Chapter 2 Generation of high voltages A fundamental knowledge about generators and circuits which are in use for the generation of high voltages belongs to the background of work on h.v. technology. Generally commercially available h.v.generators are applied in routine testing laboratories;they are used for testing equipment such as transformers, bushings,cables,capacitors,switchgear,etc.The tests should confirm the effi- ciency and reliability of the products and therefore the h.v.testing equipment is required to study the insulation behaviour under all conditions which the apparatus is likely to encounter.The amplitudes and types of the test voltages, which are always higher than the normal or rated voltages of the apparatus under test,are in general prescribed by national or international standards or recommendations,and therefore there is not much freedom in the selection of the h.v.testing equipment.Quite often,however,routine testing laboratories are also used for the development of new products.Then even higher volt- ages might be necessary to determine the factor of safety over the prospective working conditions and to ensure that the working margin is neither too high nor too low.Most of the h.v.generator circuits can be changed to increase the output voltage levels,if the original circuit was properly designed.There- fore,even the selection of routine testing equipment should always consider a future extension of the testing capabilities. The work carried out in research laboratories varies considerably from one establishment to another,and the type of equipment needed varies accordingly. As there are always some interactions between the h.v.generating circuits used and the test results,the layout of these circuits has to be done very carefully. The classes of tests may differ from the routine tests,and therefore specially designed circuits are often necessary for such laboratories.The knowledge about some fundamental circuits treated in this chapter will also support the development of new test circuits. Finally,high voltages are used in many branches of natural sciences or other technical applications.The generating circuits are often the same or similar to those treated in the following sections.It is not the aim,however,of this introductory text to treat the broad variations of possible circuits,due to space limitation.Not taken into account are also the differing problems of electrical power generation and transmission with high voltages of a.c.or d.c.,or the
Chapter 2 Generation of high voltages A fundamental knowledge about generators and circuits which are in use for the generation of high voltages belongs to the background of work on h.v. technology. Generally commercially available h.v. generators are applied in routine testing laboratories; they are used for testing equipment such as transformers, bushings, cables, capacitors, switchgear, etc. The tests should confirm the effi- ciency and reliability of the products and therefore the h.v. testing equipment is required to study the insulation behaviour under all conditions which the apparatus is likely to encounter. The amplitudes and types of the test voltages, which are always higher than the normal or rated voltages of the apparatus under test, are in general prescribed by national or international standards or recommendations, and therefore there is not much freedom in the selection of the h.v. testing equipment. Quite often, however, routine testing laboratories are also used for the development of new products. Then even higher voltages might be necessary to determine the factor of safety over the prospective working conditions and to ensure that the working margin is neither too high nor too low. Most of the h.v. generator circuits can be changed to increase the output voltage levels, if the original circuit was properly designed. Therefore, even the selection of routine testing equipment should always consider a future extension of the testing capabilities. The work carried out in research laboratories varies considerably from one establishment to another, and the type of equipment needed varies accordingly. As there are always some interactions between the h.v. generating circuits used and the test results, the layout of these circuits has to be done very carefully. The classes of tests may differ from the routine tests, and therefore specially designed circuits are often necessary for such laboratories. The knowledge about some fundamental circuits treated in this chapter will also support the development of new test circuits. Finally, high voltages are used in many branches of natural sciences or other technical applications. The generating circuits are often the same or similar to those treated in the following sections. It is not the aim, however, of this introductory text to treat the broad variations of possible circuits, due to space limitation. Not taken into account are also the differing problems of electrical power generation and transmission with high voltages of a.c. or d.c., or the
Generation of high voltages 9 pure testing technique of h.v.equipment,the procedures of which may be found in relevant standards of the individual equipment.Power generation and transmission problems are treated in many modern books,some of which are listed within the bibliography of an earlier report.(1) This chapter discusses the generation of the following main classes of volt- ages:direct voltages,alternating voltages,and transient voltages. 2.1 Direct voltages In h.v.technology direct voltages are mainly used for pure scientific research work and for testing equipment related to HVDC transmission systems.There is still a main application in tests on HVAC power cables of long length,as the large capacitance of those cables would take too large a current if tested with a.c.voltages (see,however,2.2.2:Series resonant circuits).Although such d.c.tests on a.c.cables are more economical and convenient,the validity of this test suffers from the experimentally obtained stress distribution within the insulating material,which may considerably be different from the normal working conditions where the cable is transmitting power at low-frequency alternating voltages.For the testing of polyethylene h.v.cables,in use now for some time,d.c.tests are no longer used,as such tests may not confirm the quality of the insulation.(50) High d.c.voltages are even more extensively used in applied physics (accelerators,electron microscopy,etc.),electromedical equipment (X-rays), industrial applications(precipitation and filtering of exhaust gases in thermal power stations and the cement industry;electrostatic painting and powder coating,etc.),or communications electronics (TV,broadcasting stations). Therefore,the requirements on voltage shape,voltage level,and current rating, short-or long-term stability for every HVDC generating system may differ strongly from each other.With the knowledge of the fundamental generating principles it will be possible,however,to select proper circuits for a special application. In the International Standard IEC 60-12)or IEEE Standard.4-19953)the value of a direct test voltage is defined by its arithmetic mean value,which will be designated as V.Therefore,this value may be derived from v-vod. (2.1) where T equals a certain period of time if the voltage V(t)is not constant,but periodically oscillating with a frequency of f =1/T.Test voltages as applied to test objects then deviate periodically from the mean value.This means that *Superscript numbers are to References at the end of the chapter
Generation of high voltages 9 pure testing technique of h.v. equipment, the procedures of which may be found in relevant standards of the individual equipment. Power generation and transmission problems are treated in many modern books, some of which are listed within the bibliography of an earlier report.1Ł This chapter discusses the generation of the following main classes of voltages: direct voltages, alternating voltages, and transient voltages. 2.1 Direct voltages In h.v. technology direct voltages are mainly used for pure scientific research work and for testing equipment related to HVDC transmission systems. There is still a main application in tests on HVAC power cables of long length, as the large capacitance of those cables would take too large a current if tested with a.c. voltages (see, however, 2.2.2: Series resonant circuits). Although such d.c. tests on a.c. cables are more economical and convenient, the validity of this test suffers from the experimentally obtained stress distribution within the insulating material, which may considerably be different from the normal working conditions where the cable is transmitting power at low-frequency alternating voltages. For the testing of polyethylene h.v. cables, in use now for some time, d.c. tests are no longer used, as such tests may not confirm the quality of the insulation.50 High d.c. voltages are even more extensively used in applied physics (accelerators, electron microscopy, etc.), electromedical equipment (X-rays), industrial applications (precipitation and filtering of exhaust gases in thermal power stations and the cement industry; electrostatic painting and powder coating, etc.), or communications electronics (TV, broadcasting stations). Therefore, the requirements on voltage shape, voltage level, and current rating, short- or long-term stability for every HVDC generating system may differ strongly from each other. With the knowledge of the fundamental generating principles it will be possible, however, to select proper circuits for a special application. In the International Standard IEC 60-12 or IEEE Standard. 4-19953 the value of a direct test voltage is defined by its arithmetic mean value, which will be designated as V . Therefore, this value may be derived from V D 1 T T 0 Vt dt. 2.1 where T equals a certain period of time if the voltage Vt is not constant, but periodically oscillating with a frequency of f D 1/T. Test voltages as applied to test objects then deviate periodically from the mean value. This means that Ł Superscript numbers are to References at the end of the chapter.
10 High Voltage Engineering:Fundamentals a ripple is present.The amplitude of the ripple,8V,is defined as half the difference between the maximum and minimum values,or V =0.5(Vmax -Vmin). (2.2) The ripple factor is the ratio of the ripple amplitude to the arithmetic mean value,or 8V/V.For test voltages this ripple factor should not exceed 3 per cent unless otherwise specified by the appropriate apparatus standard or be necessary for fundamental investigations. The d.c.voltages are generally obtained by means of rectifying circuits applied to a.c.voltages or by electrostatic generation.A treatment of the generation principles according to this subdivision is appropriate. 2.1.1 A.C.to D.C.conversion The rectification of alternating currents is the most efficient means of obtaining HVDC supplies.Although all circuits in use have been known for a long time, the cheap production and availability of manifold solid state rectifiers has facilitated the production and application of these circuits fundamentally.Since some decades,there is no longer a need to employ valves,hot cathode gas- filled valves,mercury pool or corona rectifiers,or even mechanical rectifiers within the circuits,for which the auxiliary systems for cathode heating,etc., have always aggravated their application.The state of the art of such earlier circuits may be found in the work of Craggs and Meek,(4)which was written in 1954.All rectifier diodes used now adopt the Si type,and although the peak reverse voltage is limited to less than about 2500V,rectifying diode units up to tens and hundreds of kVs can be made by series connections if appropriate means are applied to provide equal voltage distribution during the non-conducting period.One may treat and simulate,therefore,a rectifier within the circuits-independently of the voltage levels-simply by the common symbol for a diode. The theory of rectifier circuits for low voltages and high power output is discussed in many standard handbooks.Having the generation of high d.c. voltages in mind,we will thus restrict the treatment mainly to single-phase a.c.systems providing a high ratio of d.c.output to a.c.input voltage.As, however,the power or d.c.output is always limited by this ratio,and because very simple rectifier circuits are in use,we will treat only selected examples of the many available circuits. Simple rectifier circuits For a clear understanding of all a.c.to d.c.conversion circuits the single-phase half-wave rectifier with voltage smoothing is of basic interest (Fig.2.1(a)). If we neglect the leakage reactance of the transformer and the small internal
10 High Voltage Engineering: Fundamentals a ripple is present. The amplitude of the ripple, υV, is defined as half the difference between the maximum and minimum values, or υV D 0.5Vmax Vmin. 2.2 The ripple factor is the ratio of the ripple amplitude to the arithmetic mean value, or υV/V. For test voltages this ripple factor should not exceed 3 per cent unless otherwise specified by the appropriate apparatus standard or be necessary for fundamental investigations. The d.c. voltages are generally obtained by means of rectifying circuits applied to a.c. voltages or by electrostatic generation. A treatment of the generation principles according to this subdivision is appropriate. 2.1.1 A.C. to D.C. conversion The rectification of alternating currents is the most efficient means of obtaining HVDC supplies. Although all circuits in use have been known for a long time, the cheap production and availability of manifold solid state rectifiers has facilitated the production and application of these circuits fundamentally. Since some decades, there is no longer a need to employ valves, hot cathode gas- filled valves, mercury pool or corona rectifiers, or even mechanical rectifiers within the circuits, for which the auxiliary systems for cathode heating, etc., have always aggravated their application. The state of the art of such earlier circuits may be found in the work of Craggs and Meek,4 which was written in 1954. All rectifier diodes used now adopt the Si type, and although the peak reverse voltage is limited to less than about 2500 V, rectifying diode units up to tens and hundreds of kVs can be made by series connections if appropriate means are applied to provide equal voltage distribution during the non-conducting period. One may treat and simulate, therefore, a rectifier within the circuits – independently of the voltage levels – simply by the common symbol for a diode. The theory of rectifier circuits for low voltages and high power output is discussed in many standard handbooks. Having the generation of high d.c. voltages in mind, we will thus restrict the treatment mainly to single-phase a.c. systems providing a high ratio of d.c. output to a.c. input voltage. As, however, the power or d.c. output is always limited by this ratio, and because very simple rectifier circuits are in use, we will treat only selected examples of the many available circuits. Simple rectifier circuits For a clear understanding of all a.c. to d.c. conversion circuits the single-phase half-wave rectifier with voltage smoothing is of basic interest (Fig. 2.1(a)). If we neglect the leakage reactance of the transformer and the small internal
Generation of high voltages 11 V-(t) RL(load) h.t. transformer (a) F2.8V v(t) (t a.T v_(t) T=1/f ( Figure 2.1 Single-phase half-wave rectifier with reservoir capacitance C. (a)Circuit.(b)Voltages and currents with load RL impedance of the diodes during conduction-and this will be done throughout unless otherwise stated-the reservoir or smoothing capacitor C is charged to the maximum voltage +Vmax of the a.c.voltage V~(t)of the h.t.transformer, when D conducts.This is the case as long as V<V~(t)for the polarity of D assumed.If I=0,i.e.the output load being zero (R=oo),the d.c.voltage across C remains constant (+Vmax),whereas V~(t)oscillates between +Vmax. The diode D must be dimensioned,therefore,to withstand a peak reverse voltage of 2Vmax. The output voltage V does not remain any more constant if the circuit is loaded.During one period,T=1/f of the a.c.voltage a charge o is transferred to the load RL,which is represented as e=/aou=元人o=r=子 (2.3)
Generation of high voltages 11 (a) (b) V~(t) V~(t) V (t) t a.T V max V min D C h.t. transformer V c 2.d V a i L (t) RL i (t) (load) i (t) T = 1/f Figure 2.1 Single-phase half-wave rectifier with reservoir capacitance C. (a) Circuit. (b) Voltages and currents with load RL impedance of the diodes during conduction – and this will be done throughout unless otherwise stated – the reservoir or smoothing capacitor C is charged to the maximum voltage CVmax of the a.c. voltage V¾t of the h.t. transformer, when D conducts. This is the case as long as V<V¾t for the polarity of D assumed. If I D 0, i.e. the output load being zero RL D 1, the d.c. voltage across C remains constant CVmax, whereas V¾t oscillates between šVmax. The diode D must be dimensioned, therefore, to withstand a peak reverse voltage of 2Vmax. The output voltage V does not remain any more constant if the circuit is loaded. During one period, T D 1/f of the a.c. voltage a charge Q is transferred to the load RL, which is represented as Q D T iLt dt D 1 RL T Vt dt D IT D I f. 2.3
12 High Voltage Engineering:Fundamentals I is therefore the mean value of the d.c.output i(t),and V(t)the d.c.voltage which includes a ripple as shown in Fig.2.1(b).If we introduce the ripple factor 8V from egn(2.2),we may easily see that V(t)now varies between Vmax≥V(t)≥Vmin;Vmin=Vmax-2(8V). (2.4) The charge O is also supplied from the transformer within the short conduction time te=aT of the diode D during each cycle.Therefore,O equals also to e-idr=. (2.5) JaT As aT<T,the transformer and diode current i(t)is pulsed as shown idealized in Fig.2.I(b)and is of much bigger amplitudes than the direct current it1. The ripple 8V could be calculated exactly for this circuit based upon the expo- nential decay of V(t)during the discharge period T(1-a).As,however,for practical circuits the neglected voltage drops within transformer and rectifiers must be taken into account,and such calculations are found elsewhere,(3)we may assume that a=0.Then 8V is easily found from the charge O transferred to the load,and therefore IT I Q=28C=1T;8V=2元=2f元· (2.6) This relation shows the interaction between the ripple,the load current and circuit parameter design values f and C.As,according to eqn (2.4),the mean output voltage will also be influenced by 8V,even with a constant a.c.voltage V~(t)and a lossless rectifier D,no load-independent output voltage can be reached.The product fC is therefore an important design factor. For h.v.test circuits,a sudden voltage breakdown at the load (RL0) must always be taken into account.Whenever possible,the rectifiers should be able to carry either the excessive currents,which can be limited by fast, electronically controlled switching devices at the transformer input,or they can be protected by an additional resistance inserted in the h.t.circuit.The last method,however,increases the internal voltage drop. Half-wave rectifier circuits have been built up to voltages in the megavolt range,in general by extending an existing h.v.testing transformer to a d.c. current supply.The largest unit has been presented by Prinz,(5)who used a 1.2- MV cascaded transformer and 60-mA selenium-type solid state rectifiers with an overall reverse voltage of 3.4 MV for the circuit.The voltage distribution of this rectifier,which is about 12 m in length,is controlled by sectionalized parallel capacitor units,which are small in capacitance value in comparison with the smoothing capacitor C (see Fig.2.14).The size of such circuits, however,would be unnecessarily large for pure d.c.supplies. The other disadvantage of the single-phase half-wave rectifier concerns the possible saturation of the h.v.transformer,if the amplitude of the direct current
12 High Voltage Engineering: Fundamentals I is therefore the mean value of the d.c. output iLt, and Vt the d.c. voltage which includes a ripple as shown in Fig. 2.1(b). If we introduce the ripple factor υV from eqn (2.2), we may easily see that Vt now varies between Vmax ½ Vt ½ Vmin; Vmin D Vmax 2υV. 2.4 The charge Q is also supplied from the transformer within the short conduction time tc D ˛T of the diode D during each cycle. Therefore, Q equals also to Q D ˛T it dt D T iLt dt. 2.5 As ˛T − T, the transformer and diode current it is pulsed as shown idealized in Fig. 2.l(b) and is of much bigger amplitudes than the direct current iL ¾D I. The ripple υV could be calculated exactly for this circuit based upon the exponential decay of Vt during the discharge period T1 ˛. As, however, for practical circuits the neglected voltage drops within transformer and rectifiers must be taken into account, and such calculations are found elsewhere,3 we may assume that ˛ D 0. Then υV is easily found from the charge Q transferred to the load, and therefore Q D 2υVC D IT; υV D IT 2C D I 2fC. 2.6 This relation shows the interaction between the ripple, the load current and circuit parameter design values f and C. As, according to eqn (2.4), the mean output voltage will also be influenced by υV, even with a constant a.c. voltage V¾t and a lossless rectifier D, no load-independent output voltage can be reached. The product fC is therefore an important design factor. For h.v. test circuits, a sudden voltage breakdown at the load RL ! 0 must always be taken into account. Whenever possible, the rectifiers should be able to carry either the excessive currents, which can be limited by fast, electronically controlled switching devices at the transformer input, or they can be protected by an additional resistance inserted in the h.t. circuit. The last method, however, increases the internal voltage drop. Half-wave rectifier circuits have been built up to voltages in the megavolt range, in general by extending an existing h.v. testing transformer to a d.c. current supply. The largest unit has been presented by Prinz,5 who used a 1.2- MV cascaded transformer and 60-mA selenium-type solid state rectifiers with an overall reverse voltage of 3.4 MV for the circuit. The voltage distribution of this rectifier, which is about 12 m in length, is controlled by sectionalized parallel capacitor units, which are small in capacitance value in comparison with the smoothing capacitor C (see Fig. 2.14). The size of such circuits, however, would be unnecessarily large for pure d.c. supplies. The other disadvantage of the single-phase half-wave rectifier concerns the possible saturation of the h.v. transformer, if the amplitude of the direct current
Generation of high voltages 13 is comparable with the nominal alternating current of the transformer.The biphase half-wave (or single-phase full-wave)rectifier as shown in Fig.2.2 overcomes this disadvantage,but it does not change the fundamental effi- ciency,considering that two h.v.windings of the transformer are now avail- able.With reference to the frequency f during one cycle,now each of the diodes Di and D2 is conducting for one half-cycle with a time delay of T/2.The ripple factor according to egn (2.6)is therefore halved.It should be mentioned that the real ripple will also be increased if both voltages Vi and V2~are not exactly equal.If V2max would be smaller than (Vi max-28V) or Vmin,this h.v.winding would not charge the capacitance C.The same effect holds true for multiphase rectifiers,which are not treated here. V-) D h.t. transformer D2 V2-(t) Figure 2.2 Biphase half-wave rectifier circuit with smoothing capacitor C Thus single-phase full-wave circuits can only be used for h.v.applications if the h.t.winding of the transformer can be earthed at its midpoint and if the d.c.output is single-ended grounded.More commonly used are single-phase voltage doublers,a circuit of which is contained in the voltage multiplier or d.c.cascade of Fig.2.6,see stage 1.Although in such a circuit grounding of the h.v.winding is also not possible,if asymmetrical d.c.voltages are produced,the potential of this winding is fixed.Therefore,there is no danger due to transients followed by voltage breakdowns. Cascade circuits The demands from physicists for very high d.c.voltages forced the improve- ment of rectifying circuits quite early.It is obvious that every multiplier circuit in which transformers,rectifiers and capacitor units have only to withstand a fraction of the total output voltage will have great advantages.Today there are many standard cascade circuits available for the conversion of modest a.c.to high d.c.voltages.However,only few basic circuits will be treated
Generation of high voltages 13 is comparable with the nominal alternating current of the transformer. The biphase half-wave (or single-phase full-wave) rectifier as shown in Fig. 2.2 overcomes this disadvantage, but it does not change the fundamental effi- ciency, considering that two h.v. windings of the transformer are now available. With reference to the frequency f during one cycle, now each of the diodes D1 and D2 is conducting for one half-cycle with a time delay of T/2. The ripple factor according to eqn (2.6) is therefore halved. It should be mentioned that the real ripple will also be increased if both voltages V1¾ and V2¾ are not exactly equal. If V2 max would be smaller than V1 max 2υV or Vmin, this h.v. winding would not charge the capacitance C. The same effect holds true for multiphase rectifiers, which are not treated here. V1∼(t) V2∼(t) D1 D2 h.t. transformer C V RL Figure 2.2 Biphase half-wave rectifier circuit with smoothing capacitor C Thus single-phase full-wave circuits can only be used for h.v. applications if the h.t. winding of the transformer can be earthed at its midpoint and if the d.c. output is single-ended grounded. More commonly used are single-phase voltage doublers, a circuit of which is contained in the voltage multiplier or d.c. cascade of Fig. 2.6, see stage 1. Although in such a circuit grounding of the h.v. winding is also not possible, if asymmetrical d.c. voltages are produced, the potential of this winding is fixed. Therefore, there is no danger due to transients followed by voltage breakdowns. Cascade circuits The demands from physicists for very high d.c. voltages forced the improvement of rectifying circuits quite early. It is obvious that every multiplier circuit in which transformers, rectifiers and capacitor units have only to withstand a fraction of the total output voltage will have great advantages. Today there are many standard cascade circuits available for the conversion of modest a.c. to high d.c. voltages. However, only few basic circuits will be treated.
14 High Voltage Engineering:Fundamentals In 1920 Greinacher,a young physicist,published a circuit(6)which was improved in 1932 by Cockcroft and Walton to produce high-energy positive ions.(7)The interesting and even exciting development stages of those circuits have been discussed by Craggs and Meek.(+)To demonstrate the principle only,an n-stage single-phase cascade circuit of the 'Cockcroft-Walton type', shown in Fig.2.3,will be presented. HV output open-circuited:I =0.The portion 0-n'-V(t)is a half-wave rectifier circuit in which Cr charges up to a voltage of +Vmax if V(t)has reached the lowest potential,-Vmax.If Cn is still uncharged,the rectifier D conducts as soon as V(t)increases.As the potential of point n'swings up to +V2max during the period T=1/f,point n attains further on a steady potential of +2Vmax if V(t)has reached the highest potential of +Vmax.The part n'-n-0 is therefore a half-wave rectifier,in which the voltage across D can be assumed to be the a.c.voltage source.The current through D that D H.V.output D C D2 D D Dn- (n-1y' n-1) C-1 D- Cn D'n 0 V(t);Vmax (a) 吉 Figure 2.3 (a)Cascade circuit according to Cockroft-Walton or Greinacher.(b)Waveform of potentials at the nodes,no load
14 High Voltage Engineering: Fundamentals In 1920 Greinacher, a young physicist, published a circuit6 which was improved in 1932 by Cockcroft and Walton to produce high-energy positive ions.7 The interesting and even exciting development stages of those circuits have been discussed by Craggs and Meek.4 To demonstrate the principle only, an n-stage single-phase cascade circuit of the ‘Cockcroft–Walton type’, shown in Fig. 2.3, will be presented. HV output open-circuited: I D 0. The portion 0 n0 Vt is a half-wave rectifier circuit in which C0 n charges up to a voltage of CVmax if Vt has reached the lowest potential, Vmax. If Cn is still uncharged, the rectifier Dn conducts as soon as Vt increases. As the potential of point n0 swings up to CV2 max during the period T D 1/f, point n attains further on a steady potential of C2Vmax if Vt has reached the highest potential of CVmax. The part n0 n 0 is therefore a half-wave rectifier, in which the voltage across D0 n can be assumed to be the a.c. voltage source. The current through Dn that ∼ V(t); Vmax C′ n Cn D′ n Dn C′ n−1 Cn−1 D′ n−1 Dn−1 n′ n 4′ 3′ D3 D2 D1 C1 C2 C3 2′ 1′ 1 (n−1)′ (n−1) C3 ′ D3 ′ C2 ′ D2 ′ C1 ′ D1 ′ 2 3 4 0 I H.V. output (a) Figure 2.3 (a) Cascade circuit according to Cockroft–Walton or Greinacher. (b) Waveform of potentials at the nodes, no load
Generation of high voltages 15 1=H.V.output Stages(n-2)to 3 D1...Dn Conducting Di....Dn (b) Conducting Figure 2.3 (continued) charged the capacitor C was not provided by D,but from V(t)and C.We assumed,therefore,that C was not discharged,which is not correct.As we will take this into consideration for the loaded circuit,we can also assume that the voltage across C is not reduced if the potential n'oscillates between zero and +2Vmax.If the potential of n',however,is zero,the capacitor C is also charged to the potential of n,i.e.to a voltage of +2Vmax.The next voltage oscillation of V(t)from -Vmax to +Vmax will force the diode Dn-1 to conduct,so that also C will be charged to a voltage of +2Vmax. In Fig.2.3(b)the steady state potentials at all nodes of the circuit are sketched for the circuit for zero load conditions.From this it can be seen,that: the potentials at the nodes 1'.2'...n'are oscillating due to the voltage oscillation of V(t); the potentials at the nodes 1,2...n remain constant with reference to ground potential; the voltages across all capacitors are of d.c.type,the magnitude of which is 2Vx across each capacitor stage,except the capacitor C which is stressed with Vmax only;
Generation of high voltages 15 1 = H.V. output 1′ 2′ V0 = n . 2V max (n −1) Stages (n−2) to 3 (n −1)′ 2 n n′ 2Vmax V(t) t 0 t1 t2 D1...Dn Conducting D1 ′ . .. Dn ′ Conducting Vmax (b) Figure 2.3 (continued) charged the capacitor Cn was not provided by D0 n, but from Vt and C0 n. We assumed, therefore, that C0 n was not discharged, which is not correct. As we will take this into consideration for the loaded circuit, we can also assume that the voltage across Cn is not reduced if the potential n0 oscillates between zero and C2Vmax. If the potential of n0 , however, is zero, the capacitor C0 n 1 is also charged to the potential of n, i.e. to a voltage of C2Vmax. The next voltage oscillation of Vt from Vmax to CVmax will force the diode Dn 1 to conduct, so that also Cn 1 will be charged to a voltage of C2Vmax. In Fig. 2.3(b) the steady state potentials at all nodes of the circuit are sketched for the circuit for zero load conditions. From this it can be seen, that: ž the potentials at the nodes 10 , 20 ...n0 are oscillating due to the voltage oscillation of Vt; ž the potentials at the nodes 1, 2 ...n remain constant with reference to ground potential; ž the voltages across all capacitors are of d.c. type, the magnitude of which is 2Vmax across each capacitor stage, except the capacitor C0 n which is stressed with Vmax only;
16 High Voltage Engineering:Fundamentals every rectifier D,D...D,D'is stressed with 2Vmax or twice a.c.peak voltage;and the h.v.output will reach a maximum voltage of 2n Vmax. Therefore,the use of several stages arranged in this manner enables very high voltages to be obtained.The equal stress of the elements used is very convenient and promotes a modular design of such generators.The number of stages,however,is strongly limited by the current due to any load.This can only be demonstrated by calculations,even if ideal rectifiers,capacitors and an ideal a.c.voltage source are assumed. Finally it should be mentioned that the lowest stage n of the cascade circuit (Fig.2.3(a))is the Cockcroft-Walton voltage doubler.The a.c.voltage source V(t)is usually provided by an h.t.transformer,if every stage is built for high voltages,typically up to about 300kV.This source is always symmet- rically loaded,as current is withdrawn during each half-cycle (t and t2 in Fig.2.3(b)).The voltage waveform does not have to be sinusoidal:every symmetrical waveform with equal positive and negative peak values will give good performance.As often high-frequency input voltages are used,this hint is worth remembering. H.V.output loaded:I>0.If the generator supplies any load current I,the output voltage will never reach the value 2nVmax as shown in Fig.2.3(b). There will also be a ripple on the voltage,and therefore we have to deal with two quantities:the voltage drop AVo and the peak-to-peak ripple 28V.The sketch in Fig.2.4 shows the shape of the output voltage and the definitions of 2n Vmax (no load) 28V Vo max Vo (t)with load 0 -V(t) 一T=1/f Figure 2.4 Loaded cascade circuit,definitions of voltage drop AVo and ripple 8V
16 High Voltage Engineering: Fundamentals ž every rectifier D1, D0 1 ...Dn, D0 n is stressed with 2Vmax or twice a.c. peak voltage; and ž the h.v. output will reach a maximum voltage of 2nVmax. Therefore, the use of several stages arranged in this manner enables very high voltages to be obtained. The equal stress of the elements used is very convenient and promotes a modular design of such generators. The number of stages, however, is strongly limited by the current due to any load. This can only be demonstrated by calculations, even if ideal rectifiers, capacitors and an ideal a.c. voltage source are assumed. Finally it should be mentioned that the lowest stage n of the cascade circuit (Fig. 2.3(a)) is the Cockcroft–Walton voltage doubler. The a.c. voltage source Vt is usually provided by an h.t. transformer, if every stage is built for high voltages, typically up to about 300 kV. This source is always symmetrically loaded, as current is withdrawn during each half-cycle (t1 and t2 in Fig. 2.3(b)). The voltage waveform does not have to be sinusoidal: every symmetrical waveform with equal positive and negative peak values will give good performance. As often high-frequency input voltages are used, this hint is worth remembering. H.V. output loaded: I > 0. If the generator supplies any load current I, the output voltage will never reach the value 2nVmax as shown in Fig. 2.3(b). There will also be a ripple on the voltage, and therefore we have to deal with two quantities: the voltage drop V0 and the peak-to-peak ripple 2υV. The sketch in Fig. 2.4 shows the shape of the output voltage and the definitions of 2n Vmax (no load) V0 max V0 (t) with load 2 δ V ∆V0 +Vmax t 1 0 t 2 V(t) T = 1/f t Figure 2.4 Loaded cascade circuit, definitions of voltage drop V0 and ripple υV
Generation of high voltages 17 AVo and 28V.The time instants t and t2 are in agreement with Fig.2.3(b). Therefore,the peak value of Vo is reached at ti,if V(t)was at +Vmax and the rectifiers D1...D just stopped to transfer charge to the 'smoothing column' C1...C.After that the current I continuously discharges the column,inter- rupted by a sudden voltage drop shortly before t2:this sudden voltage drop is due to the conduction period of the diodes D...D,during which the 'oscillating column'C1...C is charged. Now let a charge g be transferred to the load per cycle,which is obviously =1/f=IT.This charge comes from the smoothing column,the series connection of Ci...Cn.If no charge would be transferred during T from this stack via D...D to the oscillating column,the peak-to-peak ripple would merely be 2V=IT∑/C,). i=1 As,however,just before the time instant t2 every diode D...D transfers the same charge g,and each of these charges discharges all capacitors on the smoothing column between the relevant node and ground potential,the total ripple will be v=（+++…） (2.7) Thus in a cascade multiplier the lowest capacitors are responsible for most ripple and it would be desirable to increase the capacitance in the lower stages.This is,however,very inconvenient for h.v.cascades,as a voltage breakdown at the load would completely overstress the smaller capacitors within the column.Therefore,equal capacitance values are usually provided, and with C=C1=C2...Cn,eqn (2.7)is 8=xnm+1) X fc (2.7a) 4 To calculate the total voltage drop AVo,we will first consider the stage n. Although the capacitor C at time ti will be charged up to the full voltage Vmax,if ideal rectifiers and no voltage drop within the a.c.-source are assumed, the capacitor Cn will only be charged to a voltage ng (V)mat-2Vma-Cr-2Vm-V. as C has lost a total charge of (ng)during a full cycle before and C has to replace this lost charge.At time instant 2.Cn transfers the charge gto C
Generation of high voltages 17 V0 and 2υV. The time instants t1 and t2 are in agreement with Fig. 2.3(b). Therefore, the peak value of Vo is reached at t1, if Vt was at CVmax and the rectifiers D1 ...Dn just stopped to transfer charge to the ‘smoothing column’ C1 ...Cn. After that the current I continuously discharges the column, interrupted by a sudden voltage drop shortly before t2: this sudden voltage drop is due to the conduction period of the diodes D0 1 ...D0 n, during which the ‘oscillating column’ C0 1 ...C0 n is charged. Now let a charge q be transferred to the load per cycle, which is obviously q D I/f D IT. This charge comes from the smoothing column, the series connection of C1 ...Cn. If no charge would be transferred during T from this stack via D0 1 ...D0 n to the oscillating column, the peak-to-peak ripple would merely be 2υV D ITn iD1 1/Ci. As, however, just before the time instant t2 every diode D0 1 ...D0 n transfers the same charge q, and each of these charges discharges all capacitors on the smoothing column between the relevant node and ground potential, the total ripple will be υV D 1 2f 1 C1 C 2 C2 C 3 C3 C ... n Cn . 2.7 Thus in a cascade multiplier the lowest capacitors are responsible for most ripple and it would be desirable to increase the capacitance in the lower stages. This is, however, very inconvenient for h.v. cascades, as a voltage breakdown at the load would completely overstress the smaller capacitors within the column. Therefore, equal capacitance values are usually provided, and with C D C1 D C2 ...Cn, eqn (2.7) is υV D I fC ð nn C 1 4 . 2.7a To calculate the total voltage drop V0, we will first consider the stage n. Although the capacitor C0 n at time t1 will be charged up to the full voltage Vmax, if ideal rectifiers and no voltage drop within the a.c.-source are assumed, the capacitor Cn will only be charged to a voltage Vcn max D 2Vmax nq C0 n D 2Vmax Vn as Cn has lost a total charge of nq during a full cycle before and C0 n has to replace this lost charge. At time instant t2, Cn transfers the charge q to C0 n 1