**3. Basic principle of gyrotron**

In a gyrotron, the electron beam, which is normally in the shape of a thin hollow cylinder, is injected into a region with strong axial magnetic field and passed through a cylindrical cavity or waveguide region containing an electromagnetic wave with an azimutal component of electric field [1–3]. The rotational velocities of the electrons are normally 1.2 to 2 times the axial velocity. So, majority of the electron energy is rotational.

Because the magnetic field is very large, the orbit diameter for the electrons is very small. As a result, the thickness of the hollow electron beam is several times the diameter of the electron orbit as shown in **Figure 4**, and in effect, the hollow electron-beam contains a large number of small beams, referred as beamlets [2, 6]. **Figure 4** shows the thickness of the hollow electron-beam as twice the diameter of beamlet.

The basic operating mechanism of gyrotron can be explained by considering the interaction of a single beamlet of electrons with the electric field. In **Figure 5** it is assumed that electrons in a single beamlet are initially uniformly distributed along a single helical path prior to interaction with the RF electric field. The electrons are assumed to rotate in the counter clockwise direction as they move through the RF field. The rotational frequency of electrons is the cyclotron frequency, which is given by

$$
\omega\_c = \frac{e}{m} B\_0 \tag{1}
$$

Where, *B0* is the externally applied axial magnetic field, *e* is the electron-charge, and m is the mass of the electron. In a gyrotron, the velocity of electrons is a significant fraction of the free-space velocity of light, hence, the relativistic effect comes into play. So, the mass of the electrons is significantly greater than their rest-mass, i.e.,

**Figure 4.** *Gyrotron cross section showing electron trajectories.*

*Gyrotron: The Most Suitable Millimeter-Wave Source for Heating of Plasma in Tokamak DOI: http://dx.doi.org/10.5772/intechopen.98857*

**Figure 5.** *Bunching of electrons in a Gyrotron.*

$$m = \gamma m\_0 \tag{2}$$

Where *m0* is the rest-mass of the electron and *γ* is the relativistic mass factor, which is given by

$$\gamma = \frac{1}{\sqrt{1 - \left(v/c\right)^2}}\tag{3}$$

Where, *v* is the velocity of electron and *c* is the free space velocity of light. The value of *γ* increases when an electron is placed in an accelerating field.

The radius of the gyrating orbit, alternatively known as Larmor radius (*rL*), may be obtained by the following equation

$$
\rho\_{\mathcal{C}} \mathbf{r}\_L = \mathbf{v}\_t \tag{4}
$$

Where, *vt* is the transverse velocity of the electron-beam.

Now, referring again to **Figure 5(a)**, when the electric field is such that it tends to accelerate electrons (top of the orbits), the electron mass is increased and so the cyclotron frequency (*ωc*) decreases for these electrons. Similarly, when the electric field tends to decelerate electrons (bottom of the orbits), the electron mass is decreased, and so *ω<sup>c</sup>* is increased. Since the rate of rotation is decreased for some electrons and is increased for others, orbital bunching occurs if the electrons are permitted to drift, as indicated at the right-hand side of **Figure 5(a)**.

If the cyclotron frequency (ωc) is somewhat lower than the frequency of the electromagnetic wave (ω), then the position of the bunches along the helical orbit is delayed with respect to the phase of the applied field as indicated in **Figure 5(b)**. Hence, the bunched electrons face a decelerating field and give-up their kinetic energy to the field. As the electron bunches rotate in near synchronism with the alternating RF-field, they continue to give-up energy on each half-cycle of rotation.

The interaction that has just been described for the electrons in a single beamlet in a gyrotron also takes place in the other beamlets. Thus, the electron distribution becomes as indicated in **Figure 6**. As the direction of the electric field alternates, the direction of motion of electron also alternates, and so the electrons throughout the transit period of the electron-beam give-up energy on each half cycle of operation. This is how the beam-wave interaction happens.

**Figure 6.** *Electron motion in relation to direction of electric field in a gyrotron.*
