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