# Space vector representation

The term space vector representation is understood in the electrical display of physical quantities in three-phase system as a pointer in a complex plane . The main area of ​​application of the space vector representation is the description of magnetic fields , voltages and currents in rotary field machines .

In alternating current technology , the simple pointer representation is used instead of the space vector representation .

## Space vector

Stator coils in the αβ coordinate system fixed to the stator. Each coil generates a magnetic field with a predetermined direction. The resulting space vector
B g is obtained by superimposing these

The basic idea of ​​the space vector representation is based on the fact that the zero condition is fulfilled: The star point (if present) of the load is not connected to the neutral conductor of the three-phase system. This means that the sum of the phase currents is always zero. Then one can infer the third by knowing two quantities. So such a three-phase system is fully described by two quantities.

If you consider induction machines, three coils are arranged there at an angle of 120 °. A separate magnetic field is generated in each of these coils when current flows through the machine . If the machine is now connected to a three-phase system, a different magnetization of the coils results for each instantaneous value . Since the three magnetic fields are superimposed, there is a non-uniform magnetic flux density distribution in the air gap between the stator and rotor . The distribution of the flux density along the circumference of the air gap has a maximum at a certain point. For each instantaneous value of the phase currents, the magnetic flux density has a specific orientation in the machine. This geometric orientation can now be represented by the two values ​​of ( real part and imaginary part ) as a space vector.

## Calculation of the space vector

In general, the space vector can be calculated from the three individual quantities using the following relationship, it being assumed that the coordinate system is arranged in such a way that the winding U has the same phase position as the real axis.

${\ displaystyle {\ vec {B}} = {\ tfrac {2} {3}} \ cdot (B_ {U} + a \ cdot B_ {V} + a ^ {2} \ cdot B_ {W})}$

Where a and a² represent the rotation operators.

${\ displaystyle a = e ^ {\ mathrm {j} 120 ^ {\ circ}} = - {\ tfrac {1} {2}} + \ mathrm {j} {\ tfrac {\ sqrt {3}} {2 }}}$
${\ displaystyle a ^ {2} = e ^ {- \ mathrm {j} 120 ^ {\ circ}} = - {\ tfrac {1} {2}} - \ mathrm {j} {\ tfrac {\ sqrt { 3}} {2}}}$

Since it is assumed that the null condition is satisfied, is represented by the space vector representation of the three-phase winding system with a two-phase winding system consisting of two mutually perpendicular coils replaced.

The representation as a space vector is not only limited to the magnetic flux density, but can also be used analogously for all other electrical quantities such as voltage, current and flux .

## Coordinate system

Overview of the coordinate systems. The dq-coordinate system fixed to the rotor rotates around the αβ-coordinate system fixed to the stator.

In the previous consideration it was assumed that the coordinate system is stationary and connected to the stator . It was also assumed that the real axis of the coordinate system coincides with the winding axis of the U winding.

The space vector does not necessarily have to be represented in the previously described αβ coordinate system with the Clarke transformation . For special applications, such as field-oriented control , it is necessary to let the coordinate system rotate with the rotor of the induction machine. The dq coordinate system fixed to the rotor thus rotates with the mechanical angular velocity Ω rotor around the αβ coordinate system fixed to the stator. The space vector in the dq coordinate system can be calculated from the phase quantities using the dq0 transformation .

Regardless of this, any coordinate system can be selected that is based on the stator flux, the air gap flux or the rotor flux, for example.

## application

The space vector display is mainly used in electrical drive technology for the control of induction machines. Frequency converters partly work internally with space vectors, which are output to control the electrical machine with the help of space vector modulation .

## literature

• Dierk Schröder: Electric drives - basics. 1st edition, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009, ISBN 978-3-642-02989-9 .
• Joachim Specovius: Basic course in power electronics. 2nd edition, Vieweg, 2008, ISBN 978-3-8348-0229-3 .