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# Voltage calculation of unbalanced three-phase line

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#### Mithun_K_Das

We know that the voltage calculation for phase to phase is line voltage x root(3).
But what formula we can use for unbalanced voltage. Such as,

Phase A: (N-P) 220V,
Phase B: (N-P) 210V,
Phase C: (N-P) 230V.

Then what will be the Line - Line voltage calculation? I mean the equation.

You can calculate the voltage under the assumption of 3x120° phase angle, but due to asymmetrical voltage drop it's most likely not fulfilled.

Hi,

If you want it to be exact, you need the phase angles information of the three line voltages. Treat them as vectors. Then subtract the vectors as you like.

For a raw estimation maybe averaging the two line voltages before multiplying with sqrt(3l is sufficient.

I don't know how you get the values of L1, L2, L3.
* It may be a DVM: Then you don't have the chance to get the angle, but you have tha chance to connect the DVM directly to L1 and L2 fir example.
Mind that a DVM is slow, it does not see fast variation, nor noise. And the voltage will vary!
So the values you see are valid only for a short time. Let's say you measured L1 first, then L2 .... can you be sure that L1 did't change?
What I want to say: if the measured voltages are not reliable and stable, then there is no need for an overly complicated calculation, because the result will not be more reliable than the inputs.

* if it is a scope: use the add/subtract function

* if it is an ADC: use synchronous sampling then subtract two input channels v1(t) - v2(t)

Klaus

Here's an example with unbalanced resistive load and Xs = 5*Rs

#### Attachments

• asymmetrical 3phase.zip
1.1 KB · Views: 156

Hi,

If you want it to be exact, you need the phase angles information of the three line voltages. Treat them as vectors. Then subtract the vectors as you like.

For a raw estimation maybe averaging the two line voltages before multiplying with sqrt(3l is sufficient.

I don't know how you get the values of L1, L2, L3.
* It may be a DVM: Then you don't have the chance to get the angle, but you have tha chance to connect the DVM directly to L1 and L2 fir example.
Mind that a DVM is slow, it does not see fast variation, nor noise. And the voltage will vary!
So the values you see are valid only for a short time. Let's say you measured L1 first, then L2 .... can you be sure that L1 did't change?
What I want to say: if the measured voltages are not reliable and stable, then there is no need for an overly complicated calculation, because the result will not be more reliable than the inputs.

* if it is a scope: use the add/subtract function

* if it is an ADC: use synchronous sampling then subtract two input channels v1(t) - v2(t)

Klaus
I do not need to calculate 100% preciously. As with the DMM, near value is enough. Using a DMM, I can measure the Line-Line voltage. But I need the equation so that I can calculate the line-line voltage measuring the Neutral-Phase voltage.

Your average method may work. But I'm sure. I did not find any equation that can calculate the V(L-L) measuring V(N-P) while the phased have different voltages. We know the equation for all same V(N-P) value.

Please review the simulation in previous post. It clarifies why the average attempt (or any similar approach) doesn't work. You get actual phase-phase voltages between 350 and 406 V for a similar situation due to phase shift.

Please review the simulation in previous post. It clarifies why the average attempt (or any similar approach) doesn't work. You get actual phase-phase voltages between 350 and 406 V for a similar situation due to phase shift.
But that we also find using DMM while measuring voltage Line to Line. How can we find the equation of that reading? Yes we know that taking average will not give actual value. But what equation we should use?

There's no simple equation, you need to find the actual phase angle between the three voltages. It can be calculated from voltage measurements, e.g. V1 = 230, V2 = 210, V12=350 gives alpha12 = 105.3 degree using cosine law.

Hi,

Please understand the the voltage is not enough of information.

You may have three equal line voltages, all = 230V.
But still unknown Line-to-line voltages depending on phase angle. In best case it is multiplied with sqrt(3).
But this is only true if the phase angles are all exactly 120°.

In FvM´s simulation you see that phase angle may change thus you may get three different line-to-line voltages.

--> you need the phase angle informations.

Klaus

just from mind: I´d use this formula (adding vectors)
V_12 = sqrt( (V1 * sin(phi1) - V2 * sin(phi2) )^2 +(V1 * cos(phi1) - V2 * cos(phi2) )^2 )

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In common 3 phase panel meters (volt) the lines and the neutral are connected to the meter. The GND of the internal circuit is common with Neutral. Or with a series resistor. And the other 3 phase lines are connected through some resistors. Simple voltage divider circuit.

But how do they calculate the phase to phase voltage just by measuring Phase to Neutral voltage? Of course, they use some kind of equation. Maybe that is not accurate enough but they measure it. And no negative cycle is measured there. If the N-Ph voltage is different, line-line voltage is also different.

Also, they use simple microcontrollers that are not suitable for complex math.

That is why I'm looking for that equation. The equation that can calculate the line to line voltage measuring neutral to phase voltage.
--- Updated ---

I don't know if it is right but as I measured with DMM, I found an equation that is exactly what we measure using DMM.

Say,
V1, V2, V3 are the Neutral to Line voltages.

Then,

Line to Line voltage:
V[1][/2] = Root([V[1]][/2] + V[2]][/2] +V[1]V[2])
--- Updated ---

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Hi,

You ignore rules for physics and math.

How can we help?

Klaus

In common 3 phase panel meters (volt) the lines and the neutral are connected to the meter. The GND of the internal circuit is common with Neutral. Or with a series resistor. And the other 3 phase lines are connected through some resistors. Simple voltage divider circuit.

But how do they calculate the phase to phase voltage just by measuring Phase to Neutral voltage? Of course, they use some kind of equation. Maybe that is not accurate enough but they measure it. And no negative cycle is measured there. If the N-Ph voltage is different, line-line voltage is also different.
If they calculate phase-to-phase voltage instead of measuring directly, they are adding/substracting instantaneous values, not rms values.

I notice that you didn't mention the project your are working on, used measurement hardware etc. Respectively it's not clear which kind of measurements is available and which calculations are feasible for you.

Hi,

You ignore rules for physics and math.

How can we help?

Klaus
No, it's your misunderstanding. I know the basic math and physics of 3phase. But you are misunderstanding me. Yes, it's complex to calculate. But I found many 3phase meters at extremely low cost. Inside the MCUs are also lower level which is not suitable for the basic math of calculating vectors, phase angles, and so on. But they are measuring. That is what I explained in my previous post. Also explained what I'm practically getting.

If they can measure, that means they are using a suitable equation to calculate. If so, then what is that?
--- Updated ---

If they calculate phase-to-phase voltage instead of measuring directly, they are adding/substracting instantaneous values, not rms values.
Does that mean at a certain moment, they are comparing the difference between two voltage signals and then calculating from that difference? As physically the MCU is not capable of measuring the negative half.
--- Updated ---

I notice that you didn't mention the project your are working on, used measurement hardware etc. Respectively it's not clear which kind of measurements is available and which calculations are feasible for you.
I just want to know how they are calculating the line-line voltage measuring the line-neutral voltage while the individual phase is not equal in voltage. Nothing else. Also, it doesn't need to be 100% accurate.

Hi
No, it's your misunderstanding. I know the basic math and physics of 3phase. But you are misunderstanding me. Yes, it's complex to calculate. But I found many 3phase meters at extremely low cost. Inside the MCUs are also lower level which is not suitable for the basic math of calculating vectors, phase angles, and so on. But they are measuring. That is what I explained in my previous post. Also explained what I'm practically getting.

If they can measure, that means they are using a suitable equation to calculate. If so, then what is that?
Maybe it´s a misunderstanding. Maybe because we still don´t know your situation. Sometimes you talk about DMM sometimes microcontroller. It is different as already written. All is already written.

Given just the values of your post #1:
The line-to-line voltages can be any value between 0V and 450V depending on phase angle. You need to understand this. There is no way to calculate line voltage without phase angle.
--> you need to know the phase angles.

But then you talk about a microcontroller. And you say they are not able to do complicated math. This simply is not true. From my experience every microcontroller I know is able to do this math. Complex math, vector math .. it´s just a matter of the software.

If you have a microcontroller then what exactly is the input to the mictrocontroller?
If they don´t get the RMS_value as input but the instantaneous_values, then the math is a s simple as "subtraction". As already written in post#3.

--> give clear informations and get detailed answers. You need to cooperate.

A sketch about your situaton is always a useful and clear information.

Klaus

I just want to know how they are calculating the line-line voltage measuring the line-neutral voltage while the individual phase is not equal in voltage. Nothing else. Also, it doesn't need to be 100% accurate.

We need to talk about instrument specifications and measurement circuits. In my view, it's yet an unsubstantiated assumption that instruments with the claimed behaviour operate without proper measurement of instantaneous values. It's just technically infeasible.
--- Updated ---

Alternatively, phase-to-phase voltages can be calculated from average rms or rectified values plus phase angle measurements. As you say, you know about ac power mathematics. So you can easily find it out yourself.
--- Updated ---

I don't know if it is right but as I measured with DMM, I found an equation that is exactly what we measure using DMM.

Say,
V1, V2, V3 are the Neutral to Line voltages.

Then,

Line to Line voltage:
V[1][/2] = Root([V[1]][/2] + V[2]][/2] +V[1]V[2])
--- Updated Today at 7:41 AM ---
You get 381 V for line voltages of 230 and 210 V, it's the correct result for phase angle of 120°. In contrast the simulation shows 350 V with the assumed unbalanced load and a resulting phase angle of 105°.

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