Short Circuit Ratio of a Synchronous Machine

Short Circuit Ratio

The ratio of the field current required to generate rated voltage on open-circuit to the field current required to circulate rated armature current on short circuit is called as the short-circuit ratio (SCR) of a synchronous machine.

Calculation of SCR

The short circuit ratio of a synchronous machine can be calculated from the open-circuit characteristic (O.C.C.) and the short-circuit characteristic (S.C.C.) of the synchronous machine, which are shown in the figure.

From the figure, the SCR is given by,

$$\mathrm{SCR \:=\: \frac{I_{f} \: \text{ for rated open circuit voltage}}{I_{f}\:\text{ for rated short circuit current}} \:=\: \frac{Oa}{Od} \:\: \dotso \: (1)}$$

Since the triangles "Oab" and "Ode" are similar triangles,

$$\mathrm{SCR \:=\: \frac{Oa}{Od} \:=\: \frac{ab}{de} \:\: \dotso \: (2)}$$

Also, from the figure, for a field current (I_f) equal to Oa,

$$\mathrm{X_{{d}(\Omega)} \:=\: \frac{ac}{ab} \:\: \dotso \: (3)}$$

Where, X_d is the direct-axis synchronous reactance of the machine and is defined as the ratio of open-circuit voltage for a given field current to the armature short-circuit current for the same value of the field current.

Eq. (3) gives the value of X_d in ohms. The per-unit value of the direct-axis synchronous reactance (X_d) is given by,

$$\mathrm{X_{d\:(pu)} \:=\: \frac{X_{d\Omega}}{Base\:impedance\: (Z_{base})} \:\: \dotso \:(4)}$$

Where,

$$\mathrm{Base \: impedace, \:\: Z_{bae} \:=\: \frac{\text{Rated voltage per phase}}{\text{Rated armature current per phase}}}$$

$$\mathrm{\Rightarrow \: Z_{bae} \:=\: \frac{V_{rated}/phase}{I_{a(rated)}/phase} \:=\: \frac{ac}{de} \:\Omega \:\: \dotso \: (5)}$$

Therefore, the per-unit value of X_d will be,

$$\mathrm{X_{d\:(pu)} \:=\: \frac{X_{d\Omega}}{Z_{bae}} \:=\: \frac{\left(\frac{ac}{ab}\right)} {\left (\frac{ac}{de} \right)} \:=\: \frac{de}{ab} \:\: \dotso \: (6)}$$

Hence, from eqns. (2) & (6), the short circuit ratio (SCR) is

$$\mathrm{SCR \:=\: \frac{ab}{de} \:=\: \frac{1}{X_{d\:(pu)}} \:\: \dotso \: (7)}$$

It is clear from Eq. (7) that the short-circuit ratio (SCR) is equal to the reciprocal of the per-unit value of the direct-axis synchronous reactance of the synchronous machine.

It should also be noted that the short-circuit ratio is single-valued since it belongs to the rated voltage on the open-circuit characteristic and rated armature current on the short-circuit characteristic.