Pulse shape modification for capacitor driven pulsed

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Pulse shape modification for capacitor driven pulsed magnets

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INSTITUTE OF PHYSICS PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 14 (2003) 1075–1082

PII: S0957-0233(03)61185-9

Pulse shape modification for capacitor driven pulsed magnets K Rosseel1 , W Boon and F Herlach Laboratorium voor Vaste-Stoffysica en Magnetisme, K U Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium E-mail: [email protected]

Received 21 March 2003, accepted for publication 14 May 2003 Published 17 June 2003 Online at stacks.iop.org/MST/14/1075 Abstract Two special design features of capacitor banks for driving pulsed magnets are proposed; these are dimensioned by Laplace transforms. The stray capacitance and the inductance of the wiring can result in sharp voltage pulses with amplitude of up to twice the initial charging voltage. A simple RC filter is designed which efficiently suppresses these overvoltages. The second feature is a circuit that modifies the pulse shape in order to facilitate magnetization measurements on superconducting samples where the magnetic response not only depends on the field but also on the rate of the field sweep. This is achieved by adding a suitable combination of inductors and capacitors to the capacitor bank circuit. Keywords: magnetic field strength, pulsed magnetic fields, capacitor banks,

pulse shaping, magnetization, high-Tc superconductors

1. Introduction Pulsed magnets energized by capacitor banks are now used for scientific research in many laboratories [1]. In a capacitor bank installation for driving pulsed field magnets, these are usually connected to the switchgear by coaxial cables, while the wiring of the bank generally may consist of a combination of open busbars and coaxial cables. The stray capacitance and inductance of these components gives rise to transient voltages that can result in voltage peaks of up to twice the charging voltage. These overvoltages may cause damage to the pulsed field coil and/or circuit components. In this paper, we calculate the transient overvoltages and propose a practical solution to suppress them. We also propose a modification of the general circuit in order to optimize the pulse shape of the magnetic field for pulsed field magnetization measurement (PFMM) on systems in which the magnetization is strongly dependent on the sweep rate of the magnetic field. At the K U Leuven Pulsed Field Laboratory [2], magnetic fields in a typical range of 50–60 T are generated using a 5 kV, 500 kJ capacitor bank. For the highest fields (>60 T) and for magnet testing, a 10 kV, 625 kJ capacitor bank is used. The capacitor banks can be discharged using either a 1 Author to whom any correspondence should be addressed.

mechanical switch or a thyristor stack. The mechanical switch consists of two brass blocks that are pulled together by a small electromagnet. This switch allows very high current densities and is therefore used for testing new magnets, sometimes up to destruction of the magnet. For regular measurements it is not well suited; it is rather noisy due to the arcing and evaporation of the brass blocks, that are easily degraded and have to be cleaned and realigned before each pulse. The thyristor-switch provides a very clean current pulse and is therefore applied for user magnets. However, thyristors are very sensitive to overrating regarding voltage, current and rise time. It is common practice to protect the thyristors by series inductances that limit the current. At the K U Leuven, the switching arrangement for the 10 kV bank, which will be considered in this paper, consists of three parallel stacks of four ABB 5STP 33L2800 thyristors. Each stack is connected in series with a 50 µH protection inductance to limit the shortcircuit current in each stack to 45 kA.

2. Basic equations The general circuit considered in this paper is shown in figure 1. In this circuit, C0 represents the capacitor bank; L 0 and R0 are the self-inductance and resistance of the pulsed field magnet.

0957-0233/03/071075+08$30.00 © 2003 IOP Publishing Ltd Printed in the UK

1075

K Rosseel et al R

L

i(t)

RC1

RC i1(t)

C0

C1

R0

i2(t) C

L0

Figure 1. Basic circuit of a capacitor bank; L, C, R and RC represent the wiring or added components. The circuit with RC1 and C1 is added for the overvoltage suppression.

R, L and C represent the cabling and the parasitic impedance of the capacitor bank (section 3), or represent added components to adjust the pulse shape for PFMM (section 4). The circuit formed by C1 and RC1 that suppresses overvoltages will only be considered and designed in section 3; the resistor RC is only of importance for section 4. The differential equations for the currents i 0 (t), i 1 (t) and i 2 (t) in the different branches of the circuit of figure 1 were solved using Kirchhoff’s laws for the Laplace transforms [3] of the voltages over the different circuital elements. Initially, the bank C0 is charged to a voltage V0 , while C is not charged. Due to the presence of the self-inductances, all currents are zero at the start of the bank discharge. The currents in the circuit of figure 1 (without the C1 –RC1 branch) are then given in their Laplace representation as a2 s 2 + a1 s + a0 I0 (s) = (1) V0 N (s) I1 (s) I0 (s) (R0 + L 0 s)C0 Cs (R0 + L 0 s)C0 Cs = = (2) V0 V0 a2 s 2 + a1 s + a0 N (s) I2 (s) 1 (Rc Cs + 1)C0 = (I0 (s) − I1 (s)) = (3) V0 V0 N (s) with I (s) = L[i(t)], where s is the complex frequency s = γ + jω, with γ representing a damping factor and ω the angular frequency. The denominator is N (s) = b4 s 4 + b3 s 3 + b2 s 2 + b1 s + b0

(4)

and the coefficients in equations (1)–(4) are given by a2 = C0 L 0 C a1 = (R0 + Rc )C0 C a0 = C0 b4 = C LC0 L 0 b3 = CC0 (L(R0 + Rc ) + L 0 (R + Rc )) b2 = CC0 (R R0 + (R + R0 )Rc ) + C0 (L 0 + L) + C L 0 b1 = C0 (R + R0 ) + C(R0 + Rc ) b0 = 1. The voltage over the magnet can be derived from Um (s) I2 (s) = (L 0 s + R0 ). (5) V0 V0 The induced voltage in a pick-up coil located in the centre of the pulsed magnet is given by Uind (s) ∝ s I2 (s). The currents and voltages in the time domain can be determined by decomposing their Laplace transforms 1076

into partial fractions and then applying the inverse Laplace transformation [3]; however, this is not always straightforward. Several numerical methods can be applied; a general and elegant method of calculating the inverse Laplace transform can be found in [4]. This method is easy to program and the inverse Laplace transform can be derived with any desired accuracy. Alternatively, numerical solutions can be found using specialized software packages. For the simple circuit considered here, all the currents and voltages can be calculated analytically by determining the roots s0 , s¯0 , s1 and s¯1 , with si = γi + jωi , of the fourth order polynomial N (s) (see appendix A) and then calculating the partial fractions (see appendix B). The currents are then given as
 i k (t) Bk,0 − γ0 Ak,0 = e−γ0 t Ak,0 cos ω0 t + sin ω0 t V0 ω0
 B − γ k,1 1 A k,1 sin ω1 t (6) + e−γ1 t Ak,1 cos ω1 t + ω1 with k = 0, 1, 2. In expression (6), ω0 and γ0 are the angular frequency and damping factor related to the capacitor bank C0 and the pulsed magnet L 0 , R0 , while ω1 and γ1 are mainly determined by L , C and R. Similar expressions are obtained for the voltages in the circuit of figure 1.

3. Transient overvoltages in a capacitor bank A capacitor bank circuit with its parasitic impedances from the wiring is represented in figure 1 by putting RC = 0. The resistance of the bank and of the wiring is lumped into the resistor R. For the results described in this section, the parameters for the pulsed field coil are L 0 = 805 µH, R0 = 22 m and the capacitance of the bank C0 = 1.2 mF. From short circuit tests we found that the self-inductance of the bank in this configuration and using the mechanical switch is L = 6 µH, with a resistance of R = 15 m. The relevant parasitic capacitance is C = 1 nF. This yields the following frequencies and damping factors: γ1 = 1.24 × 103 s−1 ; γ0 = 36.4 s−1 ; ω1 = 1.30 × 107 rad s−1 ; ω0 = 1.01 × 103 rad s−1 . The voltage variation over the magnet can be calculated from equation (5) as u m (t) = e−γ0 t (A0 cos ω0 t + B0 sin ω0 t) V0 + e−γ t (A1 cos ωt + B1 sin ωt)

(7)

with A0 = 0.993 = − A1 ; B0 = −4.43 × 10−3 ; B1 = −9.36 × 10−5 . When using the thyristor switch we find L = 21 µH and R = 37 m, yielding γ1 = 8.59 × 102 s−1 ; γ0 = 22.8 s−1 ; ω1 = 6.99 × 106 rad s−1 ; ω0 = 1.00 × 103 rad s−1 , A0 = 0.975 = − A1 ; B0 = −8.14 × 10−3 and B1 = −1.14 × 10−4 . As the self-inductance L 0 of the pulsed magnet and the capacitance of the bank C0 are much larger than the parasitic components L and C, respectively, the voltage variation over the magnet for the first few microseconds of the pulse is mainly determined by L and C and equation (7) can be well approximated by a simple exponentially damped function

Pulse shape modification for capacitor driven pulsed magnets

u m (t) ≈ 1 − e−γ1 t (cos ω1 t + B1 sin ω1 t) V0

(8)

with B1 ≈ γ1 /ω1 . Equation (8) describes the voltage variation over an initially uncharged C connected through a self-inductance L with a capacitor C0 , charged to an initial voltage of V0 . At times t = (2n + 1)π/ω, with n = 0, 1, 2, . . . , the voltage over the magnet reaches a maximum, with u m (π/ω1 )/V0 ≈ 2V0 . The presence of L and C produces oscillations in the megahertz range; the amplitude of these can be double the initial voltage on the capacitor bank. The current through the magnet shows a small ripple of order 0.01% of the current amplitude; this may somewhat reduce the resolution of measurements on samples that are sensitive to the sweep rate of the magnetic field. Fortunately, due to the skin effect and radiation losses, the effective initial resistance of the wiring is two orders of magnitude higher than its dc resistance, such that these highfrequency oscillations become strongly damped and become negligible approximately 10 µs after the start of the capacitor discharge. As an illustration, in figure 2 the voltage over the magnet branch is shown for a pulse with V0 = 320 V using the mechanical switch. To account for the measured pulse shape, we assume that the voltage drop over the switch can be described as an exponential function

(a)

(b)

Vs (t) ≈ V0 e−t/τ with τ the time constant describing the voltage decay over the switch. Equation (8) is then modified to u m (t) ≈ 1 + K 1 e−at + e−γ1 t (K 2 cos ω1 t + K 3 sin ω1 t) V0

(9)

with γ12 + ω12 a(2γ1 − a) , K2 = 2 2 (a − γ1 ) + ω1 (a − γ1 )2 + ω12 2 a γ1 (γ1 − a) + ω1 K3 = a = 1/τ. ω1 (a − γ1 )2 + ω12

K1 = −

Equation (9) was used to fit the data shown in figure 2(a), with a, γ1 and ω1 as free parameters. To determine γ1 and ω1 , we fitted the last 5 µs of the data, as the first part of the curve is determined by conductance variations and bouncing of the mechanical switch. From the fit we find for the damping factor γ1 = 9 × 105 s−1 , while the angular frequency is ω1 = 1.32 × 107 rad s−1 , in very good agreement with the calculated frequency. The decay time for the switch τ was determined from fitting over the data range 0–9 µs while keeping γ1 and ω1 constant. From the fit we find τ = 0.1 µs. For comparison, the resulting curve is plotted in figure 2(a); it describes our data reasonably well—taking into account the noisy character of the discharge when using the mechanical switch. In the inset we have plotted the resulting curve for τ = 0 µs. The decay time of the switch, in combination with a large damping factor, effectively limits the overvoltage to a factor of 1.5 instead of 2. To improve the quality of the data, the remainder of the measurements described in this section were performed using thyristors. A typical pulse (for V0 = 506 V) is shown in

Figure 2. Normalized voltage over the coil branch for (a) a 320 V pulse using the mechanical switch and (b) for a 506 V pulse, using thyristors. The dots denote the measured data, while the full curve shows a fit using equation (9). In the inset the response is shown using the parameters extracted from the fits but with the switch decay time τ = 0.

figure 2(b). Also in this case, the response can be described by equation (9), albeit by considering K 1 as a free parameter and incorporating some time shift u m (t) → u m (t −t0 ) to account in a phenomenological way for the thyristor I –V characteristic. From the fit, we find a decay time of τ ≈ 0.4 µs, which leads to an even lower overvoltage (∼1.2–1.4, depending on C0 and initial voltage V0 ), while the presence of the protection inductances decreases the overvoltage frequency from 2.07 to 1.1 MHz. Nevertheless, to avoid damage to e.g. the crowbar diodes and to improve the field resolution at the initial stages of the field pulse, it is necessary to eliminate the oscillations due to the parasitic components L and C. To achieve this goal, an RC branch consisting of a resistor RC1 and a capacitor C1 is added to the basic circuit of figure 1. To determine suitable values for RC1 and C1 , we analysed the current behaviour in the different branches. To simplify calculations, we neglect any delay effects from the switches. Furthermore, the initial current in the pulsed field magnet is neglected, as the transient overvoltages occur in the first few microseconds of the discharge. 1077

K Rosseel et al

The Laplace form of the current through L and R is then I (s) ≈ C0 (RC1 C1 Cs + C + C1 ){LC0 CC1 RC1 s 3 V0 + [LC0 (C + C1 ) + R RC1 C0 C1 C]s 2 + [RC0 (C + C1 ) + RC1 C1 (C + C0 )]s + C + C1 + C0 }−1 . Because C, C1  C0 , we can further simplify equation (10) to I (s) ≈ (RC1 C1 Cs + C + C1 ){[LC RC1 C1 s 3 V0 + [L(C + C1 ) + RC RC1 C1 ]s 2 + [R(C + C1 ) + RC1 C1 ]s + 1]}−1 . Suitable values for C1 and RC1 can be determined from the condition that the poles of I (s) are real, such that no damped oscillations will occur in the initial microseconds of the bank discharge. Assuming L(C1 + C)  RC RC1 C1 and √ R(C + C1 )  RC1 C1 , and taking n = C1 /C and y = LC/(3RC1 C), this condition can be calculated to be (see appendix C)
 
3 (n + 1) 3 4 (8 − 20n − n 2 ) 2 1 = 0. y + y + n n 12n 2 27

(11)

For equation (11) to have real roots the condition n 3 − 24n + 192n − 512 = (n − 8)3
0 has to be fulfilled; this is satisfied when n
8. For n = 8 and C = 1 nF we have C1 = 8 nF and y = 0.51. With typical values for our 10 kV bank of L = 6 and 21 µH, without and with the thyristors, respectively, this yields RC1 = 50.3 and 98.5 . With the bank capacitance C0 = 1.2 mF and resistance R = 15 m, the assumptions C0  C1 , L(C1 + C)  RC RC1 C1 and R(C + C1 )  RC1 C1 are fulfilled. This is even valid for any practical n: if we take e.g. n = 1000 we have C1 = 1 × 10−6 F and solutions y = 5.27, yielding RC1 = 9.2  and y = 0.67, yielding RC1 = 72.6 . The conditions on the extra components C1 and RC1 are thus not very stringent. In practice, one will go for a low capacitance–high resistance solution, as this is the easiest to implement. With the addition of the RC suppression circuit, the normalized voltage over C—and thus over the magnet—is then given by

z

(10)

Figure 3. Suppression of overvoltage over the magnet branch for several values of n = C1 /C. The main figure shows curves for τ = 0, calculated using equation (12). In the inset the results are given for τ = 0.4 µs, calculated using equation (13).

A maximum suppression of ∼1.1V0 can be achieved, albeit for an inconveniently large n. Moreover, taking into account the voltage decay over the thyristor switch, in practice a more efficient damping can be expected for the low n solutions. The influence of the switches is easily calculated: in this case the voltage over the magnet branch is given by 5 u m (t)  = K i esi t V0 i=1

2

UC (s) I0 (s) RC1 C1 s + 1 Um (s) = = V0 V0 V0 s(C + C1 + RC1 C1 s) 4  s/(C L) + 1/(LC RC1 C1 ) Ki ≈ = s(s − s1 )(s − s2 )(s − s3 ) s − si i=1   [3]. with the coefficients K i = lims→si (s − si ) UmV(s) 0 The voltage in the time domain can then be directly derived as 4 u m (t)  = K i esi t . (12) V0 i=1 In figure 3 the voltage over the pulsed field magnet in the initial microseconds of the pulse, calculated using equation (12), is shown for several values of n. The large initial oscillations due to the parasitic impedances C and L are effectively suppressed to a maximum of u m,max < 1.25V0 . 1078

(13)

with a = 1/τ , K i = K i si a+a for i :1 . . . 4 and K 5 =

1 − C1 RC1 a 1 . (a + s1 )(a + s2 )(a + s3 ) LCC1 RC1

In the inset of figure 3, the results are shown for the thyristor switch, with τ = 0.4 µs. It is evident that even this short decay time can further suppress the overvoltage significantly for low n. A low n value (small C1 , large RC1 ) is thus adequate. In our case we have taken n = 10, resulting in C1 = 10 nF and RC1 = 86 . In figure 4, the experimental results are presented, showing the influence of the RC1 C1 branch for n = 10 and various values of RC1 . The results were obtained for 500 V pulses using the thryristor switch. In comparison with the predicted results from equation (13), the initial slope is very well reproduced, while the damping itself is more pronounced: even for RC1 = 80  the RC suppression circuit acts as a very efficient low pass filter. In the inset of figure 4, the pick-up voltage in the magnet is shown; it is evident that the suppression of the overvoltages has as a beneficial effect on the quality of the pick-up signal in the magnet, as small modulations in the derivative di 2 /dt of the current through the magnet are eliminated.

4. Suitable pulse shape for magnetization measurements of high TC superconductors 4.1. Motivation We have shown [5, 6] that inductive measurements of the magnetization in pulsed fields can be an interesting tool for

Pulse shape modification for capacitor driven pulsed magnets

4.2. Addition of a magnet with a ferromagnetic core A first simple solution to reduce the initial sweep rate is the insertion of an inductance with a ferromagnetic core in series with the pulsed magnet. As long as the inductance with the ferromagnetic core is much larger than the inductance of the pulsed magnet, the initial current remains low. Once the core becomes saturated, the current rises more rapidly in the circuit. The generated maximum field is somewhat reduced due to the inductance with the saturated core. Although the initial rise of the current i 2 (t) in the pulsed magnet for this modified circuit is more gradual, di 2 (t)/dt = 0 at t = 0, so that the initial response of dM/dt is still rather large. Also, spikes were observed in the dM/dt signal when the ferromagnetic core starts to saturate. As the circuit only modifies the initial stages of the field pulse, the asymmetries in |dH /dt | are retained. Figure 4. Voltage over the magnet with damping circuit, for different values of RC1 . In the inset the signal from the pick-up coil in the centre of the pulsed field magnet is plotted, showing that the damping circuit also eliminates parasitic oscillations in the derivative of the current through the magnet.

studying vortex dynamics in high temperature superconductors (HTSs) over a broad range of the parameter space of applied field H (t), field sweep rate dH /dt and temperature T . However, accurate measurements of the magnetic properties of HTSs in a pulsed magnetic field are hampered by two effects: (i) the Meissner effect and (ii) the time dependence of the field sweep rate dH /dt during a field pulse. In an AC field, a sample generates in the measuring dH = dM , pick-up coil a voltage that is proportional to dM dt dH dt where M is the magnetization. At the start of the field pulse, a superconducting sample is in the Meissner state and dM/dH = −1. Also, for the basic RLC circuit, |dH /dt| has a maximum at t = 0. The Meissner state only lasts for a very short time (∼10 µs) because of the high sweep rate |dH /dt| ≈ 103 –104 T s−1 and the small lower critical field µ0 Hc1 ≈ 0.1 T for the HTS. Once the superconductor is in the mixed state, the induced voltage can become two orders of magnitude lower than the initial signal that is related to the Meissner state. To record the dM/dt signal in a precise way it is thus necessary to use a transient recorder with a very large dynamic range, a high resolution and a large bandwidth. Another particular feature when using pulsed fields is that not only the field magnitude H but also the sweep rate dH /dt varies during a field pulse. As the current–voltage curves for HTSs are highly non-linear [5, 7], changes in the sweep rate are immediately reflected in the shape of the hysteresis curves [6, 8]. In principle, to extract the critical current density from the hysteresis curves, |dH /dt | should be the same at a given field for the rising (0 → Hmax ) and lowering (Hmax → 0) branches of the field pulse. In particular, at very high fields this condition is not fulfilled, as these fields can only be reached when a diode crowbar is added to the bank circuit (otherwise the coil would overheat). This involves the large disadvantage that the difference in sweep rate in the rising and lowering branches of the pulse can differ by more than an order of magnitude. In an ideal set-up for PFMM, the pulse form should thus be such that |dH /dt | rises gradually from zero value at t = 0, while H (t) is symmetric around the maximum field.

4.3. Use of a storage capacitor C and storage inductor L A second approach consists in transferring some of the bank energy in an auxiliary capacitor, such that the current in the pulsed magnet rises more gradually, with |dH /dt| = 0 at t = 0. In terms of the current through the high field magnet i 2 (t)in its Laplace transform, this requirement leads to the condition lim

t→0

di 2 (t) = lim s(s I2 (s)) = 0. s→∞ dt

(14)

To this end, one could add a large capacitance in parallel with the magnet. Instead of representing the capacitance of the wiring, in this discussion C will thus represent a ‘storage’ capacitance. The resistor RC is then added to the circuit of figure 1 in order to account for the resistance in the wiring of C. RC can be expected to be small, of the same order as R, the resistance of wires and bus bars of the main part of the capacitor bank. Inspection of the general equations (1)–(3) shows that equation (14) is fulfilled if the self-inductance L is present in the general circuit of figure 1. Starting from this configuration, we then determine the magnitude of the additional components L and C that is needed to obtain a symmetrical field pulse. Also, energy losses should be minimized in order not to reduce the peak field too much. Of course, to avoid high frequency components that will cause rapid oscillations in i 2 (t) and to reduce the difference in dH /dt in the branches of the H (t) curve, the added L and C will have to be of the same order of magnitude as C0 and L 0 . In order to realize this condition with the available capacitors, we have C = Cmax − C0 , with C  C0 and Cmax = 10.2 mF. We want to keep the capacitance C0 as large as possible, in order not to reduce the initially stored energy too much. To obtain suitable values for L and C we consider the most basic case: no thyristors, no crowbar and all the resistances are zero. In this ideal case, γ0 = γ1 = 0 and the fourth order polynomial N (s) of equation (4) is reduced to N (s) = b4 s 4 + b2 s 2 + 1 = b4 (s 2 + ω02 )(s 2 + ω12 ).

(15)

Then equation (6) for the currents in the circuit simplifies to i k (t) Bk,0 Bk,1 = sin ω0 t + sin ω1 t, V0 ω0 ω1

with k = 0, 1, 2. (16) 1079

K Rosseel et al

For ω1 = 3ω0 , the current through the magnet is symmetrical and has a very smooth initial rise. From this condition and equation (15), and with α = C/C0 and β = L/L 0 , this leads to the condition 9(1 + β + α)2 = 100αβ. Solving this equation for β, we obtain real solutions when α
36/64 ≈ 0.563. The angular frequencies ω0 and ω1 in terms of α and β are then given by 1 1 ω02 = √ 3 αβ C0 L 0

and

1 3 ω12 = √ . (17) αβ C0 L 0

Equation (16) for the currents i 1 (t)—the current through the storage capacitor C—and i 2 (t)—the current through the magnet—further simplifies to   1 1 3 2 i 1 (t) = sin ω0 t 1 − sin ω0 t (18) V0 ω0 β L 0 2  1 1 1 β i 2 (t) sin3 ω0 t. = (19) V0 ω0 β L 0 2 α In comparison with a basic LC circuit with the same capacitance C0 as the bank, the maximum current through the magnet i 2,max is modified by a factor √ 34 1 i 2,max = i ref . 2 αβ In this expression, i ref denotes the maximum current through the magnet for the reference LC circuit; in this case it is possible to obtain the same current for the modified circuit as for our reference LC circuit: we have i 2,max = i ref for α = β = 0.75. These are, however, not the optimum values. If we take C0 = Cmax for the reference LC circuit and denote the maximum obtainable current through the magnet for this case as i ref,max , i 2,max can be written as √ 34 1 1 √ i 2,max = i ref,max . 2 αβ 1 + α √ The current i 2,max will thus be further reduced by a factor 1/1(1 + α). In figure 5, the evolution of i 2,max is shown as a function of α and β. We have as optimum values α = 0.64 and β = 0.93; this results in C0 = 6.22 mF. The maximum current in the magnet is then i 2,max = 0.77i ref ,max . In the interval 0.75  β  1.25, however, we have i 2,max
0.75i ref ,max . To put these results more in perspective, we note that in day-today use of our 10 kV bank we use typically a total capacitance of C0 = 8.4 mF, which gives α ≈ 0.21. Actually, the current through the magnet is thus reduced by roughly 15% in comparison to this standard setting for the 10 kV bank. These results show that the addition of a large extra L and C does not have to result in a large reduction of the obtainable fields. In figure 6 the currents (figure 6(a)) and voltages (figure 6(b)) in the different branches of the circuit of figure 1 are plotted for the ideal case (no resistance) and α = 0.64 and β = 0.93. The voltages over the storage inductor L, storage capacitor C (and thus over the magnet as RC = 0) and bank C0 are zero at current maximum, as is the case for a standard LC circuit. 1080

Figure 5. Maximum current through the magnet, as a function of α and β. As a reference we take the maximum current obtained for a basic LC circuit, with C0 = Cmax . The black triangles denote the current through the magnet for the LC reference circuit, as a function of the capacitance of the bank C0 = Cmax /(1 + α). The inset shows the solutions for β as a function of α. (a)

(b)

Figure 6. Currents and voltages through the circuit of figure 1 (without RC1 –C1 branch), for α = 0.64 and β = 0.925 and with all resistors equal to zero.

In figure 7 we have plotted the current through the magnet for α = 0.64 and β = 0.93 and with the addition of typical resistor values in the different branches of the circuit of figure 1: Rc = 15 m, R = 30 m and R0 = 22 m.

Pulse shape modification for capacitor driven pulsed magnets

5. Conclusions The simple RC filter that was designed and tested efficiently suppresses the overvoltages caused by the stray inductance and capacitance of the wiring in the capacitor bank. These overvoltages could amount to twice the charging voltage. The filter has a beneficial effect on the quality of the pick-up signal in the magnet, as small modulations in the derivative of the current di 2 /dt through the magnet are eliminated. The addition of suitable self-inductances and capacitors to the discharge circuit results in a symmetrical pulse shape without sharp breaks that facilitates magnetization measurements on samples in which the magnetic response not only depends on the field but also the field sweep rate. Figure 7. Current through the magnet for the reference LC circuit, with C0 = 8.4 mF, L 0 = 805 µH, R0 = 22 m and R = 15 m, and for the modified circuit, for α = 0.64 and β = 0.925, C0 = 6.22 mF, L 0 = 805 µH, C = 3.98 mF, L = 745 µH and typical resistor values Rc = 15 m, R = 30 m and R0 = 22 m.

The resistance R represents the resistance of the wiring and of the self-inductance L. As the latter is only needed to store the energy, we can use a Brooks type of coil and cool it to liquid nitrogen temperature in order to minimize its resistance. Although the capacitor C is not charged initially, our calculations show that no overvoltages are generated: for all reasonable values of α and β, the voltage over C is always smaller than V0 . This modified circuit has many advantages for magnetization measurements on superconductors. In the best case there are no oscillations in the pulse, as compared to the standard L RC circuit including parasitic inductance and capacitance. At the start of the pulse and at the half period both the magnetic field and the field sweep rate dH /dt are zero. This will greatly reduce the dynamical range needed for accurately recording the dM/dt signal. The field pulse has a good symmetry and the maximum and the minimum of dH /dt have nearly the same absolute value. This means that at the same value of the field H nearly the same sweep rate |dH /dt| is used for the superconductors, which facilitates extraction of the critical current. In comparison with a standard L RC circuit, the modified circuit proposed here may also allow for an elegant way to disconnect the coil after a half period. In the standard L RC circuit (without the crowbar), virtually the full bank voltage would be over the magnet at this point. In the modified circuit proposed here, both the voltage as well as the current through the magnet is practically zero at half period. This means that switching off the magnet branch and the branch containing C and RC could be done at this point. This is by no means a perfect situation, though, as the capacitors on the bank still experience a large voltage reversal (of order 80%). If this is no issue, one might consider a situation in which the magnet is switched off after half a period and a damping circuit could be switched on, allowing the capacitor bank to be discharged in a critically damped fashion over the storage inductor L. As the maximum currents over L are much lower than the maximum current through the magnet, heating in the storage inductor is not expected to be an issue.

Acknowledgments This work is supported by the FWO-Vlaanderen, the Flemish GOA, project VIS no 99/001 and the Belgian IUAP programs. The authors thank A Lagutin and J Vanacken for stimulating discussions and a critical reading of this manuscript and F Gentens and P Muylaert for technical assistance. This paper is dedicated to the memory of Professor Willy Boon.

Appendix A. Calculation of ω and γ To find the roots of the fourth order polynomial P(s) = N (s)/b4 , we start by seeking a real root u 1 for the cubic equation u 3 − c2 u 2 + (c1 c3 − 4c0 )u − (c12 + c0 c32 − 4c0 c2 ) = 0, with ci = bi /b4 . Once u 1 is determined, 

we can write P(s) = c2

P0 (s)P1 (s), with P0,1 (s) = s 2 + c23 ∓ 43 + u 1 − c2 s + u21 ∓

u 21 − c0 [9]. 4 As all the roots of P(s) are imaginary for typical values of the components C0 , L 0 , R0 , C, L and R, we can also write P(s) = (s + s0 )(s + s¯0 )(s + s1 )(s + s¯1 ), with si = γi + jωi , so 2 2 + ω0,1 , from which that we arrive at P0,1 (s) = s 2 + 2γ0,1 s + γ0,1 ω0,1 and γ0,1 can readily be derived.

Appendix B. Explicit expressions for the currents i, i1 and i2 As the roots are imaginary, we can write N (s) = b4 (s + s0 ) (s + s¯0 )(s + s1 )(s + s¯1 ), with sl = γl + jωl . The currents Ik (s)in the circuit of figure 1 (without the C1 –RC1 branch) can then be written as Ik (s) tk,2 s 2 + tk,1 s + tk,0 = V0 (s + s0 )(s + s¯0 )(s + s0 )(s + s¯1 ) Ak,0 s + Bk,0 Ak,1 s + Bk,1 = + . 2 2 (s + γ0 ) + ω0 (s + γ1 )2 + ω12 For the inverse Laplace transform we then find
 Bk,0 − γ0 Ak,0 i k (t) −γ0 t Ak,0 cos ω0 t + =e sin ω0 t V0 ω0
 Bk,1 − γ1 Ak,1 sin ω1 t + e−γ1 t Ak,1 cos ω1 t + ω1 1081

K Rosseel et al

with the coefficients determined by [3] Ak,l = − Im[Q k,l ], with Q k,l =

Bk,l − γl Ak,l = Re[Q k,l ], ωl

  1 Ik (s) . lim (s + sl )(s + s¯l ) ωl s→−sl V0

Explicit expressions for these coefficients can easily be derived. For l = 0 we have Ak,0 = −[(2tk,2 γ0 − tk,1 )((γ1 − γ0 )2 + (ω12 − ω02 )) + 2(γ1 − γ0 )(tk,2 (γ02 − ω02 ) − tk,1 γ0 + t0 )] × [((γ1 − γ0 )2 + (ω12 − ω02 ))2 + 4ω02 (γ1 − γ0 )2 ]−1 1 Bk,0 − γ0 Ak,0 = [(tk,2 (γ02 − ω02 ) − tk,1 γ0 + t0 )((γ1 − γ0 )2 ω0 ω0 + (ω12 − ω02 )) − 2ω02 (γ1 − γ0 )(2tk,2 γ0 − tk,1 )] × [((γ1 − γ0 )2 + (ω12 − ω02 ))2 + 4ω02 (γ1 − γ0 )2 ]−1 . For l = 1 we find

Bk,1 − γ1 Ak,1 1 = [(tk,2 (γ12 − ω12 ) − tk,1 γ1 + t0 )((γ0 − γ1 )2 ω1 ω1 + (ω02 − ω12 )) − 2ω12 (γ0 − γ1 )(2tk,2 γ1 − tk,1 )] × [((γ0 − γ1 )2 + (ω02 − ω12 ))2 + 4ω12 (γ0 − γ1 )2 ]−1 .

Appendix C. Calculation of the real root condition We seek a condition to have real roots for the third order polynomial

(20)

When equation (20) can be written in the form g(s) = s 3 + a1 s + a0 , real roots are found when (21)

This can be accomplished by using the transform s → 1 )+RC RC1 C1 , yielding s − α, with α = L(C+C 3LCC1 RC1 g(s) = LCC1 RC1 s 3
(L(C + C1 ) + RC RC1 C1 )2 + − 3LCC1 RC1  + R(C + C1 ) + RC1 C1 s (L(C + C1 ) + RC RC1 C1 )(R(C + C1 ) + RC1 C1 ) 3LCC1 RC1 2(L(C + C1 ) + RC RC1 C1 )3 + . (22) 27(LCC1 RC1 )2

+1−

1082

Multiplying by (LC)3 and taking C1 = nC and y = this can be further written as

LC , 3RC1 C





 n+1 2 1 3 2−n n+1 3 2 − +  0. y + y+ y n 3 2n n
 
3 n+1 3 4 8 − 20n − n 2 2 1 y + y +  0. n n 12n 2 27

+ 2(γ0 − γ1 )(tk,2 (γ12 − ω12 ) − tk,1 γ1 + t0 )] × [((γ0 − γ1 )2 + (ω02 − ω12 ))2 + 4ω12 (γ0 − γ1 )2 ]−1

a02 a13 +  0. 4 27

√

This can be rearranged in powers of y to yield

Ak,1 = −[(2tk,2 γ1 − tk,1 )((γ0 − γ1 )2 + (ω02 − ω12 ))

f (s) = LCC1 RC1 s 3 + (L(C + C1 ) + RC RC1 C1 )s 2 + (R(C + C1 ) + RC1 C1 )s + 1.

It is to be expected that in general the following conditions be met: L(C1 + C)  RC RC1 C1 and R(C + C1 )  RC1 C1 , such that after normalization equation (22) can be simplified to 
(C + C1 )2 1 s + g(s) = s 3 + − 3(CC1 RC1 )2 LC 2C − C1 2(C + C1 )3 + + . 2 3LC C1 RC1 27(CC1 RC1 )3 The real root condition (21) can then be written as 3
(C + C1 )2 1 − + (3CC1 RC1 )2 3LC 
2C − C1 (C + C1 )3 2 + +  0. 6LC 2 C1 RC1 (3CC1 RC1 )3

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