1.1. HES based on pulse transformer charging
In the fields of electrical discipline, power electronics and pulsed power technology, the common used modes of energy transferring and energy storage include mechanical energy storage (MES), chemical energy storage (CHES), capacitive energy storage (CES), inductive energy storage (IES) and the hybrid energy storage (HES) [1-3]. The MES and CHES are important ways for energy storage employed by people since the early times. The MES transfers mechanical energy to pulse electromagnetic energy, and the typical MES devices include the generator for electricity. The CHES devices, such as batteries, transfer the chemical energy to electrical energy. The energy storage modes aforementioned usually combine with each other to form an HES mode. In our daily life, the MES and CHES usually need the help of other modes to deliver or transfer energy to drive the terminal loads. As a result, CES, IES and HES become the most important common used energy storage modes for users. So, these three energy storage modes are analyzed in detail as the central topics in this chapter.
The CES is an energy storage mode employing capacitors to store electrical energy [3-5]. As Fig. 1(a) shows,
The IES is another energy storage mode using inductive coils to generate magnetic fields for energy storage. As shown in Fig. 1(b), the basic IES cell needs matched operations of the opening switch (
In many applications, CES combining with IES is adopted for energy storage as a mode of HES. Fig. 1(c) shows a typical HES mode based on CES and IES. Firstly, the energy source charges
Generally speaking, a system can be called as HES module if two or more than two energy storage modes are included in the system. In this chapter, the centre topics just focus on CES, IES and the HES based on the CES and IES, as they have broad applications in our daily life. The CES and IES both have their own advantages and defects, but the HES mode based on these two achieves those individual advantages at the same time. In applications, a lot of facilities can be simplified as the HES module including two capacitors and a transformer shown in Fig. 2 [9-16]. Switch
1.2. Applications of HES based on pulse transformer charging
The HES based on pulse transformer charging is an important technology for high-voltage boosting, high-power pulse compression, pulse modification, high-power pulse trigger, intense electron beam accelerator and plasma source. The HES cell has broad applications in the fields such as defense, industry, environmental protection, medical care, physics, cell biology and pulsed power technology.
The HES based on pulse transformer charging is an important way for high-power pulse compression. Fig. 3(a) shows a high-power pulse compression facility based on HES in Nagaoka University of Technology in Japan, and its structure is shown in Fig. 3(b). The Blumlein pulse forming line plays as the load capacitor in the HES cell, and two magnetic switches respectively control the energy transferring. The pulse compression system can compress the low voltage pulse from millisecond range to form high voltage pulse at 50ns/480kV range.
The HES based on pulse transformer charging is an important way for high-power pulse trigger. Fig. 4(a) shows a solid state pulse trigger with semiconductor opening switches (SOS) in the Institute of Electrophysics Russian Academy of Science [10-11]. Fig. 4(b) presents the schematic of the pulse trigger, which shows a typical HES mode based on pulse transformer charging. SOS switch and IGBT are employed as the switches controlling energy transferring. The pulse trigger delivers high-voltage trigger pulse with pulse width at 70ns and voltage ranging from 20 to 80kV under the 100Hz repetition. And the average power delivered is about 50kW.
The HES cell based on pulse transformer charging is also an important component in intense electron beam accelerator for high-power pulse electron beams which are used in the fields of high-power microwave,plasma, high-power laser and inertial fusion energy (IFE). Fig. 5(a) shows the “Sinus” type accelerator in Russia, and it also corresponds to the HES mode based on transformer charging shown in Fig. 2. The pulse transformer of the accelerator is Tesla transformer with opened magnetic core, while spark gap switch controls energy transferring. The accelerator has been used to drive microwave oscillator for high-power microwave. Fig. 5(b) presents a high-power KrF laser system in Naval Research Laboratory of the U. S. A., and the important energy storage components in the system just form an HES cell based on transformer charging [13-14]. The HES cell drives the diode for pulse electron beams to pump the laser, and the laser system delivers pulse laser with peak power at 5GW/100ns to theIFE facility.
The HES based on pulse transformer charging also have important applications in ultra-wideband (UWB) electromagnetic radiation and X-ray radiography. Fig. 6 shows an ultra-wideband pulse generator based HES mode in LoughboroughUniversity of the U. K. . The air-core Tesla transformer charges the pulse forming line (PFL) up to 500kV, and spark gap switch controls the energy transferring form the PFL to antenna. The “RADAN” series pulse generators shown in Fig. 7 are portable repetitive high-power pulsers made in Russia for X-ray radiography . The “RADAN” pulser which consists of Tesla transformer and PFL are also based on the HES mode shown in Fig. 2. The controlling switches are thyristors and spark gap.
Besides, the HES cell is also used in shockwave generator , dielectric barrier discharge , industrial exhaust processing , material surface treatment , ozone production , food sterilization , cell treatment and cell mutation .
2. Parametric analysis of pulse transformer with closed magnetic core in HES
Capacitor and inductor are basic energy storage components for CES and IES respectively, and pulse transformer charging is important to the HES mode shown in Fig. 2. So, it is essential to analyze the characteristic parameters of the common used high-power pulse transformer, and provide theoretical instructions for better understanding of the HES based on transformer charging.
There are many kinds of standards for categorizing the common used pulse transformers. From the perspective of magnetic core, pulse transformers can be divided into two types, such as the magnetic-core transformer [24-25] and the air-core transformer . In view of the geometric structures of windings, the pulse transformer can be divided to many types, such as pulse transformer with closed magnetic core, solenoid-winding transformer, curled spiral strip transformer , the cone-winding Tesla transformer [16, 27], and so on. The transformer with magnetic core is preferred in many applications due to its advantages such as low leakage inductance, high coupling coefficient, high step-up ratio and high efficiency of energy transferring. Russian researchers produced a kind of Tesla transformer with cone-like windings and opened magnetic core, and the transformer withhigh coupling coefficient can deliver high voltage at MV range in repetitive operations . Usually, pulse transformer with closed magnetic core, as shown in Fig.8, is the typical common used transformer which has larger coupling coefficient than that of Tesla transformer. The magnetic core can be made of ferrite, electrotechnical steel, iron-based amorphous alloy, nano-crystallization alloy, and so on. The magnetic core is also conductive so that the core needs to be enclosed by an insulated capsule to keep insulation from transformer windings.
Paper  presents a method for Calculation on leakage inductance and mutual inductance of pulse transformer. In this chapter, the common used pulse transformer with toroidal magnetic core will be analyzed in detail for theoretical reference. And a more convenient and simple method for analysis and calculation will be presented to provide better understanding of pulse transformer [24-25].
The typical geometric structure of pulse transformer with toroidal magnetic core is shown in Fig. 9(a). The transformer consists of closed magnetic core, insulated capsule of the core and transformer windings. The cross section of the core and capsule is shown in Fig. 9(b). Transformer windings are formed by high-voltage withstanding wires curling around the capsule, and turn numbers of the primary and secondary windings are
Define the geometric parameters in Fig. 9(b) as follows. The height, outer diameter and inner diameter of the closed magnetic core are defined as
2.1. Inductance analysis of pulse transformer windings with closed magnetic core
2.1.1. Calculation of magnetizing inductance
Define the permittivity and permeability of free space as
Define the inner and outer circumferences of magnetic core as
2.1.2. Leakage inductance of primary windings
The leakage inductances of primary and secondary windings also contribute to the total inductances of windings. The leakage inductance
Define the magnetic field intensity generated by
When the magnetic core works in the linear district of its hysteresis loop, the magnetic energy
From the geometric structure in Fig. 10, , the leakage magnetic energy
So, the total leakage magnetic energy
2.1.3. Leakage inductance of secondary windings
Usually, the simple and typical layout of the secondary windings of transformer is also the single layer structure as shown in Fig. 11(a). The windings are in single-layer layout both at the inner wall and outer wall of insulated capsule. As
In this chapter, the single-layer layout and “quasi-single-layer ” layout shown in Fig. 11 (a) and (b) respectively are both analyzed to provide reference for HES module. And the multi-layer layout  can also be analyzed by the way introduced in this chapter.
Define the current flowing through the secondary windings as
Firstly, the single-layer layoutshown in Fig. 11 (a) is going to be analyzed. The analytical model is similar to the model analyzed in Fig. 10. If (
In view of that , the leakage inductance of single-layer layout of the secondary windings is as
As to the “quasi-single-layer ” layout shown in Fig. 11 (b), it also can be analyzed by calculating the leakage magnetic energy firstly. Under this condition, the length of leakage magnetic pass enclosed by the secondary windings is revised as . The leakage magnetic energy
Finally, the leakage inductance of the “quasi-single-layer ” layout is obtained by the same way of (14) as
2.1.4. The winding inductances of pulse transformer
Define the total inductances of primary windings and secondary windings as
2.2. Distributed capacitance analysis of pulse transformer windings
The distributed capacitances of pulse transformer include the distributed capacitances to ground , capacitance between adjacent turns or layers of windings [29-32], and capacitance between the primary and secondary windings [32-33]. It is very difficult to accurately calculate every distributed capacitance. Even if we can do it, the results are not liable to be analyzed so that the referential value is discounted. Under some reasonable approximations, lumped capacitances can be used to substitute the corresponding distributed capacitances for simplification, and more useful and instructive results can be obtained . Of course, the electromagnetic dispersion theory can be used to analyze the lumped inductance and lumped capacitance of the single-layer solenoid under different complicated boundary conditions [34-35]. In this section, an easier way is introduced to analyze and estimate the lumped capacitances of transformer windings.
2.2.1. Distributed capacitance analysis of single-layer transformer windings
In the single-layer layout of transformer windings shown in Fig. 11(a), the equivalent schematic of transformer with distributed capacitances is shown in Fig. 12.
In the single-layer layout shown in Fig. 11(a), define the lengths of single coil turn in primary and secondary windings as
If the relative permittivity of the dielectric between adjacent coil turns is
Actually, the whole long coil wire which forms the primary or secondary windings of transformer can be viewed as a totality. The distributed capacitances between adjacent turns are just formed by the front surface and the back surface of the wire totality itself. In view of that, lumped capacitances
2.2.2. Distributed capacitance analysis of inter-wound “quasi-single-layer” windings
Usually, large turn number
If the coil turns are tightly wound, the average distance between two adjacent coil turns is
The non-adjacent coil turns have large distance so that the capacitance effects are shielded by adjacent coil turns. In the azimuthal direction of the inner layer wires, small angle d
In view of that
According to the equivalent lumped schematic in Fig. 13(b), the total lumped capacitance
2.3. Dynamic resistance of transformer windings
Parasitic resistance and junction resistance of transformer windings cause loss in HES cell. Define the resistivity of winding wires under room temperature (20oC) as
When the working frequency
3. Pulse response analysis of high power pulse transformer in HES
In HES cell based on pulse transformer charging, the high-frequency pulse response characteristics of transformer show great effects on the energy transferring and energy storage. Pulse response and frequency response of pulse transformer are very important issues. The distributed capacitances, leakage inductances and magnetizing inductance have great effects on the response pulse of transformer with closed magnetic core [36-39]. In this Section, important topics such as the frequency response and pulse response characteristics to square pulse, are discussed through analyzing the pulse transformer with closed magnetic core.
3.1. Frequency-response analysis of pulse transformer with closed magnetic core
The equivalent schematic of ideal pulse transformer circuit is shown in Fig. 14(a).
3.1.1. Low-frequency response characteristics
Define the frequency and angular frequency of the pulse source as
In Fig. 15(a),
An example is provided as follows to demonstrate the analysis above. In many measurements, coaxial cables and oscilloscope are used, and the corresponding terminal impedance is about
The conclusion is that low-frequency response capability of pulse transformer is mainly determined by
3.1.2. High-frequency response characteristics
When the transformer responds to high-frequency pulse signal (
In Fig. 16(a), when
Select the amplitude of the periodical pulse signal
The conclusion is that high-frequency response characteristics of transformer are mainly determined by distributed capacitance
3.2. Square pulse response of pulse transformer with closed magnetic core
In Fig. 14(b),
3.2.1. Response to the front edge of square pulse
Define the voltage of
If the factor for Laplace transformation is as
Define the amplitude and pulse duration of square voltage pulse source as
Equations (30) can be solved by Laplace transformation and convolution, and there are three states of solutions such as the over dumping state, the critical dumping state and the under dumping state. In the transformer circuit, the resistors are always small so that the under dumping state usually appears. Actually, the under dumping state is the most important state which corresponds to the practice. In this section, the centre topic focuses on the under dumping state of the circuit.
The under dumping state solution of (30) is as
The load current
From (33) and (36), the rise time of the response signal
The conclusion is that the rise time of the front edge of response pulse can be improved by minimizing the capacitance
3.2.2. Pulse droop analysis of transformer response
In Fig.17, when the front edge of pulse is over,
The initial conditions are as
3.3.3. Response to the back edge of square pulse
When the flat top of square pulse is over, all the reactive components in Fig. 17 have stored certain amount of electrical or magnetic energy. Though the main pulse of the response signal is over, the stored energy starts to deliver to the load through the circuit. As a result, high-frequency resonance is generated again which has a few differences from the resonance at the front edge of pulse. In Fig. 17,
The circuit equations are presented in (39) with initial condition as
The under dumping solution of (39) is calculated as
The responses to front edge and back edge of square pulse have differences in essence, as the exciting sources are different. Define functions
The response signal
According to (40) and (42),
Fig. 21(b) shows an impression of the effect of
4. Analysis of energy transferring in HES based on pulse transformer charging
As an important IES component, the pulse transformer is analyzed and the pulse response characteristics are also discussed in detail. The analytical theory aforementioned is the base for HES analysis based on pulse transformer charging in this section. In Fig.2, the HES module based on capacitors and transformer operates in three courses, such as the CES course, the IES course and the CES course. Actually, the IES course and the latter CES course occur almost at the same time. The pulse transformer plays a role on energy transferring. There are many kinds of options for the controlling switch (
The pulse signals in the HES module are resonant signals. According to the analyses from Fig. 15 and Fig.16, the common used pulse transformer shown in Fig. 9(a) has good frequency response capability in the band ranging from several hundred Hz to several MHz. Moreover,
In Fig. 22(a),
The voltages of
In view of Fig. 22(b), the circuit equations of HES module can also be established as
The initial conditions are as
Define the effective coupling coefficient of the HES module based on transformer charging as
Equations (47) have general forms of solution as
4.1. The lossless method
The first method employs lossless approximation. That’s to say, the parasitic resistances in the HES module are so small that they can be ignored. So, the HES module has no loss. Actuallyin many practices, the “no loss” approximation is reasonable. As a result, equation (50) can be simplified as
In (51), it is easy to get the two independent characteristic solutions defined as
In (52), the voltages of energy storage capacitors have phase displacements in contrast to the currents. All of the voltage and current functions have two resonant angular frequencies as
4.2. The “little disturbance” method
The “little disturbance” method was introduced to analyze the Tesla transformer with open core by S. D. Korovin in the Institute of High-Current Electronics (IHCE), Tomsk, Russia. Tesla transformer with open core has a different energy storage mode in contrast to the transformer with closed magnetic core. Tesla transformermainly stores magnetic energy in the air gaps of the open core, while transformer with closed core stores magnetic energy in the magnetic core. So, the calculations for parameters of these two kinds of transformer are also different. However, the ideaof “little disturbance” is still a useful reference for pulse transformer with closed core [24-25]. So, the “little disturbance” method is introduced to analyze the pulse transformer with closed magnetic core for HES module.
The “little disturbance” method employs two little disturbance functions Δ
The solutions of (50) are as
Define two effective quality factors of the double resonant circuit of HES module as
From (58), all of the voltage and current functions have two resonant frequencies. In many situations of practice, the terms in (58) which include cos(
Obviously, when the switch of
Under the condition
Usually, semiconductor switch such as thyristor or IGBT is used as the controlling switch of
Actually, the efficiency of energy transferring is also determined by the charge time of
5. Magnetic saturation of pulse transformer and loss analysis of HES
5.1. Magnetic saturation of pulse transformer with closed magnetic core
Transformers with magnetic core share a communal problem of magnetic saturation of core. The pulse transformer with closed magnetic core consists of the primary windings (
The total magnetic flux in the magnetic core is
As parameters such as Δ
According to (63), some methods are obtained to avoid saturation of core as follows. Firstly, Δ
Generally speaking, it is quite difficult to increase Δ
5.2. Loss analysis of HES
The loss is a very important issue to estimate the quality of the energy transferring module. In Fig. 22(a), the main losses in the HES module based on pulse transformer charging include the resistive loss and the loss of magnetic core of transformer. The resistive loss in HES module consists of loss of wire resistance, loss of parasitic resistance of components, loss of switch and loss of leakage conductance of capacitor. Energy of resistive loss corresponds to heat in the components. The wire resistance is estimated in (27), and the switch resistance and leakage conductance of capacitor are provided by the manufacturers. According to the currents calculated in (58), the total resistive loss defined as Δ
5.2.1. Hysteresis loss analysis
In the microscope of the magnetic material, the electrons in the molecules and atoms spin themselves and revolve around the nucleuses at the same time. These two types of movements cause magnetic effects of the material. Every molecule corresponds to its own magnetic dipole, and the magnetic dipole equates to a dipole generated by a hypothetic molecule current. When no external magnetic field exists, large quantities of magnetic dipoles of molecule current are in random distribution. However, when external magnetic field exists, the external magnetic field has strong effect on these magnetic dipoles in random distribution, and the dipoles turn to the same direction along the direction of external magnetic field. The course is called as magnetizing, in which a macroscopical magnetic dipole of the material is formed. Obviously, magnetizing course of the core consumes energy which comes from capacitor
Define the electric field intensity, electric displacement vector, magnetic field intensity and magnetic induction intensity in the magnetic core as, , andrespectively. The total energy density of electromagnetic field . As the energy density of electric field is the same as which of magnetic field, the total energy density
The magnetizing current which corresponds to
According to (65) and (66), the hysteresis loss of magnetic core of transformer is obtained as
In some approximate calculations, the loss energy density is equivalent to the area enclosed by the hysteresis loop. If the coercive force of the loop is
5.2.2. Eddy current loss analysis
When transformer works under high-frequency conditions, the high-frequency current in transformer windings induces eddy current in the cross section of magnetic core. Define the eddy current vector as, magnetic induction intensity of eddy current as, magnetic field intensity of eddy current as, magnetic induction intensity of ip(t) as, and magnetic field intensity of ip(t) as. As shown in Fig. 25, the direction of is just inverse to the direction of ip(t), so the eddy current field weakens the effect of . The eddy current heats the core and causes loss of transformer, and it should be eliminated by the largest extent when possible.
In order to avoid eddy current loss, the magnetic core is constructed by piled sheets in the cross section as Fig. 25 shows. Usually, the sheet is covered with a thin layer of insulation material to prevent eddy current. However, the high-frequency eddy current has “skin effect”, and the depth of “skin effect” defined as δ is usually smaller than the thickness hof the sheet. As a result, the eddy current still exists in the cross section of core. Cartesian coordinates are established in the cross section of core as shown in Fig. 25, and the unit vectors are as , and . To a thin sheet, its length () and width () are both much larger than the thickness h (). So, approximation of infinite large dimensions of sheet in and directions is reasonable. That’s to say, and . The “little disturbance” theory aforementioned before still can be employed to calculate the field generated by eddy current.
The total magnetic induction intensity in the core is as , and . generated by eddy current can be viewed as the variable of “little disturbance”. According to Maxwell equations,
From (68), it is easy to obtain the formula, while (Ex, Ey, Ez) and (Hx, Hy, Hz) corresponds to vectorsand. Through integration,
Define the conductivity of the sheet in magnetic core as σ. From the second equation in (68), . It demonstrates that infinitesimal conductivity is the key factor to prevent eddy current. When working frequency is f, the depth of “skin effect” of the sheet is calculated as δ=(πfμσ)-1/2. According to (69), the “little disturbance” field of eddy current in isotropic magnetic material is presented as
Through averaging the field along the thickness direction () of sheet,
As the electric energy and magnetic energy of the eddy current field are almost the same, the eddy current loss defined as Wloss2 is calculated as
From (72), Wloss2 is proportional to the conductivity σof the core, and it is also proportional to (h/σ)2. As a result, Wloss2 can be limited when h<<δ.
5.2.3. Energy efficiency of the HES module
As to the HES module based on transformer charging shown in Fig. 22(a), the energy loss mainly consists of ΔWR, Wloss1 and Wloss2. Total energy provided from C1 is as . In practice, the energy stored in C1 can not be transferred to C2 completely, though the loss of the module is excluded. In other words, residue energy defined as W0r exists in C1. Define the allowed maximum efficiency of energy transferring from C1 to C2 as ηmax. So, ηmax of the HES module is as
From (62),ηa, ηe and ηmaxhave relation as.
This work wassupported by the National Science Foundationof China under Grant No.51177167. It’s also supported by the Fund of Innovation, Graduate School of National University of Defense Technology under Grant No.B100702.
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