NONLINEAR MAGNETIC SWITCHES FOR PULSE GENERATION by SUSAN E. BLACK, B.S. in E.E. A THESIS I N ELECTRICAL ENGINEERING Submitted to the Graduate .FacuIiy of Texas Tech University in Partial Ful fi IIment of the Requirements for the Degree of MASTER OF SCIENCE I N ELECTRICAL ENGINEERING ^ppr:ov^d Accepted May, 1980
This is an exhaustive thesis by Susan Black on the engineering physics of saturable reactor design. It should be useful to those studying or designing magnetic pulse compression circuits for lasers and other applications.
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by
A THESIS
Partial Fulfi IIment of
ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to Dr. T. R.
Burkes
for his invaluable guidance in this project and resulting thesis.
I
would like to thank Dr. John P. Craig and Dr. Wayne T. Ford for
their
helpful comments while serving on my committee. Finally, I would
like
to extend my appreciation to Greg Hill for his suggestion
concerning the
use of saturable inductors and to my fellow graduate students at
the
High Voltage/Pulsed Power Lab for their help and support.
I
OF
C O N T E N T S
A C K N O W L E D G E M E N T S
I
L I S T OF T A B L E S
L I S T
I I
MAGNETIC CORE RESET 17
GEOMETRICAL CONSIDERATIONS 34
INDUCTOR LOSSES 64
MAGNETIC MATERIALS "79
M A G N E T I C S W I T C H D E S I G N 88
LIST OF REFERENCES 103
Magnetic Materials Suitable for Use in
Saturable Inductors 84
1-1 A Block Diagram for a Typical Pulsed Power Network
Shown with the Circuit Energy Flow vs. Time 2
1-2 A Typical B-H Curve for a Magnetic Material Suitable
for Use in Saturable Inductors 4
I 1-1 A Simple L-R Circuit Illustrating the Switching
Action
of a Saturable Inductor with the Voltage and Current for
the Inductor Shown vs. Time "7
11-2 A B-H Curve Used to Illustrate the Need for Magnetic
Core Reset 8
Inductor and PFN Voltage and Current Shown vs. Time 12
11-4 A Circuit Utilizing a Saturable Inductor as Discharge
Switch Shown with Inductor Voltage and Current vs. Time.
An Alternative Placement of the Saturable Inductor
is Also Shown in (c) 14
I I
Indicating the Approximate Change in Induction Available
for a Given Pre-Switch Condition 18
I I 1-2 A B-H Curve Used to I I lustrate the Effect
of dc Bias
on Switching Action 24
I I
1-3 For an L-C Circuit, the Effect of Bias on the
Inductor
Voltage and Current Is Shown on Varying Time
Sea
I
es 26
I I 1-4 A Circuit Realization of a dc Constant Current
Supply for Reset Purposes 31
I I
1-5 A Circuit Providing a Reset Current Pulse After
Energy
Transfer with the Effect of the Reset Pulse on the
Inductor, PFN, Reset Resistor, and Diode Voltage
Current Shown vs. Time 32
IV-1 Two Typical Core Forms Used In Saturable Inductors 36
IV-2 The Cross-Section of an Inductor with one Winding
Shown with the Radial Dependence of the Magnetic
Intensity in the Core and Winding shown for (b)
a solenoid and (c) a toroid 39
vs.
IV-4 The Inductive Geometry Factor for a Toroidal Core
vs. Winding Thickness for Various Core Radii 4 3
IV-5 The Cross-Section of a Saturable Inductor Shown with
the Magnetic Intensity vs. Radius for the Bias
Winding 45
IV-6 The Coefficient of Coupling for a C-Core Inductor vs.
Winding Thickness for Various Core Radii 49
IV-7 The Coefficient of Coupling for a Toroidal Inductor vs.
Winding Thickness for Various Core Radii 50
IV-8 Saturated Inductance for a Toroidal Inductor vs.
Winding Thickness for Various Core Radii 55
IV-9 A Representative Function for the Number of Turns
Scaled with Stand-off Voltage, E, and rms Conduction
Current,
rms 60
Inductance Scaled with Stand-off Voltage, E, and rms
Current, I 61
IV-12 Core Volume Scaled with the Stand-off Voltage, E, and
rms Conduction Current, I 6 2
rms
to Switch Operation 6 6
V-2 A Typical Lamination in a Laminated Core with Width
w and Thickness d Shown with the Effect of the Eddy
Current Magnetic Intensity on the Exciting Magnetic
Intensity and Magnetizing Magnetic Intensity 69
V-3 The Hysteresis Function vs. the Ratio of the Pulse
Duration, t. Over the Lamination Time Constant, 75
V-4 The Eddy Current Loss Function vs. the Ratio of the
Pulse Duration, t, over the Lamination Time
Constant, T 76
Materials for Use in Saturable Inductors 31
VI1-1 The Design Circuit Utilizing a Saturable Inductor
as Switch Delay 89
VI 1-2 The dc B-H Curve for Silicon Steel 93
VI1-3 Oscillograms Showing the PFN Voltage and Inductor
Current for a Saturable Inductor Used as Charge
Delay Designed to Delay 4 0 ysec at 3 kV. The Stand
off voltages applied to the Inductor are (a) 3 kV,
(b) 2 kV, (c) 1 kV 99
VI
The power requirements of some electrically " pulsed" systems
such as radars and lasers involve the delivery of large amounts
of
energy in short pulses. The general method of achieving this
pulsed
power is by slowly storing energy in a storage element and then sw
itch
ing the stored energy to the load so that a short, high power
pulse
Is obtained. A block diagram for a typical pulsed power network
is
shown in Figure l-la; indicated In Figure l-lb is the energy flow w
ith
respect to time for this network.
Any nonlinear electrical element which exhibits a drastic
change
in impedance may be loosely considered as a switch. Switches
appli
cable to a pulse form of energy transfer must close quickly and
conduct
large amounts of current with reliable pulse-to-pulse
repeatability.
Typical discharge or " closing" switches used in pulsed power
applica
tions are thyratrons and spark gaps; the " closing"
action of these
devices may be characterized as a transition from a high to low
Impe
dance,
in the open state, these switches withstand or "hold off "
large
static voltages; closure is obtained on command with a trigger
pulse.
Inductors utilizing the nonlinear properties of ferromagnetic
materials may also be made to perform as switches. These switches
offer
several advantages in certain applications over the classical
switch.
The nonlinear Inductor is rugged, has a long lifetime, and Is
compara
tively inexpensive.
Nonlinear inductors achieve their switching action by
changing
> LOAD
(a)
(b)
Figure 1-1 A Block Diagram for a Typical Pulsed Power Network
Shown with the Circuit Energy Flow vs. Time
inductor saturates; thus the nonlinear inductor is commonly called
a
saturable inductor or magnetic switch. The high unsaturated
Induc
tance of a saturable inductor corresponds to an open switch while
the
low saturated inductance corresponds to the closed condition.
The
hysteresis characteristic of a ferromagnetic material is shown
in
Figure 1-2 where induction, B, is a function of magnetic intensity,
H.
The B-H curve of Figure 1-2 indicates that the operation of the
Induc
tor core Is cyclic and that the switching action of the saturable
in
ductor is dynamic in that the transition to a closed state Is
accom-
pllshed by the inductor and not by a trigger pulse. This implies
that
the switching action of a saturable inductor Is that of a delayed
switch
rather than that of a triggered switch.
The use of a saturable Inductor imposes several design
considera
tions and operational constraints necessary for satisfactory
perfor
mance as a switch for pulse power applications. Reliable
pulse-to-
pulse repeatability requires that the magnetic core be In the
same
pre-pulse state before each application of voltage to the
inductor.
This Initial conditioning is achieved by magnetically resetting
the
core to a point such as (a) In Figure 1-2. In addition to the
switch
ing winding, an auxiliary winding may be added to the Inductor for
re
set purposes.
A detailed description of the operation of saturable
Inductors
Is provided in Chapter II along with design considerations and
several
basic applications suited to saturable inductors. Methods for
resetting
the magnetic core are examined in Chapter III. The effect of
physical
Cow inductance)
F i g u r e , - 2 A T y p i c a l B -H C u r v e f o r
S u i t a b l e f o r Usf
a M a g n e t i c M a t e r
\a\
' n S a t u r a b l e I n d u c t o
r s
5
are presented In Chapter IV along with the effects of scaling for
high
power handling capabilities based on geometry and volume
constraints.
Chapter V presents a detailed description of inductor losses
Including
eddy current and hysteresis losses in the core. Ferromagnetic
materials
suitable for use in saturable Inductors are examined in Chapter VI
with
design constraints based on available materials. Chapter VII
presents
a practical application of a saturable inductor with the design
proce
dure and experimental results of the operation of this design. A
sum
marization of the theory of saturable Inductors and conclusions
are
presented in Chapter V I M .
SWITCHING PERFORMANCE OF SATURABLE INDUCTORS
The sw itching action of a saturable Inductor Is achieved by
utili
zing the noni inearity of the hysteresis characteristic of
ferromagnetic
materials. This nonI inearity leads to two sets of equations
describing
the inductive switch. One set pertains to the unsaturated, open
switch
operation of the inductor while the other set describes the
saturated,
closed switch operation.
As a result of the hysteresis effects, the inductor switch
Inherent
ly operates in three
switch delay, energy transfer, and reset.
These modes may be Illustrated with the circuit of Figure 11-1,
The
hysteresis curve for a ferromagnetic material Is shown in Figure
11-2,
where the pre-switch condition for the Inductor s assumed at point
(a).
At time t = 0, the stepped dc supply voltage drops across the
saturable
inductor so that the inductor operates in the switch delay mode,
which
corresponds to the high permeability region of the B-H curve. The
high
permeability provides a high Inductance for low power during the
switch
delay period. Upon application of the supply voltage to the
Inductor,
the change in flux density in the magnetic core is given by:
t
r
J
core,
N Is the number of
turns in the inductor winding, and V is the voltage applied to the
in
. = .
^
^BHC..veUse.toM,ust. 3 tet.eNeed
air and switching action is Initiated. The saturated inductance
is
typically two to three orders of magnitude lower than the
unsaturated
inductance under pulsed conditions.
During saturation, the magnetic core operates In the energy
trans
fer mo de , characterized by low permeability and low inductance.
The
low inductance is necessary for fast energy transfer. Once
saturation
occurs, the magnetic intensity, H, of the magnetic core
begins to in
crease with the Increase in current that accompanies energy
transfer.
Afte r the energy transfer is complet e, the current in the
inductor and
H in the core go to zero; the magnetic core then operates at point
(b)
In Figure 11-2. In order to recover the switching ability of the
induc
tor, the magnetic core must be reset to the pre-switch
condition (point
(a). Figure 11-2). Reset may be achieved by
inducing a negative magne
tic intensity (reverse current) In the inductor, or may be
induced
through the use of a "bias" winding. A saturable
Inductor used as a
switch mi ght then include a reset or bias winding as well as the
switch
ing winding, similar to a two-winding transformer.
A given but arbitrary switching delay, t,, may be achieved
through
the saturable inductor design. If the voltage applied to the
Inductor
is constant for the duration of the switch delay, typically the
case in
most pulsed pow er applications, then the relationship between time
de
lay and stand-off voltage is approximately
t = ^ ^ ^ ^ (11-2)
^d V.
where V is the voltage applied to the inductor and AB Is the change
in
L
It is assumed that the switching winding is wound
tightly to the Inductor core so that the inductor area. A,
corresponds
to the cross-sectional area of the magnetic core.
The magnetic core is sometimes laminated to limit eddy
current los
ses (see Chapter V ) . The effective cross-sectional
area of the ferro
magnetic material is reduced due to spaces between the
laminations.
Therefore, the magnetic core area becomes
A = A'S (11-3)
where A Is the magnetic area. A' is the gross core area, and S Is
the
stacking factor. The stacking factor accounts for area reduction
due
to laminating the core.
N^y u A
L = ^r -2 - (11-4)
U X,
where y is the relative permeability of the unsaturated core, y
is
the permeability of air, and I is the magnetic length
of the core. It
Is assumed n Equation 11-4 that y is large enough that most of
the
flux density produced by the switching winding is contained in the
mag
netic core. Upon saturation, the inductance of the switch
becomes
N^y y AG
sat
36
where y is the saturated permeability and G Is a multiplying factor
due
s
to winding geometry. For a magnetic core with a relatively square
B-H
11
the inductor behaves as an air core inductor and the assumption
that all
of the flux is concentrated in the magnetic core may no longer
hold.
The inductance due to the flux in the winding and the core may be
great
er than the inductance due to just the flux in the saturated
magnetic
core.
The factor, G, accounts for the discrepancy in Inductance
and
is discussed in detail in Chapter IV.
Initial conditioning of the magnetic core, or reset, is
achieved
by applying a negative flux to the core. The negative flux is
produced
by a negative current in either the switching winding or the bias
win
ding. The amount of current required to reset the core may be
deter
mined as:
I = - ^ (11-6)
r N
where H refers to the magnetic intensity of the pre-switch initial
con
dition.
Depending on the magnetic material and application, H
may
differ from the coercive force, H , of the material, indicated in
Figure
11-2. The effect of core reset on switching action,
applications of
saturable inductors requiring reset of the magnetic core, and
methods
to achieve reset are discussed In Chapter M l .
The performance of a saturable inductor may be illustrated by
analy
zing its response in several typical applications. Two
applications
that may be used as examples that Involve saturable inductors are
charge
delay and discharge delay.
The saturable inductor used as charge delay is shown in Figure 11-3
.
As described in reference [ 1] , the purpose of the charge delay is
to act
Figure 11-3 A Charge Delay Utilizing
a Saturable Inductor with
Inductor and PFN Voltage and Current
Shown vs. Time
13
before application of the charging voltage to the pulse forming
network,
PFN. The saturable inductor voltage and current as functions
of time
are shown in Figure ll-3b. The charging voltage initially drops
across
the saturable inductor. The inductor withstands the voltage for a
time,
then saturates, allowing the PFN to resonantly charge. The amount
of
time the inductor withstands the voltage before saturating is the
delay
time of the inductor, t . In this application, the delay time
should
correspond to the amount of time required by the discharge switch
to re
cover. The effect of the switching action of the saturable inductor
on
the PFN charging voltage and current is shown in Figure Il-3c. As
indi
cated, the switch by the Inductor to a lower inductance allows
faster
charging and consequently higher pulse repetition rates than
conventional
inductive charging while still allowing the discharge switch
adequate
recovery time.
Core reset for the saturable inductor used as a charge delay
may
be achieved through two methods. The first method allows the
reverse
bias current from the diode of the circuit in Figure I I-3a to
reset the
core.
This method work s well for designs using a core with a very
low
coercive force, H , so that a smaI
I
' c'
For cores requiring larger bias currents, application of the reset
cur
rent through a bias winding provides the necessary negative flux
bias.
The use of a bias winding also provides more control over the
exact
pre-switch condition of the magnetic core, thus reducing variation
in
switch delay, commonly referred to as jitter.
Figure 11-4 A Circuit Utilizing a Saturable
Inductor as Discharge
Switch Shown with Inductor Voltage and Current
vs. Time,
An Alternative Placement of the Saturable
Inductor is
Also Shown In (c)
15
before application of the current pulse to the triggered or main
switch
[2]. This delay reduces anode heating for a gaseous discharge
type of
switch and increases di/dt capabilities for most solid state
switches.
The inductor voltage and current as functions of time are shown
in
Figure ll-4b. When the main discharge switch is closed, the PFN
begins
to discharge. The voltage of the discharge pulse initially drops
across
the saturable inductor, maintaining a low Initial current through
the
main switch. After the time delay, the inductor saturates, the
switch
conducts the current pulse, and the energy stored in the PFN is
trans
ferred to the load. This application requires a very low saturated
in
ductance to keep the inductive effect on the discharge pulse to
a mini
mum. Core reset for a discharge delay may be achieved through a
bias
winding. Reset automatically occurs when the Inductor is placed in
the
circuit so that the PFN charging current resets the core, as shown
in
Figure Il-4c.
The illustrations of a saturable inductor as charge delay
or dis
charge delay involve the use of one inductive switch stage per
applica
tion.
The cascading of these saturable inductors in parallel or
series
combinations may be utilized to achieve pulse compression. The
design
of multiple stages of saturable inductors is discussed by
Busch, et.al.
[3], Coates and Swain [ 4 ] , and Melville [5] , along with
several other
applications involving saturable inductors.
Therefore, a saturable inductor may be utilized in systems
which
require or allow a switch delay. From a desired switch delay and
"hold-
off"
16
of the core material and the number of turns may be used to
determine
the unsaturated and saturated inductances in Equations (M- 4 ) and
(11-5)
The amount of reset current required may be determined from the
number
of turns and the characteristics of the magnetic core. These
design
values and constraints determine the overall electrical performance
of
the saturable inductor.
The need for pulse-to-pulse repeatability in an inductive
switch
requires that the inductor core be reset to the same pre-switch
condi
tion before each application of voltage to the inductor. Reset
is
achieved by applying a negative flux to the
core.
The reset flux may
be produced by a reverse current flowing in either the switching
winding
or an auxi Ilarybias w inding. If a bias winding is used for core
reset,
then the presence of the winding and the negative bias of the core
wiI I
affect the switching action of the saturable inductor. For
instance,
variations from pulse-to-pulse in the pre-switch condition achieved
by
the bias current will result in jitter.
The length of the switch delay may be varied by varying the
amount
of bias flux applied to the core, as illustrated in Figure
lll-l.
With
out the aid of reset, the core wi M relax to point 1. If a
magnetic
intensity of -H ^ is applied to the core, the magnetic core will
reset
to point 2, allowing a switching time delay of
AB^
where V Is the voltage applied to the inductor during switch delay
and
AB refers to the positive change in flux density experienced by
the
magnetic core before saturating, as indicated in Figure
111-1. In
order to provide maximum switch delay, a reset magnetic intensity
of
-H should be induced in the core, allowing the magnetic core to
cycle
over the entire hysteresis loop.
17
Indicating the Approximate Change in Induction Available
for a Given Pre-SwItch Condition
19
The dependence of the switch delay, t , on the pre-switch
condition
of the magnetic core may be determined in general by examining
Figure 1.
Switch delay as a function of change in induction, AB, may be
expressed
a s :
N AAB
t^ - ^ ^ (111-2)
where N is the number of turns in the switching winding and A Is
the
cross-sectional area of the magnetic core. The change in induction
may
also be expressed as:
^ r
Th
e reset magnetic intensity of -H is produced by the reset
current.
I ; i.e.,
where
2 .
is the magnetic length of the core and N^ refers to the
number
of turns on the winding providing the reset current. This winding
may
be either the switching winding or an auxiliary bias winding.
Therefore,
N Ay y N I
s o r r r / ,•, cv
* d = v l • 5 '
It should be noted that the maximum delay of a saturable inductor
is
limited by the magnetic characteristics of the core such that
N A2B
V
20
where B Is the maximum induction of the magnetic material that may
be
achieved before saturation.
Core reset may be achieved either with a constant dc bias
current
or with a reverse current pulse that occurs after the energy
transfer
is complete. Switch operation is influenced by the method of reset
em
ployed. Reset achieved by a reverse current pulse might induce a
pre-
switch condition corresponding to point (4) in Figure
lll-l, while a
constant dc current could maintain a pre-switch condition of point
(3 ).
Assume a constant dc current is applied to the bias winding
contin
uously. Before application of voltage to the inductor, the
initial
condition of the core corresponds to the magnetic intensity
produced by
the constant dc current, as indicated in Equation
II
1-4. Upon applica
tion of voltage to the switching inductor, positive current begins
to
flow in the switching winding. The Induced switching flux
counteracts
the bias flux, allowing positive magnetic intensity to build up in
the
magnetic core as the flux density in the core increases. When the
flux
density in the magnetic core reaches B , the core saturates and
energy
is transferred to the load. As the current begins to decrease In
the
switching winding, the magnetic intensity in the core begins to
decrease
and point (1) on the B-H curve of Figure lll-l is approached. At
this
point, the magnetic intensity induced by the switching current
cancels
the magnetic intensity induced by the bias current for a net H of
zero
in the core. As the switching current decreases further, a net
negative
magnetic intensity is induced in the core so that the core begins
to
reset. The pre-switch condition of point (3) is achieved when
the
switching current goes to zero.
Core reset occurs simultaneously with the cessation of current
in
the switching winding if the dc bias current is provided by a
constant
current supply. A constant current supply may be simulated by a
dc
voltage supply In series with a large inductance. This
configuration
allows a large voltage spike to be induced across the bias winding
when
the current in the switching winding ceases, resetting the
core.
Reset may also be achieved by the application of a reverse
flux
pulse to the core after energy transfer is complete. In this
case,
the pre-switch magnetic intensity is zero so that the pre-switch
condi
tion of the magnetic core might correspond to point (4) In
Figure lll-l.
As before, voltage is applied to the inductor, the inductor
saturates,
and energy is transferred to the load. When the current in the
switch
ing winding ceases after the energy transfer, the magnetic
intensity in
the core goes to zero so that the core operates at point CI) on the
B-H
curve. If the voltage is reapplied to the inductor while the
magnetic
core is operating at point CI ), no switch delay would occur;
instead,
the core would saturate immediately. To reset the core for
switching
operation, a negative magnetic intensity should be induced in the
core.
Core reset in the instance of a reverse current pulse after
energy
transfer is similar to the switch delay. Initially, the inductor
re
ceives a current pulse; the di/dt of the current pulse induces a
nega
tive voltage across the inductor. This negative voltage Induces
a
decrease in flux density while the current pulse induces a negative
mag
netic intensity resetting the core. This form of core reset
inherently
creates a reset time delay; this time delay may be determined by
recog
nizing that:
r
where t is the reset time and i(t) is the instantaneous current
that
r >.r
produces the reset magnetic intensity.
The presence of the bias winding has several effects. The
addition
of a bias winding increases the size and weight of the saturable
induc
tor.
For high voltage applications, the need for an insulation
layer
between the bias and switching winding also increases the winding
size
of the saturable inductor. The Inclusion of the bias winding and
insu
lation layer decreases the maximum amount of core window area that
may
be filled by the switching winding. The effects influence the size
of
the core chosen for use In a saturable inductor.
Because the switching and bias windings are magnetically
coupled,
the saturable inductor behaves as a transformer. It is desirable
to
minimize the transfer of energy to the bias winding for efficient
switch
ing. This implies that the coefficient of coupling between the bias
and
switching windings should be small during energy transfer. During
satu
ration, the core permeability approaches the permeability of air,
auto
matically reducing the coupling between the switching windings.
Methods
for reducing the coefficient of coupling to lower values are
discussed
in Chapter IV along with the effect of the bias winding on core
size
and geometry.
teristics of the inductive switch by affecting the initial
permeability
of the magnetic c o r e . For a pre-switch magnetic
force of H , shown in
Figure
1-2, the permeability of the magnetic core will remain
constant
during the switch delay. This implies that the delay inductance
will
remain constant so that the Inductor voltage and current during
switch
delay wi II be as shown In Figure I Il-3b for the cIrcuit of Figure
II I-3a.
The permeability of the core does not remain constant for a
pre-switch
magnetic intensity of H^, In this c a s e , the
pre-switch magnetic perme-
ability remains low until H = H . At this point, the core
"unsaturates",
y reverts to its unsaturated value, and the switch becomes capable
of
withstanding voltage. The change in switch inductance corresponds
to
the change in permeability; i.e., the inductance starts low then
unsatu
rates to a larger value for switch delay.
The amount of time the Inductor operates In the pre-delay
saturation
mode is relatively short compared to the switch delay t i m
e . This pre-
delay t i m e , t ,, may be determined from the change
in Induction, AB ,,
pd po
experienced by the core during operation in the pre-delay m
o d e , indicated
in Fi gure 1 Il-2a :
NAAB
t
pd V
2± (111-9)
Even though the inductor Is initially saturated, the switch does
not be
have as if it were a conducting switch; rather, it behaves as if it
were
a comparatively small inductance. This implies that the voltage
across
the saturable inductor does not appreciably change during the
pre-delay
Figure M 1-2 A-B-H Curve Used to Illustrate the Effect
of dc Bias on Switching Action
25
the circuit of Figure lll-3a, the effect of the pre-delay
saturation of
the core on the switching delay voltage and current are shown in
Figure
111-3?.
The use of a dc bias current to reset the magnetic core will
influ
ence the energy transfer operation. When the magnetic intensity In
the
core reaches the value of -H Indicated in Figure
I
ll-2|t, the core unsa
turates in the reverse direction so that the value of the switch
induc
tance becomes L . At this time, t , positive current may still be
flow-
u ' u
ing in the switching winding. The voltage and current of the
saturable
inductor In the circuit of Figure
I
Il-3a are affected as indicated in
Figure Ill-3d. Figure 1 M-3 d also shows the effect of
the use of a dc
bias on the overall performance of the saturable inductor by
presenting
the pre-delay, the switch delay, energy transfer, and reverse
unsatu-
ration in perspective.
The reverse unsaturation of the Inductive switch increases the
time
required to transfer energy to the load. The amount of time
increase is
dependent upon the application of the saturable inductor. As an
example,
the reverse unsaturation time, t , wi
II
I 1
Figure I Il-3a indicated an inductively charged
capacitor; the ini
tial charging current in this application will be:
I = /f^ Vosin (ujt)
I I
1-3 For an L-C Circuit, the Effect of Bias on the
Inductor
Voltage and Current is Shown on Varying Time Scales
27
where L is the unsaturated inductance, C is the value of
capacitance
u
being charged, and V is the supply voltage, as indicated in
Figure
IIl-3a.
At t = t ,, the core saturates. Since current through the
Inductor
cannot change Instantaneously, it can be shown that
I = / — ^ V si n( / = —^ ^ (t+t'-t,)) (t ,< t < t
)
^ -/^sat ^ A s a ^ ^ ^
the current flowing through the Inductor
when the core saturates.
The core unsaturates
(I
I
1-13)
^
^ sat sat
The factor t" - t accounts for the current flowing
through the inductor
winding when the core unsaturates.
The time at which the core unsatura
tes t may be determined by recalling
that at t = t^, H =-H^ (see
' u'
Figure
assuming
the
flux
N c N
Therefore, when the core unsaturates, the switching current at t =
t is
^ u
o
At t = T, the switching current goes to zero. Therefore, it can be
shown
from Equation 111-13 that
u / u
(I I 1-18)
when 1 = 0 . The reverse unsaturation t ime, t ,
may be expressed as
c ru
ru u
unsaturation time may be written as
/—;::> . / u . , ' ( / sat u r
( 1 I 1-20)
29
The inductor voltage and current for the circuit of Figure II I-3a
are
indicated in Figure lll-3d with t ,, t,, t , and t show n.
pd d u ru
For an inductor design Implementing a dc bias, the
maximum repeti
tion rate, or rep-rate, at which the inductor may be operated is
limited
by the dc bias. The maximum rep-rate, f , may be written
as
max ^
d et ru
where t is the time required for energy transfer for an inductor
reset
with dc current. The pre-delay unsaturation, t , occurs during
the
switching delay time because the change In induction during the
pre-
delay, A B . , Is considered part of the AB determined for design
purposes.
An inductor design employing a reverse bias pulse for reset incurs
the
same form of rep-rate limitation. In this case, the maximum
rep-rate
would be
d et "»-
where t' is the time required for energy transfer for an inductor
that
et
is reset with a reverse current bias pulse.
The time required for switch delay and energy transfer Is set
by
the application of the inductor and resulting inductor design. With
the
dc bias, the reverse unsaturation time is also inherent
in the inductor
30
the time required to achieve the reverse current maximum required
to
reset the core, as indicated by Equation
(MI-8).
A dc bias current may be supplied to a bias winding with the
circuit
of Figure
I I
1-4. The bias winding and the switching winding couple
to
gether to act as a transformer. Therefore, any voltage or current
pulse
applied to the switching winding will be transformed to the bias
winding.
For most saturable inductors, N « N , so the transformed voltage
pulse
will be relatively small. The Inductors of the bias circuit are
added
to approximate a constant current supply as discussed previously,
and
to protect the supply from the current pulse transformed to the
bias dur
ing energy transfer.
A reverse current pulse for core reset may be automatically
provided
by the system in which the saturable inductor is utilized. One such
cir
cuit is shown in Figure
I
1l-5a. The voltage and current of the PFN, sa
turable Inductor, and resistor are shown in Figure M l-5b. During
the
transfer of energy to the PFN, the PFN Is charged to approximately
twice
the supply voltage, V . After the voltage across the PFN reaches 2V
,
s -3
the PFN starts to discharge through the resistor and
inductor.
The reverse bias leakage current of the diode mav be sufficient
to
reset the magnetic core: if so. the resistor across the diode is
not
required. If a larger current is required for reset than the diode
will
orovlde.
V V N
r r
The time required to reset the core, t^, corresponds to the time
con
Figure II1-4 A Circuit Realization of a dc Constant Current
Supply for Reset Purposes
t
V
( b )
( c )
( d )
e )
Figure 111-5 A Circuit Providing a Reset Current Pulse After
Energy
Transfer with the Effect of the Reset Pulse on the
Inductor, PFN, Reset Resistor, and Diode Voltage
Current vs. Time
H 2V )
r s'
N R
If the reverse current from the diode Is used to reset the core,
then
the time required to reset the core corresponds to the recovery
time
of the diode.
From these forms of reset, several bias schemes for producing
a
desired pre-switch condition have been devised. By determining
the
effect of reset on the switching inductor and the system in which
the
inductor is to be utilized, the most effective form of reset for
an
application may be selected.
The physical configuration of a saturable inductor directly
affects
the operation of the inductor as a switch. Use of a saturable
inductor
results in a switch delay followed by a relatively fast energy
transfer.
The minimum time required for the energy transfer is determined In
part
by the saturated seIf-inductance L ,, which is affected by the
inductor
s a t ' ^
geometry. A bias winding used in conjunction with the switching
winding
implies the existence of a coefficient of coupling between the two
win
d i n g s , which is also affected by the inductor geometry.
The coefficient
of coupling In turn affects the amount of energy transformed to the
bias
circuit, thus affecting the switch efficiency. The geometry of the
in
ductor includes the winding configuration and the shape of the
ferro
magnetic core.
The primary geometrical factors are window area, core
cross-section
al area, core volume, magnetic length of the core, the thicknesses
of
the bias and switching windings, and the amount of insulation
between
the two windings. The window area refers to the area of the hole
in
the core. For a toroid, this area may be expressed as
W = Trr., (IV-1)
a id
where r is the inner radius of the core. The thickness of the
switch-
id
ing and bias windings refers to the depth of the windings on the
inside
of the core in the core window, measured radially from the core
toward
the center of the core window.
34
This chapter investigates the effect of inductor geometry on
the
speed of energy transfer, switch efficiency, and scaling of the
inductor
design to accommodate different stand-off voltages and conduction
cur
rents. The speed of energy transfer is limited by the
saturated self-
inductance of the switch. The saturated self-inductance, L ^, is
af-
' sat
fected by the core cross-sectional area and the magnetic length of
the
core, as indicated in Equation
(11-5).
The switch efficiency is affec
ted by the coupling coefficient, k, between the bias and switching
win
dings.
The coupling coefficient Is dependent upon the thickness of
the
switching and bias windings and the thickness of any insulation
layer
between the two windings, along with the core radius and the radius
of
the core window. The scaling proportions of the inductor are found
to
be dependent upon the stand-off voltage and conduction current in a
si
tuation where the ratio between the radius of the core window and
the
radius of the core is fixed.
Two core shapes commonly used in saturable inductors are the
toroid
and C-core, shown in Figure lV-1. For the toroid, it Is assumed
that
the wire is wound over the entire length of the toroid, thus
utilizing
al I of the core material. The C-core consists of two
C-chaped pieces of
ferromagnetic material placed together to form a square core. For
the
C-core, it is assumed that the wire is wound on just one leg of the
core.
This allows the C-core to be approximated as a solenoid in any
calcula
tion w here the winding shape has an affect.
Under saturated conditions, the relative permeability of the
core,
y , approaches unity. Indicating that a saturated inductor behaves
as
an air core inductor. As such, the saturated self-inductance, L^g^.
is
approximately
sat i^
where I is the current in the switching winding and H is the
magnetic
field intensity induced In the "air" core. The Integral is taken
over the
volume of the field. The saturated inductance is determined in this
in
stance for an inductor with one winding.
It is assumed that the length of the solenoid is large compared
to
the radius of the magnetic core, and the inner radius of the toroid
is
large compared to the radius of the magnetic core. Therefore, the
mag
netic intensity has only radial dependence for the solenoid so
that
H(r) =
(IV-3)
where f(r) is a unitless function describing the radial dependence
of
H ( r ) . As shown in Figure IV-2a by the cross-section of a
one-winding
inductor with a circular core, the radius, r, of Equation (IV-3 )
increa
ses from the center of the magnetic core to the outer edge of the
Induc
tor winding. Equation (IV-3 ) may also be used to approximate the
mag
netic Intensity for a toroid.
It may be assumed without major error that flux is
distributed
u n i
formly radially across the magnetic core. The radial dependence of
the
magnetic Intensity is shown for a solenoid In Figure IV-2b and for
a
toroid in Figure IV-2c. The radial dependence, f(r), may be
determined
from the winding distribution for a solecoid (C-core) as:
^ 2 C C S
F o r a t o r o i d , f ( r ) becom es
1 0 < r < r
V ^ ' = I X ' (IV-5)
' ^ - ^ c X ^ - V - i d ' ^ ^ s ' ^ ' - i d - ^ ' r
< r < r t a
a (2r. ,-a )
c c s
where r is the radius of the core , r. , Is the inner
radius of the toroid
c id
wi ndow, and a is the thick ness of the switching winding, as
Indicated
in Figure IV-2a.
The parabolic shape of H(r) for a toroid Is due to the
winding
dis
trib utio n. The winding on an inductor is normally layered. The
number
of turns in a layer is proportional to the circumference of the
window
area:
N^
° lirr (IV-6 )
wh ere N. is the number of turns in the first layer, and r^ is
equal to
r As more layers are wound, the available window area obviously
de-
id'
n n
switching
winding
magnetic
core
Figure lV-2 The Cross-Section of an Inductor with One Winding Shown
with
the Radial Dependence of the Magnetic Intensity In the Core
4 0
in number of turns per layer in a toroid implies that f(r) is
parabolic
as shown in Figure IV-2c and described by Equation
(IV-5).
self-inductance may be written as
2 2-n i r
^sat " ~ ^ ' ' ' ^^^^^ ^ ^^ ^^ ^® • (IV-8)
Since the magnetic intensity does not depend upon £ or 9, the
saturated
inductance may be expressed as
y N 2Tr /• ^
sat Z
From Equation (IV-9) and the radial dependence of magnetic
intensity ex
pressed in Equations (IV-4 ) and
(IV-5),
the following expressions for the
saturated self-inductance of a solenoid (C-core) and toroid may be
deter-
mlned:
C 61 s c s c
2
I = 9-L |r l(a +r )^+ — - ^ A-^ a -
^a (r - r . ,) - 2r. . r ) +
h I 2 s c ^2r. ,-a ) 2 s 3 s c id id c
id s
41
where Lp is the self-inductance of the solenoid
switching w inding, Ly
is the self-Inductance of the toroid switching
winding, and N is the
number of turns in the switching winding.
In general, the saturated self-inductance may
be simply expressed
as
sat I
where G Is a dimensionless factor accounting for
the effect of winding
geometry and A Is the cross-sectional area of
the magnetic core. The
factor G may be determined from Equation (IV-IO) for
a solenoid as
G^ = — V (a + 4r a + 6 r^) . (IV-13)
C 2 s c s c
6r
c
G = -4r ( (a +r + -r^—^ r (^a^-fa (r -
r. ,)-2r. ,r ) +
T 2 2 s c (2r. .-a ) 2 s 3 s c id id c
r
c
( 1V-14)
a 1 7 1 9 4 2
+ rA-^ a -Ta r -2 r . ,) + a ir. -r
) r - r. , r .
(2r.,-a )2 6 s 5 s c id s id c 3 id c
d s
for a toroid.
The change of Q>^ with
respect to winding
thickness is shown in
Figure IV-3; G^ as a function of a
is indicated In Figure IV-4. Due
to core geometry, the maximum winding thickness for
a toroid is r. and
II II II
a o o
4 4
normalized to one. The width of the C-core window, D, is chosen so
that
the window area and magnetic length of the C-core and toroid are
the
same for a specific core radius. A uniform window area implies that
the
same number of turns are wound on the two inductors at a specific
winding
thick ness. By maintaining a similar number of turns and magnetic
length,
any differences between Gp and G y at a specific core
radius are due to
core and winding geometry alone.
For a specific core radius, window area, and magnetic length,
the
increase of G with winding depth is less for the C-core inductor
than for
the toroidal Inductor, as indicated In Figures IV-3 and IV-4 . This
im
plies that the saturated inductance is less for the inductor wound
on a
C-core for specific winding dimensions. As indicated in Figure IV-4
, a
low saturated inductance may be achieved for a toroid from a
geometry
requiring a core radius that is small compared to the radius of the
win
dow, with the thickness of the switching winding less than half the
radius
of the window.
A bias winding wound over the switching winding would link the
same
flux as the switching winding, forming a simple two winding
transformer.
The amount of energy transformed to the bias winding during the
energy
transfer mode reduces the total energy transfer, thus affecting
the
effi
ciency of the switch. One way to maximize the switch efficiency
(neglec
ting losses) would be to minimize the coefficient of coupling
between the
two windings.
k = ^ < 1 (IV-15)
2 c s D
Figure IV-5 The Cross-Section of a Saturable Inductor Shown
With
the Magnetic Intensity vs. Radius for the Bias Winding
4 6
where L is the self-inductance of
the switching windinq, L^ . is the
saT
^ ^* bsat
self-inductance of the bias winding, and M Is
the mutual inductance [ 6 ] .
The mutual inductance may be expressed as
M =
current
the corresponding bias magnetic Intensity.
The cross-section of a saturable Inductor with bias
winding is shown
in Figure IV-5a; the radial dependence of H (r)
is Indicated in Figure
lV-5b.
The
thickness
as the
toroid bias
indlng. The radial dependence of H (r) may
be expressed as:
w
c s
1 — ^ r + a + < r < r + a + A + a .
a.
(lV-17)
where a^ is the thickness of the bias
winding, and A is the thickness
b
of
the
saturated self-inductance
of the
bias winding
may be determined in a manner similar to
the saturated self-inductance
of the switching self-inductance:
47
2
y_N
k . . . = - 2 . ^ 2 - I ^ a ^ + ^ a. ( r +a +A ) + ( r +a + A ) ^ 1
. ( I V - 1 8 )
D s a t 36 6 b 3 b c s c s
The saturated self-inductance of the switching winding
remains the same
as determined for a one-winding inductor.
Therefore, the mutual induc
tance for the inductor wound on C-core may
be determined using Equation
( I V - 1 6 ) :
(— a^ + r a + r
IV-19)
^ i 3 s c s c
The mutual inductance for the toroidal inductor may
be expressed as
y N N, , ^ a . ^
MT = 5 I ^ (a + r ) + — V
(-r a +
id s
3 s c id c id
From the mutual
Inductances, the self-inductances of the bias
windings,
and the self-inductances of the switching
windings, the coefficient of
coupling for the C-core, kp, and
the toroid, k^, may be determined:
[| a^ + r a + r^]
k^ =
(IV-21)
^
[ y ( a + r ^ )^+ :r—^ ) ( j a A l a ( r - 2 r . ,
) - r r . . ) ]
2 s c 2 r j ( j - a 5 4 s 3 s c i d c i d
[ ^ ^ a . ^ + | - a . ( r ^ + a +A) + ( r + a + A ) ^ ) ( ^ ( a + r
)
+
^ s 1 2 2 ^Q 1 ^ 1 9
( 2 r j . -a ) 2 s 3 s c id id c ( 2 r . , -a )^ 6 s 5 s c id
l a s I d s
+ a r . . ( r . , - r ) + T ' - J I ' ) ) ]
^
s id I d c 3 Id c
( I V - 2 2 )
The coupling coefficient for a C-core inductor
is shown In Figure
IV-6 as a function of switching winding
thickness. As before, t he width
of the C-core window, D, varies with
the core radius to achieve the same
window area and magnetic length as
the toroid. The width of the insula
tion layer and the bias w inding
are assumed to be .ID.
The coupling co
efficient for a toroidal Inductor
Is shown as a function of a
in Figure
IV-7. where r.^ is normalized
to one. For the toroid, a. and A are a s
-
I d D
Figure IV-6 indicates that for a solenoid
approximation, the coef
ficient of coupling for a particular core
geometry varies little with
the winding thickness or a change in core
radius. The coefficient of
coupling for a toroidal inductor. Indicated
in Figure IV-7, shows a
configurations, the coefficient of coupling will drop from near
unity
while the core is unsaturated to .2 upon saturation.
The general performance capabilities of saturable inductors as
high
power switches can be evaluated in part by Investigation of the
geometri
cal constraints imposed on inductor design by the peak current,
stand
off voltage, and switch delay required. This evaluation may be
obtained
by scaling the inductor design for various stand-off voltages and
conduc
tion currents while maintaining a constant switch delay. Several
factors
that may be used to determine core performance are number of turns
in the
switching w inding, saturated inductance, switching Dl/dt, and core
volume.
The geometrical factors that will be affected by scaling are
window
area, core cross-sectional area, magnetic length of the core, and
core
volume. The minimum window area is specified by the number of turns
and
wire cross-section required for a specific conduction current. The
core
cross-sectional area is specified in part by the stand-off voltage.
The
core cross-sectional area and window area determine the magnetic
length
while the core volume may be determined from the magnetic length
and
cross-sectional area of the core. Therefore, the magnetic length
and
volume of the core are affected by the stand-off voltage and
conduction
current.
The number of turns may be expressed in terms of conduction
current
and winding geometry by recognizing the physical limitations
presented
by the window area and conductor material. The number of turns may
be
written as
52
where A^ is the area of conducting wire in the switching w inding
and
r^ is the radius of a single conductor. The area, A , may be
determined
from the area of the switching w inding and the area lost to the "
pack ing"
factor and insulation. The packing factor arises from the use of
round
wire and reduces the available area for current conduction such
that
A . = .75 A (IV-24)
WI re s
where A^r^g 's the area actually filled by wire and A Is the area
of
the switching winding. The amount of insulation on the wire will
depend
upon the stand-off voltage and number of turns; assume that the
insula
tion of the conductor accounts for 1/3 of the winding area so
that
A = .5 A . (IV-25)
c s
The area of the switching winding is determined in the plane of the
core
window . In terms of the thick ness of the switching winding, a ,
and the
radius of the window, the area of the switching winding may be
expressed
as
.5 a (2r. , - a )
53
It should be noted that the thickness of the switching winding is
less
than the radius of the core window due to the presence of the bias
win
ding and insulation layer.
For a given rms current, the wire radius may be determined from
the
allowable current density. A typical rms current density for pulse
power
applIcations Is
' max = 2.35 (10^) A/m^ (IV-28)
assuming a copper conductor [ 8]. This value for J is chosen for
safe-
ma x
ty reasons and may vary, depending upon the application and
conductor ma
t e r i a l . Based on the maximum allowable rms current
density, the conduc
tor area may be determined as
2 . /,
7T
w / rms
Therefore, the number of turns may be expressed in terms of the rms
cur
rent as
I
rms
The core radius and switching winding thickness may be specified
In
terms of the window radius by recognizing the desirability of
maintaining
54
saturation. The saturated inductance for
a toroidal inductor as a func
tion of core radius and switching winding
thickness is shown In Figure
4 5
w I d ^
a o a < r ^
,-^)M2-^r ^y
.
-
id
As the thickness of the switching
winding is increased, the satura
ted inductance also increases.
However, the coefficient of coupling de
creases with an increase in switching winding
thickness, as indicated In
Figure IV-7. A
low coefficient of coupling implies that a =
.8 r. , while
low saturated Inductance requires that a = .1 r. .
A compromise between
the desire for low k and low saturated
Inductance may be obtained by
choosing
r =
(IV-33)
Based on Equation (IV-31) and
the values for r^ and a^,
the number
of turns may be determined in terms of the
rms current and the window
radius:
E t^
N = ( IV-35)
ABA
where the area of the core. A, may
be written as
A = (.25 r . J ^ . (IV-36)
I d
From Eq ua t i on ( IV - 3 6 , t h e w indow ra d iu s may be w r i
t t e n i n t e rms o f N and
I so t h a t th e number o f tu rn s in Eq ua t ion ( I V -3 5)
becomes
J^
N = / -2 / . ( IV -3 7)
A B / I
The saturated inductance of the switching
winding may be written as
N^y A G
. CIV-38)
sat „
The relationship for N, L ,, dl/dt, and V may now
he written as
^ sat
N = - ' - '^^
N = 2. (IV-35)
ABA
where the area of the core. A, may be written as
A = (.25 r. j 2 . (IV-36 )
I d
From Equation (IV-36), the window radius may be written
in terms of N and
I so that the number of turns in Equation (IV-35) becomes
rms ^
/ A B / I
The saturated inductance of the switching winding may be written
as
N^y A G^
sat o
The relationship for N, L ,, dl/dt, and V may now be written
as
saT
-8 F^/^
L , = 1.7(10 °) ^
sat I
58
The expression for N Is shown in Figure IV-9 as a function of
stand
off voltage, E, and rms current, I . As expected, the number of
turns
rms ^
increases as the voltage and current increase, as shown in Figure
IV-9.
For the case where the voltage and current are scaled at the same
rate,
the number of turns remains constant. This is due to the fact that
the
increase in voltage requires an increase in core area to maintain
the same
switch delay for the same core material. This increase in core area
is off
set by the increase In core window area necessary for higher
currents.
The saturated Inductance Increases as the stand-off voltage is
in
creased, as indicated in Figure I V - 1 0 . This
implies that the dl/dt
capability of the switch decreases with an Increase in
stand-off
v o l
I V - 1 1 .
the inductor Is scaled, the relationship between voltage and
current
must be such that
lV-45)
The constant oc is added for the purpose of balancing units.
Figure IV-12 Indicates the change in core volume with respect
to
3/2
current and voltage. By specifying that al >. E , an increase
in
core volume occurs as indicated. The large increase in volume
required
to maintain a constant or increasing dl/dt with a scale to larger
cur
rents or voltages indicates that dl/dt vs. volume is a major
considera
tion In inductor design.
Figures IV-9 through IV-12 represent the scaling of an
inductor
for the case where
X J
c
d
The scaling relations are approximate and are not good over an
arbitra
ry range. The results obtained are general in that a change of
these
variables will affect only the constant of proportionality in
Equation
(IV-44). The exponentiona1 powers of E and I are Independent
of the
val ues of r , a , and t_,.
c s d
characteristics of the inductor in several
ways.
The winding depth
of the switching winding limits the dl/dt capabilities of the
switch
by increasing the saturated Inductance of the switch. The
percentage
of coupling between the switching and bias windings may effect the
effi
ciency of switch operation. Overall switch performance is not
main
tained by scaling E and I In a similar manner. If the inductor
Is
3/2
scaled in size to correspond with ai >. E , the dl/dt capability
of
the switch either remains constant or increases with an increase
in vol
tage and current.
design. The thermal considerations include core cooling and volume
of
the core required to prevent excessive temperature rise. The
inductor
losses include core and winding losses. The energy dissipated In
the
magnetic core Is comprised of hysteresis and eddy current losses.
The
2
winding losses consist primarily of I R losses in the conductor.
Total
energy losses may be represented in joules per pulse for a
particular
Inductor design. The joules/pulse losses may be used to determine
the
switch efficiency by comparing the energy loss with the amount of
energy
transferred.
The core losses may be used to specify the minimum core volume
re
quired to limit the core heating. It Is necessary to maintain a
tempe
rature in the core that Is lower than the Curie temperature [9] .
At the
Curie temperature, a ferromagnetic material becomes paramagnetic
[10].
The change from ferrcmagnetism to paramagnetism is also accompanied
by
a rise in the resistivity of the magnetic material and a decrease
in In
duction. By maintaining temperatures somewhat less than the Curie
tem
perature, resistivity and induction may be held approximately
constant
with respect to change In temperature. The core temperature
during
operation may be determined by calculating the amount of heat
produced
by the losses in the core and by eonsldering the manner in which
the
heat flows from the center of the core to the surface.
Thermodynamic
6 4
65
considerations form an Important part of inductor design but are
beyond
the scope of this thesis and will not be considered here
[11].
The losses experienced by the magnetic core during a cycle
of ope
ration may be explained with the aid of the B-H curve of Figure
V-1.
Assume .that point (a) corresponds to the pre-switch condition.
Upon
application of voltage, the flux density in the core begins to
increase
as previously expressed in Equation 11-1. Eddy currents are induced
in
the core in response to the time rate of change of B:
^ = ^ ^ ^ (V-1)
dt NA
where e(t) is the voltage applied to the inductor. When the core
satu
rates at point (b ), the relative permeability of the magnetic
material
approaches unity, switching occurs, and the current In the winding
ra
pidly Increases. Simultaneously, a decrease in the voltage across
the
inductor occurs, thus the eddy current losses decrease as indicated
by
Equation (V-1). Since the eddy current losses are low
and the winding
2
current Is large, the I R losses of the switching winding dominate
du
ring saturation.
A magnetic material may be considered as consisting of many
small
magnetic domains [12 ]. When a magnetic field is applied to the
core,
the magnetic domains tend to align themselves with the field.
Physical
movement of these domains generates heat due to the friction
Incurred by
realignment. The area of region 1 of Figure 1 corresponds to the
energy
required to machanically align the magnetic domains in the
"forward" di
rection. " Forward" in this case is a matter of convention and
refers to
V-1 A R u o
- C _ n . 3 t . t . , C o . e C o s s e 3 . , . , . 3 p e c .
+0 Switch Operation
6 7
the opposite direction and the electrical energy (Region 2, Figure
V-1)
expended In aligning the domains is released in the form of heat.
The
energy loss during a complete cycle due to the hysteresis effect
Is
determined by:
W^ = Vol / H dB (V-2)
where Vol is the volume of the core, and the B-H loop is taken at
opera
ting frequency [13 .
Eddy current losses arise from the currents induced in the core
to
oppose the establishment of flux in the core. An estimation of
eddy
current loss for laminated cores under pulsed conditions has been
made
by W. S. Melville [ 14 ] . Melville assumes that
(a) the core material does^ not experience a
rapidly
changing permeability,
(b) the material at the surface of a lamination does
not experience a B-H cycle that is appreciably
different in characteristic from the interior of
lami
nation.
The first assumption implies that the core does not saturate. For
satu
rable inductors, Melville's estimation may be used to characterize
eddy
current losses before saturation. The second assumption Implies
that
the time delay, t,, is greater than the time constant of the
core lami
nation; i.e., the flux has sufficient time to penetrate the
lamination
during switch delay. The time constant of the core lamination Is
ex
plained in more detail later In this chapter.
A laminated magnetic core usually consists of a thin
lamination
wound spirally In some predetermined form. The eddy currents
circulate
in the cross-section plane of the lamination; this plane
corresponds to
the plane perpendicular to the flux. The magnetic intensity
produced
by the eddy currents tends to reduce the effect of the exciting
mag
netic intensity applied to the lamination. By taking an
average
exci
ting magnetic intensity within the core and considering the effect
that
the eddy currents have on this average H, an estimation of the
eddy
current losses may be obtained.
The cross-section of a magnetic lamination Is shown in Figure
V-2a.
An exciting magnetic intensity, H,,^, exists external to the
lamination;
inside the lamination, the exciting magnetic intensity consists of
eddy
current and magnetizing components. Eddy currents flow in the
lamina
tion to resist the change of flux. The effect of the eddy currents
on
the magnetizing force, H , Is more pronounced in the center of
the lami
nation, creating a skin effect as shown in Figure V-2b. The
magnetizing
intensity averaged over the width of the lamination may be
expressed as:
d/2
m d/2 / X X
where H is the net magnetizing force within the lamination. The
aver-
X
age value, H , Is shown in Figure V-2c. With respect to H^, the
average
value for the eddy current magnetic intensity Is
d/2
69
(a)
Figure V-2 A Typical Lamination in a Laminated Core with Width w
and
Thickness d Shown with the Effect of the Eddy Current
Magnetic Intensity on the Exciting Magnetic Intensity and
Magnetizing Magnetic Intensity
70
where H is the opposing magnetic Intensity due to eddy
currents.
ex
At the surface of the lamination (x = d/2. Figure
V-2a), the exciting
magnetic Intensity consists of eddy current and magnetizing
components
as Indicated in Figure V-2c. Thus, the exciting magnetic
Intensity
may be expressed as
d/2 e m
where H Is the eddy current component of the magnetic Intensity at
the
surface of the lamination. Since the magnetic intensity due to
the
eddy currents opposes the magnetizing H, it follows that at some
depth
within the lamination
X m ex
where H and H are at some distance x from the center of the
lamina-
X ex
tlon surface, as Indicated in Figure V-2c.
The voltage in an incremental strip of width Ax at a distance
x
from the lamination center (see Figure V-2a) is Induced by the
flux
between the strip and the laminar center. This voltage may be
expressed
as:
' X X
where w is the width of the lamination, and y is constant. The
eddy
current, i , in an incremental strip Ax wide may be expressed
as
i = ®^ Ax (V-9)
71
where I is the length of the lamination so that M x is
the cross-section
al area that the eddy current flows through and w Is the length of
the
current path which corresponds to the width of the lamination. This
im
plies that
1 ® ^ = — (V-11)
dx pw
w h e r e A x ->• 0 . T h e r e f o r e , an e x p r e s s i o n
f o r H a s a f u n c t i o n o f x may
be d e t e r m i n e d f ro m E q u a t i o n s ( V - 6 ) , ( V - 8
) , a nd ( V - 9 ) :
_ § > i = l i - i _ I (H - H ) d x . ( V - 1 2 )
dx p 3 t / m ex
By t a k i n g t h e L a p l a c e t r a n s f o r m , t h i s e q
u a t i o n becom es
2
^ ^ 2 p m e x
The solution to Equation (V-13 ) in the s-domain Is
/^{/v^A
H = H . .
ex m 1
where 3 . and ^^ ^^Y ^® f u n c t i o n s o f s .
The functions ^^ determined from boundary
condi
tions of the lamination. At the center of
the lamination where x = 0,
3H
dx
due to the spatial symmetry of the eddy
currents within the lamination.
By differentiation Equation (V-14) with respect to x
and applying the
boundary condition of Equation ( V - 1 5 )
, It can be seen that 6 = 0.
This imp
ex m 1
S u b s t i t u t i o n o f t h e e x p r e s s i o n f o r H i n t
o E q u a tio n ( V -4 ) y i e l d s
ex '
^f2
JH^
^ 1 = . ' y ^ 6 , ^m • ^^ -18 )
s i n h C / ^ s j )
T e r e f o re , t he mag ne t i c i n te n s i t y due to eddy cu
r re n t s may be exp ressed
as:
S ' n h ( / - s j )
72
For an applied unit step voltage, the magnetizing component of
the
exciting magnetic intensity is
ex ~ yNA s U
1- (V-21)
sinh(/fs f)
The time domain solution may be obtained by taking the inverse
Laplace
transform of Equation
( V - 2 1 ) :
ex = ^ / <* Z - ^ : = ; - (V22)
o / p a 4 ^ / p 2
where a is a constant. The value of a is determined
from the condition
sin h (/ ^a-^) = 0 so that
p 2
2 . 2
-n p47r
^x = IJA Z -2 •
73
At X = d/2, cos(-p-x) becomes (-1)" and H
becomes H ,^/^., implying
d ex e(d/2) ^ ' ^
2 2 ,
3 T
12p
The constant, T, Is usually referred to as
the lamination time constant.
From Equation ( V - 1 ) , it may be shown for
a constant applied voltage that
k
- f •
eddy current component may now be
expressed in terms of the ratio
t/x'-
^ -n27T2 t
e " y t ^ , ^ o
The eddy current losses may
be determined by integrating Equation
(V-29) over the change in induction during
switch delay:
AB
74
where Vol is the volume of the magnetic material in the core. A
con
stant applied voltage implies that
dB = ^ dt (V-32)
We = ^ I H^ dt . (V-32)
By substituting the magnetic intensity due to the eddy currents
expres
sed in Equation (V-29) Into Equation (V-32) and manipulating the
result,
the following solution may be obtained:
00
2_2
AB^
W =
n=l 4
n
In general, the eddy current magnetic Intensity and losses may be
ex
pressed as
e = y t T
The function ^p is graphed in Figure V-3 and $ in Figure
V-4 with res
pect to t/x.
The total losses experienced by an Inductor during one cycle
con-
2
I e h I
W_ = ^R t ^ (V-37 )
I et
where I is the average switching current over one pulse, t is
the
duration of the energy transfer pulse, and R is the resistance of
the
switching conductor. The total may be written as:
2
As discussed previously, the losses experienced during switch
delay
and saturation are eddy current, hysteresis, and winding losses.
During
reset,
the primary losses are due to hysteresis and eddy currents.
The
eddy current losses derived in Equation (V-3 3) are a function of
the
length of pulse applied to the inductor. This pulse duration would
cor
respond to the switch delay for the delay mode of operation and to
the
reset time for the reset mode. The hysteresis loss experienced by
the
core occurs partially during switch delay and partially during
reset.
For magnetic materials with a B-H curve such as Figure V-1, half of
the
hysteresis loss would occur during switching and the other half
during
reset. Therefore, the loss incurred during delay and energy
transfer,
W , may be expressed as
W = W (t^) + I W + I^R t^^ (V-39)
s e d 2 h et
^r " ^e^^r^
From the energy transferred during switching and the energy
loss/pulse,
the switch efficiency, n, may be determined
W + W
n = 1
(V-41)
where W is the energy transferred to the load per pulse by the
switch.
The foregoing analysis provides a procedure for determining
the
loss per unit volume of the ferromagnetic material and allows the
de
termination of switching efficiency for any particular design.
The
loss/unit volume along with appropriate thermal analysis will
verify a
design for temperature limitations and cooling requirements. The
elec
trical switch efficiency may be used to verify performance of a
design
for utilization in pulse power applications.
CHAPTER VI
MAGNETIC MATERIALS
The response of a saturable Inductor as a high power switch
is
closely related to the magnetic characteristics of the core
material.
The choice of core material for a switch application is dependent
upon
the desired switch behavior. A wide variety of magnetic materials
and
types of core construction that are suitable for use in saturable
in
ductors are currently available. By examining the characteristics
of
these cores with respect to the desired switching properties, the
suit-
ability of a material for a specific saturable inductor may be
deter
mined. Critical parameters that may affect material choice are
stand
off voltage, required efficiency, easy reset, etc.
Figure VI-1 illustrates the B-H curve of a material suitable
for
use in saturable inductor cores. The unsaturated permeability, y
,
provides a high unsaturated inductance for low energy transfer
during
the switch delay. The saturated permeability should be low (y = 1
)
to allow a low saturated inductance for a relatively fast energy
trans
fer during conduction. A saturated permeability of approximately
unity
also allows the bias and switching windings to effectively decouple
for
some designs during energy transfer Increasing switch efficiency
In
some applications (see Chapter IV).
The saturated relative permeability of the magnetic material
will
in most cases approximate unity for high currents during energy
trans
fer. The squareness ratio Indicates the amount of current
(magnetic
intensity) required after saturation of the magnetic material to
force
79
80
the permeability to one. The squareness
ratio is the ratio of residual
induction to saturated induction.
The closer the squareness ratio is
to unity, the less conduction
current is required to force y to
one
after the core has saturated.
The "knee" of the B-H curve
should be square; the " knee"
refers to
the transition region from unsaturated to saturated
operation on the
B-H curve. A square " knee" implies an abrupt
transition between " open"
and " closed" states of a saturable inductor.
The saturated inductance is affected by
the change in induction,
AB, required to saturate the core.
The number of turns in the switching
winding ig inversely proportional to the change
induction, AB , so that
for a step applled voltage.
N = ^ . (VI-1)
2
The saturated inductance is directly proportional to N so that
from
Equations I 1-5) and Vl - l ) ,
2 2
L ^ = ^ ^ - ^ . VI-2)
^^ ^ AAB^ I
Therefore, a large available
change In induction
implies a relatively low
saturated inductance for a given inductor geometry.
The available change
In Induction for delay
purposes Is limited to the linear
portion of the
B-H curve where y is large. The maximum
induction before saturation, B^,
is an approximate Indication of
the change in Induction for large
y^. The
value for B Is usually determined at some point
above the knee of the
B-H curve (see Figure VI-1). If the knee of
the curve is rounded, then
some value of Induction lower
than B^ must be used to determine
the AB
available for switch delay.
Figure VI-1 A B-H Curve Illustrating Characteristics of a
Magnetic
Material that May Be Used in Comparison of Core
Materials for Use in Saturable Inductors
82
For small hysteresis losses, the coercive force, H , of the
mag-
c
netic material should be low; a low coercive force also allows
easy
reset. A high resistivity, p. Indicates a low eddy current loss
because
the magnitude of the eddy currents in the material are directly
affec
ted by the electrical resistivity of the material.
The Curie temperature, T , of the magnetic material affects
core
volume requirements. A high Curie temperature Indicates that a
large
energy may be released in the core In the form of heat without
seriously
affecting the magnetic properties of the material. This indicates
that
the minimum volume required for a saturable inductor designed for
a
specific application is limited by the core losses and by the
Curie
temperature of the magnetic material. The temperature restrictions
of
the winding insulation may limit the internal temperature of the
induc
tor to an even lower value.
Magnetic materials are manufactured in a variety
of ways. Magnetic
materials suited for use in saturable inductors usually consist of
iron
or iron oxides combined with other materials such as silicon,
nickel,
or cobalt in varying percentages. The presence of other elements
in
small percentages may dramatically change the magnetic properties
of
the material [15 ].
mini
mize possible eddy current losses In the core. "Tape wound" or
"fer-
rite" cores are examples of cores constructed for different
operational
requirements. Other types of cores are available, but their
character
istics are not as suited for use in saturable inductors as the
tape
wound or ferrite core.
83
A tape wound core is made from a magnetic alloy that can be
rolled
into a continuous strip. The core is formed by winding a narrow
width
of the tape material Into a predetermined shape, usually toroidal [
16 ] .
The thinner the tape Is rolled, the less area the eddy currents
have in
which to circulate. This implies that a core with a small tape
thick
ness would have a relatively low eddy current loss. A
small tape thick
ness also indicates that flux penetration to the center of the tape
may
be achieved In shorter times. Two forms of alloys are manufactured
in
tape form: the metallic alloy, and the amorphous alloy. The
metallic
alloy has a crystalline atomic structure while the amorphous alloy
has
a random atomic structure similar to glass.
A ferrite core consists of a mixture of crystals of iron oxide
with
various other metallic oxides. The additional metallic oxide might
be
magnesium oxide, nickel oxide, or zinc oxide. The ferrite core is
a
uniform, solid body similar in texture and mechanical properties
to
oxide or silicate bodies [ 17 ] .
A comparison of the basic magnetic characteristics for several
tape
wound cores of metallic and amorphous alloy and a typical ferrite
core
is presented in Table Vl-J. The characteristics compared are
unsatura
ted permeability, y , maximum induction, B^, residual induction, B
,
saturation induction, B , the squareness ratio, coercive force, H.
,
s ^
resistivity, p. Curie temperature, T^, and average watts/kg loss
for
6 0 cycle operation. The values of Table VI-1 are average values
taken
from several manufacturer's specifications.
The watts/kg rating presented in Table VI-1 s useful only as
a
comparative value for the materials presented. Since the
frequencies
•si- ^ ^ c n r ^
85
at which the saturable inductor is operated are high, the losses
indu
ced in the core during switching will be higher than the losses
Induced
at 60 cycles. The Initial permeability may also be used only as a
means
for comparison because the y^ values for the tape wound cores were
de
termined at 4 00 Hz by using the constant current flux reset, CCFR,
test
method [18 ]. The initial permeability of some cores tends to
decrease
at higher frequencies. To determine the actual initial or
saturated
permeability for design purposes, a pulse magnetization curve for
the
desired operational pulse width (switch delay) should be
examined.
As indicated in Table Vl-l, the amorphous materials have a
lower
coercive force and core loss than the metallic or the ferrite
materials.
However, the squareness ratio and initial permeability are also
lower
than the average metallic alloy. The metallic alloys have a higher
max
imum induction and squareness ratio than either the ferrite or
amorphous
materials.
The ferrite materials have a very high resistivity.
Indica
ting a very low eddy current loss. However, the maximum
induction, B ,
is low in comparison to the amorphous and metal Iic materials. The
ini-
permeabillty of the ferrite material does not tend to decrease as
much
as the tape wound cores for operation at higher frequencies.
Cores made of ferrite material have low losses and therefore may
be
operated at higher rep-rates than most tape wound cores. The tape
thick
ness of tape wound cores limits the maximum rep-rate at which the
core
may be operated. This limitation Is due in part to excessive
heating
from eddy currents. The loss ratings of the three types of
materials
86
the ferrite or metallic tape cores. However, the Curie temperature
of
the amorphous material indicates that a core made of this material
can
not tolerate as high a temperature rise as either the ferrite or
metal-
I
ic tape cores. Therefore, the reduction in volume obtained by
low
loss in amorphous materials Is offset in part by the low Curie
tempera
ture. •
Three types of cores are considered suitable for use in
saturable
inductors. They are ferrite cores and tape wound cores made of
amor
phous or metallic tape. The magnetic characteristics of these
cores
used for comparison are Initial permeability, maximum
induction, resi
dual induction, saturation Induction, squareness ratio, coercive
force,
resistivity. Curie temperature, and average watts/kg loss for 6 0
cycle
operation.
Based on magnetic characteristics, the response of these
materials
as cores In saturable Inductors may be determined in general.
The
watts/kg rating In conjunction with the Curie temperature indicates
the
volume requirements for a desired stand-off voltage and conduction
cur
rent for the saturable Inductor. The resistivity of the material
may
be used as a rough indication of the eddy current loss.
A squareness
ratio near unity implies that the saturated permeability rapidly
ap
proaches one during saturation. The relative permeability
partially
determines the amount of energy transfer to the load during switch
de
lay. The maximum induction before saturation determines the number
of
turns for a specific application, thus affecting the saturated
induc
tance.
will determine which of these magnetic characteristics are most
criti
cal. Based on the preferred characteristics, a material may
be chosen
which best suits the application.
V I I
MAGNETIC SWITCH DESIGN
The design of a saturable inductor for use as a switch is
dependent
upon several factors. The initial design constraints are
stand-off
voltage and switch delay. Based on these values and the desired
switch
performance, the core material and core geometry may be chosen.
Core
parameters that affect design and consequently characteristics of
the
switch are window area of the core, cross-sectional area of the
magnetic
material in the core, magnetic length of the core, maximum
induction of
the magnetic material, unsaturated and saturated permeability, and
co
ercive force. These parameters allow the determination of the
number of
turns in the switching winding and the required reset current.
The
method of achieving reset also affects* the desl'gn and
characteristics
of switch performance as discussed in Chapter 111.
An example design may be useful in illustration of the manner
in
which core material and geometry are determined and switch
performance
evaluated. Figure VI1-1 indicates the circuit in which the
saturable
inductor is to be utilized. In this application, the saturable
induc
tor is utilized as a charge delay switch (see Chapter 11).
The performance characteristics of the magnetic switch that
are
of importance in this application are the switch delay, the
current
during switch delay or hold-off current, and the energy transfer
time.
The choice of switch delay is dependent upon the recovery time of
the
switch (hydrogen thyratron, SCR, etc.) and should be long enough
to
prevent discharge switch reclosure during switch delay. The
hold-off
Inductor as Switch Delay
To choose
discharge
switch and the length of the discharge
pulse of the PFN should be taken
into account. Because
d
PFN rec
where Xppj^ i s the len gth o f th e d isc ha rge pu ls e and t Is
the rec ove ry
t i m e f o r t h e t h y r a t r o n u se d a s d i s c h a r g e
s w i t c h . F or t h e s w i t c h d e l a y ,
an a p p r o p r i a t e v a l u e f o r a s t a n d - o f f v o l
t a g e o f 3 kV m i g h t b e :
t , = 40 ysec .
m
commonly called silicon steel. Indicated in Table
Vl-l. As shown in
Equation (VI-2), the saturated inductance
is
tively large cross-sectional area is desired. Since
cores are generally
constructed
in
chosen.
The lamination thickness of the magnetic tape in
the core may be
determined from
t e
V, the
switch delay
time constant as expressed in Equation (V-27) is
,2
.2
d</— (VI1-3)
y y
^o^r
where d Is the lamination thickness. As presented in Table VI-1,
the
resistivity for silicon steel is p = 5(10~ )Q-fr\. The
value for y is
r
approximately 3500 for a pulse duration of 4 0 ysec. Therefore,
the
lamination thickness may be determined as
d < 2.3(10""^) m
-5
The lamination thickness for this application is chosen at 2.54(10
) m
to insure flux penetration of the lamination. As a result, a
silicon
steel core is chosen for use in the saturable inductor with the
follow
ing physical dimensions:
A = 13.1(10""^) m^
-5
92
The dc B-H curve for silicon steel is shown In
Figure VI1-2. At
pulse widths of 40 ysec, the B-H curve will be considerably
different
since y^ is
However, this B-H curve does provide an indication
of the response of the magnetic material In terms of maximum
induction,
saturated permeability, reset magnetic intensity, etc.
Due to the round knee, the maximum Induction that Is useful
for
switch delay is approxim