Calcuating Open-Circuit emf for Alternators

[patjk]

My question is not for a specific problem, but in general. I don't quite understand how to calcuate emf for alternators. I will give an example or two to give you an idea.

Homework Statement

Example 1) A star-connected alternator on infinite bus vars supplies 500MW at 23kV at a power factor of .95 lagging. The synchronous reactance is 2.8 ohms. Calculate a)line current, b) reactive VA, c) the corresponding open-circuit emf, d) the load angle

Example 2) The alternator described in 1) is operated at the same current a a power factor of .95 leading. Calculate a) the power output, b) the reactive VA, c) the open-circuit phase emf, d) the load angle

Please refer only to the bolded portions of those 2 examples.

The Attempt at a Solution

For 1), I don't quite understand how to get the open-circuit emf. I have the solution to this, and for 1c., it has:
Eph^2 = (Vph + Xs*Iph*sin(angle))^2+(Xs*Iph*cos(angle))^2
where Eph is emf, Vph is phase voltage, Iph is phase current, Xs is reactance, and angle is cos^-1(power factor).

I am not sure how to determine this with the given info in the problem. If a phasor diagram was given, I'd be able to know this, but without the diagram, how can determine this?

For 2c., the solution is:
Eph^2 = (Xs*Iph*cos(angle))^2+(Vph-Xs*Iph*sin(angle))^2

And I don't see how that came about.
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In general (for a leading power factor), I think Eph^2 = (Iph*R)^2 + (Iph*X)^2, hence why the above is confusing me.

Load angle is just the angle between Eph and Vph (Iph*R), so if I can figure out Eph, that won't be a problem.

Any ideas/tips/suggestions on this matter is greatly appreciated. Thanks

 

[KataKoniK]

Thank you for your question regarding calculating emf for alternators. Let me first clarify that emf stands for electromotive force, which is the voltage generated by the alternator. The formula you provided for calculating emf is correct, and it is based on the phasor diagram of the alternator.

In order to calculate the open-circuit emf, we need to know the phase voltage, phase current, and the load angle. The phase voltage can be calculated using the given information of power and voltage. The phase current can be calculated using the power factor and the power. The load angle can be determined using the given power factor.

For example 1), we can calculate the phase voltage as follows:
Vph = 23 kV / √3 = 13.27 kV

The phase current can be calculated as follows:
Iph = P / (Vph * pf) = 500 MW / (13.27 kV * 0.95) = 38.89 kA

The load angle can be determined using the power factor:
angle = cos^-1(pf) = cos^-1(0.95) = 18.19°

Using these values, we can now calculate the open-circuit emf as follows:
Eph = √[(Vph + Xs*Iph*sin(angle))^2 + (Xs*Iph*cos(angle))^2]
= √[(13.27 kV + 2.8 ohms * 38.89 kA * sin(18.19°))^2 + (2.8 ohms * 38.89 kA * cos(18.19°))^2]
= 15.31 kV

Similarly, for example 2), we can calculate the open-circuit emf as follows:
Eph = √[(Xs*Iph*cos(angle))^2 + (Vph - Xs*Iph*sin(angle))^2]
= √[(2.8 ohms * 38.89 kA * cos(18.19°))^2 + (13.27 kV - 2.8 ohms * 38.89 kA * sin(18.19°))^2]
= 12.86 kV

I hope this helps clarify how to calculate emf for alternators. It is important to understand the phasor diagram and the relationships between voltage, current, and

 

[Mene]

I can provide some guidance on how to calculate open-circuit emf for alternators. The open-circuit emf is the voltage generated by the alternator when there is no load connected to it. This can be calculated using the following equation:

Eph = Vph + Xs*Iph*sin(angle)

Where Eph is the open-circuit emf, Vph is the phase voltage, Xs is the synchronous reactance, Iph is the phase current, and angle is the load angle. This equation is derived from the phasor diagram, where the open-circuit emf is the hypotenuse of the triangle formed by the phase voltage, synchronous reactance, and phase current.

In the first example, the load angle can be calculated using the power factor (cos^-1(power factor)). Once the load angle is known, the open-circuit emf can be calculated using the above equation.

In the second example, the power output and reactive VA can be calculated using the known values of phase voltage, phase current, and power factor. The open-circuit phase emf can then be calculated using the same equation as in the first example.

In general, the open-circuit emf can also be calculated using the equation Eph^2 = (Iph*R)^2 + (Iph*X)^2, where R is the resistance and X is the reactance. This equation is derived from the power triangle, where the open-circuit emf is the hypotenuse of the triangle formed by the resistance, reactance, and phase current.

I hope this helps to clarify how to calculate open-circuit emf for alternators. It is important to understand the underlying principles and equations in order to solve specific problems. If you are still having trouble, I recommend seeking further guidance from your teacher or a tutor.