Short Circuit Ratio of a Synchronous Machine

The Short Circuit Ratio (SCR) of a Synchronous Machine

The Short Circuit Ratio (SCR) of a synchronous machine is defined as the ratio of the field current needed to generate rated voltage under open-circuit conditions to the field current required to sustain rated armature current during a short-circuit condition. For a three-phase synchronous machine, the SCR can be derived from its Open-Circuit Characteristic (O.C.C) at rated speed and Short-Circuit Characteristic (S.C.C), as illustrated in the figure below:

From the above figure, the short circuit ratio is given by the equation shown below.

Since the triangles Oab and Ode are similar. Therefore,

Direct Axis Synchronous Reactance (Xd)

The direct axis synchronous reactance X_d is defined as the ratio of the open-circuit voltage corresponding to a specific field current to the armature short-circuit current under the same field current condition.

For a field current of magnitude Oa, the direct axis synchronous reactance (in ohms) is expressed by the following equation:

Relationship Between SCR and Synchronous Reactance

From equation (7), it is evident that the Short Circuit Ratio (SCR) equals the reciprocal of the per-unit direct axis synchronous reactance X_d. In a saturated magnetic circuit, the value of X_d is contingent upon the degree of magnetic saturation.

Significance of the Short Circuit Ratio (SCR)

The SCR is a critical parameter for synchronous machines, influencing their operational characteristics, physical dimensions, and cost. Key implications include:

• Voltage Regulation Impact

• Synchronous generators with lower SCR values exhibit more pronounced terminal voltage fluctuations with load changes. Maintaining constant terminal voltage requires wide-ranging adjustments to the field current I_f).

• Stability Limitations

• A smaller SCR corresponds to reduced synchronizing power, which is essential for maintaining synchronism. This results in a lower stability limit, meaning machines with low SCR are less stable when operating in parallel with other generators.

• Trade-offs in Design

• High-SCR machines offer superior voltage regulation and enhanced steady-state stability but entail higher armature short-circuit fault currents. Additionally, they influence machine size and cost due to design trade-offs.

The excitation voltage of a synchronous machine is described by the equation:

For the same value of Tph Excitation voltage is directly proportional to the field flux per pole.

The synchronous inductance is given as:

Relationship Between SCR and Air Gap

Thus, the Short Circuit Ratio (SCR) is directly proportional to the air gap reluctance or air gap length. Increasing the air gap length elevates the SCR, though this requires a higher field magnetomotive force (MMF) to maintain the same excitation voltage (Ef). To increase the field MMF, either the field current or the number of field turns must be augmented, necessitating taller field poles and an enlarged machine diameter.

Impact on Machine Design

This leads to a key conclusion: a higher SCR inherently increases the size, weight, and cost of the synchronous machine.

Typical SCR Values by Machine Type

• Cylindrical Rotor Machines: SCR ranges from 0.5 to 0.9.

• Salient-Pole Machines: SCR falls between 1.0 and 1.5.

• Synchronous Compensators: SCR is typically 0.4.

These values reflect the design trade-offs between stability, voltage regulation, and physical dimensions in different synchronous machine configurations.