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# What is Synchronous Reactance & Synchronous Impedance?

In an alternator or synchronous generator, the actual generated voltage consists of the summation of two component voltages. One of these component voltages is the *excitation voltage * (𝐸_{exc}); it is the voltage that would be generated because of the field excitation only. The excitation voltage is the voltage that would be generated when there is no armature reaction.

The other component of the generated voltage is the voltage that must be added to the excitation voltage to take care of the effect of the armature reaction on the generated voltage. This component voltage is known as the *armature reaction voltage* and is denoted by (𝐸_{𝐴𝑅}).

Therefore, the actual generated voltage by the alternator is given by,

$$\mathrm{𝐸_{𝑎} = 𝐸_{exc} + 𝐸_{𝐴𝑅} … (1)}$$

The armature reaction voltage in a circuit caused by change in the flux by current in the same circuit and its effect is of the nature of the inductive reactance. Hence, the armature reaction voltage (𝐸_{𝐴𝑅}) is equivalent to a voltage of inductive reactance, i.e.,

$$\mathrm{𝐸_{𝐴𝑅} = −𝑗𝐼_{𝑎}𝑋_{𝐴𝑅} … (2)}$$

The inductive reactance (X_{AR}) is an imaginary reactance which results in a voltage in the armature circuit of the alternator to take care of the effect of armature reaction upon the voltage relations of the armature circuit. Therefore, the armature reaction voltage (E_{AR}) can be represented as an inductor in series with the internal generated voltage.

Apart from the effect of armature reaction, the armature winding also has winding resistance and a self-inductance.

Let

**𝑅**= Armature winding resistance_{𝑎}**𝐿**= Self inductance of armature winding_{𝑎}**𝑋**= Self inductive reactance of armature winding_{𝑎}

Thus, the terminal voltage of the alternator is given by,

$$\mathrm{𝑉 = 𝐸_{𝑎} − 𝑗𝐼_{𝑎}𝑋_{𝐴𝑅} − 𝑗𝐼_{𝑎}𝑋_{𝑎} − 𝐼_{𝑎}𝑅_{𝑎} … (3)}$$

Where,

**𝐼**= Voltage drop due to armature resistance_{𝑎}𝑅_{𝑎}**𝐼**= Voltage drop due to armature leakage reactance_{𝑎}𝑋_{𝑎}**𝐼**= Armature reaction voltage_{𝑎}𝑋_{𝐴𝑅}

The effect of armature reaction and the effect of the leakage flux in the alternator both are represented by inductive reactances. Therefore, they can be combined into a single reactance and this single reactance is known as **synchronous reactance** of the alternator and is denoted by X_{S}.

Hence,

$$\mathrm{𝑋_{𝑆} = 𝑋_{𝑎} + 𝑋_{𝐴𝑅} … (4)}$$

Now, from eqns. (3) & (4), we can write,

$$\mathrm{𝑉 = 𝐸_{𝑎} − 𝑗𝐼_{𝑎}𝑋_{𝑆 }− 𝐼_{𝑎}𝑅_{𝑎}}$$

$$\mathrm{\Rightarrow\:𝑉 = 𝐸_{𝑎} − 𝐼_{𝑎}(𝑅_{𝑎} + 𝑗𝑋_{𝑆})}$$

$$\mathrm{\Rightarrow\:𝐸_{𝑎} − 𝐼_{𝑎}𝑍_{𝑆} … (5)}$$

Where,

$$\mathrm{𝑍_{𝑆} = (𝑅_{𝑎} + 𝑗𝑋_{𝑆}) … (6)}$$

The impedance (Z_{S}) is known as the **synchronous impedance** of the alternator.

The synchronous impedance (Z_{S}) is an imaginary impedance employed to account for the voltage effects in the armature circuit of the alternator, which is produced by the actual armature resistance, the actual armature leakage reactance, and the effect of the armature reaction.

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