What are the advantages and disadvantages of different reference frames for synchronous machine analysis?

1 Physical frame

The physical frame is the most intuitive and straightforward way to represent the synchronous machine, as it corresponds to the actual geometry and construction of the machine. In this frame, the stator and rotor windings are modeled by sinusoidal voltage and current sources, and the electromagnetic torque and power are calculated by the interaction of the stator and rotor fluxes. The main advantage of this frame is that it preserves the physical meaning and interpretation of the variables and parameters, and it is easy to apply circuit analysis techniques. The main disadvantage is that the equations are nonlinear and time-varying, due to the rotation of the rotor and the periodic nature of the sinusoidal functions. This makes the analysis more complex and difficult, especially for transient and stability studies.

2 Rotating frame

The rotating frame is a common and convenient way to simplify the synchronous machine analysis, by transforming the variables and equations to a reference frame that rotates at a constant speed. The most widely used rotating frame is the d-q frame, which is aligned with the rotor flux axis (d-axis) and the quadrature axis (q-axis) of the machine. In this frame, the stator and rotor windings are modeled by constant voltage and current sources, and the electromagnetic torque and power are calculated by the product of the d- and q-axis currents. The main advantage of this frame is that it eliminates the time variation and nonlinearity of the equations, and it allows the use of linear control and state-space methods. The main disadvantage is that it loses the physical meaning and interpretation of the variables and parameters, and it requires a transformation matrix to convert between the physical and rotating frames.

3 Stationary frame

The stationary frame is another way to simplify the synchronous machine analysis, by transforming the variables and equations to a reference frame that is fixed to the stator. The most widely used stationary frame is the a-b-c frame, which is aligned with the stator phases (a-phase, b-phase, and c-phase) of the machine. In this frame, the stator and rotor windings are modeled by balanced three-phase voltage and current sources, and the electromagnetic torque and power are calculated by the power invariant transformation of the a-b-c variables. The main advantage of this frame is that it preserves the balanced three-phase nature of the system, and it does not require a transformation matrix to convert between the physical and stationary frames. The main disadvantage is that it still retains some nonlinearity and time variation of the equations, due to the rotation of the rotor and the phase shift of the sinusoidal functions. This makes the analysis less simple and elegant than the rotating frame.

4 Synchronous frame

The synchronous frame is a less common but useful way to generalize the synchronous machine analysis, by transforming the variables and equations to a reference frame that rotates at a variable speed. The most widely used synchronous frame is the d'-q' frame, which is aligned with the rotor flux axis (d'-axis) and the quadrature axis (q'-axis) of the machine. In this frame, the stator and rotor windings are modeled by variable voltage and current sources, and the electromagnetic torque and power are calculated by the product of the d'- and q'-axis currents. The main advantage of this frame is that it can accommodate any speed variation of the rotor, such as slip or frequency deviation, and it can handle unbalanced or distorted conditions of the system. The main disadvantage is that it introduces additional variables and equations to account for the speed variation and transformation of the reference frame, and it requires more computational effort and accuracy.

5 Here’s what else to consider

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