation Principal modulation methods: Venturini, Space vector, Max-mid-min, Fictitious DC Link

School of Electrical and Electronic Engineering, University of Nottingham, UK


November 2005

Presentation Outline II Design Issues (Jon Clare) • Comparison of modulation methods • Input Filter design • Matrix Converter losses and comparisons with other topologies

Two-Stage Matrix Converters (Pat Wheeler) • • • •

Basic Principle of Operation Circuit topologies and device count Comparison of Sparse Matrix Converter Topologies Modulation Schemes

Experimental Matrix Converters and applications (Pat Wheeler) • Application Examples • Industrial Products

Potential Future Application Areas (Jon Clare and Pat Wheeler)

Jon Clare



November 2005

Matrix Concept Input filter

3-phase supply

Bidirectional switch

Load

Variable frequency Variable voltage 3-phase output

Basic Ideas Switching pattern and commutation control must avoid line to line short circuits at the input Switching pattern and commutation control must avoid open circuits at the output Each output phase can be connected to any input phase at any time Switch duty cycles are modulated so that the “average” output voltage follows the desired reference (for example a sinusoidal reference) Modulation is arranged so that the “average” input current is sinusoidal when the input voltage, output reference and output current are sinusoidal



November 2005

Nomenclature Phase Labelling Convention

A B

SAa

C a

b

c

Load

Example Switching Pattern Possible arrangement SAa (on)

SCa (on)

SBa (on) tBa

tAa

tCa SCb (on)

SBb (on)

SAb (on) tAb

tBb

SAc (on)

tCb SBc (on)

tAc

tBc

SCc (on) tCc

Tseq (sequence time)

Output phase a Output phase b Output phase c

Repeats

Switching frequency = 1/Tseq Modulation strategy ensures that tAa - tCc are generated so that the average output voltage during each sequence equals the target output voltage. The sequence time is constant.



November 2005

Illustrative Output Waveforms Fin > Fout

Output line to supply neutral voltage

360 Volts

50Hz in - 25Hz out switching frequency 500Hz

240 120 0

-120 -240 -360 0

Volts

Time (ms)

40

Output line to line voltage

600

Low switching frequency shown for visual clarity

20

400 200 0 -200 -400 -600 0

10

Time (ms)

20

Illustrative Output Waveforms Fin < Fout

Output line to supply neutral voltage

360 Volts

50Hz in - 100Hz out switching frequency 1kHz

240 120 0

-120 -240 -360 0

10

Time (ms)

20

Output line to line voltage 600


Volts

400 200 0

-200 -400 -600 0


10

Time (ms)

20


November 2005

Illustrative Input Waveforms 1.2 0.8

Input current (unfiltered) 50Hz in - 25Hz out

0.4 0 -0.4 -0.8


-1.2 0

20

40

60

Time(ms) 80

0

5

10

15

Time(ms) 20

1.2 0.8

Input current (unfiltered) 50Hz in - 100Hz out

0.4 0 -0.4 -0.8 -1.2

Example Spectra 100 % 80

50Hz in - 25Hz out

Output voltage 25Hz Sidebands around multiples of the switching frequency

60 40

2kHz switching

20 0 0

Exact nature of spectra depends on modulation method

1

100 50Hz

% 80

2

3

4

kHz

5

Input Current Sidebands around multiples of the switching frequency

60 40 20 0 0


1

2

3

4

kHz

5


Modulation Control A number of modulation strategies have been proposed. All of them allow flexible control with the following features: • Continuous control of output voltage amplitude from zero up to a maximum limit • Continuous control of output frequency up to a maximum feasible limit of approximately 1/10 of the switching frequency • Control of input displacement factor: unity, leading and lagging regardless of output power factor

DC-AC and AC-DC conversion is an inherent feature by setting either the input or output frequency to zero

Matrix Converter Features Direct conversion - No DC link - “all silicon solution” No restriction on input and output frequency within limits imposed by switching frequency Inherent bi-directional power flow in all modes with 4 quadrant voltage-current characteristics at both ports “Sinusoidal” input and output currents Potential for high power density if switching frequency is high enough Output voltage limited to 87% of input voltage (for most modulation schemes) Higher semiconductor count than other AC-AC configurations


November 2005


November 2005

Alternatives Rectifier DC link Inverter

3-Phase Supply

3-Phase Load

Industry “workhorse” - made from a few kW to MW Unidirectional power flow Poor AC supply current waveforms DC link capacitor is often 30% - 50% of the power circuit volume at 20kW upwards

Alternatives “Back to Back” DC link Inverter

3-Phase Supply

3-Phase Load

Bi-directional power flow PWM control of input bridge with line inductors gives sinusoidal input currents Large DC link capacitor and line inductors Matrix converter provides the same functionality



November 2005

Perceived and Actual Limitations Voltage Transfer Ratio • Output voltage is limited to 86% of the input voltage • Only a problem if standard motors are used from a standard supply

Device Count • Normally requires 18 fully controllable switching devices for a 3-phase to 3-phase converter • Compares to 12 switching devices and large reactive components for a back-to back inverter circuit

Control Algorithms • Considered complex by some researchers • Have been reported as processor intensive • No longer really and issue

Device Count

Topology

Fully Controlled Devices

Fast Diodes

Rectifier Diodes

Large Electrolytic Capacitors

Large Inductors

Matrix Converter

18

18

0

0

0

Back-toBack Inverter

12

12

0

1

3

Inverter with Diode Bridge

6

6

6

1

0 or 1

Conventional rectifier DC Link inverter • Has poor supply current waveforms • Provides no regenerative capability • Requires a DC link capacitor


Back to back inverter • • •

Provides regenerative capability Has sinusoidal supply currents Requires a DC link capacitor


November 2005

Pat Wheeler

Presentation Outline

Power Circuit Implementation • Bi-directional switch implementation and available semiconductor device products • Current Commutation strategies • Practical circuit layout issues • Power circuit protection



November 2005

Matrix Concept

Bidirectional Switch

Motor

The Bi-directional Switch • Must be able to conduct positive and negative currents • Must be able to block positive and negative voltages

Possible Switch Configurations Diode Bridge • High conduction losses » Two diodes and a switching device conducting

• Only one switching device per switch



November 2005

Possible Switch Configurations Back to Back Switch • Two switching devices per switch • Conduction losses of only one diode and one switching device • Common Collector » Pair of switching devices arranged with collectors connected » Diodes required for reverse blocking capability

Possible Switch Configurations Back to Back Switch • Common Emitter » Pair of switching devices arranged with emitters connected » Both devices can be gated from the same isolated power supply

• Can Control Direction of Current Flow within each Switch » Useful for most current commutation strategies

• Diodes can be Si or SiC » SiC may offers lower conduction losses, depending on device rating



November 2005

Possible Switch Configurations Back to Back Switch • Reverse Blocking IGBTs » Pair of reverse blocking IGBTs » Lower conduction losses » Reverse recovery can be an issue and may lead to higher switching losses

• Simpler Power Semiconductor Module Design » Increase in theoretical reliability?

• Can Control Direction of Current Flow within each Switch

Matrix Converter Device Packaging A Bi-directional Switch in a Single Package • Two IGBTs and associated diodes • A rearranged ‘Inverter leg’ • 200Amp samples available from Dynex Semiconductors

A Matrix Converter Output Leg in a Single Package • Possible to have 3 bi-directional switches in a single package » One package per output leg of the converter » Possible advantages in the minimisation of inductance between devices

• Can be built as specials by Dynex and Semelab • Products from Fuji, IXSY and Mitsubishi using Reverse blocking IGBTs

A Complete Matrix Converter in a Single Package • Suitable for lower power levels • Eupec had a 400V, 7.5kW matrix converter ‘ECONOMAC’ module



November 2005

A Bi-directional Switch in a Single Package

Dynex 200Amp Bi-directional Module DIM200MBS12-A

Nine packages for a 3-phase to 3-phase Matrix Converter Used for larger converters, say >200Amps

Common Emitter

A Matrix Converter Output Leg in a Single Package

600V, 300A (SEMELAB)

1700V, 600A (DYNEX)

Three packages for a 3-phase to 3-phase Matrix Converter Used for medium converters, say 50Amps to 600Amps



November 2005

A Complete Matrix Converter in a Single Package

EUPEC 35 Amp Matrix Converter Module

One package for a 3-phase to 3-phase Matrix Converter Used for small converters, say >50Amps

A Complete Matrix Converter in a Single Package EUPEC 35 Amp Matrix Converter Module

A three phase to three phase matrix converter 7.5kW from a 400V supply



November 2005

The Current Commutation Problem

3-phase input

Motor

Two Rules • Do not short circuit input lines » will short circuit the supply

• Do not open circuit output lines » will open circuit inductive load

The Two Rules for Safe Current Commutation • Do not short circuit input lines 2-phase input

Load

• Do not open circuit output lines 2-phase input

Load



November 2005

Switch Cells for a 2-Phase to 1-Phase Converter

SA1

 A1 A2  B1 B2  

SA2 SB1

RL Load

SB2

2-Switch Converter Commutation Options Switch states for a 2 to 1 matrix Converter • Allowable conditions for each state is given

Commutation path just has to follow the allowable conditions

V1 V2

Io


1 1 1 1   V1=V2

1 1  1 0   V1>V2

1 0 1 1    V1V2

1 0 1 0   Io +ve

0 1 0 1   Io -ve

1 1 0 0    Any

0 0 1 1   Any

0 0  0 0    Io = 0

1 0 0 0    Io +ve 2

0 1  0 0    Io -ve

0 0  1 0   Io +ve

1 0 0 1    V1V2

1 1 0 1   V1 0

• V1 = +2.5 Volts and V2 = -1.2 Volts If IL < 0

and V2 = +2.5 Volts


Only devices which are conducting are turned on

Forms a Two Step Commutation Strategy •

• S2 and D2 are conducting • S1 and D1 are reverse biased

• V1 = -1.2 Volts

Turns off all Devices Which are Not Conducting •

• S1 and D1 are conducting • S2 and D2 are reverse biased

Direct measurement of actual current flowing Current direction information passed between cells

Minimisation of switch state change delays


November 2005

Current Direction

Current [mA]

Current Detection Circuit Output During Increasing Current 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time [ms]

Current [mA]

Current Detection Circuit Output During Decreasing Current 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 Time [ms]

Experimental Results 40

30Hz Output

30

Loa d Curre nt (A)

20

2kHz Switching

10 0 -10 -20 -30 -40 0

5

10

15

20

25 Time (ms)

30

35

40

45

50

5

10

15

20

25 Time (ms)

30

35

40

45

50

400 300

Loa d Volta ge (V)

200 100 0 -100 -200 -300 -400 0



November 2005

Input Voltage Based Commutation Uses Input Voltages to Make Current Commutation Decisions • Relies on knowledge of relative magnitude order of the input voltages • Requires accurate and balanced measurement of input voltage waveforms required

Example: 4-Step Voltage Commutation • Must avoid critical areas where input voltages are close » Prevention method » Replacement method

4-Step Voltage Based Commutation SA1

SA1

SA1 0V

SA2

SA2

SA2

SB1

SB1

SB1 100V

SB2

SB2

SB2 SA1

SA2

SA1

SA2 SB1

SB2


SB1

SB2


November 2005

4-Step Voltage Based Commutation Critical areas VB

VA

VC

Problems may occur when voltages are very close •

Critical areas

•

Could commutated via the other voltage

•

Could rearrange commutation sequence

−

−

Extra losses and unwanted pulses Waveform quality issues unless inherent in control algorithm

4-Step Voltage Based Commutation VA

VA

VB

VB

VC

VC

…A

…A – C – B – B – C – A…

B – C – A \

/

B – C… \

C

/ C

Extra states

Critical Step Prevention Method •

Rearrange commutation sequence


Critical Step Replacement Method •

Commutated via the other voltage


November 2005

Comparison of Commutation Methods Output Current Commutation Methods • Relies on measurement of the output current direction on each output leg • Output line open circuit if a commutation error occurs » Overvoltage clamp used

Input Voltage Commutation Methods • Relies on measurement of the relative input voltages • Longer commutation times • Input line short circuit is a commutation error occurs »?

Some Protection Issues Fault conditions • Overcurrent due to short circuit » Commutation failure

• Loss of supply • Output power overload

Protection strategies • No natural freewheeling paths • Have to provide energy storage in event of turning-off all devices » Overvoltage clamp » Freewheeling with the matrix converter circuit



November 2005

Matrix Converter Protection

a b c

Input filter

Cin

CClamp

Lin

Clamp circuit line

A B C

3x3 matrix of bi-directional switches

Auxiliary circuits supply unit (gate-drivers, transducers, control)

SMPS

IM 3~

motor

Capacitor is typically very small depends on nature of load

For a 3kW Matrix Converter Drive for an Aircraft Actuator (shown later) machine inductance = 1.15mH maximum output current is, say, 30Amps capacitor required is 2µF

Power Circuit Layout Minimisation of mutual inductance between input lines Inclusion of local capacitance between input lines Laminated input line bus bars • Simplifies power circuit assembly Lstray Clocal Lstray Clocal

Clocal Lstray



November 2005

IGBT Turn-off Voltage when using Laminated Input Power Planes

Device Voltage (V)

600 500 400 300 200 100 0 0

200

400

600

800

Time (ns)

Jon Clare


1000


November 2005


Modulation Algorithms • Mathematical model • Basic Modulation problem and solution • Voltage ratio limitation • Principal modulation methods » Venturini, Space vector, Max-mid-min, Fictitious DC Link

Ideal Switch Matrix vA Input

vB vC

iA iB

SAa

iC va

vb

vc

ia

ib

ic

Output

Assume voltage fed input and current sink output - inductors represent inductive load Measure all voltages with respect to a hypothetical star (wye neutral) point of the supply



November 2005

Mathematical Model Assuming instantaneous and perfect commutation v a ( t )   S Aa ( t ) v ( t )  =  S ( t )  b   Ab  v c ( t )   S Ac ( t )

S Ba ( t ) S Bb ( t )

 i A ( t )   S Aa ( t )  i (t ) = S (t )  B   Ba  i C ( t )   S Ca ( t )

S Ab ( t )

S Ca ( t )  v A ( t )  S Cb ( t )   v B ( t )  S Cc ( t )  v C ( t ) 

S Bc ( t )

S Ac ( t )   i a ( t )  S Bc ( t )   i b ( t )  S Cc ( t )   i c ( t ) 

S Bb ( t ) S Cb ( t )

where the switching

function

S Kj ( t ) is 1 when the switch

joining input line K to output line j is ON and is 0 otherwise. Voltage and current constraint

∑S

K = A ,B ,C

Ka

(t ) =

∑S

K = A ,B ,C

Kb

(t ) =

∑S

K = A ,B ,C

rules require that :

Kc

( t ) =1

Example Switching Pattern SBa =1

SAa=1 (on)

tBa

tAa SAb=1

tCa

SBb=1

tAb

SCb=1

tBb

SAa=1

tBc

Output phase a Output phase b

tCb SBc=1

tAc

Sca=1

SCc=1 tCc

Tseq (sequence time)

Output phase c Repeats

Switching frequency fsw = 1/Tseq Many different ways of sequencing the switches are possible – depends on modulation strategy Define the modulation duty cycle for each switch as mAa(t) = tAa/Tseq etc



November 2005

Low Frequency Modulation Model Switching function model gives instantaneous relationships - not immediately useful for studying modulation Assume that the input frequency and output frequency (fi, fo) a2 : input is inductive (lagging displacement factor) a1 < a2 : input is capacitive (leading displacement factor)

Assuming unity displacement factor solution, allows the switch duty cycle calculation to be reduced to: mKj =

1  2v K v j  1 + 2  3 Vim 


for K = A, B,C and j = a, b, c


November 2005

Voltage Ratio Limitation Average output voltage taken over each switching sequence equals the target voltage Target voltage must fit within input voltage envelope Input voltage envelope Target output voltages

1.2 0.8 0.4 0 -0.4 -0.8 -1.2

0

90

180

270

360

Basic algorithm has a voltage ratio limitation of 0 < q < 0.5

Optimised Voltage Ratio Modify target output voltages to use all the input volt-second area. Target voltages become:

1.2 0.8 0.4 0 -0.4 -0.8 -1.2

 cos(ωot ) − 61 cos(3ωot ) + 1 cos(3ωi t )  2 3 [vo (t )] = qVim cos(ωot + 2π / 3) − 61 cos(3ωot ) + 2 13 cos(3ωi t ) cos(ω t + 4π / 3) − 1 cos(3ω t ) + 1 cos(3ω t ) o o i  6 2 3 

Target output voltages with q=0.866

0

90

180

270

360

Input voltage envelope

Maximum voltage increased to 87% of input Added triple harmonics cancel in the output line to line voltages



November 2005

Added Voltage Cancellation Vim cos(ω i t )

Matrix Converter

qVim 6

cos(3ωot )

im − qV cos(3ωi t ) 2 3

qVim cos(ω o t )

Venturini Optimum Amplitude Method Extension to original method to allow use of the modified target waveform set Input displacement factor control is at the expense of voltage ratio Algorithm can be simplified for unity displacement factor to yield:

m Kj =

 1  2v K v j 4q + sin( ω i t + β K ) sin( 3 ω i t ) 1 + 3 Vim 2 3 3 

for K = A, B,C and j = a, b, c β K = 0,2 π/3,4 π/3 for K = A,B,C respectively and v j includes the triple harmonic addition



November 2005

Cyclic Venturini Method (1) Original Venturini method uses a “single-sided” fixed switching sequence S11 t11

S21

S12

S13

t12

t13 S23

t21

S22 t22

t23

S31

S32

S33

t31

t32

t33

tseq

S11 ≡ SAa, S12 ≡ SBa etc

Cyclic Venturini Method (2) Cyclic Venturini method uses a “double-sided” switching sequence S13

S12

S12

S11

t12/2

t12/2

t11/2 S21

S11 t11/2

t13/2 S23

S21

t23/2

S33

t33/2

t21/2 S31 t31/2

tseq/2

S22 t22/2 S32 t32/2

S22 t22/2 S32 t32/2

t21/2

S13 t13/2 S23 t23/2

S31

S33

t31/2

t33/2

tseq/2

“Cyclic” refers to the fact that the selection order of input voltages (3-12-2-1-3 above) is changed every 60O of input period. Input voltage with largest absolute magnitude (1 above) is always placed in the middle. Duty cycle calculations are identical to standard (optimum) Venturini method.



November 2005

Cyclic Venturini Method (3) Line to Line Voltage

Non-cyclic (standard)

Cyclic

Cyclic method eliminates sub-optimal vectors

Space Vector Concept Space vector concept allows a 3-phase set of quantities to be represented by a single vector on a complex plane Define space vector of (Va, Vb, Vc) as: 2 Vo (t ) =  v a (t ) + v (t )e j 2π / 3 + v c (t )e j 4π / 3  b 3 

Geometrically, this amounts to plotting the instantaneous values of the three voltages along axes displaced by 120O



November 2005

Space Vector Illustration Assume target voltages are:

v a = qVim cos(ω o t ), v b = qVim cos(ω o t + 2π / 3), v c = qVim cos(ω o t + 4π / 3) Result is that Vo(t) - the target output voltage space vector has constant length qVim and rotates at ωO when plotted in the complex plane imd0046.html Space vector of input current is defined in the same way  2 I (t ) =  ia(t ) + i (t )e j2π / 3 + ic (t )e j 4π / 3  i b 3 

Target space vector of input current is normally chosen to line up with the input voltage space vector (unity displacement factor), and rotates at ωi

Matrix Converter Space Vectors 27 possible vectors can be split into 3 groups Group I: each output line is connected to a different input line. Space vectors of output voltage rotate at ωi Space vectors of input current rotate at ωO

Group II: two output lines are connected to a common input line, the remaining output line is connected to one of the other input lines. Space vectors of output take one of 6 fixed positions (varying amplitude) Space vectors of input current take one of 6 fixed positions (varying amplitude)

Group III: all output lines are connected to a common input line. All space vectors are at the origin (zero length)

Group I is not useful, only Groups II (18 vectors) and III (3 vectors) are used



November 2005

Group II Space Vectors Vector Number +1 -1 +2 -2 +3 -3 +4 -4 +5 -5 +6 -6 +7 -7 +8 -8 +9 -9

Conducting Switches SAa SBa SBa SCa SCa SAa SBa SAa SCa SBa SAa SCa SBa SAa SCa SBa SAa SCa

SBb SAb SCb SBb SAb SCb SAb SBb SBb SCb SCb SAb SBb SAb SCb SBb SAb SCb

SBc SAc SCc SBc SAc SCc SBc SAc SCc SBc SAc SCc SAc SBc SBc SCc SCc SAc

Output Phase Voltages vb vc va vB vB vA vA vA vB vB vC vC vB vB vC vA vA vC vC vC vA vA vB vB vB vA vA vB vC vC vC vB vB vC vA vA vA vC vC vB vA vB vA vB vA vC vB vC vB vC vB vA vC vA vC vA vC

Output Line to Line Voltages vab vbc vca 0 -vAB vAB -vAB 0 vAB vBC 0 -vBC 0 -vBC vBC 0 -vCA vCA 0 -vCA vCA -vAB vAB 0 0 vAB -vAB 0 -vBC vBC 0 vBC -vBC -vCA vCA 0 0 vCA -vCA 0 -vAB vAB 0 vAB -vAB 0 -vBC vBC 0 vBC -vBC 0 -vCA vCA 0 vCA -vCA

Input Line Currents IA Ia Ib+Ic 0 0 Ib+Ic Ia Ib Ia+Ic 0 0 Ia+Ic Ib Ic Ia+Ib 0 0 Ia+Ib Ic

IB Ib+Ic Ia Ia Ib+Ic 0 0 Ia+Ic Ib Ib Ia+Ic 0 0 Ia+Ib Ic Ic Ia+Ib 0 0

IC 0 0 Ib+Ic Ia Ia Ib+Ic 0 0 Ia+Ic Ib Ib Ia+Ic 0 0 Ia+Ib Ic Ic Ia+Ib

Modulation Calculations Calculations are performed at a regular sampling frequency. Target output voltage space vector rotates, but can be assumed to be fixed at a particular magnitude and position during each sampling period. Output voltage space vectors that the converter can produce are fixed in position (or zero). Time weighted switching between adjacent vectors, produces the correct target “average” output voltage vector during each sampling period. Use of 4 (non-zero) vectors in each sampling period allows input current space vector direction to be controlled as well (for unity displacement factor). Any extra time in the sampling period not occupied by active vectors is filled with zero vectors. Sequence of the 4 active vectors is chosen to minimise commutations.



November 2005

Target Vector Synthesis ±4, ±5, ±6

±2, ±5, ±8

±7, ±8, ±9

ωo

±1, ±4, ±7

Target vector

±1, ±2, ±3

±3, ±6, ±9

Input current space vectors

Target ±1, ±2, ±3 vector

Output voltage space vectors

±3, ±6, ±9 ±4, ±5, ±6

±7, ±8, ±9

±1, ±4, ±7

ωi ±2, ±5, ±8

For any condition, using 4 vectors allows control of output voltage magnitude and angle and input current angle (displacement factor) In this case vectors are 5, 6, 8, 9

Vector Sequences S13

S11

t13/2 S23 t23/2 S33

t31/2

t33/2 01

V1

S12 S12 t12/2 t12/2 S22 S22 t22/2 t22/2 S32 S32 t32/2 t32/2

t11/2 S21 t21/2 S31 V2

02

V3 V4

03

03

S11

S12

t11/2 S21

t12/2

S22

t23/2

t21/2

t22/2

S32 t32/2

t33/2 01

V1

V2

S23 t23/2

V4 V3

02

S33

t33/2 V2

V1

01

tseq/2

t13/2 S23 S33

S13 t13/2

t21/2 S31 t31/2

tseq/2 S13

S11 t11/2 S21

V3 V4

tseq/2


02

S12 t12/2 S22 t22/2 S32 t32/2 02

S11 t11/2 S21 t21/2

V4 V3

V2

tseq/2

S13 t13/2

S23

t23/2

S33 t33/2 V1

Double sided 3-zero states V1 → V4 are active states 01 → 03 are zero states

Double sided 2-zero states

01


November 2005

Space Vector Comments Selection of vector sequence is not unique - different implementations possible Different implementations give different high frequency (distortion) characteristics at the input and output port Common mode addition to output target is inherent with space vector method → 87% voltage ratio Freedom to control input current vector position can be beneficial under distorted/unbalanced load/supply conditions

Min-Mid-Max Method Oyama et al Attempts to minimise switching loss Minimise commutations by having only 2 output phases switched in each sampling period Minimise voltage change at each commutation through optimum selection of switching sequence S11

S11 t11/2

t11/2

S21

S22

S23

S23

S22

S21

t23/2

t22/2

t23/2

t23/2

t22/2

t21/2

S31 S32 t31/2 t32/2 tseq/2


S33

S33

S32

t33/2

t33/2

t32/2 t31/2

tseq/2

S31


November 2005

Fictitious DC Link Modulation 1 Modulation considered as a two step process

[v o (t )] = ([A][v i (t )])[B ] First step - multiply by A, second step - multiply result by B [A] and [B] are given by:

cos(ω i t )    [A] = α cos(ω i t + 2π / 3) cos(ω i t + 4π / 3)

T

cos(ω o t )    [B ] = β cos(ωo t + 2π / 3) cos(ω o t + 4π / 3)

Fictitious DC Link Modulation 2 First step yields the “fictitious DC link” and is analogous to rectification 3αVim [ A][v i (t )] = 2 Second step modulates this DC constant at the output frequency and is analogous to conventional inversion using PWM 

cos(ω o t )



[A][v i (t )][B ] = 3αβVim cos(ωot + 2π / 3) 2

cos(ω o t + 4π / 3)

Theoretical maximum values of a and b are:

α MAX =

4 3 2 , β MAX = 2π π

yielding a maximum voltage transfer ratio of 1.053!



Fictitious DC Link Modulation 3 For q > 0.87 the mean output voltage in each sequence cannot equal the target voltage → Increased low frequency distortion in output and/or input As q → 1.05 input current and output voltage approach quasi-square wave For q < 0.87, method is similar to others Sparse Matrix Converter makes the distinction between [A] and [B] in hardware - but still without DC energy storage

Modulation - Observations Practical implementation of switching schemes (any of them) with a modern DSP is straightforward Switch duty cycles are normally calculated at each sampling instant based on input voltage measurement (all methods) Low frequency distortion/unbalance in input voltage does not appear at output (Instantaneous power out) = (Instantaneous power in) at all instants in a matrix converter


November 2005


November 2005

Modulation - Conclusions No restriction on input and output frequency within limits imposed by switching frequency Inherent bi-directional power flow in all modes with 4 quadrant voltage-current characteristics at both ports “Sinusoidal” input and output currents Input displacement factor can be controlled Output voltage limited to 87% of input voltage (for most modulation schemes) Schemes for which q > 0.87 have significant performance penalties

Jon Clare




Design Issues • Comparison of modulation methods • Input Filter design • Matrix Converter losses and comparisons with other topologies

Comparison - Introduction Define: • Modulation frequency (fm) = frequency at which switching pattern repeats • Sampling frequency (fsamp) = frequency at which modulation duty cycles are calculated • Switching frequency (fsw) = average frequency at which each bidirectional switch commutates

Comparison of modulation methods not straightforward since: • Often fm ≠ fsamp ≠ fsw • Ratio fm/fsw, fsamp/fsw etc depends on modulation method • Even for equal fsw, different modulation methods can give vastly different switching losses


November 2005


November 2005

Comparison (1) Comparison of output voltage weighted THD for equal commutation frequency (8kHz)

WTHD =

n max

∑

n =2

 f1  I (fn )      fn  I (f1 ) 

Sampling frequencies Vent (8kHz – single sided) SVM 3z (6kHz – double sided) SVM 2z (7kHz – double sided) MMM (9kHz – double sided)

Comparison (2)

Comparison of input current weighted THD for equal commutation frequency (8kHz)


2


November 2005

Comparison (3)

Comparison of losses for 30kW converter Balance between conduction and switching loss depends on devices chosen – relatively slow devices used in this example

Input Filter Design R

L C

Matrix Converter

C chosen to limit voltage distortion at converter terminals L chosen to limit current distortion at supply R chosen to give adequate damping • Limit overshoot on turn-on • Avoid excitation of resonance by supply or converter



November 2005

Simple Filter Analysis Iin L Assume harmonic current flows entirely in C to calculate distortion on Vin

Vin In

C

Use calculated distortion on Vin to determine distortion on Iin Enables C and L to be determined directly from weighted THD curves and target THD for Iin and Vin

∑ ((f

ITHD1 =

/ f n )I (f n ))

fn ≠ fi

I (f i )

∑ ((f

ITHD 2 =

 Power   I  C =  THD1  2    VinTHD  6π fiVll 

2

i

i

/ fn )2 I(fn )

)

 I  1  L =  THD2   3C (2π f )2  I in  THD  i 

2

I (f i )

fn ≠ fi

Simple Example 4.5

0.40

Input current weighted (1/f) THD Venturini optimum method, q =0.8

Weighted THD %

3.5

Input current weighted (1/f 2) THD Venturini optimum method, q =0.8

0.35 Weighted THD %

4.0

0.30

3.0

0.25

2.5

I THD2

0.20

I THD1

2.0

0.15

1.5 1.0

0.10

0.5

0.05

0.0

0.00

0

50

100

150

f sw /f i

200

0

50

100

150

f sw /f i 200

Example: 415V line to line input at 50Hz, 15kW power level at q=0.8, 8kHz switching frequency Target distortions: Input current THD 5%, Converter input voltage THD 5% Data from curves at fsw/fi = 160: ITHD1 = 0.35%, ITHD2 = 0.004% Component values: C = 6µF, L = 210µH Space vector or cyclic Venturini modulation would yield smaller values



November 2005

Comparison of AC to AC Converter Losses

Research programme looking at 30kW integrated matrix converter induction motor drive 3 configurations studied Rectifier PWM drive Active front-end PWM drive Matrix converter drive Conduction and commutation losses considered in detail

Voltage Source Inverter Drives Drive application supplying a 30kW induction motor is considered A 400V induction motor load is used with the inverter drives

Ls

Ls 400V 50Hz

IM

400V 50Hz

IM

≡

≡

Rectifier input PWM Inverter Drive

Active front-end Inverter Drive



November 2005

Matrix Converter Drive • Maximum voltage transfer ratio of matrix converter is 0.866 • A 340V induction motor load is therefore used for the matrix converter drive v1i

400V 50Hz

v2i v3i

i1i i2i i3i

S11

S21

S31

S12

S22

S32

S13

S23

S33

340V 30kW

≡

OR

IM

Bi-directional Switch 1200V, 200A IGBTs

Matrix Converter Drive

Device Conduction Losses • Fit curve to the IGBT and diode forward voltage drop characteristics. • Matrix Converter - output current flows through a series combination of an IGBT and a diode at all times. • Inverter – Dependant on the output fundamental displacement angle. • Diode bridge – Dependant on supply impedance.



November 2005

Device Commutation Losses • Simulations for each converter were used to identify switching instants • IGBT turn-on, turn-off losses and diode recovery energy loss included • Soft turn-on, turn-off instances due to zero current switching • Matrix Converter – switching voltage dependant upon the switching instants • A linear relationship of switching loss with voltage and current at commutation instant was assumed

Results (1) 3000

Total ) ( loss W t (w) pu t u O d et a R t a s e s s o L r et r ev n o C

DB-Inverter AFE-Inverter Venturini M.C. S VM 2z S VM 3z

2500

2000

Note:

1500

THD of SVM method < Venturini at equal sampling frequency

1000

500

0

0

5

10

15

Modulation fre que ncy (kHz)

Variation of total converter loss against sampling frequency at rated load



November 2005

75

Load Current (%)

50

25

0 0

2.5

5

7.5

10

15 12.5

Frequency (kHz)

Rectifier Input PWM Inverter

4500 4000 3500 3000 2500 2000 1500 1000 500 0 150 125 100

Total Converter Losses (W)

4500 4000 3500 3000 2500 2000 1500 1000 500 0 150 125 100



Results (2)

75

50

Load Current (%)

25

0 0

2.5

5

7.5

10

15 12.5

Frequency (kHz)

4500 4000 3500 3000 2500 2000 1500 1000 500 0 150 125 100

75

50

25

Load Current (%)

Active front-end Inverter

0

0

2.5

5

7.5

10

15 12.5

Frequency (kHz)

Matrix Converter

Total Converter Loss against load current and sampling frequency

Loss Comparison - Conclusions

• Highest efficiency obtained with diode rectifier PWM inverter • Matrix converter is more efficient than the active front-end drive that has similar characteristics



November 2005

Pat Wheeler


Two-Stage Matrix Converters (Sparse) • Basic Principle of Operation • Circuit topologies and device count reduction • Comparison of Sparse Matrix Converter Topologies • Modulation Schemes



November 2005

Two-Stage Matrix Converters ‘DC’ Link Voltage Bi-directional Switches

Output Line Voltage

3-Phase Supply

3-Phase to 2-phase Matrix Converter

3-Phase Load

Also known as the ‘Sparse’ Matrix Converter Same Functionality as a Matrix Converter Exception: rotating vectors are not possible, ie. different input phase connected to each output phase

In this form it has the same number of devices as a Matrix Converter

Two-Stage Matrix Converters Input Voltage [Volts/10] Unfiltered Input Current [Amps]

‘DC Link’ Voltage [Volts]

Output Voltage (L-N) [Volts]

Output Currents [Amps]



November 2005

Sparse Matrix Converters

Single-Stage and Two-Stage Converters a b c

Input filter

Cin

CClamp

Lin

Clamp circuit

A B C

IM 3~

3x3 matrix of bi-directional switches

Auxiliary circuits supply unit (gate-drivers, transducers, control)

SMPS

line

Clamp circuit

Lin Cin

CClamp

IM 3~ Auxiliary circuits supply unit (gate-drivers, transducers, control)

SMPS

motor

line

Both Converters need LC input filter, clamp circuit, Vout/Vin < 0.87! ☺ Save diodes for clamp circuit on load side ☺ Flexible design of rectifier stage ☺ Dead-time commutation in inversion stage ☺ Possible ZCS of rectifier stage during a zero-voltage vector ☺ Conduction losses are load dependent Cannot produce rotating vectors ZCS ⇒ Rectifier stage decrease max. voltage transfer ratio Higher conduction losses at rated power


motor


November 2005

Indirect Modulation Model Indirect modulation model for MC = two stage transformation • a rectification stage, to provide a (constant) DC-link voltage • an inversion stage, to produce the three output voltages

Rectification stage p

a b c

Inversion stage

A B C

Upn

[R]=[Sa, Sb, Sc]

n

[T] = [R]⋅[I]

[I]=[SA, SB, SC]T

Known PWM modulation methods may apply easily

Rectifier Stage SV-Modulation Combine adjacent current vectors for sharing the constant output power to the input lines ⇒ sine wave Va ab

Line c b a

REC = ca

P=c

Lin

Cclamp

Cin ac

Iδ dδ⋅Iδ

θ*in

cb

N=a

Iin

dγ⋅Iγ

Rectification Stage ⇒VPN

bc Iγ

Vc

Vb ca

ba

π  d γ = mI ⋅ sin  − θ*in  3 

( )

dδ = mI ⋅ sin θ

* in


Sector γ-sequence:

1

2

3

4

5

ac

0

bc

ba

ca

cb

ab

VP

Va

Vb

Vb

Vc

Vc

Va

VN

Vc

Vc

Va

Va

Vb

Vb

Vline- γ

Vac

Vbc

Vba

Vca

Vcb

Vab

ab

ac

bc

ba

ca

cb

VP

Va

Va

Vb

Vb

Vc

Vc

VN

Vb

Vc

Vc

Va

Va

Vb

Vline- δ

Vab

Vac

Vbc

Vba

Vca

Vcb

δ-sequence:


November 2005

Inverter Stage SV-Modulation Line

Combine adjacent voltage vectors for accurate generation of the reference voltage vector

REC = ca

c b a

P=c

INV=011

Lin 001

101

Vβ

Vout

dβ⋅Vβ θ*out

011

Cclamp

Cin

IDC

=“acc”

100

dα⋅Vα

Inversion Stage

Vα

α-sequence

β-sequence

0

100 = IA

110 = -IC

0 IA -IC IA 0

1

110 = -IC

010 = IB

0 -IC IB -IC 0

2

010 = IB

011 = -IA

0 IB -IA IB 0

3

011 = -IA

001 = IC

0 -IA IC -IA 0

4

001 = IC

101 = -IB

0 IC -IB IC 0

5

101 = -IB

100 = IA

0 -IB IA -IB 0

Sector 010

110

π  dα = mU ⋅ sin  −θ*out  3  * dβ = mU ⋅ sin θ out

(

C=c B=c A=a

N=a

)

IDC [0-α-β -α-0]

Pulse Width Generation Removing the Zero Current Vector from REC Stage = maintain dutyREC proportion Rectification stage duty-cycles

d γR = VPN =

dγ

d δR =

d γ + dδ

dδ dγ + dδ

π  dα = mU ⋅ sin  −θ *out  3   dβ = mU ⋅ sin (θ *out )

d γR⋅Vline- γ + d δR ⋅Vline- δ dγ

Rectifier Stage

-

-

d1

d0 = dγR ⋅ 1 − ( dγ + dδ ) ⋅ ( dα + d β ) 

δ -

d2

α

0

Inversion stages duty-cycles

dδ

γ d0

Inverter Stage

mU = 2 ⋅Vout VPN

-

β

d3

- d4

α

d1 = dγ ⋅ dα

0 Overflow

d 2 = (dγ + dδ ) ⋅ d β

Reload

d3 = dδ ⋅ dα

Timer Equivalent switching sequence

0

-

αγ

-


βδ

-βγ -

αγ

-0

d4 = dδR ⋅ 1 − ( dγ + dδ ) ⋅ ( dα + d β ) 


November 2005

Pat Wheeler

Matrix Converter Product The Yaskawa Matrix Converter • The first commercial Matrix Converter product • Launched in 2004 • Aimed at Lift and hoist applications • An important milestone in the development of Matrix Converter • Some circuit optimisation still required, for example in size and wieght



November 2005

Matrix Converter Modules

600V, 300A SEMELAB Leg Module

1200V, 35A EUPEC Matrix Converter Module

1200V, 200A Dynex Switch Module

Applications?

Integrated Motor Drives • No DC link capacitor • Voltage ratio not a limitation

Industrial Applications • Lifts and Hoists • Power density • Regeneration

Aerospace • Power density • Temperature tolerance

Electric Military Vehicles • Weight and volume • Bi-directional power flow


1700V, 600A DYNEX Leg Module


November 2005

An EHA using a Matrix Converter Permanent Magnet Motor Drive

Aims • Produce a 3kW Matrix Converter to drive an EHA • Demonstrate the actuator as part of the TIMES programme

Testing • Prototype EHA has been tested on 400Hz and variable frequency supplies over a range of realistic loading conditions • Converter has also been tested as a motor drive under various supply conditions found on aircraft

An EHA using a Matrix Converter Permanent Magnet Motor Drive (2) EHA Control Loops Voltage transducers

Matrix Converter

Supply

Supply Voltage

LEMs

PM Motor

Resolver

Actuator

Motor Current

Control (DSP and FPGA)

Motor Speed

Ram Position Demand


Ram Position

LVDT


November 2005

An EHA using a Matrix Converter Permanent Magnet Motor Drive (3) Matrix converter driving two 400Hz induction motor fans, V/f mode 10

24

5

20

A 16

0

12

-5

8

-10

4

-15

A

Output current (400Hz)

-20 Input

0

current

-25 (360Hz)

-4 -8 0.001

0.0015

0.002

0.0025

0.003

-30 0.004

0.0035

An EHA using a Matrix Converter Permanent Magnet Motor Drive (4)

Supply Loss Operation

Speed reversal at 9600rpm 15000

Motor shaft speed (rpm)

5000 0 -5000 -10000

Iq ref[Amps]

-15000 0.00

4 2 0 -2 -4 -6 -8 -10 -12 0.00

0.05

0.10

0.15

0.20

0.25

0.30

7500

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0.10

0.15

0.20

0.25

0.30

Phase A current

5

Iq

15 10 5 0 .0

0 .1

0 .2

0 .3

0 .4

0.10

0.15

0.20

0.25

0.30

Phase B current

Io 2 [Amps]

10 5 0 -5 -10 0.05

0.10

0.15

0.20

15

0.25

0.30

0 .6

0 .7

0 .8

Input supply voltages

5 0 -5 -10 0.05

0.10

0.15

0.20

Time [secs]


0.25

200 150 100 50 0 -50 -100 -150 -200 0.0

0.1

0.2

0.3

0.4 T ime [secs]

Phase C current

10

Input Supply [Volts]

0.05

15

Io 3 [Amps]

0 .5

-5

-10

-15 0.00

0 .8

0

0 -5

-15 0.00

0 .7

20

q-axis current 0.05

10

-15 0.00

0 .6

25

15

Io 1 [Amps]

Motor speed 8000

7000

Iq [Amps]

Speed [rpm]

10000

Motor Speed [rpm]

8500

0.30

0.5

0.6

0.7

0.8


November 2005

Integrated Electromechanical Actuator (EMA) Technology Demonstrator

Electronics

Motor

To design and build an Integrated Electro Mechanical Actuator (EMA) intended as a technology demonstrator for a rudder actuator on a large, twin-engined, civil aircraft. Need to continuously deploy rudder under some flight conditions drives thermal design (stationary motor with high torque) Natural cooling considered

Integrated EMA Technology demonstrator 30kW matrix converter integrated with ballscrewheatsink Switching Signals Gate Drive Circuits Voltage Clamp Capacitors Voltage Clamp Diodes Input Filter Capacitors

Ballscrew housing



November 2005

Integrated EMA Technology demonstrator

Bespoke PM motor designed and constructed Speed limited to 4950rpm by use of existing actuator for demonstrator

Integrated EMA Technology demonstrator



November 2005

100kW Direct Converter PM Motor Drive

Water-cooled direct power converter 100kW vector controlled PM motor 360Hz-800Hz input, dc-1200Hz output 230V phase voltage input 120kVA rating Aerospace power quality targets Bespoke semiconductor packaging Preliminary results

Dynex/Nottingham collaboration

Entire system designed and developed at Nottingham Control system Control electronics Detailed modelling Power circuit

100kW Direct Converter PM Motor Drive

Input Current [Amps]

200 150 100 50 0 -50 -100 -150 -200 0

0.002

0.004

0.006

0.008

0.01

Time [secs]

Converter on test in USA, May 2005

Input Voltage [Volts]

400 300 200 100 0 -100 -200 -300 -400 0

0.002

0.004

0.006 Time [secs]


0.008

0.01


November 2005

An Integrated Matrix Converter Induction Motor Drive (1)

Power Electronics house in the motor end plate

=

+

IGBTs, diodes and filter capacitors Redesigned end plate

Induction Motor

Matrix Converter

Integrated Motor Drive (Power Electronics housed in a redesigned End Plate)

Extra fins to cool the devices

Specially packaged devices (Dynex Semiconductors) 200 Amp Bi-directional Switch module

Integrated Drives above 7.5kW are not feasible within the same motor space envelope DC Link Capacitors form about 40% of the volume

Matrix Converter will give same functionality as a back-to-back inverter drive Regeneration to supply Input current waveform quality BUT no large capacitors or inductors

Bi-directional Switch Modules

Redesign Motor End Plate

Integrated Motor Drive

Bi-directional Switches and Output Connections Power Planes and Input Filter Capacitors

Complete Converter with Gate Drives



November 2005

An Integrated Matrix Converter Induction Motor Drive (3)

Output Voltages

Power Circuit fits in available space

2500 2000

Output Voltages [Volts]

1500

Input inductor fits into a slightly modified terminal box

1000 500 0

Cooling requirement known – design for appropriate end plate exists

-500 -1000 -1500 -2000 -2500 0

5

10

15

20

25

30

35

40

45

50

Viability of 30kW integrated drive using matrix converter has been demonstrated

Time [ms ecs ]

Output Current

Input Currents

80


60 40 20 0 -20 -40 -60 -80 0

5

10

15

20

25

30

35

40

45

50

Time [msecs]

A 130kW Matrix Converter Vector Controlled Induction Motor Drive Control Platform • Infineon C167 control platform • FPGA based Current Commutation control • Fibre-optic connections from control card to to gate drives

Power Circuit • Water cooled heat sinks • Laminated input power planes Controller Board

Gate Drivers

Work done in collaboration with the US Army Research Labs Design and construction of a large Matrix Converter power circuit

FPGA

Micro Contr.

(6)

PWM

(6)

Bidirectional Switches Current Direction (3)

Current Direction Sensor


Input voltage

(6)

Results from 150kVA tests with an Induction Motor Load under v/f control Closed loop vector control of a 150HP Induction Motor

D/A

Motor

Speed Encoder

Fiber Optic Links (27)

Desired voltage, freq.

PC Controller

Serial Link


November 2005

A 130kW Matrix Converter Vector Controlled Induction Motor Drive (2)

Results from a 600Amp, 1200V IGBT Matrix Converter

Output Currents 500 400

150HP Induction Motor Load, 480Volt supply Output Power 129kW (156kVA)

300 200

Amps

100 0 -1 0 0

Switching Frequency: 4kHz

-2 0 0 -3 0 0 -4 0 0 -5 0 0

0

5

1 0

1 5

2 0

2 5

30

35

40

4 5

5 0

Output Voltages 1750 1500 1250

134.0kW

Output Power

129.5kW

Total converter losses

1000 750 500 250

Volts

Input Power

0 -250 -500 -750 -1000 -1250 -1500 -1750 0

5

10

15

20

25

30

35

40

45

50

4530W

Output Power Factor

0.835

Efficiency

96.2%

Input Voltage (L to L)

475V

Input Current

172A

Input Power Factor

0.985

Output Voltage

362V

Output Current

256

Time, m illiseconds

A 130kW Matrix Converter Vector Controlled Induction Motor Drive (3)

Speed Demand

ωref

id

Compensation terms

*

input voltages

*

Id Current Control

vd

Iq Current Control

vq

jθ

e Speed Control

iq

*

3-Phase Supply

MICRO-CONTROLLER Infineon SAB80C167

Flux Current Demand

vα 2/3

vβ

*

va vb vc

vAB vBC

Closed Loop Vector Control of a 150HP Induction Machine

Voltage A to D Input Filter

Matrix Converter Control Algorithm

Matrix Converter Power Circuit

Gate Drives

• Natural regeneration • Low cost Micro-controller control platform

ωr i q*

ωsl

ωe

τ i d*

PWM

dt

id iq

e-jθ

iα iβ

3/2

ia ib

FPGA

Current A to D

ic 1000

ωr

A⊕B Timers

Up/Down

800

FPGA

A

Closed Loop Motor Control

Closed Loop Vector Scheme applied to the Matrix Converter Induction Motor Drive

B

motor

Speed [rpm]

Rotor Speed

600 400 200

Encode

0

800

Id, Iq [Amps]

600 400 200 0

-200 -400


600 400 200 0 -200 -400 -600 0

1

2

3 Time [secs]

Control Platform


4

5


November 2005

Field Power Supply Using a Four-Output Leg Matrix Converter 250 200

• • • • •

150

Matrix Converter Power Circuit Variable Speed Diesel Engine Permanent Magnet Generator Designed for 10kVA Load 50Hz, 60Hz or 400Hz Output Frequency

Output Line to Line Voltages [V]

Field power supply

100 50 0 -50 -100 -150 -200

DIESEL ENGINE

Matrix 10kW

Gen

Load

-250 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Time [s]

400Hz Output Voltage Waveforms FILTER

FILTER

MATRIX CONVERTER

Input Voltage

Space Vector Modulator

Output Current

Modulation D,Q, Control and Engine Demand

Engine Speed Control

Output Voltage

• • • • •

IGBT based Matrix Converter 25kHz Sampling Frequency DSP/FPGA Control Platform LC Output Filter Output Voltage Control Loop designed using a Genetic Algorithm Optimisation • A collaborative project with the US Army Research Labs

Conclusions Matrix converters can offer advantages • Size • Regenerative operation • Sinusoidal input/output Modulation control is not difficult New power devices (eg Silicon Carbide) will increase the attractiveness of matrix converters Current research is application orientated Ongoing research into derived circuits



November 2005

Book A Book entitled “Matrix Converters” is due for publication in 2006 • Authors: » Prof Jon Clare » Dr Pat Wheeler » Dr Christian Klumpner » Dr Lee Empringham

• Publisher: » Springer
