Singularity of power dissipation in fractal AC circuits Patricia - PowerPoint PPT Presentation

Singularity of power dissipation in fractal AC circuits Patricia Alonso Ruiz University of Connecticut June 18, 2017

Passive linear networks. Resistors Ohm’s law x V xy = I xy R xy . R xy G Kirchoff’s voltage law y V xy = v ( x ) − v ( y ) , ( v ( x ) , v ( y )) ∈ R 2 potential function.

Passive linear networks. Inductors and capacitors Time-dependent voltage V ( t ) and current I ( t ) functions. Inductor L V ( t ) = L d dt I ( t ) . I ( t ) = C d Capacitor dt V ( t ) . C

Frequency domain. Impedances � ∞ 1 � v ( t ) e − i ω t dt . Fourier transform: V ( ω ) = a 2 π −∞ V ( ω ) = i ω L � � I ( ω ) =: Z L � Inductor: I ( ω ) , 1 � I ( ω ) =: Z C � � Capacitor: V ( ω ) = I ( ω ) , i ω C V ( ω ) = R � � I ( ω ) =: Z R � Resistor: I ( ω ) .

Ohm’s law revisited Ohm’s law (complex-valued) x V xy ( ω ) = I xy ( ω ) Z xy ( ω ) . Z xy G Kirchoff’s voltage law y V xy ( ω ) = v ( ω, x ) − v ( ω, y ) , ( v ( ω, x ) , v ( ω, y )) ∈ C 2 potential function.

Electromotive force From now on: frequency ω is fixed, ϕ phase shift. V xy ( t ) = | V xy | e i ω t , I xy ( t ) = | I xy | e i ( ω t − ϕ ) , Z xy = | Z xy | e i ϕ . Electromotive force emf xy ( t ) = I xy ( t ) Z xy = | I xy || Z xy | e i ω t ,

Power dissipation Average energy loss � T 1 ℜ ( emf xy ( t )) ℜ ( I xy ( t )) dt = · · · = 1 2 | I xy | 2 ℜ ( Z xy ) . T 0 Power dissipation of the potential ( v ( x ) , v ( y )) ∈ C P [ v ] Z xy = 1 ℜ ( Z xy ) | Z xy | 2 | v ( x ) − v ( y ) | 2 . 2

Power dissipation in graphs Let G = ( V , E ) be a finite graph, Z = { Z xy , { x , y } ∈ E } a network on G and ℓ ( V ) = { v : V → C } . The quadratic form � P Z [ v ] = 1 ℜ ( Z xy ) | Z xy | 2 | v ( x ) − v ( y ) | 2 2 { x , y }∈ E is the power dissipation in G associated with the network Z . � ◮ If Z x , y , I xy , v real, P Z ( v ) = 1 Z xy ( v ( x ) − v ( y )) 2 . 1 2 { x , y }∈ E

Power dissipation in an infinite network. The infinite ladder Feynman’s infinite ladder network [4] x Z C y Z L If ω 2 LC < 4, the characteristic impedance of the circuit satisfies ℜ ( Z eff xy ) > 0 even though all elements in the circuit have purely imaginary impedances!

The Feynman-Sierpinski ladder Infinite network Z FS = { Z xy , { x , y } ∈ E ∞ } . 1 Capacitors Z C = i ω C , inductors Z L = i ω L .

Theorem [2]: The effective impedance of the Feynman-Sierpinski ladder has positive real part whenever √ √ 15 ) < 2 ω 2 LC < 9 ( 4 + 9 ( 4 − 15 ) (FC) (filter condition). In this case, � � � 1 144 ω 2 LC − 4 ( ω 2 LC ) 2 − 81 Z eff ( 9 + 2 ω 2 LC ) i + FS = . 10 ω C

From infinite graphs to fractals Underlying infinite graph structure G ∞ approximated by finite graphs G n = ( V n , E n ) , n ≥ 0. · · · G 0 G 0 G 1 G 2 G 3 ◮ π : G ∞ → R 2 ◮ π ( G 0 ) ⊆ π 1 ( G 1 ) ⊆ . . . ⊆ π n ( G n ) ⊆ . . .

The fractal Q ∞ The unique compact set Q ∞ ⊆ R 2 such that � Eucl Q ∞ = π ( G n ) n ≥ 0 is a fractal quantum graph.

The fractal K ∞ The set � ˚ K ∞ = Q ∞ \ π ( E n ) n ≥ 0 is the union of countable many isolated points (nodes in V ∗ ) and a Cantor dust C ∞ (accumulation points).

Observations/consequences ◮ Identify V n with π ( V n ) , ◮ V ∗ = � n ≥ 1 V n is dense in K ∞ , ◮ K ∞ is compact in the Euclidean topology.

Networks on G n

Networks on G n Z ε, n = { Z ε, xy | { x , y } ∈ E n } , Z ε, xy = Z xy + ε. Z eff Z eff ε ε Z eff ε Z ε, 0 Z ε, 1 Z ε, 2 (For completeness, Z eff n →∞ Z eff := lim ε, n . ) ε

Theorem [2]: Under (FC), the network Z ε, n approximates the Sierpinski ladder Z in the sense that n →∞ Z eff ε, n = Z eff ε → 0 + lim lim FS , where Z eff ε, n is the effective impedance of Z ε, n . ◮ Up to now, assume that (FC) holds.

Towards power dissipation in K ∞ The power dissipation in V ∗ associated with the Feynman- Sierpinski ladder is the quadratic form P FS [ v ] := lim ε → 0 + lim n →∞ P Z ε, n [ v | Vn ] , where P Z ε, n : ℓ ( V n ) → R is the power dissipation in G n associated with Z ε, n .

dom P FS := { v ∈ ℓ ( V ∗ ) | P FS [ v ] < ∞} ◮ meaningful functions in this set? ◮ extension of functions?

Harmonic functions ◮ A function h ∈ ℓ ( V ∗ ) is harmonic if for any ε > 0 P Z ε, 0 [ h | V 0 ] = P Z ε, n [ h | Vn ] for all n ≥ 0 . ◮ Notation: H FS ( V ∗ ) := { h ∈ ℓ ( V ∗ ) harmonic } . ◮ For any h ∈ H FS ( V ∗ ) P FS [ h ] = lim ε → 0 + P Z ε, n [ h | Vn ] .

Harmonic extension rule Theorem [2]: For any h ∈ H FS ( V ∗ ) , j = 1 , 2 , 3, h | Gj ( V 0 ) = A j h | V 0 , where   3 Z C + 5 Z eff 3 Z C 3 Z C FS 1     A 1 = 3 Z C + 2 Z eff 3 Z C + 2 Z eff 3 Z C + Z eff   9 Z C + 5 Z eff  FS FS FS  FS   3 Z C + 2 Z eff 3 Z C + Z eff 3 Z C + 2 Z eff FS FS FS   3 Z C + 2 Z eff 3 Z C + 2 Z eff 3 Z C + Z eff FS FS FS   1   A 2 = 3 Z C + 5 Z eff  3 Z C 3 Z C  9 Z C + 5 Z eff  FS  FS   3 Z C + Z eff 3 Z C + 2 Z eff 3 Z C + 2 Z eff FS FS FS   3 Z C + 2 Z eff 3 Z C + Z eff Z C + 2 Z eff FS FS FS 1     A 3 = 3 Z C + Z eff 3 Z C + 2 Z eff 3 Z C + 2 Z eff .   9 Z C + 5 Z eff  FS FS FS  FS   3 Z C + 5 Z eff 3 Z C 3 Z C FS

Observations ◮ A 1 , A 2 , A 3 have the same eigenvalues 3 Z eff λ 3 = 1 FS λ 1 = 1 , λ 2 = , 3 λ 2 , 9 Z C + 5 Z eff FS ◮ span { u 1 } = { constant harmonic functions } , ◮ | λ 3 | < | λ 2 | < 1. Otherwise, P FS [ h ] = P Z 0 [ A j h | V 0 ] (power dissipation concentrates in one single cell, a contradiction).

Continuity of harmonic functions Theorem (A.R.’17): Harmonic functions are continuous on V ∗ .

Harmonic extension and power dissipation Lemma: There exists r ∈ ( 0 , 1 ) such that P Z 0 [ A j h 0 ] ≤ r 2 P Z 0 [ h 0 ] ∀ j = 1 , 2 , 3 and any non-constant function h 0 ∈ ℓ ( V 0 ) .

Consequences ◮ Harmonic functions are well-defined on K ∞ , H FS ( K ∞ ) = { h : K ∞ → C | h | V ∗ harmonic on V ∗ } . ◮ Well-defined power dissipation in K ∞ , P FS [ h ] = P FS [ h | V ∗ ] , h ∈ H FS ( K ∞ ) .

Power dissipation measure Theorem (A.R.’17): For each non-constant h ∈ H FS ( K ∞ ) , power dissipation induces a continuous measure ν h on K ∞ with supp ν h = C ∞ . Define � ν h ( T w ) := lim ε → 0 + lim P Z ε, n [ h ] xy n →∞ x , y ∈ T w ∩ V n { x , y }∈ E n for each m -cell T w .

Oscillations Corollary: For any m -cell T w , ν h ( T w ) ≍ osc ( h | Tw ) 2 .

Self-similar measure on K ∞ Bernouilli measure µ on K ∞ : 3 � µ ( T w 1 ... w n ) = µ w 1 · · · µ w n , µ i = 1 . i = 1 ◮ supp µ = C ∞ , ◮ ( C ∞ , µ ) is probability space, ◮ take µ 1 = µ 2 = µ 3 = 1 3 .

Singularity of power dissipation Theorem (A.R.’17): Assume that for any non-constant h ∈ H FS ( K ∞ ) such that h | V 0 = v 0 x �→ � D P 0 M n ( x ) . . . M 1 ( x ) v 0 � is non-constant for some n ≥ 1. Then, the measure ν h is singular with respect to µ .

Summary ◮ Power dissipation on an infinite (fractal) AC network ◮ harmonic potentials are continuous ◮ (non-atomic) power dissipation measure ◮ singularity of power dissipation measure

References P. Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski ac circuits , (2017), arXiv:1701.08039. L. Anderson, U. Andrews, A. Brzoska, J. P. Chen, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew, L. G. Rogers, and A. Teplyaev, Power dissipation in fractal AC circuits , Journal of Physics A (2017), accepted. O. Ben-Bassat, R. S. Strichartz, and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals , J. Funct. Anal. 166 (1999), no. 2, 197–217. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman lectures on physics. Vol. 2: Mainly electromagnetism and matter , Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. Thank you for your attention!