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DC MACHINES

DYNAMIC ANALYSIS AND CONTROL OF AC MACHINES

ERASMUS MUNDUS MASTER COURSE on SUSTAINABLE TRANSPORTATION AND

ELECTRIC POWER SYSTEMS

Dynamic Analysis and Control of AC Machines - Lesson 2 Pagina 2

STRUCTURE OF A DC MACHINE

• EXCITATION (FIELD) WINDING in the stator • ARMATURE WINDING in the rotor • The armature is a closed winding and portions of it

are accessible through a collector and brushes

EXCITATION WINDING

• A DC current, fed to the excitation winding, allows to establish a flux density field of constant amplitude which crosses radially the airgap

• The excitation field has fixed direction (polar axis)

EXCITATION FIELD

• Field distribution induces a voltage in rotor moving conductors

• The induced voltage in each conductor is proportional to B and ω

• All the conductors under one pole are subject to voltages of the same sign

BACK-EMF

• The back-emf at armature terminals is

1iiLvBE

1iriLBEπτω

πτωravg L

2NBE =

( ) ravg LB2NE ωτπ

reEKE ωΦ=

ARMATURE WINDING

• Originally, it was a ring-type winding (Pacinotti ring) • Now, it is a drum-type winding, with two layers

TYPES OF DRUM WINDINGS

Lap winding Wave winding

LAP WINDING

WAVE WINDING

ARMATURE CURRENT

• Current is the same for all the conductors under the same pole • As a conductor crosses the interpolar axis, the current in it goes

to zero and reverses sign (commutation) • However, the commutator and brushes system keeps the

current distribution constant under a pole

POLAR AXIS

INTERPOLAR AXIS

TORQUE

• Torque derives from the combination of all the forces acting on the conductors

• Macroscopically, torque can be seen as the interaction (cross product) of the excitation and armature fields

• The commutator and brushes keep the two fields fixed in space and in quadrature

INTERPOLAR AXIS

POLAR AXIS Φe

BldIFd ×⋅=

STEADY STATE EQUATIONS

EIRV aaa +=

eee IRV =

reEKE ωΦ=

aeT IKT Φ=)I(f ee =Φ

MECHANICAL CHARACTERISTICS

EVI −=

EVKT −Φ=

reEaeT R

KVKT ωΦ−Φ=

RKKT Φ

−= ω

Tstart

aeTstart R

VKT Φ=

VOLTAGE REGULATION

RKKT Φ

−= ω

aeTstart R

VKT Φ=

Va increasing

• When the armature voltage is increased from 0 to nominal voltage, the mechanical characteristic moves parallel to itself

• The usable region is delimited by the nominal torque (constant torque)

FIELD REGULATION

RKKT Φ

−= ω

aeTstart R

VKT Φ=

• When the excitation field is decreased from nominal value to 0, the mechanical characteristic inclines itself.

• All characteristics are tangent to an hyperbola (constant power curve)

Φe decreasing

COMBINED VOLTAGE AND FIELD REGULATION

T, Φe

Va, PE

Ia = const.

Constant torque

Constant power

Φe = ΦenVa increasing

Φe decreasingVa = Van

DYNAMIC ANALYSIS OF DC MACHINES

DYNAMIC EQUATIONS

edtdiLiRv a

aaaa ++=dtdiLiRv e

eeee +=

reEKe ωϕ=

aeT iKT ϕ=)i(f ee =ϕ

IeReLeLa

dJTT ωω+=−

BLOCK DIAGRAM

∫1/Le

∫1/La

× 1/J ∫

ve ie ϕe

iava KT

dωr/dtdia/dt

die/dt+

BLOCK DIAGRAM (FIXED EXCITATON)

∫1/La

1/J ∫

iava Kt

dωr/dtdia/dt

• The model can be linearized if we consider φe = constant

eEe KK Φ= eTt KK Φ=

LAPLACE DOMAIN BLOCK DIAGRAM

• This model can be used to represent a DC machine with fixed excitation under linearity hypothesis.

1/(Ra+sLa)iava Kt

ωr+ -

+ 1/(B+sJ)

SPEED TRANSIENT

• Speed transient can be analyzed using effect superposition principle

)s(T)s(G)s(V)s(G)s( L2a1r ⋅+⋅=Ω

0)s(LTa

r1 )s(V

)s()s(G=

0)s(aVL

r2 )s(T

)s()s(G=

• Simplifying hypotheses:

0B = mJJ =

SPEED TRANSIENT

• Speed transient caused by voltage variation

sLR1K1

⋅⋅

KKJLsK

1KKsLRsJ

( )1ssK1)s(G

=τττ

mam KK

SPEED TRANSIENT

• Usually, τm >> τa

( )( )ame1 s1s1K

1)s(Gττ ++

( )amm ss τττ +≅

( )me1 s1K

1)s(Gτ+

• Applying a voltage step of amplitude ∆Va

( ) sV

s1K1)s( a

∆⋅

+≅Ω

• In time domain:

( ) )0(e1KV)t( r

ar ωω τ +−

∆≅ −

SPEED TRANSIENT

• Similarly

( )( )( )am

a2 s1s1

R)s(Gττ

+−≅

• Applying a load step of amplitude ∆TL

• In time domain:

( ) )0(e1KKRT)t( r

aLr ωω τ +−

∆−≅ −

( )( )

+⋅−=

RKKsLRsJ

sLR)s(G

( ) sT

KKR)s( L

∆⋅

+−≅Ω

CURRENT TRANSIENT

• Applying a voltage step:

( ) ( ))s(K)s(Vs11

R1)s(I rea

aaa Ω−⋅

Ω−∆

= )s(KsV

R1)s(I re

aaa τ

• In time domain (assuming no load torque):

( ) ( )

∆⋅−+−

∆+= −− m/t

aaa e1

RKVKe1

RV)0(i)t(i ττ

DC MACHINE TRANSIENTS VOLTAGE STEP

DC MACHINE TRANSIENTS LOAD STEP

DC DRIVES

FORMING A DC DRIVE

ieReLeLa=

PWMSpeed regulator+ -

Speed signal conditioning

ωr* va

TACHODC-DC

CONVERTER

• This drive structure conceptually works, since there is a direct relation between voltage and speed.

• However, in this structure the current is not controlled: – during transients current can exceed limits – torque is not controlled

ARMATURE VOLTAGE REGULATION (FIXED EXCITATION)

FORMING A DC DRIVE

• This drive structure controls also the current • It is composed by two nested loops • The inner loop must have higher bandwidth, so that

its closed loop transfer function can be considered as “instantaneous” from the outer loop

• A third, position loop can be added outside

ieReLeLa=

PWMSpeed regulator+ -

Current signal conditioning

ωr* va

TACHODC-DC

CONVERTER

ia*Current

regulator1/KtT*

Speed signal conditioning

MODELLING OF A DC DRIVE

• So far, the following models have already been derived: – DC machine

1/(Ra+sLa)iava Kt

ωr+ -

+ 1/(B+sJ)

– Transducers

• The regulators are generally formed with PID controllers

1/(1+sτ)x xmeas

KD d/dt

+KP+KI/s+sKD

MODELLING OF A DC DRIVE

• A model for the converter is needed:

DC-DC CONVERTER

MODELLING OF DC-DC CONVERTER

ton = d Ts

va,avg

va,pu*

vamax,pu*

Triangular carrier

• Different time-scales can be considered: – Instantaneous – Switching period average

• Also, non-ideal switches can be considered: – Turn on, turn off times – Dead time – On-state voltage drop

• Instantaneous model

carrier*

pua,dc

carrier*

pua,dca v vif ,V

v vif ,V)t(v

dcavga,dc

*aavg,a

,v)t(v

≤≤−

• Switching period average model

carrier*

pua,carrier

dccarrier

≤≤−

= carrierdc

pua, VVv vif =

• In general, the switching period average model is used to determine the controller parameters of the drive.

• This method neglects the harmonics at (multiples of) the switching frequency. Therefore, the armature current calculated with this model will neglect harmonics which actually exist.

• This implies a limitation in the bandwidth of the current loop.

• Can this limitation be overcome? No, because we can vary the voltage reference va

* only once (maximum twice) per switching period.

MODELLING OF DC-DC CONVERTER SWITCHING PERIOD AVERAGE MODEL

LINEARIZED SWITCHING PERIOD AVERAGE MODEL

va* va1

MODEL OF A DC DRIVE

1/(Ra+sLa)iava Kt

ωr+ -

+1/(B+sJ)Speed

regulator+ -

ωr* va

ia*Current

regulator1/KtT*

1/(1+sτ1)

1/(1+sτ2)

MOTOR MODELCONVERTER

TRANSDUCERS MODELS

• Having modeled all components, it is now possible to analyze the regulators of the two control loops.

BANDWIDTHS

log(f)

fs BW1BWfilter,1BWiBWωBWθ

• At least one decade is needed between frequencies, in order to allow enough attenuation.

• The switching frequency imposes all others, since it is usually also the sampling frequency and corresponds to the rate of change for the innermost control variable va

*. • The current ripple at switching frequency must be

filtered to avoid aliasing.

BANDWIDTHS

• Example: – Switching frequency: 10 kHz – BW of current transducer: 100 kHz – Cutoff frequency of filter: 1 kHz – BW of current regulator: 100 Hz – BW of speed regulator: 10 Hz – BW of position regulator: 1 Hz

CURRENT LOOP

1/(Ra+sLa)iava

1/(1+sτ1)

KD d/dt

1/(1+sτfilt1)

• At the end, the current loop will be represented by:

CONSIDERATIONS ABOUT CURRENT LOOP

• Inner loop has the highest bandwidth • Current loop is also a torque loop

• Current HAS to be the inner loop due to the cause and effect relationships – Voltage is the cause that makes current change – Torque is the cause that makes speed change

ia KtTia*

1/KtT*

SPEED LOOP

ia KtT

+1/(B+sJ)Speed

regulator+ -

1/KtT*

1/(1+sτfilt2)

1/(1+sτ2)

• Methods for the synthesis of the regulator parameters of the speed loop have already been investigated within the “Control of electromechanical systems” module

INCREASING CURRENT LOOP BANDWIDTH

• It is possibile to eliminate the filter by using a particular PWM technique

• The sampling must be done twice per switching period and sampling instants should be at the peaks of the triangular carrier.

• The resulting PWM is the so-called asimmetric PWM, updated twice per switching period

• It is possible to demonstrate that in this way only the average value of the current is sampled.

• Therefore the filter can be eliminated and the bandwith of the current regulator can be increased.

BW switchingsamplingmax,i =≤

CURRENT LOOP SYNTHESIS

1/(Ra+sLa)iava

∫KI +

1/(Ra+sLa)iava

1Kp+Ki/s

• Requirements: – Zero steady state error – Specified closed loop bandwith BWi

• Feasible desired closed loop transfer function:

s11)s(W *

*i τ+

2πτ =

• Ideal closed loop transfer function:

1)s(W *i =

• Hypothesis: ideal zero-pole cancellation

• Open loop transfer function

)s(G I

RK)s(F

I ⋅=

1/Ra(1+sτa)

1Ki(1+Kp/Kis)s

• Set the back-emf equal to zero

• Closed loop transfer function

1)s(F1

)s(F)s(W

• The transfer function has zero steady state error for step input and can achieve the desired bandwidth if:

aKR τ= *

iaI BW2RK ⋅⋅= π

• EXERCISE: Determine what happens if the zero-pole cancellation is not exact

• Design the current regulator with bandwidth equal to 300 Hz for a DC machine having the following parameters

EXAMPLE

Ω= m100Ra

mH3La =

Hz300BW *i =

6549.5KP =

5.188KI =V10Vdc =

)(@V5E nω=

CURRENT LOOP SYNTHESIS EXAMPLE

Current response at zero speed

• The back-emf will act as a slowly changing (mechanical time constant) disturbance to the current regulator.

• It will not affect the zero steady-state error. • If there is a back-emf, the effective voltage acting

to change the current is ∆v=(va-e). • Therefore, the regulator (in particular the integral

term) will have to produce a bigger voltage reference va

*, in order to obtain the same ∆v. • This affects the transient response.

INCLUDING THE EFFECT OF THE BACK-EMF

CURRENT LOOP SYNTHESIS INCLUDING THE EFFECT OF THE BACK-EMF

Current response at rated speed

• In order to improve the transient response, we can add a feed-forward term, based on an estimate of the back-emf

INCLUDING THE EFFECT OF THE BACK-EMF

• Even a rough estimate can significantly improve the transient

1/Ra(1+sτa)

1Ki(1+Kp/Kis)s

ωrKee^+

CURRENT LOOP SYNTHESIS INCLUDING THE EFFECT OF THE BACK-EMF

Current response with estimated back-emf FF compensation

• At high speeds, expecially if fast transients are required, the reference voltage va

* can exceed the maximum feasible voltage va

* = Vdc.

DC-DC CONVERTER NON LINEAR MODEL

CURRENT LOOP SYNTHESIS DC-DC CONVERTER NON LINEAR MODEL

• In this case, we need to consider the non-linear model of the DC-DC converter.

1/Ra(1+sτa)

Ki(1+Kp/Kis)s

ωrKee^+

• When saturation is considered, a significant oscillation in the current appears.

INTEGRATOR TERM WINDUP

• The oscillation is originated by the “integrator windup”.

• Initially, the output is saturated (current rises linearly, not with desired BW).

• When the error is zero, the output is still saturated.

• It takes time to exit saturation, which causes the oscillation.

INTEGRATOR TERM WINDUP

• This phenomenon can be avoided by using a PI regulator scheme with “anti-windup”

ANTI-WINDUP

1/Ra(1+sτa)

Ki(1+Kp/Kis)s

ωrKee^+

va,presat*

• The response is still dominated by the saturation, but no oscillation occurs.

ANTI-WINDUP

CURRENT LOOP SYNTHESIS COMPLETE SCHEME FOR CURRENT REGULATOR

Ki(1+Kp/Kis)s

va,presat*

• The scheme, made of PI regulator, FF disturbance compensation and anti-windup, is used also for all other control loops.

CURRENT LOOP SYNTHESIS – DIGITAL DOMAIN

• The scheme, made of PI regulator, FF disturbance compensation and anti-windup, is used also for all other control loops.

MOTION CONTROLLER REFERENCE MODEL FEEDFORWARD

The reference model is used to enhance transient response.

DC DRIVE WITH FIELD WEAKENING COMPLETE SCHEME

Field weakening control scheme

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