Microwave Generators

A multi-cavity klystron is a linear-beam (O-type) microwave power amplifier. Three or more resonant cavities are placed along an electron beam to get high gain and high power.

 cathode  input    idler     idler    output
  gun    cavity   cavity    cavity   cavity  collector
 |>==== [ C1 ] == [ C2 ] == [ C3 ] == [ C4 ] ====|
  beam ->  RF in                        RF out
       <---- drift spaces with focusing B ---->

Working

1. The electron gun produces a uniform beam accelerated by V₀ to v₀ = √(2eV₀/m).
2. The weak RF input at the buncher (first) cavity velocity-modulates the beam. Some electrons are sped up and some are slowed down.
3. In the drift space the fast electrons catch up with the slow ones, which forms bunches (density modulation).
4. Each intermediate (idler) cavity is excited by the bunched beam. It builds up a larger gap voltage and modulates the beam again, so the bunching becomes tighter. Each cavity adds about 15–20 dB of gain.
5. The catcher (output) cavity is placed where the bunches are tightest. Its retarding field takes kinetic energy from the bunches, and amplified RF is coupled out.
6. The spent beam is collected by the collector.

Features

• Gain of about 40–60 dB with 3–5 cavities. Peak power up to tens of MW (pulsed). Efficiency of about 35–50%.
• Synchronous tuning (all cavities at one frequency) gives the highest gain but narrow bandwidth. Stagger tuning widens the bandwidth at the cost of gain.
• An axial magnetic field (solenoid or PPM) keeps the beam focused.

Applications: UHF TV transmitters, radar transmitters, particle accelerators (linacs), and satellite uplink stations.

A cavity magnetron is a cross-field (M-type) high-power microwave oscillator. Electrons move under a radial DC electric field and an axial DC magnetic field that is perpendicular to it, and give their energy to the RF fields of resonant cavities.

Construction

• A cylindrical, heated cathode sits at the centre.
• Around it is a copper anode block with N cavities (for example 8), cut as hole-and-slot, vane or rising-sun shapes. The cavities open into the cathode–anode interaction space.
• A strong permanent magnet gives an axial magnetic field B₀. A DC voltage V₀ is applied between anode and cathode.
• Straps join alternate segments to keep the oscillation in the π-mode. A loop in one cavity couples out the power.

          anode block, N cavities
        ___   ___   ___   ___
       /   \_/   \_/   \_/   \
      |   interaction  space   |
      |       ( cathode )      |  B0 axial
      |   spokes rotate  ->    |  (out of page)
       \___/ \___/ \___/ \___/
              output loop -> RF out

Working principle

1. Without B: electrons go straight from cathode to anode.
2. With B: the force F = −e(E + v × B) bends their paths into cycloid-like curves. At the Hull cut-off field, electrons just graze the anode and return: B₀c = √(8mV₀/e) / [b(1 − a²/b²)], where a = cathode radius and b = anode radius. The magnetron is run slightly above this field.
3. Any noise excites the cavities. The RF field fringing across the slots forms a wave that travels around the interaction space.
4. Phase focusing: electrons that meet a retarding RF field give energy to the wave, move outward and finally reach the anode. Electrons that meet an accelerating field gain energy and return quickly to the cathode, which removes them from the interaction. This sorting groups the useful electrons into spokes (like a wheel), which rotate in step with the RF field (Hartree condition).
5. In the π-mode, adjacent cavities are 180° apart and there are N/2 spokes. Each spoke passes a slot when the field there is retarding, so energy is fed to the cavities each cycle and oscillations are sustained.

Features: efficiency of about 40–70%, peak powers of kW to several MW in pulsed operation, compact, frequency fixed mainly by the cavity size.

Applications: pulsed radar transmitters, microwave ovens (2.45 GHz), industrial heating, and linear accelerators.

A backward wave oscillator (BWO), also called a carcinotron, is a linear-beam (O-type) slow-wave tube that works as an electronically tunable microwave oscillator. Its beam interacts with a backward space harmonic: the wave's group velocity (energy flow) is opposite to the beam direction, while the phase velocity of that harmonic is along the beam.

Construction

• An electron gun gives a beam focused by an axial magnetic field.
• A slow-wave structure (helix, interdigital line or folded line) runs along the beam.
• The RF output is taken from the gun end of the structure. The collector end is matched with an attenuator or termination. There is no RF input.

 RF out                          matched load
   ^                                 |
 gun |>=== slow-wave structure ======| collector
       beam  ----->  (electrons)
       energy <----- (group velocity)

Working

1. Noise sets up a wave on the structure. The beam is velocity-modulated by the backward space harmonic, whose phase velocity is in step with the electrons.
2. Bunches form as the beam moves forward. The bunched beam gives energy to the wave, and that energy travels backward toward the gun.
3. Near the gun end this energy modulates the fresh beam again. This is built-in feedback, so oscillations start when the beam current exceeds a start-oscillation value.
4. Since the synchronous condition depends on beam velocity, the frequency is tuned by the beam voltage, roughly v₀ ∝ √V₀.

Features: a wide electronic tuning range (up to an octave), output of mW up to about 1 W (O-type), low efficiency (about 5–20%).

Applications: swept-frequency signal generators, local oscillators, broadband jammers, and mm-wave sources.

Proof that TM₁₁ is the lowest TM mode

For TM modes in a rectangular waveguide (a × b, a > b), H_z = 0 and E_z must satisfy the wave equation with E_z = 0 on all four walls (tangential E vanishes on a perfect conductor). Solving by separation of variables:

E_z(x,y,z) = E0 sin(mπx/a) sin(nπy/b) e^(-jβz)
kc² = (mπ/a)² + (nπ/b)²
fc  = (c/2) √[(m/a)² + (n/b)²]

• If m = 0, then sin(0) = 0 and E_z = 0 everywhere. All other transverse fields come from E_z (E_t, H_t ∝ ∇_t E_z), so the whole field is zero and no TM₀ₙ mode exists.
• If n = 0, likewise E_z = 0, so no TMₘ₀ mode exists.
• So both m ≥ 1 and n ≥ 1. The smallest values are m = n = 1, which give the lowest TM cut-off:

fc(TM11) = (c/2) √[(1/a)² + (1/b)²]

Example: for WR-90 (a = 2.286 cm, b = 1.016 cm), fc(TM₁₁) ≈ 16.2 GHz, while fc(TE₁₀) = 6.56 GHz. So TM₁₁ is the lowest TM mode, but TE₁₀ is still the dominant mode of the guide.

Circular-beam cross-field amplifier (reentrant CFA / amplitron)

A cross-field amplifier (CFA) is an M-type tube in which DC E and B fields are at right angles. The circular, reentrant type (the amplitron) is built like a magnetron, but the anode is a slow-wave structure of coupled cavities (vanes) with an RF input and an RF output.

        RF in       RF out
          \           /
       _/ \_/ \_/ \_/ \_   cavity (vane)
      /   circular beam   \  slow-wave anode
     |  ( cathode / sole ) |  B axial
      \_   spokes  ->     _/
        \_/ \_/ \_/ \_/
     drift space between in and out

Working

1. A cylindrical cathode (or sole) and the anode give a radial E field, and a magnet gives an axial B field. Electrons drift around in a circular path at the E × B drift velocity v = E/B.
2. The RF input fed into the cavity structure travels around as a slow wave. Its phase velocity is set close to the electron drift velocity (synchronous condition).
3. As in a magnetron, phase focusing makes the electrons in retarding RF fields give potential energy to the wave and move toward the anode. Electrons in accelerating fields are returned to the cathode. Spokes form and grow with the wave.
4. The wave grows as it travels from input to output, and the amplified RF is taken out. A drift region without the slow-wave structure stops the RF from feeding back, so the tube amplifies instead of oscillating.

Features: high efficiency (about 40–70%), moderate gain (about 10–20 dB), broadband (about 10%), high peak power. Used as power amplifiers in radar and satellite transmitters.

Bunching effect

Bunching is the grouping of electrons in a beam into dense packets (bunches) at regular time intervals. A uniform beam passes an RF gap and is velocity-modulated: electrons crossing during the positive half-cycle speed up and those crossing during the negative half-cycle slow down. In the following drift space the faster electrons catch up with the slower ones, so the current becomes density-modulated at the RF frequency. Bunches give energy to an RF field when they pass through a retarding field. This is the basis of klystrons and TWTs (O-type tubes).

Field effect (cross-field effect)

The cross-field effect is the motion of electrons under a DC electric field E and a DC magnetic field B that are perpendicular to each other. The Lorentz force F = −e(E + v × B) makes the electrons move in cycloidal paths that drift at velocity E/B, perpendicular to both fields. The electrons interact with an RF wave moving at that drift speed and give it their potential energy. This is used in magnetrons and cross-field amplifiers (M-type tubes).

Cavity magnetron: a high-power oscillator using both effects

The cavity magnetron uses the cross-field effect (radial E, axial B) to move electrons around the cathode. Phase focusing then groups them into spokes, which is a form of bunching.

       anode with N cavities (N = 8)
       _/ \_/ \_/ \_/ \_
      /    \ spoke  /    \
     |   --( cathode )--   |   B0 axial
     |    /  spoke   \     |   E radial
      \_/ \_/ \_/ \_/ \_/
         output loop -> RF

Working

1. The heated cathode emits electrons. The DC anode voltage V₀ gives a radial E field, and the magnet gives an axial B₀ field above the Hull cut-off value B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. Without RF, electrons curve back to the cathode.
2. Noise excites the cavities. In the π-mode, adjacent cavities are 180° out of phase, and the fringing RF fields across the slots form a wave rotating around the interaction space.
3. Field effect: electrons move around the cathode at the E × B drift speed. When this matches the angular velocity of the RF wave (Hartree condition), the electrons keep a constant phase relative to the wave.
4. Bunching (phase focusing): electrons in a retarding RF field lose energy, move outward and give energy to the cavities. Electrons in an accelerating field gain energy and are sent back to the cathode. So the useful electrons gather into N/2 rotating spokes.
5. Each spoke passes a slot just when its field is retarding, so energy is fed to the cavities every cycle and the oscillation is sustained. Power is taken out with a coupling loop. Strapping keeps the π-mode separate from other modes.

Features: efficiency of about 40–70%, kW to MW pulsed power. Used in radar transmitters and microwave ovens.

A cross-field (cavity) magnetron is an M-type tube in which a radial DC electric field and an axial DC magnetic field act at right angles on the electrons. The electrons give their potential energy to a travelling RF wave carried by the cavities of the anode.

Mode and number of cavities for 60° phase difference

For an anode with N cavities, the phase difference between adjacent cavities in mode n is:

φ = 2πn / N
60° = 360°·n / N   =>  N = 6n
N = 6  -> n = 1  (one spoke)
N = 12 -> n = 2  (two spokes)

So a 6-cavity anode in its n = 1 mode (or a 12-cavity anode in its n = 2 mode) has 60° between adjacent cavities. The RF field pattern repeats every 360°/60° = 6 cavities. The number of electron spokes equals n. (The usual π-mode has 180° between cavities and N/2 spokes.)

   6-cavity anode, 60° between cavities
          0°
     300°  _   60°
       \ / \ /
   ----( cath )----   B0 axial (out of page)
       / \_/ \        E radial
     240°     120°
         180°
   one spoke rotating at ω (angular)

Working principle

1. The heated cathode emits electrons. The anode voltage V₀ gives a radial E field. An axial B₀ above the Hull cut-off value, B₀c = √(8mV₀/e) / [b(1 − a²/b²)], bends the electron paths so that, without RF, they curve back to the cathode.
2. Noise excites the cavities. In this mode each cavity's fringing field is 60° behind its neighbour, so the RF pattern forms a wave that rotates around the cathode at angular velocity ω/n.
3. The DC fields make electrons drift around the cathode at the E × B speed. V₀ and B₀ are chosen (Hartree condition) so that this drift matches the wave's angular velocity.
4. Phase focusing: electrons that meet a tangential retarding RF field lose energy to the wave, slow down and move outward step by step to the anode. Electrons that meet an accelerating field gain energy and are returned to the cathode, which removes them early.
5. The useful electrons form n rotating spokes (one spoke for N = 6). Each spoke passes the cavity slots in step with the retarding phase, so energy flows into the cavities every cycle. The oscillation builds up and RF is taken out with a coupling loop.

Remarks

• Modes other than π-mode are close in frequency. Strapping or a rising-sun anode is used to choose and hold the wanted mode.
• Efficiency is about 40–70%, because the electrons hit the anode with little leftover kinetic energy.
• If an RF signal is injected into the cavity chain (as in a cross-field amplifier), the same interaction amplifies it.

Bunching effect

Bunching is the grouping of electrons in a beam into dense packets. When a uniform beam meets an RF field it is velocity-modulated: electrons in the accelerating half-cycle speed up and those in the retarding half-cycle slow down. As the beam drifts on, fast electrons catch up with slow ones, so the beam becomes density (current) modulated. Bunches that pass through a retarding field give their kinetic energy to the RF wave, which amplifies it.

Traveling Wave Tube (TWT)

A TWT is a linear-beam (O-type), broadband microwave amplifier. The beam interacts continuously with an RF wave travelling on a slow-wave structure (usually a helix).

Construction

       RF in                      RF out
         |                          |
 gun |>==[ helix slow-wave structure ]===| collector
  beam ->  ~~~~~~[atten]~~~~~~~~
        magnet / PPM focusing along tube

• Electron gun: cathode, focusing electrode and anode give a thin beam at velocity v₀ = √(2eV₀/m).
• Helix: a wire spiral that slows the RF wave. Its axial phase velocity is v_p ≈ c·p / √(p² + (πd)²) ≈ c·p/(πd), where p = pitch and d = diameter. This makes v_p close to the beam velocity.
• Attenuator near the middle of the helix absorbs reflected waves and stops oscillation.
• Input and output couplers (coax or waveguide) at the two ends of the helix.
• Focusing magnet: a solenoid or periodic permanent magnet (PPM) keeps the beam narrow.
• Collector gathers the spent beam. It is often depressed (lower voltage) to improve efficiency.

Operation

1. The input RF travels along the helix with an axial E field whose phase velocity is about equal to the beam velocity. The beam is set slightly faster than the wave (v₀ ≈ v_p).
2. Electrons in accelerating regions speed up and those in retarding regions slow down, so bunches form continuously along the tube.
3. Because the beam is slightly faster, the bunches drift into the retarding phase of the wave and give energy to it. The larger wave bunches the beam more strongly, so the wave grows exponentially along the helix.
4. The amplified signal is taken out at the output end. The attenuator absorbs backward waves but passes the bunched beam, which carries the signal on.
5. The power gain is about A_p = −9.54 + 47.3·C·N dB, where C is the gain parameter and N is the helix length in wavelengths.

Features: high gain (30–60 dB), very wide bandwidth (up to an octave, since the helix has no resonant cavity), low noise, efficiency of about 20–40%. Used in satellite transponders, radar, ECM and microwave links.

Cross-section of a magnetron with 45° between adjacent cavities

In mode n, the phase difference between adjacent cavities of an N-cavity anode is φ = 2πn/N. For φ = 45°, N = 8n, so an 8-cavity anode in the n = 1 mode gives 45°. The pattern repeats every 8 cavities, so there is one rotating spoke. (A 16-cavity anode with n = 2 gives two spokes.)

   8 cavities, phase of each cavity's RF field
              0°
       315° _/ \_ 45°
          /       \
    270° | (cathode) |  90°     B0 axial
          \  spoke  /          E radial
       225° \_ _/ 135°
             180°
   RF in -> cavity 0°      RF out <- cavity 315°

Functioning as a power amplifier

A magnetron is normally an oscillator. It works as a power amplifier (as in a cross-field amplifier, amplitron type) when an RF input is injected into the cavity chain and the drive conditions are set so that it does not oscillate by itself.

1. Fields: a heated cathode at the centre and the anode block give a radial DC field E (anode voltage V₀). A magnet gives an axial field B₀, slightly above the Hull cut-off B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. Electrons drift around the cathode at the E × B speed.
2. Input: the weak RF signal is coupled into one cavity. The cavities are coupled through the interaction space, so the signal moves from cavity to cavity with a 45° phase step. This forms a slow wave that rotates around the cathode at angular velocity ω/n.
3. Synchronism: V₀ and B₀ are adjusted (Hartree condition) so that the electron drift velocity equals the wave's phase velocity.
4. Phase focusing and energy transfer: electrons in the retarding phase of the RF field give potential energy to the wave and move toward the anode. Electrons in the accelerating phase gain energy and return to the cathode. The useful electrons form a spoke that stays locked to the wave and keeps giving it energy.
5. Growth and output: the wave grows as it travels from the input cavity to the output cavity. The amplified signal is coupled out with a loop. A drift section (or a non-resonant gap) between output and input stops the amplified signal from reaching the input again, so no self-oscillation occurs.

Performance: efficiency of about 40–70%, gain of about 10–20 dB, bandwidth of about 10%, high peak power. Used in radar transmitters.

Bunching effect

Bunching is the grouping of electrons into dense packets at regular intervals. In O-type tubes (klystron, TWT) it comes from velocity modulation followed by drift: fast electrons catch up with slow ones. In magnetrons the same idea appears as phase focusing. Electrons in favourable (retarding) RF phase stay in the interaction, while unfavourable ones are removed, so electrons gather into spokes. The bunched electrons give energy to the RF field in step with it.

High-frequency limits of conventional tubes

Conventional tubes (triodes, tetrodes, pentodes) lose output and gain at microwave frequencies because of three main effects:

1. Inter-electrode capacitance: small capacitances (C_gk, C_gp, C_pk) exist between electrodes. Their reactance X_C = 1/(ωC) falls as frequency rises, so at GHz frequencies they short-circuit the input and output. They also give feedback through C_gp and limit the gain–bandwidth product (GBW ∝ g_m / C).
2. Lead inductance: the leads connecting the electrodes to the outside have inductance L. Their reactance X_L = ωL rises with frequency and drops part of the signal across the leads. Lead L with inter-electrode C forms a resonant circuit that sets an upper frequency limit f ≈ 1/(2π√(LC)). Cathode lead inductance also gives negative feedback that raises the input conductance.
3. Transit-time effect: electrons need a finite time τ = d/v to cross from cathode to grid or anode. When τ is comparable to the RF period, the grid voltage changes while electrons are still in transit. The plate current then lags the grid voltage, the grid takes power from the source, and the input conductance rises roughly as g ∝ g_m ω²τ². The input is heavily loaded and efficiency falls.

Making the tube smaller reduces these effects, but it also cuts the power and heat-handling ability. That is why microwave tubes use transit time on purpose, as in klystrons, TWTs and magnetrons.

Traveling Wave Tube (TWT)

A TWT is an O-type broadband microwave amplifier. The electron beam interacts continuously with an RF wave on a slow-wave helix.

       RF in                      RF out
         |                          |
 gun |>==[ helix slow-wave structure ]===| collector
  beam ->  ~~~~~~[atten]~~~~~~~~
        solenoid / PPM magnet

Construction: an electron gun, a helix slow-wave structure (axial phase velocity v_p ≈ c·p/(πd), about c/10), a lossy attenuator in the middle, input and output couplers, a focusing magnet and a collector.

Working

1. The input RF on the helix moves along the axis at a phase velocity close to the beam velocity v₀ = √(2eV₀/m). The beam is kept slightly faster than the wave.
2. The axial RF field speeds up and slows down electrons, so the beam is velocity-modulated and bunches form continuously along the helix.
3. Since the beam is a little faster, the bunches sit in the retarding phase and give kinetic energy to the wave. The wave grows, bunching becomes stronger, and the signal rises exponentially.
4. The attenuator absorbs reflected waves (preventing oscillation) while the bunched beam carries the signal past it.
5. The amplified wave is coupled out at the output end, and the spent beam goes to the collector.

Features: gain of 30–60 dB, very wide bandwidth (non-resonant structure), efficiency of 20–40%. Used in satellite transponders, radar and EW systems.

Cross-field effect

The cross-field effect is the motion of electrons in a DC electric field E and a DC magnetic field B that are perpendicular to each other. The force F = −e(E + v × B) bends the electron path into a cycloid that drifts at speed v = E/B, at right angles to both fields. Electrons moving in step with an RF wave give it their potential energy. This principle is used in M-type devices such as the magnetron, CFA and amplitron.

Multi-cavity klystron amplifier

A multi-cavity klystron is an O-type (linear-beam) microwave power amplifier. It has an electron beam passing through three or more re-entrant cavities separated by drift spaces.

Construction

 cathode  input    idler     idler    output
  gun    cavity   cavity    cavity   cavity  collector
 |>==== [ C1 ] == [ C2 ] == [ C3 ] == [ C4 ] ====|
          ^RF in      drift spaces       v RF out
      <------- axial focusing magnet ------->

• Electron gun (cathode, focusing electrode, anode) gives a uniform beam at v₀ = √(2eV₀/m) ≈ 0.593×10⁶ √V₀ m/s.
• Input (buncher) cavity, one or more intermediate (idler) cavities, and an output (catcher) cavity. Each is a re-entrant resonator with a narrow gap (grids) that the beam passes through.
• Drift tubes between cavities, with no field inside.
• A solenoid or PPM magnet keeps the beam focused.
• A collector to collect the spent beam.

Principle of operation

1. Velocity modulation: the input RF sets up a gap voltage V₁ sin ωt in the buncher. An electron crossing at time t₁ leaves with v = v₀[1 + (β_i V₁ / 2V₀) sin ωt₁], where β_i is the beam coupling coefficient.
2. Bunching: in the drift space, faster electrons catch up with slower ones that left earlier, so the beam becomes density-modulated. The bunching parameter is X = β_i V₁ θ₀ / (2V₀), where θ₀ = ωL/v₀ is the drift transit angle.
3. Intermediate cavities: the bunched current excites each idler cavity. Since the cavity has a high Q, the induced gap voltage is much larger than V₁. It re-modulates the beam and gives tighter bunches. Each extra cavity adds about 15–20 dB of gain.
4. Energy extraction: the catcher sits where the bunches are tightest. The fundamental RF current is i₂ = 2I₀J₁(X), which is largest at X = 1.841 (J₁ = 0.582). The bunches induce a current in the catcher, and its gap field slows them, so kinetic energy is converted to amplified RF. The output is coupled out through a loop or waveguide.
5. The spent electrons hit the collector.

Performance: total gain of 40–60 dB, peak power up to tens of MW, efficiency of 35–50%. Synchronous tuning gives the most gain; stagger tuning gives more bandwidth. Used in radar, UHF TV transmitters and accelerators.

Transit time effect

Transit time is the time an electron needs to travel between two electrodes: τ = d/v. At low frequencies τ is much shorter than the signal period and can be ignored. At microwave frequencies τ is comparable to the period, so the electrode voltage changes while the electron is still travelling. In conventional tubes this causes phase lag, loading of the input (input conductance ∝ ω²τ²) and lower gain and efficiency. Microwave tubes such as klystrons and magnetrons use transit time on purpose to bunch electrons.

Magnetron with 45° phase difference between adjacent cavities

For N cavities in mode n, the phase difference is φ = 2πn/N. For 45°, N = 8n: an 8-cavity anode in mode n = 1 (one spoke), or a 16-cavity anode in mode n = 2 (two spokes).

   8-cavity anode, phase of each cavity
              0°
       315° _/ \_ 45°
          /       \
    270° | (cathode) |  90°     B0 axial
          \  spoke  /          E radial
       225° \_ _/ 135°
             180°

Working principles

1. Cross fields: the cathode is at the centre and the anode block (8 cavities) around it. The anode voltage gives a radial E, and a magnet gives an axial B₀ above the Hull cut-off, B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. Without RF, electrons curve back to the cathode.
2. RF pattern: noise starts oscillation in the cavities. In this mode each cavity's slot field lags its neighbour by 45°, so the RF fields in the interaction space form a wave rotating around the cathode once per n RF cycles. Its angular velocity is ω/n.
3. Synchronism (Hartree condition): V₀ and B₀ are set so that the E × B drift of electrons around the cathode matches the angular velocity of the wave.
4. Phase focusing: electrons in the retarding phase of the tangential RF field give energy to the wave and move outward to the anode. Electrons in the accelerating phase take energy, return to the cathode and are removed.
5. Spokes and output: the useful electrons form n rotating spokes (one for 8 cavities). These keep giving energy to the cavities in step, which sustains the oscillation. Power is coupled out by a loop in one cavity.
6. Mode control: because this is not the π-mode (180°), the mode must be chosen by the anode design. Strapping favours π-mode, so a non-π mode like this needs suitable coupling or an injected signal to hold it.

Uses: pulsed radar transmitters, industrial heating and, in amplifier form (cross-field amplifier), radar power stages.

Velocity modulation is the periodic change in electron velocity caused by an RF gap voltage. Bunching is the grouping of electrons that follows when the faster electrons catch up with slower ones. A "multicavity klystron" is an amplifier and a "reflex klystron" is a single-cavity oscillator, so both are explained below, since the same velocity-modulation process drives each.

Velocity modulation (two/multi-cavity klystron)

 cathode buncher   drift space L   catcher
 |>==== [ gap 1 ] ============= [ gap 2 ] ==| collector
   V0   V1 sin ωt  bunches form ->   RF out

The beam is accelerated by V₀:

v0 = √(2eV0/m) = 0.593×10^6 √V0  m/s

An electron crossing the buncher gap (average time t₁, gap voltage V₁ sin ωt) leaves with

v(t1) = v0 [1 + (βi V1 / 2V0) sin(ω t1)]     (V1 << V0)
βi    = sin(θg/2) / (θg/2),  θg = ω d / v0   (gap coupling)

Bunching in the drift space (length L):

arrival time at catcher:  t2 = t1 + L / v(t1)
ω t2 ≈ ω t1 + θ0 − X sin(ω t1)
θ0   = ω L / v0            (DC transit angle)
X    = βi V1 θ0 / (2V0)    (bunching parameter)
i2(t) = I0 + Σ 2 I0 Jn(nX) cos[n(ω t2 − θ0)]
fundamental: I_f = 2 I0 J1(X)
max at X = 1.841, J1 = 0.582  ->  I_f = 1.16 I0
max efficiency ≈ 58% (theory)

Electrons that cross when V₁ goes from negative to positive form the centre of the bunch: earlier electrons are slowed and later ones sped up, so they meet. In a multicavity tube the intermediate cavities re-modulate the beam to sharpen the bunches.

Velocity modulation and bunching in a reflex klystron

  cathode   cavity gap        repeller (−Vr)
 |>====== [ anode ] ------------->  |
            ^   <-- electrons return|
            |  bunches return to gap
           RF out

1. The beam passes the cavity gap once and is velocity-modulated as above.
2. A negative repeller voltage V_r turns the electrons back. Faster electrons go further into the repeller field and take longer to return. Slower electrons turn back sooner. So electrons that left at different times return together as a bunch.
3. The round-trip time in the repeller region is

T' = 2 m L v(t1) / [e (Vr + V0)]

4. For oscillation, the bunch must return when the gap field is maximally retarding:

ω T' = 2πn − π/2 = 2πN,  N = n − 1/4
modes: N = 3/4, 1¾, 2¾, ...

5. The returning bunches give energy to the cavity, sustaining oscillation. Changing V_r changes T' slightly, which gives electronic tuning.

Efficiency: about 20–30% in theory, a few % in practice. Output is mW. Used as a local oscillator and in signal generators.

A cavity magnetron is a cross-field (M-type) high-power microwave oscillator. The electrons move in a radial DC electric field and an axial DC magnetic field, and they give energy to RF fields in resonant cavities cut in the anode.

Mode for 45° phase shift

For N cavities in mode n, the phase shift between adjacent cavities is φ = 2πn/N. With 45°, N = 8n. An 8-cavity anode in mode n = 1 gives 45° and one rotating spoke. A 16-cavity anode in mode n = 2 gives two spokes. (The common π-mode has 180° and N/2 spokes.)

   8-cavity anode, cavity phases
              0°
       315° _/ \_ 45°
          /       \
    270° | (cathode) |  90°   B0 axial
          \  spoke  /        E radial
       225° \_ _/ 135°
             180°  -> output loop

Working principle

1. Construction: a heated cylindrical cathode (radius a) sits inside a copper anode block (radius b) with 8 resonant cavities that open into the interaction space. A magnet gives an axial B₀, and a DC voltage V₀ is applied between anode and cathode.
2. Cross-field motion: electrons follow cycloid-like paths under E and B. If B₀ is above the Hull cut-off, B₀c = √(8mV₀/e) / [b(1 − a²/b²)], they curve back without reaching the anode.
3. RF wave: noise excites the cavities. With 45° between neighbouring cavities, the fringing fields form a wave rotating around the cathode at angular velocity ω/n.
4. Synchronism (Hartree condition): V₀ and B₀ are set so that the electron drift speed around the cathode equals the wave's angular speed.
5. Phase focusing: electrons in the retarding RF phase give energy to the wave and drift outward to the anode. Electrons in the accelerating phase gain energy and return to the cathode. The useful electrons form a spoke that rotates with the wave.
6. Output: the spoke keeps giving energy to the cavities as it passes them, so oscillation builds up. A loop in one cavity takes out the RF power.
7. Mode control: non-π modes are close in frequency to others. The anode shape (rising-sun) or straps and coupling are designed to hold the wanted mode.

Efficiency is about 40–70%, and pulsed peak power ranges from kW to several MW.

Applications

• Radar transmitters (pulsed search and weather radar).
• Microwave ovens (2.45 GHz) and industrial heating, drying and plasma generation.
• Linear accelerators for medical radiotherapy.
• Microwave diathermy and some ECM/jamming transmitters.
• In amplifier form (cross-field amplifier), high-power radar stages.

Slow (backward) wave structure

A slow-wave structure is a periodic circuit, such as a helix, folded line, interdigital line, coupled cavities or a corrugated guide. It makes the RF wave travel along the tube axis at a phase velocity much lower than c (about c/10), close to the velocity of an electron beam, so that the beam and wave can interact over a long length.

Because the structure is periodic (period p), its field is a sum of space harmonics with phase constants βₙ = β₀ + 2πn/p. Some harmonics (n < 0) have a phase velocity along the beam but a group velocity (energy flow) opposite to it. These are backward waves. A slow-wave structure designed to interact with such a harmonic is a slow backward-wave structure, as used in the BWO (carcinotron), where the energy flows back toward the gun and gives internal feedback.

 helix:  /\/\/\/\/\   pitch p, diameter d
 vp ≈ c·p/(πd)   (axial phase velocity)
 forward wave : energy ->  with beam  (TWT)
 backward wave: energy <-  against beam (BWO)

Low-noise amplifier (LNA) – low-noise TWT

Here LNA means a low-noise amplifier at the front end of a receiver. Among microwave tubes, the low-noise TWT is the standard LNA. (Solid-state LNAs using GaAs FET/HEMT follow the same aim: high gain with minimum noise figure.)

Construction

     RF in (weak)                 RF out
        |                           |
 gun |>=[  helix slow-wave line  ]=====| collector
  low-noise gun   [atten]
     PPM / solenoid focusing

• A low-noise electron gun: the beam is accelerated in several stages with special potential profiles near the cathode. This reduces velocity and current fluctuations (shot noise) in the beam.
• A helix slow-wave structure with input and output couplers and a lossy attenuator in the middle.
• Focusing magnet and collector, as in a power TWT, but with low beam current.

Working principle

1. The weak input signal travels along the helix at a phase velocity v_p ≈ c·p/(πd), about equal to the beam velocity v₀ = √(2eV₀/m). The beam is slightly faster.
2. The axial RF field velocity-modulates the beam, and bunches form continuously along the helix.
3. The bunches give energy to the wave, which grows exponentially, giving a gain of 25–40 dB.
4. The noise figure is kept low (about 3–8 dB in tubes) by low beam current, careful gun design that gives a smooth, quiet beam, and good matching at the input. The attenuator stops oscillations caused by reflections.
5. The amplified, low-noise signal is taken from the output end and passed to the mixer.

Uses: front ends of radar, satellite earth-station and microwave-link receivers. Today GaAs FET/HEMT LNAs (noise figure below 1 dB) and cooled parametric amplifiers or masers have largely replaced the low-noise TWT.

Bunching effect

Bunching is the formation of dense groups (bunches) of electrons in a beam. A uniform beam passing through an RF field is velocity-modulated: some electrons are sped up and some are slowed down. As they drift on, the fast ones catch up with the slow ones, so the beam becomes density modulated. The bunches can then give kinetic energy to an RF field by passing through it during its retarding phase. Bunching is the basic mechanism of klystrons, TWTs and BWOs.

Backward wave oscillator (BWO)

A BWO (O-type carcinotron) is a slow-wave tube oscillator. Its beam interacts with a backward space harmonic of a periodic structure. The harmonic's phase velocity is in the beam direction, but its group velocity (energy flow) is opposite to the beam.

Construction

 RF out                              matched
  ^                                 termination
  |                                     |
 gun |>==== slow-wave structure ========| collector
        (helix / interdigital line)
   beam  -------------->   v0
   energy <-------------   vg (backward)
        axial focusing magnet

• An electron gun gives a beam at velocity v₀ = √(2eV₀/m), focused by an axial magnetic field.
• A periodic slow-wave structure (helix, interdigital line or folded waveguide).
• The output is taken from the gun end. The collector end is matched so no wave reflects. There is no RF input.

Working principle

1. Any noise sets up fields on the structure. The space harmonic whose phase velocity equals the beam velocity interacts with the beam.
2. The beam is velocity-modulated near the gun and bunches form as it moves toward the collector.
3. The bunches meet the retarding phase of the backward harmonic and give energy to it. This energy flows backward along the structure toward the gun end.
4. At the gun end the strengthened wave modulates the fresh, unbunched beam even more. This gives a closed feedback loop inside the tube. When the beam current exceeds the start-oscillation current, oscillations build up and power is taken out at the gun end.
5. Electronic tuning: the oscillation frequency is the one at which the backward harmonic's phase velocity equals v₀. Changing the beam voltage V₀ changes v₀, so the frequency changes. The tuning range can be up to an octave.

Features: wide voltage tuning, low to moderate power (mW to W), low efficiency (5–20%). M-type BWOs (cross-field) give more power.

Applications: swept-frequency signal generators, local oscillators in wideband receivers, broadband jamming, and mm-wave and THz sources.

Density modulation

Density modulation (current modulation) is the periodic change in the number of electrons per unit length of a beam at the RF frequency. In klystrons it is produced by velocity modulation at the buncher gap followed by a field-free drift. Faster electrons catch up with slower ones, so the beam current at the catcher has a strong RF component: i₂ = I₀ + 2I₀ J₁(X) cos(ωt − θ₀) + …, where X is the bunching parameter.

Multi-cavity klystron oscillator

A klystron amplifier becomes an oscillator when part of the output is fed back to the input in the correct phase. The usual form is a two-cavity (or multi-cavity) klystron with a feedback loop from the catcher to the buncher.

            feedback cable (phase-adjusted)
     +-------------------------------------+
     |                                     |
 gun |>==[ buncher ]== drift ==[ catcher ]=| collector
         gap 1        space L     gap 2  -> RF out

Working principle

1. Beam: the gun gives a beam at v₀ = √(2eV₀/m), focused by a magnet.
2. Start-up: noise or switch-on transients set up a small voltage V₁ sin ωt in the buncher cavity at its resonant frequency.
3. Velocity modulation: electrons leave the buncher with v = v₀[1 + (β_i V₁/2V₀) sin ωt₁].
4. Bunching: in the drift space the velocity modulation turns into density modulation. Bunches are tightest where X = β_i V₁ θ₀ / (2V₀) ≈ 1.84.
5. Energy extraction: the bunches cross the catcher during its retarding field and induce a large RF voltage. Part of the RF goes to the load.
6. Feedback: part of the catcher output is fed back to the buncher through a coupling loop or cable. Oscillation is sustained when:
   • Phase condition: the total phase around the loop (drift transit angle + feedback path) is a multiple of 2π.
   • Gain condition: the gain of the klystron is at least equal to the loss in the feedback path.
7. Intermediate cavities, in a multi-cavity tube, add gain, so oscillation starts more easily and more power is available.

Tuning: the frequency is set mainly by the cavity tuning (all cavities tuned together). The beam voltage changes the transit angle θ₀, so it allows small electronic tuning.

Features and uses: stable, high-power CW or pulsed output (W to kW), efficiency of about 20–40%. Used in high-power transmitters and accelerators. For low-power uses, the simpler single-cavity reflex klystron (where the repeller does the feedback) is preferred.

Cross-field effect

The cross-field effect is the motion of electrons under a DC electric field E and a DC magnetic field B that are perpendicular to each other. The Lorentz force F = −e(E + v × B) bends the electrons into cycloidal paths that drift at speed v = E/B, at right angles to both fields. An RF wave travelling at this drift speed can take potential energy from the electrons. This is the basis of M-type tubes (magnetron, CFA, amplitron).

Cross-field cavity power amplifier with 90° between adjacent cavities

Design choice of cavities and mode

The phase shift between adjacent cavities in mode n of an N-cavity structure is φ = 2πn/N. For φ = 90°:

90° = 360°·n / N  ->  N = 4n
choose N = 8 cavities, n = 2  -> 2 spokes
(N = 4, n = 1 also gives 90° with 1 spoke)

An 8-cavity anode is chosen. The RF pattern repeats every 4 cavities (0°, 90°, 180°, 270°), and two electron spokes are formed.

   8-cavity anode, phases (n = 2)
              0°
       270° _/ \_ 90°
          /       \
    180° | (cathode) | 180°   B0 axial
          \ 2 spokes/        E radial
       90°  \_ _/  270°
              0°
  RF in -> cavity 1        RF out <- cavity 7
  (drift section between out and in)

Design steps (outline)

1. Choose the anode voltage V₀ and anode/cathode radii b, a.
2. Set B₀ above the Hull cut-off: B₀c = √(8mV₀/e) / [b(1 − a²/b²)].
3. Choose V₀, B₀ to satisfy the Hartree condition: the electron drift angular velocity equals the RF wave's angular velocity ω/n = ω/2.
4. Size the cavities to resonate (or form a slow-wave band) at the working frequency, and cut out the coupling between output and input with a drift section so the tube cannot oscillate.

Working principle as a power amplifier

1. The heated cathode emits electrons. They move in the cross fields and drift around the cathode.
2. The input RF is coupled into the first cavity. The cavities, coupled through the interaction space, carry it as a slow wave, with 90° phase step per cavity.
3. Since the drift speed matches the wave speed, electrons stay in a fixed phase relative to the wave.
4. Phase focusing: electrons in the retarding RF phase give potential energy to the wave and move to the anode. Those in the accelerating phase gain energy and return to the cathode. Two spokes form and rotate with the wave.
5. The wave grows from cavity to cavity as the spokes give it energy. The amplified signal is coupled out at the output cavity. The drift section stops feedback, so the device amplifies instead of oscillating.

Performance: efficiency of about 40–70%, gain of about 10–20 dB, about 10% bandwidth, high peak power. Used as radar transmitter output stages.

In a reflex klystron, bunching happens in the repeller region. The electrons pass the single cavity gap, are turned back by a negative repeller, and return to the gap as bunches.

 cathode  anode/cavity gap        repeller (−Vr)
 |>====== [ gap ] ------------------>  |
            ^ <---- returning bunch ----|
            |   fast e: goes deeper, returns later
         RF out slow e: turns back sooner

Process

1. Velocity modulation: an electron crossing the gap at time t₁ leaves with v = v₀[1 + (β_i V₁/2V₀) sin ωt₁], where v₀ = √(2eV₀/m).
2. Reflection: in the retarding field between gap and repeller, an electron with higher velocity travels farther and takes longer to return. A slower electron returns sooner. The round-trip time is

T' = 2 m L v(t1) / [e (Vr + V0)]

3. Bunch formation: electrons that left a little later (sped up) and those that left earlier (slowed down) arrive back at the gap at the same time. The bunch forms around the electron that crossed when the gap voltage was going from positive to negative (the reference electron).
4. Energy transfer: for oscillation, the bunch must return when the gap field most strongly retards it:

ω T' = 2πn − π/2,  n = 1, 2, 3, ...
modes: N = n − 1/4 = 3/4, 1¾, 2¾, ... cycles

The returning bunch gives kinetic energy to the cavity, and the oscillation is sustained.

Key points

• The bunching parameter is X' = β_i V₁ θ₀' / (2V₀), and the output is largest near X' ≈ 2.41.
• Changing V_r changes the transit time, so the frequency can be electronically tuned a little.
• The efficiency is about 20–30% in theory and much lower in practice. Output is in mW. Used as local oscillators and in lab signal sources.

Bunching effect

Bunching is the grouping of electrons in a beam into dense packets at the RF frequency. A uniform beam crossing an RF gap is velocity-modulated: electrons crossing in one half-cycle are sped up and those in the other half-cycle are slowed down. During the following drift (or reflection), the faster electrons catch up with the slower ones, so the beam becomes density-modulated. If the bunches pass through a gap when its field opposes them, they give kinetic energy to the RF field. This is the working basis of all klystrons.

Klystron oscillator (reflex klystron)

The most common klystron oscillator is the reflex klystron. It uses a single cavity for both bunching and energy extraction. A negative repeller sends the beam back through the same cavity, which gives built-in feedback.

 cathode  anode+cavity gap          repeller (−Vr)
 |>====== [ gap ] ---------------->  |
   V0      ^   <---- bunches return -|
           |       (retarding field)
        RF out (coupling loop)

Construction: an electron gun (cathode and accelerating anode at +V₀), a re-entrant cavity whose gap the beam passes, and a repeller electrode at −V_r. Output is taken by a loop in the cavity.

Working principle

1. Beam: electrons are accelerated to v₀ = √(2eV₀/m) = 0.593×10⁶ √V₀ m/s.
2. Start-up: noise excites the cavity, giving a small gap voltage V₁ sin ωt.
3. Velocity modulation: an electron crossing the gap at t₁ leaves with v(t₁) = v₀[1 + (β_i V₁/2V₀) sin ωt₁].
4. Bunching in the repeller space: the electrons enter the retarding field of the repeller. A faster electron goes deeper and returns later. A slower one returns sooner. So electrons that left at different times return to the gap together. The bunch forms around the electron that crossed when the gap voltage was passing from positive to negative. Round-trip time:

T' = 2 m L v(t1) / [e (Vr + V0)]

5. Energy transfer: the bunch must return when the gap field most strongly opposes it, so it gives up energy to the cavity:

ω T' = 2πn − π/2   (n = 1, 2, 3 ...)
N = n − 1/4 = 3/4, 1¾, 2¾ ... cycles

6. Sustained oscillation: the energy given each cycle makes up for cavity and load losses, so oscillation builds up and stays. The output power is taken through the loop.

Characteristics

• Modes: each value of n gives a mode. Output power is highest for the mode with the longest transit time that still keeps the bunches tight.
• Electronic tuning: changing V_r changes T', which shifts the frequency by a few MHz around the cavity resonance. Mechanical tuning changes the cavity size.
• Power and efficiency: output 10–500 mW, efficiency 20–30% in theory (a few % in practice), frequency range about 1–25 GHz.

Applications: local oscillators in microwave receivers, signal sources in labs, pump sources for parametric amplifiers, and FM transmitters in portable microwave links.

(A two-cavity klystron amplifier can also be made to oscillate by feeding part of the catcher output back to the buncher in correct phase, but the reflex klystron is the standard klystron oscillator.)

Bunching effect

Bunching is the formation of dense groups of electrons at regular intervals in a beam. It results when electrons of different velocities (caused by an RF field) catch up with one another, as in klystrons and TWTs. In a magnetron the same grouping comes from phase focusing: electrons in the favourable RF phase stay in the interaction, and those in the unfavourable phase are removed. So the electrons form spokes, which give energy to the RF field in step with it.

Cavity magnetron

A cavity magnetron is a cross-field (M-type) high-power microwave oscillator.

Construction

        anode block with N = 8 cavities
         _/ \_/ \_/ \_/ \_
        /                  \
       |  interaction space |
       |    ( cathode )     |   B0 axial (out of page)
       |  spokes rotate ->  |   E radial
        \_/ \_/ \_/ \_/ \_/
          straps   output loop -> RF out

• A cylindrical cathode (radius a) coated with oxide, heated from inside.
• A copper anode block (radius b) with N cavities (usually 8 to 16) of hole-and-slot, vane or rising-sun shape, opening into the interaction space.
• A permanent magnet giving an axial magnetic field B₀, and a DC anode voltage V₀ giving a radial electric field.
• Straps joining alternate anode segments to hold the π-mode, and a coupling loop in one cavity to take out power.

Operational features

1. Electron paths: with E only, electrons go straight to the anode. With crossed E and B, they follow curved (cycloid-like) paths. At the Hull cut-off field they just graze the anode: B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. The tube is run with B₀ slightly above this value.
2. RF field: noise excites the cavities. In the π-mode, adjacent cavities are 180° out of phase, and their fringing fields form a wave rotating around the cathode.
3. Hartree condition: V₀ and B₀ are chosen so that the electron drift speed around the cathode equals the angular phase velocity of this wave.
4. Phase focusing (bunching): electrons in a retarding tangential RF field lose energy to the wave and move outward toward the anode. Electrons in an accelerating field gain energy and are bent back to the cathode, which removes them. The useful electrons form N/2 spokes.
5. Energy transfer: each spoke passes a slot when its field is retarding, so it keeps giving energy to the cavities and the oscillation is sustained.
6. Performance: efficiency of 40–70% (high, because electrons reach the anode with little kinetic energy), pulsed peak power from kW to 10 MW, CW power up to several kW. Frequency is set mainly by cavity size. Frequency pushing (change with current) and pulling (change with load) occur.
7. Mode separation: strapping or a rising-sun anode separates the π-mode from nearby modes, so the tube does not jump modes.

Applications: radar transmitters, microwave ovens (2.45 GHz), industrial heating and medical linear accelerators.

A two-cavity klystron is an O-type (linear-beam) microwave amplifier. It has two cavities, a buncher and a catcher, separated by a field-free drift space.

 cathode buncher   drift space L   catcher
 |>==== [ gap 1 ] ============= [ gap 2 ] ==| collector
         ^ RF in   bunches form ->   v RF out

Working

1. The beam is accelerated by V₀ to v₀ = √(2eV₀/m) ≈ 0.593×10⁶ √V₀ m/s.
2. Velocity modulation: the input RF sets the buncher gap voltage V₁ sin ωt. An electron crossing at t₁ leaves with v = v₀[1 + (β_i V₁/2V₀) sin ωt₁].
3. Bunching: in the drift space, faster electrons catch up with slower ones that crossed earlier. Bunches form around the electron that crossed when the gap voltage went from negative to positive. The bunching parameter is X = β_i V₁ θ₀ / (2V₀), with θ₀ = ωL/v₀.
4. Output: the catcher is placed where the bunches are tightest. The fundamental beam current is 2I₀J₁(X), which is largest at X = 1.841 (J₁ = 0.582). The bunches induce RF current in the catcher, and its retarding gap field takes their kinetic energy, giving amplified RF output.

Key figures

• Maximum theoretical efficiency about 58%. Practical about 15–30%.
• Gain about 10–20 dB with narrow bandwidth, set by the cavity Q.
• Can be made into an oscillator by feeding part of the output back to the input.

Uses: UHF TV and troposcatter transmitters, radar driver stages, and particle accelerators. More cavities (multi-cavity klystron) are added for higher gain.