Micro Wave Tubes
The term microwave is used to designate electromagnetic oscillations with frequencies ranging from 300 MHz to 300 GHz. The IEEE recommended microwave band designations are as follows:
0.3 – 1 GHz
1 – 2 GHz
2 – 4 GHz
4 – 8 GHz
8 – 12 GHz
12 – 18 GHz
18 – 27 GHz
27 – 40 GHz
40 – 300 GHz
> 300 GHz
Typical applications of microwaves are as follows:
0.3 – 3 GHz
TV, radar, LOS communication, telemetry, medicine, food industry
3 – 30 GHz
Altimeter, air- and ship-borne radar, navigation, satellite communication
30 – 300 GHz
Radio astronomy, radio meteorology, space research, nuclear physics
Microwave transmission is as follows:
The conventional two-conductor line cannot be used for microwave transmission because of the losses. Hollow metallic tubes called waveguides are used for transmission.
The energy propagation in these structures is basically a reflection phenomenon.
Microwave circuit elements are as follows:
The conventional circuit elements such as resistors, inductors and capacitors do not respond well at microwave frequencies. For example, a coil of wire may be an excellent inductor at 1 MHz, but at 500 MHz it may be an equally good capacitor because of the predominating effect of inter-turn capacitance.
Circuit elements for microwave applications are constructed from sections of microwave line that offers reactances varying from - to +.
Similarly, resonant microwave line sections known as resonant cavities replace conventional resonant and anti-resonant circuits.
Generation and amplification of microwaves are as follows:
The operation of conventional vacuum tubes and solid state devices at microwave frequencies is limited by transit time effects.
A number of new principles of operation such as velocity modulation, interaction of space charge waves with electromagnetic fields, avalanche breakdown, quantum mechanical tunneling, transferred electron techniques are used for generation of microwaves.
Power coupling is carried out as follows:
The waveguide transmission of microwaves is associated with some problems such as coupling of power to another system, say from generator to line, exciting of waves in a waveguide, etc.
The three basic coupling methods evolved are:
The principle of operation of a vacuum tube at microwave frequencies (> 1 GHz) is different from electronic vacuum tubes, such as triodes, tetrodes and pentodes. The failure of these conventional electronic vacuum tubes at microwave frequencies is discussed below.
The degradation in the performance of conventional vacuum tubes at high (microwave) frequencies is due to the following three factors.
The leads from electrodes to the external terminals of the tube may have sufficient inductance at high frequencies to cause resonance effects with the interelectrode capacitances. The following figure shows the equivalent circuit of any triode.
Internal self-oscillation will occur. At this frequency the input signal is short-circuited and becomes ineffective in inducing changes in the plate. The grid lead inductance Lg could give rise to a resonance phenomenon with the electrode capacitance at:
The skin effect causes current to flow not in a solid wire, but in its outside layers. The skin depth is thus reduced causing very significant increase in series resistance and inductance.
Transit time (transit angle) effects
At low frequencies, it is possible to assume that electrons leave the cathode and arrive at the plate of the tube instantaneously. This is not true at microwave frequencies. The transit time becomes an appreciable fraction of the RF cycle.
In the conventional vacuum tube, electrons are first density-modulated and then accelerated. The density-modulation by the signal voltage is performed in the cathode-grid region where the velocity of the electron is relatively small because of the retarding effect of the negative grid potential. As a consequence, the alternately accelerating and decelerating signal potential can significantly affect the velocity of the electron. If the frequency of the signal is sufficiently high, an electron may well be subjected to alternately accelerating and decelerating forces on its way to the grid if the transit time of the electron is comparable to the signal period.
The electron transit angle is defined as
- transit time across the gap
d - separation between cathode and grid
vo - 0.593 106 is the velocity of the electron
Vo - dc voltage
It seems reasonable to conclude that accelerating and then velocity modulating the electron stream can minimize transit time effects. This is done in klystrons.
In the case of ordinary vacuum tubes, the output is usually a tuned circuit in order to maximize the output voltage. The following circuit is the equivalent of a pentode.
rp - plate resistance
R - load resistance
L - tuning inductance for the stray capacitance
The gain-bandwidth product of the circuit is
It is a function of the transconductance and stray capacitance only, it is a constant independent of frequency. This means that for a given tube an increase in bandwidth can be obtained only at the expense of a lower gain.
The two-cavity klystron is a microwave amplifier operated by the principles of velocity and current modulation. The characteristics of a two-cavity klystron amplifier are as follows:
Efficiency : ~ 40%
Power output : 500 KW (CW power) & 30 MW (pulsed power)
Power gain : 30 dB
The following is the schematic diagram of a two-cavity klystron amplifier. The cavity close to the cathode is known as the buncher cavity or input cavity, which velocity-modulates the electron beam. The other cavity is called the catcher cavity or output cavity; it catches energy from the bunched electron beam. The beam then passes through the catcher cavity and is terminated at the collector. The quantitative analysis of a two-cavity klystron can be described in four parts: (1) reentrant cavities, (2) velocity modulation process, (3) bunching process, (4) output power. The following are the assumptions:
The electron beam is assumed to have a uniform density in the cross section of the beam.
Space-charge effects are negligible.
The magnitude of the microwave signal input is assumed to be much smaller than the dc accelerating voltage.
All electrons injected from the cathode arrive at the first cavity with uniform velocity. Those electrons passing the first cavity gap at zeros of the gap voltage (or signal voltage) pass through with unchanged velocity; those passing through the positive half cycles of the gap voltage undergo an increase in velocity; those passing through the negative swings of the gap voltage undergo a decrease in velocity. As a result of these actions, the electrons gradually bunch together as they travel down the drift space. The variation in electron velocity in the drift space is known as velocity modulation.
The density of the electrons in the second cavity gap varies cyclically with time. The electron beam contains an ac component and is said to be current-modulated. The density of the electrons in the second cavity gap varies cyclically with time.
The maximum bunching should occur approximately midway between the second cavity grids during its retarding phase; thus the kinetic energy is transferred from the electrons to the field of the second cavity. The electrons then emerge from the second cavity with reduced velocity and finally terminate at the collector.
When the operating frequency is increased, both the inductance and the capacitance must be reduced to a minimum in order to maintain resonance. Ultimately, the inductance is reduced to a minimum by a short wire. Therefore, the reentrant cavities are designed for use in klystrons. A reentrant cavity is one in which the metallic boundaries extend into the interior of the cavity. Several types of reentrant cavities are shown below.
(a) Coaxial cavity (b) Radial cavity (c) Tunable cavity (d) Toroidal cavity (e) Butterfly cavity
The coaxial cavity is often used and shown below. The characteristic impedance of the coaxial line is:
- magnetic permeability
- electric permittivity
The coaxial cavity is similar to a coaxial line shorted at two ends and joined at the center by a capacitor. The input impedance to each shorted coaxial line is given by:
where l is the length of the coaxial line. The inductance and the capacitance of the cavity are given by:
At resonance, the inductive reactance of the two shorted coaxial lines in series is equal in magnitude to the capacitive reactance of the gap. That is,
where is the phase velocity in any medium.
The solution to this equation gives the resonant frequency of a coaxial cavity. Since the equation contains the tangent function, it has an infinite number of solutions with larger values of frequency. Therefore, this type of reentrant cavity can support an infinite number of resonant frequencies or modes of oscillation.
When electrons are first accelerated by the high dc voltage Vo before entering the buncher grids, their velocity is uniform:
It is assumed that electrons leave the cathode with zero velocity. When a microwave signal is applied to the input terminal, the gap voltage between the buncher grids appears as:
where V1 is the amplitude of the signal and V1 << Vo is assumed.
In order to find the modulated velocity in the buncher cavity in terms of either the entering time to or the exiting time t1 and the gap transit angle , it is necessary to determine the average microwave voltage in the buncher gap as indicated below.
Since V1 << Vo the average transit time through the buncher gap distance d is
The average gap transit angle can be expressed as:
The average microwave voltage in the buncher gap can be found as under:
Then using the trigonometric identity that ,
It is defined as
is known as the beam-coupling coefficient of the input cavity gap. The following graph shows that increasing the gap transit angle decreases the coupling between the electron beam and the buncher cavity; that is, the velocity modulation of the beam for a given microwave signal is decreased.
Immediately after the velocity modulation, the exit velocity from the buncher gap is given by:
where the factor is called the depth of velocity modulation. Using the binomial expansion under the assumption , the above equation becomes:
This is the equation of velocity modulation. The above equation can also be given as:
Once the electrons leave the buncher cavity, they drift with a velocity given above along in the field-free space between the two cavities. The effect of velocity modulation produces bunching of the electron beam – current modulation. The electrons that pass the buncher at Vs = 0 travel through with unchanged velocity vo and become the bunching center.
Those electrons that pass the buncher cavity during the positive half cycles of the microwave input voltage Vs travel faster than the electrons that passed the gap when Vs = 0. Those electrons that pass the buncher cavity during the negative half cycles of the voltage Vs travel slower than the electrons that passed the gap when Vs = 0. At a distance of along the beam from the buncher cavity, the beam electrons have drifted into dense clusters.
The distance from the buncher grid to the location of dense electron bunching for the electron at tb is
Similarly, the distances for the electrons at ta and tc are:
The equation of velocity modulation is given by:
The vmin & vmax can be calculated from the above equation by taking the minimum and maximum values that the sin function can take. The sin term can take a minimum value of –1 and a maximum value of +1. Therefore, the values of vmin & vmax are given by:
The necessary condition for those electrons at ta, tb, and tc to meet at the same distance is:
The distance given by the above equation is not the one for a maximum degree of bunching. The following figure shows the distance-time plot or Applegate diagram.
Since the drift region is field free, the transit time for an electron to travel a distance of L is given by:
is the dc transit angle between cavities,
is the bunching parameter of a klystron.
We know that
is the buncher cavity departure angle and is the catcher cavity arrival angle. Now, differentiating t2 with respect to to results in:
At the buncher gap a charge passing through at a time interval is given by:
where Io is the dc current. From the principle of conservation of charges this same amount of charge dQo also passes the catcher at a later time interval dt2. Hence
where i2 is the current at the catcher gap. The current arriving at the catcher cavity is then given as:
The beam current at the catcher cavity is a periodic waveform of period about dc current. Therefore, the current i2 can be expanded in a Fourier series and so
where n is an integer, excluding zero. The series coefficients are given by:
By using the trigonometric functions
The two integrals involve cosines and sines of a sine function. Each term of the integrand contains an infinite number of terms of Bessel functions. These are:
If these series are substituted into the integrals, the coefficients are:
where is the nth-order Bessel function of the first kind. The beam current is:
The fundamental component of the beam current at the catcher cavity has a magnitude:
The fundamental component has its maximum amplitude at
The maximum bunching should occur approximately midway between the catcher grids. The phase of the catcher gap voltage must be maintained in such a way that the bunched electrons, as they pass through the grids, encounter a retarding phase. When the bunched electron beam passes through the retarding phase, its kinetic energy is transferred to the field of the catcher cavity. When the electrons emerge from the catcher grids, they have reduced velocity and are finally collected by the collector.
Since the current induced by the electron beam in the walls of the catcher cavity is directly proportional to the amplitude of the microwave input voltage V1, the fundamental component of the induced microwave current in the catcher is given by:
- the beam coupling coefficient of the catcher gap. If the buncher and catcher cavities are identical, then . The fundamental component of current induced in the catcher cavity then has a magnitude
The following figure shows an output equivalent circuit in which Rsho represents the wall resistance of catcher cavity, RB the beam loading resistance, RL the external load resistance, and Rsh the effective shunt resistance. The output power delivered to the catcher cavity and the load is given as:
Rsh - total equivalent shunt resistance of the catcher cavity
V2 - fundamental component of catcher gap voltage.
Efficiency of klystron
The electronic efficiency of the klystron amplifier is defined as the ratio of the output power to the input power:
If the coupling is perfect, , the maximum beam current approaches:
and the voltage V2 is equal to Vo. Then the maximum electronic efficiency is about 58%. In practice, the electronic efficiency of a klystron amplifier is in the range of 15 to 30%. The efficiency is a function of the catcher gap transit angle , as shown below.
Exercise 1: A klystron amplifier is operated with a beam voltage of 3 KV. If the coupling coefficient is 0.9 and the magnitude of the signal voltage at the input cavity gap is 100 V, find the velocities of the electrons leaving the input gap.
Exercise 2: A two-cavity klystron amplifier has beam current 400 mA, an output cavity coupling coefficient 0.95 and the magnitude of the voltage across the output cavity gap is 75 V. Calculate the magnitude of the fundamental induced current and the maximum power output.
Exercise 3: A two-cavity klystron amplifier is operated at 10 GHz with Vo = 1200 V, Io = 30 mA, d = 1 mm, L = 4 cm and Rsh = 40 k. Neglecting beam loading, calculate (a) input RF voltage V1 for a maximum output voltage, (b) voltage gain, (c) efficiency, (d) beam loading conductance, (e) the electron velocity, and (f) the dc electron transit time.
Exercise 4: A two-cavity klystron amplifier has voltage gain of 20 dB. If the input power is 1 mW, Rsh = 30 k for both cavities and loading evidence is 40 k , determine (a) the input rms voltage, (b) output rms voltage and (c) power delivered to the load.
Exercise 5: A two-cavity klystron amplifier has Vo = 1000 V, Io = 25 mA, Ro = 40 k, and f = 3 GHz. The other parameters are: d = 1 mm, L = 4 cm, and Rsh = 30 k. Calculate (a) the input gap voltage to give maximum voltage, (b) the voltage gain, neglecting the beam loading in the output cavity, (c) the efficiency of the amplifier, neglecting beam loading, and (d) the beam loading conductance and show that neglecting it was justified in the preceding calculations.
Exercise 6: The parameters of a two-cavity klystron amplifier are: Vo = 1200 V, Io = 28 mA, f = 8 GHz, d = 1 mm, L = 4 cm, and Rsh = 40 k (excluding beam loading). Find (a) the input microwave voltage V1 in order to generate a maximum output voltage, (b) the voltage gain, (c) the efficiency of the amplifier, and (d) the beam loading conductance.
Exercise 7: A two-cavity klystron amplifier has a voltage gain of 15 dB, input power of 5 mW, total shunt impedance of input cavity 30 k, total shunt impedance of output cavity 40 k, and load impedance at output cavity 40 k. Determine (a) the input voltage, (b) the output voltage, and (c) the power delivered to the load.
Exercise 8: A two-cavity klystron has beam voltage 900 V, beam current 30 mA, frequency 8 GHz, gap spacing in either cavity 1 mm, spacing between centers of cavities 4 cm, and effective shunt impedance 40 k. Find (a) the electron velocity, (b) the dc electron transit time, (c) the input voltage for maximum output voltage, and (d) the voltage gain in dB.
If a fraction of the output power is fed back to the input cavity and if the loop gain has a magnitude of unity with a phase shift of multiple , the klystron will oscillate. A two-cavity klystron oscillator is not constructed because, when the oscillation frequency is varied, the resonant frequency of each cavity and the feedback path phase shift must be readjusted for a positive feedback. The reflex klystron is a single-cavity klystron that overcomes the disadvantages of the two-cavity klystron oscillator.
output power ~ 500 mW
frequency ~ 1 to 25 GHz
efficiency ~ 20 to 30%
This type is used in:
the laboratory for microwave measurements
the microwave receivers as local oscillators
A schematic diagram of the reflex klystron is shown below. The electron beam injected from the cathode is first velocity-modulated by the cavity-gap voltage. Some electrons accelerated by the accelerating field enter the repeller space with greater velocity than those with unchanged velocity. Some electrons decelerated by the retarding field enter the repeller region with less velocity. All electrons turned around by the repeller voltage then pass through the cavity gap in bunches that occur once per cycle.
On their return journey, the bunched electrons pass through the gap during the retarding phase of the alternating field and give up their kinetic energy to the electromagnetic energy of the field in the cavity. Oscillator output energy is then taken from the cavity. The walls of the cavity or other grounded metal parts of the tube finally collect the electrons.
The electron entering the cavity gap from the cathode at z = 0 and time to is assumed to have uniform velocity:
The same electron leaves the cavity gap at z = d at time t1 with velocity:
The same electron is forced back to the cavity z = d and time t2 by the retarding electric field E, which is given by:
This retarding field E is assumed to be constant in the z direction. The force equation for one electron in the repeller region is:
where is assumed. Integrating the above equation yields:
at t = t1, , then
Integrating again, we get
at t = t1, z = d = K2; then
On the assumption that the electron leaves the cavity gap at z = d and time t1 with a velocity of and returns to the gap at z = d and time t2, then, at t = t2 , z =d,
The round-trip dc transit time in the repeller region is given by:
where is the round-trip dc transit time of the center-of-the-bunch electron. Multiplying the above equation with the radian frequency results in:
where is the round-trip dc transit angle of the center-of-the-bunch electron and
is the bunching parameter of the reflex klystron oscillator.
Power output and efficiency
In order for the electron beam to generate a maximum amount of energy to the oscillation, the returning electron beam must cross the cavity gap when the gap field is maximum retarding. In this way, a maximum amount of kinetic energy can be transferred from the returning electrons to the cavity walls. It can be seen from the following Applegate diagram that for a maximum energy transfer, the round-trip transit angle, referring to the center of the bunch, must be given by:
where n = any positive integer for cycle number, and is the number of modes. The beam current injected into the cavity gap from the repeller region flows in the negative z direction. Consequently, the beam current of a reflex klystron oscillator can be written as:
The fundamental component of the current induced in the cavity by the modulated electron beam is:
is neglected as a small quantity compared to . The magnitude of the fundamental component is:
The dc power supplied by the beam voltage Vo is:
and the ac power delivered to the load is given by:
We know that:
Therefore, the electronic efficiency of a reflex klystron oscillator is defined as:
has a maximum value at X’ = 2.408, as shown below.
For a given beam voltage Vo, the relationship between the repeller voltage and cycle number n required for oscillation is found as under:
We also know that
The power can be expressed in terms of the repeller voltage, Vr. That is,
It can be seen from the above two equations that, for a given beam voltage Vo and cycle number n, the center repeller voltage Vr can be determined in terms of the center frequency. Then the power output at the center frequency can be calculated. When the frequency varies from the center frequency and the repeller voltage about the center voltage, the power output will vary accordingly, assuming a bell shape, as shown below.
Exercise 1: A reflex klystron is operated at 5 GHz with an anode voltage 1000 V and the cavity gap 2 mm, calculate the gap transit angle. Find the optimum length of the drift region.
Exercise 2: A reflex klystron operates at the peak mode of n = 2 with Vo = 280 V, Io = 22 mA, and V1 = 30 V. Determine (a) the input power, (b) the output power, and (c) the efficiency.
Exercise 3: A reflex klystron operates at 8 GHz at the peak of n = 2 mode with Vo = 300 V, Rsh = 20 k, and L = 1 mm. If the gap transit time and beam loading are neglected, find the (a) repeller voltage, (b) beam current necessary to obtain a RF gap voltage of 200 V, and (c) the electronic efficiency.
Exercise 4: A reflex klystron operates at the peak of n = 1 mode with dc power input of 30 mW and . Determine (a) the efficiency, and (b) total output power.
Exercise 5: A reflex klystron is oscillating at the peak of the n = 2, Vo = 600 V, L = 1 mm, Rsh = 15 k, and fr = 9 GH. Find (a) the value of the repeller voltage, (b) the direct current necessary to give a microwave gap voltage of 200 V, and (c) the electronic efficiency.
Exercise 6: A reflex klystron operates at the peak of n = 2 mode with beam voltage 300 V, beam current 20 mA and the signal voltage 40 V. Determine (a) the input power, (b) the output power, and (c) the efficiency.
Exercise 7: A reflex klystron operates at the peak of n = 2 with Vo = 500 V, Rsh = 20 k, fr = 8 GHz, and L = 1 mm. Find (a) the value of repeller voltage, (b) the direct current necessary to give microwave gap voltage of 200 V, and (c) the electronic efficiency.
Exercise 8: A reflex klystron operates at the peak of n = 2 mode with dc power input of 40 mW and . If 20% of the power delivered by the beam is dissipated in the cavity wass, find the power delivered to the load.
Traveling wave tube (TWT)
TWTs are used for broadband applications. The microwave circuit is non-resonant and the wave propagates with the same speed as the electrons in the beam. There are two major differences between the TWT and the klystron:
The interaction of the electron beam and the RF field in the TWT is continuous over the entire length of the circuit, but the interaction in the klystron occurs only at the gaps of a few resonant cavities.
The wave in the TWT is a propagating wave; the wave in the klystron is not.
The general characteristics of the TWT are:
Frequency range : 3 GHz
Efficiency : up to 40%
Output : up to 10 kW (average)
Power gain : up to 60 dB
The ability of many slow-wave / periodic structures to support a wave having a phase velocity much less than that of light is basic importance for TWT circuits. In a TWT, efficient interaction between the electron beam and the electromagnetic field is obtained only if the phase velocity is equal to the beam velocity. This requires considerable slowing down of electromagnetic wave. Several slow-wave structures, as shown below, are designed for producing large gain over a wide bandwidth.
The commonly used slow-wave structure is a helical coil of the types shown below. The retardation process can be understood qualitatively by considering an electromagnetic wave propagating along the turns of the helix with a velocity c.
The applied signal propagates around the turns of the helix and produces an electric field at the center of the helix, directed along the helix axis. The axial electric field progresses with a velocity that is very close
to the velocity of light multiplied by the ratio of the helix pitch to helix circumference. It can be shown that
where c is the velocity of light in free space, p is helix pitch, d is diameter of the helix and is the pitch angle. For a very small pitch angle, the phase velocity along the coil in free space is approximately represented by
Electromagnetic waves have two velocities:
Phase velocity the velocity with which phase changes
Propagation velocity the velocity with which wave propagates
While is proportional to , the phase velocity vp is constant, and therefore all component waves making up a signal will be transmitted at the same velocity, vp. This is distortionless transmission condition.
The situation can occur where is not proportional to . Component sine waves of a signal will be transmitted with different velocities, and the question then is, at what velocity does the signal wave travel?
The composite signal is seen to consist of high frequency waves modulated by a low frequency envelope. Detailed analysis shows that the envelope travels along the line with a velocity called group velocity (the velocity with which the group of two sine waves travels). If varies rapidly with , serious distortion will result.
This is Brillouin diagram for a helical slow-wave structure. The helix - diagram is useful in designing a helix slow-wave structure. Once is found, vp can be computed for a given dimension of the helix. The group velocity of the wave is merely the slope of the curve as given by
The fundamental requirement of a slow-wave structure is that it must possess an axial component of electric field having phase velocity less than the velocity of light. The axial electric field is of the form:
where f(z) is periodic in L (fundamental period). We may expand the field distribution f(z) in a Fourier series of fundamental period L:
The negative and positive values of n in the series represent forward and backward traveling waves. The individual terms in the Fourier series are the spatial harmonics of the field. All the terms of the series vary in phase velocities with the same frequency. The phase velocity of the nth harmonic is
The phase velocity is small for large n. The group velocity
Thus the group velocity is the same for all the spatial harmonics and depends on frequency alone.
In the tube, an electron gun assembly consisting of heater, cathode and control anode produces abeam of initially uniform velocity electrons that pass through the helix to the collector. A source of microwave energy connected to the input waveguide excites a wave that moves along the helix to the output waveguide. The RF input propagating on the helix produces a longitudinal component of the electric field on the axis of the helix, which propagates axially with reduced velocity.
Interaction of the longitudinal component of the electric field of the helix with the beam of electrons results in reduction of the average velocity of the electrons and in conversion of the kinetic energy lost by the electrons into microwave energy of the wave. The energy of the wave is delivered to the output,
In order that the electron beam shall completely pass through the helix, which may be a foot or more in length, a longitudinal magnetic focusing field of a few hundred gausses is used. An attenuator placed near the center of the helix reduces all the waves traveling along the helix to nearly zero so that the reflected waves from the mismatched loads can be prevented from reaching the input and causing oscillation.
The axial space harmonic phase velocity is synchronized with the beam velocity for possible interaction between the electron beam and electric field. That is vpn = vo. In general, the dc velocity of the electrons is adjusted to be slightly greater than the axial velocity of the electromagnetic wave for energy transfer.
The electromagnetic wave traveling along the helix axis, has a longitudinal component of electric field as shown below. These electric fields exert force F = - e E on the electrons.
An electric field directed against the electron flow accelerates the electrons and that directed along the electron flow decelerates the electrons. Thus a density modulation in accordance with the input signal frequency will occur to form bunches.
As the dc velocity of the beam is maintained slightly greater than the phase velocity of the traveling wave, more electrons face the retarding field than the accelerating field, and a great amount of kinetic energy is transferred from the beam to the electromagnetic field. Thus the field amplitude increases forming more compact bunch and a larger amplification of the signal voltage appears at the output end of the helix as shown in the following figure.
The motion of electrons in the helix-type traveling-wave tube can be quantitatively analyzed in terms of the axial electric field. If the traveling wave is propagating in the z-direction, the z component of the electric field can be expressed as:
where E1 is the magnitude of the electric field in the z direction. If t = to at z = 0, the electric field is assumed maximum. Note that p = /vp is the axial phase constant of the microwave, and vp is the axial phase velocity of the wave.
The equation of motion of the electron is given by
Assume that the velocity of the electron is
vo dc electron velocity
ve magnitude of velocity fluctuation in the velocity-modulated electron beam
e angular frequency of velocity fluctuation
e phase angle of the fluctuation
Now the equation of motion becomes
The distance traveled by the electron in the helix is
Comparison of the left and right hand sides shows that:
It can be seen that the magnitude of the velocity fluctuation of the electron beam is directly proportional to the magnitude of the axial electric field.
In order to determine the relationship between the slow-wave circuit and the electron beam quantities, the convection current induced in the electron beam by the axial electric field and the microwave axial field produced by the beam must first be developed.
When the space-charge effect is considered, the electron velocity, the charge density, the current density, and the axial electric field will perturb about their averages or dc values. These quantities are expressed as:
where is the propagation constant of the axial waves. The minus sign is attached to Jo so that it may be positive in the negative z direction. For a small signal, the electron beam current density can be written as:
If an axial electric field exists in the structure, it will perturb the electron velocity according to the force equation. Hence the force equation can be written as:
In accordance with the law of conservation of electric charge, the continuity equation can be written as:
We know that
If the magnitude of the axial electric field is uniform over the cross-sectional area of the electron beam, the spatial ac current I will be proportional to the dc beam current Io with the same proportionality constant for J1 and Jo. Therefore the convection current in the electron beam is given by:
Since , the phase constant of the velocity-modulated electron beam and , then
This equation is called the electronic equation, for it determines the convection current induced by the axial electric field. If the axial field and all parameters are known, the convection current can be found by means of the above equation.
The convection current in the electron beam induces an electric field in the slow-wave structure. This induced field adds to the field already present in the circuit and causes the circuit power to increase with distance. The coupling relationship between the electron beam and the helix is shown below.
A distributed lossless transmission line represents the helix. The electron beam current i flow adjacent to a lossless transmission line, in which the line current and voltage are I and V, respectively. Considering the closed cylindrical surface about the electron beam can show the physical mechanism by which the current is induced on the line.
The current flowing into the left-hand end of the surface is i. The current flowing out of the right-hand end is , where the length of the cylinder dz is assumed short compared with a wavelength of the wave along the transmission line. If is positive, more convection current comes out of the cylinder than went in and the alternating charge density inside the cylinder will be negative. Field lines from the transmission line to this negative charge density will constitute a displacement current. The value of this displacement current could be calculated, but this can be avoided by applying Kirchhoff’s law to the total current (displacement + convection) into the cylinder.
Kirchhoff’s voltage law around the closed loop gives:
Substituting and in the above two equations:
If the convection current is not present, the above reduces to a typical wave equation of a transmission line. When i = 0, the propagation constant is defined from the above equation is:
and the characteristic impedance of the line can be determined from the above equations as follows:
When the convection current is present, the above equation can be written as:
We know that and , therefore,
Since , the axial electric field is given by
This equation is called the circuit equation because it determines how the spatial ac electron beam current affects the axial electric field of the helix.
Solving the electronic and circuit equations for the propagation constants can determine the wave modes of TWT. Each solution for the propagation constants represents a mode of traveling wave in the tube. The equation for the convection current i is:
The above is an equation of fourth order in and thus has four roots. This means that there are four modes of traveling wave in the helix-type TWT. The exact solutions for the above equation can be obtained with numerical methods. However, the approximate solutions may be found by equating the dc electron beam velocity to the axial phase velocity of the traveling wave, which is equivalent to setting:
Then the above equation reduces to:
where K is the TWT gain parameter and is defined as:
There are three forward traveling waves corresponding to and one backward traveling wave corresponding to . Let the propagation constant of the three forward traveling waves be
where it is assumed that . Rewriting the previous equation in terms of above, we get:
Since , the quantity in the first set of parentheses on the left-hand side of the last equation is approximately j2, and the quantity in parentheses on the right-hand side of the equation is approximately . The last equation then reduces to:
( n = 0, 1, 2 )
The first root 1 at n = 0 is:
The second root at n = 1 is:
The third root at n = 2 is:
The fourth root corresponding to the backward traveling wave can be obtained by setting:
Thus the values of the four propagation constants are given by:
forward wave and grows exponentially
forward wave and decays exponentially
forward wave with constant amplitude
backward wave with no change in amplitude
The first wave travels with a velocity slower than electron beam and energy transfer takes place from beam to traveling wave. The second wave travels with same velocity and the energy flows from wave to beam. The third wave travels slightly at higher velocity and no energy exchange takes place.
For simplicity, it is assumed that the structure is perfectly matched so that there is no backward traveling wave. Thus the total circuit voltage is the sum of three forward voltages corresponding to the three forward traveling waves. This is equivalent to:
The input current can be found from the earlier equation:
On substituting and , we get
We can rewrite (since K << 1), the above equation becomes:
The input fluctuating component of velocity of the total wave may be found from the earlier equation:
[ , and ]
To determine the amplification of the growing wave, the input reference point is set at z = 0 and the output reference point is taken at z = d. It follows that at z = 0, the voltage, current and velocity at the input point are given by
The simultaneous solution of the above three equations with and is
Since the growing wave is increasing exponentially with distance, it will predominate over the total voltage along the circuit. When the length d of the helix is sufficiently large, the output will be almost equal to the voltage of the growing wave. Substituting the value of in the above voltage equation gives the output voltage as
The factor is written as , where N is the circuit length in electronic length,
The amplitude of the output voltage is then given by
The output power gain in dB is defined as
= -9.54 + 47.3 NK
where NC is a numerical number. The output power gain indicates an initial loss at the circuit input of 9.54 dB. This loss results from the fact that the input voltage splits into three waves of equal magnitude. It can also be seen that the power gain is proportional to the length N and the gain parameter K of the circuit.
Exercise 1: A helix TWT is operated with a beam current of 300 mA, beam voltage of 5 kV, and characteristic impedance of 20 . What length of the helix will be selected to give a output power gain of 50 dB at 10 GHz?
Exercise 2: A TWT has Vo = 3 kV, Io = 3 mA, f = 10 GHz, Zo = 25 and normalized circuit length N = 50. Calculate (a) the gain parameter C, and (b) dB power gain.
Exercise 3: A TWT has Vo = 2 kV, Io = 4 mA, f = 8 GHz, Zo = 20 and normalized circuit length N = 50. Calculate (a) the gain parameter C, and (b) dB power gain.
Exercise 4: A TWT has Vo = 2.5 kV, Io = 50 mA, f = 8 GHz, Zo = 6.75 and normalized circuit length N = 45. Calculate (a) the gain parameter C, (b) dB power gain, (c) all four propagation constants, and (d) the wave equations for all four modes.
Exercise 5: A TWT has Vo = 3 kV, Io = 30 mA, f = 10 GHz, Zo = 10 and normalized circuit length N = 50. Calculate (a) the gain parameter C, (b) dB power gain, (c) all four propagation constants, and (d) the wave equations for all four modes.
Exercise 6: An O-type TWT operates at 2 GHz. The slow-wave structure has a pitch angle of 5.7. Determine the propagation constant of the traveling wave in the tube. It is assumed that the tube is lossless.
Exercise 7: An O-type TWT operates at 8 GHz. The slow-wave structure has a pitch angle of 4.4 and an attenuation constant of 2 Np/m. Determine the propagation constant of the traveling wave in the tube.
Exercise 8: In an O-type TWT, the acceleration voltage is 3000 V. The characteristic impedance is 10 . The operating frequency is 10 GHz and the beam current is 20 mA. Determine the propagation constants of the four modes of the traveling waves.
Exercise 9: In an O-type TWT, the acceleration voltage is 4000 V and the magnitude of the axial electric field is 4 V/m. The phase velocity on the slow-wave structure is 1.1 times the average electron beam velocity. The operating frequency is 2 GHz. Determine the magnitude of velocity fluctuation.
All magnetrons consist of some form of anode and cathode operated in a dc magnetic field normal to a dc electric field between the cathode and anode. Because of the crossed field between the cathode and anode, the electrons emitted from the cathode are influenced by the crossed field to move in curved paths. If the dc magnetic field is strong enough, the electrons will not arrive in the anode but return instead to the cathode. Magnetrons can be classified into three types:
Split-anode magnetron: This type of magnetron uses a static negative resistance between two anode segments.
Cyclotron-frequency magnetron: This type operates under the influence of synchronism between an alternating component of electric field and a periodic oscillation of electrons in a direction parallel to the field.
Traveling-wave magnetrons: This type depends on the interaction of electrons with a traveling electromagnetic field of linear velocity. They are customarily referred to simply as magnetrons.
A schematic diagram of a cylindrical magnetron is shown below. It consists of a cylindrical cathode of finite length and radius a at the center surrounded by a cylindrical anode of radius b. The anode is a slow wave structure consisting of several reentrant cavities equi-spaced around the circumference and coupled together through the anode cathode space by means of slots. Radial electric field is established by dc voltage Vo in between the cathode and the anode and a dc magnetic flux denoted by Bo is maintained in the positive z direction by means of a permanent magnet or an electromagnet.
The equations of motion for electrons in an electric field in cylindrical coordinates are written as under. The cylindrical coordinates are defined as in the following figure. It can be seen that:
Z = z
are unit vectors in r, , z directions. is constant, but and are functions of .
The position vector:
In order to determine the path of an electron in an electric field, the force must be related to the mass and acceleration by Newton’s second law of motion. So
The differential equations of motion for an electron in an electric field in rectangular coordinates are:
Ex, Ey and Ez are the components of E in rectangular coordinates. The equations for electron in an electric field in cylindrical coordinates are:
The force exerted on the electron moving with a velocity v by the magnetic field of flux density B is:
The equations of motion of the electron in a magnetic field in rectangular coordinates are:
The equations of motion of the electron in a magnetic field in cylindrical coordinates are:
If both electric and magnetic fields exist simultaneously, the motion of the electron depends on the orientation of the two fields. If the two fields are in the same or in opposite directions, the magnetic field exerts no force on the electron, and the electron motion depends only on the electric field. When the electric field and the magnetic field are at right angles to each other, a magnetic force is exerted on the electron. This type of field is called a crossed field.
In a crossed-field tube such as magnetron, electrons emitted by the cathode are accelerated by the electric field and gain velocity; but the greater their velocity, the more their path is bent by the magnetic field. The Lorenz force acting on an electron because of the presence of both the electric field E and the magnetic flux B is given by:
The equations of motion for electrons in a crossed field are expressed in rectangular coordinates and cylindrical coordinates as:
Since the magnetic field is normal to the motion of electrons that travel in a cycloidal path, the outward centrifugal force is equal to the pulling force.
, where R is the radius of the cycloidal path.
The cyclotron angular frequency of the circular motion of the electron is then given by:
The equations of motion for electrons in a cylindrical magnetron can be written as:
is assumed in the positive z direction. Multiplying the above equation results in:
Integration of the above equation yields:
where K = constant. At , where a is the radius of the cathode cylinder, . Therefore,
Therefore, the above equation becomes:
Since the electrons move in direction perpendicular to the magnetic field, the kinetic energy of the electrons is given by the electric field only, so:
However, the electron velocity has r and components such as:
At , where b is the radius from the center of the cathode to the edge of the anode, V = Vo, and , when the electrons just graze the anode. The above equations become:
The electron will acquire a tangential as well as a radial velocity. Whether the electron will just graze the anode and return toward the cathode depends on the relative magnitudes of Vo and Bo. The Hull cutoff magnetic equation is obtained from the above equation. As defined earlier,
This means that if Bo > Boc for a given Vo, the electrons will not reach the anode. Conversely, the cutoff voltage is given by:
This means that if Vo > Voc for a given Bo, the electrons will not reach the anode. Voc is called the Hull cutoff voltage equation.
The trajectories of the electrons are shown for different magnetic field strengths in the figure. At zero magnetic field, the electron takes the straight path a’, by the influence of electric field only. For a given Vo if the magnetic field is increased, the electrons take curved path b’ to reach the anode.
At a critical value of the magnetic field Boc, they electrons just graze the anode surface at radius b and take the path c’ to return to the cathode for a given voltage Vo. If the magnetic field is above Boc, all the electrons return to the cathode by a typical path d’ without reaching the anode.
The period of one complete revolution can be expressed as:
Since the slow-wave structure is closed on itself, or reentrant, oscillations are possible only if the total phase shift around the structure is an integral multiple of radians. Thus, if there are N reentrant cavities in the anode structure, the phase shift between two adjacent cavities can be expressed as:
, where n is an integer indicating the nth mode of oscillation.
Magnetron oscillators are operated in the mode. That is,
The equivalent resonant circuit of a magnetron is shown below.
The following figure shows the lines of force in the mode of an eight-cavity magnetron. It is evident that in the mode, the excitation is largely in the cavities, having opposite phase in successive cavities.
The successive rise and fall of adjacent anode-cavity fields may be regarded as a traveling wave along the surface of the slow-wave structure. For the energy to be transferred from the moving electrons to the traveling field, a retarding field must decelerate the electrons when they pass through each anode cavity. If L is the mean separation between cavities, the phase constant of the fundamental mode field is:
Since each mode corresponds to a different frequency, the various modes are detuned differently from the fundamental resonant frequency of the cavities but differ very little in frequency from each other as shown below. This makes it difficult to separate the -mode from the next immediate mode for maximum excitation of the cavities. Hence it is possible for the frequency to jump from one mode to another, which is highly undesirable.
The separation of -mode frequency from other modes is commonly done by the strapping method as shown below. In this method, two metallic rings are arranged with one ring connected to the even numbered anode and the others to the odd numbered anode poles. Thus for -mode each ring is at same potential and no -mode current flows in the straps and the straps inductance has no effect. But the two rings having opposite potentials provide a capacitive loading in parallel to the capacitance C at each slot of the resonant cavities and lower the frequency of -mode.
For other modes, each ring experiences a phase difference between the successive connection points and the resulting current flow gives rise to an inductive field with reduced capacitive effect. The inductance is in parallel with the slot, thus raising the unwanted mode frequency. Therefore, the strapping method increases the frequency separation between the -mode and the higher adjacent modes.
The electrons emitted from the cathode try to travel towards the anode with the influence of dc electric field E but take a curved trajectory with the influence of dc magnetic field B. One can postulate the existence of RF oscillations in the resonant structure due to noise voltages in the dc biasing circuit. The RF electric field at the resonant frequency of the -mode structure in the slot of the cavities and fringes out in the cathode anode space is shown below.
When the dc magnetic field exceeds the cutoff value, the electrons try to return back to the cathode in the absence of the RF field. Due to fringing RF electric field, the electron a experiences a retarding electric field and electron b experiences accelerating electric field. The retarded electron a experiences reduced magnetic force due to its reduced velocity and moves towards the anode. By adjusting the dc anode voltage and dc magnetic field, the circumferential velocity component of the electron can be made such that the electron a takes approximately one half cycle of the RF oscillation to travel from one slot position to the next. This makes electron a experience a retarding field and ultimately reaches the anode surface after continuously delivering energy to the RF oscillations.
On the other hand, the RF field accelerates the electron b to return quickly to the cathode. Thus since electron b remains in the interaction space for a much shorter duration compared to the electron a, the energy absorbed by b is much smaller than the energy delivered by a to the system to sustain oscillations.
The electrons around a, such as c and d are acted upon both radial and tangential component of the RF field in such a way that the electron c moves faster than a and that d moves slower than a to form a bunch around a. Then these electrons from c to d are confined to spokes and terminated to the alternate anodes. This is called phase focusing. For -mode these spokes have angular velocity equal to two anode poles per cycle and the electrons within the spokes deliver energy to the oscillations before the anode collects them.
For strong interaction between the wave on the anode structure and the electron beam, the phase velocity of the wave should be nearly equal to the drift velocity. The oscillations for -mode start at:
VoH is known as the Hartree voltage. Here f is the operating frequency and N is the number of the resonators. A plot of Hull cut-off voltage and the Hartree voltage versus Bo is shown below. The region of oscillations is indicated by shaded area.
The frequency of a conventional magnetron can be changed by mechanically inserting a tuning element, such as a rod, into the holes of the hole-and-slot resonators to change the inductance of the resonant circuit.
A magnetron can deliver a peak power output up to 40 MW with the dc voltage of 50 kV at 10 GHz. The average power output is of the order of 800 kW. The magnetrons have a very high efficiency ranging from 40 to 70%. The magnetrons are used in radar transmitters, industrial heating, and microwave ovens. A microwave oven requires a standard power of 600 W and frequencies 915 MHz or 2450 MHz.
Exercise 1: A pulsed cylindrical magnetron is operated with the following parameters:
Anode voltage = 25 kV
Beam current = 25 A
Magnetic density = 0.34 Wb/m2
Radius of cathode cylinder = 5 cm
Radius of anode cylinder = 10 cm
Calculate (a) the angular frequency, (b) the cut-off voltage, and (c) the cut-off magnetic flux density.
Exercise 2: A cylindrical magnetron is operated at 5 GHz with a = 3 cm, b = 5 cm, N = 16, Vo = 20 kV, Bo = 0.05 T. Calculate the Hull cut-off voltage and cut-off magnetic flux density, and Hartree voltage. How do the cut-off voltage and Hartree voltage vary with Bo? Indicate the operating range.
Exercise 3: An X-band pulsed cylindrical magnetron has Vo = 30 kV, Io = 80 A, Bo = 0.01 Wb/m2, a = 4 cm, and b = 8 cm. Calculate the cyclotron angular frequency, cut-off voltage and cut-off magnetic flux density.
Exercise 4: For a magnetron a = 0.6 m, b = 0.8 m, N = 16, Bo = 0.06 T, f = 3 GHz, and Vo = 1.6 kV. Calculate the average drift velocity for electrons in the region between the cathode and anode.
Forward-wave crossed-field amplifier
The crossed-field amplifier (CFA) is an outgrowth of the magnetron. CFAs can be grouped by their electron stream source as emitting-sole or injected-beam type.
In the emitting-sole tube, the current emanated from the cathode is in response to the electric field forces in the space between the cathode and anode. The amount of current is a function of the dimension, the applied voltage, and the emission properties of the cathode. The perveance of the interaction geometry tends to be quite high, which results in a high-current and high-power capability at relative low voltage.
In the injected-beam tube (see below) the electron beam is produced in a separate gun assembly and is injected into the interaction region.
The beam-circuit interaction features are similar in both the emitting-sole and the injected-beam tubes. Favorably phased electrons continue toward the positively polarized anode and are ultimately collected, whereas unfavorably phased electrons are directed toward the negative polarized electrode.
The figure shows the pattern of the electron flow in the CFA. When the spoke is positively polarized or the RF field is in the positive half cycle, the electron speeds up toward the anode; while the spoke is negatively polarized or the RF field is in the negative half cycle, the electrons are returned toward the cathode. Consequently, the electron beam moves in a spiral path in the interaction region.
The total power generated in a given CFA is independent of the RF input power, as long as the input power exceeds the threshold value for spoke stability at the input. Only increasing the anode voltage and current can increase the power generated. Therefore, the CFA is not a linear amplifier but rather is termed a saturated amplifier. The CFA is characterized by its:
low or moderate power gain,
small in size,
low weight, and
CFAs are used in a variety of electronic systems ranging from low-power and high-reliability space communications to multimegawatt, high average power, coherent pulsed radar.
Backward-wave crossed-field oscillator (Carcinotron)
The backward-wave crossed-field oscillator or carcinotron has two configurations: linear carcinotron and circular carcinotron. A linear carcinotron is shown below. The interaction between the electrons and the slow-wave structure takes place in a space of crossed field. The slow-wave structure is in parallel with an electrode known as the sole. A dc electric field is maintained between the grounded slow-wave structure and the negative sole. A dc magnetic field is directed into the page. The electrons emitted from the cathode are bent through 90 angle by the magnetic field.
The electrons interact with a backward-wave space harmonic of the circuit, and the energy in the circuit flows opposite to the direction of the electron motion. The slow-wave structure is terminated at the collector end, and the RF signal output is removed at the electron-gun end. Its efficiency is very high, ranging from 30 to 60%.
The perturbed electrons moving in synchronism with the wave are shown below. Electrons at position A near the beginning of the circuit are moving toward the circuit, whereas electrons at position B are moving toward the sole. Farther down the circuit, electrons at position C are closer to the circuit, and electrons at position D are closer to the sole. However, electrons at position C have departed a greater distance from the unperturbed path than have electrons at position D.
Thus, the electrons have lost a net amount of potential energy, this energy having been transferred to the RF field. The reason for the greater displacement of the electrons moving toward the circuit is that these electrons are in stronger RF fields, since they are closer to the circuit. Electrons at position G have moved so far from the unperturbed position that some of them are being intercepted on the circuit. The length from position A through position G is a half cycle of the electron motion.
In the circular carcinotrons, the slow-wave structure and sole are circular and nearly reentrant to conserve magnet weight. The sole has the appearance of the cathode in a magnetron.
In the circular configurations, the delay line is terminated at the collector end by spraying attenuating material on the surfaces of the conductors. The output is taken from the gun end of the delay line, which is an Interdigital line. Clearly, in this case, the electron drift velocity has to be in synchronism with a backward-space harmonic.