Perspectives on cavity field control Olof Troeng, Dept. of Automatic Control, Lund Univeristy Low-Level RF Workshop, 2019-10-03 1
Outline Thoughts from an automatic-control (and linac) perspective 1. Complex-coefficient LTI systems 2. Energy-based parametrization of cavity dynamics 3. Normalized cavity dynamics, sensitivity to disturbances 4. Parasitic modes 2
Baseband cavity models Complex SISO representation Real TITO representation ω 1 / 2 � � ω 1 / 2 ω 1 / 2 − ∆ ω P a ( s ) = P a ( s ) = s + ω 1 / 2 − i ∆ ω (∆ ω ) 2 + ( s + ω 1 / 2 ) 2 ∆ ω ω 1 / 2 3
Baseband cavity models Complex SISO representation Real TITO representation ω 1 / 2 � � ω 1 / 2 ω 1 / 2 − ∆ ω P a ( s ) = P a ( s ) = s + ω 1 / 2 − i ∆ ω (∆ ω ) 2 + ( s + ω 1 / 2 ) 2 ∆ ω ω 1 / 2 The complex representation is well known, but not much used for control design. 3
Baseband cavity models Complex SISO representation Real TITO representation ω 1 / 2 � � ω 1 / 2 ω 1 / 2 − ∆ ω P a ( s ) = P a ( s ) = s + ω 1 / 2 − i ∆ ω (∆ ω ) 2 + ( s + ω 1 / 2 ) 2 ∆ ω ω 1 / 2 The complex representation is well known, but not much used for control design. Complex-coefficient systems are ubiquitous in communications, but rare in control: “Poles always come in conjugate pairs” 3
Baseband cavity models Complex SISO representation Real TITO representation ω 1 / 2 � � ω 1 / 2 ω 1 / 2 − ∆ ω P a ( s ) = P a ( s ) = s + ω 1 / 2 − i ∆ ω (∆ ω ) 2 + ( s + ω 1 / 2 ) 2 ∆ ω ω 1 / 2 The complex representation is well known, but not much used for control design. Complex-coefficient systems are ubiquitous in communications, but rare in control: “Poles always come in conjugate pairs” nope, they lied to you 3
Baseband cavity models Complex SISO representation Real TITO representation ω 1 / 2 � � ω 1 / 2 ω 1 / 2 − ∆ ω P a ( s ) = P a ( s ) = s + ω 1 / 2 − i ∆ ω (∆ ω ) 2 + ( s + ω 1 / 2 ) 2 ∆ ω ω 1 / 2 The complex representation is well known, but not much used for control design. Complex-coefficient systems are ubiquitous in communications, but rare in control: “Poles always come in conjugate pairs” nope, they lied to you Other control applications: FB linearization of RF amps. Vibration damping of rotating machinery 3
Relation between real and complex representations Complex SISO representation Real TITO representation � � G ( s ) = G Re ( s ) + iG Im ( s ) G Re ( s ) − G Im ( s ) G equiv ( s ) = G Im ( s ) G Re ( s ) 4
Relation between real and complex representations Complex SISO representation Real TITO representation � � G ( s ) = G Re ( s ) + iG Im ( s ) G Re ( s ) − G Im ( s ) G equiv ( s ) = G Im ( s ) G Re ( s ) Consider eigendecomposition � � � � 1 G ( s ) 0 1 1 U H , √ G equiv ( s ) = U U = . 0 G ∗ ( s ) − i i 2 Note: G ∗ ( i ω ) = G ( − i ω ). Positive and negative frequencies are intertwined in G equiv ( s ) 4
Relation between real and complex representations Complex SISO representation Real TITO representation � � G ( s ) = G Re ( s ) + iG Im ( s ) G Re ( s ) − G Im ( s ) G equiv ( s ) = G Im ( s ) G Re ( s ) Consider eigendecomposition � � � � 1 G ( s ) 0 1 1 U H , √ G equiv ( s ) = U U = . 0 G ∗ ( s ) − i i 2 Note: G ∗ ( i ω ) = G ( − i ω ). Positive and negative frequencies are intertwined in G equiv ( s ) Advantages of the complex-coefficient representation Simplifies understanding, calculations, life in general, etc Structure is implicit, good for system identification More efficient computations 4
Control theory for complex-coefficient systems Standard tools and results apply but Change A T to A H Remember to consider negative frequencies 5
Control theory for complex-coefficient systems Standard tools and results apply but Change A T to A H Remember to consider negative frequencies MatLab handles complex-coefficient okay, but some problems, e.g., nyquist 5
Control theory for complex-coefficient systems Standard tools and results apply but Change A T to A H Remember to consider negative frequencies MatLab handles complex-coefficient okay, but some problems, e.g., nyquist Illustration of Bode’s sensitivity integral (the water-bed effect) 5
Control theory for complex-coefficient systems Standard tools and results apply but Change A T to A H Remember to consider negative frequencies MatLab handles complex-coefficient okay, but some problems, e.g., nyquist Illustration of Bode’s sensitivity integral (the water-bed effect) 5
Control theory for complex-coefficient systems Standard tools and results apply but Change A T to A H Remember to consider negative frequencies MatLab handles complex-coefficient okay, but some problems, e.g., nyquist Illustration of Bode’s sensitivity integral (the water-bed effect) 5
Intuitive understanding of loop-phase adjustment Open-loop system: e i θ adj C ( s ) e − i θ P a ( s ) e − s τ = e δ L 0 ( s ) where δ := θ adj − θ . Im L ( i ω ) e i θ adj C ( s ) e − i θ P a ( s ) e − s τ Re L ( i ω ) − 1 -1 δ = 0 ◦ 6
Intuitive understanding of loop-phase adjustment Open-loop system: e i θ adj C ( s ) e − i θ P a ( s ) e − s τ = e δ L 0 ( s ) where δ := θ adj − θ . Im L ( i ω ) e i θ adj C ( s ) e − i θ P a ( s ) e − s τ Re L ( i ω ) − 1 -1 δ = 0 ◦ Loop-phase-adjustment error δ gives corresponding phase-margin reduction! 6
Intuitive understanding of loop-phase adjustment Open-loop system: e i θ adj C ( s ) e − i θ P a ( s ) e − s τ = e δ L 0 ( s ) where δ := θ adj − θ . Im L ( i ω ) e i θ adj C ( s ) e − i θ P a ( s ) e − s τ Re L ( i ω ) − 1 -1 δ = 15 ◦ Loop-phase-adjustment error δ gives corresponding phase-margin reduction! 6
Intuitive understanding of loop-phase adjustment Open-loop system: e i θ adj C ( s ) e − i θ P a ( s ) e − s τ = e δ L 0 ( s ) where δ := θ adj − θ . Im L ( i ω ) e i θ adj C ( s ) e − i θ P a ( s ) e − s τ Re L ( i ω ) − 1 -1 δ = 30 ◦ Loop-phase-adjustment error δ gives corresponding phase-margin reduction! 6
Intuitive understanding of loop-phase adjustment Open-loop system: e i θ adj C ( s ) e − i θ P a ( s ) e − s τ = e δ L 0 ( s ) where δ := θ adj − θ . Im L ( i ω ) e i θ adj C ( s ) e − i θ P a ( s ) e − s τ Re L ( i ω ) − 1 -1 δ = 45 ◦ Loop-phase-adjustment error δ gives corresponding phase-margin reduction! 6
Energy-based parametrization of cavity dynamics 7
Accelerator Cavity Modeling (1/2) Equivalent-circuit parametrization: d V dt =( − ω 1 / 2 + i ∆ ω ) V + R L ω 1 / 2 (2 I g + I b ) P g = 1 r Q Q ext | I g | 2 4 8
Accelerator Cavity Modeling (1/2) Equivalent-circuit parametrization: d V dt =( − ω 1 / 2 + i ∆ ω ) V + R L ω 1 / 2 (2 I g + I b ) P g = 1 r Q Q ext | I g | 2 4 RF drive is modeled as fictitious generator current I g . Problematic. “ One word of caution is required here: /.../ for considerations where Q ext varies /.../ or where ( R / Q ) varies /.../ the model currents cannot be considered constant; they have to be re-adapted” [Tückmantel (2011)] 8
Accelerator Cavity Modeling (1/2) Energy-based parametrization: Equivalent-circuit parametrization: d A 2 γ ext F g + α d V � dt = ( − γ + i ∆ ω ) A + 2 I b dt =( − ω 1 / 2 + i ∆ ω ) V + R L ω 1 / 2 (2 I g + I b ) � √ � A – Mode amplitude J P g = 1 r � √ Q Q ext | I g | 2 � F g – Forward wave W 4 � � � V = α A α = ω a ( r / Q ) P g = | F g | 2 Haus (1984) Waves and fields in optoelectronics plus beam loading 8
Accelerator Cavity Modeling (1/2) Energy-based parametrization: Equivalent-circuit parametrization: 2 γ ext F g + α d A d V � dt = ( − γ + i ∆ ω ) A + 2 I b dt =( − ω 1 / 2 + i ∆ ω ) V + R L ω 1 / 2 (2 I g + I b ) P g = 1 r P g = | F g | 2 Q Q ext | I g | 2 4 Advantages of energy-based parameterization: Cleaner expressions, e.g., P g = | F g | 2 States and parameters are well defined Direct connection to physical quantities of interest 8
Accelerator Cavity Modeling (1/2) Energy-based parametrization: Equivalent-circuit parametrization: 2 γ ext F g + α d A d V � dt = ( − γ + i ∆ ω ) A + 2 I b dt =( − ω 1 / 2 + i ∆ ω ) V + R L ω 1 / 2 (2 I g + I b ) P g = 1 r Q Q ext | I g | 2 4 Helpful to think of γ as both decay rate and bandwidth. The total decay rate γ = γ 0 + γ ext , not so intuitive if considered as bandwidths Common for laser cavities 8
Optimal coupling and detuning 1 � ( − γ 0 − γ ext + i ∆ ω ) A 0 + α � � | F g0 | 2 = � � Minimize 2 I b0 � � 2 γ ext � with repsect to ∆ ω and γ ext Solution: ∆ ω = − 1 Im α γ ext = γ 0 − 1 Re α 2 I b0 , 2 I b0 A 0 A 0 9
Optimal coupling and detuning 1 � � ( − γ 0 − γ ext + i ∆ ω ) A 0 + α � | F g0 | 2 = � � Minimize 2 I b0 � � 2 γ ext � with repsect to ∆ ω and γ ext Solution: ∆ ω = − 1 Im α γ ext = γ 0 − 1 Re α 2 I b0 , 2 I b0 A 0 A 0 ( − γ + i ∆ ω ) A √ 2 γ ext F g α 2 I b √ √ Terms of d Mode amplitude A [ J] dt A [ J / s] 9
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