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