ERL 03-15

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RESONANT RING FOR HIGH POWER TESTS OF RF COUPLERS

V. Veshcherevich

INTRODUCTION Under Cornell Energy Recovery Linac project (ERL) [1], a 1300 MHz CW high power input coupler is being developed for injector cavities [2]. After fabrication, each coupler will be tested up to a high power of at least 75 kW. No klystron station of the necessary power level will be available soon. Therefore, in order to be able to test couplers using a klystron of a much lower power we propose to build a traveling wave resonator, or resonant ring [3], [4], [5]. The RF components to be tested will be parts of this resonant ring.

GENERAL LAYOUT A resonant ring, also known as a traveling wave resonator, is a waveguide loop which can amplify RF power through the coupling of waves at its input. Figure 1 shows a basic resonant ring circuit.

Resonant Ring

Directional Coupler MatchedLoadPower Source

Arm 1 Arm 3

Arm 2 Arm 4

Figure 1: Resonant ring circuit. For effective power amplification, the ring should be in a state of resonance at the test frequency. In order for this to occur, the length of the ring must be equal to an integral number of guide wavelengths of coupled wave. In this case, the waves coupled into the ring through the directional coupler add constructively. This creates a power gain in the ring.

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RF components to be tested should be parts of the resonant ring, and the resonance should be achieved with these components. The input coupler of an ERL injector cavity consists of a coaxial vacuum part, two ceramic windows, and a coaxial-to-waveguide transition [2]. According to that, for building a resonant ring we need two couplers of similar design with a coupling device between them. The couplers can be either identical ones, or the second coupler can have a simplified design, but it must withstand full test power. As a coupling device between two couplers, it is the most reasonable to use a cavity with very strong coupling. It is convenient to pump the vacuum parts of the couplers through the coupling device. The remaining part of the resonant ring will be made of pieces of rectangular waveguide (see Figure 2).

CouplingCavityCoaxial

Coupler

Rectangular Waveguide

Figure 2: Scheme of the resonant ring with couplers.

COUPLING CAVITY The coupling cavity in our scheme is a spherical copper cavity. As the ERL input couplers are antenna type couplers, it is natural to position them symmetrically on the cavity axis. Figure 3 presents the geometry of the coupling cavity with two couplers. There is a strong dependence of the resonance frequency of the cavity and the coupling strength on the antennae insertions. Figures 4 to 6 show the reflection and transmission parameters of the coupling cavity as functions of the gap between antennae tips. One can see that the cavity can provide a very wide bandwidth (250 MHz at S11 < –30 dB and 70 MHz at S11 < –40 dB). There is a strong dependence of cavity frequency on the gap. However, ERL couplers have a wide tuning range for antenna position (15 mm), and it is easy to adjust the right frequency. This tuning has only a very small influence on the electrical length of the traveling wave resonator (see Figure 6).

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Ø22.8 Ø62.0Ø220.0

2g

120.0

Figure 3: Geometry of the coupling cavity.

Figure 4: Reflection from the coupling cavity for different gaps.

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Figure 5: Transmission through the coupling cavity as a function of the gap.

Figure 6: Phase of the signal transmitted through the coupling cavity.

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Field maps (Figures 7 and 8) show that the fields in the coupling cavity are low in comparison with the fields in the coaxial lines (e. g. magnetic fields on most of the cavity surface are about two times lower than the field on the outer tube of the coaxial line). Therefore the loss in the cavity is also very low. In fact, it is lower than the loss in a part of the coaxial line of the same length as the cavity.

Figure 7: Electric fields in the coupling cavity with couplers (g = 18 mm).

Figure 8: Magnetic fields in the coupling cavity with couplers (g = 18 mm).

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The cavity geometry has been optimized for obtaining a wide frequency band. It turned out that the optimum occurs when resonance frequencies of E010 and E011 modes of the cavity (with different boundary conditions on the ends of coaxial lines) coincide (see Figure 9). In traveling wave mode the fields in the cavity are a mixture of E010 and E011 fields. See, for example, the distribution of the electric field along the z axis in the cavity gap presented in Figure 10.

a) Mode E010 b) Mode E011

Figure 9: Electric fields lines in the coupling cavity with electric (a) and magnetic (b) boundary conditions on the end of coaxial line.

Figure 10: Electric field in the gap between antenna tips in coupling cavity.

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POWER GAIN The ring performance depends on several variables: coupling coefficient, attenuation and reflection in the ring, transmission coefficient, and electrical length. For a resonant ring in resonance, when the phase shift φ of the wave along the whole ring is equal to 2πn (n is an integer), the power gain M is:

2

2222

222

)1(1121111

−+−−−

−−−=

−−

−

CerCerCeCM

LL

L

αα

α

,

where C is the coupling coefficient of the directional coupler; r is the reflection coefficient in the ring; e–αL is the attenuation in the circuit; α is the effective attenuation coefficient; and L is the effective length of the ring. The lower the attenuation and the reflection in the resonant ring are, the higher is the power gain. Figure 11 illustrates the behavior of the M parameter for the ring with no reflection (r = 0).

0.1

1

10

100

1000

-30 -25 -20 -15 -10 -5

Coupling Coefficient, dB

Pow

er G

ain

aL = 0.001 aL = 0.002 aL = 0.005 aL = 0.01 aL = 0.02 aL = 0.05 aL = 0.1

Figure 11: Power gain of a resonant ring with no reflections. For each attenuation value, there exists the optimal coupling coefficient, for which the power gain is maximal. For no reflection case, this optimal coupling coefficient is

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Lopt eC α21 −−= ,

and the maximal power gain is

Lopt eC

M α22max 111−−

== .

In our case, the resonant ring consists of a waveguide part (WR650 waveguide) and a coaxial part; the latter includes pieces of 60 Ohm and 46 Ohm lines, ceramic windows, and a coupling cavity. Attenuation coefficient for a rectangular waveguide is:

2

2

0

21

221

−

+

=

wh

wwh

ZRsw

wλ

λ

α ,

where Rsw is the surface resistance of the waveguide wall material, Z0 ≈ 377 Ohm is the free space impedance, h is the height of the waveguide, w is its width, λ is the wave length in free space. For WR650 waveguide (w = 6.5" = 185.1 mm, h = 3.25" = 82.55 mm) with rough aluminum walls (equivalent conductivity σAl = 2.8×107 Ohm–1m–1, Rsw = 0.0135 Ohm), we have αw = 9.0×10–4 Np/m or 7.8×10–3 dB/m. Attenuation coefficient for a coaxial line is:

+=

i

oo

i

o

scс

rr

r

rr

ZR

ln2

1

0

α ,

where Rsc is the surface resistance of the coaxial line material, ro and ri are the radii of outer and inner conductors of coaxial line. For the 46 Ohm coaxial line with ro =31 mm, ri = 14.4 mm and rough copper (equivalent conductivity σCu = 4.6×107 Ohm–1m–1, Rsc = 0.0106 Ohm), we have αc = 1.9×10–3 Np/m or 1.6×10–2 dB/m. For 60 Ohm coaxial line with ro =31 mm, ri = 11.4 mm, we have αc = 1.7×10–3 Np/m or 1.46×10–2 dB/m. We will use the bigger value of 1.6×10–2 dB/m for the whole coaxial line. The total length Lc of coaxial part of the resonant ring is actually the length of two ERL couplers, it equals approximately 1.8 m. Therefore, the length Lw of waveguide part will be around 2.6 m. So, the total attenuation in transmission lines of the resonant ring is

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αL = αc Lc + αw Lw = 5.7×10–3. In addition to attenuation due to wall loss in the transmission lines, there is also attenuation due to power loss in ceramic windows. Making an optimistic assumption that the windows loss is roughly the same as the wall loss, we double the number above and get the overall attenuation in the resonant ring

αL = 0.0114. For this attenuation value the optimal coupling coefficient of the directional coupler Copt = 0.15 i. e. –16.5 dB, and the maximal power gain Mmax = 44.4 i. e. 16.5 dB. However, as one can see in Figure 11, the curves of power gain have very broad peaks. We calculated the coupling coefficient and the power gain assuming ideal matching in the traveling wave resonator. In fact, reflections strongly affect the parameters of resonant rings. Figure 12 illustrates this effect for the resonant ring with αL = 0.0114.

0

5

10

15

20

25

30

35

40

45

-30 -25 -20 -15 -10 -5Coupling Coefficient

Powe

r Gai

n

r = 0 r = 0.002 r = 0.005 r = 0.01 r = 0.02 r = 0.05 r = 0.1

Figure 12: Power gain of a resonant ring with reflections (αL = 0.0114).

Figure 13 shows dependence of power gain on reflection coefficient for the resonant ring with αL = 0.0114 and C = –16.5 dB. It is obvious that the reflection coefficient in the resonant ring should be below –40 dB (below 0.01) if we want to obtain a considerable power magnification.

10

0

5

10

15

20

25

30

35

40

45

-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10

Reflection Coefficient, dB

Powe

r Gai

n

Figure 13: Power gain of a resonant ring with reflections (αL = 0.0114).

0

20

40

60

80

100

120

140

160

-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10Attenuation, dB

Powe

r Gai

n

r = 0

r = 0.005

r = 0.01

Figure 14: Power gain of a resonant ring with coupling coefficient C = –16.5 dB.

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Figure 14 presents dependence of the power gain on circuit attenuation for the resonant ring with given coupling and different reflection values. There are other reasons for keeping reflection in the resonant ring low. Due to resonant effects, magnitudes of reflected waves in the circuit become M times higher. Also, when r > αL, the power gain achieves its maximum at φ ≠ 2πn. However, at r = αL the input reflection is already excessively high (S11 ≈ –6dB) and therefore it should be r < αL.

MAIN DIRECTIONAL COUPLER For coupling the traveling wave resonator to the main transmission line, we need a directional coupler with a coupling coefficient of C = –16 dB that has low reflection and a relatively wide frequency band. It can be a multi-hole coupler of Chebyshev type. For calculation of the directional coupler we used the approach and tables from [6]. Then the model of the coupler was checked by the 3D computer code CST Microwave Studio® [7], and geometry of the coupler was slightly corrected for obtaining a better performance. The final geometry of a 16 dB four-hole Chebyshev type coupler with the ratio fmax/fmin = 1.3 is shown in Figure 15. Scattering coefficients of that directional coupler calculated by CST Microwave Studio are presented in Figures 16 to 20. One can see that, according to simulations, the coupler properties are close to the expected ones. From Figure 21 we see that directivity (S41 – S21) of the directional coupler is better than 40 dB.

80.55 80.87 80.55

82.55

2.03 R1.02

Ø66.41Ø46.91

Ø66.41

Figure 15: Geometry of directional coupler.

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Figure 16: Reflection in the main transmission line with the directional coupler.

Figure 17: Parasitic coupling (in wrong direction) of the directional coupler.

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Figure 18: Transmission in the main transmission line with the directional coupler.

Figure 19: Coupling coefficient of the directional coupler.

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Figure 20: Phases of S-parameters of the directional coupler.

10

20

30

40

50

60

1 1.1 1.2 1.3 1.4 1.5 1.6

Frequency, GHz

Dire

ctiv

ity, d

B

Figure 21: Directivity of the directional coupler.

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RESONANT RING FOR COUPLER TESTS A 3D view of resonant ring for coupler tests is shown in Figure 22. Besides the components mentioned above, additional items for tuning the ring and for monitoring have to be included in the circuit. As it was explained, matching in the resonant ring is highly important. A three-stub transformer is included in the circuit for matching purposes. We also need instrumentation directional couplers in the resonant ring as well as in the main waveguide. Waveguide shims can be used for coarse tuning of the traveling wave resonator, whereas fine tuning can be made by moving the coupler antennae. The couplers are designed for superconducting cavities and some parts of them will operate at a cold temperature. It is a good idea to test couplers at conditions close to the real ones. Therefore, it is reasonable to keep the coupling cavity and adjacent parts of couplers including “cold” ceramic windows at the temperature of liquid nitrogen. A simple nitrogen cryostat can be built for this purpose.

1 2 6 5 3 4 7

Figure 22: 3D view of resonant ring for coupler tests. 1: main directional coupler; 2: 3-stub transformer; 3: instrumentation directional coupler; 4, 6: couplers under test; 5: coupling cavity; 7: vacuum pump.

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THREE-STUB TRANSFORMER Three-stub impedance transformers are often used for matching imperfect loads in transmission lines. We use transformers of this type in CESR RF system [8]. In the case of traveling wave resonator, we need very fine compensation of small reflections. Therefore, unlike CESR design, we chose thin stubs with small stokes and a choke type design that is shown in Figure 23. The choke design gives a possibility to place a finger contact in a location where surface currents are very small. That is illustrated in Figures 24 and 25, where the picture of magnetic field and magnetic field distribution along the plunger are presented. For the geometry shown, the magnetic field (i. e. surface current) in the area of RF contact is two orders of magnitude lower than near the waveguide wall.

58.053.0

Ø10.0Ø14.0Ø24.0

R2.0 Ø28.0

2.0R1.0

Figure 23: Movable plunger with a choke in the waveguide. Figure 26 shows the magnitudes of S11 parameter of a waveguide with a stub calculated by CST Microwave Studio for different insertions of the plunger into the waveguide, and Figure 27 shows a normalized admittance of the stub, calculated from S-parameters.

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Figure 24: Magnetic field in the area of stub with choke.

-50

0

50

100

150

200

250

300

350

400

-20 -10 0 10 20 30 40 50 60 70

x, mm

H, A

/m

Figure 25: Magnetic field distribution along the plunger at the power in the

waveguide P = 75 kW. Direction and origin of the x axis are shown in Figure 23. RF contact is located at x = 63 mm. Insertion of the plunger into the waveguide is 20 mm.

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-80

-70

-60

-50

-40

-30

-20

-10

-5 0 5 10 15 20 25 30

Insertion, mm

S11

, dB

Figure 26: S11 parameter of a stub in the waveguide.

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

-5 0 5 10 15 20 25 30

Insertion, mm

Stub

Adm

ittan

ce, n

orm

aliz

ed

Figure 27: Normalized admittance of a stub in the waveguide.

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A three-stub transformer can be described by three normalized stub admittances Y1 = jb1, Y2 = jb2, and Y3 = jb3 located in a transmission line with offsets of a quarter wave length. For matching a load with normalized admittance YL = gL + jbL in the plane of the stub #1, the stub admittances should be [8]:

LL

L bg

gb −−= 111 , 11

2 −=Lg

b , b3 = 0 .

In the case of low reflection (|S11| ^ 1), YL = 1 – aL + jbL where aL ^ 1, bL ^ 1, and

2211 2

1 baS +≈ . For the given |S11| value the best scenario is when aL = 0, bL ≈ 2|S11|.

Then for matching we need b1 = –bL, b2 = 0, b3 = 0, i. e. the magnitude of reflection produced by a stub is equal to the magnitude of reflection from the load. The worst scenario is when bL = 0, aL = 2|S11|. In that case we need 1121 2 Sabb L ≈≈≈ , b3 = 0,

i. e. two stubs should produce reflections 1121 S≈ which are significantly higher than

the reflection from the load. For instance, for matching the load with a reflection |S11| = –20 dB, we may need to produce reflections by single stubs up to –12 dB. The three-stub transformer can also be used for phase tuning, though its main function is compensation of reflections in the resonant ring. Figure 28 shows the phase shift produced by a stub in the waveguide.

0

2

4

6

8

10

12

14

-5 0 5 10 15 20 25 30

Insertion, mm

Del

ta p

hi, d

egre

e

Figure 28: Additional phase shift due to the stub in the waveguide.

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CONCLUSION A physical design of a traveling wave resonator (a resonant ring) is proposed for high power tests of input couplers for injector cavities of Cornell ERL project. The resonant ring will provide a power gain around 40, and therefore an inexpensive klystron amplifier of a power as low as 2 kW will suffice for full power tests (up to at least 75 kW) of ERL couplers.

REFERENCES [1] G. Hoffstaetter et al. “The Cornell ERL Prototype Project”. In: Proc. of the 2003

Particle Accelerator Conf. (Portland, OR). Web page: http://warrior.lbl.gov:7778/ pacfiles/papers/TUESDAY/AM_ORAL/TOAC005.PDF.

[2] V. Veshcherevich et al. “Input Coupler for ERL Injector Cavities”. In: Proc. of the

2003 Particle Accelerator Conf. (Portland, OR). Web page: http://warrior.lbl.gov: 7778/pacfiles/papers/TUESDAY/AM_POSTER/TPAB009/TPAB009.PDF.

[3] S. J. Miller. “The Traveling Wave Resonator and High-Power Microwave Testing”.

The Microwave Journal, September, 1960, pp. 50-58. [4] E. Gerken et al. “Resonant Ring for Testing of Accelerator RF Windows”. In: Proc.

of PAC '97, pp. 3725-3727. [5] V. D. Shemelin. “Development of Accelerating Structure and Tests of VLEPP

Microwave Components”. Novosibirsk, 1995. [6] A. L. Feldshtein et al. “Handbook on Microwave Techniques”. Moscow, 1973. [7] CST GmbH, Darmstadt, Germany. [8] V. Veshcherevich and S. Belomestnykh. Correction of the Coupling of CESR RF

Cavities to Klystrons Using Three-Post Waveguide Transformers. Cornell report SRF 020220-02, 2002.