I wound up doing an impromptu magnetics (as it applies to transformers) review during a recent SqWr meeting. This summarizes, re-orders, and maybe expands on some quadrille paper scribbling, so that if I ever do it again I’ll have a better starting point. Searching on the obvious terms will produce a surfeit of links; Wikipedia may be helpful for diagrams.

Corrections and further points-to-ponder will be gratefully received…

~~Magnetizing force H comes from amp-turns (amps I, turns N) around the core, which produces flux phi = Φ = NI.~~ **Edit: that’s not quite right. Thanks to Martin Bosch for catching the units mismatch!**

Magnetomotive force ℱ comes from ampere-turns (amps I, turns N) around the core: mmf = ℱ = NI.

The magnetizing force H is the mmf per unit length of the solenoid or core: H = ℱ / L.

Flux density B comes from permeability (mu = μ) times H, which is a DC relationship that doesn’t care about frequency in the least: B = μH. For an air-core inductor or transformer, μ is the mu-sub-zero = μ_{0} of free space, but if there’s a core involved, then you use the permeability of the material near the conductor, which will be the material’s dimensionless relative permeability μ_{R} times μ_{0}.

Total flux Φ is the integral of flux density B over all the little areas covering the surface you’re interested in, oriented in a consistent manner using the right-hand rule.

If you have an iron(-like) core inside the coil, then essentially *all* the flux is in the core, so the integral reduces to B times the area (call it a) of the core at right angles to the flux: Φ = Ba = μaH. In this case, μ is the relative permeability of the core times μ_{0} of free space.

You can plot BH curves (B for various H values) using a straightforward circuit and an oscilloscope. The X axis voltage is proportional to the winding current I and the Y axis voltage is proportional to Φ. The trick is the integrator on the secondary that converts EMF = dΦ / dt into a voltage directly proportional to Φ. The same trick works on inductors if you add a few turns to act as a secondary.

All that’s true for DC as well as AC, but transformers only work on AC, as summarized by Lenz.

The induced EMF is proportional to flux change through the secondary windings, which number n turns: EMF = – n dΦ / dt. That’s obviously proportional to frequency: higher frequency = higher EMF. Flux is all in the transformer core, so it’s still μaH. Note that these are *secondary* turns, so it’s n rather than N. Air-core transformers exist, but coupling the flux poses a problem; looking up variometer or variocoupler may be instructive.

The negative sign says the induced EMF creates a current that creates a magnetic field that points the other way, so as to oppose the original field change. In effect, the induced EMF works to cancel out the field you’re creating.

Knowing how much EMF you need in the secondary for the purposes of your circuit, you know the product of five things:

- n – secondary turns
- μ – (relative) core permeability
- a – area
- f – frequency
- H – from primary

Now you get to pick what’s important, but they all have gotchas:

The ratio n:N seems easy to control, but it tops out at a few hundred. If you care about the voltage ratio, then that fixes the turns ratio.

Choose different core material to increase μ, but then you hit core saturation in B as H increases. Practical core materials may give you permeability over two or three orders of magnitude, but with significant side effects.

Reduce B for a given Φ by using a larger core area a, which obviously requires a bigger core that may not fit the application.

Increase frequency f to get more EMF and thus H, but it may be limited by your application and other losses and effects. Higher frequency = more traverses of that BH curve with hysteresis = more core losses = can’t use lossy metals.

Increase primary H, but again you hit core saturation in B.

The circuit driving the primary must be able to handle the total load, which means it must be able to drive the impedance presented by the transformer + secondary load. That determines the primary inductance (to get the reactance high enough that the transformer presents the *secondary* load to the primary circuit), which determines the core + turns at the operating frequency.

The core must support the flux required to drive the load without saturation, which constrains the material and the area. For heavy loads (i.e., “power” transformers), output power also constrains the secondary turns and wire size, which constrains the minimum core opening and thus overall size.

#1 by

Red County Peteon 2013-06-11 - 11:25Good summary! Transformers never quite made it in IC technology, so I didn’t do much with magnetics in my paid career. I did play with them a bit as an intern at Motorola in the early days of the magnetic bubble memory boomlet, and a couple of friends worked briefly for Intel Magnetics, but the technology proved to be a dead end. I suspect some of the materials research ended up in hard drive applications, though.

In my unpaid career, I might have a use or three…

#2 by Ed on 2013-06-11 - 13:13

That whole mu

_{R}thing just didn’t scale well to atomic dimensions.I still want to use that homebrew Hall effect current sensor for something, klunky though it may be. Maybe a smaller ferrite toroid will work just barely well enough for what I need…