Magnetic circuit analysis for microwave circulators using different techniques, selection of permanent magnets, temperature compensation

1. Th. Lingel 1
Magnetic Biasing Techniques for Circulators,
Analysis and Design Considerations
Thomas Lingel
lingel@IEEE.org

2. Th. Lingel 2
Outline
• Motivation
• Review Units and Magnetic Materials
• Hysteresis & Demagnetization curve of a
Permanent Magnet
• Magnetic Circuit Analysis
• Permanent Magnet Materials
• Temperature Compensation
• Conclusion
This presentation is mainly geared towards biased above ferromagnetic resonance designs for
circulators /isolators although concepts are generally applicable to other devices as well.

3. Th. Lingel 3
Circulator Magnetic Components

4. Th. Lingel 4
• Establish DC bias field inside the ferrite(s) for proper
RF operation over temperature
• Guaranty operation after exposure to extreme
temperature conditions (storage, reflow, etc.)
• Minimize volume of necessary components while
providing efficient shielding
• Cost efficient designs with the right selection of
materials and dimensions
Motivation

5. Th. Lingel 5
• Biot-Savart Law and Ampere’s Law
2
00
4 r
rldI
Bd
rr
r ×
=
π
µ
∫∫ = AdJldH
L
rrrr
NI=
Magnetic Term Symbol SI unit CGS unit conversion factor
Magnetic Induction B Tesla (T) Gauss (G) 1 T = 104 G
Magnetic Field Strength H A/m Oersted (Oe) 1 A/m =4π/103 Oe
Magnetization M A/m emu/cm3 1 A/m = 10-3 emu/cm3
Magnetic Moment m Am2, J/T emu 1 Am2 = 103emu
Flux Φ Wb (Vs) or Tm2 Maxwell or Gcm2 1Wb= 10-8 Mx
Permeability of
free space
µ0 H/m dimensionless 4πx10-7 H/m = 1 (cgs)
Overview Magneto-Static Analysis
• Two common unit Systems

6. Th. Lingel 6
Ferromagnetic Elements (“Iron Triad”)
Currie
Temp.[K]
4πMs[G]
@ 20°C
Fe 1043 21580
Co 1388 17900
Ni 627 6084
Note: Magnetic Materials
typically contain one or
more ferromagnetic
Element
Sm2Co17

7. Th. Lingel 7
H
B
HB 0µ=
B
H
)(0 MHB += µ
Permanent Magnetic Materials
M
H
)(HM
Material Contribution
Soft-Magnetic
Hard-Magnetic
B
H
+ =

8. Th. Lingel 8
B/µ0M
H
HcHci/HcJ
Br
BHmax
intrinsic
normal
H
HHB rr
rrr
)1(0
0
χµ
µµµ
+=
==
)(0 MHB
rrr
+= µ
SMHB
rrr
π4+=
JHB
rrr
+= 0µ
Permanent Magnetic Materials
cgs:
SI: Sommerfeld
SI: Kennely

9. Th. Lingel 9
0
1000
2000
3000
4000
5000
6000
7000
-40 -20 0 20 40 60 80 100
Alloy
Ferrite
0
1000
2000
3000
4000
5000
6000
7000
0 100 200 300 400 500
H [Oe]
B[G]
1500
1600
1700
1800
1900
2000
2100
2200
2300
0 50 100 150 200
H [Oe]
B[G]
Soft-Magnetic Materials
Properties of Return path
material (typically Steel) will
also have to be included
4πMs[G]
Temperature [°C]

10. Th. Lingel 10
Demagnetizing Field
0=Gl
RBB =
0=H
RB
Gl
RMG BBB <=
B
RMG BBB <=
0=∫ ldH
rr
GGMM lHlH −=
MH
GH
Gl
N S
Permanent Magnet
Old concept to illustrate that a
closed magnetized toroid does
not have an internal magnetic
field strength; this changes once
an air gap is introduced

11. Th. Lingel 11
Magnetic Circuit Analysis
0=⋅= ∫ ldNI H
( )M
G
M
G H
l
l
H −=
0=+ GGMM lHlH
MΘ GΘ
Gl
Ml
GA
MA
• Field Strength in Steel Yoke neglected
• Fringing neglected
Permanent
Magnet
Steel Yoke
Air-Gap

12. Th. Lingel 12
GM Φ=Φ GG HB 0µ= ( )M
G
M
G H
l
l
H −=
G
G
M
MGGMM A
l
l
HABAB 0µ−==
GM
MG
M
M
lA
lA
H
B
−=
0µ
“Load Line” or Permeance Coefficient:
GM
MG
M
M
lA
lA
k
k
H
B






−=
2
1
0µ
Leakage coefficient k1 and Loss
Factor k2 can be used to account
for non-ideal models
Isolated Permanent Magnet,
demagnetization factor N
determined by Geometry
N
N
H
B
m
M −
−=
1
0µ

13. Th. Lingel 13
Bm
Hm
GM
MG
lA
lA
0Pc:Slope µ−=
)(0 MHBM += µ
Operating point
Scaled B-H curve of the air-
gap, mirrored on the B-axis
Permeance Coefficient or Loadline
( )M
G
M
G H
l
l
H −=
Open
Short
Energy
Bd
Hd

14. Th. Lingel 14
Intrinsic Permeance coefficient
Additional magneto-motive force (m.m.f.)
GM
MG
M
MM
lA
lA
H
MH
−=
+
0
0 )(
µ
µ
MB
Hm
Bm
1+== cci
M
M
PP
H
M
ml
Ni
Note: Permeance coefficients are usually
defined as positive numbers
/µ0M

15. Th. Lingel 15
• RF specifications dictate ferrite size and DC bias level
• Magnet size has to be determined and Material selected:
Bias level must be achieved with margin for tuning
Magnet Volume is minimized, Operation at a high
Energy level without risking demagnetization at
extreme temperatures
Magnet is producible (aspect ratios, minimal height)
Magnet fits all other design constraints
(housing size, cost)
Design Approach
Different concepts are presented: Analytical/load-line approach, graphical solution, equivalent network approach

16. Th. Lingel 16
Ferrite
Fl
Ml
FA
MA
Permanent
Magnet
Steel Yoke
Loadline Approach with Ferrite
Bm
Hm
FM
MF
lA
lA
0µ−
M
F
F
A
A
M0µ
0=⋅∫ ldH GM Φ=Φ
Slope:
CP

17. Th. Lingel 17
0
2000
4000
6000
8000
10000
12000
-12000 -10000 -8000 -6000 -4000 -2000 0
H [Oe]
B[G]
Source
Load
Example
Ferrite
Diameter: 20mm
Height: 2mm
4πMs: 2000G
HDC: 1000Oe
Magnet SmCo Ceramic
HM [Oe] 5000 2000
BM [G] 5000 2000
Height [mm] 0.4 1
Diameter [mm] 15.49 24.49
Operating point
from Source line
Input
M
FF
M
H
lH
l −=
M
FFF
M
B
AMH
A
)(0 +
=
µ
Comparison: What size magnet do I need
to achieve 1000Oe internal field strength
for a given ferrite using a SmCo or a
Ceramic magnet, both operated on an
idealized demagnetization curve at
maximum energy output
turns out that energy product times
volume has to be the same!

18. Th. Lingel 18
B, H and M do not need to
be parallel/anti-parallel to
each other!
Maxwell-2D BOR-model
]Oe[zH
z
z
H
B
0µ
Ceramic
Magnet
Ferrite
Magnet appears effectively
~2mm smaller in diameter
because of fringing fields
This is an FEM model of the
ceramic magnet case from the
previous slide
Fringing is causing the magnet to
look effectively smaller in diameter,
resulting in a steeper loadline and
lower field strength within the
ferrite

19. Th. Lingel 19
H
B Φ
Θ
Multiply by area
Multiply by height
∫=Θ ldH
rr
∫=Φ AdB
rr
Graphical Solution with “Magnetic
Voltages and Currents”
Graphical solutions can
take nonlinearities into
account

20. Th. Lingel 20
Magnetic Circuit Analysis
Permanent Magnet
Soft-magnetic Material
A
l
RM
µ
=
l
A
PM
µ
=
lHcM =Θ
A
l
RM
µ
=
AMSM π4=Φ
Electrical circuit analysis
tools can be an efficient
way to analyze magnetic
circuits

21. Th. Lingel 21
Scalar Magnetic Potential (Voltage) [A]:
∫=Θ ldH
rr
∫=Φ AdB
rr
Magnetic Flux (Current) [Wb, Vs]:
Φ
Θ
=MRMagnetic Reluctance [A-turn/Wb]:
A
l
RM
µ
=
Hl=Θ
BA=Φ
Also Magnetomotive Force m.m.f [A-turns]
Permeance [H]
Θ
Φ
=P
l
A
P
µ
=
Θ−∇=⇒=×∇ HH
rr
0
Equivalent Networks

22. Th. Lingel 22
0
Magnet_Flux
Source2
Ferrite_Flux
Source1 Magnet_Reluctance
Ferrite_Reluctance
Airgap_Reluctance V Theta_Magnet
+A
PHI_Magnet
V Theta_Ferrite
ReturnPath_Reluctance
V
+A
V
[Oe] [mG] [Oe] [MGOe] [mG]

23. Th. Lingel 23
Operating point
H
B
Ferrite is modeled like a permanent
magnet, one has to ensure that the
operating point is in the saturated
area of the First Quadrant of the
Hysteresis
• Based on the dimensions of the ideal model a numerical
model can be generated, taking fringing and all material
properties into account
• Energy Product and Permeance Coefficient are
varying within the magnet volume !
This is the trick which was used in the
circuit model on the previous slide
Rather than working with the nonlinear
curve of the softmagnetic ferrite we
assume a linear BH characteristics of a
permanent magnet operated in the first
quadrant

24. Th. Lingel 24
Demagnetization at Temperature Extremes
Recoil on minor
hysteresis,
irreversible
change
- Temperature +
Note: The higher reluctance of ferrites at elevated temperatures reduces the operating
temperature range even further.
Load line is passing the
knee point at elevated
temperature
This leads to
irreversible field loss

25. Th. Lingel 25
Comparison of Magnetic Materials
NdFeB
SmCo
Ceramic
AlNiCo

26. Th. Lingel 26
Permanent Magnetic Materials
Energy
Prod.
[MGOe]
~ ~ Tc[°C] µ-recoil
AlNiCo ~1.4-10 -0.02 +0.01 ~900 grade dependent (~2..5)
Ceramic ~2.7-4 -0.2 +0.27 ~450 ~1.05-1.15
Sm2Co17 ~18-32 -0.035 -0.2 ~820 ~1.05-1.1
NdFeB ~10-48 -0.12 -0.65 ~350 ~1.05-1.1
Numbers shown are only guidelines, many different materials are available.
A Higher Energy Product is usually traded for lower Hci values.
The change of Br and Hci is not linear, therefore numbers are only rough guidelines.
All sintered magnets are brittle, another alternative are bonded magnets which typically
have lower Energy Products.




°∆
∆
CTB
B
r
r %




°∆
∆
CTH
H
ci
ci %
α β
Reversible Changes

27. Th. Lingel 27
Measurement Results of NdFeB at
different Temperatures
25°C
75°C 100°C 130°C
Limit of the Measurement Equipment

28. Th. Lingel 28
B/µ0M
H
Measured Demagnetization Curve
Tuning
Tuning is necessary in most cases to account for material and mechanical tolerances.
The Operating point in this case will be on a minor hysteresis loop.
Measured Demagnetization
curves at different
demagnetization levels
This is what happens during
calibration “knock-down” of a
circulator, we start at saturation
and find the right minor
hysteresis for the specified
frequency range

29. Th. Lingel 29
Temperature Compensation Elements









 −
=
100
0
0
µκ
κµ
µ j
j
P
t
22
0
0
1
ωω
ωω
µ
−
+= m
22
0 ωω
ωω
κ
−
= m
The Bias field needs to be reduced in the above
Resonance operation if the Saturation Magnetization
decreases with increasing temperature, however on-
and off- diagonal elements can not be kept constant
simultaneously by only adjusting the bias level.
Adjustment of µeff can be used as guideline, but the
frequency response has to be the criteria.
Significant change in Br (Ceramic Magnets, NdFeB) make it
easier to temperature compensate when biased above
resonance. Other materials need more/additional temperature
compensation components.

30. Th. Lingel 30
Temperature
Frequency
soft magnetic Flux limiting Airgap
Bandwidth
Nickel content
Less More MoreLess
Operation of Temperature Compensation Alloys
)(thicknessf=α
Summary on how temperature
compensation elements (typically
disks) in a series configuration work
First knee point is related to when
the temperature compensation
material gets saturated, slope of the
center frequency vs temperature
depends on the thickness, second
knee point relates to the Curie
temperature of the temperature
compensation material
Temperature compensation alloys
are typically binary NiFe alloys with
about 30%-32% Nickel content
This plot is a contour plot of center frequency and upper/lower frequency limit vs temperature, it
provides insight on the temperature compensation design and what handles can be adjusted

31. Th. Lingel 31
Conclusion
• The DC bias design is essential for proper RF
performance
• With increasing material costs the optimization of
magneto-static components becomes more important
• Simple circuit models of the magneto-static problem
help to get a basic understanding and to define
starting structures for numerical simulations
• There is no one-fits-all design. Specific material
selection and geometry are driven by actual RF-
specifications and mechanical constraints.