Jun 22, 2001 - The MUltifrequency Spectral Eulerian (MUSE) model is ideally suited ... Previous MUSE model did not include dispersion or space charge, and.
PPPS 2001 Pulsed Power and Plasma Science Las Vegas, Nevada 18-22 June, 2001 Paper O2B4
Finite Bandwidth and Space Charge Effects in the MUSE Modela,b ¨ hlbier, John H. Booske, John G. Wo Ian Dobson
University of Wisconsin-Madison a See
Friday morning mini-course poster session for more detail and discussion.
b This work was supported in part by AFOSR Grant 49620-00-1-0088 and by DUSD(S&T) under the Innovative Microwave Vacuum Electronics Multidisciplinary University Research Initiative (MURI) program, managed by the Air Force Office of Scientific Research under Grant F49620-99-1-0297.
W¨ ohlbier et al. – PPPS 2001 – 1
Abstract We have recently developed a new one-dimensional nonlinear model of a Traveling Wave Tube (TWT) being driven by an input signal with multiple frequencies. The MUltifrequency Spectral Eulerian (MUSE) model is ideally suited to predict the pre-saturation evolution of drive, harmonic, and intermodulation frequencies when there are many closely-spaced drive frequencies. The primary benefits of MUSE include reduced computation time for dense input spectra and enhanced analytic tractability over the conventional “disk” model. Since MUSE uses an Eulerian description of the electron beam it can not predict saturation accurately. However, the model is of practical use since TWTs are often driven with backed off input levels during multifrequency excitation so that they do not saturate. The first formulation of MUSE did not incorporate a frequency dependent cold circuit phase velocity, frequency dependent interaction impedance, or beam space charge. In this paper we present an enhanced MUSE model that accounts for these physical effects. Several test cases are presented and compared with a disk model. Additionally, we discuss the foundation of the MUSE model and comment on some physical insights elucidated by the model. W¨ ohlbier et al. – PPPS 2001 – 2
Overview • Introduction to MUSE Model and “what’s new ” ⇒ 1-D nonlinear helix TWT model • Comparison of MUSE Model to “Disk ” Model • Effect of dispersion and space charge on output spectrum as predicted by MUSE Model • Summary
W¨ ohlbier et al. – PPPS 2001 – 3
MUSE Model • MUltifrequency Spectral Eulerian Model ⇒ Eulerian 1-D “pre-linearized Pierce ” TWT equations ⇒ Spectral steady-state analysis for MUltifrequency inputs • Previous MUSE model did not include dispersion or space charge, and was “small C ” (Pierce gain parameter) ⇒ Model now includes dispersion, space charge, and is for arbitrary C • Why? ⇒ Computational tool for tubes run with “backed-off ” input levels ⇒ Analytically tractable model ⇒ Equation structure yields physical insight into nonlinearities
W¨ ohlbier et al. – PPPS 2001 – 4
MUSE Basics ∂V ∂z ∂I ∂z
∂Esc ∂z ∂v ∂v + v ∂t ∂z ∂ρ ∂t
+ v ∂ρ ∂z
= =
= =
=
˜ K(t) ∂I v ˜ph (t) ∗ ∂t 1 ∂V ∗ ˜ ∂t K(t)˜ vph (t) − A ∂ρ ∂t ρ 0 ˜ eK(t) − me v˜ph (t) ∗ ∂I ∂t ˜ ∗ Esc + mee R(t) −ρ ∂v ∂z
• Spectral Analysis, e.g. ⇒ V (z, t) =
Telegrapher s Equations Gauss Law Newton s Law
Continuity Equation ∞
=−∞
V˜ (z)eif ω0 t
W¨ ohlbier et al. – PPPS 2001 – 5
MUSE Basics, continued • Get nonlinear ordinary differential equations for Fourier coefficients of all quantities, e.g. dV˜ dz
= flinear (V˜ , I˜ , E˜ , v˜ , ρ˜ ) + gnonlinear (V˜m,n , I˜m,n , E˜m,n , v˜m,n , ρ˜m,n )
• Nonlinear terms in time domain become convolutions in frequency domain • Eulerian beam cannot predict electron overtaking, so does not agree with Disk model at saturation
W¨ ohlbier et al. – PPPS 2001 – 6
MUSE Model dV˜ dz dI˜ dz ˜ dE dz d˜ v v˜ ∗ dz
v˜ ∗
d˜ ρ dz
=
−
if ω0 K(f ω0 ) ˜ if ω0 ˜ I V − u0 vph (f ω0 )
(1)
=
−
if ω0 ˜ if ω0 V˜ − I + if ω0 A˜ ρ K(f ω0 )vph (f ω0 ) u0
(2)
=
−
=
=
ρ˜ if ω0 ˜ E + u0 0 if ω0 eK(f ω0 ) ˜ e ˜ + if ω0 v˜ I + R(f ω0 )E me vph (f ω0 ) me ω0 − v˜ ∗ if v˜ u0 ω0 d˜ v if ω0 v˜ ∗ if ρ˜ − ρ˜ ∗ v˜ . if ω0 ρ˜ − + u0 dz u0
(3)
(4) (5)
W¨ ohlbier et al. – PPPS 2001 – 7
Disk Model Model equations (see for example [1]) dx dy
=
−if
2 Cmax
x −
b Cmax
(2 + bCmax ) f2 a
4K() (1 + bCmax ) 2 1 + f Kmax Cmax T0 da dy ˜ dE dy ∂Γ ∂y
= =
T0
e−if ψ(y,ψ0 ) dψ0 1 + Cmax Γ(y, ψ0 )
x I0 1 if ˜ E + − Cmax Cmax Aω0 0 T0
=
[1 + Cmax Γ]−1
−
∞
1
2
T0
e−if ψ(y,ψ0 ) dψ0 1 + Cmax Γ(y, ψ0 )
(Cmax x + if a ) eif ψ(y,ψ0 )
=−∞
e ˜ eif ψ(y,ψ0 ) + R()E 2 me u0 ω0 Cmax ∞
=−∞
∂ψ ∂y
=
Γ 1 + Cmax Γ W¨ ohlbier et al. – PPPS 2001 – 8
Disk Model, continued with y
=
ψ
=
V (y, ψ)
=
x
=
Cmax ω0 z u0 ω0 uz0 − t ∞ Kmax I0 4Cmax =−∞ da dy
if ψ
a (y)e
∞
˜ (y)eif ψ E =−∞
E(y, ψ)
=
∂z ∂t u0 vph
=
u0 [1 + Cmax Γ]
=
1 + bCmax
W¨ ohlbier et al. – PPPS 2001 – 9
MUSE vs. Disk • Hughes L-Band 8537 TWT: drive frequency & 2nd harmonic Disk MUSE
10000
V V* (Volts 2)
1
1.6GHz
Pierce Parameters
1E-4
3.2GHz
C1.6GHz = 0.11 C3.2GHz = 0.006 QC 1.6GHz = 0.15 QC 3.2GHz = 0.37 b1.6GHz = 0.95 b3.2GHz = 4.09
1E-8
1E-12
1E-16
1E-20 0
10
20
30
40
50
z (cm)
W¨ ohlbier et al. – PPPS 2001 – 10
Hughes L-Band 8537 Space TWT [2] • Power: 80W nominal • Bandwidth: 1.530GHz - 1.548GHz (but can be as large as 1.4GHz 1.75GHz) • Periodic permanent magnet (PPM) focussing with guide field approximately 1.5 times Brillouin field ⇒ largely 1-D electron motion • Three support rods; no vanes • Simulations here for one stage of 8537
W¨ ohlbier et al. – PPPS 2001 – 11
Effects of Dispersion and Space Charge • For notched input spectrum, calculate output spectra for model with ⇒ No dispersion, No space charge ⇒ Dispersion, No space charge ⇒ Dispersion and space charge
W¨ ohlbier et al. – PPPS 2001 – 12
Input Spectrum
10 1
V V* (Volts 2)
0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz) W¨ ohlbier et al. – PPPS 2001 – 13
Input No effects 10 1
V V* (Volts 2)
0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz) W¨ ohlbier et al. – PPPS 2001 – 15
Input No effects Add dispersion
10 1
V V* (Volts 2)
0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
W¨ ohlbier et al. – PPPS 2001 – 16
Input No effects Add dispersion Add space charge
10 1
V V* (Volts 2)
0.1 0.01 1E-3 1E-4 1E-5 1E-6 1E-7 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
W¨ ohlbier et al. – PPPS 2001 – 17
Summary • MUSE is an alternative multifrequency 1-D TWT model • MUSE agrees well with disk model before saturation • MUSE has potential for speedy calculation of many, densely-packed tones • Require further study to see if saturation predictable
W¨ ohlbier et al. – PPPS 2001 – 18
References 1. A. J. Giarola, “ A theoretical description for the multiple-signal operation of a TWT ”, IEEE Trans. Electron Devices, Vol. ED-15, No. 6, June 1968, pp. 381-395. 2. D. K. Abe et al. “ A comparison of L-band helix TWT experiments with CHRISTINE, a 1-D multifrequency helix TWT code ”, IEEE Trans. on Plasma Science, Vol. 28, No. 3, June 2000, pp. 576-587.
W¨ ohlbier et al. – PPPS 2001 – 19
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W¨ ohlbier et al. – PPPS 2001 – 20
