# Maximum Reactive Power for a Synchronous Generator or Alternator

For a salient-pole synchronous generator or alternator, the per phase reactive power is given by,

$$\mathrm{π_{1π} =\frac{ππΈ_{π}}{π_{π}}cos\:πΏ −\frac{π^{2}}{2π_{π}π_{π}}\lbrace{(π_{π} + π_{π} ) − (π_{π} − π_{π})\:cos\:2\delta}\rbrace … (1)}$$

Where,

• V is the terminal voltage per phase.

• Ef is the excitation voltage per phase.

• $\delta$ is the per phase angle between Ef and V.

• Xd is the direct-axis synchronous reactance.

• Xq is the quadrature-axis synchronous reactance.

For reactive power to be maximum,

$$\mathrm{\frac{ππ_{1π}}{π\delta}= 0}$$

$$\mathrm{\Rightarrow\:\frac{π}{π\delta} \left(\frac{ππΈ_{π}}{π_{π}}cos\:\delta −\frac{π^{2}}{2π_{π}π_{π}}\lbrace(π_{π} + π_{π}) − (π_{π} − π_{π}) cos\:2\delta\rbrace \right)= 0}$$

$$\mathrm{−\frac{ππΈ_{π}}{π_{π}}sin\:\delta −\frac{2π^{2}}{2π_{π}π_{π}}(π_{π} − π_{π})sin\:2\delta = 0}$$

$$\mathrm{\Rightarrow\:πΈ_{π}\:sin\:\delta +\frac{π}{π_{π}}(π_{π} − π_{π})(2\:sin\:\delta\:cos\:\delta) = 0}$$

$$\mathrm{\Rightarrow\:cos\:\delta = −\frac{πΈ_{π}π_{π}}{2\:π(π_{π} − π_{π} )}… (2)}$$

By putting the value of from eq. (2) in eq. (1), we have,

$$\mathrm{π_{1π\:πππ₯} =\frac{ππΈ_{π}}{π_{π}}\left(−\frac{πΈ_{π}π_{π}}{2\:π(π_{π} − π_{π} )}\right)−\frac{π^{2}}{2π_{π}π_{π}}{(π_{π} + π_{π})+\frac{π^{2}}{2π_{π}π_{π}}}{(π_{π} - π_{π})}(2 cos^{2}\:\delta − 1)}$$

$$\mathrm{\Rightarrow\:π_{1π\:πππ₯}=−\frac{{πΈ^{2}_{π}}π_{π}}{2\:π_{π}(π_{π}−π_{π})}-\frac{π^{2}}{2π_{π}π_{π}}{(π_{π} + π_{π})+\frac{π^{2}}{2π_{π}π_{π}}}{(π_{π} - π_{π})}\left(\frac{2{πΈ^{2}_{π}}π^{2}_{π}}{4 π^{2}(π_{π}− π_{π})^{2}}-1\right)}$$

$$\mathrm{\Rightarrow\:π_{1π\:πππ₯}=-\frac{π^{2}}{2π_{π}π_{π}}\lbrace{(π_{π} + π_{π})-(π_{π} - π_{π})}\rbrace−\frac{πΈ^{2}_{π}π_{π}}{2\:π_{π}(π_{π} − π_{π})}+\frac{πΈ^{2}_{π}π^{2}_{π}}{4 π_{π}(π_{π} − π_{π} )}}$$

$$\mathrm{\Rightarrow\:π_{1π\:πππ₯}=\frac{π^{2}}{π_{π}}-\frac{{πΈ^{2}_{π}}π^{2}_{π}}{4 π_{π}(π_{π} − π_{π} )}… (3)}$$

Equation (3) gives the maximum value of the reactive power per phase for the salient-pole alternator.

Again, for a cylindrical rotor alternator,

$$\mathrm{π_{π} = π_{π} = π_{π }}$$

$$\mathrm{∴\:π_{1π} =\frac{ππΈ_{π}}{π_{π }}cos\:\delta −\frac{π^{2}}{π_{π }}}$$

$$\mathrm{\Rightarrow\:π_{1π} =\frac{π}{π_{π }}(πΈ_{π}\:cos\:\delta − π) … (4)}$$

From Eq. (4), it can be seen that when $πΈ_{π}\:cos\:\delta = π$ i.e. under normal excitation, then $π_{1π}$ = 0 and the alternator operates at unity power factor.

• When πΈπ cos $\delta$ > π, i.e., the alternator is over-excited, the reactive power is positive. Therefore, the alternator supplies reactive power to the busbars.

• When πΈπ cos $\delta$ < π, i.e., the alternator is under-excited, the reactive power is negative. Therefore, the alternator absorbs reactive power.

Updated on: 01-Oct-2021

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