Tesla Coil Design for Electron Gun Application

TESLA COIL DESIGN FOR ELECTRON GUN APPLICATION M. Parafiev%, C. Gough, S. Ivkovic

Paul Scherrer Institute, Accelerator Division 5232 Villigen PSI, Switzerland

Abstract

describe our results of optimizing the parameters of the resonant air-core transformer.

The current project is to build an electron gun for X-ray Free Electron Laser (XFEL) application. The electron

II. RESONANT TRANSFORMER

gun will utilize field emission and extreme accelerating

gradient to achieve very low emittance. However for long-term study of cathode characteristics, a stable pulsed voltage in the megavolt range is needed. The first project phase is to design and test a 500kV pulser using a resonant air-core transformer (Tesla coil). Detailed results of simulations with Microwave Studio® and PSpice® for various coil geometries, tuning and coupling factors are given, and the optimum values for this application are given. In addition, experimental results are given for the most promising geometries.

The electrical circuit of a lossless resonant pulse transformer is shown in Figure 1., where Sw is a switch, L and Ls are primary and secondary inductances, C and Cs are primary and secondary capacitances, and K is coupling factor. To derive the differential equations describing the electrical behavior of the circuit, the equivalent circuit shown in Figure 2. is used. K

I. INTRODUCTION

T I

The requirement to sustain very high anode-cathode gradient (0.2... lGV/m) spoke in favor of having the shortest possible high voltage pulse. The cathode field emission current I is given by Fowler-Nordheim law shown below: 2 (O 3/2) I=AclF expLc2Y

SC,

c Figure 1.

()A

L

L

Simplified circuit of a resonant pulse transformer U3

B B

where A is emitting area, F is local field strength, y work u, 34 U2 I iS function of the emitting material, cl and c2 are constants. L1 C L2 C2 I I Equation (1) shows that the emitted current is strongly Figure 2. Equivalent circuit of a resonant pulse dependant on the extracting field. In order to compare different field emitters, the pulser should have good transformer. All values are referred to the primary side. stability and repeatability. The resonant air-core transformer technology was chosen because it is fast (z 200 ns), stable, fully linear where i1 to i5 are the currents in branches with the shown and scalable. In the frequency domain, the resonant directions, ul and u2are respectively the voltages in frequencies of the primary and secondary LC circuits primary and secondary side andU3 is the voltage across L3, which represents the coupling. All values are referred separate with increased coupling. There is a particular coupling factor, namely 0.6, when one resonant frequency to the primary side. The values of the components in the is exactly twice the other. This case is unique with its equivalent circuit are the following: asymmetric voltage waveform and with its capability to (2) reach the maximum voltage within one cycle (Figure 3.). Cl = C2 = C, This particular case of the resonant transformer we will call critically coupled. In the rest of this paper, we -= = L(K+1), (3)

4 email: martin.paraliev @psi.ch

0-7803-9189-6/05/$20.00 ©C2005 IEEE.

1085

l+ k1u - k2u2 = 0

(7)

where U is the initial voltage of primary capacitor C. Going back to the original variables we find the solutions for primary voltage ul and secondary one u2 ] (21) Ul 2 (22) =-[cos(OS(t) - cos(cO2t)]. 2 The solution consists of two cosine functions. In order to have an asymmetric output signal (Figure 3.) we are interested in the particular case when the second frequency 02 is exactly twice the first one o . Using equations (17) and (18) the critical coupling KC can be calculated:

U2

(8)

2coil =2

L3= L

1-K2

(4)

U-[Colt)+COS(01

Applying Kirchoff's rule for junctions A and B we can write: (5) 1 =13 +14 i5 =2 +i3 (6)

Substituting the currents and making some equivalent mathematical transformations, the following system of two second order differential equations can be obtained:

where

212 | 2+ k1u2 -k2ul =0 I 12

L=

k2=

L(I

K 2)'

2

K

By adding and subtracting equations (7) and (8), and using substitutions (11) and (12), the equations in the system can be decoupled [1]. U+U2 =z1

1

K

(10)

K

LC(1-K 2)

(24)

=0.6

(25)

1.2

0.8

u2

(1)

~~~(12)

81-82 Z= 2

°Q 0

As a result, two independent homogeneous second order-0 differential equations are obtained: z1 + (k1 - k2)z1 =0,'

(13)

+k2z2= 0.

(14)

Z2(l

1

(23)

-.

''\

'' '

VV 0

2

1

3

Cycles

Figure 3. Normalized primary (ul) E

and secondary (u2) voltages

The solutions of (13) and (14) are shown below:

z12Acos(co1t2y1),

(15)

III. PARAMETRIC STUDY

Z= Bcos(co2t +y2),

(16)

In order to be able to evaluate the behavior of the transformer, two criteria have been chosen to be maximized: output peak voltage and negative to positive amplitude ratio. The equivalent circuit was simulated with PSpice. All values are normalized to the critical coupling case (K =0.6), and primary and secondary sides covered tuned to toone and the the influence same frequency. Two oncases Inwere check of damping the

where A, B, ql and y2 are integration constants and 1

k

I2LC(1 Bcos(

(17)

k1 ) t

02

0t)2 =-/ LC(11 - K)

(16) *t(18)

Using the initial conditions the final solutions for z,

andOz2 canbefound:

Z= u cos(a)lt) 1J2C= U cos(ma2t)

(19)

(18) (20)

circuit behavior. For the first case the damping factor was

small (

.-l1

;

L

V. SIMULATION AND EXPERIMENTAL

°55 Coupling 1A0.40 Factor K

RESULTS

Microwave Studio® model of 20 turns spiral coil, with outer diameter 370mm, span 3.6mm, made out of strip

Figure 5. Negative to positive peak voltage ratio of secondary voltage as function of coupling and primary capacitance with 30% loss factor

conductor 5mm wide and 0.9mm thick, is shown in Figure 7.

The graphs show that the negative to positive peak voltage ratio is more sensitive to the parameters change than the peak amplitude. Table 1. summarizes the parametric study results. Table 1. Summarized results of the parametric study