Infinite AC Ladder with a "Twist"
Quan M. Nguyen, Linh K. Nguyen, Tung X. Tran, Chinh D. Tran, Truong H. Cai, Trung Phan
aa r X i v : . [ phy s i c s . e d - ph ] O c t Infinite AC Ladder with a “Twist”
Quan M. Nguyen, Linh K. Nguyen,
2, 3
Tung X. Tran,
1, 3
Chinh D. Tran, Truong H. Cai, and Trung V. Phan ∗ Hanoi-Amsterdam High School, 01 Hoang Minh Giam Str.,Trung Hoa Nhan Chinh, Cau Giay Dist., Hanoi 100000, Vietnam. Lam Son High School, 307 Le Lai Str.,Dong Son Dist., Thanh Hoa 440000, Vietnam Massachusetts Institute of Technology, Cambridge, MA 02139. Hung Vuong High School, 70 Han Thuyen Str.,Tan Dan, Viet Tri, Phu Tho 35000, Vietnam. School of Engineering, Brown University, Providence, RI 02912, USA. Department of Physics, Princeton University, Princeton, NJ 08544, USA. (Dated: October 14, 2020)
Abstract
The infinite AC ladder network can exhibit unexpected behavior. Entangling the topology bringseven more surprises, found by direct numerical investigation. We consider a simple modificationof the ladder topology and explain the numerical result for the complex impedance, using linearalgebra. The infinity limit of the network’s size corresponds to keeping only the eigenvectors ofthe transmission matrix with the largest eigenvalues, which can be viewed as the most dominantmodes of electrical information that propagate through the network. . A CURIOUS FINDING FROM AN INFINITE AC LADDER WITH A “TWIST” FIG. 1: (A) The infinite AC ladder network. (B) An infinite AC ladder network with a“twist”. We call this topology the symmetric twisted ladder.The infinite AC network with ladder topology given in Fig. 1A is well-known, appearsin many introductory physics courses and standard textbooks as a model of a transmis-sion line . It was even introduced to high school students in 18th International PhysicsOlympiad (East Germany 1987). For the network consisting of identical inductors L and ca-pacitors C , by adding one more unit cell and assuming convergence, we obtain a consistencyequation : Z AB ( ω ) = 2 iωL + Z AB ( ω ) /iωCZ AB ( ω ) + 1 /iωC , (1)in which the complex impedance can be solved: Z AB ( ω ) = ωL (cid:0) i + p − /ω LC (cid:1) , (2)where ω is the AC frequency. This result exhibits two distinct behaviors in different rangesof ω : (i) when ω > p /LC , Z AB ( ω ) is purely imaginary; (ii) when ω ≤ p /LC , Z AB ( ω )has both non-zero imaginary and real components, which is strange since all elements in thenetwork have imaginary impedances. In fact, the consistency equation requires Z AB ( ω ) toconverge as the size of the networks goes to infinity N → ∞ , which is wrong in this rangeof frequency. However, by adding a very small resistance to every element in this network,2e (cid:0) Z AB ( ω ) (cid:1) emerges and convergence reappears, which is shown elegantly in a numericalinvestigation by Van Enk et al and later studied in greater theoretical detail .FIG. 2: A numerical investigation on the symmetric twisted ladder consists of inductors L = 1H and capacitors C = 1F. (A) The complex impedance versus the network size N ,for frequency ω = 2rad/s and all AC elements have no resistance. (B) The compleximpedance versus the network size N , for frequency ω = 2rad/s and all AC elements haveresistance R = 0 . N ∈ [450 , R = 0 . N → ∞ and R → and fractaltopologies . Here we consider a much simpler topology with as much complexity if notmore: a modification of the ladder topology by “adding a twist”, which we call a symmetrictwisted ladder topology (see Fig. 1B). Understanding more about the properties of theladder-like AC networks are of relevant to not only electrical engineering (eg. transmissionline designs) but also biophysics, for example as a cochlear model and a model of neuralionic-channels . We have found many surprises: not only is there no frequency range where Z AB ( ω ) converges, but also adding infinitesimal resistances to all AC elements can make3 AB ( ω ) converges to a positive real value, completely eliminating the imaginary component: Z AB ( ω ) (cid:12)(cid:12)(cid:12) ω ≥ / √ LC ∈ R + , lim ω →∞ Z AB ( ω ) = p L/C . (3)Our numerical finding is shown in Fig. 2. A theoretical explanation in detail for this curiousbehavior will be provided Section II.
II. AN EXPLANATION WITH THE TRANSMISSION MATRIX METHOD
For linear-linking resistive networks, the voltages and currents of nodes in consecutiveunit cells can be related by a transmission matrix . We can apply this linear algebra methodto find Z AB ( ω ) of the symmetric twisted ladder.We label the nodes as shown in Fig 3A. We split every point into two, then label thecurrents as shown in Fig 3B. The unit cells are now separated into blocks of A j A ′ j B ′ j B j asshown in Fig 3C. The inlet current is I A and the outlet current is I B , I A = I B . Dueto symmetry, we have the currents I A j = I B j , I A ′ j = I B ′ j and the voltages V A j = − V B j , V A ′ j = − V B ′ j . If an unit amplitude current comes into node A and goes out from node B ,then I A = I B = 1, I A ′ = I B ′ = 0 and V A = Z AB ( ω ) / × T ]:[ E j ] = [ T ][ E j +1 ] , [ E j ] = V A j Z I A j V A ′ j Z I A ′ j , (4)The transmission matrix can be found by applying Ohm’s law and Kirchoff’s laws to a unitcell, then solving for the components of [ E j ]. Denoting z = − Z /Z , we obtained:[ T ] = − z − − z −
11 0 0 01 1 1 − z . (5)The transmission matrix [ T ]’s eigenvalues λ and eigenvectors [ t ] satisfy:[ T ][ t ] = λ [ t ] ⇒ λ + zλ + 2(1 − z ) λ + zλ + 1 = 0 . (6)4IG. 3: (A) We label the nodes with A , A , A , ... and B , B , B , ... in which A ≡ A and B ≡ B . (B) We split every points A j , B j into A j and A ′ j − , B j and B ′ j − connectedby a wire of no resistance. Then we label the currents in those wires with I A j , I B j and thecurrents flow in the diagonal resistors with I A ′ j and I B ′ j . (C) An unit cell is a block of A j A ′ j B ′ j B j .There are four complex solutions λ , λ , λ , λ to this quartic polynomial, correspondingto four eigenvectors [ t ], [ t ], [ t ], [ t ]. We will not write their expressions explicitly here,because not only are they very long and complicated but also we do not need their analyticalforms to carry on the analysis. Without loss of generality, we assume that | λ | ≥ | λ | ≥ | λ | ≥ | λ | . (7)The components of the eigenvectors are in the same order of magnitude. From the symmetricproperties of the polynomial, it can be deduced that λ λ = 1 and λ λ = 1. For apure LC network with Z = iωL and Z = 1 /iωC , λ = λ ∗ and [ t ] = [ t ] ∗ . When ω ≥ / √ LC we even have all eigenvalue amplitudes to be equal, λ = λ ∗ and [ t ] = [ t ] ∗ .If we add infinitesimally small resistance 0 + Ω to every circuit elements then z obtains a5ositive imaginary part Im(z) = 0 + and clear splitting between the eigenvalue amplitudesappears as shown in Fig. 4: | λ | > | λ | > > | λ | > | λ | . (8)FIG. 4: The semilog plots represent how eigenvalue amplitudes of transmission matrix [ T ]for the symmetric twisted ladder AC network depend on frequency ω . (A) No resistancepresented in the network. (B) A small resistance is added in series to every circuitelements.For a network of finite size N there should be no current coming in from the end:[ E N ] = V A N V A ′ N , [ E ] = Z ( N ) AB / Z V A ′ . (9)Given that the eigenvalues are distinct, there is always a unique way to decompose [ E N ] intoelectrical information modes [ t ] : [ E N ] = X k =1 c k [ t k ] , (10)where c , c , c , c are complex numbers. Thus,[ E ] = [ T ] N [ E ] = [ T ] N [ E N ] = X k =1 c k λ Nk [ t k ] . (11)6olving the boundary conditions X k =1 c k [ t k ] = 0 , X k =1 c k λ Nk [ t k ] = Z , X k =1 c k [ t k ] = X k =1 c k λ Nk [ t k ] = 0 , (12), where [ t k ] i denotes the i-th component of the k-th eigenvector, gives us the coefficients c , c , c , c , which can then be used to obtain Z ( N ) AB = 2 Z P k =1 c k λ Nk [ t k ] P k =1 c k λ Nk [ t k ] . (13)The infinite network’s complex impedance is defined from taking the limit of N → ∞ : Z AB = lim N →∞ Z ( N ) AB , (14)which only makes sense when Z ( N ) AB is convergent. A. On the Emergence of Convergence
To understand how convergence emerges, we start with a set of four eigenvalues withdistinct amplitudes | λ | > | λ | > | λ | > | λ | , (15)At large N → ∞ limit, the solution to the boundary conditions (12) satisfies | c | ≪ | c | ∼| c | ∼ | c | and | c λ N | ∼ | c λ N | ≫ | c λ N | ≫ | c λ N | , (16)where the sign ∼ represents the estimation within an order of magnitude. In more detail,the ratio c λ N /c λ N ≈ − [ t ] / [ t ] (17)and the ratio c /c , c /c are independent of N : c /c c /c ≈ − [ t ] [ t ] [ t ] [ t ] − [ t ] [ t ] . (18)From the separation of scale (16), the complex impedance can be approximated as Z ( N ) AB ≈ Z [ t ] [ t ] − [ t ] [ t ] [ t ] [ t ] − [ t ] [ t ] (cid:18) α λ N λ N (cid:19) , (19)7here α is given by: α = c c (cid:18) [ t ] [ t ] − [ t ] [ t ] [ t ] [ t ] − [ t ] [ t ] − [ t ] [ t ] − [ t ] [ t ] [ t ] [ t ] − [ t ] [ t ] (cid:19) . (20)Given that | λ | > | λ | , Z ( N ) AB converges to Z AB = 2 Z [ t ] [ t ] − [ t ] [ t ] [ t ] [ t ] − [ t ] [ t ] . (21)Note that this value only depends on the eigenvectors [ t ] , [ t ] associated with the two eigen-values that have the highest amplitudes. Physically speaking, the complex impedance ofour infinite AC network comes from the two most dominant electrical modes propagatingthrough the infinite series of unit cells.The equation (20) shows that the convergence of Z ( N ) AB has an exponential decay rate γ and an oscillating frequency | κ | which can be found from the identifications e − γN ∼ (cid:12)(cid:12) λ N /λ N (cid:12)(cid:12) ⇒ γ = ln (cid:0) | λ /λ | (cid:1) ,e iκN ∼ e i arg( λ N /λ N ) ⇒ | κ | = (cid:12)(cid:12) arg( λ /λ ) (cid:12)(cid:12) . (22)Such behavior can be seen from numerical investigation as shown in Fig. 5A. The circuitinvestigated was the symmetric twisted ladder at frequency ω = 2rad/s, with inductors of L = 1H and capacitors of C = 1F, each connected in series with a resistance R = 0 . | arg( λ /λ ) | ≈ . | κ | ≈ | arg( λ /λ ) | ≈ . γ = (6 . ± . × − , which is not far off fromln( | λ /λ | ) ≈ . × − .When | λ | = | λ | , the convergence vanishes. When all four eigenvalues have the samemagnitude, the situation becomes even worse because now there is no dominant electricalmode. Those two are the cases with our symmetric twisted ladder, ω < / √ LC and ω ≥ / √ LC correspondingly, thus no convergence for all frequency ω .By adding an infinitesmal resistors R = 0 + Ω in series with every AC elements: Z = iωL → R + iωL ,Z = 1 /iωC → R + 1 /iωC , (23)all eigenvalues will have different magnitudes. In more detail, for ω < / √ LC : | λ | = 1 + | ǫ | , | λ | = 1 − | ǫ | , | ǫ | ∝ R , (24)8IG. 5: (A) The real component of the complex impedance Re (cid:0) Z ( N ) AB (cid:1) gradually convergesto a fixed value. (B) Fourier transformation of Re (cid:0) Z ( N ) AB (cid:1) in the range N ∈ [800 , (cid:0) Z ( N ) AB (cid:1) in the range N ∈ [800 , ω ≥ / √ LC : | λ | = 1 + | ǫ | , | λ | = 1 − | ǫ | , | ǫ | ∝ R , | λ | = 1 + | ǫ ′ | , | λ | = 1 − | ǫ ′ | , | ǫ ′ | ∝ R , (25)where the sign ∝ represents a proportional relation. From (22), the rate of convergence γ = 2 | ǫ | ∝ R , disappears when R = 0 as expected. B. On the Emergence of Real Impedance
Using equation (21), we arrive at the analytical result: Z AB ( ω ) = lim R → lim N →∞ Z ( N ) AB ( ω ) = p L/C × q ω LC + 2 iω √ LC √ − ω LC , (26)9here the square-root value √· lies on the right half plane. This yields a good agreement innumerical investigation, as shown in Fig. 2B, under the condition: N ≫ min( ωL, /ωC ) /R ≫ . (27)When ω ≥ / √ LC , the complex impedance has no imaginary component and thus becomespurely real: Z AB ( ω ) (cid:12)(cid:12)(cid:12) ω ≥ / √ LC = p L/C × q ω LC − ω √ LC √− ω LC . (28)At a very large frequency ω → ∞ , the complex impedence converges with a power law decay: Z AB ( ω ) ≈ p L/C (1 + 1 / ω LC ) → p L/C , (29)which can also be written as:lim ω →∞ lim R → lim N →∞ Z ( N ) AB ( ω ) = p L/C . (30)It is important to note that condition (27) indicates N → ∞ , R → ω → with the averagingtime T , the kinematic viscosity ν and the stirring length scale L :lim L→∞ lim ν → lim T →∞ S ( l ) = − ǫl/ , (31)where S is the third order longitudinal structure function, l is the distance between pointsof interests and ǫ is the energy dissipation per unit mass. III. DISCUSSION
This paper is our attempt to scratch the surface of a potentially deep topic. Not onlycan the complex impedance of an infinite AC network gain a real component by addinginfinitesimal resistances, we now know that its imaginary component can be completelyeliminated by entangling the topology. 10IG. 6: (A) The asymmetric twisted ladder. (B) The average complex impedance in therange N ∈ [450 , L = 1H, C = 1F, and each isconnected in series to a resistance R = 0 . N → ∞ and R → Z AB ( ω ) = lim R → lim N →∞ Z ( N ) AB ( ω ) = p L/C × q ω LC + iω √ LC √ − ω LC . (32)The result of our numerical investigation on this topology is given in Fig. 6B. Here wecan see all three distinct behaviors corresponding to regions I, II and III of the frequency ω : (I) when ω < ω , Z AB ( ω ) has both non-zero imaginary and real components; (II) when ω ≤ ω ≤ ω , Z AB ( ω ) has only a real component; (III) when ω ≤ ω , Z AB ( ω ) has only animaginary component. The frequencies ω , ω are: ω = p / LC , ω = q (1 + √ / LC . (33)We summarize the ranges and trends of the complex impedances for some circuits in Fig.7, using Nyquist plots – a method often used in electrochemical impedance spectroscopy .We have not observed any infinite real impedance but only observed infinite imaginary parts.11IG. 7: The Nyquist plots of different infinite AC ladder networks, where the arrowindicates the evolution of Z AB ( ω ) in the complex plane as the frequency goes from low tohigh ω = 0 → ∞ .Here, we also link some features of these plots to some features in the topologies of thecorresponding circuits based on our limited data, and attempt some qualitative explanationswhich we find useful intuitively. The complex impedance remains finite in the top-center,bottom-left and bottom-right circuits in Fig. 7 even when the frequencies approach infinity.We notice that these three circuits differ from the rest in the presence of “capacitor chains”:there are many ways ( ∼ N ) to connect the two terminals by capacitors with only a fewinductors in between. When the frequency becomes large, the impedance of the capacitors12anishes, and the impedances of those chains are of the order of jωL + N/jωC + N R , ordersof magnitude smaller than the jN ωL of “inductor chains”.Hence, using the random-walk analogy of electrical circuits , the two circuits can beviewed as containing ∼ N paths with large probabilities between the two terminals comparedto other circuits. Therefore, the overall probability connecting to the circuit should also belarger. A very rough estimate would add all the probabilities of such paths together. Afterconverting escape probability to impedance, it gives jωL/N + R . We clearly overdid it, butat least this estimation shows a hint of why circuits of these types are expected to havefinite complex impedances in the limit of interests. Note that the original analogy is for realimpedances only, but here we extend it using naive analytical continuation.There are many unanswered questions and features of infinite AC networks that we stilldo not fully understand: the ties to graph theory and how order of magnitude estimates ofthe limiting behaviors of the impedance can be made just based on the features of the circuit,how the network’s topology decides (conclusively) which behavior of Z AB will be shown atwhat range of frequency ω , how the convergence rate γ depends on the network’s topology,can the choice of adding infinitesimal resistance to every AC elements always lead to Z ( N ) AB convergence, can the choice of posing boundary conditions at the infinitely far end change Z AB in a drastic manner, can a negative real component of complex impedance Re( Z AB ) < ACKNOWLEDGMENTS
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