State space equation 2

[zolka]

Helo!
Can anyone tell me how to transform this equation into state-space equation?
I managed at the motor, but here are the capacitors, which confuses me!
The currents are the state variables!
Thanx

(The equation is here:

http://files.engineering.com/getfil...93-4b66-ac73-b172f5b12c9f&file=ind.gener..JPG

 

[7anoter4]

1) I think you asked, first of all, how to solve the matrix:
Vds=(R1+L1p+1/pC+Mp-wM]*ids
Vqs=(R1+L1p+1/pC+wM+Mp)*iqs
Vdr=(Mp+R2+L2p-wL2)*idr
Vqr=(Mp+wL2+R2+L2p)*iqr
Rearranging per rank:
Vds=[R1-wM+(L1+M)p+1/pC]*ids
Vqs=[R1+wM+(L1+M)p+1/pC]*iqs
Vdr=[R2-wL2+(L2+M)p]*idr
Vqr=[R2+wL2+(L2+M)p]*iqr
2)From here you'll find the current[still in Laplace mode]:
Let's say: Vds=VDSeff*sin(w*t+psids), for instance
So
ids=VDSeff*sin(w*t+psids)/[R1-wM+(L1+M)p+1/pC]
Further let's split ids=idsa+idst where
idsa=unattenuated sinusoidal current
idst=attenuated sinusoidal current
If Z={(R1-wM)^2+[w(L1+M)-1/wC]^2} then :
idsa=VDSeff*sin(w*t+psids-fi)/Z where tan(fi)=(w(L1+M)-1/wC)/(R1-wM)
Don't ask me how to calculate idst! It's a very boring calculation and the result is something like this:
idst=-VDSeff/Z*exp(-alpha*t)*Function(psi,fi,L,C,w,alpha,wo)
where :
alpha= (R1-wM)/(L1+M)/2 and wo= wo=sqrt(1/(L1+M)/C-alpha^2)
Good Luck!

 

[xnuke]

7anoter4's response doesn't even remotely approach the correct way to multiply a matrix by a vector.

Why don't you examine the state-space representation provided for the RLC-loaded SEIG (eq. 3.31) in your book and leave out the last two rows in each matrix/vector? Since those rows apply to having a load on the system, they don't appear in any of the other entries, and you're trying to represent an unloaded SEIG. It looks like it would work.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[ijl]

Hint: Introduce a new state variable, the voltage over the capacitor.

 

[7anoter4]

Thank you, xnuke for your remarks. I did not recognize the simple symbolic representation of the V=Z*I equation!
So the correct development will be:
Vds=(R1+L1p+1/pC)*ids+Mp*idr
Vqs=(R1+L1p+1/pC)*iqs+Mp*iqr
Vdr=Mp*ids+wM*iqs+(R2+L2p)*idr+wL2*iqr
Vqr=-wMids+Mp*iqs-wL2*idr+(R2+L2p)*iqr
Since it is a square matrix [determinant] you don't need to linearize this but to use the determinants ratio in order to find the currents.
The main determinant is the "impedance matrix". If you change the column vector corresponding
to a certain current-for instance first column is for ids-with voltage column you'll get the nominator of the ratio and divided by "impedance determinant" you'll get the current.
Unfortunately, all the software I know needs to feed numbers not symbols. If you want to stay in the
symbols world you need to solve manually the determinants.
I think could be a mistake -instead of "w" would be "p" connected with L and M.
Also it seems to me is something wrong in the Vdr and Vqr equation.
I wander what kind of motor -or generator -it is about?
Why somebody will need to use Laplace for a usual calculation?

 

[zolka]

Helo!
When I said I wanted the state space equations, I didn’t mean to multiply the 4*4 vector with i (current).
Here is an example for the motor(attached)! On the left you see something similar to what I attached first, except here is no capacitor. On the right side you see the state space equation which is suitable for simulink simulation in my case. I don’t know how they transformed it, but I need a similar model derived from what I attached first (ind. gen.)!
I hope now you understand what my problem is.
I think if any of You have a little more experience (than I) in mathematics, this should be no problem! This is pure mathematics, I think!

http://files.engineering.com/getfil...4-d419-43fd-9d4c-c73ec9fb66ff&file=motorr.JPG

 

[ijl]

Hint: Introduce a new state variable, the voltage over the capacitor, in order to get rid of the 1/p-terms. Then solve the p-terms.

 

[electricpete]

I think that's correct. Let me try to start thinking through it.

New state variables for capacitance: Vdc and Vqc (along with the 4 currents)
d/dtVdc = C * Id.
d/dtVqc = C * Iq This assumes the same current flowing in the machine flows in the caps. If there is more to the circuit then modification is required.

Now you have Vdc and Vdq available for use in your calculations of the remaining terms. That 1/[pC] * Ids term (when the matrix is expanded) can be replaced with Vdc and 1/[pC] * Iqs can be replaced with Vcq.

If the caps were in parallel then Vdc and Vdq are also the same as Vds and Vqs. That's not a problem since they remain available for calculation.

I've got to run and havne't checked it. Can you please describe the circuit connection of the caps and is there external current to a load other than the caps?


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[electricpete]

Whoa. Those emoticons invaded my post and I didn't put them there.
Replace [pC] with [ignore][p*C][/ignore]

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[zolka]

Helo!

The capacitors are paralell with the stator windings, like on the picture!
At first, I want to assemble a model without load!

Thanx for your efforts!

Best regards

 

[zolka]

Helo!
Here is the modified model!(attached)
Question: Is it correct?
I tried to do it according to your suggestions!
Regards

 

[ijl]

No. You must have the voltage(s) over the capacitor(s) on the right hand side, as unknowns. Additionally, some 'p':s and some elements are missing in the matrix.

The idea is to use i = CdV/dt instead of V = C*integral(i) That is, write V = (R+Lp+1/pC)i as
V = (R+Lp)i+Vc, and add a new equation i = C pVc, where Vc is the voltage over the capacitor.

 

[electricpete]

For a balanaced linear system with wye connections for machine and caps, the cap voltages will be same as machine stator voltages - i.e. cap voltage can be represented by stator votlage: Vds and Vqs.

So we will end up with six state variables: Vqs, Vds, along with four currents. There is a very lot of algebra required to get it into state variable form with or without the caps.... for example to derive the form you show in your post 16 Jul 09 5:11.

My book (Krause) uses flux linkage as the state variables rather than currents. That reduces the algebra when modeling the machine connected to a fixed sinusoidal voltage source (not sure about when caps are involved).

If it is a very simple setup you are modeling,I'll bet we can find it already in the proper form in some reference book. But I can't figure out what you are trying to model since your voltage equations don't look like any I have seen. It looks to me like you are modeling in the stationary reference frame? (since there are no speed voltages in the stator variables). Is it an induction machine?



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[zolka]

Helo Electricpete!

The matrix equation in my last post is not correct, I just wanted to get rid of 1/pC, obviously I didn't do it the right way! It is suppossed to be an induction generator! The basic equation is in my first post (15 Jul 09 4:55)!