**Subject :**Microwave Engineering

**Unit :**Microwave tubes

## Linear magnetron

The schematic diagram of a linear magnetron is shown in Figure . In the linear magnetron as shown in Figure, the electric field E_{x} is assumed in the positive x direction and the magnetic flux density B, in the positive z direction. The differential equations of motion of electrons in the crossed-electric and magnetic fields can be written as

**Figure: Schematic diagram of a linear magnetron**

..(1)

..(2)

..(3)

In general, the presence of space charges causes the field to be a nonlinear function of the distance x, and the complete solution of Eqs. (1) through (3) is not simple. Equation (2), however, can be integrated directly. Under the assumptions that the electrons emit from the cathode surface with zero initial velocity and that the origin is the cathode surface, Eq. (2) becomes

..(4)

Equation (4) shows that, regardless of space charges, the electron velocityparallel to the electrode surface is proportional to the distance of the electron from

the cathode and to the magnetic flux density Bi . How far the electron moves from the cathode depends on B, and on the manner in which the potential V varies with x, which in turn depends on the space-charge distribution, anode potential, and electrode spacing. If the space charge is assumed to be negligible, the cathode potential zero, and the anode potential Vo, the differential electric field becomes

.(5)

where Vo = anode potential in volts

d = distance between cathode and anode in meters

Substitution of Eq. (5) into Eq. (1) yields

..(6)

Combination of Eqs. (4) and (6) results in

..(7)

Solution of Eq. (7) and substitution of the solution into Eq. (4) yield the following equations for the path of an electron with zero velocity at cathode (origin point) as

..(8)