A combined compensation method for the output voltage of an insulated core transformer power supply L. Yang, J. Yang, K. F. Liu, B. Qin, and D. Z. Chen Citation: Review of Scientific Instruments 85, 063302 (2014); doi: 10.1063/1.4884340 View online: http://dx.doi.org/10.1063/1.4884340 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetostrictive vibrations model of a three-phase transformer core and the contribution of the fifth harmonic in the grid voltage J. Appl. Phys. 115, 17A316 (2014); 10.1063/1.4863931 Core loss behavior in high frequency high power transformers—I: Effect of core topology J. Renewable Sustainable Energy 4, 033112 (2012); 10.1063/1.4727910 Static voltage distribution between turns of secondary winding of air-core spiral strip transformer and its application Rev. Sci. Instrum. 82, 094704 (2011); 10.1063/1.3625280 Effect of the change in the load resistance on the high voltage pulse transformer of the intense electron-beam accelerators Rev. Sci. Instrum. 80, 115110 (2009); 10.1063/1.3263902 Analytical expression for the output voltage of the triple resonance Tesla transformer Rev. Sci. Instrum. 76, 104702 (2005); 10.1063/1.2093764
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 063302 (2014)
A combined compensation method for the output voltage of an insulated core transformer power supply L. Yang,a) J. Yang,b) K. F. Liu, B. Qin, and D. Z. Chen State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
(Received 20 March 2014; accepted 8 June 2014; published online 25 June 2014) An insulated core transformer (ICT) power supply is an ideal high-voltage generator for irradiation accelerators with energy lower than 3 MeV. However, there is a significant problem that the structure of the segmented cores leads to an increase in the leakage flux and voltage differences between rectifier disks. A high level of consistency in the output of the disks helps to achieve a compact structure by improving the utilization of both the rectifier components and the insulation distances, and consequently increase the output voltage of the power supply. The output voltages of the disks which are far away from the primary coils need to be improved to reduce their inhomogeneity. In this study, by investigating and comparing the existing compensation methods, a new combined compensation method is proposed, which increases the turns on the secondary coils and employs parallel capacitors to improve the consistency of the disks, while covering the entire operating range of the power supply. This method turns out to be both feasible and effective during the development of an ICT power supply. The non-uniformity of the output voltages of the disks is less than 3.5% from no-load to full-load, and the power supply reaches an output specification of 350 kV/60 mA. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884340] I. INTRODUCTION
Medium and low-energy irradiation accelerators are widely used in radiation processing,1–3 environmental protection,4–7 etc. The high-voltage power supply plays an important role of providing energy for the electron beam in accelerators. With the merits of high efficiency (>85%),8, 9 high power, and reliability,4, 10, 11 an insulated core transformer (ICT) power supply is becoming an optimal choice for high-current, medium, and low-energy irradiation accelerators.11–13 Research and development on high-voltage electron accelerators has been carried out at the Huazhong University of Science and Technology (HUST).14 With the progress in electrical technology, leading insulation technology and new electrical materials have been designed and developed a high-performance ICT power supply (HUST-ICT) to meet the requirements for future markets. The basic principle of the ICT power supply was proposed by Van De Graaf in 1965.15 An ICT consists of two yokes and several insulated cores (Fig. 1). Each segmented core corresponds to a secondary coil and related rectifier circuit.16 The rectifier outputs are connected in series to produce a high voltage.17 The high magnetic flux leakage caused by the structure of segmented cores leads to a decrease in the induced voltage of the secondary coils, especially for the upper layers. Moreover, the magnetic leakage flux increases the output reactance of the secondary coils. Thus, the output
a) Present address: College of Electrical & Electronic Engineering, HUST,
Luoyu Road 1037, Wuhan, Hubei 430074, People’s Republic of China.
b) Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0034-6748/2014/85(6)/063302/6/$30.00
voltages of different layers exhibit non-uniform distribution. Generally, there are three ways to solve this problem. (1) Van De Graaf put forward the I-shaped core to improve the induced voltage of the secondary coils to some extent,18 but the core processing is complicated and costly. (2) The traditional method is to increase the turns of the secondary coils according to the magnetic flux leakage distribution of each layer.19 This method makes output reactance of the more remote secondary coils increase. As a result, the output voltage of the remote coil drops more when the power supply output changes from no-load to full-load. (3) The leakage flux is compensated by generating a current that has the same phase as the excitation current from the capacitors in parallel across the coil, and maintains the same number of turns for all secondary coils.19, 20 The simulated model and analytic method for a general ICT power supply are introduced in Sec. I. Based on this, the compensation effect of existing methods (increasing turns of the secondary coils and using parallel capacitors) on a 350 kV/60 mA ICT power supply are analyzed in Sec. II. Simulation results indicate that when the increasing turns method is used, the load capacity of disks becomes weaker as their distance from the primary coil increases and the load capacity becomes stronger when using parallel capacitors. The distribution of each disk’s load capacity is the opposite when the above two methods are adopted. The implication is that by integrating the two methods in an ICT, a better result will be obtained. A combined method is proposed, and the uniformity and consistency of all disks agree with the simulation results. Finally, experimental results are given that verify the feasibility and advancement of the combined method.
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© 2014 AIP Publishing LLC
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FIG. 1. Structure of the HUST-ICT. The six-layer (1#–6#) secondary cores and coils are used for producing the high voltage. “Pri” represents the primary cores and coils.
II. SIMULATION OF THE ICT POWER SUPPLY
To analyze the compensation methods more effectively, a simulation model of the power supply is established. The magnetic core of the HUST-ICT is shown in Fig. 1, which consists of two annular yokes (on the top and at the bottom), primary cores, and seven layer secondary cores. The top layer secondary cores and their coils are used as a filament power supply, and the coils can be ignored in the simulation model because their output power is much smaller than other layers. The core’s limb is made of silicon-steel laminations 0.27 mm thick, with a radius of 89 mm and six steps. Considering the filling rates (about 92%),21 the effective radius of the magnetic section is about 85 mm. Each disk has three secondary coils with corresponding double voltage rectifier circuits. All the rectifier disks produce high voltages in series. The mutual inductance matrix of the seven single-turn coils on the same phase was obtained using the finite element method (FEM) code ANSYS and is shown in Table.I. The actual mutual inductance matrix is calculated using Eq. (1): Mxy = Mxy (1) nx ny ,
According to the results of the magnetic field calculations for the HUST-ICT, the magnetic flux intensity was much less than the saturation value.14 When the core saturation was not considered, the mutual inductance matrix was able to represent all the characteristics of the transformer. Therefore, the simulation model was established based on the mutual inductance matrix. In addition, only one phase of the ICT needs to be simulated because of the symmetry in the magnetic circuit of the transformer. A double voltage rectifier circuit was connected to each secondary coil. The current waveform in the secondary coil was complex when the power supply was under load. The power supply was simulated with MATLAB/Simulink software. Based on the mutual inductance matrix calculated by the FEM, the one phase simulation model was established and is shown as Fig. 2. Because of the long time-constant of the rectifier, it takes a considerable time to obtain stable simulation results under the no-load condition. On the other hand, after the power supply reaches the stable state, the input current of the rectifier goes to zero and the current waveform of the secondary coils becomes sinusoidal. The loads of the secondary coils are only the compensation capacitors. In this case, the output voltages and currents of coils satisfy the transformer
(1)
where x and y represent the number of coils, p is the number of primary coils and s (s = 1. . . 6) is the number of secondary coils, Mxy is the mutual inductance between coil x and coil y, Mxy (1) is the mutual inductance between single-turn coil x and y, and nx , ny are the turns of the coils x and y.
TABLE I. Mutual inductance of single-turn coils (unit: μH). Pri, 1#–6# present one primary coil and six secondary coils belong to one phase. It is convenient to calculate the actual mutual inductance of coils when the number of turns is changed. Mxy (1)
Pri
1#
2#
3#
4#
5#
6#
Pri 1# 2# 3# 4# 5# 6#
3.18 2.50 2.22 2.01 1.86 1.77 1.72
2.50 2.62 2.28 2.03 1.87 1.77 1.71
2.22 2.28 2.42 2.14 1.94 1.82 1.76
2.01 2.03 2.14 2.33 2.09 1.94 1.87
1.86 1.87 1.94 2.09 2.33 2.14 2.04
1.77 1.77 1.82 1.94 2.14 2.44 2.29
1.72 1.71 1.76 1.87 2.04 2.29 2.64
FIG. 2. (a) is the double voltage rectifier module. (b) is the simulation model of the ICT power supply, which is based on the mutual inductance module. The mutual inductance matrix calculated by Eq. (1) is used in the mutual inductance module. The disks in (b) are the rectifier circuits as shown in (a). C1 –C6 are the compensation capacitors of the secondary coils.
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equation (2): U = jωMI + RI, (2) T ˙ p, U ˙1 ···U ˙ 6 ]T and I = ˙Ip , ˙I1 · · · ˙I6 are the where U = [U separate voltages and currents of the coils. The elements in the vectors U and I are in plural form, M is the mutual inductance matrix of the coils, ω is the angular frequency of the power supply and its value is 100π rad/s, and R is the resistance matrix of the coils, which can be ignored because the impedance of the capacitors is much larger than the resistance of the coils. The voltages and currents of the secondary coils satisfy Ohm’s law: ˙ s= U
1 ˙Is . jωCs
(3)
The secondary output voltages Us (rms) are proportional to the input voltage Up and can be obtained using Eqs. (2) and (3). The maximum value of the output voltage of each rectifier disk is set at 60 kV taking into consideration the insulation limits. There are three secondary coils and double voltage rectifier circuits in a disk, and the maximum value of √ the 2 voltage of the secondary coils is then 60 kV ÷ 3 ÷ 2 ÷ √ = 5 2 kV. Initially, the input voltage is set at Up [0] , and the secondary coils output voltages (Us [0] ) and their maximum values (Umax [0] ) can be obtained. The real output voltages can then be obtained using Eq. (4): √ 5 2 Us [0] . (4) Us = Umax [0] Thus, the general simulation model of the ICT power supply has been established and it can be used to analyze each compensation method. III. EFFECT OF THE EXISTING COMPENSATION METHODS ON THE HUST-ICT
The first method was to increase the turns of the secondary coils, which were remote from the primary coils, to increase their output voltages. The turns of the primary coils (np ) of the HUST-ICT were optimized as 92 and the input rated voltage (rms) was set at 380 V. When the input voltage was √ 380 V, the output voltage of each secondary coil should be 5 2 kV. Therefore, the turns of the secondary coils were calculated using Eq. (5): ns =
Us Lp (1) np k, Up Mps (1)
(5)
where k is the unified compensation coefficient and used to compensate for the voltage drop of silicon rectifier stacks, and Lp (1) is the self-inductance of the single-turn primary coil. The turns of the secondary coils calculated using Eq. (5) are shown in Table II. TABLE II. Turns of secondary coils if the increasing turns method is adopted. Coil No.
1(n1 )
2(n2 )
3(n3 )
4(n4 )
5(n5 )
6(n6 )
Turns
2736
3080
3400
3671
3863
3974
FIG. 3. Output voltages of the disks at full-load and no-load if the method of increasing the turns is adopted. By increasing the turns of the secondary coils, the output voltages are consistent for the no-load condition. However, when the power supply is at full-load, the output voltages of the top layer disks drop even more.
The mutual inductance matrix of the coils whose turns are shown in Table II can be obtained using Eq. (1). Using this compensation method, the output voltages of the disks achieve the same under no-load condition. The output voltages of the disks can be obtained using the simulation model and are shown in Fig. 3. The simulation results indicate that as the distance between the disks and primary coils increases, the output voltages of the disks under full-load decrease, and the nonuniformity is about 21.3%. It is shown that the load capacity of the disks becomes weaker with increasing distance from the primary coil. Another method is to use capacitors to compensate for the voltage, assuming all the secondary coil turns are the same. There is a capacitive load current in the secondary coils. The magnetic flux excited by this load current has the same phase as the main magnetic flux. In other words, the compensation capacitors can increase both the magnetic flux through the secondary coils and the induced voltage. If the capacitor compensation method is adopted, the turns of the secondary coils can be set at 2853, which is calculated using Eq. (5), where Mps (1) needs to be replaced by its average value Mps (1) . The compensation capacitor was calculated using Eq. (6):19 Ccom =
l0 , ns 2 (2π f )2 μ0 Sc
(6)
where l0 = 2 mm is the gap between cores, f = 50 Hz is the frequency of the input power supply, μ0 is the permeability of vacuum, and Sc is the effective magnetic cross section area. The compensation capacitor calculated using Eq. (6) was 87 nF. The maximum output voltage of the disks was set at 60 kV by adjusting the input voltage. The output voltages of disks with and without load are shown in Fig. 4. The simulation result indicates that the no-load output voltages of the disks reduce with increasing distance between each disk and the primary coils. However, under full-load the situation is just the opposite. The non-uniformity is about 11.4% under full-load. Conversely, when the capacitor compensation method was adopted, the more remote the disk, the stronger their load capacity.
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FIG. 4. Output voltages of the disks under full-load and no-load when the capacitor compensation method was adopted. The output voltages were relatively consistent under the no-load condition. However, when the power supply was at full-load, the output voltages of the bottom layer disks dropped even more.
FIG. 5. Output voltages of the disks when the compensation capacitor (C6 ) values change. C6 can improve the voltage of each layer. When the value of the compensation capacitor is larger, the output voltage of this layer increases more. It will also affect other layers: the output voltages of layers closer to it increase more.
IV. COMBINED COMPENSATION METHOD
Based on the definition of the self-inductance, com can be expressed by Eq. (9):
The simulation results indicate that the uniformity of the rectifier disks is inadequate when the methods of increasing turns or capacitor compensation are adopted on their own. In addition, an important characteristic was found from the simulation results: the distribution of the load capacities of the disks was the opposite when the two compensation methods were adopted. Therefore, we combined the two methods in the HUST-ICT to obtain a better result. The steps for the combined compensation method follow. The turns of the secondary coils were corrected using Eq. (7): ns = ns +κ(ns −n1 ),
(7)
where ns are turns of the secondary coils calculated using Eq. (5) and listed in Table II, and κ is the turns compensation coefficient. A different κ implies a different compensation design. We selected a compensation coefficient, and then adjusted the compensation capacitors to make the output characteristics of each disk as similar as possible. The capacitors were different if κ took on different values, and the best uniformity of the output voltages of the disks was also different. We continued trying different values for κ until the uniformity met the requirements. When κ was selected as 0.75, the turns for the secondary coils are listed in Table III, as calculated using Eq. (7). The mutual inductance matrix for the design of the secondary coils was obtained using Eq. (1). The output voltages of disks as shown in Fig. 5 were obtained using the simulation model when the power supply was under full-load. The output voltage of disk 6# (V6 ) was lower than disk 1# (V1 ) at 15.5 kV. The compensation voltage (rms) of coil 6# was set at √ Ucom = 15.5 kV ÷ 2 ÷ 3 ÷ 2 = 1.8 kV. The compensation magnetic flux (rms) is given by com = Ucom /n6 /ω.
(8)
TABLE III. Turns of secondary coils when κ is 0.75. Coil No. Turn
1#
2#
3#
4#
5#
6#
2736
2994
3234
3437
3581
3665
com = L6 Icom /n6 = n6 L6 (1) Icom .
(9)
The compensation current can be obtained from Eqs. (8) and (9): Icom =
Ucom . ωn6 2 L6 (1)
(10)
The compensation capacitor can be obtained using Eq. (11): C6 =
1 = 72 nF. ωU6 /Icom
(11)
Under full-load, the output voltages of the disks are as shown in Fig. 5 when the compensation capacitor of coil 6# (C6 ) is 72 nF. The simulation result shows that V6 is higher than other disks when its capacitor is 72 nF, which means the capacitance is too large. V1 and V6 are the same when C6 is 51 nF and the output voltages of other disks are as shown in Fig. 5. V3 and V4 are lower than the voltages of the other disks, and the non-uniformity is about 8.1% under full-load. The uniformity of the disks is unsatisfactory when compensating coil 6# alone, while it can be improved by compensating the other coils. Unfortunately, the relationship between the output voltages of the disks and the compensation capacitors is very complex and difficult to predict analytically. It is very difficult to calculate the value of the compensation capacitors from some equations. Nonetheless, some useful relationships between the output voltages of the disks and the compensation capacitors have been found by simulation: (1) the output voltage will rise when the compensation capacitor in the same layer increases; (2) the more remote the disk from the primary coils, the greater the change in the voltages of the disks for the condition of the same compensation capacitor. Based on those relationships, the uniformity of the disks can be improved by taking steps to adjust the compensation capacitors. The minimum output voltage appears in the middle of the disks (3#, 4#) when C6 is 51 nF (solid line in Fig. 6). Therefore, C3 and C4 need to be adjusted (increased). Figure 6 shows the adjustment process for the compensation capacitors to improve the uniformity of the disks. The difference between the maximum voltage and minimum voltage was
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FIG. 6. Improvement of the output voltages of the disks by adjusting compensation capacitors, according to the distribution of the voltage of each layer.
reduced when C3 and C4 increased to 20 nF (dotted line). At the same time, V6 was higher than V1 , so C6 needed to decrease (dashed line); the voltages of the middle disks were still lower than those for the upper and lower disks, C3 and C4 , which needed to continue to increase. This adjustment process is repeated and the compensation capacitors of each layer modified until the uniformity of the disks meets the requirements. Finally, the non-uniformity decreased to 1.8% when the compensation capacitors combination was C2 = C3 = C4 = C6 = 30 nF, and C5 = 15 nF (dashed-dotted line). The uniformity of the output voltages of the disks under full-load has been well optimized. However, the uniformity under no-load has not yet been analyzed. In fact, because of the differences in the induced voltages and leakage inductances of each secondary coil, the output voltage drops of each disk are not matched by changes in output currents. In other words, the uniformity is related to the output current. The design was not acceptable unless the uniformity was adequate from no-load to full-load. Based on Eqs. (2)–(4), the output voltages of the disks were obtained under no-load and are shown in Fig. 7. It was found that the non-uniformity was about 4.3% under no-load when the combined compensation capacitors were C2 = C3 = C4 = C6 = 30 nF, and C5 = 15 nF (dotted line). In addition, V6 was higher than the other volt-
FIG. 7. Improvement in the output voltage of the disks under no-load and full-load. The indicator (no) represents the results under no-load. By adjusting compensation capacitors, the uniformities under these two conditions both reached a better value.
FIG. 8. The ICT power supply developed in HUST. Sulfur hexafluoride was used as the insulating medium for this power supply. This photograph was taken before covering the steel drum.
ages under both no-load and full-load, so C6 needed to be decreased. The output voltages of the disks are shown in Fig. 7 when C6 decreased to 27 nF. The non-uniformity of the disks was about 3.2% under no-load and about 2.4% under full-load (dashed-dotted line and dashed line). A better combination of capacitors may be found if the adjusting process is continued. However, this result was acceptable for engineering practice. A better combination may not be significant or necessary. Two reasons for this are: (1) the actual mutual inductance matrix will change a little because of the mechanical error in installing cores; (2) there are some differences between the FEM simulation and the actual model. Finally, the compensation coefficient needed to be reselected if there was no combination of capacitors that met the uniformity requirement. V. EXPERIMENTAL RESULTS
Because of the current limit of the actual coils, the compensation capacitors cannot be too large. It turns out that the maximum acceptable capacitance for the HUST-ICT was
FIG. 9. Improvement in uniformity of the disks in practice. The indicator (no) represents the results under no-load.
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filter capacitors and silicon stacks. Improving the uniformity can increase the average output voltage of the disks and the maximum output voltage of the power supply. The proposed combined compensation method was applied to an ICT power supply, and the test results demonstrate its effectiveness and feasibility. ACKNOWLEDGMENTS
This work was supported by National Nature Science Foundation of China No. 11305068. 1 J.
FIG. 10. Test results compared with the simulation results. “Sim” represents the simulation results and “test” represents the test results.
40 nF, and the compensation capacitors cannot exceed this, in practice. High voltage ceramic capacitors were used as the compensation capacitors in our project. The power supply used to test the combined compensation method is shown in Fig. 8. The search process for the combination of compensation capacitors, in practice, was the same as that in simulation, which was to adjust the capacitors according to the output voltages of the disks, as shown in Fig. 9. The non-uniformity was about 3.5% under full load and about 2.0% under no-load when the capacitors’ combination was C2 = 2 nF, C3 = C4 = 30 nF, C5 = 5 nF, and C6 = 35.7 nF. Figure 10 shows the test results and simulation results when the combination of capacitors was C2 = 2 nF, C3 = C4 = 30 nF, C5 = 5 nF, and C6 = 35.7 nF. It was found that the trends for the voltage distribution of the disks in the experiment were similar to those in the simulation, and the maximum difference between them was about 2.5%. Considering the errors of the mechanical installation of the cores, the test results confirm the validity of the simulation model and the feasibility of the combined compensation method. VI. CONCLUSIONS
With the aim of improving the consistency in the output voltages of the rectifier disks, the proposed combined compensation method is summarized by the following steps: (1) calculate the mutual inductance matrix for “single-turn” coils; (2) select a proper turns compensation coefficient (κ) and calculate the number of turns for the coils; (3) calculate the mutual inductance matrix for this design of the number of turns; (4) search for the capacitors’ combination under both no-load and full-load to meet the uniformity requirement; (5) repeat Steps 2–4 if Step 4 is not achieved. Generally, the nonuniformity under both no-load and full-load can reach 4% when the turns compensation coefficient (κ) is set between 0.3 and 0.8. The uniformity of disks is an important factor for an ICT power supply, because it ensures the homogeneity of the electric field and the utilization of rectifier components such as
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 063302 (2014)
A combined compensation method for the output voltage of an insulated core transformer power supply L. Yang,a) J. Yang,b) K. F. Liu, B. Qin, and D. Z. Chen State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
(Received 20 March 2014; accepted 8 June 2014; published online 25 June 2014) An insulated core transformer (ICT) power supply is an ideal high-voltage generator for irradiation accelerators with energy lower than 3 MeV. However, there is a significant problem that the structure of the segmented cores leads to an increase in the leakage flux and voltage differences between rectifier disks. A high level of consistency in the output of the disks helps to achieve a compact structure by improving the utilization of both the rectifier components and the insulation distances, and consequently increase the output voltage of the power supply. The output voltages of the disks which are far away from the primary coils need to be improved to reduce their inhomogeneity. In this study, by investigating and comparing the existing compensation methods, a new combined compensation method is proposed, which increases the turns on the secondary coils and employs parallel capacitors to improve the consistency of the disks, while covering the entire operating range of the power supply. This method turns out to be both feasible and effective during the development of an ICT power supply. The non-uniformity of the output voltages of the disks is less than 3.5% from no-load to full-load, and the power supply reaches an output specification of 350 kV/60 mA. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884340] I. INTRODUCTION
Medium and low-energy irradiation accelerators are widely used in radiation processing,1–3 environmental protection,4–7 etc. The high-voltage power supply plays an important role of providing energy for the electron beam in accelerators. With the merits of high efficiency (>85%),8, 9 high power, and reliability,4, 10, 11 an insulated core transformer (ICT) power supply is becoming an optimal choice for high-current, medium, and low-energy irradiation accelerators.11–13 Research and development on high-voltage electron accelerators has been carried out at the Huazhong University of Science and Technology (HUST).14 With the progress in electrical technology, leading insulation technology and new electrical materials have been designed and developed a high-performance ICT power supply (HUST-ICT) to meet the requirements for future markets. The basic principle of the ICT power supply was proposed by Van De Graaf in 1965.15 An ICT consists of two yokes and several insulated cores (Fig. 1). Each segmented core corresponds to a secondary coil and related rectifier circuit.16 The rectifier outputs are connected in series to produce a high voltage.17 The high magnetic flux leakage caused by the structure of segmented cores leads to a decrease in the induced voltage of the secondary coils, especially for the upper layers. Moreover, the magnetic leakage flux increases the output reactance of the secondary coils. Thus, the output
a) Present address: College of Electrical & Electronic Engineering, HUST,
Luoyu Road 1037, Wuhan, Hubei 430074, People’s Republic of China.
b) Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0034-6748/2014/85(6)/063302/6/$30.00
voltages of different layers exhibit non-uniform distribution. Generally, there are three ways to solve this problem. (1) Van De Graaf put forward the I-shaped core to improve the induced voltage of the secondary coils to some extent,18 but the core processing is complicated and costly. (2) The traditional method is to increase the turns of the secondary coils according to the magnetic flux leakage distribution of each layer.19 This method makes output reactance of the more remote secondary coils increase. As a result, the output voltage of the remote coil drops more when the power supply output changes from no-load to full-load. (3) The leakage flux is compensated by generating a current that has the same phase as the excitation current from the capacitors in parallel across the coil, and maintains the same number of turns for all secondary coils.19, 20 The simulated model and analytic method for a general ICT power supply are introduced in Sec. I. Based on this, the compensation effect of existing methods (increasing turns of the secondary coils and using parallel capacitors) on a 350 kV/60 mA ICT power supply are analyzed in Sec. II. Simulation results indicate that when the increasing turns method is used, the load capacity of disks becomes weaker as their distance from the primary coil increases and the load capacity becomes stronger when using parallel capacitors. The distribution of each disk’s load capacity is the opposite when the above two methods are adopted. The implication is that by integrating the two methods in an ICT, a better result will be obtained. A combined method is proposed, and the uniformity and consistency of all disks agree with the simulation results. Finally, experimental results are given that verify the feasibility and advancement of the combined method.
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FIG. 1. Structure of the HUST-ICT. The six-layer (1#–6#) secondary cores and coils are used for producing the high voltage. “Pri” represents the primary cores and coils.
II. SIMULATION OF THE ICT POWER SUPPLY
To analyze the compensation methods more effectively, a simulation model of the power supply is established. The magnetic core of the HUST-ICT is shown in Fig. 1, which consists of two annular yokes (on the top and at the bottom), primary cores, and seven layer secondary cores. The top layer secondary cores and their coils are used as a filament power supply, and the coils can be ignored in the simulation model because their output power is much smaller than other layers. The core’s limb is made of silicon-steel laminations 0.27 mm thick, with a radius of 89 mm and six steps. Considering the filling rates (about 92%),21 the effective radius of the magnetic section is about 85 mm. Each disk has three secondary coils with corresponding double voltage rectifier circuits. All the rectifier disks produce high voltages in series. The mutual inductance matrix of the seven single-turn coils on the same phase was obtained using the finite element method (FEM) code ANSYS and is shown in Table.I. The actual mutual inductance matrix is calculated using Eq. (1): Mxy = Mxy (1) nx ny ,
According to the results of the magnetic field calculations for the HUST-ICT, the magnetic flux intensity was much less than the saturation value.14 When the core saturation was not considered, the mutual inductance matrix was able to represent all the characteristics of the transformer. Therefore, the simulation model was established based on the mutual inductance matrix. In addition, only one phase of the ICT needs to be simulated because of the symmetry in the magnetic circuit of the transformer. A double voltage rectifier circuit was connected to each secondary coil. The current waveform in the secondary coil was complex when the power supply was under load. The power supply was simulated with MATLAB/Simulink software. Based on the mutual inductance matrix calculated by the FEM, the one phase simulation model was established and is shown as Fig. 2. Because of the long time-constant of the rectifier, it takes a considerable time to obtain stable simulation results under the no-load condition. On the other hand, after the power supply reaches the stable state, the input current of the rectifier goes to zero and the current waveform of the secondary coils becomes sinusoidal. The loads of the secondary coils are only the compensation capacitors. In this case, the output voltages and currents of coils satisfy the transformer
(1)
where x and y represent the number of coils, p is the number of primary coils and s (s = 1. . . 6) is the number of secondary coils, Mxy is the mutual inductance between coil x and coil y, Mxy (1) is the mutual inductance between single-turn coil x and y, and nx , ny are the turns of the coils x and y.
TABLE I. Mutual inductance of single-turn coils (unit: μH). Pri, 1#–6# present one primary coil and six secondary coils belong to one phase. It is convenient to calculate the actual mutual inductance of coils when the number of turns is changed. Mxy (1)
Pri
1#
2#
3#
4#
5#
6#
Pri 1# 2# 3# 4# 5# 6#
3.18 2.50 2.22 2.01 1.86 1.77 1.72
2.50 2.62 2.28 2.03 1.87 1.77 1.71
2.22 2.28 2.42 2.14 1.94 1.82 1.76
2.01 2.03 2.14 2.33 2.09 1.94 1.87
1.86 1.87 1.94 2.09 2.33 2.14 2.04
1.77 1.77 1.82 1.94 2.14 2.44 2.29
1.72 1.71 1.76 1.87 2.04 2.29 2.64
FIG. 2. (a) is the double voltage rectifier module. (b) is the simulation model of the ICT power supply, which is based on the mutual inductance module. The mutual inductance matrix calculated by Eq. (1) is used in the mutual inductance module. The disks in (b) are the rectifier circuits as shown in (a). C1 –C6 are the compensation capacitors of the secondary coils.
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equation (2): U = jωMI + RI, (2) T ˙ p, U ˙1 ···U ˙ 6 ]T and I = ˙Ip , ˙I1 · · · ˙I6 are the where U = [U separate voltages and currents of the coils. The elements in the vectors U and I are in plural form, M is the mutual inductance matrix of the coils, ω is the angular frequency of the power supply and its value is 100π rad/s, and R is the resistance matrix of the coils, which can be ignored because the impedance of the capacitors is much larger than the resistance of the coils. The voltages and currents of the secondary coils satisfy Ohm’s law: ˙ s= U
1 ˙Is . jωCs
(3)
The secondary output voltages Us (rms) are proportional to the input voltage Up and can be obtained using Eqs. (2) and (3). The maximum value of the output voltage of each rectifier disk is set at 60 kV taking into consideration the insulation limits. There are three secondary coils and double voltage rectifier circuits in a disk, and the maximum value of √ the 2 voltage of the secondary coils is then 60 kV ÷ 3 ÷ 2 ÷ √ = 5 2 kV. Initially, the input voltage is set at Up [0] , and the secondary coils output voltages (Us [0] ) and their maximum values (Umax [0] ) can be obtained. The real output voltages can then be obtained using Eq. (4): √ 5 2 Us [0] . (4) Us = Umax [0] Thus, the general simulation model of the ICT power supply has been established and it can be used to analyze each compensation method. III. EFFECT OF THE EXISTING COMPENSATION METHODS ON THE HUST-ICT
The first method was to increase the turns of the secondary coils, which were remote from the primary coils, to increase their output voltages. The turns of the primary coils (np ) of the HUST-ICT were optimized as 92 and the input rated voltage (rms) was set at 380 V. When the input voltage was √ 380 V, the output voltage of each secondary coil should be 5 2 kV. Therefore, the turns of the secondary coils were calculated using Eq. (5): ns =
Us Lp (1) np k, Up Mps (1)
(5)
where k is the unified compensation coefficient and used to compensate for the voltage drop of silicon rectifier stacks, and Lp (1) is the self-inductance of the single-turn primary coil. The turns of the secondary coils calculated using Eq. (5) are shown in Table II. TABLE II. Turns of secondary coils if the increasing turns method is adopted. Coil No.
1(n1 )
2(n2 )
3(n3 )
4(n4 )
5(n5 )
6(n6 )
Turns
2736
3080
3400
3671
3863
3974
FIG. 3. Output voltages of the disks at full-load and no-load if the method of increasing the turns is adopted. By increasing the turns of the secondary coils, the output voltages are consistent for the no-load condition. However, when the power supply is at full-load, the output voltages of the top layer disks drop even more.
The mutual inductance matrix of the coils whose turns are shown in Table II can be obtained using Eq. (1). Using this compensation method, the output voltages of the disks achieve the same under no-load condition. The output voltages of the disks can be obtained using the simulation model and are shown in Fig. 3. The simulation results indicate that as the distance between the disks and primary coils increases, the output voltages of the disks under full-load decrease, and the nonuniformity is about 21.3%. It is shown that the load capacity of the disks becomes weaker with increasing distance from the primary coil. Another method is to use capacitors to compensate for the voltage, assuming all the secondary coil turns are the same. There is a capacitive load current in the secondary coils. The magnetic flux excited by this load current has the same phase as the main magnetic flux. In other words, the compensation capacitors can increase both the magnetic flux through the secondary coils and the induced voltage. If the capacitor compensation method is adopted, the turns of the secondary coils can be set at 2853, which is calculated using Eq. (5), where Mps (1) needs to be replaced by its average value Mps (1) . The compensation capacitor was calculated using Eq. (6):19 Ccom =
l0 , ns 2 (2π f )2 μ0 Sc
(6)
where l0 = 2 mm is the gap between cores, f = 50 Hz is the frequency of the input power supply, μ0 is the permeability of vacuum, and Sc is the effective magnetic cross section area. The compensation capacitor calculated using Eq. (6) was 87 nF. The maximum output voltage of the disks was set at 60 kV by adjusting the input voltage. The output voltages of disks with and without load are shown in Fig. 4. The simulation result indicates that the no-load output voltages of the disks reduce with increasing distance between each disk and the primary coils. However, under full-load the situation is just the opposite. The non-uniformity is about 11.4% under full-load. Conversely, when the capacitor compensation method was adopted, the more remote the disk, the stronger their load capacity.
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FIG. 4. Output voltages of the disks under full-load and no-load when the capacitor compensation method was adopted. The output voltages were relatively consistent under the no-load condition. However, when the power supply was at full-load, the output voltages of the bottom layer disks dropped even more.
FIG. 5. Output voltages of the disks when the compensation capacitor (C6 ) values change. C6 can improve the voltage of each layer. When the value of the compensation capacitor is larger, the output voltage of this layer increases more. It will also affect other layers: the output voltages of layers closer to it increase more.
IV. COMBINED COMPENSATION METHOD
Based on the definition of the self-inductance, com can be expressed by Eq. (9):
The simulation results indicate that the uniformity of the rectifier disks is inadequate when the methods of increasing turns or capacitor compensation are adopted on their own. In addition, an important characteristic was found from the simulation results: the distribution of the load capacities of the disks was the opposite when the two compensation methods were adopted. Therefore, we combined the two methods in the HUST-ICT to obtain a better result. The steps for the combined compensation method follow. The turns of the secondary coils were corrected using Eq. (7): ns = ns +κ(ns −n1 ),
(7)
where ns are turns of the secondary coils calculated using Eq. (5) and listed in Table II, and κ is the turns compensation coefficient. A different κ implies a different compensation design. We selected a compensation coefficient, and then adjusted the compensation capacitors to make the output characteristics of each disk as similar as possible. The capacitors were different if κ took on different values, and the best uniformity of the output voltages of the disks was also different. We continued trying different values for κ until the uniformity met the requirements. When κ was selected as 0.75, the turns for the secondary coils are listed in Table III, as calculated using Eq. (7). The mutual inductance matrix for the design of the secondary coils was obtained using Eq. (1). The output voltages of disks as shown in Fig. 5 were obtained using the simulation model when the power supply was under full-load. The output voltage of disk 6# (V6 ) was lower than disk 1# (V1 ) at 15.5 kV. The compensation voltage (rms) of coil 6# was set at √ Ucom = 15.5 kV ÷ 2 ÷ 3 ÷ 2 = 1.8 kV. The compensation magnetic flux (rms) is given by com = Ucom /n6 /ω.
(8)
TABLE III. Turns of secondary coils when κ is 0.75. Coil No. Turn
1#
2#
3#
4#
5#
6#
2736
2994
3234
3437
3581
3665
com = L6 Icom /n6 = n6 L6 (1) Icom .
(9)
The compensation current can be obtained from Eqs. (8) and (9): Icom =
Ucom . ωn6 2 L6 (1)
(10)
The compensation capacitor can be obtained using Eq. (11): C6 =
1 = 72 nF. ωU6 /Icom
(11)
Under full-load, the output voltages of the disks are as shown in Fig. 5 when the compensation capacitor of coil 6# (C6 ) is 72 nF. The simulation result shows that V6 is higher than other disks when its capacitor is 72 nF, which means the capacitance is too large. V1 and V6 are the same when C6 is 51 nF and the output voltages of other disks are as shown in Fig. 5. V3 and V4 are lower than the voltages of the other disks, and the non-uniformity is about 8.1% under full-load. The uniformity of the disks is unsatisfactory when compensating coil 6# alone, while it can be improved by compensating the other coils. Unfortunately, the relationship between the output voltages of the disks and the compensation capacitors is very complex and difficult to predict analytically. It is very difficult to calculate the value of the compensation capacitors from some equations. Nonetheless, some useful relationships between the output voltages of the disks and the compensation capacitors have been found by simulation: (1) the output voltage will rise when the compensation capacitor in the same layer increases; (2) the more remote the disk from the primary coils, the greater the change in the voltages of the disks for the condition of the same compensation capacitor. Based on those relationships, the uniformity of the disks can be improved by taking steps to adjust the compensation capacitors. The minimum output voltage appears in the middle of the disks (3#, 4#) when C6 is 51 nF (solid line in Fig. 6). Therefore, C3 and C4 need to be adjusted (increased). Figure 6 shows the adjustment process for the compensation capacitors to improve the uniformity of the disks. The difference between the maximum voltage and minimum voltage was
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FIG. 6. Improvement of the output voltages of the disks by adjusting compensation capacitors, according to the distribution of the voltage of each layer.
reduced when C3 and C4 increased to 20 nF (dotted line). At the same time, V6 was higher than V1 , so C6 needed to decrease (dashed line); the voltages of the middle disks were still lower than those for the upper and lower disks, C3 and C4 , which needed to continue to increase. This adjustment process is repeated and the compensation capacitors of each layer modified until the uniformity of the disks meets the requirements. Finally, the non-uniformity decreased to 1.8% when the compensation capacitors combination was C2 = C3 = C4 = C6 = 30 nF, and C5 = 15 nF (dashed-dotted line). The uniformity of the output voltages of the disks under full-load has been well optimized. However, the uniformity under no-load has not yet been analyzed. In fact, because of the differences in the induced voltages and leakage inductances of each secondary coil, the output voltage drops of each disk are not matched by changes in output currents. In other words, the uniformity is related to the output current. The design was not acceptable unless the uniformity was adequate from no-load to full-load. Based on Eqs. (2)–(4), the output voltages of the disks were obtained under no-load and are shown in Fig. 7. It was found that the non-uniformity was about 4.3% under no-load when the combined compensation capacitors were C2 = C3 = C4 = C6 = 30 nF, and C5 = 15 nF (dotted line). In addition, V6 was higher than the other volt-
FIG. 7. Improvement in the output voltage of the disks under no-load and full-load. The indicator (no) represents the results under no-load. By adjusting compensation capacitors, the uniformities under these two conditions both reached a better value.
FIG. 8. The ICT power supply developed in HUST. Sulfur hexafluoride was used as the insulating medium for this power supply. This photograph was taken before covering the steel drum.
ages under both no-load and full-load, so C6 needed to be decreased. The output voltages of the disks are shown in Fig. 7 when C6 decreased to 27 nF. The non-uniformity of the disks was about 3.2% under no-load and about 2.4% under full-load (dashed-dotted line and dashed line). A better combination of capacitors may be found if the adjusting process is continued. However, this result was acceptable for engineering practice. A better combination may not be significant or necessary. Two reasons for this are: (1) the actual mutual inductance matrix will change a little because of the mechanical error in installing cores; (2) there are some differences between the FEM simulation and the actual model. Finally, the compensation coefficient needed to be reselected if there was no combination of capacitors that met the uniformity requirement. V. EXPERIMENTAL RESULTS
Because of the current limit of the actual coils, the compensation capacitors cannot be too large. It turns out that the maximum acceptable capacitance for the HUST-ICT was
FIG. 9. Improvement in uniformity of the disks in practice. The indicator (no) represents the results under no-load.
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filter capacitors and silicon stacks. Improving the uniformity can increase the average output voltage of the disks and the maximum output voltage of the power supply. The proposed combined compensation method was applied to an ICT power supply, and the test results demonstrate its effectiveness and feasibility. ACKNOWLEDGMENTS
This work was supported by National Nature Science Foundation of China No. 11305068. 1 J.
FIG. 10. Test results compared with the simulation results. “Sim” represents the simulation results and “test” represents the test results.
40 nF, and the compensation capacitors cannot exceed this, in practice. High voltage ceramic capacitors were used as the compensation capacitors in our project. The power supply used to test the combined compensation method is shown in Fig. 8. The search process for the combination of compensation capacitors, in practice, was the same as that in simulation, which was to adjust the capacitors according to the output voltages of the disks, as shown in Fig. 9. The non-uniformity was about 3.5% under full load and about 2.0% under no-load when the capacitors’ combination was C2 = 2 nF, C3 = C4 = 30 nF, C5 = 5 nF, and C6 = 35.7 nF. Figure 10 shows the test results and simulation results when the combination of capacitors was C2 = 2 nF, C3 = C4 = 30 nF, C5 = 5 nF, and C6 = 35.7 nF. It was found that the trends for the voltage distribution of the disks in the experiment were similar to those in the simulation, and the maximum difference between them was about 2.5%. Considering the errors of the mechanical installation of the cores, the test results confirm the validity of the simulation model and the feasibility of the combined compensation method. VI. CONCLUSIONS
With the aim of improving the consistency in the output voltages of the rectifier disks, the proposed combined compensation method is summarized by the following steps: (1) calculate the mutual inductance matrix for “single-turn” coils; (2) select a proper turns compensation coefficient (κ) and calculate the number of turns for the coils; (3) calculate the mutual inductance matrix for this design of the number of turns; (4) search for the capacitors’ combination under both no-load and full-load to meet the uniformity requirement; (5) repeat Steps 2–4 if Step 4 is not achieved. Generally, the nonuniformity under both no-load and full-load can reach 4% when the turns compensation coefficient (κ) is set between 0.3 and 0.8. The uniformity of disks is an important factor for an ICT power supply, because it ensures the homogeneity of the electric field and the utilization of rectifier components such as
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