I Some questions about how electrical transformers work

[saadm]

Hi all,
I just read this article about how electrical transformers work and have some questions.

https://www.dfliq.net/blog/the-basics-of-electrical-transformers/#:~:text=The%20three%20important%20components%20of,magnetic%20flux%20is%20initially%20produced.

i). What exactly is magnetic flux linkage as oppose to just magnetic flux?
ii). What is this electric source that the primary transformer winding (or section) is connected to?
ii). Why should there any reluctance maintained at all between the primary and the secondary transformer windings or between one of the windings and the electrical source? Should the magnetic flux be maximized if there is zero reluctance (by eliminating the air gap completely)?

Thanks,
Saad

 

[dlgoff]

@saadm
This is a very old link I have. Hope it still works.
https://www.electronics-tutorials.ws/transformer/transformer-basics.html

 

[berkeman]

@saadm
This is a very old link I have. Hope it still works.
https://www.electronics-tutorials.ws/transformer/transformer-basics.html

Link still works, but the flux direction looks backwards to me. That shouldn't be a problem for the OP's questions, but it looks odd to me...

 

[dlgoff]

... but the flux direction looks backwards to me.

You're right. I never noticed that.

 

[berkeman]

I'll try to address your questions so far. Feel free to ask follow-up questions as needed. Also, is your main emphasis for 50/60Hz power transformers, or medium-to-high frequency communication transformers? There are important design ideas involved with each type...

i). What exactly is magnetic flux linkage as oppose to just magnetic flux?

The magnetic flux $\Phi$ is related to the magnetic field $B$ and the cross-sectional area $A$ that that field is present in. The simple equation is $\Phi = BA$

The difference between total flux and flux linkage may be due to the geometry of the magnetic core, where almost all of the flux is contained. If you have a multi-leg transformer core, like an E-E or E-I core, then the windings on the bobbins on the multiple legs may not receive all of the flux from the primary source coil. If you have an E-E core transformer with one primary bobbin and two secondary bobbins, then each or the two secondary bobbin windings will receiver about half of the total flux generated by the primary coil, so the flux linkage would be about half for each of those secondary coils.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html

Also, there is an important distinction between the "Magnetizing Inductance" $L_m$ and the "Leakage Inductance" $L_k$ as seen by the primary coil winding when it is driven by the source. The $L_m$ is the inductance that involves the flux that is coupled to and through the magnetic core of the transformer, linking the primary winding to the secondary winding(s). The $L_k$ is the (hopefully small) inductance generated by the leakage magnetic field that does not couple well to the core. Transformer coils have spatial extent with some windings not right up against the core, so some of the magnetic field generated by those coils is just coupled to the air around them, and not totally contained in the core. So those $B$ field components do not contribute to the linked flux to the other coils on the magnetic core.

ii). What is this electric source that the primary transformer winding (or section) is connected to?

The voltage source or current source that is connected to the primary transformer winding is the source of the energy that is being coupled through the transformer. So for a power transformer, it would be the AC Mains circuit that you plug your device into to turn it on. For a communication transformer it would be the AC voltage or current source that is transmitting the communication waveform through the transformer onto the twisted pair network (like Ethernet).

ii). Why should there any reluctance maintained at all between the primary and the secondary transformer windings or between one of the windings and the electrical source? Should the magnetic flux be maximized if there is zero reluctance (by eliminating the air gap completely)?

With pretty much all transformer cores except toroids, there will be joining surfaces that create slight "air gaps". For many transformers we try to minimize the gaps by polishing the mating surfaces (like in E-E and E-I cores), to minimize the reluctance. Explicit air gaps are used (in some pot cores for example) to limit the maximum flux in the core below the magnetic core's saturation flux density if all other performance parameters of the transformer are met.

Hope that helps.

 

[vanhees71]

Link still works, but the flux direction looks backwards to me. That shouldn't be a problem for the OP's questions, but it looks odd to me...

View attachment 314702

How can a flux, which is a scalar quantity, have a direction? I think a lot of confusion comes from not clearly distinguishing the scalars appearing in the integral Maxwell equations from the corresponding fields/densities in the differential Maxwell equations. One should also note that the integral form needs much more care with all the definitions of directions in the line- and surface integrals than the differential form, which give the same physics in terms of fields at any point in space.

 

[berkeman]

How can a flux, which is a scalar quantity, have a direction?

Interesting point. I guess my intuition is lazy, but the flux clearly has a direction with respect to the cross-sectional area of the magnetic core. Otherwise, there would be no way to calculate the voltages and currents in transformers.

The scalar flux $\Phi$ is defined by the vector dot product equation involving the vector $B$ and the vector area element(s) $dA$, so the sign of the flux direction is determined by those vectors, and hence the sign/direction of the little $\Phi$ arrows in the figure that I commented on was wrong. Would you agree with that part?

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

 

[berkeman]

After thinking more about this, it seems like even though $\Phi$ is a scalar, it has an implied direction in the case of a magnetic core (at least for the flux that contributes to $L_m$ and not $L_k$). Just as the scalar speed of a vehicle has an implied direction when traveling on a road...

 

[hutchphd]

even though Φ is a scalar

I think the issue here devolves to consistently defining B and currents relative to a flux surfaces using the right hand rule. One could change all to the left hand rule without a ripple. Is there another issue here?

 

[vanhees71]

Interesting point. I guess my intuition is lazy, but the flux clearly has a direction with respect to the cross-sectional area of the magnetic core. Otherwise, there would be no way to calculate the voltages and currents in transformers.

The scalar flux $\Phi$ is defined by the vector dot product equation involving the vector $B$ and the vector area element(s) $dA$, so the sign of the flux direction is determined by those vectors, and hence the sign/direction of the little $\Phi$ arrows in the figure that I commented on was wrong. Would you agree with that part?

View attachment 314738
https://en.wikipedia.org/wiki/Magnetic_flux

As a scalar a flux has a sign (not a direction!), which is relative to the arbitrary choice of the two possible directions of the surface-normal vectors. It's most intuitive for real fluxes like the flux of electromagnetic charge. In the fundamental Maxwell equations it's described by the current density $\vec{j}$. It's meaning is that for a given oriented surface the current (i.e., the flux of the current density),
$$I=\int_A \mathrm{d}^2 \vec{f} \cdot \vec{j}$$
is the charge per unit time flowing through the surface, with the sign of the contribution of any surface element $\mathrm{d}^2 \vec{f}$ determined by the choice of the direction of this surface-normal-vector element.