Peter N. Ostroumov and Ali Nassiri (ANL) PHY862 Accelerator Systems Microwave Engineering for Large Accelerators Peter N. Ostroumov and Ali Nassiri (ANL)

Content Literature Shunt resonant circuit model RF power coupling to cavities Equivalent circuit for a resonant cavity system Equivalent circuit for a cavity coupled to two waveguides Transient behavior of a resonant cavity system Wave description of a waveguide to cavity coupling Glossary of terms related to RF RF power sources Performance and efficiency Coaxial transmission lines and waveguides Thomas Wangler, RF Linear Accelerators, Chapter 5 David M. Pozar, Microwave Engineering, Chapter 3 “transmission lines and waveguides” Literature P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

RF System for Accelerating Cavity Major sub-systems: Master oscillator, RF amplifier with power supply, Resonator, Circulator, Low Level RF (LLRF), transmission line P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Shunt resonant circuit model Current generator supplies current I(t) at frequency  V is the axial voltage in the resonator Resonant frequency Stored energy Dissipated power Quality factor Shunt resistance. The accelerator shunt resistance (impedance) is twice of circuit shunt resistance (impedance) See also V. Yakovlev, Lecture 6, p.53 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Steady state solution of the damped driven oscillator equation Assume I0 , V0 are real amplitudes and  is the phase of resonator voltage with respect to the driving current Steady state solution of the damped driven oscillator Assume I0 , V0  const Oscillator equation Use P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Solution Use the following properties of complex values Fall 2019 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Shunt impedance Now, we have Because V0 is a real value, detuning factor Circuit shunt impedance is At resonance, therefore, the impedance is real P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Resonance curve Voltage vs frequency deviation, Resonator bandwidth P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

General solution of damped oscillator equation (9-1) c1 and c2 are constants and depend from initial conditions Resonant frequency for damped oscillator and time constant We choose , assume which results in We are looking for transient change of the voltage while resonator is driven with external current at the same frequency: Therefore: , and P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Voltage transients Therefore the real part of the equation on p. 9 becomes After an interval of several time constants, the voltage is at the steady-state value of Now, we assume what happens if we turn off the external current at arbitrary phase  at Initial conditions are satisfied by choosing Voltage decay is P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Coupling of electromagnetic energy to a cavity (resonator) a magnetic-coupling loop at the end of a coaxial transmission line a hole or iris in a cavity wall to which a waveguide is connected an electric-coupling probe or antenna, using an open-ended center conductor of a coaxial transmission line Main condition: electric or magnetic field distribution created with the coupler should overlap with the field distribution created by unperturbed mode of oscillations in the cavity Similar to a conventional transformer See also V. Yakovlev, Lecture 6, p.51-52 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

RF power transmission lines Most popular transmission lines in accelerators are: coaxial lines and rectangular waveguides Main requirement: low power losses, high electrical conductivity material P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Coaxial transmission line Circuit analysis: physical dimensions of a network are much smaller than wavelength Transmission lines: a distributed network, voltages and currents can vary in magnitude and phase over its length. The size of transmission line can be many wavelengths P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Voltage and current over z (b) Lumped-element equivalent circuit. (a) Voltage and current definitions. Voltage and current definitions and equivalent circuit for an incremental transmission line. The series inductance L represents the total self-inductance of the two conductors, and the shunt capacitance C  is due to the close proximity of the two conductors. The series resistance R represents the resistance due to the finite conductivity of the individual conductors, and the shunt conductance G is due to dielectric loss in the material between the conductors. R and G, therefore, represent loss. A finite length of transmission line can be viewed as a cascade of sections of the form shown in Figure above P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Telegrapher equations Kirchhoff’s voltage law can be applied to give Kirchhoff’s current law leads to Divide by z and taking the limit as z → 0 gives the following differential equations: P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

For the sinusoidal steady-state condition P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

For the sinusoidal steady-state condition These equations result in two wave equations Where  is the complex propagation constant, function of frequency P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Traveling wave solution where the e−γz term represents wave propagation in the +z direction, and the eγz term represents wave propagation in the −z direction The characteristic impedance is (Voltage / Current) P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Lossless transmission line In practice the loss in the transmission line is vey small Zero shunt conductance equivalent to perfect isolation without power losses Which results in Voltage and current can be written as The wavelength the phase velocity P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Reflections in terminated lossless transmission line Voltage and current in the transmission line already includes reflected wave At z=0 we have Solving for gives 𝑉 0 − P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Reflection coefficient Very important parameter for loaded transmission lines We always try to minimize reflection Maximum possible power goes into the load: accelerating cavity in our case P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Characteristic impedance of coaxial line, C Calculate capacitance per unit length as charge per unit length divided by voltage between the inner and outer conductor Apply Gauss Law [Farads/m] P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Characteristic impedance of coaxial line, L We will calculate inductance per unit length Apply Ampere Law [Henry/m] P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Parameters of coaxial line Rs is the surface resistance, depends from the frequency and conductance of material The conductance of isolator is close to zero In the case of air 1 The characteristic impedance of the transmission line is P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Equivalent Circuit for a Resonant-Cavity System Block diagram of RF system components and the equivalent circuit Describes steady state behavior of the system Circulator: transmits forward waves going into the cavity but absorbs all backward waves in the matched load Z0 Circuit impedance P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Equivalent circuit transformed into the generator circuit RF coupler acts as a step-up transformer At resonance: P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Stored energy, power The resonator stored energy is Average power dissipated in the resonator Power dissipated in the external load is Unloaded quality factor P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Intrinsic and external quality factor The loaded quality factor is Relationship between QL, Qex and Q0 Transmission line-cavity coupling strength,  See also V. Yakovlev, Lecture 6, p.52 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Coupling strength <1 – undercoupled >1 – overcoupled =1 – critically coupled. In this case Relationship between  and n P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Cavity time constant Voltage change in the cavity when generator is turned on or off Time constant P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Maximum average power to the load (cavity) To deliver maximum power, the load should be matched, i.e. the load resistance = transmission line impedance This condition is satisfied if =1 and (driving in resonance) then Maximum average power to the cavity is If   1, power will be reflected from the load and power delivered to the cavity will be less than Pmax P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Resonator voltage as a function of coupling strength  Equivalent circuit transformed into the resonator circuit to determine maximum voltage in the cavity then P+ is the forward power from the generator to resonator Cavity at resonance: is the load impedance Steady state voltage on the cavity at resonance Resonator shunt impedance is twice of the circuit shunt impedance, R, Therefore P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Maximum voltage in the resonator as a function of  Calculate Critical coupling provides maximum voltage in the cavity at given forward power, P+ P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Arbitrary N-port microwave network The ports may be any type of transmission line or transmission line equivalent of a single propagating waveguide mode At a specific point on the nth port, a terminal plane, tn, is defined along with equivalent voltages and currents for the incident (V+n, I+n) and reflected (V−n, I−n) waves. The terminal planes are important in providing a phase reference for the voltage and current phasors. At the nth terminal plane, the total voltage and current are given by P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Summary of Lecture 1 “Microwave topics” Equivalent circuit of the resonator Relationship between accelerating cavity (resonator) and circuit parameters Resonator impedance which contains only real part when in resonance Damped oscillator equation; partial and general solution, voltage as a function of time Telegrapher’s equation Relationship between available RF power, accelerating voltage, forward and reflected wave, transmission line with and without losses, reflection coefficient, maximum power in the resonator, characteristic impedance Coaxial lines and waveguide Characteristic impedance of coaxial line Coupling strength,  Under coupled, critically coupled, over coupled P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Not critically coupled resonator,   1 Equivalent circuit transformed into the current generator circuit For a line terminated in its own characteristic impedance there is no reflection. Reflection coefficient is not zero For =1, =0. Thus, critical coupling results in a matched cavity load as seen from the waveguide or transmission line As β → 0,  →-1 , which corresponds to a short-circuit load, and as β →,  →+1, which is an open-circuit load P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Power in cavity as a function of coupling strength If P+ is the incident power from the generator, the power propagating back from the input coupler toward the generator is From energy conservation, the power delivered into the cavity through the input power coupler is As we already showed, Pc is maximum if =1 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Scattering matrix The scattering matrix, or [S] matrix, is defined in relation to incident and reflected voltage waves as P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Network analyzer Measurement of the scattering matrix elements S11, S12, S21 S22 (two-port measurements , usually, driving frequency is swept) S11 input reflection coefficient , S11 =-1 – full reflection S12 reverse transmission coefficient S21 forward transmission coefficient (resonance curve), pickup signal S22 output reflection coefficient Impedance, matching, Q-factor,… P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

S11 and S21 Resonant frequency is 160.93 MHz Frequency span is 30 kHz Critically coupled, =1 QL= 7527 Q0= 27527=15054 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Electromagnetic Spectrum All the electromagnetic waves travel with the same velocity (i.e. 3  108 m/s) in the free space with different frequencies. The arrangement of electromagnetic radiations according to wavelength or the frequency is referred as electromagnetic spectrum. As we know electromagnetic spectrum has no definite upper or lower limit and various regions of EM spectrum do not have sharply defined boundaries. The electromagnetic spectrum types, their frequency, wavelength, source and applications have been outlined in the table below. As mentioned EM waves include electric wave, radio wave, microwave, infrared, visible light, ultra violet, X-rays, gamma rays and cosmic rays. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Electromagnetic Spectrum EM Wave Type Frequency (Hz) Wavelength (m) Source Applications Electric wave 50 to 60 5 x 106 to 6 x 106 Weak radiation from AC circuit Lighting Radio wave 3 x 104 to 3 x 109 1 x 10-1 to 104 Oscillating circuits Radio communication, TV Microwave 3 x108 to 3 x 1011 1 x 10-3 to 1 Oscillating current in special vacuum tubes, Gunn, IMPATT, Tunnel diodes Radar, TV, Satellite communication, remote sensing. Infrared 1 x 1013 to 4 x 1014 7.5 x 10-7 to 3 x 10-5 Excitation of atoms and molecules Gives information on the structure of molecules and of external atomic electron shells, remote sensing. Visible light 4 x 1014 to 8 x 1014 3.75 x 10-7 to 7.5 x 10-7 Excitation of atoms and vacuum spark Gives information on the structure of molecules and of external atomic electron shells, remote sensing Ultra violet 8 x 1014 to 1 x 1016 3 x 10-8 to 3.75 x 10-7 X-ray 1 x 1016 to 3 x 1019 1 x 10-10 to 3 x 10-8 Bombardment of high atomic number target by electrons X-rays therapy, industrial radiography, medical radiography, crystallography Gamma Ray 3 x 1019 to 5 x 1020 6 x 10-15 to 1 x 10-10 Emitted by radioactive substances Gives information about the structure of atomic nuclei Cosmic Ray > 1020 < 10-11 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Glossary of terms as relates to RF Frequency range Klystrons are dominant and used above 300 MHz Other devices such as IOT, Tetrode and Solid State Amplifiers are below 300 MHz Peak power Is related to energy gain in an accelerating structures. High peak power typically results in arcing within the accelerating structures. Average power Is defined as the product of peak power and Duty Factor in pulsed systems For CW systems, the output power is equal to the average power. Defines the amount of heat produced by the system (amplification) Gain Klystrons, in general, have a high gain ~50 dB ( i.e. less drive power). IOTs are low gain devices- ~20 dB ( i.e., more drive power). Phase Stability Klystron is a voltage driven device and the rf phase is stable if the voltage is stable. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Glossary of terms as relates to RF Decibel (dB) 𝑑𝐵𝑚=10 𝐿𝑜𝑔10("Power in mW") 𝑑𝐵=10 𝐿𝑜𝑔10(𝑃1/𝑃2) 𝑑𝐵=20 𝐿𝑜𝑔10(𝑉1/𝑉2) 𝑑𝐵𝑉=20 𝐿𝑜𝑔10("Voltage Vrms") 𝑑𝐵µ𝑉=20 𝐿𝑜𝑔10(𝑉oltage µ𝑉𝑟𝑚𝑠) 𝑑𝐵𝑐=10 𝐿𝑜𝑔10(𝑃𝑐𝑎𝑟𝑟𝑖𝑒𝑟/𝑃𝑠𝑖𝑔𝑛𝑎𝑙) dBm, W (𝑥𝑑𝐵𝑚/10) 𝑃 = 10 𝑥 = 10 𝐿𝑜𝑔 P 𝑑𝐵𝑚 10 mW 𝑚𝑊 dBm = 1 mW 30 W 60 kW 90 MW P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Glossary of terms as relates to RF 3 dB  factor of 2100.3 — 𝑃∕𝑃𝑟𝑒𝑓 [dB]=10 (𝑥/10) 𝑥𝑑𝐵=10 𝐿𝑜𝑔10 (𝑃/𝑃𝑟𝑒𝑓) P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

RF Power Sources Two principal classes of microwave vacuum devices are in common use today: o Linear-beam tubes o Crossed-field tubes Linear Beam Devices Klystrons Hybrid O-type TWT Twystron Multi- cavity Two - cavity Helix Ring-bar TWT Helix BWO Reflex Laddertron Coupled cavity TWT 46 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Linear Beam Devices In a linear-beam tube, as the name implies, the electron beam and the circuit elements with which it interacts are arranged linearly. In such a device, a voltage applied to an anode accelerates electrons drawn from a cathode, creating a beam of kinetic energy. Power supply potential energy is converted to kinetic energy in the electron beam as it travels toward the microwave circuit. A portion of this kinetic energy is transferred to micro-wave energy as RF waves slow down the electrons. The remaining beam energy is either dissipated as heat or returned to the power supply at the collector. Because electrons will repel one another, there usually is an applied magnetic focusing field to maintain the beam during the interaction process. 47 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Cross-Field Devices The magnetron is the pioneering device of the family of crossed-field tubes. Although the physical appearance differs from that of linear-beam tubes, which are usually circular in format, the major difference is in the interaction physics that requires a magnetic field at right angles to the applied electric field. Whereas the linear-beam tube sometimes requires a magnetic field to maintain the beam, the crossed-field tube always requires a magnetic focusing field. Crossed-field devices Distributed emission Injected beam tube MBWO Carcinotron Magnetron Crossed-field amplifier Voltage tunable magnetron Crossed field amplifier P.N. Ostroumov PHY862 "Accelerator Systems" 48 Fall 2019

Magnetron P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Commercial RF Sources 10000 peak < 1 Power kW per single tube 1000 Tetrodes & Diacrodes available from industry 10000 peak < 1 Power kW per single tube 1000 ms 100 10 100 200 300 Frequency MHz 400 500 PHY 862 Accelerator Systems 12 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Amplifier Class Class A Class B Class C C AB B A A AB C Fall 2019 13 Operative curve Class B Class C Operative curve Operative curve Output Signal Output Signal Less than 180⁰ Unsused area Output Signal Unsused area Input Signal Input Signal Input Signal Amplifier Class Description Class-A Full cycle 360⁰ of conduction Class-AB More than 180⁰ of conduction Class-B Half cycle 180⁰ of conduction Class-C Less than 180⁰ of conduction C Efficiency 100% 75% 50% 25% 0% AB B 180⁰ π Co A A AB C 360⁰ 270⁰ 3π/4 0⁰ nduction Angle 0 90⁰ 2π PHY 862 Accelerator Systems 13 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Grid Vacuum Tubes The physical construction of a vacuum tube causes the output power and available gain to decrease with increasing frequency. The principal limitations faced by grid-based devices include the following: Physical size. Ideally, the RF voltages between electrodes should be uniform, but this condition cannot be realized unless the major electrode dimensions are significantly less than 1/4 wavelength at the operating frequency. This restriction presents no problems at <1 GHz, but as the operating frequency increases into the microwave range, severe restrictions are placed on the physical size of individual tube elements. Electron transit time. Inter electrode spacing, principally between the grid and the cathode, must be scaled inversely with frequency to avoid problems associated with electron transit time. Possible adverse conditions include: 1) excessive loading of the drive source, 2) reduction in power gain, 3) back-heating of the cathode as a result of electron bombardment, and 4) reduced conversion efficiency. Voltage standoff. High-power tubes operate at high voltages. This presents significant problems for microwave vacuum tubes. For example, at 1 GHz the grid-cathode spacing must not exceed a few mils. This places restrictions on the operating voltages that may be applied to the individual elements. Circulating currents. Substantial RF currents may develop as a result of the inherent inter electrode capacitances and stray inductances/capacitances of the device. Significant heating of the grid, connecting leads, and vacuum seals may result. Heat dissipation. Because the elements of a microwave grid tube must be kept small, power dissipation is limited. 52 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Tetrode Vacuum tube based on intensity modulation of a electron beam Typical parameters: Frequency: accelerator applications up 300 to 400MHz Finite electron drift time limits the achievable gain at higher frequencies Limiter gain of ~15 dB mean that thigh power tetrode amplifiers need 2-3 stage of amplification, which drives up the cost and results in complicated amplifier systems. Grounded Grid Grounded Cathode P.N. Ostroumov PHY862 "Accelerator Systems" 53 Fall 2019

IOT Inductive Output Tube (IOT) Klystrode IOT developed for accelerators [Thales, CPI]:80 kW CW at 470 –760 MHz High efficiency (70%) operation in class B Intrinsic low gain ( 20– 25 dB) ⇒Pin= 1 Kw Less gain than klystrons but higher than tetrodes No need for a long drift space ( like klystrons) More compact and cost efficient IOTs ( and all gridded tubes) are limited in their frequency reach by the distance of the control grid from the cathode. The RF period has to be smaller than the time of flight from cathode to this grid. Frequency of IOTs is limited to ~ 1.3 GHz The max. power of a single beam IOTs is limited to ~100 kW. Less Amplitude/Phase sensitivity to HV ripples Compact, external cavity ⇒easy to handle Low unit power ⇒power combiners 1.3 GHz for cw XFEL linacs and ERLs o 16 to 20 kW CW, efficiency 55 to 65% ( CPI, E2V, Thales) As a tetrode As a klystron CERN SPS TH795 IOT transmitters. Two transmitters (4 tubes) deliver 18 480 kW at 801 MHz. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Klystron Principle of klystrons was published by Oskar and Agnessa Heil (Germany) in 1935. During the same time, W. W. Hansen at Stanford was investigating “ a scheme for producing high-voltage electrons” for use in X-ray spectroscopy. In the process, he invented the microwave cavity , “ Rhumbatron.” Working with Hansen Varian brothers (Russell and Sigurd) developed klystron Electron gun Continuous beam Beam arrives at 1st cavity Bunched beam Beam arrives at 2nd cavity Collector P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Performance parameters of different RF sources P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Efficiency P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Overall Efficiency of RF System PRFin ≃ 1 to 5 % PRFout (Gain is usually high) ηRF/DC ≃ 65 % (including overhead) η PAC/PDC ≃ 95 % to 98 % Amplifier cooler ≃ 15 % PRFout Building cooler ≃ 30 % PRFout 𝐎𝐯𝐞𝐫𝐚𝐥𝐥 𝐞𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐜𝐲 Amplifier Cooler Building Cooler Heat out RF power in RF power out Amp DUT (Device Under Test) DC power in AC power in AC/DC 𝐏𝐑𝐅𝐨𝐮𝐭 = 𝐏𝐑𝐅𝐢𝐧 + 𝐏 𝐀𝐂𝐢𝐧 + 𝐏 𝐜𝐨𝐨𝐥𝐞𝐫𝐬 𝑶𝒗𝒆𝒓𝒂𝒍𝒍 𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 𝐏 ≃ 𝑷𝑹𝑭𝒐𝒖𝒕 ≃ 45 % ≃ 𝐑𝐅𝐨𝐮𝐭 𝐏𝑹𝑭𝒐𝒖𝒕 (𝟎. 𝟎𝟓 + 𝟏. 𝟔𝟐 + 𝟎. 𝟒𝟓) 𝑷𝑨𝑪𝒊𝒏+𝑷𝑹𝑭𝒊𝒏 +𝑷𝒄𝒐𝒐𝒍𝒆𝒓𝒔 ≃ 𝟒𝟓 % P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

RF Power Distribution Consider a simple high power rf layout: RF power source Klystron, etc. Circulator DUT (Cavity) Driver Amplifier Size Outer conductor Inner conductor Outer diamet er Inner diamet er 7/8" 22.2 mm 20 mm 8.7 mm 7.4 mm 1 5/8" 41.3 mm 38.8 mm 16.9 mm 15.0 mm 3 1/8" 79.4 mm 76.9 mm 33.4 mm 31.3 mm 4 1/2" 106 mm 103 mm 44.8 mm 42.8 mm 6 1/8" 155.6 mm 151.9 mm 66.0 mm 64.0 mm Coaxial line Waveguide Waveguide Coaxial Lines Load (Termination) Coaxial cables are often with PTFE foam to keep concentricity 𝐷 𝑑 Flexible lines have spacer helicoidally placed all along the line 60 Rigid lines are made of two rigid tubes maintained concentric with supports Zc = ε ln 𝑟 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Rectangular Waveguide Waveguides are usable over certain frequency ranges For very lower frequencies the waveguide dimensions become impractically large For very high frequencies the dimensions become impractically small & the manufacturing tolerance becomes a significant portion of the waveguide size λ λ𝑔 = Wavelength 1−( λ )2 2𝑎 Cut-off frequency dominant mode Cut-off frequency next higher mode Usable frequency range c 2𝑎 c 𝑎 b f = a c f c2 = 1.3 fc to 0.9 fc2 Waveguide name Recommended frequency band of operation (GHz) Cutoff frequency of lowest order mode (GHz) Cutoff frequency of next mode (GHz) Inner dimensions of waveguide opening (inch) EIA RCSC IEC WR2300 WG0.0 R3 0.32 — 0.45 0.257 0.513 23.000 × 11.500 WR1150 WG3 R8 0.63 — 0.97 1.026 11.500 × 5.750 WR340 WG9A R26 2.20 — 3.30 1.736 3.471 3.400 × 1.700 WR75 WG17 R120 10.00 — 15.00 7.869 15.737 0.750 × 0.375 WR10 WG27 R900 75.00 — 110.00 59.015 118.03 0.100 × 0.050 WR3 WG32 R2600 220.00 — 330.00 173.571 347.143 0.0340 × 0.0170 60

Reflection 𝑉 0 + 𝑉 0 − Forward and reflected waves are in phase Zc Source Line = Zc Full reflection in phase: When the waves are 180 out of phase Full reflection out of phase: 𝑉 0 + 𝑉 0 − ZL 𝑉𝑚𝑎𝑥 1 + Г = 1 − Г V𝑆𝑊𝑅 = 𝑉𝑚𝑖𝑛 61 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Reflection In case of a full reflection (Pmax equivalent to 4 Pf) One needs to protect the RF power amplifiers if Pr > Prmax o Not always possible or may not be desirable since it may impact operation Pf P r Fast protection if Pr Pf forward power Pr reflected power 62 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Circulator Protect from this reflected power passive non-reciprocal three-port device signal entering any port is transmitted only to the next port in rotation The best place to insert it is close to the reflection source Lines between circulator and DUT will see 4xPf if fully reflected A load for Pf is needed on port 3 to absorb Pr 𝑉 0 + 𝑉 0 + 𝑉 0 − 𝑉 0 − P.N. Ostroumov PHY862 "Accelerator Systems" 63 Load Fall 2019

Circulator Even in case of full reflection Vmax = 2 Vf (Pmax equivalent to 4 Pf) RF power amplifiers will not see reflected power and will not be affected Lines between circulator and DUT MUST at least be designed for 4 Pf Loads must be designed for Pf Pf 4 Pf Pf DUT = Device under test Load 64 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Limitations of microwave tubes Performance is limited by a number of factors including: Heat dissipation Voltage breakdown Output window failure Multipactor discharge The dimensions of the RF structures and the windows of microwave tubes generally scale inversely with frequency. The maximum CW or average power that can be handled by a particular type of tube depends upon the maximum temperature that the internal surfaces can be allowed to reach. This temperature is independent of the frequency, so the power that can be dissipated varies inversely with the frequency. The power is also limited by the power that can be generated by an electron gun and formed into a beam. The beam diameter scales inversely with frequency and the beam current density is determined by the maximum attainable magnetic focusing field. Since the field is independent of frequency the beam current scales inversely with the square of the frequency 65 P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Limitations of Microwave Tubes The beam voltage is related to the current by the gun perveance range of 1.0 to 2.0 for power tubes. The maximum gun voltage is limited by the breakdown field in the gun and therefore varies inversely with frequency for constant perveance. The maximum power obtainable from a tube is 𝑓−2.5 𝑡𝑜 −3.0 depending on the assumptions made. The efficiencies of tubes tend to fall with increasing frequency. This is partly because the RF losses increase with frequency and partly because of the design compromises that must be made at higher frequencies. The maximum power obtainable from a pulsed tube is often determined by the power-handling capability of the output window. The output window of an external cavity klystron is in the form of a cylinder within the cavity and close to the output gap. This arrangement is limited to powers of about 70 kW. At higher power levels integral cavities are used and the power is brought out through waveguide or coaxial line windows. Very high power klystrons commonly have two windows in parallel to handle the full output power. Windows can be destroyed by excessive reflected power, by arcs in the output waveguide, by X-ray bombardment, and by the multipactor discharges described in the next Section. The basic cause of failure is overheating and it is usual to monitor the window temperature and to provide reverse power and waveguide and cavity arc detectors.

Power Handling – Coaxial Line The coaxial transmission line supports a TEM mode which has no cut-off frequency, that is, coax can be used down to d.c. This mode, in which the electric field is radial and the magnetic field azimuthal, has phase velocity and characteristic impedance in Ohm given by Coaxial transmission lines for high-power transmission are commonly available in 50  and 75  characteristic impedances, the former representing a compromise between breakdown field strength and power handling capacity, and the latter being selected for minimum attenuation. The ratio b/a is fixed by the characteristic impedance of the line at 2.3 for the 50  line and 3.49 for the 75  line. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Power handling – coaxial line The average power carried on a coaxial line depends on the peak electric field As a reference, the breakdown electric field strength in dry air at standard pressure and temperature is 3 MV/m. Higher-order modes (TE and TM modes) can propagate in coax at higher frequencies, and one wants to avoid these modes because mode conversion from TEM to TE or TM modes represents a source of power loss. The cut off wavenumber kc for the mode with the lowest cut off frequency, the TE11 mode, is approximately given by P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Standard waveguide characteristics Attenuation constant for a 14 inch coaxial line and WR1800 WG. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Standard waveguide characteristics Attenuation constant vs frequency for standard WGs. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019

Rectangular waveguide At >500 MHz, choosing a waveguide over a coax is obvious. In order to avoid higher order waveguide mode losses, the diameter of the coax must be reduced, leading to higher attenuation and lower power-handling capacity as shown above. For short distances of transmission, however, the higher losses of coaxial transmission lines may be acceptable. P.N. Ostroumov PHY862 "Accelerator Systems" Fall 2019