Hybrid Energy Storage and Applications Based on High Power 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.


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][2][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][4][5]. As Fig. 1(a) shows, C0 is the energy storage component in CES, and the load of C0 can be inductors, capacitors and resistors respectively. Define the permittivity of dielectric in capacitor C0 as ε, the electric field intensity of the stored electrical energy in C0 as E. The energy density WE of CES is as 2 1 .
Usually, WE which is restricted to ε and the breakdown electric field intensity of C0 is about 10 4~1 0 5 J/m 3 . The traditional Marx generators are in the CES mode [4][5].
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 (Sopen) and the closing switch (Sclose) [6][7], while L0 is as the energy storage component. When the charging current of L0 reaches its peak, Sopen becomes open and Sclose becomes closed at the same time. As the instantaneously induced voltage on L0 grows fast, the previously stored magnetic energy in the magnetic field is delivered fast to the load through Sclose. The load of L0 also can respectively be inductors, capacitors and resistors. The explosive magnetic flux compression generator is a kind of typical IES device [7]. The coil winding of pulse transformer which has been used in Tokamak facility is another kind of important IES device [8]. Define the permeability of the medium inside the coil windings as μ, the magnetic induction intensity of the stored magnetic energy as B. The energy density WB of IES is as Usually, WB restricted by μ and B is about 10 7 J/m 3 . IES has many advanced qualities such as high density of energy storage, compactness, light weight and small volume in contrast to CES. However, disadvantages of IES are also obvious, such as requirement of high power opening switches, low efficiency of energy transferring and disability of repetitive operations. 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 C1 in CES mode. Secondly, Sclose1 closes and the energy stored in C1 transfers to L0 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 [12], 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 highpower 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 the IFE facility. The HES based on pulse transformer charging also have important applications in ultrawideband (UWB) electromagnetic radiation and X-ray radiography. Fig. 6 shows an ultrawideband pulse generator based HES mode in Loughborough University of the U. K. [15]. 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 [16]. 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.

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 [26]. 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 [26], 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 conelike windings and opened magnetic core, and the transformer with high coupling coefficient can deliver high voltage at MV range in repetitive operations [27]. 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 [28] 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 N1 and N2, respectively. Usually, transformer windings have a layout of only one layer of wires as shown in Fig. 9(a), which corresponds to a simple structure. In other words, this simple structure can be viewed as a single-layer solenoid with a circular symmetric axis in the azimuthal direction. The transformer usually immerses in the transformer oil for good heat sink and insulation. 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 lm, D4 and D3 respectively. The height, outer diameter and inner diameter of the insulated capsule are defined as l0, D2 and D1 respectively. The thicknesses of the outer wall, inner wall and side wall of insulated capsule are defined as d1, d2 and d5 in order. The distances between the side surfaces of capsule and magnetic core are d3 and d4 shown in Fig. 9(b). Define diameters of wires of the primary windings and secondary windings as dp and ds respectively. The intensively wound primary windings with N1 turns have a width about N1dp.

Calculation of magnetizing inductance
Define the permittivity and permeability of free space as ε0 and μ0, relative permeability of magnetic core as μr, the saturated magnetic induction intensity of core as Bs, residue magnetic induction intensity of core as Br, and the filling factor of magnetic core as KT. The cross section area S of the core is as Define the inner and outer circumferences of magnetic core as l1 and l2, then l1 = πD3 and l2 = πD4. The primary and secondary windings tightly curl around the insulated capsule in separated areas in the azimuthal direction. In order to get high step-up ratio, the turn number N1 of primary windings is usually small so that the single-layer layout of primary windings is in common use. Define the current flowing through the primary windings as ip, the total magnetic flux in the magnetic core as Φ0, and the magnetizing inductance of transformer as Lµ. According to Ampere's circuital law,

Leakage inductance of primary windings
The leakage inductances of primary and secondary windings also contribute to the total inductances of windings. The leakage inductance Llp of the primary windings is caused by the leakage magnetic flux outside the magnetic core. If μr of magnetic core is large enough, the solenoid approximation can be used. Through neglecting the leakage flux in the outside space of the primary windings, the leakage magnetic energy mainly exists in two volumes. As Fig.10 shows, the first volume defined as V1 corresponds to the insulated capsule segment only between the primary windings and the magnetic core, and the second volume defined as V2 is occupied by the primary winding wires themselves. The leakage magnetic field in the volume enclosed by transformer windings can be viewed in uniform distribution. The leakage magnetic energy stored in V1 and V2 are as Wm1 and Wm2, respectively. Define the magnetic field intensity generated by ip from the N1-turn primary windings as Hp in V1. According to Ampere's circuital law, Hp≈ip/dp. From Fig. 10

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 D2 is much larger than D1, the density of wires at the inner wall is larger than that at the outer wall. However, if the turn number N2 becomes larger enough for higher step-up ratio, the inner wall of capsule can not provide enough space for the single-layer layout of wires while the outer wall still supports the previous layout, as shown in Fig. 11(b). We call this situation as "quasi-single-layer " layout.
In the "quasi-single-layer " layout shown in Fig. 11 (b), the wires at the inner wall of capsule is in two-layer layout. After wire 2 curls in the inner layer, wire 3 curls in the outer layer next to wire 2, and wire 4 curls in the inner layer again next to wire 3, then wire 5 curls in the outer layer again next to wire 4, and so on. This kind of special layout has many advantages, such as minor voltage between adjacent coil turns, uniform voltage distribution between two layers, good insulation property and smaller distributed capacitance of windings.
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 multilayer layout [29] can also be analyzed by the way introduced in this chapter. Define the current flowing through the secondary windings as is, the two volumes storing leakage magnetic energy as Va and Vb, the corresponding leakage magnetic energy as Wma and Wmb, the total leakage magnetic energy as Wms, wire diameter of secondary windings as ds, and the leakage inductance of secondary windings as Lls.
Firstly, the single-layer layout shown in Fig. 11 (a) is going to be analyzed. The analytical model is similar to the model analyzed in Fig. 10. If (D2-D1)<<D1, the length of leakage magnetic pass enclosed by the secondary windings is as In volume Vb which is occupied by the secondary winding wires themselves, Wmb can be estimated as 22 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 Finally, the leakage inductance of the "quasi-single-layer " layout is obtained by the same way of (14) as 22 22

The winding inductances of pulse transformer
Define the total inductances of primary windings and secondary windings as L1 and L2 respectively, the mutual inductance of the primary and secondary windings as M, and the effective coupling coefficient of transformer as Keff. From (5), (11), (14) or (16), When μr>>1, M and Keff are presented as Hybrid Energy Storage and Applications Based on High Power Pulse Transformer Charging 189

Distributed capacitance analysis of pulse transformer windings
The distributed capacitances of pulse transformer include the distributed capacitances to ground [30], capacitance between adjacent turns or layers of windings [29][30][31][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 [29]. 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.

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. CDpi is the distributed capacitance between two adjacent coil turns of primary windings, and CDsi is the counterpart capacitance of the secondary windings. Cpsi is the distributed capacitance between primary and secondary windings. Common transformers have distributed capacitances to the ground, but this capacitive effect can be ignored if the distance between transformer and ground is large. If the primary windings and secondary windings are viewed as two totalities, the lumped parameters CDp, CDs and Cps can be used to substitute the "sum effects" of CDpis, CDsis and Cpsis in order, respectively. And the lumped schematic of the pulse transformer is also shown in Fig. 12. Cps decreases when the distance between primary and secondary windings increases. In order to retain good insulation for high-power pulse transformer, this distance is usually large so that Cps also can be ignored. At last, only the lumped capacitances, such as CDp and CDs, have strong effects on pulse transformer. In the single-layer layout shown in Fig. 11(a), define the lengths of single coil turn in primary and secondary windings as ls1 and ls2 respectively, the face-to-face areas between two adjacent coil turns in primary and secondary windings as Sw1 and Sw2 respectively, and the distances between two adjacent coil turns in primary and secondary windings as Δdp and Δds respectively. According to the geometric structures shown in Fig. 10 and Fig. 11(a), ls1=2l0+4dp+D2-D1，ls2=2l0+4ds+D2-D1, Sw1=dpls1 and Sw2=dsls2. Because the coil windings distribute as a sector, Δdp and Δds both increase when the distance from the centre point of sector increases in the radial direction. Δdp and Δds can be estimated as 1 2 12 If the relative permittivity of the dielectric between adjacent coil turns is εr, CDpi and CDsi can be estimated as 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 CDp and CDs can be used to describe the total distributed capacitive effect. As a result, CDp and CDs are calculated as 2 11 1 2 22 From (21), CDp or CDs is proportional to the wire length ls1 or ls2, while larger turn number and smaller distance between adjacent coil turns both cause larger CDp or CDs

Distributed capacitance analysis of inter-wound "quasi-single-layer" windings
Usually, large turn number N2 corresponds to the "quasi-single-layer " layout of wires shown in Fig. 13(a). In this situation, distributed capacitances between the two layers of wires at the inner wall of capsule obviously exist. Of course, lumped capacitance CLs can be used to describe the capacitive effect when the two layers are viewed as two totalities, as shown in Fig. 13(b). Define CDs1 and CDs2 as the lumped capacitances between adjacent coil turns of these two totalities, and CDs is the sum when CDs1 and CDs2 are in series. As a result, the lumped capacitances which have strong effects on pulse transformer are CDp, CDs1, CDs2 and CLs. If the coil turns are tightly wound, the average distance between two adjacent coil turns is ds. The inner layer and outer layer at the inner wall of capsule have coil numbers as 1+N2/2 and N2/2-1 respectively. According to the same way for (21) 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 dθ corresponds to the azimuthal width of wires as dl and distributed capacitance as dCLs. Then, 02 1 0 . If the voltage between the (n-1)th and nth turn of coil (n ≤ N2) is ΔU0, the inter-wound method of the two layers aforementioned retains the voltage between two layers at about 2ΔU0. So, the electrical energy WLs stored in CLs between the two layers is as Energy Storage -Technologies and Applications 192 In view of that WLs=CLs(2ΔU0) 2 /2, CLs can be calculated as According to the equivalent lumped schematic in Fig. 13 When the working frequency f is high, the "skin effect" of current flowing through the wire corss-section becomes obvious, which has great effects on Rw0. Define the depth of "skin effect" as Δdw, and the dynamic parasitic resistance of winding wires as Rw(f, Tw). As Δdw=(ρ / π f μ0) 0. 5 , Rw(f, Tw) is presented as

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][37][38][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.

Frequency-response analysis of pulse transformer with closed magnetic core
The equivalent schematic of ideal pulse transformer circuit is shown in Fig. 14(a). Llp and Lls are the leakage inductances of primary and secondary windings of transformer calculated in (11), (14) and (16). Lumped capacitances Cps, CDP and CDS represent the "total effect" of the distributed capacitances of transformer, while CDP and CDS are calculated in (21) and (25). Lµ is the magnetizing inductance of pulse transformer calculated in (5). Define the sum of wire resistance of primary windings and the junction resistance in primary circuit as Rp, the counterpart resistance in secondary circuit as Rs, load resistance as RL, the equivalent loss resistance of magnetic core as Rc, and the sinusoidal/square pulse source as U1. Usually, Cps is so small that it can be ignored due to the enough insulation distance between the primary and secondary windings. In order to simplify the transformer circuit in Fig.  14(a), the parameters in the secondary circuit such as CDs, Lls, Rs and RL, can be equated into the primary circuit as CDs0, Lls0, Rs0 and RL0, respectively. And the equating law is as 22

Low-frequency response characteristics
Define the frequency and angular frequency of the pulse source as f and ω0. When the transformer responds to low-frequency pulse signal (f<10 3 Hz), Fig. 14(b) can also be simplified. In Fig. 14(b), CDp is in parallel with CDs0, and the parallel combination capacitance of these two is about 10 -6~1 0 -9 F so that the reactance can reach 10k~1M. Meanwhile, the reactance of Lµ is small. As a result, CDp and CDs0 can also be ignored. Reactances of Lls0 and Llp (10 -7 H) are also small under the low-frequency condition, and they also can be ignored. Usually, the resistivity of magnetic core is much larger than common conductors to restrict eddy current. In view of that Rs0<< RL0<<Rc, the combination of Rs0, RL0 and Rc can be substituted by R0. Furthermore, R0  RL0. Finally, the equivalent schematic of pulse transformer under low-frequency condition is shown in Fig.15(a).  Fig. 15(a), Lµ and R0 are in parallel, and then in series with RP which is at m range. R0 is usually very small due to the equating process from (28). When ω0 of the pulse source increases, reactance of Lµ also increases so that ω0Lµ>>R0. In this case, the Lµ branch gets close to opening, and an ideal voltage divider is formed only consisting of RP and R0. At last, the pulse source U1 is delivered to the load R0 without any deformations. And the response voltage pulse signal U2 of transformer on the load resistor is as When RP << R0, U1= U2 which means the source voltage completely transfers to the load resistor. On the other hand, if ω0Lµ<<R0, Lµ shares the current from the pulse source so that the current flowing through R0 gets close to 0. In this situation, the pulse transformer is not able to respond to the low-frequency pulse signal U1.
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 RL=50. So, the R0 may be at m range when it is equated to the primary circuit. Select conditions as follows: Rp=0.09, Lµ=12.6µH, and U1 is the periodical sinusoidal voltage pulse with amplitude at 1V. The low-frequency response curve of pulse transformer is obtained from Pspice simulation on frequency scanning, as Fig. 15(b) shows. When f of U1 is larger than the second inflexion frequency (100Hz), response signal U2 is large and stable. However, when f is less than the first inflexion frequency (10Hz), response signal U2 gets close to 0. And the cut-off frequency fL is about 10Hz.
The conclusion is that low-frequency response capability of pulse transformer is mainly determined by Lµ, and the response capability can be improved through increasing Lµ calculated in (5).

High-frequency response characteristics
When the transformer responds to high-frequency pulse signal (f>10 6 Hz), conditions "ω0Lµ>>R0" and "ω0Lµ>>Rp" are satisfied so that the branch of Lµ seems open. In Fig. 14(b), the combination effect of Rs0, RL0 and Rc still can be substituted by R0. Substitute Llp and Lls0 by Ll, and combine CDs0 and CDp as CD. The simplified schematic of pulse transformer for high-frequency response is shown in Fig. 16(a).
In Fig. 16(a), when ω0 of pulse source increases, reactance of Ll increases while reactance of CD decreases. If ω0 is large enough, ω0Ll>>R0>>1/(ω0CD) and the response signal U2 gets close to 0. On the other hand, condition 1/(ω0CD)R0 is satisfied when ω0 decreases. The pulse current mainly flows through the load resistor R0, and the good response of transformer is obtained. Especially, when ω0Ll<<Rp, Ll also can be ignored. Under this situation, Rp is in series with R0 again, and the response signal U2 which corresponds to the best response still conforms to (29).
Select the amplitude of the periodical pulse signal U1 at 1V. If Rp, Ll and CD are at ranges of m, 0.1µH and pF respectively, the high-frequency response curve of transformer is also obtained as shown in Fig.16(b) from Pspice simulation. When f is less than the first inflexion frequency (about 300kHz), response signal U2 is stable. When f is larger than the second inflexion frequency (about 10MHz), response signal U2 gets close to 0. And the cut-off frequency fH is about 10MHz. The conclusion is that high-frequency response characteristics of transformer are mainly determined by distributed capacitance CD and leakage inductance Ll. The high-frequency response characteristics can be obviously improved through restricting CD and Ll , or increasing Lµ.

Square pulse response of pulse transformer with closed magnetic core
In Fig. 14(b), Rs0<< RL0<<Rc, and the combination effect of Rs0, RL0 and Rc can be substituted by R0. Combine CDs0 with CDp as CD. The simplified schematic of pulse transformer circuit for square pulse response is shown in Fig. 17. U1 and U2 represent the square voltage pulse source and the response voltage signal on the load respectively. The total current from the pulse source is i(t), while the branch currents flowing through R0, CD and Lµ are as i1, i2 and i3 respectively.

Response to the front edge of square pulse
Usually, Lµ ranges from 10 -6 H up to more than 10 -5 H, and the square pulse has front edge and back edge both at 100ns~1µs range. So, when the fast front edge and back edge of square pulse appear, reactance of Lµ is much larger than the equated load resistor R0. Under this condition, i3 is so small that the effect of Lµ on the front edge response can be ignored.
Define the voltage of CD as Uc(t). As aforementioned, Lµ has little effect on the response to the front edge of square pulse. Through Ignoring the Lµ branch, the circuit equations are presented in (30) with initial conditions as i(0)=0, i1(0)=0 and Uc(0)=0.

U t i t R L d i td tLd i td ti t R L d it d t it R it d tC
If the factor for Laplace transformation is as p, the transformed forms of U1(t) and i1(t) are defined as U1(p) and I1(p). Firstly, four constants such as α, β, γ and λ are defined as The under dumping state solution of (30) is as The load current i1(t) =U2(t)/R0. From (34), response voltage pulse U2(t) on load consists of an exponential damping term and a resonant damping term. The resonant damping term which has main effects on the front edge of pulse contributes to the high-frequency resonance at the front edge. Constant a defined in (33) is the damping factor of the pulse droop of square pulse U2(t), b is the damping factor of the resonant damping term, and ω is the resonant angular frequency. Substitute λR0Us by U0, and define two functions f1(t) and f2(t) as f1(t) is just the resonant damping term divided from (34), while f2(t) is the pure resonant signal divided from f1(t). If pulse width T0=5µs, the three signals U2(t), f1(t) and f2(t) are plotted as Curve 1, Curve 2 and Curve 3 in Fig. 18 respectively. In the abscissa of Fig. 18, the section when t < 0 corresponds to the period before the time when the square pulse appears.  The conclusion is that the rise time of the front edge of response pulse can be improved by minimizing the capacitance CD and leakage inductance Llp and Lls0 of the transformer. The waveform of the response voltage signal can be improved through increasing the damping resistor of the circuit in a proper range.

Pulse droop analysis of transformer response
In Fig.17, when the front edge of pulse is over, Uc(t) of CD and the currents flowing through Llp and Lls0 all become stable. And these parameters have little effects on the response to the flat top of square pulse. During this period, load voltage signal U2(t) is mainly determined by Lµ. So, the simplified schematic from Fig.17 is shown as Fig.19 (a). The circuit equations are as The initial conditions are as i3(0)=0 and U2(0)=R0Us/(R0+Rp). The load voltage U2(t) is obtained as (38) through solving equations in (37).
In (38), τ is the constant time factor of the pulse droop. When Lµ increases which leads to an increment of τ, the pulse droop effect is weakened and the pulse top becomes flat. If U20=R0Us/(Rp+R0), the response signal to the flat top of square pulse is shown in Fig. 19(b). When pulse duration T0 is short at µs range, the pulse droop effect (0<t<T0) of U2(t) is not obvious at all. However, when T0 ranges from 0.1ms to several milliseconds, time factor τ has great effect on the flat top of U2(t), and the pulse droop effect of the response signal is so obvious that U2(t) becomes an triangular wave.

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, U1 and Rp have no effects on the pulse tail response when the main pulse is over. CD which was charged plays as the voltage source. Combine Llp and Lls0 as Ll. The equivalent schematic for pulse tail response of transformer is shown in Fig. 20.
i1(0) and i3(0) are determined by the final state of the pulse droop period. There are also three kinds of solutions, however the under dumping solution usually corresponds to the real practices. So, this situation is analyzed as the centre topic in this section. Define six constants α1, β1, γ1, α2, β2 and γ2 as (40).
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 f3(t) and f4(t) as (43), according to (41).
The response signal U2(t) in (41) also consists of an exponential damping term f3(t) and a resonant damping term f4(t).
Define B1+B2 as ' 0 U . In order to help to establish direct impressions, a batch of parameters are selected (CD=2.14µF, Lµ=12.6µH and Ll=1.09µH) for plotting the response pulse curves. According to (41) and (43), signals U2(t), f3(t) and f4(t) are plotted as Curve 1, Curve 2 and Curve 3 respectively in Fig. 21(a) for example. Because the damping factor a1 defined in (42) is large, the amplitude of f3(t) which corresponds to Curve 2 is very small with slow damping. The resonant damping term f4(t) which is damped faster determines the resonant angular frequency ωs. The resonant parts of U2(t) and f4(t) are also in superposition at the back edge of pulse. When f4(t) is damped to 0, U2(t) becomes the same as f3(t). The half of the resonant period td is as According to (40) and (42), R0 has effects on the damping factors of f3(t) and f4(t). The resonant frequency is mainly determined by leakage inductance, magnetizing inductance and distributed capacitance of transformer. Fig. 21(b) shows an impression of the effect of Lµ on the tail of response signal. When Lµ changes from 0.1µH to 1mH while other parameters retain the same, the resonant waveforms with the same frequency do not have large changes. So, the conclusion is that, td and the resonant angular frequency ωs are not mainly determined by Lµ. Fig. 21(c) shows the effect of leakage inductances of transformer on the pulse tail of response signal.
When Ll is small at 10nH range，the back edge of pulse (Curve 2) is good as which of standard square pulse. When Ll increases from 0.01µH to 1µH range, the resonances become fierce with large amplitudes. If Ll increases to 10µH range, the previous under damping mode has a transition close to the critical damping mode (Curve 4). The fall time td of the pulse tail increases obviously. Fig. 21(d) shows the effect of distributed capacitances of transformer on the pulse tail of response signal. The effect of CD obeys similar laws obtained from Lµ. So, the conclusion is that the pulse tail of the response signal can be improved by a large extent through minimizing the leakage inductances and distributed capacitances of transformer windings. Paper [24] demonstrated the analysis above in experiments.

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 (S1) of C1, such as mechanical switch, vacuum trigger switch, spark gap switch, thyristor, IBGT, thyratron, photo-conductive switch, and so on. S1 has double functions including opening and closing. S1 ensures the single direction of HES energy transferring, from C1 and transformer to C2. In this section, the energy transferring characteristics of HES mode based on transformer charging is analyzed in detail.
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, C1 and C2 in HES module are far larger than the distributed capacitances of pulse transformer. So, the distributed capacitances can be ignored in HES cell. In the practical HES module, many other parameters should be considered, such as the junction inductance, parasitic inductance of wires, parasitic inductance of switch, parasitic resistance of wires, parasitic resistance of switch, and so on. These parameters can be concluded into two types as the parasitic inductance and parasitic resistance. As a result, the equivalent schematic of the HES module is shown in Fig. 22(a). In Fig. 22(a), C1 and C2 represent the primary energy-storage capacitor and load capacitor respectively. Lpl and Lsl represent the parasitic inductances in the primary circuit and secondary circuit, while Rp and Rs stand for the parasitic resistances in the primary circuit and secondary circuit respectively. L1, L2 and M of transformer are defined in (17) and (18). ip(t) and is(t) represent the current in the primary and secondary circuit. The pulse transformer with closed magnetic core has the largest effective coupling coefficient (close to 1) in contrast to Tesla transformer and air-core transformer. Under the condition of large coupling coefficient, the transformer in Fig. 22(a) can be decomposed as Fig. 22(b) shows. Lµ, Llp and Lls are defined in (5), (11) and (14) The voltages of C1 and C2 are Uc1(t) and Uc2(t), respectively. According to Fig. 22(a), the circuit equations of HES module are as In view of Fig. 22(b), the circuit equations of HES module can also be established as 2 2' The initial conditions are as ip(0)=0, is(0)=0, Uc1(0)=U0 and Uc2(0)=0. In view of that ip(t)=-C1dUc1(t)/dt and is(t)=-C2dUc2(t)/dt, Equations in (45) can be simplified as 22  In (47), ωp and ωs are defined as the resonant angular frequencies in primary and secondary circuits, while k p and ks are defined as the coupling coefficients of the primary and secondary circuits respectively. These parameters are presented as 22  x in the characteristic equation (50) represents the characteristic solution. As a result, x, D1 and D2 should be calculated before the calculations of Uc1(t) and Uc2(t). Obviously, the characteristic solution x can be obtained through the solution formula of algebra equation (50), but x will be too complicated to provided any useful information. In order to reveal the characteristics of the HES module in a more informative way, two methods are introduced to solve the characteristic equation (50) in this section.

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. Actually in many practices, the "no loss" approximation is reasonable. As a result, equation In (51), it is easy to get the two independent characteristic solutions defined as x±. Uc1(t)=D1e xt and Uc2(t)=D2e xt can also be calculated combining with the initial circuit conditions. Finally, the most important four characteristic parameters such as Uc1(t), Uc2(t), ip(t) and is(t), are all obtained as 10  In (52), LΣ represents the sum of the parasitic inductances and leakage inductances, while ω+ and ωstand for the two resonant angular frequencies existing in the HES module (ω+>>ω-). Parameters such as T, LΣ, ω+ and ωare as 22 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 ω+ and ωat the same time, which demonstrates that the HES module based on transformer with closed magnetic core is a kind of double resonant module. The input and output characteristics and the energy transferring are all determined by (52).

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 transformer mainly 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 idea of "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 Δx± to rectify the characteristic equation (50) or (51). That's to say, the previous characteristic solutions x± are substituted by x±+Δx±. In HES module, the parasitic resistances which cause the energy loss still exist, though they are very small. So, the parasitic resistances also should be considered. Define j as unit of imaginary number, and variable xj as -jx/ωs. Equation (50) In (57), ρ1 represents the characteristic impedance of the resonant circuit, and ρ1=[LΣ(1+T)/C1] 1/2 . According to (55), the general solutions of (49) (Uc1(t)=D1e xt and Uc2(t)=D2e xt ) are clarified. When the initial circuit conditions are considered, the important four characteristic parameters such as Uc1(t), Uc2(t), ip(t) and is(t) are obtained as (58). In (58), β±=|Δω±|=|Δx±|ωs, coefficients such as G1, G2 and G3 are defined as 22  Usually, semiconductor switch such as thyristor or IGBT is used as the controlling switch of C1. However, these switches are sensitive to the parameters of the circuit such as the peak current, peak voltage, and the raising ratios of current and voltage. The raising ratio of Uc1(t) and ip(t) (dUc1(t)/dt and ip(t)/dt) can also be calculated from (58), which provides theoretical instructions for option of semiconductor switch in the HES module.
Actually, the efficiency of energy transferring is also determined by the charge time of C2 in practice. Define the charge time of C2 as tc, the maximum efficiency of energy transferring on C2 as ηa, and the efficiency of energy transferring in practice as ηe. If the core loss of transformer is very small, the efficiencies of HES module based on pulse transformer charging are as

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 (N1 turns) and the secondary windings (N2 turns), and it works in accordance with the hysteresis loop shown in Fig.24. Define the induced voltage of primary windings of transformer as Up(t), and the primary current as ip(t). If the input voltage Up(t) increases, the magnetizing current in primary windings also increases, leading to an increment of the magnetic induction intensity B generated by ip(t). When B increases to the level of the saturation magnetic induction intensity Bs, dB/dH at the working point (H0, B0) decreases to 0 and the relative permeability μr o f m a g n e t i c c o r e d e c r eases to 1. Under this condition, magnetic characteristics of the core deteriorate and magnetic saturation occurs. Once the magnetic saturation occurs, the transformer is not able to transfer voltage and energy. So, it's an important issue for a stable transformer to improve the saturation characteristics of magnetic core and keep the input voltage Up(t) at a high level simultaneously.