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Emf and regulation of 1500 kVA 13 kV alternator

Short answer

Rated Vph=7505.6 V, I=66.62 A. Unity pf: E=√((V+IRa)²+(IXs)²) = 7584.3 V/phase, regulation 1.05%. 0.8 pf lagging: E=√((Vcosφ+IRa)²+(Vsinφ+IXs)²) = 7883.0 V/phase, regulation 5.03%.

A three phase, star connected 1500 kVA, 13 kV alternator has armature resistance of 0.9 ohm per phase and synchronous reactance of 8 ohm per phase. In each of the following cases the alternator is supplying rated full load current at rated terminal voltage. Calculate emf generated and voltage regulation in each following cases: i) unity P.F. ii) 0.8 P.F. lagging

Given: star, 1500 kVA, 13 kV, Ra=0.9 ΩR_a = 0.9\ \Omega, Xs=8 ΩX_s = 8\ \Omega per phase; rated current at rated voltage.

Vph=130003=7505.55 VI=1500×1033×13000=66.62 AIRa=66.62×0.9=59.96 V,IXs=66.62×8=532.94 V\begin{aligned} V_{ph} &= \frac{13000}{\sqrt3} = 7505.55\ \text{V} \\ I &= \frac{1500\times10^3}{\sqrt3\times13000} = 66.62\ \text{A} \\ IR_a &= 66.62\times0.9 = 59.96\ \text{V},\quad IX_s = 66.62\times8 = 532.94\ \text{V} \end{aligned}

(i) Unity power factor

E=(V+IRa)2+(IXs)2=7565.512+532.942=7584.26 V per phase(EL=13136 V)%Reg=7584.26−7505.557505.55×100=1.05%\begin{aligned} E &= \sqrt{(V + IR_a)^2 + (IX_s)^2} = \sqrt{7565.51^2 + 532.94^2} \\ &= 7584.26\ \text{V per phase}\quad (E_L = 13136\ \text{V}) \\ \%\text{Reg} &= \frac{7584.26 - 7505.55}{7505.55}\times100 = 1.05\% \end{aligned}

(ii) 0.8 pf lagging

Vcos⁡ϕ+IRa=6004.44+59.96=6064.40 VVsin⁡ϕ+IXs=4503.33+532.94=5036.27 VE=6064.402+5036.272=7882.95 V per phase(EL=13654 V)%Reg=7882.95−7505.557505.55×100=5.03%\begin{aligned} V\cos\phi + IR_a &= 6004.44 + 59.96 = 6064.40\ \text{V} \\ V\sin\phi + IX_s &= 4503.33 + 532.94 = 5036.27\ \text{V} \\ E &= \sqrt{6064.40^2 + 5036.27^2} = 7882.95\ \text{V per phase}\quad (E_L = 13654\ \text{V}) \\ \%\text{Reg} &= \frac{7882.95 - 7505.55}{7505.55}\times100 = 5.03\% \end{aligned}
CaseE per phase (V)E line (V)Regulation
Unity pf7584.26131361.05%
0.8 pf lagging7882.95136545.03%

Answer: unity pf: E = 7584.3 V/phase, regulation 1.05%; 0.8 lag: E = 7883.0 V/phase, regulation 5.03%.

✓ Answer checked by Claude Sonnet 5.5

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