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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.

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,

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

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

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

Also, from the figure, for a field current (𝐼_{𝑓}) equal to Oa,

$$\mathrm{𝑋_{{𝑑}(Ω)} =\frac{𝑎𝑐}{𝑎𝑏}… (3)}$$

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,

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

Where,

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

$$\mathrm{\Rightarrow\:𝑍_{𝑏𝑎𝑠𝑒} =\frac{𝑉_{rated}/phase}{𝐼_{𝑎(rated)}/phase}=\frac{𝑎𝑐}{𝑑𝑒}Ω … (5)}$$

Therefore, the per-unit value of $𝑋_{𝑑}$ will be,

$$\mathrm{𝑋_{𝑑\:(𝑝𝑢)} =\frac{𝑋_{𝑑Ω}}{𝑍_{𝑏𝑎𝑠𝑒}}=\frac{(\frac{𝑎𝑐}{𝑎𝑏})}{(\frac{𝑎𝑐}{𝑑𝑒})}=\frac{𝑑𝑒}{𝑎𝑏}… (6)}$$

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

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

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.

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