Hysteresis and Frequency Tunability of Gyrotrons

Abstract

We present the first devoted theoretical and experimental study of the hysteresis phenomenon in relation to frequency tunability of gyrotrons. In addition, we generalize the theory describing electron tuning of frequency in gyrotrons developed earlier to arbitrary harmonics. It is found that theoretical magnetic and voltage hysteresis loops are about two times larger than experimental loops. In gyrotrons whose cavities have high quality factors, hysteresis allows one only little to broaden the frequency tunability range.

Appendix

1.1 Electron Tuning of Frequency

To estimate the role of hysteresis in frequency tunability, we use the formalism of electron tuning of frequency developed in [7], generalizing it at the same time to arbitrary harmonics. According to this formalism, for a constant operating current, the frequency shift due to change of electron beam parameters is given by the expression

$$ \frac{\delta f}{f}=\frac{1}{2{Q}_{tot}}\cdot \frac{\partial \kappa }{\partial \varDelta}\cdot \delta \varDelta $$

(1)

where

$$ \kappa =-\frac{\mathrm{Re}\left(\chi \right)}{\mathrm{Im}\left(\chi \right)} $$

(2)

The dielectric susceptibility of the electron beam χ is given by the expression

$$ \chi =\frac{1}{\pi F}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left[{\displaystyle \underset{0}{\overset{\zeta_{out}}{\int }}{p}^n\left(\zeta, \vartheta \right){\overline{f}}^{*}\left(\zeta \right)d\zeta}\right]d{\vartheta}_0} $$

(3)

The imaginary part of susceptibility is normalized as follows:

$$ \mathrm{I}\mathrm{m}\left(\chi \right)=\frac{1}{I} $$

(4)

where I is the dimensionless current

$$ I=0.238\cdot {10}^{-3}{I}_A{Q}_{tot}\frac{J_{m\pm n}^2\left(\frac{\nu {R}_b}{R_{cav}}\right)}{\left({\nu}^2-{m}^2\right){J}_m^2\left(\nu \right)}\cdot \frac{\lambda }{L}\left(\frac{n^n}{2^nn!}\right)\frac{\beta_{\perp}^{2\left(n-3\right)}}{\gamma_{rel}} $$

(5)

Here, I _A is the current in amperes, m is the azimuthal index of the mode, n is the harmonic number, ν is the eigenvalue of the mode, R _b is the electron beam radius, R _cav is the cavity radius, λ is the wave length, L is the cavity length, β _⊥ is the normalized perpendicular velocity, and γ _rel is the relativistic factor.

The normalized electron perpendicular momentum p can be found by solving the following differential equation:

$$ \frac{dp}{d\zeta }+\frac{i}{n}p\left(\varDelta +{\left|p\right|}^2-1\right)=\overline{if}{\left({p}^{*}\right)}^{n-1}F, $$

(6)

where

$$ \zeta =\left({\beta}_{\perp}^2{\omega}_{cyc}/2{\beta}_{||}c\right)z $$

(7)

is the dimensionless longitudinal coordinate

$$ \varDelta =\frac{2}{\beta_{\perp}^2}\left(\frac{\omega_r-n{\omega}_{cyc}}{\omega_r}\right) $$

(8)

is the frequency mismatch. The dimensionless amplitude of stationary oscillations F is given by the expression

$$ F={\left(0.47\cdot {10}^{-3}\cdot {Q}_{tot}\cdot {P}_{out}\frac{J_{m\pm n}^2\left(\frac{\nu }{R_{cav}}\cdot {R}_b\right)}{\gamma_{rel}U{\eta}_{el}{\beta}_{||}{\beta}_{\perp}^{2\left(2-n\right)}\left({\nu}^2-{m}^2\right){J}_m^2\left(\nu \right){\displaystyle \underset{0}{\overset{\zeta_{out}}{\int }}{\left|\overline{f}\left(\zeta \right)\right|}^2d\zeta }}\right)}^{\frac{1}{2}}\cdot \left(\frac{n^{n+1/2}}{2^nn!}\right) $$

(9)

where P _out is the output power, U is the voltage, ${\eta}_{el}=1/\left(1+{\left(\frac{\beta_{\left|\right|}}{\beta_{\perp }}\right)}^2\right)$. The field profile $\overline{f}\left(\zeta \right)$ which enters into Eqs. (6) and (8) can be found by solving the nonuniform string equation

$$ \frac{d^2\overline{f}}{d{\zeta}^2}+{\gamma}^2\overline{f}=0 $$

(10)

where

$$ \gamma =2\frac{\beta_{||}}{\beta_{\perp}^2}\frac{c}{\omega}\sqrt{\frac{\omega^2}{c^2}-\frac{\nu^2}{R^2}} $$

(11)

Equation (6) has to be suplemented by the initial condition

$$ p\left({\zeta}_{in}\right)= \exp \left(i\frac{\vartheta_0}{n}\right)\kern2em \left(0\le {\vartheta}_0<2\pi \right) $$

(12)

and Eq. (9) by the boundary conditions

$$ d\overline{f}/d{\zeta}_{|\zeta =0}=i\gamma \overline{f} $$

(13)

$$ d\overline{f}/d{\zeta}_{|\zeta ={\zeta}_{out}}=-i\gamma \overline{f} $$

(14)

1.2 Extension of Tunability Range due to Hysteresis in the Case of Magnetic Tuning

As seen in Fig. 6, the hysteresis loop lies between B ≈ 8.50 T and B ≈ 8.51 T. Corresponding values of ∆ and κ are given in Table 2.

Table 2 ∆ and κ as a function of magnetic field

Thus,

$$ \frac{\partial \kappa }{\partial \varDelta }=\frac{-0.84+0.93}{0.141-0.135}=15 $$

Due to hysteresis, the frequency pulling is

$$ \delta f=460.33\cdot \frac{1}{2\cdot 16217}\cdot 15\cdot \left(0.141-0.085\right)\cdot 1000=11.9\; MHz $$

This value is significantly smaller than the theoretical prediction shown in Fig. 3. For example, the increase of magnetic field from 8.58 to 8.59 T increases the frequency by ∼140 MHz.

1.3 Extension of Tunability Range due to Hysteresis in the Case of Voltage Tuning

As seen in Fig. 6, the hysteresis loop lies between U ≈ 19 kV and U ≈ 19.3 kV. Corresponding values of ∆ are given in Table 3.

Table 3 ∆ as a function of voltage

Due to hysteresis, the frequency pulling is

$$ \delta f=460.33\cdot \frac{1}{2\cdot 16217}\cdot 15\cdot \left(0.112-0.085\right)\cdot 1000=5.7\;\mathrm{MHz} $$

This value also is significantly smaller than the theoretical prediction shown in Fig. 5. For example, the variation of voltage between 15 and 15.3 kV results in frequency variation on the order ∼100 MHz.

1.4 Triode Gun

In this gyrotron, a triode gun is used with the following parameters: the magnetic compression b = 43.6, the distance between the cathode and the anode d = 3.8 mm, the cathode angle θ _c  = 39.1°, and the cathode radius R _c  = 6 mm. Although the electron gun has been carefully designed with the E-Gun code to form a laminar electron beam, the velocity pitch factor is evaluated with an analytic expression in the simulation calculations. The dimensionless perpendicular electron velocity is given by the standard expression:

$$ {\beta}_{\perp }=\frac{1}{\gamma_{rel}c}\cdot \frac{b^{3/2}}{B}\cdot \frac{{\left( \cos {\theta}_c\right)}^2}{R_c \ln \left[1+\left(d \cos {\theta}_c\right)/{R}_c\right]}\cdot {U}_{\mod{}} $$

(15)

Here, γ _rel  = 1 + U/511 B is the magnetic field in the cavity, and U _mod is the modulation voltage. In this particular gun, U _mod = 8.7 kV.