Synchronous Reactance

The rotating held in the air gap of a synchronous machine is generally considered to be free of space harmonics, when the basic operation of the machine is being considered. In an actual machine there are space harmonics present in the air gap, more in salient pole machines than a cylindrical rotor machine, see for example References 4 and 6. It is acceptable to ignore the effects of space harmonics when considering armature reaction and the associated reactances. Therefore the flux wave produced by the rotating field winding can be assumed to be distributed sinusoidally in space around the poles of the rotor and across the air gap. 

When the rotor pole axis coincides with the axis of the stator coils the magnetic circuit seen by the stator has minimum reluctance. The reactance corresponding to the armature reaction in this rotor position is called the direct axis synchronous reactance Xs/- If the stator winding leakage reactance, Xa, is deducted from Xgd the resulting reactance is called the direct axis reactance X/. 

A similar situation occurs when the rotor pole axis is at right angles to the axis of the stator coils. Here the magnetic reluctance is at its maximum value due to the widest part of the air gap facing the stator coils. The complete reactance in this position is called the quadrature axis synchronous reactance Xsq . Deducting Xa results in the quadrature axis reactance X. ... 

The damper bars or winding act in a manner very similar to an induction motor and provide a breaking torque against the transient disturbances in shaft speed. To be effective the damper needs to have a steep torque versus shp characteristic in the region near synchronous speed. The equivalent impedance of the damper requires a low resistance and a high reactance. High conductivity copper bars are embedded into the pole face to provide a low reluctance path for the leakage flux. 

Most practical generators have an armature resistance Ra that is much less in value than the synchronous reactances Xj and Xq. Consequently the equations in sub-section 3.5.2 can be further simplified without incurring a noticeable error. They become,... 

Figure 3.5 D-axis synchronous reactance versus generator MVA rating. 11.0...

Find the value of the gain Ga for an AVR fitted to a generator that has a synchronous reactance of... 

The emf feeds a series circuit consisting of the load plus the synchronous reactance Xs. It can be shown that the emf E is,... 

If the above sequence is repeated for different values of synchronous reactance then appropriate values of the AVR gain G can be found, as shown in Table 4.1. 

A synchronous generator (and a synchronous motor) can be represented by many inductances and reactances to account for transformer-type induction, rotational induction, mutual coupling between windings, leakage and self-induction, magnetising and excitation induction and the effects of the pole-face damper windings. Extremely complex equivalent circuits have been developed for synchronous machines, see References 1 and 2 as examples. 

E shows the synchronous emf behind the synchronous reactance Xsd- Used for calculating the steady state fault current, which will then be fully symmetrical, since all the sub-ffansient and transient effects will have decayed to zero. The emf E will be the ceiling voltage of the exciter since the AVR will have seen a severe depression in terminal voltage and will have forced the exciter to give its maximum possible output. See also sub-sections 7.2.8 and 12.2.2.1. 

The sub-transient impedance determines the initial decay, i.e. in the first cycle or so. Therefore the emfs E" and E, together with the reactances Xj and X j, need to be used for calculating the fault currents. In a similar way to induction motors, the synchronous motors will contribute to fault-making dnty reqnirements. However, they will also contribute towards the fault-breaking duty because of the transient effects. 

Induction motors can be represented by the 2-axis theory, by using the derivations for synchronous machines but deleting the field winding. In this case some of the reactances become zero, and the field resistance is infinity. Hence, the derived reactances X ... 

Steady state stability relates to the ability of the synchronous source (generators) to transfer power to the synchronous sink (motors and/or other generators). This may be explained by simplifying the synchronous power system as a transmission link (cable or overhead line) of reactance X and zero resistance, a synchronous source (generator at the sending end of the link) and a synchronous sink (load at the receiving end). 

Since the reactance X consumes no power, the receiving end power must equal the sending end power. (If the end voltages are not in steady state synchronism then the system is regarded as being unstable.)... 

As an example, consider two offshore platforms, each with its own generators and loads, operating in synchronism through an interconnecting power cable of reactance X (as shown in Figure 11.14. Assume the resistance of the interconnecting cable is zero. 

In an interconnected power system there will be two or more synchronous machines (or groups of machines). These machines will be coupled through their own internal reactances and through... 

The synchronous generators and motors are represented by their sub-transient, transient and synchronous reactances and time constants in both the d and the q axes, hence saliency is... 

The derived reactances are those most frequently used to specify synchronous generators and motors. They are the synchronous, transient and sub-transient reactances in the d and g-axes. The most convenient method of deriving these is from the application of a three-phase short circuit at the terminals of the unloaded machine, whether it be a generator or a motor. For a motor the testing procedure is more complicated as described in sub-section 5 of Reference 23. The factory tests. The g-axis are usually taken as their design values because the necessary factory tests are more difficult to perform. The tests are described in for example IEEE standard 112 and BS4296. 

The absence of the field winding can be used to convert the mathematical model of the synchronous machine into one for an induction machine. In addition the mutual inductance in the < -axis is made equal to mumal inductance in the d-axis, i.e. the machine becomes symmetrical in both axes. The matrix equations (20.6) to (20.16) are modified as shown below. In these equations the mutual inductances Mj and become M, Lim and Lihq become L/j., Rjut and R/aj become Rk. All the derived reactances and time constants for an induction machine are equivalent to those appUcable to the g-axis of the synchronous machine. 

Application of a three-phase short circuit to the terminals of an unloaded induction motor is not a practical factory test, especially for a large high-voltage motor, because the motor can only be excited at its stator windings from the power supply. A three-phase short circuit at or near the stator terminals can occur in practice e.g. damaged supply cable, damage in the cable terminal box. The parameters of the stator and rotor windings can be obtained from other factory tests. However, the derived reactance can be defined in the same manner as those for the synchronous machine, but with... 

The harmonic losses developed in the motor by the drive are very small in a typical mine hoist low speed synchronous motor powered from a Variable Speed Drive. In this type of configuration the Voltage Source Inverter (VSI) and Active Rectifier Unit (ARU) typically increase the motor losses by the order of 1-1.5% of the total motor losses. The low harmonic losses in this case is due to the motor s high reactance and low base frequency, the inverter technology and the switching frequency. 

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