A simple analytical model of the magnetron driven by a resonant signal

1. A simple analytical model of the magnetron driven by a resonant signal G. Kazakevich*, V. Lebedev** *Muons, Inc., Batavia, 60510 IL, USA **Fermilab, Batavia, 60510 IL, USA IPAC 2017, Muons, Inc - Fermilab

2. Simple model of the charge drift approximation for a magnetron driven by a resonant signal We discuss a N-cavities (N is an even number) conventional CW magnetron with a constant uniform magnetic field H, operating in the mode with the RF electric field shifted by p in the neighbour cavity gaps. The magnetron operates at the frequency w, being loaded by a matched load with a negligible reflected signal. In the drift approximation the motion of a charge is described by motion of Larmor orbit centre. It implies that the Larmor radius, rL, is sufficiently small and the motion can be averaged over the cyclotron frequency, w. Both conditions are satisfied in a typical magnetron. We also neglect the impact of space charge and the azimuthal non-uniformity of the static electric field. Drift of the Larmor orbit with azimuthal angular velocity W in the uniform magnetic field is determined by superposition of the static electric field described by the static electric potential F0 and the RF field of the synchronous wave induced by the magnetron current and the injected resonant RF signal. In the quasi-static approximation the synchronous wave can be determined by a scalar potential F. IPAC 2017, Muons, Inc - Fermilab

3. The drift of the Larmor center can be described in the polar frame by the equations: For the static electric field: and Er=gradF0, , therefore, Ej(r)=0. Here U is the magnetron voltage, and r1 and r2 are the magnetron cathode and anode radii, respectively. In a conventional magnetron (w <<pc/r1, pc/r2), the quasi-static approximation can be used to describe the rotating synchronous wave in the magnetron space of interaction. In this case, the scalar potential F can be presented in the following form: Here is the amplitude of k-th harmonic of radial RF electric field at r=r1. The form of the potential was chosen so that the azimuthal electric field at the cathode is zero. Coefficients are determined to have zero azimuthal electric field at the anode everywhere except the coupling slits of the cavities. At n=k we have a resonant interaction. To further simplify consideration we leave only resonant term of the rotating wave. Its frequency coincides with operating frequency of magnetron, w. IPAC 2017, Muons, Inc - Fermilab

4. In the coordinate frame rotating with the synchronous wave for and for an effective potential one obtains the system of the drift equations: Substituting the potential FS into the equations and denoting: one can get: Here: and E1 is the static electric field at r=r1. • The drift of charge towards the anode is possible at with a period of 2p, i.e., only in "spokes“.The condition e≥1 does not allow operation of the magnetron. • The second equation describes the charge grouping in “spokes”. • In magnetrons rL<<2pc/w, therefore we consider drift of the Larmor circles IPAC 2017, Muons, Inc - Fermilab

5. For a typical magnetron with N =8, r1=5mm, r2/r1 =1.5, rS/r1=1.2, at r ≥r1+rL at the time interval of the drift, t: (1/pw ≤t ≤ 5/pw) for various magnitudes e of the RF field in the synchronous wave was computed part of the charge attained the magnetron anode. In magnetrons the phase grouping is manifested in the variation of the azimuthal component of the drift velocity of the charge caused by the variation of the phase-dependent radial electric field of the synchronous wave. Since the radial drift velocity is insignificantly varied over the phase range about of 1 rad, one can consider the variations of azimuthal component of the drift velocity vaz(njS)= vaz (y) in units of c for various e at the given radius r. The plots show variation the modulus of the phase-dependent azimuthal drift velocity at e =0.1, 0.2 and 0.3, respectively, at r =6.5 mm for the typical magnetron model. The arrows A and B show deceleration and acceleration of the azimuthal velocity of charge by the synchronous wave, respectively; the first one is necessary for radiation of charge. IPAC 2017, Muons, Inc - Fermilab

6. The first plot demonstrates absence of the phase grouping of the charge at low e. Deterioration of the phase grouping of the charge is caused by a decrease of magnitude of the electric field in the synchronous wave. This causes deterioration of coherency in magnetron oscillation. The phenomenon is well known as a noise frequently called as a ”pre-oscillation” in free-running tubes when the feeding voltage is approaches to the Hartree voltage. In the considered typical magnetron model a part the drifting charge at e ≤0.25 migrates from the phase interval (±p/2) allowed for a “spoke”. The emigrated electrons absorb the RF energy from the synchronous wave and hit the cathode. The loss of charge attained the anode reduces the magnetron current in a “spoke” and magnitude of the induced synchronous wave making the phase grouping insufficient. A resonant RF driving signal injected in the magnetron in accordance with the energy conservation law increases the RF energy stored in the space of interaction and in the magnetron cavities. Since the RF energy in the magnetron is determined by the static electric field, the injected resonant signal is equivalent to an increase of the magnetron feeding voltage. Thus a sufficient injected resonant signal will allow the magnetron start-up even if the magnetron feeding voltage is somewhat less than the threshold of self-excitation. In this case the magnetron current is less than the minimum current allowable in free run. This explains stable operation of the magnetron driven by a sufficient resonant signal in an extended range of current (power) control. A lack of RF voltage in the synchronous wave induced by the lower magnetron current is compensated by the injection-locking signal providing stable operation of the tube. IPAC 2017, Muons, Inc - Fermilab

7. The minimum free running magnetron power, Pmin, is usually ~1/3 of the nominal power, Pnom, and corresponds to the threshold voltage. For good magnetrons the energy of the RF field in the interaction space is approximately in 3 times less than the energy stored in the magnetron RF system including the interaction space and cavities; half of the energy in interaction space is associated with the RF field in the synchronous wave. Thus injection of the resonant driving wave with power PD is equivalent to decrease of U by: Then the minimum current, IminD, of the driven magnetron with the dynamic impedance ZD=DU/DI one estimates as: IminD ~Imin -DU/ZD, where Imin is the minimum current in free run. The power range, RD, allowed for current regulation in the driven magnetron one estimates at the nominal magnetron current, Inom, as: RD~ (Inom/ImunD). For the magnetron model at PD~ -10 dB of the nominal power one obtains: RD~ 10 dB. Extended range of power control measured with 2.5 GHz, 1 kW magnetrons and obtained by the rough estimations. IPAC 2017, Muons, Inc - Fermilab

8. Experimental verification of the proposed kinetic model V-I characteristic of 1.2 kW measured at PLock =100 W. The solid line (B-spline fit) shows available range of current with stable operation of the 2.45 GHz CW tube at given PLock. Relative magnetron efficiency vs. range of power control for various methods of control. Deterioration of coherency in generation of a magnetron at low power at various injected resonant signal. The magnetron operates below the threshold of self-excitation. Offset of the carrier frequency at various power levels of the magnetron, PMag, and the locking signal, PLock. IPAC 2017, Muons, Inc - Fermilab

9. PMag=1000 W The noise spectral density of 2.5 GHz magnetron type 2M137-IL driven by -10 dB resonant signal and fed by a switching power supply with a current feedback loop. PLock=100 W PMag=100 W PLock=30 W PLock=100 W PLock=10 W PLock=50 W PLock=100 W; HV is OFF PLock=30 W IPAC 2017, Muons, Inc - Fermilab

10. Highly-efficient magnetron transmitter Dynamic power control in magnetrons 1- the low power magnetron, 2- the high-power magnetron, 3- the low-power HV power supply controlled within the feedback loop, 4- the main uncontrolled HV power supply, 5- the current/voltage controller within the LLRF system for the low-power supply, 6- the phase controller within the LLRF system. Modelling of a dynamic power control by a wide-range management of magnetron current with a harmonic signal controlling the HV switching power supply with a current feedback loop. Summary A simplified kinetic model utilizing the charge drift approximation for the magnetron driven by a resonant (injection-locking) signal has been developed and verified in experiments with CW, 2.45 GHz, 1 kW magnetrons. The model considering the resonant grouping in the tube by a synchronous wave predicts and explains stable operation of magnetrons driven by a sufficient resonant signal and being controlled in power in a wide range (up to 10 dB) by the management of magnetron current at low noise and highest efficiency. Thank you! IPAC 2017, Muons, Inc - Fermilab