Question

The dc-offset component of the three-phase short-circuit current of an unloaded synchronous machine is different in the three phases and its exponential decay is dictated by _______.

Short answer

The dc-offset component of the three-phase short-circuit current of an unloaded synchronous machine is different in the three phases and its exponential decay is dictated by armature time constant.

Step-by-step solution

Step 1: The exponential decay of dc-offset current

During a short circuit test the synchronous machine is not loaded and thus the no load voltages are displaced by 120 degree from each other and so the ac fault current flowing through the armature is also displaced by 120 degree from each other.

This three phase ac fault current also has a dc component in it which is termed as dc offset current. It is dc component with reference to which ac current wave flows.

Exponential Decay of dc-offset current

The formula of dc offset current for any one of the three phases is shown below:

$$\begin{gathered} {\mathrm{i}}_{\mathrm{dc}\max }\left(\mathrm{t}\right)=\frac{\sqrt{2{\mathrm{E}}_{\mathrm{g}}}}{{{\mathrm{X}}_{\mathrm{d}}}^{\text{'}\text{'}}}{\mathrm{e}}^{-\mathrm{t}/{\mathrm{T}}_{\mathrm{A}}} \\ =\sqrt{2\mathrm{l}}\text{'}\text{'}{\mathrm{e}}^{-\mathrm{t}/{\mathrm{T}}_{\mathrm{A}}} \end{gathered}$$

Here, $i_{dcmax}\left(t\right)$is the maximum dc offset in any one phase,$E_g$ is the rms line-to-neutral terminal voltage of the unloaded synchronous machine, ${X_d}^{\text{'}\text{'}}$is direct axis sub transient reactance, $I^{\text{'}\text{'}}$is the rms sub transient fault current, $T_A$is called the armature time constant, and we can see the exponential decay of dc offset current is dictated by $T_A$(armature time constant).

Therefore, “The complete statement is“The dc-offset component of the three-phase short-circuit current of an unloaded synchronous machine is different in the three phases and its exponential decay is dictated by armature time constant.”