THE NUMERICAL MODEL OF 2G YBCO SUPERCONDUCTING TAPE IN THE WINDINGS OF THE TRANSFORMER

Computer assisted calculations consists in applying software for the simulation of models of certain devices and later analysing their behavior under given conditions corresponding to real working conditions in a specific environment.The paper proposes a circuit model of 2G YBCO superconducting tape created in the PSpice program. The model consists of passive blocks and active user blocks of analogue behavioural modelling (ABM). ABM blocks calculate the conductances, the currents of the individual layers of the superconducting tape, its thermal capacity, the heating power, the cooling power and resulting temperature of the tape. The model uses table of thermal power density passed to the liquid nitrogen vs. temperature. Smooth transition of the YBCO superconductor layer into the resistive state is described by Rhyner’s power law. The developed model was used for generating waveforms of thermal and electrical quantities.


INTRODUCTION
Superconductors enable to combine magnetic and electrical properties of materials, which can be noticed thanks to the loss of electrical resistance and some peculiar magnetic phenomena under specific conditions.Superconductors are in the superconducting state when the working point determined by temperature, current density and magnetic field intensity is below the critical surface characteristic for each superconducting material.
A general classification of superconductors is introduced due to the critical temperature value: • LTS -Low Temperature Superconductor, • HTS -High Temperature Superconductor.
The critical temperature TC = 25 K is the point separating LTS and HTS superconductors (Tinkham, 2004).

. The structure of 2G YBCO superconducting tape
Second generation high temperature superconducting tapes (2G HTS) are called layered or coated conductors.These tapes are usually based on yttrium high-temperature superconductor YBCO.Second generation tapes begin to replace the tapes and wires of the first generation (1G HTS), which were based on bismuth superconductor BSCCO.Characteristic about the second generation tapes in superconducting state is that they conduct current of hundreds amperes at very low losses.Additionally, when the second generation superconductor obtains the resistive state, an unlaminated tape has a relatively high resistance.
High temperature superconducting tapes of second generation are made of several relatively thin layers.A base layer is responsible for electrical and mechanical parameters and in many cases consists of non-magnetic Hastelloy (Ni -57,00%, Mo -16,00%, Cr -15,50%, Fe -5,50%, W -4,00%, Co -2,50%).The thickness of this layer is about 50 µm (Table 1).An optional layer of stabilizer which determines the performance of thermal and also mechanic parameters of the tape, is located at the top and bottom of the tape and has the thickness of about 20 µm.2G YBCO tape comprises a silver layer having thickness of about 2 µm, YBCO superconducting layer having thickness of about 1 µm, a buffer layer LaMnO3 (LMO) having thickness of 30 nm, a homoepitaxial layer MgO having thickness of 30 nm and a base layer MgO having thickness of 10 nm.Tapes produced by American Superconductor (AMSC) are covered with laminate coating of stainless steel, copper or brass.In turn, SuperPower company produces tapes without a stabilizer (SF series) and tapes with copper stabilizer (SCS series).While comparing these two types of second generation superconducting tapes (Fig. 1 and Fig. 2), it can be predicted that tapes without a stabilizer show many times higher resistance than tapes with a stabilizer above the temperature TC (Majka & Kozak, 2009;Tinkham, 2004).Second generation superconducting tapes are usually made of YBCO material.It is a chemical compound comprising yttrium oxide, barium oxide and copper oxide.The general chemical formula of this material is YBa2Cu3O6+x (0 < x < 1). Figure 3 illustrates the single structure diagram of the superconducting powder.

Equivalent circuit of 2G YBCO superconducting tape
The modelling object is 2G YBCO superconducting tape.Figure 4 presents the equivalent circuit of the tape.Mathematical model of the superconducting tape consists of three non-linear and one linear resistor in parallel.These resistors include: Hastelloy, a silver layer, a copper layer and a superconducting layer (Czerwinski, Jaroszynski, Majka, Kozak, and Charmas, 2016;Jaroszynski & Janowski, 2014).In comparison to circuit modelling finite element method (FEM) subdivides a large problem into smaller, simpler parts that are called finite elements.Due to the high geometric aspect-ratio of superconducting tape, this method leads to the much greater complexity of the calculations than circuit modelling in PSpice program.The second approach seems to be quite effective for the analysis of states when the transport current is higher than Ic (Czerwinski, Jaroszynski, Janowski, Majka, and Kozak, 2014;Janowski, Wojtasiewicz, and Jaroszynski, 2016;Jaroszynski et al., 2014).
PSpice program enables to examine the system for changes in such quantities as: voltage, current, power, temperature, additional parameters of the model.

Details of the simulation
The circuit model of 2G YBCO superconducting tape is designed in PSpice program.The model, presented in Figure 6, consists of a passive block PARAM containing the electrical, thermal and geometrical parameters of the tape, and active user blocks of the ABM (analogue behavioural modelling).Voltage blocks calculate the relative temperature (ABM1) in relation to the temperature of liquid nitrogen, the cooling (ABM2) and heating power (ABM3) of the superconducting tape, and the current block calculates the current of the tape (ABM4).For the purpose of current detection in the circuit a secondary DC source (Vpr) with zero voltage (EMF) was used.The model uses tables with the values of power density passed to the liquid nitrogen as the function of temperature (GeStCieLN) and two hierarchical blocks.The first hierarchical block represents thermal capacity (Cth), which is the sum of the thermal capacity of the layer of copper, silver, Hastelloy and the YBCO superconductor.The second hierarchical block calculates the resultant conductance of the tape (G).This block allows for a smooth transition of the YBCO superconductor layer into the resistive state in which the current is described by Rhyner's power law (1) (Czerwinski et al., 2016;Janowski et al., 2016;Jaroszynski et al., 2014).(1) where: Eelectric field intensity, V/m, Eccritical electric field intensity for HTS, 10 -4 V/m, Jcurrent density, A/m 2 , Jccritical current density, A/m 2 , nexponent depending on the temperature.
The relation between electric field intensity and current density in power law depends on the actual temperature.The temperature affects the critical value of the current and the value of the exponent n.This exponent influences the steepness in rise of current-voltage characteristics and is defined for the purposes of the circuit modelling as: where: T0reference temperature, K, Ttemperature of tape, K, n0exponent in T0 temperature (for YBCO n0 =15 ÷ 40, T0 = 77 K).
Assuming homogenous distribution of current density and electric field intensity, the equation ( 1) can be written as: After multiplying components of the equation (3) the following dependence can be described as: The dependence (4) can be presented in the form of resistance: where: RYBCOthe resistance of YBCO layer, Ω.
Critical current Ic(T) in the equation ( 5) can be illustrated by the linear relationship: where: Tccritical temperature, K, Ic0critical current in T0 temperature, A.
After substitution into the equation ( 5) the dependence (6) of critical current Ic(T) the equation is: After including the critical electric field intensity, the equation is: where: Llength of tape, m.
The conductance of the superconductor is the reciprocal of resistance: where: GYBCOthe conductance of YBCO layer, S, Rrezresidual resistance (a small value higher than zero needed to perform the convergent calculations, in this program equals 10 -15 Ω).

Simulation results
The simulation takes into the account the following factors: the sinusoidal voltage source with the amplitude of 4 V, frequency 50 Hz and internal resistance 2•10 -3 Ω, the length of tape 0.5 m, the width of tape 4•10 -3 m, the thickness of a silver layer 2•10 -6 m, the thickness of a copper layer 4•10 -5 m, the thickness of Hastelloy 5•10 -5 Ω, the thickness of YBCO layer 10 -6 m, the critical current of the tape IC0 = 100 A, the critical temperature of superconductor TC = 93 K, the temperature of liquid nitrogen T0 = 77 K, Rhyner's law exponent n0 = 40.

Thermal capacity of 2G YBCO tape
The hierarchical block Cth (Fig. 8) enables to calculate the thermal capacity of 2G YBCO tape.This block comprises thermal capacity of a copper layer CthCu, a silver layer CthAg, Hastelloy CthHST, YBCO superconductor CthYBCO.The calculations are based on the dependence of the thermal capacity for individual layers of the tape as a function of temperature, as can be seen in Fig. 9, Fig. 10 and Fig. 11.This data allows to create tables with specific heats of copper layer CwCu, silver layer CwAg, Hastelloy CwHST.The constant value of 185 J/(kg•K) is accepted as the specific heat for the superconducting layer CwHST.The calculations also include the density ϒ of the individual layers in the superconducting tape, which are presented in Table 2.In this model the conductance of 2G YBCO tape is calculated using the hierarchical block G (Fig. 6).The calculations are based on the dependence in the resistivity ρ of copper and silver layers as a function of temperature, which are presented in Fig. 12 and Fig. 13.The resistivity for Hastelloy is of constant value and equals 1.24•10 -6 Ωm.The calculations also include the thickness of individual layers, as well as the length and the width of the superconducting tape.The developed conductance model of 2G YBCO tape was used for generating conductance waveforms of copper layer GCu, silver layer GAg, Hastelloy GHST (Fig. 14). Figure 15 presents the conductance waveforms of YBCO superconducting layer vs. time.The conductance of the superconducting layer GYBCO reaches the maximum about 8 MS when the current in the circuit tends to zero.

The current of copper, silver, Hastelloy and YBCO layer
The controlled current source was used for each layer of the superconducting tape, which allowed to calculate the instantaneous current of individual layer.
Figure 16 illustrates the initial current waveforms in the copper layer ICu, silver layer IAg, Hastelloy IHST, YBCO IYBCO and total current of superconducting tape IHTS.Tape current IHTS in a longer span is also depicted in Figure 17.After setting the supply voltage in a period of time (0 -0.2 ms) the current flows only through YBCO layer.The maximum of current IYBCO is about 120 A and then, starts to decrease due to the fast heating of the tape.The copper layer allows for the flow of current ICu at the maximum 300 A. The peak current of superconducting tape IHTS is 405 A. This value is reached in 4 ms.After 300 ms current IHTS proceeds to the quasi-steady state, the amplitude is less than 100 A. The silver and Hastelloy layers conduct relatively small currents of about a few amperes.

Temperature, heating and cooling power of superconducting tape
Figure 18 presents a six-element table GeStCieLN with the values of heat flux density (W/m 2 ) passed to the liquid nitrogen vs. temperature difference ΔT.The table was based on the dependence of the heat flux density passing through the metal surface and the liquid nitrogen vs. temperature difference ΔT (Fig. 19).The temperature of superconducting tape increases and after 1 s it reaches the highest value T = 293 K.The maximum cooling power of the superconducting tape equals 400 W. The peak heating power of the tape is 1.45 kW, and during the experiment decreases to about 0.57 kW.

CONCLUSIONS
The rapid growth in information technology, the development of software and the computer calculations make the computer simulations one of The basic tools to examine and analyse the physical devices and phenonema.
PSpice program allows to design computer models of superconducting elements.Blocks of the ABM (analogue behavioural modelling) in electronic circuit simulation enable to analyse the transients of the superconducting tape which is used in the windings of superconducting transformers.
Thanks to PSpice program and active user blocks of behavioural modelling, the electrical circuit simulation seems to be a reliable tool for the analysis of superconducting materials in transitional conditions.
Fig. 1.The structure of second generation superconducting tape by AMSC (American Superconductor, 2016)

Fig. 7 .
Fig. 7. ABM voltage block calculating the conductance of the YBCO superconducting layer

Fig. 16 .
Fig. 16.Current waveforms in cooper layer ICu, silver IAg, Hastelloy IHST, superconductor IYBCO and the total current of superconducting tape IHTS vs. time

Fig. 18 .
Fig. 18.Six-element table with heat flux density passedto the liquid nitrogen vs. temperature difference ΔT

Fig. 20 .
Fig. 20.Temperature of superconducting tape as a function of time