3 Phase Instantaneous Voltage Conversion from Line to Phase and Vis-Versa 1

[dzt]

Hi,

I have encountered three equations which convert from 3 phase line to line voltages to phase voltages. I mean the complete instantaneous voltages not the magnitude which is calculated by the square-root 3. The equations are:
Va = 2/3Vab + 1/3Vbc
Vb = 2/3Vbc + 1/3Vca
Vc = 2/3Vca + 1/3Vab

These equations were used in a 3 phase inverter software model. My problem is that I would like to know how these equations have been derived and also to derive the reverse equations from phase to line voltages. I have search books and internet but to no avail. Any help would be really appreciated, please.

 

[waross]

Va = 2/3Vab + 1/3Vbc
Vb = 2/3Vbc + 1/3Vca
Vc = 2/3Vca + 1/3Vab

In a balanced circuit Va=Vb=Vc
and
Vbc=Vca=Vca
Therefor
Va=Vb=Vc=Vbc=Vca=Vca
You have proven conclusively that the instantaneous voltage is always equal to the instantaneous voltage and that the line to line voltage is equal to the phase voltage. This is only possible if the complete circuit includes a flux capacitor.
Congratulations.

Bill
--------------------
"Why not the best?"
Jimmy Carter

 

[dzt]

Hi,
Waross, I did not really understand the last part of your post. Va, Vb and Vc are the instantaneous phase voltage waveforms and Vab, Vbc and Vca are the instantaneous line to line voltage waveforms. The magnitude is equal for the phase voltages and equal for the line voltages, but 120 degrees out of phase.
Va is given by Va = Vapeak*sin(2*pi*f)t
Vb is given by Vb = Vbpeak*sin((2*pi*f)t - 2*pi/3)
Vc is given by Vc = Vcpeak*sin((2*pi*f)t + 2*pi/3)
The same for the line voltages.

 

[waross]

Sorry, I misunderstood your equations. Electricpete can probably help. Wait and see if he posts in.

Bill
--------------------
"Why not the best?"
Jimmy Carter

 

[Skogsgurra]

It looks like those equations are used to create a reference point (zero) from the three phase voltages.

Gnuplot looks like this - a horizontal line at zero level.

http://gke.org/pub/files/sumofsines gnuplot 1.png

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[dzt]

The equations are used to obtain the phase voltage waveforms from the line voltage waveforms. What I cannot figure out is how are those equations derived mathematically. And how to derive other equations to obtain the line waveforms from the phase waveforms.

 

[electricpete]

I haven't seen that form before.

I was able to recreate your equations (see attached) by solving 4 equations below for the three unknowns (Va, Vb, Vc):
Eq1:=Vab=Va-Vb;
Eq2:=Vbc=Vb-Vc;
Eq3:=Vca=Vc-Va;
Eq4:=Va+Vb+Vc=0

The first three equations are simply the definitions of Vab, Vbc, Vca in terms of phase voltages (referenced to any neutral you choose… doesn’t matter).

The fourth equation seems somwhat system dependent. One situation where it seems it might arise is a power system which is ungrounded except at the neutral. If the three phase-to-ground capacitances are similar, then the phases would float such that the sum of the phase to ground (=phase to neutral in this case) voltages are zero.


=====================================
(2B)+(2B)' ?

 

[Skogsgurra]

Nor can I

I just put the equations through gnuplot to see if something useful came out of it. It didn't.

I had two thoughts:
1. If your system is a PWM system, which it probably is, then it is not always possible to have a good zero in the switched system. But I am not so sure that the equations shown will help either.
2. If there were third tones, it is a common method to get a higher line-line voltage than would be possible otherwise (third tone injection). But I don't see any third tones.

We hope that CSWilson or Electricpete sees this.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[electricpete]

Revisiting my previous post, equations 1,2,3 are not independent (a linear combination of these equations sums to zero), so eq1,2,3 represents only two independent equations.

In order to find a solution for Va, Vb, Vc, it requires one more equation. The required additional equation to give the equations of op is Equation 4. Equation 4 represents an assumption. The assumption would be true as I outlined above for otherwise ungrounded system where the measurement[.b] neutral (neutral of three phase-neutral-connected PT's) is connected to ground and the system capacitances to ground on each phase are identical. A simpler case where it would also be true is a balanced grounded-wye system.

=====================================
(2B)+(2B)' ?

 

[dzt]

Hi Electricpete and Skogsgurra,

Yes, the equations are used in a PWM context. Basically the equations are used in Simulink (Matlab). The line to line voltages are measured from the 3 phase inputs of a rectifier modeled in Simulink. These line to line voltage waveforms are then given as inputs to the three equations above to obtain the phase voltage waveforms. These phase voltages are then converted to the alpha-beta frame to continue with the control system of the rectifier.

 

[electricpete]

Without pondering your specific situation, I'll add one more condition that satisfies equation 4.
Simply define the voltage reference (which Va, Vb, Vc are defined respect to) as
Vref = (Van+Vbn+Vcn)/0.

In this case, equation 4 amounts to a choice of voltage reference for specifying Va, Vb, Vc.


=====================================
(2B)+(2B)' ?

 

[Skogsgurra]

So, Bill was on the right track in his 8 Dec 14 13:41 post. The result is nil, zero, zilch and still some.

BTW, Pete, I was busy plotting so I wasn't aware of your post.

What did we learn?

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[sibeen]

Hmm, if you are looking at PWM switching techniques are you looking at Space Vector modulation? If so you may need to be looking at Park's transformation for Space Vector control.

 

[electricpete]

What did we learn?

My take is that those equations are true whenever the reference Vr for measuring phase voltages satisfies Var+Vbr+Vcr = 0
(where we used standard notation: Var = Va-Vr, Vbr = Vb-Vr, Vcr = Vc-Vr).

This can always be satisfied by selecting Vr = (Va+Vb+Vc)
…where all four measurements Vr, Va, Vb, Vc are referenced to any single arbitrary reference.

If the voltage reference selected were ground, then I think it’s the same as saying the common mode voltage is zero.


=====================================
(2B)+(2B)' ?

 

[Skogsgurra]

Agree. So we didn't learn anything new, then.

I am not satisfied. Why those cryptic formulae if they don't have a special meaning? Just asking.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[electricpete]

To sum up my rambling comments:

The op equations provide a mapping from phase-to-phase voltages to phase-to-reference voltage.

No mapping is possible without making an assumption about either the reference voltage or the system (equations 1 through 3 are not independent and cannot be solved for 3 unknowns without an additional equation / assumption).

The assumption required for this particular mapping is that the sum of the phase voltages with respect to their reference is zero. Or that common mode voltage is zero.

=====================================
(2B)+(2B)' ?

 

[dzt]

Hi,

First of all thanks to all for the information given. I have noticed that some did not manage to obtain waveforms from the equations I have provided. So I modeled the waveforms in Simulink. I have generated the three balanced phase voltage waveforms (Va, Vb, Vc) and obtained the line to line voltage waveforms (Vab, Vbc, Vca). Then I have given as inputs the three line to line waveforms to the equations and re-obtained the three balanced phase voltage waveforms (Va, Vb, Vc). So it seems that the equations or correct but still I did not figure out, till now, how to derive them.
The below figures show the Simulink model and the resulting waveforms. Top waveforms are the input phase voltages, the middle waveforms are the line to line voltages and the bottom waveforms are the output.

 

[dzt]

The images I am talking about in my previous post are the following. I did not manage to upload two images in the post.

http://files.engineering.com/getfil...ea3e01cff3db&file=Voltages_Function_Model.png

http://files.engineering.com/getfil...af11-4699-806b-c0e153eb8fd5&file=Voltages.png

 

[LiteYear]

The equations are true provided the three phase voltages are balanced, in both phase and magnitude. That is, Va = V<0°, Vb = V<-120°, Vc = V<120°, where '<' is the polar angle symbol.

This is easily verified by linear vector operations. I'll illustrate the first one:

LHS = Va = V<0°.
RHS = (2/3)*Vab + (1/3)*Vbc
= (2/3)*(V<0° - V<-120°) + (1/3)*(V<-120° - V<120°)
= (2/3)*(sqrt(3)V<30°) + (1/3)*(sqrt(3)V<-90°)
= (2/sqrt(3))*V<30° + (1/sqrt(3))*V<-90°
= V<0°
= LHS

The others follow similarly.

How have they been derived? Well since Vab, Vbc and Vca are linearly independent, a linear combination of any two of them can form any other vector. Then it's just a matter of picking combinations Va = x*Vab + y*Vbc, etc., and solving for x and y. Turns out that those combinations produce quite a neat set of equations.

Perhaps where the conceptualisation has gone wrong is to overlook the fact that Va, Vab, etc. are all vectors. Thoughts of PWM, sine waves and common mode voltages is over-complicating it. That's the great thing about vectors - they insulate us from such distractions!

 

[Skogsgurra]

"they insulate us from such distractions!"

Yes, right.

But then you must make sure that those "distractions" are not going to show their nasty faces when you try to use the math IRL. The distractions are very real in a set of PWM motor voltages. There, the sum actually never is zero.

In this case, I still don't understand why all the fuzz about something so self-evident. And, as I said before: Bill (8 Dec 14 13:41) was correct in pointing out, somewhat sarcastically, that the OP proves that the sum of three sines 120 degrees apart is zero.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.