Revalidation of magnetic circuit. (Field Coil Fullrange Unit) [AUDIO-2]

In my past blog article, I validated the magnetic circuit of my field coil full-range unit.
In article, I used fixed value for Magnetic permeability of Pole piece and Yoke.
I used this fixed value for calculation.
To express correctly, Magnetic permeability is not fixed value.
Magnetic permeability is the value that is equivalent to gradient of B-H hysteresis curve.
So, it will be changed according to the strength of the magnetic field.
By taking into account the point, I want to refine my article with new calculation method.
About previous article, I want to remain them with correction line.

Basic specs of magnetic circuit
Pole Piece
Round bar shape with 28mm diameter, and 102mm length
Material is Structural steel bar called SS400
Structural steel plate (SS400) with 9mm thickness.
Top and Bottom plate size are 125mm*110mm
Side plate size is 90mm*110mm
2 round disks with 3mm thickness are installed so as to sandwich the Top Plate.
(Material is SS400)
With these disks, Total thickness of top plate is 15mm.
Hole size (diameter) of Top Plate is 31mm
Pole piece diameter is 28mm
So, Travel distance of gap is (31-28)/2=1.5mm
Width of gap is 15mm.
 Voice coil diameter is φ29.5mm

Field coil
Diameter of Polyurethane wire is 0.18mm
Length of Polyurethane wire is 4,180m
DCR of Polyurethane is 2,700 to 2,800 ohms (measured value)
Coil turns is 24,265 (Calculated value)

Other important constant
Magnetic permeability of vacuum = 4π×(E-7)=1.25664×(E-6) (H/m)
Saturation magnetic flux density of SS400 = 1.8T (18,000 Gauss)

I made excel sheet for magnetic simulation.

First thing to calculate is Maximum magnetic flux density of Gap.
When incresing field coil current, magnetic flux will follow until saturation.
For this unit, I use the material called SS400 for all magnetic circuit.
That means, first point to saturate is the place with most small size of cross-sectional area.
Check the cross-sectional area size thru magnetic circuit.
Pole Piece
120*9*2/1,000,000=0.00216(㎡) ※Magnetic flux is devided into 2 stream.
Pole piece's cross-sectional area is smaller than Yoke's.
It means Pole piece is the first place to hit magnetic saturation.
When pole piece hit magnetic saturation, then Gap's magnetic flux density reaches maximum value.

Calculate total magnetic flux when pole piece saturate.
Φ is calculate as a result of maltiplitation of cross-sectional area and Saturation magnetic flux density

This total magnetic flux (Φ=Wb)is constant through the magnetic circuit.
Calculate Gap's maximum magnetic flux density.

Vioce coil size is Φ29.5mm, Gap width (Thickness) is 15mm
Cross-sectional area size of gap at voice coil (Svc) is
Svc= 29.5*15*π/1,000,000=0.00139(㎡)
Maximum magnetic flux density (Bgpmax) is
It's about 8,000(Gauss)

This is the maximum value of Gap's magnetic flux density that I can get with this circuit.

Next, I want to calculate field supply power.
This time I use Gap's magnetic flux density value as 0.76(T).
It's slightly smaller than maximum value. It means pole piece almost hit saturation level.

Magnetic flux is created by Magnetomotive force by the supplied power to the field coil.
This magnetic flux is closed circuit.
Consider this magnetic flux as same as current in electric circuit.
Total number of magnetic flux is same through the circuit as long as without branch circuit.

What I want to say is
Magnetic flux of Pole piece = Magnetic flux of gap = Magnetic flux of Yoke

I use this formula to calculate each part's magnetic resistance.

Magnetic resistance of gap (Rgp) is
Rgp=Lgp(Gap width)/(Magnetic Permeability of vacuum* Cross-sectional area of gap)

Gap area is not metal then I can use constant Magnetic Permeability, calculation is simple.
Pole piece and yoke uses metal, then I have to consider Hysteresis curve.
I need to manage to get result with some new idea or method.

I calculated as below.

First thing to calculate is Total Magnetic flux (Φ)
Φ(Wb) =Bgp×Sgp=0.76(T) ×0.001390155(㎡)=0.001056518(Wb)

Calculate each part's magnetic fluxdensity B(T)
Pole Piece (Main Part)
※Bpp value is slightly smaller than SS400 material's Saturation magnetic flux density.

Pole Piece (Top end)
※This time I designed pole piece as just a round bar,
but pole piece with wider top end part (T-shape from side view) is very common.
I calculate top part of pole piece.

Bpt[T] =Φ/Spt

Yoke frame

Alreadey get each part's magnetic flux density,
then next to get is each part's magnetic field strength (H)

Magnetic field strength H(A/m)
Magnetic Permeability B(H/m)
Relative magnetic Permeability
will be following.

Pole piece (main)
Magnetic field strength
Bpp=1.715816(T)  ⇒ Hpp=8090.816327(A/m)
Magnetic Permeability=Bpp/Hpp=0.00021207(H/m)
Relative magnetic Permeability
=Magnetic Permeability/Magnetic Permeability of vacuum
Maximum Magnetic Permeability of SS400 is several thousand level.
For Pole piece, Magnetic Permeability value is very small about few hundred level.
It's very far from maximum value.
It means using saturating area, then gradiation of B-H curve's gradient value is small.

Pole piece (Top end)
Magnetic field strength
Bpt=0.800714286(T)  ⇒ Hpt=88.96825397(A/m)
Magnetic Permeability=Bpt/Hpt=0.009(H/m)
Relative magnetic Permeability=7161.9724

Yoke Frame
Magnetic field strength
Byk=0.534(T) ⇒ Hyk=59.2883058(A/m)
Magnetic Permeability=Bpt/Hpt=0.009(H/m)
Relative magnetic Permeability=7161.9724

Now I can calculate magnetic resistance (R) as below.
R=Magnetic path length/(Magnetic Permeability*Cross-sectional area)

Pole Piece (Main)

Pole Piece (Top end)
(There is no Top end part for this unit)

Yoke Frame

Gap's magnetic resistance is
Then, total magnetic resistance (Rm) will be

Calculate the Magnetomotive force (NI).

NI=Total magnetic flux (Φ)*Total magneticresistance
    =1745.2527(A) (Ampare・Turn)

Actual field coil current is calculated as
Actual field coil current Ifc(A)=NI/Number of coil turn
At this time, Coil voltage Vc is 198(V), Power consumption is 14.23(W).

During unit production, I used temporary setting of Field coil current.
When field coil current is 62(mA), Field coil voltage was 210(V)
How did it operate?

Use excel sheet to simulate this condition.
As a result, when field coil current is 62(mA), Field coil voltage shoud be 170(V)
40(V) difference between actual result and simulation result.

Check the Simulator's parameter settings, and found some reasons.

Number of Field coil turn is just estimation.
Coil resistance is depend on temperature.
And biggest problem is SS400's B-H curve is not acculate,
because I took this data from somebody's website.
It might have big difference to actual material.

To get acculate result, I need to take the statistics with many times of trial.

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