1. Dynamically controlled devices. Klystron.

   1. Calculate the drift angle at which an electron entering the maximum accelerating voltage in the first resonator will overtake an electron accelerating by π-radians. The calculation is carried out in the kinematic approximation. In the calculations, assume a constant accelerating voltage (N_group + 1)[kV], gap length (interaction area) (N_group + 1)/ N_student [mm], depth of velocity modulation 0.01* N_student, operating frequency

N_group +3[GHz]. The problem is solved using normalized parameters: modulation depth, flight angle, grouping parameter.

2. Analyze how the grouping process will change when taking into account the space charge forces if the beam current is (N_group *100)[mA], and the beam diameter N_group[mm]? At what distance from the center of the modulating gap will the maximum grouping be achieved? Please comment on the result.

Given:

Constant accelerating voltage

Gap length

Depth of velocity modulation

Operating frequency

The beam current

The beam diameter

2. Solving:

   1. Параметр группировки

The electron in question is delayed by a phase equal to π. To catch up with the next electron, it must pass through this phase. The bunching parameter is a value that reflects the largest change in the electron phase. Let's consider the extreme case in which the electron catches up with the next one at the end of the drift space. The bunching parameter: [7],[8]

Then the angle of flight of electrons in the drift space:

2. Изменение группировки

To find the change in grouping in a klystron, we use the formula: [8]

Where is the reduction parameter:

Plasma oscillation frequency:

Electron beam density:

S – cross-section of an electron beam. Let the cross-section have the shape of a circle:

Then:

Changing the grouping in a klystron taking into account repulsion:

Answers:

Angle of flight:

Changing grouping: 1.25 балл

Task 4

1. Dynamically controlled devices. Bwt.

Calculate the parameters of a spiral slow-wave structure (SWS) for a BWT.

Determine the following quantities and parameters:

• Electron beam current and power – I₀, P₀

• Electron beam velocity

• Spiral pitch SWS – p, at which the conditions for beam and electromagnetic wave synchronism are met

• Spiral SWS length that provides a given gain value

• Draw and explain the typical frequency response and amplitude characteristic of a BWT. How will these characteristics change with changing coupling resistance? Draw them on the same graph.

Given:

Powe gain value

Beam accelerating voltage

Electron beam perveance

The beam radius

The ratio of the beam radius to the helix radius

Frequency

Coupling resistance

2. Solving:

From the meaning of perveance:

Power of electronic beam:

The gain is calculated using the formula [1]

Where: [1]

Then the length of the slow-wave system, expressed in wavelengths:

The speed of an electron in the field of the system:

Synchronicity occurs under the condition v_p = v₀, then the system slowdown coefficient [9]

Radius of the spiral:

Then the pitch of the spiral slow-motion system can be obtained from the formula: [9]

Calculating with software MathCad 15:

Step of the spiral of the slowing system: p = 0.006699 m