closed-loop control of dc drives with controlled rectifier by mr.m.kaliamoorthy department of...

Closed-loop Control of DC Drives with Controlled RectifierByMr.M.KaliamoorthyDepartment of Electrical & Electronics EngineeringPSNA College of Engineering and Technology

Solid State Drives 1

OutlineClosed Loop Control of DC DrivesClosed-loop Control with Controlled Rectifier –

Two-quadrantTransfer Functions of SubsystemsDesign of Controllers

Closed-loop Control with Field Weakening – Two-quadrant

Closed-loop Control with Controlled Rectifier – Four-quadrant

References


Closed Loop Control of DC Drives

• Closed loop control is when the firing angle is varied automatically by a controller to achieve a reference speed or torque

• This requires the use of sensors to feed back the actual motor speed and torque to be compared with the reference values


Controller Plant

Sensor

+

Referencesignal

Outputsignal

Closed Loop Control of DC Drives

Feedback loops may be provided to satisfy one or more of the following:ProtectionEnhancement of response – fast response with small

overshootImprove steady-state accuracy

Variables to be controlled in drives:Torque – achieved by controlling currentSpeedPosition


Closed Loop Control of DC Drives• Cascade control structure

– Flexible – outer loops can be added/removed depending on control requirements.

– Control variable of inner loop (eg: speed, torque) can be limited by limiting its reference value

– Torque loop is fastest, speed loop – slower and position loop - slowest


Closed Loop Control of DC Drives• Cascade control structure:

– Inner Torque (Current) Control Loop:• Current control loop is used to control torque via armature

current (ia) and maintains current within a safe limit• Accelerates and decelerates the drive at maximum permissible

current and torque during transient operations


Torque (Current)

Control Loop

Closed Loop Control of DC Drives• Cascade control structure

– Speed Control Loop:• Ensures that the actual speed is always equal to reference speed *• Provides fast response to changes in *, TL and supply voltage (i.e. any

transients are overcome within the shortest feasible time) without exceeding motor and converter capability


Speed Control

Loop

Closed Loop Control with Controlled Rectifiers – Two-quadrant

• Two-quadrant Three-phase Controlled Rectifier DC Motor Drives


Current Control Loop

Speed Control Loop


• Actual motor speed m measured using the tachogenerator (Tach) is filtered to produce feedback signal mr

• The reference speed r* is compared to mr to obtain a speed error signal• The speed (PI) controller processes the speed error and produces the

torque command Te*

• Te* is limited by the limiter to keep within the safe current limits and the armature current command ia* is produced

• ia* is compared to actual current ia to obtain a current error signal

• The current (PI) controller processes the error to alter the control signal vc

• vc modifies the firing angle to be sent to the converter to obtained the motor armature voltage for the desired motor operation speed



• Design of speed and current controller (gain and time constants) is crucial in meeting the dynamic specifications of the drive system

• Controller design procedure:1. Obtain the transfer function of all drive subsystems

a) DC Motor & Loadb) Current feedback loop sensorc) Speed feedback loop sensor

2. Design current (torque) control loop first3. Then design the speed control loop


Transfer Function of Subsystems – DC Motor and Load

• Assume load is proportional to speed

• DC motor has inner loop due to induced emf magnetic coupling, which is not physically seen

• This creates complexity in current control loop design


mLL BT


• Need to split the DC motor transfer function between m and Va

(1)

• where(2)

(3)

• This is achieved through redrawing of the DC motor and load block diagram.


sVsI

sIsω

sVsω

a

a

a

m

a

m

mt

b

sTBK

1sI

sω

a

m

21

1a

a

111

sVsI

sTsTsTK m

Back


• In (2),- mechanical motor time constant: (4)

- motor and load friction coefficient: (5)• In (3),

(6)

(7)

Note: J = motor inertia, B1 = motor friction coefficient, BL = load friction coefficient


tm B

JT

Lt BBB 1

a

b

a

tat

a

at

a

a

JLK

JLBR

JB

LR

JB

LR

TT

22

21 41

211,1

tab

t

BRKBK

21

Back

Transfer Function of Subsystems – Three-phase Converter

• Need to obtain linear relationship between control signal vc

and delay angle (i.e. using ‘cosine wave crossing’ method)(8)

where vc = control signal (output of current controller)

Vcm = maximum value of the control voltage• Thus, dc output voltage of the three-phase converter

(9)


cm

c

Vv1cos

crccm

m

cm

cmmdc vKv

VV

VvVVV

L,L

L,L L,L

3coscos3cos3 1

Transfer Function of Subsystems – Three-phase Converter

Gain of the converter

(10)

where V = rms line-to-line voltage of 3-phase supplyConverter also has a delay

(11) where fs = supply voltage frequencyHence, the converter transfer function

(12)


rr

sTK

1

sG r

cmcmcm

mr V

VVV

VV

K 35.1233

L,L

ssr ffT 1

1211

36060

21

Back

Transfer Function of Subsystems – Current and Speed Feedback

Current FeedbackTransfer function:No filtering is required in most casesIf filtering is required, a low pass-filter can be included (time

constant < 1ms).Speed Feedback

Transfer function:(13)

where K = gain, T = time constantMost high performance systems use dc tacho generator and low-

pass filterFilter time constant < 10 ms


sTK

1

sGω

cH

Design of Controllers – Block Diagram of Motor Drive

Control loop design starts from inner (fastest) loop to outer(slowest) loop

Only have to solve for one controller at a timeNot all drive applications require speed control (outer loop)Performance of outer loop depends on inner loop


Speed Control Loop

Current Control Loop

Design of Controllers– Current Controller

PI type current controller: (14)Open loop gain function:

(15)

From the open loop gain, the system is of 4th order (due to 4 poles of system)


c

cc

sTsTK

1sGc

r

mc

c

crc

sTsTsTssTsT

THKKK

111

11sGH21

1ol

DC Motor & LoadConverterController


• If designing without computers, simplification is needed.• Simplification 1: Tm is in order of 1 second. Hence,

(16)Hence, the open loop gain function becomes:

i.e. system zero cancels the controller pole at origin.Solid State Drives 19

mm sTsT 1

c

mcrc

r

c

r

mc

c

crc

r

mc

c

crc

TTHKKKK

sTsTsTsTK

sTsTsTssTsT

THKKK

sTsTsTssTsT

THKKK

1

21ol

21

1

21

1ol

where111

1sGH

1111

11111sGH

(17)


• Relationship between the denominator time constants in (17):• Simplification 2: Make controller time constant equal to T2

(18) Hence, the open loop gain function becomes:

i.e. controller zero cancels one of the system poles.


12 TTTr

2TTc

c

mcrc

r

r

r

c

TTHKKKK

sTsTK

sTsTsTsTK

sTsTsTsTK

1

1ol

21

2

21ol

where11

sGH

1111

111

1sGH


• After simplification, the final open loop gain function:(19)

where (20)

• The system is now of 2nd order.• From the closed loop transfer function: , the closed loop characteristic equation is:

or when expanded becomes: (21)


rsTsTK

11sGH

1ol

c

mcrc

TTHKKKK 1

KsTsT r 11 1

sGH1

sGHsGol

olcl

rr

rr TT

KTTTTssTT

11

121

1


• Design the controller by comparing system characteristic equation (eq. 21) with the standard 2nd order system equation:

• Hence,

• So, for good dynamic performance =0.707 – Hence equating the damping ratio to 0.707 in (23) we get


22 2 nnss

(23) 12

1

1

1

r

r

r

TTKTTTT

(22) 1

1

2

rn TT

K

23

12

707.0

1

1

1

r

r

r

TTKTTTT

Squaring the equation on both sides

r1

2

1

1

r1

2

1

1

2

1

1

1

TT1K x 2

1

TT1K x 2 x 2

0.5 12

5.0

r

r

r

r

r

r

r

TTTT

TTTT

TTKTTTT

rTT

rTTKrTTX

rTT

rTT

rTT

rTT

rTT

K12

211

2

1

2

1

11K

1

2

2

1

1

1

24

rTT

rTTKrTTX

rTT

rTT

rTT

rTT

rTT

K12

211

2

1

2

1

11K

1

2

2

1

1

1

An approximation K >> 1 & rTT 1 Which leads to

rr TT

TTTK

221

1

21

Equating above expression with (20) we get the gain of current controller

rc

mcrc

TT

TTHKKK

211

mcrr

cc THKKT

TTK1

1 12

Back

Design of Controllers– Current loop 1st order approximation • To design the speed loop, the 2nd order model of current loop

must be replaced with an approximate 1st order model• Why?• To reduce the order of the overall speed loop gain function


2nd order current loop

model

Design of Controllers– Current loop 1st order approximation • Approximated by adding Tr to T1

• Hence, current model transfer function is given by:

(24)


i

c

m

c

m

sTK

sTTTHKKK

sTTTKKK

i

crc

rc

11

11

11

sIsI

3

1

3

1

*a

a

rTTT 13

Full derivation available here.

1st order approximation of current loop


• After simplification, the final open loop gain function:


rrr TTsTTsK

sTsTK

12

11ol 111

sGH

c

mrc

TTKKKK 1

rTTsTsK

12

3ol 1

sGH

31 TTT r Since

and since rTT 1 3

ol 1sGH

sTK

Therefore

Where

Design of Controllers– Current loop 1st order approximation where (26)

(27)

(28)

• 1st order approximation of current loop used in speed loop design.

• If more accurate speed controller design is required, values of Ki and Ti should be obtained experimentally.


c

mcrcfi T

THKKKK 1

fii K

TT

1

3

fic

fii KHK

K

1

1

Design of Controllers– Speed Controller

• PI type speed controller: (29)

• Assume there is unity speed feedback: (30)


s

sss sT

sTK

1sG

11

sGω

sTH

DC Motor & Load

1st order approximation of current

loop


Open loop gain function:

(31)

From the loop gain, the system is of 3rd order.If designing without computers, simplification is needed.


1

mi

s

st

isB

sTsTssT

TBKKK

11

1sGH

DC Motor & Load

1st order approximation of current

loop


• Relationship between the denominator time constants in (31):(32)

• Hence, design the speed controller such that:(33)

The open loop gain function becomes:

i.e. controller zero cancels one of the system poles.Solid State Drives 31

mi TT

ms TT

st

isB

i

mi

m

st

isB

mi

s

st

isB

TBKKKK

sTsK

sTsTssT

TBKKK

sTsTssT

TBKKK

where

1sGH

111

111sGH


• After simplification, loop gain function:(34)

where (35)

• The controller is now of 2nd order.• From the closed loop transfer function: ,

the closed loop characteristic equation is:

or when expanded becomes: (36)Solid State Drives 32

isTsK

1

sGH

st

isB

TBKKKK

KsTs i 1

sGH1

sGHsGcl

iii T

KT

ssT 12


• Design the controller by comparing system characteristic equation with the standard equation:

• Hence:(37)

(38)

• So, for a given value of :– use (37) to calculate n

– Then use (38) to calculate the controller gain KS


22 2 nnss

n2

2n

Closed Loop Control with Field Weakening – Two-quadrant

Motor operation above base speed requires field weakening

Field weakening obtained by varying field winding voltage using controlled rectifier in:single-phase orthree-phase

Field current has no ripple – due to large LfConverter time lag negligible compared to field time

constant Consists of two additional control loops on field circuit:

Field current control loop (inner)Induced emf control loop (outer)




Field weakening


Solid State Drives 36dtdiLiRVe a

aaaa Induced emf

controller (PI-type with

limiter)

Field weakening

Field current

controller(PI-type)

Field current reference

Estimated machine -induced emf

Induced emf reference


• The estimated machine-induced emf is obtained from:

(the estimated emf is machine-parameter sensitive and must be adaptive)• The reference induced emf e* is compared to e to obtain the induced emf

error signal (for speed above base speed, e* kept constant at rated emf value so that 1/)

• The induced emf (PI) controller processes the error and produces the field current reference if*

• if* is limited by the limiter to keep within the safe field current limits• if* is compared to actual field current if to obtain a current error signal• The field current (PI) controller processes the error to alter the control

signal vcf (similar to armature current ia control loop)• vcf modifies the firing angle f to be sent to the converter to obtained the

motor field voltage for the desired motor field fluxSolid State Drives 37

dtdiLiRVe a

aaaa

Closed Loop Control with Controlled Rectifiers – Four-quadrant

• Four-quadrant Three-phase Controlled Rectifier DC Motor Drives


Closed Loop Control with Controlled Rectifiers – Four-quadrant

• Control very similar to the two-quadrant dc motor drive.• Each converter must be energized depending on quadrant of operation:

– Converter 1 – for forward direction / rotation– Converter 2 – for reverse direction / rotation

• Changeover between Converters 1 & 2 handled by monitoring– Speed– Current-command– Zero-crossing current signals

• ‘Selector’ block determines which converter has to operate by assigning pulse-control signals

• Speed and current loops shared by both converters• Converters switched only when current in outgoing converter is zero (i.e.

does not allow circulating current. One converter is on at a time.)Solid State Drives 39

Inputs to ‘Selector’ block

References• Krishnan, R., Electric Motor Drives: Modeling, Analysis and

Control, Prentice-Hall, New Jersey, 2001.• Rashid, M.H, Power Electronics: Circuit, Devices and

Applictions, 3rd ed., Pearson, New-Jersey, 2004.• Nik Idris, N. R., Short Course Notes on Electrical Drives,

UNITEN/UTM, 2008.


DC Motor and Load Transfer Function - Decoupling of Induced EMF Loop

• Step 1:

• Step 2:


DC Motor and Load Transfer Function - Decoupling of Induced EMF Loop

• Step 3:

• Step 4:


Back

Cosine-wave Crossing Control for Controlled Rectifiers


Vm

Vcmvc

0 2 3 4

Input voltageto rectifier

Cosine wave compared with control voltage vc

Results of comparison trigger SCRs

Output voltageof rectifier

Vcmcos() = vc

cm

c

Vv1cos

Cosine voltage

Back

Design of Controllers– Current loop 1st order approximation


i

cc

c

c

m

c

m

sTK

KT

s

KHK

KsTHK

sTK

sTHK

sTTTHKKK

sTTTKKK

i

fi

fi

fi

fi

fi

fi

fi

crc

rc

11

1

11

1

111

11

111

11

sIsI

33

3

3

3

1

3

1

*a

a

Back

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closed-loop control of dc drives with controlled rectifier by mr.m.kaliamoorthy department of...

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