What does the 'power' vs 'time' graph look like for a 3 phase AC supply?

[k.udhay]

TL;DR Summary

An answer in Quora said that the power curve of a 3 phase supply is a flat line (like a DC supply). Want to see if there is a mathematical proof for it.

Hi,

For years I wanted to understand why we have a 3 phase supply and not a 2 phase supply in AC. In Quora I found an interesting answer and was convinced about the purpose mentioned:

Answer to Why there is no two phase electrical inputs instead of three phase and single phase? by Paul Grimshaw https://www.quora.com/Why-there-is-...&share=a178a9bb&srid=uXqtw&target_type=answerIn short, a 3 phase supply can give a constant power output while maintaining the net current at any point of time as zero. I was checking in google for an evidence to claim that the net power of a 3 phase supply is a constant value, unfortunately I failed to find one.

Can someone pl. confirm that the power output is indeed a flat line with perhaps a calculation? Thank you!

 

[Baluncore]

Can someone pl. confirm that the power output is indeed a flat line with perhaps a calculation? Thank you!

That is correct for 3PH with a balanced resistive load.
The power from each line falls to zero at each zero crossing of the voltage.
The power delivered by each line is an offset sinewave at twice the frequency.
The sum of the power in the three lines has a constant value.

 

[k.udhay]

That is correct for 3PH with a balanced resistive load.
The power from each line falls to zero at each zero crossing of the voltage.
The power delivered by each line is an offset sinewave at twice the frequency.
The sum of the power in the three lines has a constant value.

Hi @Baluncore - Thanks for your explanation. Do you know a website where this is shown with some pictures?

 

[Baluncore]

This is for 400 V 3PH = 230 Vrms per phase. Load is 100 ohm per phase

Interpret horizontal ms as phase in degrees.
Magenta is red line power. Green is total power.

 

[k.udhay]

This is for 400 V 3PH = 230 Vrms per phase. Load is 100 ohm per phase
View attachment 303856

Interpret horizontal ms as phase in degrees.
Magenta is red line power. Green is total power.
View attachment 303857

Thank you!

 

[k.udhay]

This is for 400 V 3PH = 230 Vrms per phase. Load is 100 ohm per phase
View attachment 303856

Interpret horizontal ms as phase in degrees.
Magenta is red line power. Green is total power.
View attachment 303857

Hi Baluncore,

I also just did a small plot using a website called "desmos" and found the understanding to be correct:

Thank you!

 

[Baluncore]

I also just did a small plot using a website called "desmos" and found the understanding to be correct:

Oh ye of little faith.
You might download a free copy of SPICE and run the simulation for yourself.

https://en.wikipedia.org/wiki/LTspice

https://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html

 

[DaveE]

The math looks like this:

 

[Windadct]

There are certainly a few caveats - esp considering how the question was asked: Primarily this requires a symmetric and linear LOAD, the definition of "AC Supply" will not fully define the power delivered.

 

[anorlunda]

There are certainly a few caveats - esp considering how the question was asked: Primarily this requires a symmetric and linear LOAD, the definition of "AC Supply" will not fully define the power delivered.

The word ideal should suffice.

 

[Windadct]

I only wanted to make this point because I have come across many EEs, even with advanced degrees, that seem to have an "ideal" model in their heads, accepted as true in all cases.

 

[waross]

Sorry, it is the voltage that sums to zero, not the current.

If the voltage does not sum to zero, there will be circulating currents in delta windings of motors and transformers.
An unloaded delta transformer may overheat to the point of failure by as little as 10% difference in supply phase voltages due to the circulating currents caused by the unbalanced voltages.
The current depends on the load on each phase and often does not sum to zero.
Look at a graph of three phase voltages such as that posted by k.udhay. Post #6
Three sine waves displaced by 120 electrical degrees.
Draw a vertical line anywhere on the graph and scale the three voltages. They will sum to zero.

In the real world, you may not have sine waves.
Many generators do not produce a perfect sine wave of voltage.
An unloaded wye/wye transformer bank will develop a horrendous wave form.
It has been many years since I saw this on a scope, but comparing the phase to phase voltage with the phase to neutral voltage one of the voltages is badly distorted.

How bad?
I was our company representative at the start-up of a flood water pumping station.
One engineer wanted to check the voltages before energizing the pumps.
The phase to neutral voltage and the phase to phase voltages were definitely not in the in the accepted ratio of 1.73:1 Not even close.
This was a long time ago when true RMS meters were very rare. The Voltmeters in common use measured average values. A form factor was used to indicate RMS values rather than average values. (Form factor: The ratio between the average value of a wave form and the RMS value of a wave form. In this case a sine wave form.)

The form factor is dependent on the type of wave form. RMS, square, saw tooth etc.
The form factor of the distorted wave form was enough different to the form factor of a sine wave that the error was readily apparent.
I had to explain this to three engineers before they would start the pumps.
As soon as a load was placed on the transformer bank, the Voltmeter indicated the expected voltage.

 

[Baluncore]

Sorry, it is the voltage that sums to zero, not the current.
If the voltage does not sum to zero, there will be circulating currents in delta windings of motors and transformers.

The current depends on the load on each phase and often does not sum to zero.

The same number of electrons must flow backwards as flow forwards.
Unless there is a ground fault, the currents in the three lines must sum to zero.
When a neutral is present, the three lines and neutral must sum to zero.
The regional magnetic field, due to the total transmission line current, will be zero.

 

[waross]

90 Degree two phase versus 120 degree three phase.
First there are several ways to deliver two phase power:
Three wire systems.
Four wire systems.
and
Five wire systems.
A three wire two phase system has a common wire that carries 1.414 more current than the other two wires.
A three phase system will transmit a given amount of power with less copper than a three wire two phase system.
Four and five wire systems require even more copper to transmit a given amount of power.