ABC TO DQ

ABC TO AND DQ TransformationsINTRODUCTION: The directquadraturezero (or dq0 or dqo) transformation or zerodirectquadrature (or 0dq or odq) transformation is a mathematical transformation that rotates the reference frame of three-phase systems in an effort to simplify the analysis of three-phase circuits. The dqo transform is often referred to as Parks transformation. In the case of balanced three-phase circuits, application of the dqo transform reduces the three AC quantities to two DC quantities. Simplified calculations can then be carried out on these DC quantities before performing the inverse transform to recover the actual three-phase AC results. It is often used in order to simplify the analysis of three-phase synchronous machines or to simplify calculations for the control of three-phase inverters. In analysis of three-phase synchronous machines the transformation transfers three-phase stator and rotor quantities into a single rotating reference frame to eliminate the effect of time varying inductances.GENERAL EQUATIONS:The power-invariant, right-handed dqo transform applied to any three-phase quantities (e.g. voltages, currents, flux linkages) is shown below in matrix form.

The inverse transform is:

The transform applied to three-phase currents

where is a generic three-phase current sequence and is the corresponding current sequence given by the transformation . The inverse transform is:

The above Clarke's transformation preserves the amplitude of the electrical variables which it is applied to. Indeed, consider a three-phase symmetric, direct, current sequence

where is the RMS of , , and is the generic time-varying angle that can also be set to without loss of generality. Then, by applying to the current sequence, it results

where the last equation holds since we have considered balanced currents. As it is shown in the above, the amplitudes of the currents in the reference frame are the same of that in the natural reference frame.

Power invariant transformationThe active and reactive powers computed in the Clark's domain with the transformation shown above are not the same of those computed in the standard reference frame. This happens because is not unitary. In order to preserve the active and reactive powers one has, instead, to consider

which is a unitary matrix and the inverse coincides with its transpose. In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame,

Finally, the inverse trasformation in this case is

Since in a balanced system and thus one can also consider the simplified transform

which is simply the original Clarke's transformation with the 3rd equation thrown away, and

Geometric InterpretationThe dqo transformation is two sets of axis rotations in sequence. We can begin with a 3D space where a, b, and c are orthogonal axes.

If we rotate about the a axis by -45, we get the following rotation matrix:,which resolves to.With this rotation, the axes look like

Then we can rotate about the new b axis by about 35.26

,which resolves to.

When these two matrices are multiplied, we get the Clarke transformation matrix C:

This is the first of the two sets of axis rotations. At this point, we can relabel the rotated a, b, and c axes as , , and z. This first set of rotations places the z axis an equal distance away from all three of the original a, b, and c axes. In a balanced system, the values on these three axes would always balance each other in such a way that the z axis value would be zero. This is one of the core values of the dqo transformation; it can reduce the number relevant variables in the system.The second set of axis rotations is very simple. In electric systems, very often the a, b, and c values are oscillating in such a way that the net vector is spinning. In a balanced system, the vector is spinning about the z axis. Very often, it is helpful to rotate the reference frame such that the majority of the changes in the abc values, due to this spinning, are canceled out and any finer variations become more obvious. So, in addition to the Clarke transform, the following axis rotation is applied about the z axis:.Multiplying this matrix by the Clarke matrix results in the dqo transform:.The dqo transformation can be thought of in geometric terms as the projection of the three separate sinusoidal phase quantities onto two axes rotating with the same angular velocity as the sinusoidal phase quantities. The two axes are called the direct, or d, axis; and the quadrature or q, axis; that is, with the q-axis being at an angle of 90 degrees from the direct axis.The transformation can be thought of as the projection of the three phase quantities (voltages or currents) onto two stationary axes, the alpha axis and the beta axis.

The transform as applied to three symmetrical currents flowing through three windings separated by 120 physical degrees. The three phase currents lag their corresponding phase voltages by . The - axis is shown with the axis aligned with phase 'A'. The current vector rotates with angular velocity . There is no component since the currents are balanced.

This is the dqo transform as applied to the stator of a synchronous machine. There are three windings separated by 120 physical degrees. The three phase currents are equal in magnitude and are separated from one another by 120 electrical degrees. The three phase currents lag their corresponding phase voltages by . The d-q axis is shown rotating with angular velocity equal to , the same angular velocity as the phase voltages and currents. The d axis makes an angle with the A winding which has been chosen as the reference.The currents and are constant DC quantities.

Simulated Circuit

Figure 1:Linear R Load(current)

Waveform For Input Voltage And Current Figure 2:Input Voltage and Current

abc to for current waveform Figure 3:abc to for current

abc to dq for current waveform

Figure 4:abc to dq for currentabc to and abc to dq for voltageThe abc to and abc to dq for voltage can be obtained by means of following equation.The equation arev =2/3[va-0.5vb-0.5vc] v =2/3[3/2 vb-3/2 vc] v0=2/3[1/2 va+1/2 vb+1/2 vc]

Simulated Circuit

Figure 5:Linear R Load(voltage)abc to for voltage waveform Figure 6:abc to for voltage

abc to dq for voltage waveformFigure 7:abc to dq for voltage2.Linear RL LoadSimulated circuit

Figure 8:Linear RL Load(voltage)

abc to for current

Figure 9: abc to and abc to dq for current abc to dq for current

Figure 10: abc to dq for current

Simulated Circuit

Figure 11:Linear RL Load(current)

3.Non-Linear LoadAC VOLTAGE REGULATOR[R Load] FOR CURRENT

Figure 12 :AC voltage regulator[R load]

Waveform for Input Voltage and current

Figure 13:Input voltage and currentabc to for current

Figure 14: abc to for currentabc to dq for current


AC VOLTAGE REGULATOR[R LOAD] FOR VOLTAGE

Figure 16: AC VOLTAGE REGULATOR[R LOAD] FOR VOLTAGE

abc to for voltage

Figure 17: abc to for voltage

abc to dq for voltage

Figure 18: abc to dq for voltage

AC VOLTAGE REGULATOR [RL LOAD]-CURRENT

Figure 19: AC VOLTAGE REGULATOR [RL LOAD]CURRENT

abc to for current


Figure 21: abc to dq for currentabc to dq for voltage

Figure 22: abc to dq for voltageAC VOLTAGE REGULATOR [RL LOAD]-VOLTAGE

Figure 23:AC VOLTAGE REGULATOR [RL LOAD]-VOLTAGEabc to for voltage

BRIDGE RECTIFIER-R LOAD FOR CURRENT

Figure 24: BRIDGE RECTIFIER-R LOAD FOR CURRENT

abc to for current



BRIDGE RECTIFIER-R LOAD FOR VOLTAGE

abc to for voltage


abc to dq for current

CONTROLLED RECTIFIER R LOAD-VOLTAGE

abc to for voltage


abc to dq for voltage


CONTROLLED RECTIFIER R LOAD-CURRENT

Waveform Input Voltage and current

abc to For CURRENT

abc to dq For CURRENT

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ABC TO DQ