Control of AC drives 2019/2020 1
Chair of Electric Drive Systems
Prof. R. Leidhold
Space Vector representation
The Space Vector representation for three-phase variables in AC-machines
allows for a simpler analysis of the machine and the design of control
strategies.
The space vector is an instantaneous value. The angle of the vector
represents a spacial orientation (e.g. of the magnetic field).
Do not confuse it with Phasor, which represents one-phase AC variables in
steady state. Here the angle of the complex number represents the time
delay of a sinusoidal wave with respect to a reference wave.
Control of AC drives 2019/2020 2
Chair of Electric Drive Systems
Prof. R. Leidhold
Magnetic field due to Phase A
The winding of phase A contributes
with a flux component A ~ iA(t) in
direction 0° (or 180° if iA<0).
This flux component is represented
with the vector A.
ϕ A ∝ ψA ∝i SA
i SA
Control of AC drives 2019/2020 3
Chair of Electric Drive Systems
Prof. R. Leidhold
Magnetischer Feld der Phase B
Similarly, for the phase B in direction
120° (or 300° if iB<0).
ϕ B ∝ ψB ∝i SB
i SB
Control of AC drives 2019/2020 4
Chair of Electric Drive Systems
Prof. R. Leidhold
Magnetischer Feld der Phase C
And, for the phase C in direction 240°
(or 60° if iC<0).
i SC
ϕC ∝ψC ∝i SC
Control of AC drives 2019/2020 5
Chair of Electric Drive Systems
Prof. R. Leidhold
Feld aus den Beiträgen der drei Phasen
B Vektoraddition
ϕ=ϕ A+ ϕ B + ϕC
i SC
i SB Der Gesamtfluss, bestehend aus den
Beiträgen der Phasen A, B und C.
Der Gesamtfluss wird als Vektoraddition der
A Raumzeiger A, B und C gerechnet.
C
i SA
Control of AC drives 2019/2020 6
Chair of Electric Drive Systems
Prof. R. Leidhold
Contribution of all phases
Representation of the windings as circuit elements
B Vector addition B
ϕ=ϕ A+ ϕ B + ϕC
i SC i SB
i SB
A A
i SA
i SC
C C
i SA
Control of AC drives 2019/2020 7
Chair of Electric Drive Systems
Prof. R. Leidhold
Clarke-Transformation: from phase variables to a stationary reference frame (αβ)
β
B Vector addition
iC
i
iB A α
iA
C i A=0.74
i B=0.21
i C=−0.95
iA iB iC
1
t
Edith Clarke (1883 - 1959) was the first woman to be
professionally employed as an electrical engineer in the United
States, and the first female professor of electrical engineering in
the country.
Control of AC drives 2019/2020 8
Chair of Electric Drive Systems
Prof. R. Leidhold
Clarke-Transformation of a symmetrical three-phase AC variable
1 iA iB iC
t0 t1 t 2 t3
t0 t1 t2 t3
iC
i i
iC
i
iC iA
iB
B B B B
A i A A A
iA iC iA iA
C C C C
iB
Control of AC drives 2019/2020 9
Chair of Electric Drive Systems
Prof. R. Leidhold
Clarke-Transformation: from phase variables to a stationary reference frame (αβ)
β
B Projection of the vectors on the -Axis
iC
i
iB A α
iA
3/ 2 i α
C i A=0.74
i B=0.21
i C=−0.95
iA iB iC
1
t
Control of AC drives 2019/2020 10
Chair of Electric Drive Systems
Prof. R. Leidhold
Clarke-Transformation: from phase variables to a stationary reference frame (αβ)
β
B 3/2 i β Projection of the vectors on the -Axis
iC
i
iB A α
iA
C
i A=0.74
i B=0.21
i C=−0.95
iA iB iC
1
t
Control of AC drives 2019/2020 11
Chair of Electric Drive Systems
Prof. R. Leidhold
3-Phase-System 2-Phase-System
β
B
i SB i Sβ
A α
i SA i Sα
i SC
Control of AC drives 2019/2020 12
Chair of Electric Drive Systems
Prof. R. Leidhold
Phase variables
Time function of the voltages in a symmetric three-
phase system, e.g. with constant frequency and
amplitude
Components of the space vector in the stationary reference frame Trace of the space vector in the -plane
Control of AC drives 2019/2020 13
Chair of Electric Drive Systems
Prof. R. Leidhold
Phase variables
Time function of the voltages in a three-phase
system, e.g. with a fifth harmonic in phase-A
Components of the space vector in the stationary reference frame Trace of the space vector in the -plane
Control of AC drives 2019/2020 14
Chair of Electric Drive Systems
Prof. R. Leidhold
Rotating reference frame
β
q
i
iβ
d
iq id
φK α
iα
Control of AC drives 2019/2020 15
Chair of Electric Drive Systems
Prof. R. Leidhold
Rotating reference frame: application e.g. for rotor variables
Model in phase variables Model in phase vectors
In a stationary reference frame () for the stator, and in a
rotating reference frame (dq)
β for the Rotor.
B
i SB i Sβ
q
BR AR d
i Rq
i RB
A α
i RA i Rd
i RC i SA i Sα
i SC
CR
Control of AC drives 2019/2020 16
Chair of Electric Drive Systems
Prof. R. Leidhold
Transformation from the rotating to the stationary reference frame
β
q Projection of d- and q-components
i on the -Axis
iβ
d
iq id
φK α
iα
Control of AC drives 2019/2020 17
Chair of Electric Drive Systems
Prof. R. Leidhold
Transformation vom rotierenden zum stationären Koordinatensystem Raumzeiger
β
q Projection of d- and q-components
i on the -Axis
iβ
d
iq id
φK α
iα
Control of AC drives 2019/2020 18
Chair of Electric Drive Systems
Prof. R. Leidhold
Stationary reference frame Trace of the space vector in the plane
Rotating reference frame Trace of the space vector in dq-plane
Control of AC drives 2019/2020 19
Chair of Electric Drive Systems
Prof. R. Leidhold
Stationary reference frame Trace of the space vector in the plane
Rotating reference frame Trace of the space vector in dq-plane
