EP2540145B1 - Dc high voltage source and particle accelerator - Google Patents

Description

The invention relates to a DC high voltage source and a particle accelerator with a capacitor stack of concentrically arranged electrodes.

There are many applications where a high DC voltage is needed. One application is, for example, particle accelerators, in which charged particles are accelerated to high energies. In addition to their importance for basic research, particle accelerators are also becoming increasingly important in medicine and for many industrial purposes.

So far, linear accelerators and cyclotrons are used to produce a particle beam in the MV range, which are usually very complex and expensive equipment.

One form of known particle accelerators are so-called electrostatic particle accelerators with a DC high voltage source. The particles to be accelerated are exposed to a static electric field.

It is known e.g. Cascade accelerator (also Cockcroft-Walton accelerator), in which by means of a Greinacher circuit which is repeatedly connected in series (cascaded), generates a high DC voltage by multiplying and rectifying an AC voltage, thus providing a strong electric field.

" G. Brautti et al. Tubeless Vacuum-Insulated Cockcroft-Walton accelerator. NIM A 328 (1993) 59-63 "discloses a DC high voltage source having a capacitor stack with a constant gap between the electrodes. US 2,887,599 discloses a second electrode whose distance to the outermost intermediate electrode is larger.

The invention has for its object to provide a DC high voltage source that enables a particularly high achievable DC voltage in a compact design and at the same time an advantageous FeldstärkeverBeschreibung division around the high voltage electrode allows. The invention is further based on the object to provide an accelerator for the acceleration of charged particles, which has a particularly high achievable particle energy in a compact design.

The invention is solved by the features of the independent claims. Advantageous developments can be found in the features of the dependent claims.

• einen Kondensatorstapel,
  • mit einer ersten Elektrode, welche auf ein erstes Potential gebracht werden kann,
  • mit einer zweiten Elektrode, die zur ersten Elektrode konzentrisch angeordnet ist und auf ein zweites, vom ersten Potential unterschiedlichen Potential gebracht werden kann, sodass sich eine Potentialdifferenz zwischen der ersten und der zweiten Elektrode ausbilden kann, und
  • mit mehreren konzentrisch zueinander angeordneten Zwischenelektroden, die konzentrisch zueinander zwischen der ersten Elektrode und der zweiten Elektrode angeordnet sind, und die auf eine Abfolge von anwachsenden Potentialstufen gebracht werden können, die sich zwischen dem ersten Potential und dem zweiten Potential befinden.

The DC voltage source according to the invention for providing DC voltage has:

• a capacitor stack,
  • with a first electrode, which can be brought to a first potential,
  • with a second electrode which is concentrically arranged to the first electrode and can be brought to a second, different from the first potential potential, so that a potential difference between the first and the second electrode can form, and
  • with a plurality of concentrically arranged intermediate electrodes, which are arranged concentrically with each other between the first electrode and the second electrode, and which can be brought to a sequence of increasing potential levels, which are located between the first potential and the second potential.

A switching device connects the electrodes of the capacitor stack-that is, the first electrode, the second electrode and the intermediate electrodes-and is designed such that, when the switching device is in operation, the electrodes of the capacitor stack arranged concentrically with one another are brought to increasing potential levels. The electrodes of the capacitor stack are arranged such that the distance between the electrodes of the capacitor stack decreases towards the central electrode.

The invention is based on the idea of the most efficient, ie space-saving configuration of the high voltage source to enable and at the same time to provide an electrode assembly, which makes it possible to easily charge with favorable field strength distribution in the high voltage source.

The concentric arrangement allows a total of a compact design. The high voltage electrode may be the central electrode in the concentric arrangement, while the outer electrode may be e.g. may be a ground electrode. For favorable utilization of the volume between the inner and the outer electrode, a plurality of concentric intermediate electrodes are brought to successively increasing potential levels. The potential levels can be selected such that a substantially uniform field strength results inside the entire volume.

The inserted intermediate electrodes also increase the breakdown field strength limit, so that higher DC voltages can be generated than without intermediate electrodes. This is because the breakdown field strength in vacuum is approximately inversely proportional to the square root of the electrode distances. The inserted / n intermediate electrode / n, with which the electric field in the interior of the DC voltage source is more uniform, at the same time contribute to an advantageous increase in the possible achievable field strength.

The decreasing distance between the electrodes and the center of the high-voltage source counteracts a uniform field strength distribution between the first and the second electrodes. Due to the decreasing distance, the electrodes near the center must in fact have a smaller potential difference in order to achieve a substantially constant field strength distribution around the high-voltage electrode. Lower potential differences, however, are easier to realize via the switching device interconnecting the electrodes when charging through the electrodes by the switching device. Losses when charging through the switching device can occur because the elements of the switching device itself are lossy, and the effect of increased potential levels amplified, can be intercepted by the decreasing electrode spacing.

The distances from electrode to electrode of the capacitor stack thus decrease towards the central electrode and can in particular be selected such that a substantially constant field strength is formed between adjacent electrodes. This can e.g. mean that the field strength between a pair of electrodes differ by less than 30%, by less than 20%, in particular by less than 10% or most particularly by less than 5% from the field strength of adjacent electrode pairs, in particular in the unloaded case. As a result, the electrical breakdown probability within the capacitor stack also remains essentially the same. If the relieved case ensures stable operation with a minimized probability of breakdown, it is generally also possible during operation of the DC high voltage cascade, e.g. in operation as a voltage source for a particle accelerator that ensures safe operation.

The switching device is advantageously designed such that the electrodes of the capacitor stack from the outside, in particular via the outermost electrode, are loadable by means of a pump AC voltage and thereby brought to the growing potential levels to the central electrode.

If such a DC high voltage source is e.g. is used to generate a beam of particles such as electrons, ions, elementary particles - or generally charged particles - can be achieved in a compact design, a particle energy in the MV range.

In an advantageous embodiment, the switching device comprises a high-voltage cascade, in particular a Greinacher cascade or a Cockcroft-Walton cascade. With a device of this type, the electrodes of the capacitor stack, that is to say the first electrode, the second electrode and the intermediate electrodes, can be charged to generate the DC voltage by means of a comparatively low alternating voltage. The AC voltage may be applied to the outermost electrode.

This embodiment is based on the idea of high-voltage generation, as is made possible by a Greinacher rectifier cascade, for example. Used in an accelerator, the electric potential energy serves to convert kinetic energy of the particles by applying the high potential between the particle source and the end of the acceleration path.

In one embodiment variant, the capacitor stack is divided into two separate capacitor chains through a gap extending through the electrodes. By separating the concentric electrodes of the capacitor stack into two separate capacitor chains, the two capacitor chains can be advantageously used for the formation of a cascaded switching device such as a Greinacher or Cockcroft-Walton cascade. Each capacitor chain thereby represents an arrangement of their part concentrically arranged (partial) electrodes.

In an embodiment of the electrode stack as a spherical shell stack, the separation may be e.g. through a cut along the equator, which then leads to two hemisphere stacks.

The individual capacitors of the chains can be loaded in such a circuit respectively to the peak-to-peak voltage of the primary AC input voltage, which is used to charge the high voltage source, so that at constant shell thickness above potential equilibration, a uniform electric field distribution and thus an optimal Exploitation of the isolation distance is achieved in a simple manner.

Advantageously, the switching device, which comprises a high-voltage cascade, connect the two separate capacitor chains with each other and in particular be arranged in the gap. The input AC voltage for the high voltage cascade can be applied between the two outermost electrodes of the capacitor chains, since these are e.g. be accessible from the outside. The diode strings of a rectifier circuit can then be mounted in the equatorial gap, thereby saving space.

On the basis of the embodiment in which the electrode stack is separated by the gap in two separate capacitor chains, the advantage can be explained again, which is achieved by the decreasing towards the center electrode spacing.

Essentially, the two capacitor strings represent the capacitive charge impedances of a pumping line voltage transmission line. The capacitance between the two capacitor string stacks acts as a shunt impedance, and the waveguide is also split by AC and DC distribution Transformation of the same into charging and load direct current by means of the diodes - twice damped. The alternating voltage amplitude therefore decreases towards the high-voltage electrode - and thus the DC voltage obtained per radial unit length. In this case, if a constant shell distance or electrode spacing were used, the voltages between the inner electrodes and therewith the E field would be lower and the insulation distances less effectively utilized. This can be prevented by the decreasing electrode spacing. As the distance between the electrodes and the high-voltage electrode decreases, the inner electrodes can also be exposed to a constantly high electric field strength. It can At the same time the dielectric strength of the diodes inside are reduced.

The electrodes of the capacitor stack may be shaped such that they lie on an ellipsoidal surface, in particular a spherical surface, or on a cylinder surface. These forms are physically cheap. Particularly favorable is the choice of the shape of the electrodes as in a hollow sphere or the ball capacitor. Similar shapes, e.g. in a cylinder are also possible, the latter, however, usually has a comparatively inhomogeneous electric field distribution.

The low inductance of the shell-like potential electrodes allows the use of high operating frequencies, so that the voltage drop remains limited at current consumption despite relatively small capacitance of the individual capacitors.

The central high voltage electrode may be embedded in a solid or liquid insulating material.

Another possibility is to isolate the central high voltage electrode by high vacuum. The intermediate electrodes can also be insulated from each other by vacuum. The use of insulating materials has the disadvantage that the materials are subject to stress due to a direct electrical field for the application of internal charges - which are caused in particular by ionizing radiation during operation of the accelerator. The accumulated, migrating charges cause in all physical insulators a strong inhomogeneous electric field strength, which then leads to the local crossing of the breakdown limit and thus formation of spark channels. Isolation by high vacuum avoids such disadvantages. The exploitable in stable operation electric field strength can be increased thereby. The arrangement is thus essentially - except for a few components such as the suspension of the electrodes - free of insulator materials.

The charged particle accelerator according to the invention comprises a DC high voltage source according to the invention, wherein an acceleration channel is formed, which is formed by openings in the electrodes of the capacitor stack, so that particles charged by the acceleration channel can be accelerated. The electric potential energy provided by the high voltage source is utilized to accelerate the charged particles. The potential difference is applied between particle source and target. The central high voltage electrode may include, for example, the particle source.

In an accelerator, the use of vacuum to isolate the electrodes also has the advantage that no separate jet pipe must be provided, which in turn at least partially has an insulator surface. Again, it is avoided that critical problems of wall discharge would occur along the insulator surfaces, since the acceleration channel now does not have to have insulator surfaces.

Fig. 1

eine schematische Darstellung einer Greinacherschaltung, wie sie aus dem Stand der Technik bekannt ist.

Fig. 2

eine schematische Darstellung eines Schnitts durch eine Gleichspannungs-Hochspannungsquelle mit einer Teilchenquelle im Zentrum,

Fig. 3

eine schematische Darstellung eines Schnitts durch eine Gleichspannungs-Hochspannungsquelle, die als Tandembeschleuniger ausgebildet ist,

Fig. 4

eine schematische Darstellung des Elektrodenaufbaus mit einem Stapel zylinderförmig angeordneter Elektroden,

Fig. 5

eine schematische Darstellung eines Schnitts durch eine Gleichspannungs-Hochspannungsquelle nach Fig. 2 mit zum Zentrum hin abnehmenden Elektrodenabstand,

Fig. 6

eine Darstellung der Dioden der Schaltvorrichtung, die als vakuumkolbenfreie Elektronenröhren ausgebildet sind,

Fig. 7

ein Diagramm, das den Ladungsvorgang in Abhängigkeit von Pumpzyklen zeigt, und

Fig. 8

die vorteilhafte Kirchhoff-Form der Elektrodenenden.

Embodiments of the invention will be explained in more detail with reference to the following drawing, but without being limited thereto. Show:

Fig. 1

a schematic representation of a Greinacherschaltung, as is known from the prior art.

Fig. 2

1 is a schematic representation of a section through a DC high voltage source with a particle source in the center,

Fig. 3

FIG. 2 a schematic representation of a section through a DC voltage high voltage source, which is designed as a tandem accelerator, FIG.

Fig. 4

a schematic representation of the electrode assembly with a stack of cylindrically arranged electrodes,

Fig. 5

a schematic representation of a section through a DC voltage source according to Fig. 2 with decreasing towards the center electrode gap,

Fig. 6

a representation of the diodes of the switching device, which are designed as vacuum piston-free electron tubes,

Fig. 7

a diagram showing the charging process as a function of pump cycles, and

Fig. 8

the advantageous Kirchhoff shape of the electrode ends.

Identical parts are provided in the figures with the same reference numerals.

On the diagram in the Fig. 1 Let the principle of a high-voltage cascade 9, which is constructed in accordance with a Greinacher circuit, be clarified.

At an input 11, an AC voltage U is applied. The first half-wave charges the capacitor 15 to the voltage U via the diode 13. At the following half cycle of the alternating voltage, the voltage U from the capacitor 13 is added to the voltage U at the input 11, so that the capacitor 17 is now charged via the diode 19 to the voltage 2U. This process is repeated in the subsequent diodes and capacitors, so that in the in Fig. 1 shown circuit total at the output 21, the voltage 6U is achieved. The Fig. 2 also clearly shows how each of the first set 23 of capacitors forms a first capacitor chain and the second set 25 of capacitors forms a second capacitor chain by means of the illustrated circuit.

Based on Fig. 2 Now, the principle of a DC voltage source is explained, the inventive development is then based on Fig. 5 explained.

Fig. 2 shows a schematic section through a high voltage source 31 with a central electrode 37, an outer electrode 39 and a series of intermediate electrodes 33, which by a high voltage cascade 35, the principle in Fig. 1 have been explained, are interconnected and can be loaded by this high voltage cascade 35.

The electrodes 39, 37, 33 are hollow-spherical and arranged concentrically with each other. The maximum electric field strength that can be applied is proportional to the curvature of the electrodes. Therefore, a spherical shell geometry is particularly favorable.

At the center is the high voltage electrode 37, the outermost electrode 39 may be a ground electrode. By an equatorial cut 47, the electrodes 37, 39, 33 are divided into two hemispherical stacks separated from each other by a gap. The first hemisphere stack forms a first condenser chain 41, the second hemisphere stack forms a second condenser chain 43.

The voltage U of an AC voltage source 45 is applied to the outermost electrode shell halves 39 ', 39 "in each case form the cross connections between the two capacitor chains 41, 43, the two sets 23, 25 of capacitors Fig. 1 correspond.

In the high-voltage source 31 shown here, an acceleration channel 51, which starts from a particle source 52, for example, located inside, and allows extraction of the particle flow, passes through the second condenser chain 43. The particle flow of charged particles undergoes a high acceleration voltage from the hollow-sphere high-voltage electrode 37.

The high voltage source 31 and the particle accelerator have the advantage that the high voltage generator and the particle accelerator are integrated with each other, since then all electrodes and intermediate electrodes can be accommodated in the smallest possible volume.

To insulate the high voltage electrode 37, the entire electrode assembly is isolated by vacuum insulation. Among other things, particularly high voltages of the high voltage electrode 37 can be generated, resulting in a particularly high particle energy result. But it is also conceivable in principle isolation of the high voltage electrode by means of solid or liquid insulation.

The use of vacuum as an insulator and the use of an inter-electrode distance of the order of 1 cm make it possible to achieve electric field strengths of values above 20 MV / m. In addition, the use of vacuum has the advantage that the accelerator must not be under stress during operation, since the radiation occurring during acceleration can lead to problems for insulator materials. This allows the construction of smaller and more compact machines.

Fig. 5 shows the development of the invention based on Fig. 2 explained principle of the high voltage source, in which the distance between the electrodes 39, 37, 33 decreases toward the center. As already explained, such a configuration makes it possible to compensate for the decrease in the pump AC voltage applied to the outer electrode 39 toward the center, so that a substantially identical field strength still exists between adjacent electrode pairs. As a result, a largely constant field strength along the acceleration channel 51 can be achieved.

Fig. 3 shows a further education in Fig. 2 shown high voltage source to the tandem accelerator 61. The switching device 35 off Fig. 2 is not shown for clarity, but is in the in Fig. 3 identical high voltage source shown. Based on Fig. 3 the principle of the tandem accelerator is explained. An embodiment according to Fig. 5 with decreasing towards the center electrode spacing is also applicable. In Fig. 3 However, this is not shown because it is not necessary for the explanation of the basic principle of the tandem accelerator 61.

In the example shown here, the first capacitor chain 41 also has an acceleration channel 53 which leads through the electrodes 33, 37, 39.

Inside the central high-voltage electrode 37, a carbon film 55 for charge stripping is disposed instead of the particle source. Then, negatively charged ions may be generated outside the high voltage source 61, accelerated along the acceleration channel 53 through the first capacitor chain 41 to the central high voltage electrode 37, converted into positively charged ions when passing through the carbon foil 55, and then through the acceleration channel 51 of the second Condenser chain 43 are further accelerated and exit from the high voltage source 31 again.

The outermost spherical shell 39 can remain largely closed and thus take over the function of a grounded housing. The hemispherical shell immediately below can then be the capacity of an LC resonant circuit and part of the drive connection of the switching device.

Such a tandem accelerator uses negatively charged particles. The negatively charged particles are accelerated by the first acceleration path 53 from the outer electrode 39 toward the central high-voltage electrode 37. at the central high voltage electrode 37, a charge conversion process takes place.

This can be done for example by a film 55, through which the negatively charged particles are passed, and with the aid of which a so-called charge stripping is performed. The resulting positively charged particles are further accelerated by the second acceleration path 51 from the high voltage electrode 37 to the outer electrode 39. The charge conversion can also take place in such a way that multiply positively charged particles, such as, for example, C ^4+, are formed, which are accelerated particularly strongly by the second acceleration section 51.

One embodiment of the tandem accelerator is to generate a 1 mA proton beam with an energy of 20 MeV. For this purpose, a continuous stream of particles from an H ^- particle source is introduced into the first acceleration section 53 and accelerated to the central +10 MV electrode. The particle hit a carbon charge stripper, removing both electrons from the protons. The load current of the Greinach cascade is therefore twice as large as the current of the particle beam.

The protons gain another 10 MeV of energy as they exit the accelerator through the second acceleration section 53.

For such acceleration, the accelerator may provide a 10 MV high voltage source having N = 50 stages, ie, a total of 100 diodes and capacitors. With an inner radius of r = 0.05 m and a vacuum insulation with a breakdown field strength of 20 MV / m, the outer radius is 0.55 m. In each hemisphere find 50 spaces at a distance of 1 cm between adjacent spherical shells.

A smaller number of stages reduces the number of charge cycles and the effective internal source impedance, but increases the pump charge voltage requirements.

The diodes arranged in the equatorial gap, which connect the two hemispherical stacks together, may be e.g. be arranged in a spiral pattern. The total capacity can be 74 pF according to equation (3.4) and the stored energy 3.7 kJ. A charging current of 2 mA requires an operating frequency of approximately 100 kHz.

If carbon foils are used for charge stripping, foils with a film thickness of t ≈ 15 ... 30 μg / cm ² can be used. This thickness represents a good compromise between particle transparency and effectiveness of the charge stripping.

The lifetime of a carbon stripper film can be estimated with T _foil = k _foil * (UA) / (Z ² I), where I is the beam current, A is the spot area of the beam, U is the particle energy and Z is the particle mass. Vapor deposited films have a value of kfoil ≈ 1.1 C / Vm ² .

Carbon films produced by decomposition of ethylene by means of glow discharge have a thickness-dependent lifetime constant of kfoil ≈ (0.44 t - 0.60) C / Vm ² , the thickness being given in μg / cm ² .

With a beam diameter of 1 cm and a beam current of 1 mA, a lifetime of 10 to 50 days can be expected. Longer lifetimes can be achieved by increasing the area effectively radiated, e.g. by scanning a rotating disk or a film having a linear band structure.

Fig. 4 illustrates an electrode mold in which hollow cylindrical electrodes 33, 37, 39 are arranged concentrically to one another. Through a gap, the electrode stack in split two separate capacitor chains, which with an analogous to Fig. 2 constructed switching device can be interconnected.

Again, not shown - the electrode distances to the central axis decrease, as for the spherical shape based on Fig. 5 explained.

Fig. 6 shows an embodiment of the diodes of the switching device shown. The concentrically arranged, hemispherical shell-like electrodes 39, 37, 33 are shown only for the sake of clarity.

The diodes are shown here as electron tubes 63 having a cathode 65 and an opposing anode 67. Since the switching device is disposed in the vacuum insulation, the vacuum tube of the electron tubes that would otherwise be required to operate the electrons is eliminated.

In the following, a closer explanation is made of components of the high voltage source or to the particle accelerator.

Spherical capacitor

The arrangement follows the in Fig. 1 shown principle, to arrange the high voltage electrode inside the accelerator and the concentric ground electrode on the outside of the accelerator.

A ball capacitor with inner radius r and outer radius R has the capacity $$C=4\pi {\epsilon }_0\frac{\mathit{r\; R}}{R-r}\mathrm{,}$$

The field strength at radius ρ is then $$e=\frac{\mathit{r\; R}}{\left(R-r\right){\rho }^2}U$$

erreicht. Aus Sicht der Durchbruchsfestigkeit ist dies unvorteilhaft.This field strength is quadratically dependent on the radius and thus increases strongly towards the inner electrode. For the inner electrode surface ρ = r is the maximum $$\hat{e}=\frac{\mathit{R}}{r\left(R-r\right)}U$$

reached. From the point of view of breakdown strength, this is unfavorable.

A hypothetical spherical capacitor with a homogeneous electric field would have the capacity $$\tilde{C}=4\pi {\epsilon }_0\frac{R^2+\mathit{r\; R}+r^2}{R-r}\mathrm{,}$$

mit minimaler maximaler Feldstärke ist.Because the electrodes of the capacitors of the Greinach cascade are inserted in the cascade accelerator as intermediate electrodes at a clearly defined potential, the field strength distribution is linearly adjusted over the radius, since for thin-walled hollow spheres the electric field strength is approximately equal to the flat case $$e\to \frac{U}{\left(R-r\right)}\mathrm{,}$$

with minimum maximum field strength.

The capacity of two adjacent intermediate electrodes is $$C_k=4\pi {\epsilon }_0\frac{r_k{r}_{k+1}}{r_{k+1}-r_k}\mathrm{,}$$

Hemispherical electrodes and the same electrode spacing d = (Rr) / N leads to r _k = r + kd and electrode capacitances $$C_{2k}=C_{2k+1}=2\pi {\epsilon }_0\frac{r^2+\mathit{rd}+\left(2\mathit{rd}+d^2\right)k+d^2{k}^2}{d}\mathrm{,}$$

rectifier

Modern avalanche semiconductor diodes ("soft avalanche semiconductor diodes") have very low parasitic capacitances and have short recovery times. A series circuit does not need resistors for potential equilibration. The operating frequency can be set comparatively high in order to use the relatively small interelectrode capacitances of the two Greinacher capacitor stacks.

With a pump voltage for charging the Greinacher cascade, a voltage of U _in ≈100kV, ie 70 kV _rms , can be used. The diodes must withstand voltages of 200 kV. This can be achieved by using chains of diodes with a lower tolerance. For example, ten 20 kV diodes can be used. Diodes can be, for example, diodes from the company Philips with the designation BY724, diodes from the company EDAL with the designation BR757-200A or diodes from the company Fuji with the designation ESJA5320A.

Fast lock recovery times (reverse recovery time), eg t _rr ≈100 ns for BY724, minimize losses. The dimension of the BY724 diode of 2.5mm x 12.5mm allows all 1000 diodes for the switching device to be accommodated in a single equatorial plane for the spherical tandem accelerator specified below.

Instead of solid-state diodes and electron tubes can be used in which the electron emission is used for rectification. The chain of diodes may be formed by a plurality of mesh-like electrodes of the electron tubes connected to the hemispherical shells. Each electrode acts on the one hand as a cathode, on the other hand as an anode.

Discrete capacitor stack

The central idea is to cut the concentric successively arranged electrodes on an equatorial plane. The two resulting electrode stacks represent the cascade capacitors. It is only necessary to connect the string of diodes to opposite electrodes across the cutting plane. It should be noted that the rectifier automatically stabilizes the potential differences of the successively arranged electrodes to about 2 U _in , suggesting constant electrode spacings. The drive voltage is applied between the two outer hemispheres.

Ideal capacity distribution

pro Vollwelle in die Last durch den Kondensator C₀. Jedes der Kondensatorpaare C_2k und C_2k+1 übertragen somit eine Ladung (k+1)Q.If the circuit only has the capacity of Fig. 3 stationary operation provides an operating frequency f a charge $$Q=\frac{I_{\mathrm{out}}}{f}\mathrm{,}$$

per full wave in the load through the capacitor C ₀ . Each of the capacitor pairs C _2k and C _{2k + 1} thus carry a charge (k + 1) Q.

dar. Dadurch reduziert ein Laststrom I_out die DC-Ausgangsspannung gemäß $$U_{\mathrm{out}}=2N{U}_{\mathrm{in}}-R_G{I}_{\mathrm{out}}\mathrm{.}$$

The charge pump provides a generator source impedance $$R_G=\frac{1}{2f}{\displaystyle \underset{k=0}{\overset{N-1}{\Sigma }}}\left(\frac{2k^2+3k+1}{C_{2k}}+\frac{2k^2+4k+2}{C_{2k+1}}\right)\mathrm{,}$$

This reduces a load current I _out according to the DC output voltage $$U_{\mathrm{out}}=2N{U}_{\mathrm{in}}-R_G{I}_{\mathrm{out}}\mathrm{,}$$

The load current causes an AC ripple at the DC output with the peak-to-peak value $$\mathit{.DELTA.u}=\frac{I_{\mathrm{out}}}{f}{\displaystyle \underset{k=0}{\overset{N-1}{\Sigma }}}\frac{k+1}{C_{2k}}\mathrm{,}$$

und der Spitze-zu-Spitze-Wert der AC-Welligkeit wird $$\mathit{\delta U}=\frac{I_{\mathrm{out}}}{\mathit{fC}}\frac{N^2+N}{2}\mathrm{.}$$

When all capacitors are equal to C _k = C, the effective source impedance is $$R_G=\frac{\mathrm{8th}{N}^3+9N^2+N}{12\mathit{fC}}$$

and the peak-to-peak value of AC ripple $$\mathit{.DELTA.u}=\frac{I_{\mathrm{out}}}{\mathit{fC}}\frac{N^2+N}{2}\mathrm{,}$$

For a given total energy storage within the rectifier, a capacitive imbalance in favor of the low voltage part will slightly reduce the R _G and R _R values compared to the usual choice of equal capacitors.

Fig. 7 Figure 3 shows the charging of an uncharged cascade of N = 50 concentric hemispheres, plotted over the number of pump cycles.

stray capacitances

Any charge exchange between the two columns reduces the efficiency of the multiplier circuit, see Fig. 1 , eg due to the stray capacitances c _j and the reverse recovery charge loss q _j through the diodes D _j .

$$U_{2k}^{-}=u_{2k}$$

$$U_{2k+1}^{+}=u_{2k+1}$$

$$U_{2k+1}^{-}=u_{2k+2}$$

bis zum Index 2N - 2 und $$U_{2N-1}^{+}=u_{2N-1}-U$$

$$U_{2n-1}^{-}=U\mathrm{.}$$

The basic equations for the capacitor voltages U _k ^± at the positive and negative extrema of the peak drive voltage U, neglecting the diode forward voltage drop, are: $$U_{2k}^{+}=u_{2k+1}$$

$$U_{2k}^{-}=u_{2k}$$

$$U_{2k+1}^{+}=u_{2k+1}$$

$$U_{2k+1}^{-}=u_{2k+2}$$

up to index 2N - 2 and $$U_{2N-1}^{+}=u_{2N-1}-U$$

$$U_{2n-1}^{-}=U\mathrm{,}$$

With this nomenclature is the average amplitude of the DC output voltage $$U_{\mathrm{out}}=\frac{1}{2}{\displaystyle \underset{k=0}{\overset{2N-1}{\Sigma }}}{u}_k\mathrm{,}$$

The peak-to-peak value of the DC voltage ripple is $$\mathit{.DELTA.u}={\displaystyle \underset{k=0}{\overset{2N-1}{\Sigma }}}{\left(-1\right)}^{k+1}{u}_k\mathrm{,}$$

With stray capacitances c _i parallel to the diodes D _i , the fundamental equations for the variables u _-1 = 0, U _2N = 2 U, and the tri-diagonal equation system is $$C_{k-1}{u}_{k-1}-\left(C_{k-1}+C_k\right)u_k+\left(C_k-c_k\right)u_{k+1}=\{\begin{array}{cc}Q& \forall \mathit{k}\mathrm{even}\\ 0& \forall \mathit{k}\mathrm{odd}\mathrm{,}\end{array}$$

mit η = f t_rr und Q_D für die Ladung pro Vollwelle in Vorwärtsrichtung. G1. (3.22) wird dann zu $$C_{k-1}{u}_{k-1}-\left(C_{k-1}+\left(1-\eta \right)C_k\right)u_k+\left(\left(1-\eta \right)C_k-c_k\right)u_{k+1}=\{\begin{array}{cc}Q& \forall \mathit{k}\mathrm{even}\\ 0& \forall \mathit{k}\mathrm{odd}\mathrm{.}\end{array}$$

Reverse recovery charges Finite reverse recovery times t _{rr of} the limited diodes cause a charge loss of $$q_D=\eta Q_D$$

with η = ft _rr and Q _D for the charge per full wave in the forward direction. G1. (3.22) then becomes $$C_{k-1}{u}_{k-1}-\left(C_{k-1}+\left(1-\eta \right)C_k\right)u_k+\left(\left(1-\eta \right)C_k-c_k\right)u_{k+1}=\{\begin{array}{cc}Q& \forall \mathit{k}\mathrm{even}\\ 0& \forall \mathit{k}\mathrm{odd}\mathrm{,}\end{array}$$

Continuous capacitor stack Capacitive transmission line

In Greinacher cascades, the rectifier diodes essentially pick up the AC voltage, turn it into DC voltage and accumulate it to a high DC output voltage. The AC voltage is conducted from the two capacitor columns to the high voltage electrode and attenuated by the rectifier currents and stray capacitances between the two columns.

For a high number N of stages, this discrete structure can be approximated by a continuous transmission line structure.

ein. Der Spannungsstapelung der Gleichrichter-Dioden bewirkt eine zusätzliche spezifische Stromlast J, die proportional ist zum DC-Laststrom I_out und zur Dichte der Anzapfungen entlang der Übertragungsleitung.For the AC voltage, the capacitor structure represents a longitudinal impedance with a length-specific impedance 3. Stray capacitances between the two columns result in a length-specific shunt admittance

one. The voltage stacking of the rectifier diodes causes an additional specific current load J, which is proportional to the DC load current I _out and the density of the taps along the transmission line.

$$\mathit{Uʹ}=\mathfrak{Z}I\mathrm{.}$$

The basic equations for the AC voltage U (x) between the columns and the AC series current I (x) are $$\mathit{I\; \text{'}}=\mathfrak{N}\mathit{U}+\Im$$

$$\mathit{U\; \text{'}}=\mathfrak{Z}I\mathrm{,}$$

The general equation is an extended telegraph equation $$\mathit{U"}=\frac{\mathfrak{Z\; \text{'}}}{\mathfrak{Z}}\mathit{U\; \text{'}}-\mathfrak{Z}\mathfrak{N}U=\mathfrak{Z}\Im \mathrm{,}$$

In general, the peak-to-peak ripple at the DC output is equal to the difference in AC voltage amplitude at both ends of the transmission line $$\mathit{.DELTA.u}=U\left(x_0\right)-U\left(x_1\right)\mathrm{,}$$

Two constraints are required for a unique solution of this second order differential equation.

One of the boundary conditions may be U (x ₀ ) = U _in given by the AC drive voltage between the DC low voltage ends of the two columns. The other natural constraint determines the AC current at the DC high voltage end x = x ₁ . The boundary condition for a concentrated terminal AC impedance Z ₁ between the columns is $$\mathit{U\; \text{'}}\left(x_1\right)=\frac{\mathfrak{Z}\left(x_1\right)}{Z_1}U\left(x_1\right)\mathrm{,}$$

In the unloaded case Z ₁ = ∞, the boundary condition U '(x ₁ ) = 0.

Constant electrode distance

so dass die Verteilung der AC-Spannung geregelt ist durch $$\mathit{Uʺ}=\frac{\mathfrak{Zʹ}}{\mathfrak{Z}}\mathit{Uʹ}-\mathfrak{Z}\mathfrak{N}U=\mathfrak{Z}\Im \mathrm{.}$$

For a constant electrode distance t is the specific load current $$\Im =\frac{\mathit{\iota \pi }{I}_{\mathrm{out}}}{t}.$$

so that the distribution of AC voltage is regulated by $$\mathit{U"}=\frac{\mathfrak{Z\; \text{'}}}{\mathfrak{Z}}\mathit{U\; \text{'}}-\mathfrak{Z}\mathfrak{N}U=\mathfrak{Z}\Im \mathrm{,}$$

und die DC-Spitze-zu-Spitze-Welligkeit der DC-Spannung ist $$\mathit{\delta U}=U\left(\mathit{Nt}\right)-U\left(0\right)\mathrm{.}$$

The average DC output voltage is then $$U_{\mathrm{out}}=\frac{2U_{\mathrm{in}}}{t}{\int }_0^{\mathit{nt}}U\left(x\right)\mathit{dx}$$

and the DC peak-to-peak ripple of the DC voltage $$\mathit{.DELTA.u}=U\left(\mathit{nt}\right)-\mathrm{<figure-callout\; id="0"\; label="U"\; filenames="imgf0007.png"\; state="\{\{state\}\}">U</figure-callout>}\left(0\right)\mathrm{,}$$

Optimal electrode spacing

The optimal electrode spacing ensures a constant DC electric field strength 2 E at the planned DC load current. The specific AC load current along the transmission line is position dependent $$\Im =\frac{\mathit{\iota \pi E}{I}_{\mathrm{out}}}{U}\mathrm{,}$$

The AC voltage follows $$\mathit{UU"}-\frac{\mathfrak{Z\; \text{'}}}{\mathfrak{Z}}\mathit{UU\text{'}}-\mathfrak{Z}\mathfrak{N}{U}^2=\mathfrak{Z}\mathit{\iota \pi E}{I}_{\mathrm{out}}\mathrm{,}$$

The electrode distances result from the local AC voltage amplitudes t (x) = U (x) / E.

The DC output voltage at the planned DC load current is U _out = 2Ed. Reducing the load always increases the voltages between the electrodes, so operation with little or no load can exceed the allowable E and maximum load capacity of the rectifier columns. It may therefore be advisable to optimize the design for unloaded operation.

For any given electrode distribution other than that designed for a planned DC load current, the AC voltage along the transmission line and hence the DC output voltage is regulated by the G1. (3.27).

Linear cascade

For a linear cascade with flat electrodes of width w, height h and a distance s between the columns are transmission line impedances $$Z=\frac{2}{{\mathrm{<figure-callout\; id="0"\; label="\iota \epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\iota \epsilon </figure-callout>}}_0\mathit{\omega wh}}\mathrm{,}\mathfrak{N}=\frac{{\mathrm{<figure-callout\; id="0"\; label="\iota \epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\iota \epsilon </figure-callout>}}_0\mathit{\omega w}}{s}\mathrm{,}$$

Linear Cascade - Constant Electrode Distance

The inhomogeneous telegraph equation is $$\mathit{U"}-\frac{2}{\mathit{hs}}U=\frac{I_{\mathrm{out}}}{f{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{wht}}\mathrm{,}$$

Assuming a line extending from x = 0 to x = d = Nt and operated by U _in = U (0) and a propagation constant of γ ² = 2 / (h * s), the solution is $$U\left(x\right)=\frac{\cosh \mathit{\gamma x}}{\cosh \mathit{\gamma d}}{U}_{\mathrm{in}}+\left(\frac{\cosh \mathit{\gamma x}}{\cosh \mathit{\gamma d}}-1\right)\frac{\mathit{ns}}{2f{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{dw}}{I}_{\mathrm{out}}\mathrm{,}$$

oder - explizit - $$U_{\mathrm{out}}=2N\frac{\tanh \mathit{\gamma d}}{\mathit{\gamma d}}{U}_{\mathrm{in}}+\left(\frac{\tanh \mathit{\gamma d}}{\mathit{\gamma d}}-1\right)\frac{{\mathit{N}}^2\mathit{s}}{f{ϵ}_0\mathit{dw}}{I}_{\mathrm{out}}\mathrm{.}$$

The diodes essentially tap the AC voltage, direct it and accumulate it along the transmission line. The average DC output voltage is thus $$U_{\mathrm{out}}=\frac{2}{t}{\int }_0^{d}U\left(x\right)\mathit{dx}\mathrm{,}$$

or - explicitly - $$U_{\mathrm{out}}=2N\frac{\tanh \mathit{\gamma d}}{\mathit{\gamma d}}{U}_{\mathrm{in}}+\left(\frac{\tanh \mathit{\gamma d}}{\mathit{\gamma d}}-1\right)\frac{{\mathit{N}}^2\mathit{s}}{f{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{dw}}{I}_{\mathrm{out}}\mathrm{,}$$

und $$\mathit{\delta U}\approx \frac{d^2}{\mathit{hs}}{U}_{\mathrm{in}}+\frac{N}{f}\frac{d}{2{ϵ}_0\mathit{hw}}{I}_{\mathrm{out}}\mathrm{.}$$

A series extension to the third order after γd gives $$U_{\mathrm{out}}\approx 2N{U}_{\mathrm{in}}\left(1-\frac{2d^2}{3\mathit{hs}}\right)-\frac{2N^2}{3f}\frac{\mathrm{<figure-callout\; id="0"\; label="d\; \epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">d\; </figure-callout>}}{{\epsilon }_{}}{I}_{\mathrm{out}}$$

and $$\mathit{.DELTA.u}\approx \frac{d^2}{\mathit{hs}}{U}_{\mathrm{in}}+\frac{N}{f}\frac{d}{2{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{hw}}{I}_{\mathrm{out}}\mathrm{,}$$

The load-current-related effects correspond to G1. (3.12) and (3.13).

Linear cascade - Optimal electrode spacing

The basic equation is here $$\mathit{UU"}-\frac{2}{\mathit{hs}}{U}^2=\frac{e{I}_{\mathrm{out}}}{f{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{wh}}\mathrm{,}$$

It seems that this differential equation does not have a closed analytic solution. The implicit solution that satisfies U '(0) = 0 is $$x={\int }_{U\left(0\right)}^{U\left(x\right)}\frac{\mathit{you}}{\sqrt{\frac{2}{\mathit{hs}}\left(u^2-U^2\left(0\right)\right)+\frac{e{I}_{\mathrm{out}}}{f{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{wh}}\log \frac{\mathrm{<figure-callout\; id="0"\; label="u\; U"\; filenames="imgf0007.png"\; state="\{\{state\}\}">u\; </figure-callout>}}{U}}}\mathrm{,}$$

Radial cascade

Assuming a stack of concentric cylindrical electrodes having a radius independent height h and an axial gap s between the columns as in FIG Fig. 4 shown are the radial-specific impedances $$\mathfrak{Z}=\frac{1}{\mathit{\iota \pi }{\epsilon }_0\mathit{\omega rh}}.\mathfrak{N}=\frac{2\mathit{\iota \pi }{\epsilon }_0\mathit{.omega.r}}{s}\mathrm{,}$$

Radial Cascade - Constant Electrode Spacing

die allgemeine Lösung $$U\left(\rho \right)=A{\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)+B{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right)+\frac{I_{\mathrm{out}}}{4\mathit{\gamma f}{ϵ}_0\mathit{ht}}{\mathrm{L}}_0\left(\mathit{\gamma \rho }\right)\mathrm{.}$$

mit γ² = 2/(h*s). K₀ und I₀ sind die modifizierte Bessel-Funktionen und L₀ ist die modifizierte STRUVE Funktion L₀ nullter Ordnung.With an equidistant radial electrode distance t = (Rr) / N has the basic equation $$\mathit{U"}+\frac{1}{\rho }\mathit{U\; \text{'}}-\frac{2}{\mathit{hs}}U=\frac{I_{\mathrm{out}}}{{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{\omega ht\rho }}$$

the general solution $$U\left(\rho \right)=A{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0007.png"\; state="\{\{state\}\}">K</figure-callout>}}_0\left(\mathit{\gamma \rho }\right)+B{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right)+\frac{I_{\mathrm{out}}}{4\mathit{\gamma f}{\epsilon }_0\mathit{<figure-callout\; id="0"\; label="ht\; \; L"\; filenames="imgf0007.png"\; state="\{\{state\}\}">ht\; </figure-callout>}}{\mathrm{L}}_{}{\mathrm{}}_0\left(\mathit{\gamma \rho }\right)\mathrm{,}$$

with γ ² = 2 / (h * s). K ₀ and I ₀ are the modified Bessel functions and L ₀ is the modified STRUVE function L _{0 of} zeroth order.

$$B=\frac{U_{\mathrm{in}}{\mathrm{K}}_1\left(\mathit{\gamma r}\right)-\frac{I_{\mathrm{out}}}{4{\mathit{\gamma fϵ}}_0\mathit{ht}}\left[{\mathrm{K}}_1\left(\mathit{\gamma r}\right){\mathrm{L}}_0\left(\mathit{\gamma R}\right)+{\mathrm{K}}_0\left(\mathit{\gamma R}\right)\left({\mathrm{L}}_1\left(\mathit{\gamma r}\right)+\frac{2}{\pi }\right)\right]}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}$$

sodass $$\begin{array}{ll}U\left(\rho \right)& =U_{\mathrm{in}}\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}\\ & +\frac{I_{\mathrm{out}}}{4\mathit{\gamma f}{ϵ}_0\mathit{ht}}[{\mathrm{L}}_0\left(\mathrm{\gamma \rho }\right)-{\mathrm{L}}_0\left(\mathit{\gamma R}\right)\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}\\ & -\left({\mathrm{L}}_1\left(\mathit{\gamma r}\right)+\frac{2}{\pi }\right)\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)-{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}]\mathrm{.}\end{array}$$

The boundary conditions U '(r) = 0 at the inner radius r and U (R) = U _in at the outer radius R determine the two constants $$A=\frac{U_{\mathrm{in}}{\mathrm{I}}_1\left(\mathit{\gamma r}\right)-\frac{I_{\mathrm{out}}}{4{\mathit{\gamma f\epsilon }}_0\mathit{ht}}\left[{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{L}}_0\left(\mathit{\gamma R}\right)-{\mathrm{I}}_0\left(\mathit{\gamma R}\right)\left({\mathrm{L}}_1\left(\mathit{\gamma r}\right)+\frac{2}{\pi }\right)\right]}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}$$

$$B=\frac{U_{\mathrm{in}}{\mathrm{K}}_1\left(\mathit{\gamma r}\right)-\frac{I_{\mathrm{out}}}{4{\mathit{\gamma f\epsilon }}_0\mathit{ht}}\left[{\mathrm{K}}_1\left(\mathit{\gamma r}\right){\mathrm{L}}_0\left(\mathit{\gamma R}\right)+{\mathrm{K}}_0\left(\mathit{\gamma R}\right)\left({\mathrm{L}}_1\left(\mathit{\gamma r}\right)+\frac{2}{\pi }\right)\right]}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}$$

so that $$\begin{array}{ll}U\left(\rho \right)& =U_{\mathrm{in}}\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}\\ & +\frac{I_{\mathrm{out}}}{4\mathit{\gamma f}{\epsilon }_0\mathit{ht}}[{\mathrm{L}}_0\left(\mathrm{\gamma \rho }\right)-{\mathrm{L}}_0\left(\mathit{\gamma R}\right)\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}\\ & -\left({\mathrm{L}}_1\left(\mathit{\gamma r}\right)+\frac{2}{\pi }\right)\frac{{\mathrm{I}}_0\left(\mathit{\gamma \rho }\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)-{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_0\left(\mathit{\gamma \rho }\right)}{{\mathrm{I}}_0\left(\mathit{\gamma R}\right){\mathrm{K}}_1\left(\mathit{\gamma r}\right)+{\mathrm{I}}_1\left(\mathit{\gamma r}\right){\mathrm{K}}_0\left(\mathit{\gamma R}\right)}]\mathrm{,}\end{array}$$

K ₁ and I ₁ are the modified Bessel functions and L _{1 is} the modified Struve function L ₁ = L ' _0-2 / n, all first order.

The DC output voltage is $$U_{\mathrm{out}}=\frac{2}{t}{\int }_r^{R}U\left(\rho \right)\mathit{d\rho }\mathrm{,}$$

Radial Cascade - Optimal Electrode Spacing

The optimal local electrode spacing is t (ρ) = U (p) / E, and the basic equation becomes $$\mathit{UU"}+\frac{1}{\rho }\mathit{UU\text{'}}-\frac{2}{\mathit{hs}}{U}^2=\frac{e{I}_{\mathrm{out}}}{{\mathrm{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0007.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0\mathit{\omega h\rho }}$$

It seems that this differential equation does not have a closed analytical solution, but it can be solved numerically.

electrode shapes equipotential

A compact machine needs to maximize the electric breakdown field strength. Generally smooth surfaces with low curvature should be chosen for the capacitor electrodes. The breakdown electric field E scales roughly with the inverse square root of the interelectrode distance, so that a large number of closely spaced equipotential surfaces with lower voltage differences than a few large distances with large voltage differences are preferable.

Minimal E-field electrode edges

$$y=\frac{b}{2}+\frac{1-A^2}{2\pi }\left(\arctan \frac{2A}{1-A^2}-\arctan \frac{2A\sin \vartheta }{1-A^2}\right)\mathrm{.}$$

in Abhängigkeit der Parameter ϑ ∈ [0, n/2]. Die Elektrodenform ist in Fig. 8 gezeigt. Die Elektroden verfügen über einen normalisierten Einheitsabstand und eine asymptotische Dicke 1 - A weit weg von der Kante, die sich stirnseitig zu einer vertikalen Kante mit der Höhe $$b=1-A-\frac{2-2A^2}{\pi }\arctan A\mathrm{.}$$

verjüngt.For a substantially planar electrode construction with equidistant spacing and a linear stress distribution, the optimum edge shape is known as the KIRCHHOFF shape (see below), $$x=\frac{A}{2\pi }\ln \frac{1+\cos \theta }{1-\cos \theta }-\frac{1+A^2}{4\pi }\ln \frac{1+2A\cos \theta +A^2}{1-2A\cos \theta +A^2}$$

$$y=\frac{b}{2}+\frac{1-A^2}{2\pi }\left(\arctan \frac{2A}{1-A^2}-\arctan \frac{2A\sin \theta }{1-A^2}\right)\mathrm{,}$$

depending on the parameter θ ∈ [0, n / 2]. The electrode shape is in Fig. 8 shown. The electrodes have a normalized unit spacing and an asymptotic thickness 1 - A far away from the edge, which is frontally to a vertical edge with the height $$b=1-A-\frac{2-2A^2}{\pi }\arctan A\mathrm{,}$$

rejuvenated.

The parameter 0 <A <1 also represents the inverse E field peak due to the presence of the electrodes. The thickness of the electrodes can be arbitrarily small without introducing noticeable E field distortions.

A negative curvature, z. At the orifices along the beam path, further reduce the E-field amplitude.

This positive result is due to the fact that the electrodes cause only a local disturbance of an already existing E-field.

The optimum shape for freestanding high voltage electrodes are ROGOWSKI and BORDA profiles, with a peak in the E-field amplitude of twice the undistorted field strength.

Drive voltage generator

The drive voltage generator must provide high AC voltage at high frequency. The usual approach is to boost an average AC voltage through a high isolation output transformer.

Disturbing internal resonances caused by unavoidable winding capacitances and stray inductances make designing a design for such a transformer a challenge.

An alternative may be a charge pump, ie a periodically operated semiconductor Marx generator. Such a circuit provides an output voltage with a change between ground and a high voltage of a single polarity, and efficiently charges the first capacitor of the capacitor chain.

Dielectric strength in vacuum d ^-0.5 law

mit konstantem A in Abhängigkeit vom Elektrodenmaterial (siehe unten). Es scheint, dass für die Felder von E ≈ 20 MV/m momentan verfügbare Elektrodenoberflächenmaterialien eine Elektrodenabstandsentfernung von d ≤ 10^-2 m erfordern.There is a wealth of evidence, but not a definitive explanation, that for electrode spacings above d≈10 ^-3 m, the breakdown voltage is approximately proportional to the square root of the distance. The breakthrough E-field therefore scales according to $$e_{\mathrm{Max}}=\sigma d^{-0.5}$$

with constant A depending on the electrode material (see below). It appears that for the fields of E≈20 MV / m currently available electrode surface materials require an electrode gap distance of d ≤ 10 ^-2 m.

surface materials

steel 3.85 SS 316LN 3.79 3.16 Ni 3.04 V 2.84 Ti 2.70 Mo 1.92 Monel 1.90 Ta 1.34 Al 1.30 0.45 Cu 1.17 0.76 Abhängigkeit von der ElektrodenflächeThe flashover between the electrodes in vacuo depends strongly on the material surface. The results of the CLIC study ( A. Descoeudres et al. "DC Breakdown Experiments for CLIC", Proceedings of EPAC08, Genoa, Italy, p.577, 2008 ) show the breakthrough coefficients material $\sigma \cdot \mathrm{in}\left[\frac{\mathit{MV}}{\sqrt{m}}\right]$

steel 3.85 SS 316LN 3.79 3.16 Ni 3:04 V 2.84 Ti 2.70 Not a word 1.92 Monel 1.90 Ta 1:34 al 1.30 12:45 Cu 1.17 0.76 Dependence on the electrode surface

für Kupfereletroden-Oberflächen und 2*10^-2 mm Elektrodenabstand. Für planare Elektroden aus rostfreiem Stahl mit 10^-3 m Abstand gilt: $$E_{\max }\approx 57.38\cdot {10}^6\frac{V}{m}{\left(\frac{A_{\mathrm{eff}}}{1{\mathit{cm}}^2}\right)}^{-0.12}$$

Form des elektrostatischen FeldesThere is evidence that the electrode area has a significant impact on the breakdown field strength. The following applies: $$e_{\mathrm{Max}}\approx 58\cdot {10}^6\frac{V}{m}{\left(\frac{A_{\mathrm{eff}}}{1{\mathit{cm}}^2}\right)}^{-\mathrm{12:25}}$$

for copper electrode surfaces and 2 * 10 ^-2 mm electrode spacing. For planar electrodes made of stainless steel with a distance of 10 ^-3 m, the following applies: $$e_{\mathrm{Max}}\approx 57.38\cdot {10}^6\frac{V}{m}{\left(\frac{A_{\mathrm{eff}}}{1{\mathit{cm}}^2}\right)}^{-\mathrm{12:12}}$$

Shape of the electrostatic field

Dielectric efficiency

It is generally accepted that homogeneous E fields allow the highest voltages. The dielectric SCHWAIGER efficiency factor η is defined as the inverse of the local E field peak due to field inhomogeneities, i. the ratio of the E field of an ideal flat electrode array and the peak surface E field of the geometry, considering equal reference voltages and spacings.

It represents the use of the dielectric in terms of E-field amplitudes. For small distances d <6 * 10 ^-3 m, inhomogeneous E-fields appear to increase the breakdown voltage.

Curvature of the electrode surface

Since the E field inhomogeneity maxima occur at the electrode surfaces, the relevant measure for the electrode shape is the mean curvature H = (k1 + k2) / 2.

There are several surfaces that fulfill the ideal of vanishing local mean curvatures over large areas. For example, catenoids are H = 0 rotation surfaces.

Any purely geometric measure such as η or H can only approximate the actual breakthrough behavior. Local E-field inhomogeneities have a nonlocal impact on the breakthrough limit and may even improve the overall overall field strength.

Constant E-field electrode surfaces

Fig. 8 shows KIRCHHOFF electrode edges at A = 0.6 for a vertical E-field. The field increase within the electrode stack is 1 / A = 1.6. The front sides are flat.

An electrode surface represents an equipotential line of the electric field analogous to a free surface of a flowing liquid. A stress-free electrode follows the flow field line. With the complex spatial coordinate z = x + iy, every analytic function w (z) satisfies the POISSON equation. The boundary condition for the free flow area is equivalent to a constant size of the (conjugate) derivative v of a possible function w $$\tilde{\upsilon }=\frac{dw}{dz}\mathrm{,}$$

Each possible function w (ν̅) over a flow velocity ν̅ or a hodograph plane leads to an z-mapping of the plane $$z=\int \frac{\mathit{dw}}{\tilde{\upsilon }}=\int \frac{1}{\tilde{\upsilon }}\frac{\mathit{dw}}{d\tilde{\upsilon }}d\tilde{\upsilon }\mathrm{,}$$

Without limiting the generality, the size of the derivative on the electrode surface can be normalized to one, and the height DE can be referred to as A in comparison to AF (see Fig. 6 ). In the ν̅-plane the curve CD then maps to arc i → 1 on the unit circle.

The points in Fig. 8 A and F 1 / A, B correspond to the origin, C i, D and E correspond to 1. The complete flow pattern is mapped in the first quadrant of the unit circle. The source of the streamlines is 1 / A, that of the sink 1.

Two reflections on the imaginary axis and the unit circle extend this flow pattern over the entire complex ν̅-plane. The potential function ω is thus defined by four sources on ν̅ positions + A, -A, 1 / A, -1 / A and two sinks of magnitude 2 to ± 1. $$w=\log \left(\tilde{\upsilon }-A\right)+\log \left(\tilde{\upsilon }+A\right)+\log \left(\tilde{\upsilon }-\frac{1}{A}\right)+\log \left(\tilde{\upsilon }+\frac{1}{A}\right)-2\log \left(\tilde{\upsilon }-1\right)-2\log \left(\tilde{\upsilon }+1\right)\mathrm{,}$$

und so $$z-z_0=\int \frac{1}{\bar{\upsilon }}\left(\frac{1}{\bar{\upsilon }-A}+\frac{1}{\bar{\upsilon }+A}+\frac{1}{\bar{\upsilon }-\frac{1}{A}}+\frac{1}{\bar{\upsilon }+\frac{1}{A}}-\frac{2}{\bar{\upsilon }-1}-\frac{2}{\bar{\upsilon }+1}\right)d\bar{\upsilon }$$

Its derivative is $$\frac{dw}{d\tilde{\upsilon }}=\frac{1}{\tilde{\upsilon }-A}+\frac{1}{\tilde{\upsilon }+A}+\frac{1}{\tilde{\upsilon }-\frac{1}{A}}+\frac{1}{\tilde{\upsilon }+\frac{1}{A}}-\frac{2}{\tilde{\upsilon }-1}-\frac{2}{\tilde{\upsilon }+1}$$

and so $$z-{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="imgf0007.png"\; state="\{\{state\}\}">z</figure-callout>}}_0=\int \frac{1}{\tilde{\upsilon }}\left(\frac{1}{\tilde{\upsilon }-A}+\frac{1}{\tilde{\upsilon }+A}+\frac{1}{\tilde{\upsilon }-\frac{1}{A}}+\frac{1}{\tilde{\upsilon }+\frac{1}{A}}-\frac{2}{\tilde{\upsilon }-1}-\frac{2}{\tilde{\upsilon }+1}\right)d\tilde{\upsilon }$$

mit z₀ = i b der Punkt C. Eine analytische Integration liefert G1. (3.54).At the free boundary CD flow velocity is ν̅ = e ^iφ , so that dν̅ = iν̅dφ and $$z-{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="imgf0007.png"\; state="\{\{state\}\}">z</figure-callout>}}_0={\int }_{-\frac{\pi }{2}}^{-0}\frac{\iota }{e^{\mathit{\iota \phi }}-A}+\frac{\iota }{e^{\mathit{\iota \phi }}+A}+\frac{\iota }{e^{\mathit{\iota \phi }}-\frac{1}{A}}+\frac{\iota }{e^{\mathit{\iota \phi }}+\frac{1}{A}}-\frac{2\iota }{e^{\mathit{\iota \phi }}-1}-\frac{2\iota }{e^{\mathit{\iota \phi }}+1}\mathit{d\phi }$$

with z ₀ = ib the point C. An analytic integration provides G1. (3.54).

LIST OF REFERENCE NUMBERS

9

High voltage cascade

11

entrance

13

diode

15

capacitor

17

capacitor

19

diode

21

output

23

first set of capacitors

25

second set of capacitors

31

High voltage source

33

intermediate electrode

35

High voltage cascade

37

central electrode

39

outer electrode

39 ', 39 "

Electrode shell half

41

first condenser chain

43

second condenser chain

45

AC voltage source

47

equatorial section

49

diode

51

Acceleration channel through the second condenser chain

52

particle

61

tandem accelerator

53

Acceleration channel through the first condenser chain

55

carbon films

63

electron tubes

65

cathode

67

anode

81

High voltage source

Claims (13)

1. DC high-voltage source (81) for providing DC voltage, having:

   a capacitor stack

   - with a first electrode (37), which can be brought to a first potential,

   - with a second electrode (39), which is concentrically arranged with respect to the first electrode and can be brought to a second potential that differs from the first potential,

   - with a plurality of intermediate electrodes (33) concentrically arranged with respect to one another, which are concentrically arranged between the first electrode (37) and the second electrode (39) and which can be brought to a sequence of increasing potential levels situated between the first potential and the second potential,

   a switching device (35), to which the electrodes (33, 37, 39) of the capacitor stack are connected and which are embodied such that, during operation of the switching device (35), the electrodes (33, 37, 39) of the capacitor stack concentrically arranged with respect to one another can be brought to increasing potential levels, characterized in that

   the spacing of the electrodes (33, 37, 39) of the capacitor stack reduces toward the central electrode (37).
2. DC high-voltage source (81) according to Claim 1,
   wherein the switching device (35) is embodied such that the electrodes (33, 37, 39) of the capacitor stack can be charged from the outside, more particularly via the outermost electrode (39), with the aid of a pump AC voltage and thereby be brought to the increasing potential levels.
3. DC high-voltage source (81) according to Claim 1 or 2, wherein the spacing of the electrodes (33, 37, 39), which decreases toward the central electrode (37) of the capacitor stack is selected such that a substantially unchanging field strength forms between adjacent electrodes.
4. DC high-voltage source (81) according to one of Claims 1 to 3,
   wherein the switching device comprises a high-voltage cascade (35), more particularly a Greinacher cascade or a Cockcroft-Walton cascade.
5. DC high-voltage source (81) according to one of Claims 1 to 4,
   wherein the capacitor stack is subdivided into two mutually separate capacitor chains (41, 43) by a gap (47) which runs through the electrodes (33, 37, 39).
6. DC high-voltage source (81) according to Claim 5,
   wherein the switching device comprises a high-voltage cascade (35), which interconnects the two mutually separated capacitor chains (41, 43) and which, in particular, is arranged in the gap (47).
7. DC high-voltage source (81) according to Claim 6,
   wherein the high-voltage cascade (35) is a Greinacher cascade or a Cockcroft-Walton cascade.
8. DC high-voltage source (81) according to one of the preceding claims,
   wherein the switching device (35) comprises diodes (49).
9. DC high-voltage source according to one of the preceding claims,
   wherein the electrodes (33, 37, 39) of the capacitor stack are formed such that they are situated on the surface of an ellipsoid, more particularly on the surface of a sphere, or on the surface of a cylinder.
10. DC high-voltage source according to one of the preceding claims,
    wherein the central electrode (37) is embedded in solid or liquid insulation material.
11. DC high-voltage source (81) according to one of Claims 1 to 9,
    wherein the central electrode (37) is insulated by a high vacuum.
12. Accelerator for accelerating charged particles, with a DC high-voltage source (81) according to one of the preceding claims,
    wherein there is an acceleration channel (51), which is formed by openings in the electrodes (33, 37, 39) of the capacitor stack such that charged particles can be accelerated through the acceleration channel (51).
13. Accelerator according to Claim 12,
    wherein the particle source (52) is arranged within the central electrode (37).

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