- Email: [email protected]
Graphical methods for determining the maximum energy product of magnets
Journal of Magnetism and Magnetic Materials 88 (1990) 365-368 North-Holland
365
GRAPHICAL M E T H O D S F O R D E T E R M I N I N G THE M A X I M U M ENERGY P R O D U C T OF M A G N E T S H.H. S T A D E L M A I E R Department of Materials Science and Engineering North Carolina State University, Rale:gh, NC 27695-7907, USA
and E.-Th. H E N I G Max-Planck-lnstitut J~r Metallforschungo lnstttut ]'dr Werkstoffwis~enschaft, D - 7000 Stuttgart 80, Fed Rep. Germany
Received 3 October 1989; in revised form 2 January 1990
The graphical construction in which an optimum load line B = (BJHo)H, H~ < 0, defines the maxtmum energy product of a permanent magnet, is rigorously correct when B(H) in the second quadram fits the branch cf a single hyperbola. It fails as an approximation for some magnets with square ( B - ~t0H ) vs. H loops, and alternative procedures for dealing with such magnets are described.
1. Introduction The desire to assess quickly the energy product of a permanent magnet from the second-quadrant B ( H ) curve has led to the wide-spread use of "energy contour paper" in which ( B H ) of an operating point can be read directly from the intercept between B ( H ) and a family of plotted curves ( B H ) = coast. An approximate scheme to obtain the energy product maximum without using special paper has been described as follows by Parker and Studders [1]: "A straight line drawn between the point with the coocdinates B~, He mad the or/gin will intersect the demagnetization curve at the point of maximum energy". An unsupported statement in the textbook of Schfler and Brinkrnann [2] specifies that "this construction is applicable only when the demagnetization curve can be approximated by a second degree curve ". Actually, this is a good approximation for most Alnico magnets but not for square magnetization loops. At the limit of linear demagnetiza-
tion curves for which /%He = B~ it also predicts correctly (BH)max=Br2/41.to . We shall examine the conditions under which this appro:dmation is correct and which graphical procedures can replace it for square loops.
2. Analysis
For the meaning of the symbols, refer to fig. 1. The units, in SI, have been chosen to pro,ride an identical scale on both axes, i.e., the induction B and the field /t0H are both in tesla. By setting t
------!-.--'-
A!
.
.
.
.
.
i:A~
A . . . . . .
~
~.-¢3
~0 = l, t h e s£fflle ¢Llli~ly~l~ i ~ l ~ a p p ~ t ~ t o ~auoa a ~ u
oersted. The condition tor m.~G~qu~l~ energy density (BH) is that the dcmagnetLzation curve B ( H ) and a curve of constant (BH), wh/ch normally intersect in two points, touch with a common tangent. To conform to our units, the curve of constant (BH) [Jm -3] shall be replaced by
B ~ o H = - C 1 [T2],
0304-8853/90/$03.50 © 1990 - ElsevierSciencePublishers P.V. (North-Holland)
(1)
H.H. StedHmaier E.-Th. Henig / Maximum energy products of magnets
366
-B ~oH=C1--"t
i n d e p e n d e n t of Cl. T h e a s y m p t o t e s H = - a a n d B = b are directly c o n n e c t e d with Hc a n d Br in a simple m a n n e r : It follows f r o m eq. (2) for H = 0 (B=Br) that B r - b = - C 2 / # o a and, for B = 0 ( H = H c < 0), that H c + a = + C J # o b. T h e r e f o r e
/
,
\/
t
g
B r / H e = - ( b - C2/Fo a ) / ( a - C2/l~ob ) = - b / a ,
(5)
t~_ blk~{H÷o~,~BrIHc]H qB- _hick2' H . o ~ I
/
}J°He JJoHIT]
ructed
0'
Fig. 1. Second quadrant of B vs. ~toH showing demagnetization curve (B - b)/~o(H + a) = - (72, load line B = (Br/He)H,
la)
and constant energy contour BlxoH = -Ci. The demagnetization curve is a h)~aerbola branch BH = - C2 in the fourth quadrant shifted by H = a and B = - b.
/
/
./
where C~ is positive. The analysis proceeds from the assumption that the demagnetization curve in
"
I \l/oox, oo
\
I true
JuoHe
-Br
0
"---- -poll
poBH
---
the second quadrant can be written ( B-
b)lxo( H + a ) = - C 2 ,
BIIB~
(2)
w h e r e a, b and C2 are positive c o n s t a n t s and ( - a ) < H ~< 0, b > B >/0. Eq. (2) represents the b r a n c h of a h y p e r b o l a with a s y m p t o t e s H = - a a n d B = b. The slopes, positive for the second quadrant, are for eq. (1),
dB/ o dH
= C,/
oH 2
=
(la) -Br JuoHc "--- -pcH
for eq. (2),
0
J.,IoBH---,--
C:/l~2o( u ÷ a ) 2
dB/lz o dH=
= -(B-b)/l~o(H+a). T h e positive slope in (2a) follows f r o m B H + a > 0. The c o n d i t i o n for a c o m m o n that (la) must equal (2a), i.e. - B / H = b)/(H+a), a n d because, according to B H is finite, one o b t a i n s B/H = -b/a.
(2a) < b and slope is -(Beq. (1),
B-(7 "" "" ~ e t e d
(c)
7" ! maximum
(3)
which is the ratio of the coordinates at the point o f ma,,fmum energy p r o d u c t (BH)ma.:. C o m b i n i n g ( l a ) and (3) yields the slope for m a x i m u m ( B H ) as
d B/d H = b/a,
it,,,
----- - ~ H
0
~BH ----,,-
Fig. 2. Idealized two-segment demagnetizatmn curves and en-
(4)
ergy product ttoBH for (a) /t o I Hc I < Br/2, (b) ~to I He I > Br/2, and (c) admixture of a magnetically softer phase.
H.H. Stadelmater E.-Th. Hemg / Maxtrnum energy products of magnets
367
Table 1 M a x i m u m energy products of commercial magnets obtained by the graphical methods (i), (ii) a n d (ii0 of section 4 c o m p a r e d with the true values. N u m b e r s in parentheses are b a s e d o n an inappropriate m e t h o d and are included for comparison on!y Magnet type
M a x i m u m energy product ( M G O e ) (i) old rule
True
(ii) intercept
(rio at knee
~alue
with B/l~o H = - 1 Alnico 4
4.0
-
-
4.0
Alnico 5
5.4
-
-
5.5
Alnico 6 Alnico 9 Sr ferfite, - 60 * C Sr ferrite, + 20 o C Sr ferrite, + 60 o C CthSm C%Sm F e - N d - B , 28 M G O e F e - N d - B , 31 M G O e F e - N d - B , 20 * C F e - N d - B , 120 o C
3.64 10.5 (4.6) 4.0 3.6 14.0 18.0 27.8 31.4 38.4 25.6
(3.0) 4.3 3.6 14.0 18.0 28 31.6 38.5 (22.1)
establishing the link between the u n k n o w n asymptotes and known Br/H ~. By eliminating b / a from eq. (3) with the help of eq. (5), one obtains the equation of the load line B//n = - b//a = + nr//Hc,
n = ( B r / H c) H ,
(6) that intersects the demagnetization curve at the point of m a x i m u m energy density. Finally, the slope of the c o m m o n tangent can be determined by combining eqs. (4) and (5), as
dS/dH
=
-
BJHo.
It turns out to be the negaUve of the slope of the load line (6). Written in the form of eq. (6), the optimum load line does not d e p e n d on the parameters a, b and (72, which control the position and curvature of the hyperbola representing B ( H ) . Therefore the demagnetization curve can have any curvature, including zero (a straight line for which a, b, C 2 ~ oo). The latter condition also implies saturation of B - H and occurs when #0H~ = - B~. It is obvious that the graphical construction yields (BH)m,x = B~/41Xo under these circumstances.
3. Discussion; alternative methods When the demagnetization curve can be fitted to a hyperbola, the graphical construction is valid
5.1 (3.9) (2.5) 26.0
3.65 10.5 5.1 4.3 3.6 14.0 18.0 28 3!.6 38.8 26.0
for the simple reason that the identity (B/H)ma, = - b / a = B r / H c holds. This places the coordinate pair (B, ~oH)ma.~ on a line between the origin and (B~, ~oHc), and it also ties the hidden parameters a and b to B~ and ~toHc. Caution is indicated when the second quadrant curve deviates in any obvious way from a smooth m o n o t o m c course. The except~oas can still be handled graphically by recognizing the nature of the deviation and referring to the examples given for idealized square ( B - / ~ o H ) vs. /~oH loops, fig. 2. The (BH)ma ~ derived from the graphical methods are compared with the correct values and compiled in table 1.
(a) Alnico.
Table 1 shows the good agreement of the graphical result with the actual energy product, applied to a variety of Alnicos. This is not too surprising because they have a simple curve shape in the second quadrant.
(b) Anisotropic ferrites.
To discuss their magnetization curves, which have high loop squareness, consider the idealized R vs..~oH curves of figs. 2a and b w~h a break in slope at the knee. This could be one and the same ferrhe at low (fig. 2a) and high (fig. 2b) temperatures. When the knee of the crave is high (~o I H¢ [ < Br)' the true maximum of ( B H ) would be found at the knee, fig. 2a. When the knee is low (/1 o I Hc I > B~), the true maximum
368
H.H. Stadelmaier E.-Th. Henig / Maximum energy products of magnets
energy product is Br2/4/to, fig. 2b. For a more realistic curve of the type seen in fig. 2b, a very good approximation is obtained with values B, ~oH found at the intercept of the demagnetization curve and a load line B/t%H = - 1. Examples are the Sr ferrites at + 20 and + 60 °C in table 1. The reason for this good agreement is that for the linear demagnetization curve of fig. 2b the maximum of (BH) is quite broad, and a small error in the slope of the load line will not be detected in
(c) Admixture of a magnetically softerphase. There is an inflection near the top of the second quadrant curve, here simplified by the superposition of two square (B - / x o H ) vs. ~oH loops. It is readily seen from fig. 2c that the correct B, for the graphical construction of section 2 would have to be the extrapolated value of the hard component, whereas using the (higher) composite Br would reduce the estimated (BH)m~x below the true value.
(d) Rare earth permanent magnets. CosSm magnets approach the linear demagnetization curve with a slope of + 1 and therefore have a (BH)max near Br2/41to. For the remaining small deviation from linearity (so that Br is slightly higher than ~oHc) the graphical construction of section 2 is satisfactory. Commercial sintered F e - N d - B has a loop aquareness that can approach that of the ferfites. Therefore, the comments made in connection with the anisotropic ferrites are also true here, but the temperature dependence associated with figs. 2a and b is reversed from that of the ferrites. In particular, the use of the intercept with the line B/t%H = - 1 works quite well when /% I,ncl >
4. Summary The foregoing discussion suggests the following three rules. (i) For a smooth second quadrant curve, as found in Alnicos, the old rule is satisfactory. This includes the near-linear curves that also satisfy the next rule. (ii) For high-ill¢ magnets, whose ( B - ~oH) is almost saturated in the second quadrant or which have a prominent knee below the coordinates B =/.t o I H I, a good value of (BH)ma~ is the product of the B and H found at the intercept with the load line B/lxoH = - 1. (iii) For a curve with a prominent knee at or abooe the coordinates B = ~o I H I, the best value of (BH)ma,, is the product of the coordinates of the secondquadrant B vs. ~oH curve taken at the middle of the knee. These rules are evident in section 2 and figs. 2a and b and are supported by the examples given in table 1.
Acknowledgement H.H.S. is grateful to the Max-Planck-Gesellschaft for a stay as a visiting scientist at the Max-Planck-Institut f~ir Metallforschung, Stuttgart, FRG.
References [1] R.J. Parker and R.J. Studdets, Permanent Magnets and Their Applications (Wiley, New York, 1962) p. 32 [2] K. Schueler and K. Brinkmann, Dauermagnete (Springer, Berlin, 1970) p. 19.
365
GRAPHICAL M E T H O D S F O R D E T E R M I N I N G THE M A X I M U M ENERGY P R O D U C T OF M A G N E T S H.H. S T A D E L M A I E R Department of Materials Science and Engineering North Carolina State University, Rale:gh, NC 27695-7907, USA
and E.-Th. H E N I G Max-Planck-lnstitut J~r Metallforschungo lnstttut ]'dr Werkstoffwis~enschaft, D - 7000 Stuttgart 80, Fed Rep. Germany
Received 3 October 1989; in revised form 2 January 1990
The graphical construction in which an optimum load line B = (BJHo)H, H~ < 0, defines the maxtmum energy product of a permanent magnet, is rigorously correct when B(H) in the second quadram fits the branch cf a single hyperbola. It fails as an approximation for some magnets with square ( B - ~t0H ) vs. H loops, and alternative procedures for dealing with such magnets are described.
1. Introduction The desire to assess quickly the energy product of a permanent magnet from the second-quadrant B ( H ) curve has led to the wide-spread use of "energy contour paper" in which ( B H ) of an operating point can be read directly from the intercept between B ( H ) and a family of plotted curves ( B H ) = coast. An approximate scheme to obtain the energy product maximum without using special paper has been described as follows by Parker and Studders [1]: "A straight line drawn between the point with the coocdinates B~, He mad the or/gin will intersect the demagnetization curve at the point of maximum energy". An unsupported statement in the textbook of Schfler and Brinkrnann [2] specifies that "this construction is applicable only when the demagnetization curve can be approximated by a second degree curve ". Actually, this is a good approximation for most Alnico magnets but not for square magnetization loops. At the limit of linear demagnetiza-
tion curves for which /%He = B~ it also predicts correctly (BH)max=Br2/41.to . We shall examine the conditions under which this appro:dmation is correct and which graphical procedures can replace it for square loops.
2. Analysis
For the meaning of the symbols, refer to fig. 1. The units, in SI, have been chosen to pro,ride an identical scale on both axes, i.e., the induction B and the field /t0H are both in tesla. By setting t
------!-.--'-
A!
.
.
.
.
.
i:A~
A . . . . . .
~
~.-¢3
~0 = l, t h e s£fflle ¢Llli~ly~l~ i ~ l ~ a p p ~ t ~ t o ~auoa a ~ u
oersted. The condition tor m.~G~qu~l~ energy density (BH) is that the dcmagnetLzation curve B ( H ) and a curve of constant (BH), wh/ch normally intersect in two points, touch with a common tangent. To conform to our units, the curve of constant (BH) [Jm -3] shall be replaced by
B ~ o H = - C 1 [T2],
0304-8853/90/$03.50 © 1990 - ElsevierSciencePublishers P.V. (North-Holland)
(1)
H.H. StedHmaier E.-Th. Henig / Maximum energy products of magnets
366
-B ~oH=C1--"t
i n d e p e n d e n t of Cl. T h e a s y m p t o t e s H = - a a n d B = b are directly c o n n e c t e d with Hc a n d Br in a simple m a n n e r : It follows f r o m eq. (2) for H = 0 (B=Br) that B r - b = - C 2 / # o a and, for B = 0 ( H = H c < 0), that H c + a = + C J # o b. T h e r e f o r e
/
,
\/
t
g
B r / H e = - ( b - C2/Fo a ) / ( a - C2/l~ob ) = - b / a ,
(5)
t~_ blk~{H÷o~,~BrIHc]H qB- _hick2' H . o ~ I
/
}J°He JJoHIT]
ructed
0'
Fig. 1. Second quadrant of B vs. ~toH showing demagnetization curve (B - b)/~o(H + a) = - (72, load line B = (Br/He)H,
la)
and constant energy contour BlxoH = -Ci. The demagnetization curve is a h)~aerbola branch BH = - C2 in the fourth quadrant shifted by H = a and B = - b.
/
/
./
where C~ is positive. The analysis proceeds from the assumption that the demagnetization curve in
"
I \l/oox, oo
\
I true
JuoHe
-Br
0
"---- -poll
poBH
---
the second quadrant can be written ( B-
b)lxo( H + a ) = - C 2 ,
BIIB~
(2)
w h e r e a, b and C2 are positive c o n s t a n t s and ( - a ) < H ~< 0, b > B >/0. Eq. (2) represents the b r a n c h of a h y p e r b o l a with a s y m p t o t e s H = - a a n d B = b. The slopes, positive for the second quadrant, are for eq. (1),
dB/ o dH
= C,/
oH 2
=
(la) -Br JuoHc "--- -pcH
for eq. (2),
0
J.,IoBH---,--
C:/l~2o( u ÷ a ) 2
dB/lz o dH=
= -(B-b)/l~o(H+a). T h e positive slope in (2a) follows f r o m B H + a > 0. The c o n d i t i o n for a c o m m o n that (la) must equal (2a), i.e. - B / H = b)/(H+a), a n d because, according to B H is finite, one o b t a i n s B/H = -b/a.
(2a) < b and slope is -(Beq. (1),
B-(7 "" "" ~ e t e d
(c)
7" ! maximum
(3)
which is the ratio of the coordinates at the point o f ma,,fmum energy p r o d u c t (BH)ma.:. C o m b i n i n g ( l a ) and (3) yields the slope for m a x i m u m ( B H ) as
d B/d H = b/a,
it,,,
----- - ~ H
0
~BH ----,,-
Fig. 2. Idealized two-segment demagnetizatmn curves and en-
(4)
ergy product ttoBH for (a) /t o I Hc I < Br/2, (b) ~to I He I > Br/2, and (c) admixture of a magnetically softer phase.
H.H. Stadelmater E.-Th. Hemg / Maxtrnum energy products of magnets
367
Table 1 M a x i m u m energy products of commercial magnets obtained by the graphical methods (i), (ii) a n d (ii0 of section 4 c o m p a r e d with the true values. N u m b e r s in parentheses are b a s e d o n an inappropriate m e t h o d and are included for comparison on!y Magnet type
M a x i m u m energy product ( M G O e ) (i) old rule
True
(ii) intercept
(rio at knee
~alue
with B/l~o H = - 1 Alnico 4
4.0
-
-
4.0
Alnico 5
5.4
-
-
5.5
Alnico 6 Alnico 9 Sr ferfite, - 60 * C Sr ferrite, + 20 o C Sr ferrite, + 60 o C CthSm C%Sm F e - N d - B , 28 M G O e F e - N d - B , 31 M G O e F e - N d - B , 20 * C F e - N d - B , 120 o C
3.64 10.5 (4.6) 4.0 3.6 14.0 18.0 27.8 31.4 38.4 25.6
(3.0) 4.3 3.6 14.0 18.0 28 31.6 38.5 (22.1)
establishing the link between the u n k n o w n asymptotes and known Br/H ~. By eliminating b / a from eq. (3) with the help of eq. (5), one obtains the equation of the load line B//n = - b//a = + nr//Hc,
n = ( B r / H c) H ,
(6) that intersects the demagnetization curve at the point of m a x i m u m energy density. Finally, the slope of the c o m m o n tangent can be determined by combining eqs. (4) and (5), as
dS/dH
=
-
BJHo.
It turns out to be the negaUve of the slope of the load line (6). Written in the form of eq. (6), the optimum load line does not d e p e n d on the parameters a, b and (72, which control the position and curvature of the hyperbola representing B ( H ) . Therefore the demagnetization curve can have any curvature, including zero (a straight line for which a, b, C 2 ~ oo). The latter condition also implies saturation of B - H and occurs when #0H~ = - B~. It is obvious that the graphical construction yields (BH)m,x = B~/41Xo under these circumstances.
3. Discussion; alternative methods When the demagnetization curve can be fitted to a hyperbola, the graphical construction is valid
5.1 (3.9) (2.5) 26.0
3.65 10.5 5.1 4.3 3.6 14.0 18.0 28 3!.6 38.8 26.0
for the simple reason that the identity (B/H)ma, = - b / a = B r / H c holds. This places the coordinate pair (B, ~oH)ma.~ on a line between the origin and (B~, ~oHc), and it also ties the hidden parameters a and b to B~ and ~toHc. Caution is indicated when the second quadrant curve deviates in any obvious way from a smooth m o n o t o m c course. The except~oas can still be handled graphically by recognizing the nature of the deviation and referring to the examples given for idealized square ( B - / ~ o H ) vs. /~oH loops, fig. 2. The (BH)ma ~ derived from the graphical methods are compared with the correct values and compiled in table 1.
(a) Alnico.
Table 1 shows the good agreement of the graphical result with the actual energy product, applied to a variety of Alnicos. This is not too surprising because they have a simple curve shape in the second quadrant.
(b) Anisotropic ferrites.
To discuss their magnetization curves, which have high loop squareness, consider the idealized R vs..~oH curves of figs. 2a and b w~h a break in slope at the knee. This could be one and the same ferrhe at low (fig. 2a) and high (fig. 2b) temperatures. When the knee of the crave is high (~o I H¢ [ < Br)' the true maximum of ( B H ) would be found at the knee, fig. 2a. When the knee is low (/1 o I Hc I > B~), the true maximum
368
H.H. Stadelmaier E.-Th. Henig / Maximum energy products of magnets
energy product is Br2/4/to, fig. 2b. For a more realistic curve of the type seen in fig. 2b, a very good approximation is obtained with values B, ~oH found at the intercept of the demagnetization curve and a load line B/t%H = - 1. Examples are the Sr ferrites at + 20 and + 60 °C in table 1. The reason for this good agreement is that for the linear demagnetization curve of fig. 2b the maximum of (BH) is quite broad, and a small error in the slope of the load line will not be detected in
(c) Admixture of a magnetically softerphase. There is an inflection near the top of the second quadrant curve, here simplified by the superposition of two square (B - / x o H ) vs. ~oH loops. It is readily seen from fig. 2c that the correct B, for the graphical construction of section 2 would have to be the extrapolated value of the hard component, whereas using the (higher) composite Br would reduce the estimated (BH)m~x below the true value.
(d) Rare earth permanent magnets. CosSm magnets approach the linear demagnetization curve with a slope of + 1 and therefore have a (BH)max near Br2/41to. For the remaining small deviation from linearity (so that Br is slightly higher than ~oHc) the graphical construction of section 2 is satisfactory. Commercial sintered F e - N d - B has a loop aquareness that can approach that of the ferfites. Therefore, the comments made in connection with the anisotropic ferrites are also true here, but the temperature dependence associated with figs. 2a and b is reversed from that of the ferrites. In particular, the use of the intercept with the line B/t%H = - 1 works quite well when /% I,ncl >
4. Summary The foregoing discussion suggests the following three rules. (i) For a smooth second quadrant curve, as found in Alnicos, the old rule is satisfactory. This includes the near-linear curves that also satisfy the next rule. (ii) For high-ill¢ magnets, whose ( B - ~oH) is almost saturated in the second quadrant or which have a prominent knee below the coordinates B =/.t o I H I, a good value of (BH)ma~ is the product of the B and H found at the intercept with the load line B/lxoH = - 1. (iii) For a curve with a prominent knee at or abooe the coordinates B = ~o I H I, the best value of (BH)ma,, is the product of the coordinates of the secondquadrant B vs. ~oH curve taken at the middle of the knee. These rules are evident in section 2 and figs. 2a and b and are supported by the examples given in table 1.
Acknowledgement H.H.S. is grateful to the Max-Planck-Gesellschaft for a stay as a visiting scientist at the Max-Planck-Institut f~ir Metallforschung, Stuttgart, FRG.
References [1] R.J. Parker and R.J. Studdets, Permanent Magnets and Their Applications (Wiley, New York, 1962) p. 32 [2] K. Schueler and K. Brinkmann, Dauermagnete (Springer, Berlin, 1970) p. 19.