Mean square current vs. Average current

[kmarinas86]

Let's say the current in an inductor goes up like this:

amps
0
6.32
8.64576
9.50163968
9.816603402
9.932510052

And down like this:

amps
10
3.68
1.35424
0.49836032
0.183396598
0.067489948

With an average of 5 amps

The squared current (and therefore instantaneous power) rises as:

amps^2
0
39.9424
74.74916598
90.28115661
96.36570236
98.65475593

And drops as:

amps^2
100
13.5424
1.833965978
0.248363009
0.033634312
0.004554893

So the mean square of the amperage is 42.97134159 amps^2 [NOT RMS].

We have two equations:

Transmitted Power=VI
Power Loss=RI^2

Applying these, we would have the following for average:

Transmitted Power=V*5 amps
Power Loss=R*42.97134159 amps^2

If Average transmitted Power > Average power Loss:

V*5 amps > R*42.97134159 amps^2

V/R > 8.594268318 amps

I > 8.594268318 amps

Which is not the case...

Please help.

 

[The Electrician]

I suggested in another of your posts that you would help those who would help you if you would provide a schematic and a good description of where you get your numbers.

In this case, I think I can reverse engineer your problem. It appears that you have an ideal inductor in series with a resistance, and you have applied a voltage for a period of time, and then you have applied zero volts for the same time period.

The numbers are consistent with a 1 henry inductor in series with a 1 ohm resistor, with an applied voltage of 10 volts. Of course, it could also be 2 henries, 2 ohms and 20 volts, etc., but I'll stick with 1 henry, 1 ohm, and 10 volts.

It looks like you applied the 10 volts for 5 seconds, and then applied 0 volts for 5 seconds, taking measurements at intervals of 1 second (1 time constant).

Your numbers look ok, but when you get down to here you make an error of interpretation.

We have two equations:

Transmitted Power=VI
Power Loss=RI^2

Applying these, we would have the following for average:

Transmitted Power=V*5 amps
Power Loss=R*42.97134159 amps^2

If Average transmitted Power > Average power Loss:

V*5 amps > R*42.97134159 amps^2

V/R > 8.594268318 amps

I > 8.594268318 amps

Which is not the case...

Please help.

You need to remember that V is the applied voltage, and if you substitute 10 for V, your inequalities will look like this:

10 volts * 5 amps > R*42.97134159 amps^2

10 volts/1 ohm > 8.594268318 amps

10 amps > 8.594268318 amps

The problem is that you have interpreted the I on the left side of your inequality:

I > 8.594268318 amps

to be the average current. It isn't the average current; it's the (constant) current that would flow in the 1 ohm resistor with 10 volts applied if there were no inductor.

 

[kmarinas86]

I suggested in another of your posts that you would help those who would help you if you would provide a schematic and a good description of where you get your numbers.

In this case, I think I can reverse engineer your problem. It appears that you have an ideal inductor in series with a resistance, and you have applied a voltage for a period of time, and then you have applied zero volts for the same time period.

The numbers are consistent with a 1 henry inductor in series with a 1 ohm resistor, with an applied voltage of 10 volts. Of course, it could also be 2 henries, 2 ohms and 20 volts, etc., but I'll stick with 1 henry, 1 ohm, and 10 volts.

It looks like you applied the 10 volts for 5 seconds, and then applied 0 volts for 5 seconds, taking measurements at intervals of 1 second (1 time constant).

Your numbers look ok, but when you get down to here you make an error of interpretation.



You need to remember that V is the applied voltage, and if you substitute 10 for V, your inequalities will look like this:

10 volts * 5 amps > R*42.97134159 amps^2

10 volts/1 ohm > 8.594268318 amps

10 amps > 8.594268318 amps

The problem is that you have interpreted the I on the left side of your inequality:

I > 8.594268318 amps

to be the average current. It isn't the average current; it's the (constant) current that would flow in the 1 ohm resistor with 10 volts applied if there were no inductor.

Wow that's a great answer. Amazing. Thanks :D

 

[kmarinas86]

I have another question.

Let's consider the following:

From t=0, to t=1
I=3 amps

From t=1, to t=5
I=-1 amp

The average of I as a function of t is -0.2

A net current in the backwards direction.

The average of I^2 * abs(I)/I is 1

Which is the same result we would get if I=1 for all t.

My biggest question is, does magnetic field energy correspond to I^2 * abs(I)/I (that is, the current squared and its either clockwise or counter clockwise rotation)? If so, it appears that energy of given magnetic field can correspond current regardless of the direction of the net current. If so, then what gives?

 

[The Electrician]

It's true that the energy in a static magnetic field doesn't depend on the polarity of the field.

I don't know what you're asking when you say, "what gives?".

 

[kmarinas86]

It's true that the energy in a static magnetic field doesn't depend on the polarity of the field.

I know, but I still don't know what term would indicate both the energy of the magnetic field and it's polarity (+ or -) simultaneously. Is there one? Because I believe this has a relationship with the rotor, and polarity certainly has an effect on that.

 

[kmarinas86]

I don't know what you're asking when you say, "what gives?".

When I say "what gives", I mean that I can't justify why time average current can go one way yet the time average magnetic field energy can have the opposite polarity than if current was constant in the same direction the net current is. Maybe because it's false - I don't know.

 

[NoTime]

The squared current (and therefore instantaneous power) rises as:

This might be the source of your conceptual problems.

It is not power. You might call it stored energy, but not power.
There is a difference.

Power=EI.
In this case you need to account for the phase angle of the voltage with respect to current.
When you do account for the phasing then you will find that EI=0.

 

[The Electrician]

I know, but I still don't know what term would indicate both the energy of the magnetic field and it's polarity (+ or -) simultaneously. Is there one? Because I believe this has a relationship with the rotor, and polarity certainly has an effect on that.

The energy in a static (or varying slowly enough that radiation effects are negligible) magnetic field is proportional to the volume integral of B dot H over the volume where the field exists. It is a scalar and has no direction or polarity.

On the other hand, forces associated with a magnetic field do have direction.

 

[The Electrician]

When I say "what gives", I mean that I can't justify why time average current can go one way yet the time average magnetic field energy can have the opposite polarity than if current was constant in the same direction the net current is. Maybe because it's false - I don't know.

For the kind of situations you've been describing the "time average magnetic field energy" has no polarity. It's just a scalar; a positive number.