Is synchronous reactance constant?

[cnh1995]

In electrical machinery and power systems, the synchronous reactance is treated as a constant reactance in series with the generated emf and is independent of load condition. These threads were very helpful for me in understanding this concept (thanks to @jim hardy!).
https://www.physicsforums.com/posts/5190822/
https://www.physicsforums.com/posts/5211074/

However, I recently read something new in a book and I am a bit confused. According to the discussions we had here on PF, the armature reaction drop can be modeled as a constant (internal) series reactance and the math works out fine.Here in this book, the author says the almost same thing that was discussed in the above mentioned threads, but in the last line, he says the synchronous reactance is contsant for a given field excitation.

How can changing the field excitation change the synchronous reactance? Isn't X_s a machine parameter (at rated frequecny)? Am I missing something again?

 

[jim hardy]

How can changing the field excitation change the synchronous reactance? Isn't Xs a machine parameter (at rated frequecny)? Am I missing something again?

See if this makes any sense... ...

Let's add to your linked image my old single turn armature thought tool, and draw in its mmf.
I made them orange.

Armature reactance has two values does it not ? X_Direct and X_Quadrature ?
They differ because of the differing air gap at differing rotor angles as your excellent choice of illustration shows .

So armature reactance has an infinite number of values - X_D X_Q and everything in between.

Armature reaction we calculate (or measure) at rated excitation and publish it as a machine paramter.
Think about measuring that - we measure open circuit voltage then short circuit the machine and measure current.
Z_synchronous we calculate by ratio E_{open circuit}/ I_{short circuit} .

Now , if we repeat that test at some other value of excitation we'll get a different E_{open circuit} because of the machine's saturation curve.
But I_{short circuit} will still be determined by the field&armature amp-turns balance at zero flux. .(remember it's short circuited).
So I_{short circuit} should be real close to N_{Field Turns} X Field Amps / N_{Armature Turns} .
Aha ! There's the answer - we take the ratio E_{open circuit}/ I_{short circuit}
whose numerator includes the machine's B-H curve with its effects of saturation, and whose denominator, being measured at zero flux is oblivious to that prior saturation.

I think this author did a better job with his explanation than either i or your author...
http://people.ucalgary.ca/~aknigh/electrical_machines/synchronous/design/xs.html

Z_synchronous = f(I_field, B-H curve) / f(I_field , turns ratio) <<<<<< linear denominator, nonlinear numerator.

old jim.

 

[cnh1995]

Armature reactance has two values does it not ? XDirect and XQuadrature ?
They differ because of the differing air gap at differing rotor angles as your excellent choice of illustration shows .

Ok, let's consider a cylindrical rotor so that Xd=Xq.

Now , if we repeat that test at some other value of excitation we'll get a different Eopen circuit because of the machine's saturation curve.

Right, and this graph describes this behaviour.

Now, as per this graph, as long as we stay in the linear region, the unsaturated synchronous reactance (considering Xd=Xq) is (almost)constant no matter how much the excitation is. Is that correct?

So when we work out numerical problems on synchronous generator in class, we "assume" linear OC characteristic curve for the generator so that Xs is constant, right?

And when my author said, " the variation of armature reaction and armature reactance with power factor is difficult to calculate", it is only because of the saturation?

 

[jim hardy]

Now, as per this graph, as long as we stay in the linear region, the unsaturated synchronous reactance (considering Xd=Xq) is (almost)constant no matter how much the excitation is. Is that correct?

Right

So when we work out numerical problems on synchronous generator in class, we "assume" linear OC characteristic curve for the generator so that Xs is constant, right?

Yes. At least we did when i went through school.

And when my author said, " the variation of armature reaction and armature reactance with power factor is difficult to calculate", it is only because of the saturation?

For your smooth rotor case that's the only complication I'm aware of. Salient machine of course includes that varying air gap.

 

[cnh1995]

For your smooth rotor case that's the only complication I'm aware of. Salient machine of course includes that varying air gap.

Ok, but that's for theoretical analysis.
But in practice, does the machine ever go into saturation? How is the synchronous impedance determined for the real-world analysis of the power system?

 

[jim hardy]

At my level all i ever needed was nameplate X_synchronous. which was given on our nameplate as Short Circuit Ratio. .

The protective relaying and power system stabilizer folks used transient and subtransient reactances , and their work is a mysterious art to me. I had the good fortune to be around them but cannot claim fluency in that fascinating skillset .

So I'll have to defer your question to @anorlunda or @Babadag or any other genuine power system analysts here. Wish i could answer it.

I can tell you that when figuring voltage regulator settings my mentor and i did once use a Potier triangle which is a graphic method of accounting for saturation. I barely remember that though, it was about 1973.
http://circuitglobe.com/potier-triangle-or-zero-power-factor-method.htmlIf you are offered a course in Power System Analysis , take it.
That is going to be a good field of work as we try to squeeze more megawatts out of less resources. It needs people like you who delight in understanding the fundamentals.

old jim

 

[cnh1995]

If you are offered a course in Power System Analysis , take it.
That is going to be a good field work

I just graduated in EE a couple of months ago, and in all the power system related subjects I took, the generator was assumed to be unsaturated while solving problems.

I'll wait for the power engineers to weigh in here.
Thanks a lot for your help and patience! (I know I've troubled you a lot with this topic in the past two years..)

 

[jim hardy]

(I know I've troubled you a lot with this topic in the past two years..)

Not at all. I enjoyed the refresher and have vicariously enjoyed your progress .

You phrase your questions well and you have contributed a lot of good answers here on PF.
Wherever you work you will become 'The Explainer Of Things" that people seek out.

I just graduated in EE a couple of months ago, and in all the power system related subjects I took, the generator was assumed to be unsaturated while solving problems.

I'm relieved to hear my training isn't completely obsolete at undergrad level - Thanks !Congratulation to the new grad! Now your 'on the job training' begins .
I hope they assign you to an older mentor . Mine was the biggest single influence in my adult life.

old jim

 

[cnh1995]

But in practice, does the machine ever go into saturation? How is the synchronous impedance determined for the real-world analysis of the power system?

@jim hardy, I think I've found the answer to this. It turns out that the machine does go into saturation even during normal operation. This article mentions the concept of 'adjusted synchronous reactance' to account for saturation effect. So the nameplate short-circuit ratio is actually the reciprocal of this adjusted synchronous reactance.
https://www.google.co.in/url?sa=t&s...aAoYQFghOMAQ&usg=AOvVaw3P6sKydztxcilLUL367uVx.

 

[jim hardy]

Nice explanation there.

I think he should have worded this sentence better

The straight-line part of the OCC, if extended as shown dotted in Fig. 8.12, is called the air-gap line and would indeed be the OCC if iron did not get saturated. (show nonlinearity as it approaches saturation - jh )

He did better in this line

Since a synchronous machine under operating conditions works in a somewhat saturated region of the magnetization characteristic,

We do operate the machine with peak flux up almost to the knee of its BH curve.
If one says a synchronous machine is "saturated" he usually means it's overexcited to the point it's in danger of overheating its core. We once melted the core of a big generator by 'overfluxing' ie saturating it. Molten iron ran out its end.

So, yes normal operating flux is high enough to show effects of approach to saturation . But if you tell a generator supplier you saturated his machine you might give him a heart attack. A GE engineer told me in mid 1980's that about once a year somebody someplace in the world wrecks a generator by overfluxing it.
The dangerous time is just before synchronizing the machine when you're trying to match generator voltage with grid. If your generator voltmeter is reading low it's natural to apply more field current to bring the two meters up to same level. That's what bit us.

I honestly had forgot that short circuit ratio is calculated by ratio of
field currents to make [rated voltage at open circuit / rated current short circuit ]
so is reciprocal of adjusted Z_synchronous.
Thank You ! I'd got rusty.

That is a very informative and practical link you found. Thanks for posting it. Print a copy , fold and stash in back cover of your textbook ? Ten years from now it might come in handy.

old jim

 

[cnh1995]

That is a very informative and practical link you found. Thanks for posting it. Print a copy , fold and stash in back cover of your textbook ?

I strongly suspect it is copied from a book I later found on google books (the wording is the exact same), and that book is available in the store a few blocks away from my home!
I knew what the formula for short circuit ratio was, but didn't know it is defined so as to account for saturation effect.

Thank you for your help and encouragement!