All units are in the SI

I'm doing my task from vacuum electronics - for the given klystron with a toroidal resonator[ug]


enter image description here

I have to find the $h$ parameter, that will satisfy given conditions:



$$electrons\space stream\space diameter\space 2a=6 mm=0,006 m$$

that can be found from the given toroidal resonator eigenfrequency formula, considering, $$b=2a$$ and \$c\$, as far, as I understand equals \$0,73\$: $$f_{\text{res}}=\frac{c}{\pi a \sqrt{\frac{2 h}{d} \ln \frac{b}{a}}}$$

\$d\$ can be found from the next equation, considering \$\theta=0,75\pi\$

$$\theta_{i}=\omega \tau=2 \pi f\left(d / v_{0}\right) \quad v_{0}=5,93 \times 10^{5} \sqrt{U_{\text{res}}}=9376153,262\space mps$$

When I put all this stuff in one, I got

$$d=v_0\dfrac{0,75\pi}{2\pi f}=\dfrac{0,75*3,14*9376153,262 }{2*3,14*10000000000}=0,000351606\space m$$ that seems correct

$$h=\dfrac{\dfrac{d(\dfrac{c}{f\pi 0,003})^2}{ln2}}{2}=\dfrac{\dfrac{0,000351606(\dfrac{0,73}{10000000000*3,14*0,003})^2}{ln2}}{2}=1,5*10^{-20}\space m$$ that is obviously wrong

If I put \$d\$ and \$h\$ in $$f_{\text{res}}=\frac{c}{\pi a \sqrt{\frac{2 h}{d} \ln \frac{b}{a}}}$$, I get 10000000000, that means, that I've calculated everything correctly.

I've been checking all steps, all formulas, maybe I had missed something, but no, seems that the problem is in resonator frequency formula - it's wrong. I've tried to find it in the internet but get failed.

Please, provide a specified toroidal resonator frequency formula, or, if the formula, given in my task correct, point me, where am I wrong.


enter image description here I got the next solution of my issue


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