Center-tap Transformer Voltages

[mivey]

Concerning the nature of voltages, directions, and electromagnetic induction.

There have been a few threads discussing the directions used for positive voltages. Of particular interest were the smaller voltages taken from the center-tapped winding of a single-phase transformer.

Some contend that the positive direction for both voltages must be in the same direction and that using the center-tap as a reference point does not produce "real" voltages. Several reasons are given, including that it is easier, that the single flux in the core mandates it, and that doing otherwise will result in breaking electrical laws, physics laws, and will result in voltages that are math tricks instead of real voltages.

I contend that the positive direction is arbitrary because: By definition, the reference point of a voltage is arbitrary. Also, the "positive" direction of an AC voltage is not defined in one direction and it is we who arbitrarily pick a positive and negative direction.

I contend that we can pick either direction as positive and that doing so will result in real voltages without breaking any laws of electrics, physics, etc.



Some basics:

Potential energy and zero reference.
Potential energy is the capacity to do work. Potential energy is an integral function of force and as such, has a constant of integration. This constant can be set to any value needed to establish a desired zero point for the potential energy. Because this constant is arbitrary, the point of zero potential is completely arbitrary, although we usually pick the zero point by convenience or some logical choice. Because the point of zero potential is arbitrary, it is the difference in potential that has physical meaning to us.


Electric potential and zero reference
Electric potential energy per unit charge is also called electric potential and is measured in volts. Voltage is the difference in electric potential between two points. If an electric field exists between points A & B, and is directed from A to B, the electric potential rise from B to A is equal to the work it would take to move a positive unit charge from point B to point A. Because of the arbitrary constant of integration, the point of zero electric potential is completely arbitrary, although we usually pick the zero point by convenience or some logical choice.


Instantaneous direction for an electric field.
An electric field exerts a force on both positive and negative charges. We call the force that the field exerts on a positive charge to be the positive force direction, and thus the direction of the electric field. With this definition of the electric field direction, we say an electric field is radially outward from a positive charge and radially inward towards a negative charge.


Direction for an AC field.
The voltage with the AC we are most familiar with is represented by an alternating sinusoidal wave and the direction of voltage rise changes every 1/2 cycle. A point charge in the alternating electric field will experience a force in one direction for 1/2 cycle, then a force in the opposite direction for 1/2 cycle. The zero reference point for the measurement of the electric potential for that point charge is arbitrary.

For this potential, the capacity to do work increases to a maximum, then decreases back to zero during each 1/2 cycle with no set positive or negative direction for the full cycle. A positive or negative direction is a choice that we make.

We can attempt to use the instantaneous direction given by magnetic induction to say that one direction is positive and one direction is negative. That is mis-representing what the directions from induction are telling us because those directions only tell us what is happening with the forces at any given instant. The magnetic induction will tell us that the instantaneous forces in the winding are all "in-phase" and creating forces in the same direction. It does not tell us what direction is permanently defined to be positive or negative.

There are no laws that define what direction we have to use as a positive voltage direction and what direction we have to use as a negative voltage direction for all time intervals.

There are no laws that tell us what point is the "one true reference point" for a voltage. By its very nature, voltage is a relative value and the zero reference is completely arbitrary.

More on "the laws" next.

 

[mivey]

Now on to what "the laws" really tell us.

Instantaneous magnetic field directions.
Magnetic fields can be produced by magnets. By convention, the direction of the magnetic field is from the North pole to the South pole of the magnet. Magnetic fields can also be produced by electric currents. By convention, the direction of the magnetic field is given by the direction of current according to the right hand curl rule (see next).


Currents, magnetic field lines, and right hand curl rule.
Current in a wire creates a magnetic field with field lines at each point along the wire in circular rings (assuming no interference) around the wire. At each point along the wire, these rings are in a plane that is perpendicular to the wire.
The direction of the magnetic field lines in these perpendicular circles are directed such that if you gripped the wire with your right hand, with your thumb extended in the direction of current, your fingers would curl in the direction of the field. We also call these lines of force by another name: flux.

Electromagnetic induction by conductor motion.
If a conductor is moved lengthwise through a magnetic field, the magnetic field lines of force are cut by the conductor. The magnitude of the voltage is determined by the strength of the field, how fast the lines of force are cut, the angle of the conductor relative to the angle of the field, and the length of the conductor cutting the field lines. The cutting of field lines creates a voltage in the direction determined by the right hand rule.

The right hand rule.
The right-hand rule uses your right hand and fingers to map the relationship between the motion of the conductor, the direction of the force lines, and the direction of the induced voltage.

With the wrist held straight, the traditional right-hand rule (Fleming's Right Hand Rule) has the middle finger pointed in the direction that the palm is facing, the index finger points straight out from the wrist in a direction 90? to the middle finger, the remaining fingers curl into the palm, and the thumb extends away from the palm and 90? to both the index and middle finger (like the x, y, and z axis directions). With the index finger pointing in the flux direction, and the thumb pointing in the direction of conductor motion (or equivalent motion), the middle finger will point in the direction of voltage rise.

I find the three-finger technique is not as easy to produce in some orientations as the flat-hand technique.

With the flat-hand technique, you hold your right hand with the fingers straight out with your palm and fingers defining a plane. The thumb is extended out from the palm but in the same plane as the palm, like you were approaching someone for a handshake. The fingers point in one direction of the plane, and the thumb extends in the other direction of the plane.

Let the magnetic field hit your palm in a perpendicular direction like you were going to "catch the flux". Then let your thumb point in the direction of the wire's motion through the field (perpendicular to the conductor's length), then the fingers will point in the direction of the voltage rise in the conductor.


Electromagnetic induction by flux motion (transformers).
Another method of induction is to have the conductor in a fixed position but let the magnetic field expand and push through the conductor. If we increase the current in a conductor, the strength of the flux also increases.

Suppose we have a conductor located next to a current carrying conductor and we have the current increasing and decreasing. As the current increases and decreases, so does the flux. As the current (thus flux) increases, flux lines expand out from the current-carrying conductor and push through the second conductor. As the current (thus flux) decreases, the flux lines contract and pull back through the second conductor.

A voltage is produced in the second conductor because the action is the same as if we moved the second conductor through a fixed magnetic field as described above. If we use the right hand rule as before, with the relative motion of the conductor cutting the flux lines being the direction of motion, our palm again points in the direction of rising potential. The magnitude of the voltage is determined by the strength of the field, how fast the lines of force push through the conductor (flux wave frequency), the angle of the conductor relative to the angle of the field, and the length of the conductor cutting the field lines.

The force needed to induce a voltage in an open circuit
Without a current in the "induced" conductor, there is no significant energy required to create the voltage force (ignore eddy currents, motion friction resistance, etc). This would be the case with an open circuit. With an open circuit, no current exists to create magnetic fields and poles, and without the opposing magnetic poles, there are no competing forces of attraction or repulsion between poles.

The force needed to induce a voltage in a current-carrying circuit (Lenz's Law).
If the conductor in which we are trying to induce a voltage is carrying a current (like with a closed circuit) then it takes a force to move the conductor through the fixed flux. The reason is the following:

When we move a closed-loop conductor through the fixed field, a voltage is created and this voltage will cause a current to flow in the closed loop. This current in the conductor then creates a magnetic field.

The magnetic field lines at each point of a current-carrying conductor are in a plane that is perpendicular to the length. If this current-carrying wire is moved perpendicular through a fixed magnetic field (or, alternatively, the flux pushes through the conductor), the field lines produced by the current will interact with the fixed magnetic field and create a force that tries to resist the change.

The force is directional at that instant because the magnetic field produced by the perpendicular current has field lines on one side of the conductor that are in the same direction as the fixed field lines, and field lines on the other side of the conductor that are in a direction opposite of the fixed field lines. Therefore, this interaction will be different on one side of the conductor than it will be on the other. This unequal force interaction distorts the lines of flux and creates a force that tries to straighten the flux lines.

In Lenz's words, the induced current has a direction such that its magnetic field will oppose the motion that created the current (Lenz's Law).

The next topic is polarity.

 

[mivey]

Now let's discuss polarity.

What happens when two "in-phase" voltages are combined?
Each 1/2 cycle, each voltage will reach a maximum capacity to do work. While the direction of force is arbitrary, we have picked positive directions for both voltages such that the direction we call "positive" maximum capacity for both sources is in the same direction.

We have defined our voltages such that the rise in potential in each conductor is in the same direction, and the capacity to do work will double across both sources. That is what polarity is about: making sure that we align our forces like we want them.


So what happens when two voltages are combined that are out of phase by 180??
Each 1/2 cycle, each voltage will reach a maximum capacity to do work. While the direction of force is arbitrary, we have picked positive directions for both voltages such that the directions we call "positive" maximum capacity are in opposite directions.

We have defined our voltages such that the potential rises in each conductor are in opposite directions. However, for the "opposite" voltage, the capacity to do work in the "negative" direction is still there. This "negative" capacity to do work will combine with the "positive" capacity to do work of the other voltage. The capacity to do work will double across both sources. Again, that is what polarity is about: making sure that we align our forces like we want them.


Each two-terminal AC source can have a "positive" direction assigned because direction is arbitrary. That goes back to the fundamental definition of electric potential. The voltage produces a force in both directions and it is our assignment of "positive" and "negative" directions that make the difference. Polarity only helps us to align the forces, whether or not we have called them positive or negative.

Why is a 180? difference in the opposite direction not the same as taking the negative of a DC voltage like a battery?
A constant DC voltage does not change direction. The capacity to do work always increases in one direction. Because the electric potential always increases in one direction, there is no phase associated with the DC voltage.

The AC voltage, on the other hand, has a change in direction every 1/2 cycle. The AC voltage has a phase in addition to the assigned positive and negative.

Let's place ourselves between two points that have a difference in potential.

Suppose we look in the direction of a voltage rise on a DC source. Then suppose we turn and look in the opposite direction. In that opposite direction we will always see a decrease in electric potential.

Suppose we look in the direction of a voltage rise on an AC source. Then suppose we turn and look in the opposite direction. In that opposite direction, at that instant, we will see a decrease in electric potential. But, if we wait until 180? later, we will see an increase in electric potential. We can make use of the voltage rise in either direction because the choice of direction of voltage rise is just that: a choice.

We can prove this by combining two sources with the positive voltage rises defined in opposite directions (two voltage sources with a 180? displacement). The force created by the fall of one voltage combines with the force created by the rise of the other. The currents and flux created by these will be the exact same as those created by combining two sources with positive voltage rises defined in the same direction.

It is our assignment of positive and negative that is arbitrary because the AC source has no defined positive and negative direction.

And finally...

 

[mivey]

If you want to think of it like vehicles:

Consider two cars racing down a track and then turning around and racing back. Let this continue. You can put devices on either side of the track to take force from the vehicles. The direction you pick as "positive" is completely arbitrary.

You can take a left-to-right force from both vehicles on the same side of the track. This would be calling the direction of both vehicles positive in the same direction.

You can take a left-to-right force from one vehicle on one side of the track and a left-to-right force from the other vehicle from the other side of the track. This would be calling the direction of the vehicles positive in opposite directions.

Our assignment of direction is arbitrary and does not impact the creation of the force. In other words, we can assign the directions either way and we will not violate any electric laws or physics.

Either direction is valid and produces real voltages. If there is anything that might be called "not real" it is the assignment of one direction in the circuit to be positive and one direction to be negative. The assignment of this arbitrary direction does not invalidate the voltages that already exist by physical reality.

 

[mivey]

...

Electromagnetic induction by flux motion (transformers).
...If we use the right hand rule as before, with the relative motion of the conductor cutting the flux lines being the direction of motion, our palm again points in the direction of rising potential.

should read "...our fingers again point in the direction..."

 

[rattus]

Here we go again!

Here we go again!

Mivey, you trying to start an argument?

Well, I am with you. It never occurs to me to consider the flux in a xformer when defining a voltage.

It is conventional in electronic circuits to define the supply voltages referenced to "ground". So we quite often end up with a positive supply and a negative supply. Why not do the same with AC? Then we have two voltages one the inverse of the other. That is they are 180 degrees out of phase with each other. To argue otherwise or some arcane reason is just plain silly.

 

[mivey]

Mivey, you trying to start an argument?

Nope. Just trying to finish several.

These arguments have been spread over several threads. Some threads have closed so I can't respond there. In others, some posters would rather not follow the logic of my counter-arguments but write them off as non-related. Even though my arguments encompass things beyong a single winding, and are not restricted to some silly DC comparison, my arguments are related to the voltages in the center-tap winding.

It never occurs to me to consider the flux in a xformer when defining a voltage.

There are some here that say if you do not consider the flux, you can wind up with voltages that are not real but just mathematical models. They contend that the voltages only appear to be real because of what your meter is telling you, or some such nonsense. Hard to believe, but these are comments from posters who should know better.

It is conventional in electronic circuits to define the supply voltages referenced to "ground". So we quite often end up with a positive supply and a negative supply.

That's why I don't get the fuss. Ground is probably the most common reference used in the world. I understand it is perfectly fine to use a different reference, but to say one is valid and the other is not is just crazy talk.

Why not do the same with AC? Then we have two voltages one the inverse of the other. That is they are 180 degrees out of phase with each other. To argue otherwise or some arcane reason is just plain silly.

You would think, but it happens anyway.

 

[jjkind]

Some contend that the positive direction for both voltages must be in the same direction and that using the center-tap as a reference point does not produce "real" voltages. Several reasons are given, including that it is easier, that the single flux in the core mandates it, and that doing otherwise will result in breaking electrical laws, physics laws, and will result in voltages that are math tricks instead of real voltages.

I contend that the positive direction is arbitrary because: By definition, the reference point of a voltage is arbitrary. Also, the "positive" direction of an AC voltage is not defined in one direction and it is we who arbitrarily pick a positive and negative direction.

I contend that we can pick either direction as positive and that doing so will result in real voltages without breaking any laws of electrics, physics, etc.

I agree with Mivey's contention that the reference point for a voltage is always arbitrary, in the same way that 10 - 7 = 3 just like 3 - 0 = 3.

I think you have to be careful, however, when you say that "positive direction is arbitrary. It is arbitrary, but not in the same way the voltage amplitude reference point is arbitrary.

Here is a quote from the Wikipedia page defining voltage or electric potential (used synonymously): "The voltage between two ends of a path is the total energy required to move a small electric charge along that path, divided by the magnitude of the charge. Mathematically this is expressed as the line integral of th electric field and the time rate of change of magnetic field along that path."

Voltage is path-dependent. Voltage is a scalar quantity (not a vector, like force or magnetic field) with positive defining potential to move an electron in one direction along the path and negative defining potential to move an electron the other direction along the path. Let's call these directions D1 and D2. It isn't important if you call the voltage that moves an electron towards D1 positive or negative - you just have to account for the direction and differentiate it from a voltage that moves an electron towards D2. Zero voltage is defined when the magnetic flux stops changing (magnetic field reaches its peak and starts decreasing in absolute strength). This makes sense, since Faraday's law says that the partial differential of magnetic flux with respect to time is equal to voltage. When this differential is zero, it means that magnetic flux reaches an inflection point.

So how is a reference point for voltage always arbitrary?

Well, let's start by looking at the single-phase, center-tapped transformer that you mentioned. Let's call the center tap X2 and the other two taps X1 and X3. We know that the voltages between X1-X3 and X2 are 120V and the voltage between X1 and X3 is 240V. We could easily add another tap - X4 - between X2 and X1 and define all of the voltages from that point. Doing this we could change the maximum amplitudes of the voltage (which depends on the number of wire turns between the two points being referenced), but we could not change the relative voltages (as they vary from 0-100% of maximum voltage). These windings are connected by a single core (that defines the system or path that we are analyzing) and the relative voltage depends on the amplitude of the magnetic flux field in that core that induces said voltage. A negative voltage defines flux either growing/decreasing and a positive voltage is defined by the opposite, and again, it's not important if we call 'growing' magnetic flux positive and 'decreasing' magnetic flux negative or vice versa.

So I repeat that you have to be careful when you say that positive is arbitrary (voltage direction is not arbitary, persay). Deciding what you call 'positive' and what you call 'negative' is one thing - 0 volts in an AC system is going to occur when the flux (related to magnetic field and a vector quantity) stops changing.

Hopefully this makes sense...

As a side note, I find it very confusing working with ICs in circuits that require negative voltage supplies. If you don't have a negative voltage reference, you have to create a "virtual ground" to give you a negative voltage, but you definitely cannot use this virtual ground as a normal ground for other components on a board, as it has a potential when compared to an earth ground....unless you try to define everything based on your virtual ground, which can be very confusing.

 

[mivey]

I think you have to be careful, however, when you say that "positive direction is arbitrary. It is arbitrary, but not in the same way the voltage amplitude reference point is arbitrary.

Not arbitrary for all voltages. It is not arbitrary for DC voltages (that is why the straight DC comparison does not work). With an AC voltage, the "positive" direction is completely arbitrary because force directions change every 1/2 cycle.

Here is a quote from the Wikipedia page defining voltage or electric potential (used synonymously): "The voltage between two ends of a path is the total energy required to move a small electric charge along that path, divided by the magnitude of the charge. Mathematically this is expressed as the line integral of th electric field and the time rate of change of magnetic field along that path."

Voltage is path-dependent.

Voltage is location dependent, not general path dependent. You can take any path between the two ends. I think you mean the path in a straight line, or something like that.

...It isn't important if you call the voltage that moves an electron towards D1 positive or negative - you just have to account for the direction and differentiate it from a voltage that moves an electron towards D2.

That is the instantaneous voltage. The instantaneous voltage has a force direction defined like we would define it for a DC voltage. The direction of force at any instant is not arbitrary because it is a physical phenomenon and the direction is based on our conventions for current flow, etc. We do not change our conventions in each 1/2 of the cycle. What does change is the direction of the physical phenomenon.

We can look at the instantaneous direction of force. That is what we do when we consider polarity. The instantaneous direction of voltage force is in the same direction for all segments along the length of the winding.

What we do not have by looking at the instantaneous direction is a definition of a positive direction for both halves of the cycle. For any AC voltage, the direction changes and we can pick either direction to be the "positive" direction for both cycle halves because that designation is arbitrary.

So I repeat that you have to be careful when you say that positive is arbitrary (voltage direction is not arbitary, persay). Deciding what you call 'positive' and what you call 'negative' is one thing - 0 volts in an AC system is going to occur when the flux (related to magnetic field and a vector quantity) stops changing.

It appears you are saying, and I would agree, that the instantaneous direction of force, and its relative direction to other forces at that instant, is not arbitrary. That is what we consider when we consider polarity.

The directions at any instant are indeed defined by the physical reality. What does not match physical reality is when we pick a positive direction for all times. The labeling of permanently positive and negative directions is a choice. We can make these choices for convenience or some logical reasons, but they are still choices we have to make. These choices are not made for us by the physical world because the physical world shows both directions experience a "positive" force.

 

[gar]

111109-1041 EST

As I said before a voltage is measured as a difference in potential between two points. If the two points are identical, then the voltage difference is zero.

From a practical measurement perspective there is also the issue of instantaneous vs some form of averaging. Transient vs steady-state, magnitude alone, and with the addition of phase.

Consider an ordinary DC meter. This is a polarity (phase) sensitive device on DC and provides information on both the voltage magnitude, and polarity. Connect a bridge rectifier at the input and this meter looses its ability to measure polarity on a DC voltage and it becomes only a magnitude meter. You can run this experiment with a Simpson 260. DC and AC positions both read voltage magnitude in the AC position. In DC it also provides phase information by the needle direction as a result of the fixed unidirectional magnetic field in which the meter coil is positioned. Thus, the magnet is the phase reference. In the AC position when reading a DC voltage it only reads magnitude because of the internal bridge rectifier. All readings are upscale. The calibration is not correct, but that is only a scale factor. In an electrodynamometer meter the calibration is correct for both DC and RMS AC and is only a magnitude meter. The electrodynamometer meter is a phase sensitive meter when connected in the wattmeter configuration.

Now consider an AC voltage source. Measuring only two points the only information the Simpson provides is magnitude. Always an upscale reading. This meter can not provide polarity or phase information in this case. Something else has to enter the picture to get some polarity information.

Most normal AC meters do not read the instantaneous voltage but an average of some sort. An oscilloscope is one instrument that can provide instantaneous voltage measurement.To get phase information some other reference is needed separate from at least one of the measured voltage points.

If you build a special kind of AC voltmeter, then with a phase reference signal it is possible to get both magnitude and phase of the AC voltage being measured. I build such a circuit, a phase sensitive voltmeter, for linear mechanical measurements from an LVDT (Linear Variable Differential Transformer). Thus, as the LVDT plunger goes from full extension, to null, and then full compression, the output meter goes from +full scale to 0 to -full scale. From one side of null to the other the output phase changes 180 degrees relative to its input excitation. Using a standard AC voltmeter the reading would go from +Full scale to near 0 (never actually 0) back to +Full scale.

.

 

[rattus]

Poser

Poser

If we define the L1 and L2 voltage in a residential system as

V1n = 120Vrms@0

and

V2n = 120Vrms@-180.

Would V2n not shock you since it is regarded by some as a mathematical trick?

 

[jjkind]

Here's something to think about...

The induced magnetic field in the core of a transformer (the single phase transformer that we are discussing in this case)....

A.) oscillates between some values -B and B (that is, the alignment of the field changes direction from up to down/clockwise to counterclockwise).

B.) oscillates between 0 and B, with the magnetic field always oriented in one direction.

??

 

[gar]

111109-1758 EST

jjkind:

To make it simple consider a single air core coil.

If I apply a steady DC input, then B is a constant value.

If the input is a pulsating DC (never negative), then B never changes sign.

If the input is a balanced AC, then B is a balanced +/-B.


Next add a second coil magnetically coupled by air to the first coil. There is no DC coupling.

For the steady DC input there will be a short pulse output from the secondary at the transistion of the input. After steady-state the output is 0.

For the unidirectional pulsating DC after steady state the secondary is an AC voltage with an avetrage value of 0. Meaning the areas above and below 0 are equal.

The third case is the same result except that the voltage waveform is balanced.

.

 

[jjkind]

Thanks gar. I wanted to make a point about polarity in AC systems and why it's intuitively hard to understand, for the exact reason you described - in an AC system everything is balanced. hmmm...not as helpful as I would have liked...

 

[gar]

111109-2004 EST


jjkind:

If a steady-state AC waveform has no DC component, meaning a DC meter connected to that waveform reads zero, then the area under the positive side of the curve must equal the area under the negative side.

If you look at the steady-state output of the secondary of a transformer it must contain no DC component, because there is no path to provide that current. The insulation resistance of the transformer is extremely high, and with nothing but the meter connected to the secondary there is no path.

Put a diode in series with the meter and a resistive load and there will be a large secondary DC current. Flux rebalancing must occur for this to happen in a ferromagnetic transformer and it causes an unbalanced saturation of the core material.

I do not know what this has to do with the objective of this thread.

.

 

[jjkind]

Miscommunication. I was merely trying to point out why I think these concepts - regarding voltage 'direction' and polarity specifically - are hard to understand. I was actually trying to strip away a lot of the complexity of things like mutual inductance, etc. and look just at the magnetic circuit to show that hey...here's what's going on and it looks the same from either direction. Failed attempt to bring more focus and precise language to the thread on my part.

 

[__dan]

If you want me to read all that, my fee is a nickel per word. I will be charitable and not charge a markup for obfuscation.

Somewhere in your dissertation you have neglected to mention the fact that the apparant phase reversal is in fact caused by the load being connected in a reversed polarity arrangement. The source is two identically matched windings on one iron core. Winding turn direction is identical and so output polarity matches instantaneously. From the source looking to the load, you may by choice attach two 120 volt loads with a reversal of the leads at one load and reversal of the instantaneous voltage potential present at the otherwise two identical 120 volt loads. This gives load A a positive forward voltage potential when B is negative and B a positive forward voltage potential when A is negative. For diodes is series with the loads, they would conduct on opposite half cycles.

Note that the reversal of instantaneous voltage present between loads is caused by the reversal of the leads from source to load and not by what voltage is offered by the transformer with no leads attached. With no leads or wiring attached at the transformer, the only voltage offered is two 120 volt sources identical in every way, place, and time.

It may benefit to contemplate your audience. If you build your paradigm from the underlying physical reality up towards the explanation, you may be able to save yourself a lot of nickel charges.

The source is not arbitrary. It is fixed by the unit selected and built by the manufacturer. The connection of the load is arbitrary if the customer's demand is ignored.

The primary is a single winding. Secondary voltage is determined by the turns ratio. To get 240 volt from two 120 volt windings, you are adding winding turns in series (underlying physical reality), doubling the number of turns and doubling the voltage. No fancy phase reversal footwork is happening. The same flux directon, induced voltage potential direction, and current flow direction is present instantaneously at each identical winding turn. They are added in series.

 

[gar]

111110-1811 EST


__dan:

My guess is that you have not had extensive university level courses in AC or DC circuit analysis, or you would not resort to the arguments you are presenting. There is no need to talk about the flux once you define the voltages.

Fundamentally the center tapped secondary can be described by two voltages sources.
See my post #24 at http://forums.mikeholt.com/showthread.php?t=140766

Besoeker would like me to use ωt instead of t and that is OK because then t is in the units of seconds rather some other units of time.

.

 

[mivey]

If we define the L1 and L2 voltage in a residential system as

V1n = 120Vrms@0

and

V2n = 120Vrms@-180.

Would V2n not shock you since it is regarded by some as a mathematical trick?

I don't think they would argue that it would shock you. They would argue that they got shocked by Vnb NOT Vbn.

 

[mivey]

If you want me to read all that, my fee is a nickel per word. I will be charitable and not charge a markup for obfuscation.

I was trying to clear some of the muddy water stirred up previously. Read as much as you like, I do not charge for the lessons. However, if you send me a bill for the nickel units, I might make an exception in your case.

Somewhere in your dissertation you have neglected to mention the fact that the apparant phase reversal is in fact caused by the load being connected in a reversed polarity arrangement

I did however, mention that polarity and direction are related but are not the same thing. I will simply refer you back to my "dissertation".

The source is two identically matched windings on one iron core. Winding turn direction is identical and so output polarity matches instantaneously. From the source looking to the load, you may by choice attach two 120 volt loads with a reversal of the leads at one load and reversal of the instantaneous voltage potential present at the otherwise two identical 120 volt loads. This gives load A a positive forward voltage potential when B is negative and B a positive forward voltage potential when A is negative. For diodes is series with the loads, they would conduct on opposite half cycles.

Covered.

Note that the reversal of instantaneous voltage present between loads is caused by the reversal of the leads from source to load and not by what voltage is offered by the transformer with no leads attached. With no leads or wiring attached at the transformer, the only voltage offered is two 120 volt sources identical in every way, place, and time.

The voltages are not identical in every way. When taken in series, they do not share the same reference point.

It may benefit to contemplate your audience. If you build your paradigm from the underlying physical reality up towards the explanation, you may be able to save yourself a lot of nickel charges.

Other posters have claimed that taking the positive direction to be different for each 120 volt wave defies the physical reality. The only thing that is defying the physical reality is defining one positive direction for both halves of the wave, regardless of whether they are the same or different for each voltage. The voltage has no pre-defined positive direction for all time. The assignment of a positive direction is a choice we make and either direction is valid.

The source is not arbitrary. It is fixed by the unit selected and built by the manufacturer. The connection of the load is arbitrary if the customer's demand is ignored.

Given a set of conventions for current flow, etc, the forces are not arbitrary. It is the assignment of a positive and negative direction that is arbitrary. It is silly to argue that one assignment is right and the other is wrong. Both are valid choices.

The primary is a single winding...They are added in series.

You are trying to define the voltages we take from the source based on how they are created. How they are created is not the issue as we can create the exact same currents, flux, forces, etc with two voltages that are physically displaced by 180?. The forces are what they are.

Our trying to pick one direction to be positive for a force that changes direction every 1/2 cycle is the "fancy footwork" you seek.

To put it succinctly:

It never occurs to me to consider the flux in a xformer when defining a voltage.