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NONLINEAR MAGNETIC SWITCHES FOR PULSE GENERATION by SUSAN E. BLACK, B.S. in E.E. A THESIS IN ELECTRICAL ENGINEERING Submitted to the Graduate .FacuIiy of Texas Tech University in Partial Fulfi IIment of the Requirements for the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING ^ppr:ov^d Accepted May, 1980 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. II TABLE OF CONTENTS I I ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES NTRODUCTION SWITCHING PERFORMANCE OF SATURABLE INDUCTORS II I IV V VI MAGNETIC CORE RESET 17 GEOMETRICAL CONSIDERATIONS 34 INDUCTOR LOSSES 64 MAGNETIC MATERIALS "79 MAGNETIC SWITCH DESIGN 88 LIST OF REFERENCES 103 I I I LIST OF TABLES Table VI-1 A Comparison of the Various Characteristics of Magnetic Materials Suitable for Use in Saturable Inductors IV 84 LIST OF FIGURES Figure 1-1 1-2 I 1-1 11-2 11-3 11-4 I I 1-1 I I 1-2 I I 1-3 I I 1-4 A Block Diagram for a Typical Pulsed Power Network Shown with the Circuit Energy Flow vs. Time 2 A Typical B-H Curve for a Magnetic Material Suitable for Use in Saturable Inductors 4 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 A B-H Curve Used to Illustrate the Need for Magnetic Core Reset 8 A Charge Delay Utilizing a Saturable Inductor With Inductor and PFN Voltage and Current Shown vs. Time 12 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 The Hysteresis Characteristic for a Ferromagnetic Material Indicating the Approximate Change in Induction Available for a Given Pre-Switch Condition 18 A B-H Curve Used to I I lustrate the Effect of dc Bias on Switching Action 24 For an L-C Circuit, the Effect of Bias on the Inductor Voltage and Current Is Shown on Varying Time Sea I es 26 A Circuit Realization of a dc Constant Current Supply for Reset Purposes 31 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 I I 1-5 Figure IV-3 IV-4 IV-5 IV-6 IV-7 IV-8 The Inductive Geometry Factor for a Solenoidal Core vs. Winding Thickness for Various Core Radii 42 The Inductive Geometry Factor for a Toroidal Core vs. Winding Thickness for Various Core Radii 43 The Cross-Section of a Saturable Inductor Shown with the Magnetic Intensity vs. Radius for the Bias Winding 45 The Coefficient of Coupling for a C-Core Inductor vs. Winding Thickness for Various Core Radii 49 The Coefficient of Coupling for a Toroidal Inductor vs. Winding Thickness for Various Core Radii 50 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, I , for a limited Range of Applicability . . . . 59 IV-10 A Representative Function for the Saturated Inductance Scaled with Stand-off Voltage, E, and rms current, rms IV-11 IV-12 V-1 V-2 V-3 V-4 A Representative Function for dl/dt based on Saturated Inductance Scaled with Stand-off Voltage, E, and rms Current, I rms Core Volume Scaled with the Stand-off Voltage, E, and rms Conduction Current, I rms A B-H Curve Illustrating Core Losses with Respect to Switch Operation 60 61 62 66 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 The Hysteresis Function vs. the Ratio of the Pulse Duration, t. Over the Lamination Time Constant,! 75 The Eddy Current Loss Function vs. the Ratio of the Pulse Duration, t, over the Lamination Time Constant, T 76 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 31 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 40 ysec at 3 kV. The Standoff voltages applied to the Inductor are (a) 3 kV, (b) 2 kV, (c) 1 kV 99 VI1-1 VI CHAPTER I INTRODUCTION 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 switching 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 with 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 applications are thyratrons and spark gaps; the "closing" action of these devices may be characterized as a transition from a high to low Impedance, 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 comparatively inexpensive. Nonlinear inductors achieve their switching action by changing from a high to a low inductance as the ferromagnetic core of the STORAtit ELEMENT CHARGING SWITCH / ^ > LOAD SYSTEM (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 Inductor core Is cyclic and that the switching action of the saturable inductor is dynamic in that the transition to a closed state Is accompllshed 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 considerations and operational constraints necessary for satisfactory performance 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 reset 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. the magnetic core are examined in Chapter III. Methods for resetting The effect of physical geometry on the Inductive switch and optimizations of these geometries saturated region Cow inductance) Figure ,-2 A T y p i c a l B-H Curve f o r a Magnetic Mater \a\ S u i t a b l e f o r Usf 'n Saturable Inducto rs 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 procedure 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 . CHAPTER I I SWITCHING PERFORMANCE OF SATURABLE INDUCTORS The switching action of a saturable Inductor Is achieved by utilizing 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 Inherently operates in three modes: 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 is 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 V dt (11-1) J where A is the cross section of the magnetic core, N Is the number of turns in the inductor winding, and V is the voltage applied to the inductor. When the flux density in the magnetic core reaches the satura- tion value, B , the permeability of the core rapidly approaches that of saturable inductor jy/r^ ^ t=0 V . = . R (a) saturable inductor Figure 11-1 A Simple L-R CIrcuit I 11ustratlng the Switching Action of a Saturable Inductor with the Voltage and Current for the Inductor Shown vs. Time 8 Figure II-2 A R-w r, ^BHC..veUse.toM,ust.3tet.eNeed Por Magnetic Core Reset ] 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 transfer mode, characterized by low permeability and low inductance. low inductance is necessary for fast energy transfer. The Once saturation occurs, the magnetic intensity, H, of the magnetic core begins to increase with the Increase in current that accompanies energy transfer. After the energy transfer is complete, the current in the inductor and H in the core go to zero; In Figure 11-2. the magnetic core then operates at point (b) 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 might then include a reset or bias winding as well as the switching 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 power applications, then the relationship between time delay and stand-off voltage is approximately t =^^^^ ^d V. where V (11-2) is the voltage applied to the inductor and AB Is the change in L Induction required by the core material t o achieve s a t u r a t i o n . In 10 Equation (I 1-2), 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 losses (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. The unsaturated inductance may be determined as: N^y L u A ^r-2- = U where y (11-4) X, is the relative permeability of the unsaturated core, y is the permeability of air, and I is the magnetic length of the core. Is assumed in Equation 11-4 that y It is large enough that most of the flux density produced by the switching winding is contained in the magnetic core. Upon saturation, the inductance of the switch becomes N^y L . = sat where y y AG V ^ (11-5) 36 is the saturated permeability and G Is a multiplying factor due s to winding geometry. characteristic, y For a magnetic core with a relatively square B-H Is approximately equal to unity. This implies that 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 greater 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 winding. The amount of current required to reset the core may be deter- mined as: I = r H £ - ^ N (11-6) 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 analyzing 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 as command charge and allow the discharge switch adequate recovery time 12 L PFN jyyr\—^1 V :^ Switch . rrrry / (a) (b) V Z' PFN / / / / / (0 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 saturable inductor. The charging voltage initially drops across 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 recover. 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 works well for designs using a core with a very low coercive force, H , so that a smaI I reverse current will reset the core. ' c' For cores requiring larger bias currents, application of the reset current 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. A saturable inductor used as discharge delay is shown in Figure ll-4a. This application of a saturable inductor allows a slight delay 14 PFN mm j>[. R L t r i gger pulse (a) V. (b) I V ^ ^ L^ PFN jrYY\—^ / i\ (c) 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 transferred to the load. This application requires a very low saturated in- ductance to keep the inductive effect on the discharge pulse to a minimum. 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 discharge delay involve the use of one inductive switch stage per application. 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" voltage, the number of turns in the inductor may be specified for a given core and core material as in Equation ( 11-2). The characteristics 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. CHAPTER I I I MAGNETIC CORE RESET The need for pulse-to-pulse repeatability in an inductive switch requires that the inductor core be reset to the same pre-switch condition before each application of voltage to the inductor. achieved by applying a negative flux to the core. Reset is The reset flux may be produced by a reverse current flowing in either the switching winding or an auxi Ilarybias winding. 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. out the aid of reset, the core wi M relax to point 1. With- 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 18 Figure lll-l The Hysteresis Characteristic for a Ferromagnetic Materia 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 as: N AAB ^ ^ t^- where N (111-2) 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: AB = yH . ^ r Th e reset magnetic intensity of -H (II1-3) is produced by the reset current. Ir ; i.e., * N I = - ^ H r (111-4) I 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 sAy oyr N rI r *d = vl / ,•, cv • """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 t^ < _s m V (I 11-6) 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. ployed. Switch operation is influenced by the method of reset em- 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 continuously. 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. 21 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 condition 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 negative voltage across the inductor. This negative voltage Induces a decrease in flux density while the current pulse induces a negative magnetic intensity resetting the core. This form of core reset inherently creates a reset time delay; this time delay may be determined by recognizing that: 22 I(t) = ^ i ^ r (111-7) so that H I i(t^) = - ^ (111-8) r where t r is the reset time and i(t) is the instantaneous current that >.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 inductor. 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 switching. 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, automatically 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. 23 The pre-switch magnetic intensity affects the initial delay characteristics of the inductive switch by affecting the initial permeability of the magnetic core. For a pre-switch magnetic force of H , shown in Figure I I 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 case, 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 unsaturates 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 time. This predelay time, t ,, may be determined from the change in Induction, AB ,, pd po experienced by the core during operation in the pre-delay mode, indicated in Fi gure 1 Il-2a : t pd NAAB 2± V ' (111-9) Even though the inductor Is initially saturated, the switch does not behave 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 saturation since t . is relatively small. For the saturable Inductor in 24 H I 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 influence the energy transfer operation. core reaches the value of -H When the magnetic intensity In the Indicated in Figure I ll-2|t, the core unsa- turates in the reverse direction so that the value of the switch inductance becomes L . At this time, t , positive current may still be flowu ' 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-3d 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 unsaturation 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. the reverse unsaturation time, t As an example, , wi II be determined for the inductor in the circuit of Figure I 1 l-3a. Figure I Il-3a indicated an inductively charged capacitor; the initial charging current in this application will be: I = /f^ Vosin (ujt) ^ ^u ( I I 1-10) w = 1/ /iTc^ 26 t=o (a) '•I, V. (b) V. 'pd (c) 'pd (d) Figure 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 u is the unsaturated inductance, C is the value of capacitance 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 sin(/=—^^ (t+t'-t,)) (t ,< t < t ) ^ -/^sat ^ A s a ^ ^ ^ (111-11) where (I I 1-12) ^ u u and L ^ is the saturated inductance. The factor f - t , accounts for sat ^ the current flowing through the Inductor when the core saturates. The core unsaturates at t = t^; the current for time t > t^ may be determined in a similar manner: (I I 1-13) t" = / r r ( ^ - arcsin ( / ^ ^ i n ( ^ ^ ^ (t^+f -t^))) ) . (111-14) ^ ^ 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 unsaturates t may be determined by recalling that at t = t^, H =-H^ (see ' u' Figure I I 1-2). By assuming the flux in the bias and switching windings are completely coupled, it can be seen that: 28 4r^ = I + ^ N c N Therefore, when the core unsaturates, "f sat / (II1-15) the switching current at t = t is ^ u sat Thi s impiles that L ^ sat (H -H )£ u r o At t = T, the switching current goes to zero. Therefore, it can be shown from Equation 111-13 that - t u (I I 1-18) + t" = T T / L C ^ / u when 1 = 0 . The reverse unsaturation time, t , may be expressed as c ru t = T - t . ru u (II 1-19) By substituting Equation (111-19) In Equation (111-18), the reverse unsaturation time may be written as /—;::> ^ru = / \ F . / u . , ' ( / sat ^'""'" /^s.n(TT-arcsin(/ -^ u r ,. ^ NV^-^^^- ( 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 shown. pd d u ru For an inductor design Implementing a dc bias, the maximum repetition 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 ^ max where t t ,+ t ^ + t d et ru (I 11-21) 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 predelay, 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 max ^ t , + t' + t^ d et "»- (111-22) 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 design, and may not easily be altered. By using a reverse current pulse to reset the inductor, some control over the reset time may be obtained. In this case, the reset time, t , may be decreased simply by decreasing 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 « will be relatively small. N , so the transformed voltage pulse 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 during energy transfer. A reverse current pulse for core reset may be automatically provided by the system in which the saturable inductor is utilized. cuit is shown in Figure I 1l-5a. One such cir- 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: required. if so. the resistor across the diode is not If a larger current is required for reset than the diode will orovlde. the value of R mav be chosen so that V r V N r The time required to reset the core, t^, corresponds to the time constant determined by the resistor and the inductor such that 31 V R Figure II1-4 A Circuit Realization of a dc Constant Current Supply for Reset Purposes 32 V saturable inductor (b) PFN (c) V PFN reset resistor Diode. t (d) V. (e) diode diode 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 33 •r " R '^ i ( \, HH 2V 2V j) • r s' N R (111-24) 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. CHAPTER IV GEOMETRICAL CONSIDERATIONS 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 geometry. ,, which is affected by the inductor sat' ^ A bias winding used in conjunction with the switching winding implies the existence of a coefficient of coupling between the two windings, 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 ferromagnetic core. The primary geometrical factors are window area, core cross-sectional 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.,^ a where r id (IV-1) id is the inner radius of the core. The thickness of the switch- 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 35 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 currents. 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 windings. 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 situation 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 calculation where 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. approximately As such, the saturated self-inductance, L^g^. is 36 v_y'c (a) T 1_ (b) Figure lV-1 Two Typical Core Forms Used in Saturable Inductors 37 L ^ =% sat where I I H^ dv (IV-2) 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 magnetic intensity has only radial dependence for the solenoid so that H(r) = J- fir) (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) increases from the center of the magnetic core to the outer edge of the Inductor 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 uniformly 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: 38 0 < r < r ^"^(1-) = < (IV-4) r-r 1 - -r— r < ^2 For a t o r o i d , f ( r ) + a C S becomes 1 V^'= r < r C 0 < r < r I X ' '^-^cX^-V-id' ^^s'^'-id-^' a (2r. ,-a ) s r< c r<r (IV-5) c ta s i d s where r is the radius of the core, r. , Is the inner radius of the toroid c id window, and a is the thickness of the switching winding, as Indicated in Figure IV-2a. The parabolic shape of H(r) for a toroid Is due to the winding distribution. 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) where N. is the number of turns in the first layer, and r^ is equal to r id' As more layers are wound, the available window area obviously de- creases so that N where r oc 2irr n n (IV-7) is the radius of the window after (n-1) layers have been wound Therefore, r is less than r. and N is less than N^. This decrease 39 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 and Winding Shown for (b) a Solenoid and (c) a Toroid 40 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). By substituting Equation (IV-3) into Equation (IV-2), the saturated self-inductance may be written as 2 ^sat " ~ ^ 2-n i r ' ' ' ^^^^^ ^ ^^ ^^ ^® • (IV-8) Since the magnetic intensity does not depend upon £ or 9, the saturated inductance may be expressed as y N^2Tr L ^ = - ^ — sat Z /• ^ f (r) r dr . (IV-9) From Equation (IV-9) and the radial dependence of magnetic intensity expressed in Equations (IV-4) and (IV-5), the following expressions for the saturated self-inductance of a solenoid (C-core) and toroid may be determlned: Lo = C M2 Try N ^ fn ^ 61 s o i3^+Ara+6r) c s c 2 I = h" 9-L I |rl(a +r )^+ — - ^ "^2 s c ^2r. ,-a id s A-^ a - ^a (r -r. ,) - 2r. . r ) ) 2 s 3 s c id id c + 41 where Lp is the self-inductance of the solenoid switching winding, 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 L N ^ y AG , = -^-rr— I sat , ( IV-12) ' 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 + 6r^) . C - 2 s c s c 6r c (IV-13) From Equation (IV-11), G becomes 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 i d s c ( 1V-14) a + 1 7 19 rA-^ a -Ta id s )2^6 s (2r.,-a 5 s 4 (r -2r. ,) + a ir. c id s -r id c 2 ) "r - r. , r '. 3 id c 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 for a C-core is D, as shown in Figure lV-1. For simplicity, r.^ is 42 ^o r ^ c\j CO r-H • II f-H Q II Q A II A m II II a a A u r^ (£) i n • • • • o S-. A O LO • • II A a to (O c • t—1 r—1 CSJ II II II a o o i . %. %- [-• o i- o o cy> .. (U •o •— o -o 00 c CD fO (T "o <D 00 .. r*«. L. O (0 o 1_ CO 3 0 s— o L. S_ (D 0 <^ -1- o (0 u_ >^ ., "^ L. 4Q) E O <U O > t_ O M- Cf) 1/1 <D c o -^ J^ CD .. ^ > •— -1U 3 •o c — . . CO 0 _cz f— rO 0) cn un LO hO) c .—. •o c s — • tn > 43 o • 1—1 II II T3 •f— T3 •r— LO O • • «—• CVJ 11 II • O "O •r— T— i- &- o s- s- &U -o •p- ^ -^ CO (T3 (J &. CD L. 0 o ;i CD •o •o (U •— O S_ (r o (D L. 1— 0 (U u (/) L. 3 o »4s_ 0 +u 0 t_ fD > (U Ll. >» s_ -((D £ O CD O CD > •— s- if) CO CD c ^ o -C hO) +U c ZJ •o •o c -~- •— CD -C 1— 1 > CD u 13 cn C3 !_ 0 c s • U) > 44 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 thickness. By maintaining a similar number of turns and magnetic length, any differences between Gp and Gy 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 window, 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 efficiency 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. The coefficient of coupling is k = ^ sat < 1 bsat (IV-15) 45 bias winding insulation layer switching winding magnetic core r,=r +a +A 1 c s r«=r^+a^+A+a. 2 c s D (b) Figure IV-5 The Cross-Section of a Saturable Inductor Shown With the Magnetic Intensity vs. Radius for the Bias Winding 46 where LsaT is the self-inductance of the switching^ windinq, ^* L^ bsat. is the self-inductance of the bias winding, and M Is the mutual inductance [ 6 ] . The mutual inductance may be expressed as M = YY' I ^b ^s '^'^ where I^ Is the current in the switching winding, H tensity induced by I^, I ^ '^"^^^ is the magnetic in- is the current in the bias winding, and H is the corresponding bias magnetic Intensity. The cross-section of a saturable Inductor with bias winding is shown in Figure IV-5a; lV-5b. the radial dependence of H (r) is Indicated in Figure The thickness of the bias winding is assumed small enough that a linear approximation for H, may be used for the toroid; therefore. Figure lV-5b represents H. (r) for the C-core as well as the toroid bias windlng. The radial dependence of H (r) may be expressed as: 0 < r < r + a +A c s (lV-17) f. ( r ) = b r-ir +a +A) —^ 1 a. r+a+A<r<r+a+A+a. c s c s b where a^ is the thickness of the bias winding, and A is the thickness b of the Insulation layer between the switching and bias windings [7]. Based on the radial representation for the magnetic Intensity due to the bias current, 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 Dsat 36 6 b 3 b c s c s . (IV-18) The saturated self-inductance of the switching winding remains the same as determined for a one-winding inductor. Therefore, the mutual inductance for the inductor wound on C-core may be determined using Equation (IV-16): y NN o sb ^r = — 7 i0 ^ i ,— 1a^9+^r a +^r 2 ( (t a^ + r a + r ) . 3 s c s c ( IV-19) The mutual inductance for the toroidal inductor may be expressed as yoNsN,b ^ , / , v2 ^ , as . 2 ^ ^, ,1 MT = 5 I2 ^ (as + rc) + — T £ (2r. ,-a V) (4-r a s id + s (IV-20) + 4- a (r - 2r.,) - r r.,) J 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^] '•3 s c s c k^ = (IV-21) 48 [ y ( a + r ^ )^+(:r—^ ) ( j a A l a (r-2r.,) 2 s c 2rj(j-a5 4 s 3 s c id -rr..)] c id ^ [ ^ ^ a . ^ + | - a . ( r ^ + a +A) + ( r + a + A ) ^ ) ( ^ ( a + r ) z^b b 3 b c s c s 2 s c ^s ( 2 r j .-a ) l a s 1 2 2 s 2 3 s + c id a r..(r.,-r s id Id id c c ^Q 1 "^ 1 9 ( 2 r . ,-a )^ 6 s 5 s I d s ) +T'"-J 3 Id I' c + c id ))]^ (IV-22) The coupling coefficient for a C-core inductor is shown In Figure IV-6 as a function of switching winding thickness. As before, the 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 winding are assumed to be .ID. The coupling co- efficient for a toroidal Inductor Is shown as a function of a IV-7. where r.^ is normalized to one. Id in Figure For the toroid, a. and A are asD sumed to be •1r. , each. Figure IV-6 indicates that for a solenoid approximation, the coefficient of coupling for a particular core geometry varies little with the winding thickness or a change in core radius. coupling for a toroidal The coefficient of inductor. Indicated in Figure IV-7, shows a much larger variation under saturated conditions. For some winding 49 CO in C\J V£> m II Q II A o • A r>. r^ • II o LD • CVJ «—1 i-H II II II • • 1—t «—• II Q A O ^ 1-H II Q « •>ir) t n c\j O • II O o u s- s- &. • &- II • to (B o S-o c» .,r^ u 3 -o c — CD ..vo !_ o o1 o (U •— •— -o (U cr 0) i_ 0 o cn D i_ o M- ..ir> CD C . — C2. n o CD c u •— ^ H U O) M- CD o O CD J: H- 1 1 > CD \•D O) CO U3 LO cn OJ «—t tn M— .-— 4- cr» o M- CO CD +c .,CNJ L. CJ o c^ o •— s _ T3 > J^ c -a c — s • cn > .— 50 LO o mM C Lf) T—t II II II XJ -o •I— *r~ S- %. to (C o (30 s_ o +u 3 .. r ^ •o c ,^_ . TJ (0 Q^ fD "O .. v o O L. 0 1— (U CD U o o cn 3 1_ o >+CD C •• LO •— — o l_ CD > s_ o CL :3 O CJ My M- c ^ u cn cn CD 0 -1- cn CD x: f- U (j) •— M- • — >4- •o CD o o CD JZ 1— .. C\i 1 1 > CD U 3 U) CT> CO KO m cn CM •— c c c •_ 2 • cn > 51 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 geometrical constraints imposed on inductor design by the peak current, standoff voltage, and switch delay required. This evaluation may be obtained by scaling the inductor design for various stand-off voltages and conduction currents while maintaining a constant switch delay. Several factors that may be used to determine core performance are number of turns in the switching winding, 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 N = A /TTr ^ c w (IV-23) 52 where A^ is the area of conducting wire in the switching winding and r^ is the radius of a single conductor. The area, A , may be determined from the area of the switching winding and the area lost to the "packing" 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 WI re s (IV-24) where A^r^g 's the area actually filled by wire and A the switching winding. Is the area of The amount of insulation on the wire will depend upon the stand-off voltage and number of turns; assume that the insulation of the conductor accounts for 1/3 of the winding area so that A = .5 A c . (IV-25) s The area of the switching winding is determined in the plane of the core window. In terms of the thickness of the switching winding, a , and the radius of the window, the area of the switching winding may be expressed as A = irr.^^ - 7r(r. . - a ) ^ . s id id s (IV-26) Therefore, the number of turns may be written as .5 a (2r. , - a ) N = ^ 2 r w . (IV-27) 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 winding 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 safema x ty reasons and may vary, depending upon the application and conductor material. Based on the maximum allowable rms current density, the conduc- tor area may be determined as 7T rw2 = .Irms /, /Jmax ^( 'lV-29) ^^' so that r^ w = 3.68(10>-4") / I / rms . (IV-30) Therefore, the number of turns may be expressed in terms of the rms current as 1.175(10^) a (2r. . - a ) N = '—^ ^ . I rms (IV-31) The core radius and switching winding thickness may be specified In terms of the window radius by recognizing the desirability of maintaining a low saturated inductance and low coefficient of coupling during 54 saturation. The saturated inductance for a toroidal inductor as a func- tion of core radius and switching winding thickness is shown In Figure r 4/r. ,5 is factored out to allow the plotted function w Id ^ to be dependent upon a and r only so that IV-8. The term a o a <^ r ^ (-^)2 ( 2 - - ^ ) 2 , _ c . x 2 ^ 4 (,-^)M2-^r(^y V. y ^" r. 7 V. . ^ ^T r^ L .,( (- ^ ) = It \. _ 5 ^ sa' y r. , ^ '^ ^^ ^^ i-^! r 8(1+-^) Id (IV-32) where r. , has been normalized to one. id As the thickness of the switching winding is increased, the saturated inductance also increases. However, the coefficient of coupling decreases 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 c = .25 r. , Id (IV-33) a s = .5 r. , id 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: 2 ) r. . — ^ . Irms 7.7(10 N = 11 (IV-34) 55 i_ 0 4U r? .—. — -o (0 01 •o c — — to T7 CD L. o o cn •— 3 l_ — o o h(D l_ o o L. (H > L. o M- M- CD u c CD C -ii: o •D — sz \- to 4- -o c — •o CD -f- <u V-8 l_ 3 4(U 00 CD 1_ 3 O) to Lf) CO <\J cn cn u cn c — •o c •— s • cn > 56 The number of turns may also be expressed as: 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 Equation ( I V - 3 6 ) , the window radius may be w r i t t e n in terms of N and I so t h a t t h e number o f t u r n s in Equation (IV-35) becomes J^ N = t (10 ) /14.1E -2 / AB / I / . (IV-37) The saturated inductance of the switching winding may be written as N^y A G L — . ^ = CIV-38) „ sat The relationship for N, L ,, dl/dt, and V may now he written as ^ sat N= L ,= sat -'-'^^ -8 1.7(30 °) F^/^ ^ I ( IV-44) — dt « 5.88(10"^) c^/2 -^To V = 3.9(10"®) (EI)^''^ 57 The number of turns may also be expressed as: N = 2. ABA (IV-35) 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 ^ N t.(lO^) -^ AB / = / 14.IE . (IV-37) I The saturated inductance of the switching winding may be written as L = sat N^y A G^ 2 !_ o . (IV-38) The relationship for N, L ,, dl/dt, and V may now be written as saT N= L ,= sat ii-Ii -8 F^/^ 1.7(10 °) ^ I ( IV-44) ^ dt V { 5.88(10^) = I/E^'^2 3.9(10"®) (EI)^^"^ 58 The expression for N Is shown in Figure IV-9 as a function of standoff 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 offset by the increase In core window area necessary for higher currents. The saturated Inductance Increases as the stand-off voltage is increased, as indicated in Figure IV-10. This implies that the dl/dt capability of the switch decreases with an Increase in stand-off voltage, as shown In Figure IV-11. For a constant or Increasing dl/dt as the inductor Is scaled, the relationship between voltage and current must be such that 3/2 al >E. ( 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 currents or voltages indicates that dl/dt vs. volume is a major consideration In inductor design. Figures IV-9 through IV-12 represent the scaling of an inductor for the case where 59 (sdiUB)^^-^! cn o CM O ^ o o o to o \ \ \ \ \ •o 0) ^— (U u CO cn c i_ rj fH- 0 !_ CD J3 S 3 Z CD ^ •^ -H C CD S_ 3 QJ c 0 .,_ -1U 3 "O O QJ !_ c •— c 3 u_ CD > — I 9V N CD O) ^ CD CJ) (U +o > — -+- c ro Ql "O CD -1- •— • s . ^ o (U c t_ +CO M- V-9 z f(,oi)M o UJ 1 •o CD L. 3 (J) O M- •^ -f- < o CL < +(0 cr CM — a. _J CD in CD L£3. 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CD Q^ < _. 1 > CD S_ 3 cn S9LUU9H) :^P/IP '>' • o •— 4U ( c JZ — -o 0 — (U (J 00 62 (sdu.e)=^I O LO o o o CM o 0 0 1 in c to £ i_ 4- co o — ^ 4- +c x: i_ i_ JC 0 4- .— s •o 0 (D U 00 3 o c 0 -(u 3 •o 0 E 3 "o > 0 L. o o CN 0 1_ 3 cn C\i o ro O (^UJ) LOA '^ "vT O to o to I o c O O cn E i_ •o c (0 63 r =.25 c B r a id = 1 Tesia =.5r., s id t, = 1 usee d The scaling relations are approximate and are not good over an arbitrary 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 Clearly, the configuration of the inductor affects the switching characteristics of the inductor in several ways. of the switching winding limits the The winding depth 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 efficiency 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 voltage and current. CHAPTER V INDUCTOR LOSSES The electrical power losses experienced by the saturable inductor affect the switch efficiency and thermal considerations of Inductor design. The thermal considerations include core cooling and volume of the core required to prevent excessive temperature rise. losses include core and winding losses. The inductor 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 required 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 Induction. 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. 64 Thermodynamic 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 operation 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: ^ dt = ^^^ NA where e(t) is the voltage applied to the inductor. (V-1) 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 rapidly 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 during 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" direction. "Forward" in this case is a matter of convention and refers to alignment during switch delay. During reset, the domains are forced in 66 figure V-1 A R u o -C_n.3t.t.,Co.eCosse3.,.,.3pec. +0 Switch Operation 67 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 operating 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 lamination; 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. 68 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 magnetic 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. tion to resist the change of flux. Eddy currents flow in the laminaThe effect of the eddy currents on the magnetizing force, H , Is more pronounced in the center of the lamination, 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 H where H = -7:7 m d/2 / / H X d X is the net magnetizing force within the lamination. (V-3) 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 -\- d/2 H J ex dx = 0 (V-4) 69 (a) Figure V-2 A Typical Lamination in a Laminated Core with Width Thickness d w and Shown with the Effect of the Eddy Current Magnetic Intensity on the Exciting Magnetic Intensity and Magnetizing Magnetic Intensity 70 where H ex is the opposing magnetic Intensity due to eddy currents. 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 H^/o = H + H d/2 e m where H (V-5) 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 H X = H - H m ex where H and H are at some distance X ex x (V-6) from the center of the lamina- 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: e = -wy^^t / H ' where w is the width of the lamination, and y d X (V-8) X is constant. The eddy current, i , in an incremental strip Ax wide may be expressed as i X = ®^ Ax pw (V-9) 71 where I is the length of the lamination so that M x is the cross-sectional 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 AH = ^'x/5, (V-10) — pw (V-11) ex so that .H 1®^= dx where Ax ->• 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 as a f u n c t i o n o f x may be d e t e r m i n e d from Equations ( V - 6 ) , ( V - 8 ) , and ( V - 9 ) : _§>i = l i - i _ I dx p 3t / (H - H ) dx . m ex (V-12) By t a k i n g t h e Laplace t r a n s f o r m , t h i s e q u a t i o n becomes 2 9 H ^ ^^2 - ii- s(H - H ) = 0 . p m ex (V-13) The solution to Equation (V-13) in the s-domain Is H = H ex m +. ^. e,cosh /^{/v^A ( / ^ x ) + 6^ c i ' s h ( / ^ x ) 1 where 3. and ^^ "^^Y ^® f u n c t i o n s o f s. (V-14) 72 The functions 3^ and ^^ may be determined from the boundary conditions of the lamination. At the center of the lamination where x = 0, 3H -T^ =0 CV-15) 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-15), It can be seen that 6 = 0. This imp Iles that ^«v = ^m "^ ^1 ^osh ( / ^ x ) . ex m 1 (V-16) S u b s t i t u t i o n of the expression f o r H i n t o Equation (V-4) y i e l d s ex ' ^f2 JH^ + 3^ COS h ( / ^ X yjdx = 0, (V-17) ^1= • ^^-18) so t h a t . ' y ^ 6 , sinhC/^sj) ^m T h e r e f o r e , the magnetic i n t e n s i t y due to eddy currents may be expressed as: r/ • ^ s : r c o s h ( / ^ s x) H =H il-^l-e-^ ^ ex mi_ - u z / V i d x S ' n h ( / -sj ) ^ I . (V.20) 72 For an applied unit step voltage, the magnetizing component of the exciting magnetic intensity is H - 1 E m s yNA ' (V-20) so that u - _ i _ 1 r, ex ~ yNA s U p^2 . /m r^ sinh(/fs f) S 1- (V-21) The time domain solution may be obtained by taking the inverse Laplace transform of Equation (V-21): E ^ex r V / <^* Z = ^ o where a is a constant. /£ic°sh( / l ^ -^:=; - ^ X) e" (V22) / p a 4 ^ / p 2 The value of a is determined from the condition sin h (/ ^a-^) = 0 so that p 2 2 . 2 -n p47r a = 1 d y (V-23) and 2£ ^ (-1) c o s ( - ^ x) (1 - e ^x = IJA Z -2 n=l n ) • (V-24) 73 At X = d/2, cos(-p-x) becomes (-1)" and H becomes H ,^/^., d ex e(d/2) that implying ^ ' ^ 00 "e = -^(d/2) = — 2 S ^ ^ "-25' n= 1 The constant, a, may also be expressed as 2 2 , a = - il^ 1 3 (V-26) T 2 T = -7^^^ 12p where (V-27) 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 • From Equations (V-27) and (V-28), the eddy current component may now be expressed in terms of the ratio t/x'^ _ A B T 6 e " y t^ -n27T2 t /Cl-e ^, ^ n= I ^ ^ ^ o z — . (V-29) The eddy current losses may be determined by integrating Equation (V-29) over the change in induction during switch delay: AB W = Vol I H^ d B (V-30) 74 where Vol is the volume of the magnetic material in the core. A constant applied voltage implies that dB = ^ dt (V-32) We = ^ I H^ dt . (V-32) so that By substituting the magnetic intensity due to the eddy currents expressed in Equation (V-29) Into Equation (V-32) and manipulating the result, the following solution may be obtained: 00 /AB^ T VAB^ W = e y , 18 T t K' y 2_2 (1-e ^ V - -) TT2 t n=l ^ ^ n4 — ) . (V-33) In general, the eddy current magnetic Intensity and losses may be expressed as T'P^-^ H = — e We y VAB = y (V-34) t ^ T t . —Tt $. (,— T ). / Tc^ (v-35) The function ^p is graphed in Figure V-3 and $ in Figure V-4 with respect to t/x. The total losses experienced by an Inductor during one cycle con2 sist of eddy current, hysterisis, and I R losses such that 75 H 4-) cn 0 CO 3 CL c 0 4- in O o (T3 c O o 0 E c 0 o c in > c O LO (tj 0 u c 3 cn cn 0 0 > o i_ 0 4in I 0 c o 3 Q I Lf) 0 L. 3 cn 4J 76 0 CO 3 Q_ 0 JZ O O c (0 4- cn c O O CO 0 0 £ CO > c O u c c O to c 3 i cn 0 c 0 > O in O 0 i_ s_ 3 O >^ c O •o UJ 0 S_ 3 Q I 0 s_ 3 CJ^ H 4-> 77 W-^ = I W^ + W. + W^ . e h I (V-36) 2 The I R losses are W_ I = I^R t ^ et (V-37) 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. W T = The total may be written as: 2 ^^^^^1 <D (:t) +Vol/ HdB + I^R t y t T et . (V-38) 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-33) 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 s = W (t^) + I W + I^R t^^ e d 2 h et (V-39) 78 while the energy loss during reset, W , is ^r " ^e^^r^ ^ 1 ^h ' (V-40) From the energy transferred during switching and the energy loss/pulse, the switch efficiency, n, may be determined W n = 1- + W \ ^ (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 determination 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 inductors are currently available. By examining the characteristics of these cores with respect to the desired switching properties, the suitability of a material for a specific saturable inductor may be determined. 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 transfer 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 transfer. 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 d i r e c t l y proportional to N so that from Equations (I 1-5) and ( V l - l ) , 2 2 y yG ^ ^ - ^ AAB^ I E V L ^ = ^^^ . (VI-2) 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. 31 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 magc 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 affected 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 inductor 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]. Several techniques are used in constructing magnetic cores to minimize 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. tape form: Two forms of alloys are manufactured in 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 60 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 is useful only as a comparative value for the materials presented. Since the frequencies 84 to cn E 0 •r— i- cn CO — in to O -J 4-> "M CT> LT) t—• >?r 0 r-t 2 4«S0 4Lf) r-l ^ + 4- +in vo •^ m LO <o (Tk r-H 1—1 LO (U </> 'r^ i- <u 4-> 0 <T3 &. rtJ x: 0 to 3 0 •r~ ^ ns r> OJ 0 +J 0 3 T3 C 1—• M0 e 0 to •r— s. a. E fO LO 0 «;r •si- - o f—t >_^ r^ LO 0 LO 0 LO 0 1—4 1—I 0 LO CVJ LO CO I—J r-* . • r>^ CVJ 1—1 >,_^ 1—t LO 0 vo ^ 0 CO '^ 0 CO 0 t—1 f-H - i _ ^ CVJ CO OD LO 0 «;r cn 0 C3^ r^ r-t • cn c 0 to 1—1 o 0) ^— ja fO &3 4-> «3 00 C • i ^ cu to Z3 (1) .4-> e 1—• to s. Ln 0 S0 It0) r— JO rtJ 4-> •r— 3 OO o 3 •a CO to <u 0 c •r— (U 4-> &. <a fO s. rj cr <Tt CO 10 LO CO LO LO m <Ti CO 00 CO LO CVI r—t I-H 1—1 o LO U3 CO CVJ kHz , max to 0 •r— 4-> 'V,^' Lf) 0 353 M0 LO 0 r-t 370 c cn (O 2: 00 fXJ r*". i - to CQ (U T—» LO LO r— to f-H LO CO 00 1—t LO 03 fO CVJ CQ s= 0 0 CJ cn LO CO 3 LO CO vo cn to LO CO "•N u: 0 0 «!l- <u 6^ LO 03 10 <u U- O ID o Li- o ?f« LO LO «« •r— o LU a. <c o LO to CJ 10 OJ (J- O CO Z ?S« 0 CO 3 0 m M 0 0 ^ {/) 0 LO cn t/1 CO o ex so CO •r00 LO CO +J 03 00 CJ {= CQ t—I O <^ (U LL. •p— E OS to 0) 1— CVJ • A r^ M t E t-H . .^1^ S- •i-> a: 0 u. »+c_> CQ 'vT O 2 CO CO 0 CO (U •r— -M 03 C r^ •0 OJ c E s. •r— I— •—« ce: -a <u to 0) +J Q:: CQ c^ n: 0 LO 03 -0 0) C •r~ E s- (U 4-> OJ 0 85 at which the saturable inductor is operated are high, the losses induced 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 determined at 400 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 indicate that for a specific applied voltage and conduction current, a smaller volume of amorphous material would be required than of either 86 the ferrite or metallic tape cores. However, the Curie temperature of the amorphous material indicates that a core made of this material cannot tolerate as high a temperature rise as either the ferrite or metalI ic tape cores. Therefore, the reduction in volume obtained by low loss in amorphous materials Is offset in part by the low Curie temperature. • 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, residual induction, saturation Induction, squareness ratio, coercive force, resistivity. Curie temperature, and average watts/kg loss for 60 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 current 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 approaches one during saturation. The relative permeability partially determines the amount of energy transfer to the load during switch delay. The maximum induction before saturation determines the number of turns for a specific application, thus affecting the saturated inductance. 87 The application of the saturable inductor and design requirements will determine which of these magnetic characteristics are most critical. Based on the preferred characteristics, a material may be chosen which best suits the application. 88 CHAPTER 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 coercive 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 current should be low enough that reclosure doesn't occur due to 89 PFN T = lOysec saturable inductor Vc — diode discharge . switch f ^ \ L- 6fi Figure VI1-1 The Design Circuit Utilizing a Saturable Inductor as Switch Delay 90 current in the switch after the PFN discharge. A short energy transfer time is desired for rapid charging of the PFN. To choose the switch delay, the recovery time for the discharge switch and the length of the discharge pulse of the PFN should be taken into account. Because the supply voltage drops across the Inductor while the discharge switch is conducting, the switch delay should be tn ^ "^DCM + T d PFN (VI 1-1) rec where Xppj^ is the length o f the discharge pulse and t time f o r the t h y r a t r o n used as discharge s w i t c h . Is the recovery For the switch delay, an a p p r o p r i a t e value f o r a s t a n d - o f f voltage of 3 kV might be: t , = 40 ysec . For a relatively low saturated inductance and thus fast energy transfer. It is desirable to use a core material with a large AB. A material with a relatively high maximum Induction, B , would be 5% Si 97^ Fe, m commonly called silicon steel. Indicated in Table Vl-l. As shown in Equation (VI-2), the saturated inductance is inversely proportional to the cross-sectional area of the core, therefore, a core with a comparatively large cross-sectional area is desired. Since cores are generally constructed in standard forms, a core constructed in C-core form with A = 13.1 (10~^)m and Jl = .267 m is chosen. The lamination thickness of the magnetic tape in the core may be determined from the switch delay, resistivity of th-e material, and unsaturated permeability. As discussed in Chapter V, the switch delay should be greater than the lamination time constant to assure flux 91 penetration of the entire lamination before saturation. The lamination time constant as expressed in Equation (V-27) is ,2 d y y T = ^ 12 p Flux penetration of the lamination implies that .2 d li y t^ •d > — 712^p (VI 1-2) so that /tTrz} 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 40 ysec. Therefore, the lamination thickness may be determined as d < 2.3(10""^) m The lamination thickness for this application is chosen at 2.54(10 -5 ) 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 following physical dimensions: A = 13.1(10""^) m^ £ = .267 m d = 2.54(10"^) m The stacking factor, S, corresponding to d = 2.54(10 -5 ) m Is .89. 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 3500. 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 approximately .8 Tesla. With a reset magnetic intensity of -180 A-T/m and an unsaturated permeability of 3500, the Increase in induction would be B = y y H r^o r = .9 Tesla so that the total available change in induction for switch delay would be AB = 1.6 Tesla. Therefore, the total number of turns may be deter- mined from Equation (Vl-l) as: E t^ ^ = ^SAB = ^5 "^""'" • The saturated inductance may be determined from the number of turns and the core and winding geometries. The switching winding is wound over the entire core so that the C-core approximates a toroidal core. In this application, one layer of wire is sufficient to wind the number of turns required so that the geometry factor Is approximately unity. Insulation around the core increases the cross-sectional area to approximately 18.2(10~^) m . The squareness ratio for silicon steel as shown in Table Vl-l is .8, indicating that the saturated permeability of the core wilI not go to one. Instead, it may be assumed that 5 > u^ > 1; an average value for the purpose of calculations would be. y ^s = 4 . 93 H (A-T/m) Figure VI1-2 The dc B-H for Sil icon Stee 94 The saturated inductance may now be calculated from Equation (11-5) as: L , = sat 145 uH . The method of reset for this application Is provided by dc bias to produce a reset magnetic intensity of -210 A-T/m. If the bias winding has 12 turns, then the reset current may be determined from Equation (I I 1-4) as: H I I^r = 4 Nr = 4.0 A . As previously discussed in Chapter IV, the presence of the bias winding will possible affect switch efficiency. To evaluate this effect. It is desirable to calculate the coefficient of coupling during satura-tlon. The effective cross-sectional area during saturation Is 18.2(10 2 m . This implies a core radius such that r = .024 m. -2 From a wire area 2 of 4.4 mm and the Insulation on the wire, the winding, bias, and insulation thicknesses may be approximated as: a s Ji 2 mm a, ^ 2 mm b A — 2 mm The bias winding is wound on one leg of the C-core so that the Inductor appears as a solenoid. Therefore, Equation (IV-21) for the coupling coefficient of two windings on a solenoid core may be used to approximate the coefficient of coupling as k ^ .87 . ) 95 This value for the coefficient of coupling is rather large. However, the amount of energy transformed to the secondary of a transformer is also dependent upon the load on the secondary. In this case, two large inductances are placed in the bias circuit as shown In Figure VI 1-1. Little energy is transferred to the bias circuit during saturation because these inductors appear as an open circuit to the energy transfer pulse. Therefore, the efficiency of the inductive switch is dependent only upon core and winding losses. The hysteresis loss may be determined as In Equation (V-2): W^ = h Vol / H dB . The value for W, may be obtained by numerically integrating the curve of Figure VI1-2 so that W, h = 1 .45(10"^) J/pulse . The eddy current losses are expressed In Equation (V-33) as a function of switch delay and lamination time constant. The ratio of t^/i may be determined as t = ,/T d 296 so that the eddy current loss may be evaluated as W e = .61(10"^) J/pulse. It is necessary to determine the peak switching current and energy transfer duration to evaluate the winding losses. The switching current as a function of time is dependent upon unsaturated and saturated 96 inductances and circuit values. For the circuit of Figure Vll-1, the PFN capacitance is 1 uF. The unsaturated inductance may be determined from Equation (11-4) as L = 91 .2 mH . u The unsaturated inductance has been previously determined as L , = 141 ^ ' sat yH. The inductance of the PFN Is 36 yH so that the total inductance of the charging circuit while the magnetic switch is saturated is L c = 177 yH . The circuit of Figure Vli-1 during charging appears as the circuit of Figure 111-31. This Implies that the equations derived in Chapter 111 for the charging current apply in this case also. From Equations (111-10), (111-11), and (111-13), the switching current may be expressed as I (t) = 9.9 sin(3.3(10^) t ) s (VI 1-4) 0 < t < t , d where t i (t) = 225 sin(7.5(10'^)(t-39.9ysec)) s t ,< t < t d u (VII-5) I (t) = 9.9 sin(3.3(10 )(t+798 ysec)) s t < t < T u (Vll-6) I s t h e t i m e a t which t h e c u r e u n s a t u r a t e s . during saturation from Equation ( V I I - 5 ) I is = 225 A P The peak c u r r e n t 97 The time required to transfer energy to the load, t , may be determined et as t ^ = t - t, et u d . (VI1-7) From Equations (I I 1-7) and (VII-I 7), the energy transfer time becomes t^_^ = et 41 .75 y sec . The average current over the energy transfer pulse may be expressed as I = -^ *et I ' i (t) dt " . (VI 1-8) t By substituting Equation (VII-5) and the value for t into Equation (VII) and integrating from t , to t , the rms saturation current may be obtai ned Irms = 142.3 A The winding losses may be determined from Equation (V-37) with the rms saturation current and the winding resistance. The resistance of the winding may be calculated from the wire size and the length of wire in the winding. For wire with a cross-sectional area of 4.4 mm, the resistance per unit length is R w = 0.21 a/m . The length of wire in the switching winding, W^, may be found by multiplying the number of turns by the length of wire In one turn so that W = 11.6 m , 98 and R = .25 Q. T h e r e f o r e , t h e w i n d i n g losses a r e a p p r o x i m a t e l y Wi = i rms L Rt e^t = .21 J/pulse The t o t a l . l o s s e s i n t h e i n d u c t o r as expressed i n Equation (V-36) W T = w + W, + W^ = .213 J / p u l s e e h I ^ are , neglecting the eddy current losses during reset. The saturable inductor may be constructed and tested based on the design values derived thus far. These values are: E = 3 kV Jl = .267 m t , = 40 y sec 'd d = 2.54(10 A N = 65 Turns = 13.1(10""^) m^ -5 )m The response of the saturable Inductor in the circuit of Figure Vll-1 is shown In Figure VI1-3 for various applied voltages. The inductor operating at the design voltage of 3 kV and switch delay of 40 ysec Is shown in Figure Vll-3ft. Figure Vll-Jb represents the operation of the saturable Inductor at 2 kV for a time delay of 60 ysec while Figure VII-3C Indicates that at an applied voltage of 1 kV, the time delay is approximately 130 ysec. This Increase in switch delay with a reduction In applied voltage is predicted in Equation (11-2). 99 V PFN E CJ CVJ y^ii-vi-'j: ii.-fj.l^rt^*4r'.: ^ ® ! f :j.^^'v^>- '> •JC S2SIK « .1-' '•• ' -• .-^^'r-'<"r';.>, <: o #:*^H^^ LO ••*?- 20 usee/cm 20 usec/cm (a) V PFN > llMllI'll ••IIIHM-B • • ""'- 1 20 ysec/cm 20 usec/cm (b) V PFN L CJ CJ o CVJ 50 ysec/cm 50 ysec/cm (c) FIgure V I -3 I The Charging Current and PFN Voltage for a Saturable Inductor Designed to Delay Conduction for 40 ysec at 3 kV. The Inductor Voltages are (a) 3 kV, ^b) 2 kV, and (c) 1 kV 100 The peak current for the designed conditions of 3 kV and 40 ysec is determined from Figure Vll-3a to be I mately the predicted value of 225 A. = 200 A. This value is approxi- The discrepancy in peak current arises from a higher saturated inductance than was calculated. error may be due to poor winding techniques. The The looser the switching winding is wound, the larger the inductance is under saturated conditions. The actual value of inductance may also be due to a larger satu- rated permeability than was anticipated. The switch delay, hold-off current, and the duration of the energy transfer are the performance criteria for this saturable inductor design. The switch delay was required to be 40 ysec at 3 kV; Figure V I 1 - ^ Indicates that the switch delay was as designed within measurement error. The hold-off current at t = t , may be calculated from Equation (VI1-4) as 1.3 A, which is low enough that the discharge switch will not reclose. In Figure Vll-2a, the sensitivity is not sufficiently high to allow accurate determination of the hold-off current; it appears to be approximately 2 A. actual t The energy transfer time was calculated as 42 ysec; from Figure Vll-2a may be approximated as 44 ysec. the Therefore, the saturable Inductor performed within design limitations as predicted. CHAPTER VII I CONCLUSIONS Many advantages exist In the use of saturable inductors in repetitive pulse generation. The magnetic switch is applicable whenever a switch delay Is desired or a large transition in Inductance is useful. The saturable inductor provides a compact and inexpensive alternative to some solid state methods used to achieve switch delay. Because fail- ure mechanisms are few, inductors offer very long operational lifetimes. The design of a saturable inductor is fairly simple and performance is relatively easily predicted and reliable. Saturable inductor design may also be tailored to meet specific performance requirements; such requirements might be high dl/dt, high efficiency, accurate switch delay, low hold-off current, short duration of energy transfer, low jitter, or hi gh reliabI 1ity. Disadvantages in the use of saturable inductors arise from the nature of the switch delay. The switch delay Is controlled only by varying the reset current or the magnitude of the applied (stand-off) voltage. design. The variation in switch delay is thus limited for a particular Variations in applied voltage will result in changes in switch delay implying that the saturable inductor Is best suited to applications in which the stand-off voltage Is to remain constant. The limits of applicability of Inductor switches far exceed that achieved in the past. Scaling of simple designs indicates that the size or power handling capabilities of inductor switches can be truly gigantic. The scaling presented was based on voltage and current density constraints and mechanical forces were not included. 101 For very high 102 power pulses, the mechanical forces may be sufficient to cause damage and an analysis of mechanical stresses would be required to verify a design. Several areas need to be investigated in order to extend the characterization of the saturable Inductor as a high-power switch. A better and more comprehensive approach to the determination of core losses would be beneficial. This approach might Include cases where the core saturates and where the lamination is not fully exercised. characterization of losses In ferrite cores would also be useful. The An- other area of Investigation involves the manner of heat deposition and flow in the core. By determining the distribution of heat in the core, a core volume for a specific design may be determined which safely limits internal heating while minimizing core size. LIST OF REFERENCES S. E. Black, "Command Charge Using Saturable Inductors," Technical Papers of the Second International Pulsed Power Conference, Lubbock, Texas, June 13 -15, 1980. 2. J. E. Creedon and S. Schneider, "A Magnetic Assist for High-Power Hydrogen Thyratrons," Proceedings of the Fifth Symposium on Hydrogen Thyratrons and Modulators, Fort Monmount, N. J., May 20 - 22, pp. 145-158. 3. K. J. Busch, A. D. Hasley, and Carl Neitzert, "Magnetic Pulse Modulators," The Bell System TechnicaI Journal, September 1955, Vol. 34, pp. 943-993. 4. G. T. Coates and L. R. Swain. High Power Semiconductor-Magnetic Pulse Generators. Cambridge, Massachusetts: The M.I.T. Press. 1966. 5. W. S. Melville, "The Use of Saturable Reactors as Discharge Devices for Pulse Generators," Proceedings of the I.E.E.E., Vol. 98, pt. 3, February 1951, pp. 185-204. 6. Carl T. A. Johnk. Engineering Electromaqnetic Fields and Waves. John Wiley and Sons: New York. 1975. pp. 311-347. 7. G. N. Glasoe and J. V. Lebacquz, Ed. Publications, Inc: New York. 1965. 8. Donald G. Fink and John M. Carroll, Ed. Standard Handbook for Electrical Engineers. McGraw-Hill Book Co: New York. 1968. p. 4.86. 9. D. J. Craik. Structure and Properties of Magnetic Materials. Limited: London. 1971. pp. 37-38. 10. Samuel Robinson Williams. Magnetic Phenomenon. McGraw-Hill Book Co: New York. 1931. pp. 154-158. 11. Max Jakob. Heat Transfer. Volume I. 1949. pp. 305-312. 12. K. H. Stewart. Ferromaqnetic Domains. Cambridge, Great Britain. 1954. 13. W. S. Melville, "The Measurement and Calculation of Pulse Magnetization Characteristics of Nickel Irons from .I to 5 Microseconds," Proceedings I.E.E., 1950, Vol. 97, part II, p. 165. 103 Pulse Generators. Dover pp. 511-515. John Wiley and Sons: Pion New York. Cambridge Univ. Press: 104 14. W. S. Melville, "Theory and Design of High-Power Pulse Transformers," Journal I.E.E., 1946, Vol. 93, part IMA, p. 1063. 15. Richard M. Bozorth. Ferromaqnetism. Princeton, N.J. 1959. pp. 14-47. 16. Tape Wound Cores, TC-IOIB, The Arnold Engineering Co., 1972. 17. Harold A. (Goldsmith, "Ferromagnetic Ceramics..." ing, Apri I 1951. 18. "Toroidal Magnetic Amplifier Cores," I.E.E.E. Standards Paper # 106. D. Van Nostrand Co., Inc: Product Engineer-