Effective resistances of two dimensional resistor networks
EEffective resistances of two dimensional resistornetworks
Rajat Chandra Mishra and Himadri Barman St Stephens College, University of Delhi, New Delhi 110007, India Department of Physics, Zhejiang University, Hangzhou 310027, ChinaE-mail: [email protected]
Abstract.
We investigate the behavior of two dimensional resistor networks, withfinite sizes and different kinds (rectangular, hexagonal, and triangular) of latticegeometry. We construct the network by having a network-element repeat itself L x times in x -direction and L y times in the y -direction. We study the relationship betweenthe effective resistance ( R eff ) of the network on dimensions L x and L y . The behavior issimple and intuitive for a network with rectangular geometry, however, it becomes non-trivial for other geometries which are solved numerically. We find that R eff dependson the ratio L x /L y in all the three studied networks. We also check the consistency ofour numerical results experimentally for small network sizes. Keywords: Circuit analysis, resistor network, electrical experiment, Kirchhoff ’s laws
Submitted to:
Eur. J. Phys. a r X i v : . [ phy s i c s . c l a ss - ph ] J u l ffective resistances of two dimensional resistor networks
1. Introduction
Resistor network problems have been widely studied in various contexts, starting fromtextbook physics and competitive tests [1] to electrical engineering [2], condensed matterphysics [3], and statistical physics [4]. Since many regular electrical networks take shapesof meshes, similar to the lattices in solid state crystals, it is intriguing to find out theequivalent or effective resistance of such networks. There had been extensive studieson two-dimensional lattices in order to investigate percolation based conductivity [5]in such systems and methods like effective medium theory [5–8] and Green’s functionmethod [5, 9–11] have been formulated. However, most of these studies focus onthe infinite systems with stochastic resistance distribution (random resistor network).Although a few studies have been conducted recently for finite size networks, suchstudies either investigated equivalent resistance between two points inside the networkor for networks that do not obey the typical crystal lattice symmetries [11–16]. Hencedimensional dependence of effective resistance ( R eff ) for various geometries deservesseparate attention and it also bears an academic interest to show how R eff can simplybe estimated by solving a set of electrical equations. Network geometry dependenceof R eff brings the connection to the graph theory [17, 18] and generalization of Y -∆or star-polygon transformation can open doors of future research [19]. The networksdescribed in our paper are very straightforward and easy to solve numerically once thecorrect equations are formulated. However, such solutions do not exist in the literatureto the best of our knowledge and hence our findings are both pedagogical and researchoriented. Given resources, the network models can be constructed by students easilyand knowing the dependence of R eff on the geometry, a device can be designed whoseresistance can be controlled by tuning its dimensions.Our paper is organized in the following way. We first explain the generic resistornetwork setup and then discuss the analytical solution for the rectangular geometry.Then we discuss the numerical formulation and R eff ’s dependence on the dimensions,obtained from our numerical results for rectangular, hexagonal, and triangular resistornetworks. As a summary, we compare these results for these three different geometriesand finally we describe a small experiment to test our theoretical findings.
2. Generic resistor network configuration
We describe below our generic setup for various lattice geometries:(i) Define a two dimensional (2D) lattice. Though the geometry varies lattice to lattice,we define the size of each lattice by two Cartesian lengths L x and L y . For arectangular lattice, L x L y becomes the number of the lattice points or sites as well.(ii) Each lattice point attaches to a resistance of value R spreading along the directionof its neighborhood lattice points.(iii) Apply a bias V at one edge of the lattice (say in the direction of the length L x )and ground the other edge (hence the lattice acts like an active medium attached ffective resistances of two dimensional resistor networks i, j ) in 2D Cartesian coordinate: (cid:88) k I ij ( k ) = 0 ⇒ (cid:88) k [( V ik − V ij ) /R ik + ( V kj − V ij ) /R jk ] = 0 (1)where k denotes all nearest neighbor nodes (lattice points, voltage or groundingconnection) to the site ( i, j ). We discuss the implementation of this in the forthcomingsections, where we formulate them in the form of a matrix equation for various networkgeometries.
3. Rectangular resistor network
As the most common 2D geometry, we begin with a finite size rectangular lattice definedby lengths L x and L y . Following the setup defined in the previous section, a voltage V is applied at one end of the lattice, in the direction of the length L x (see Fig. 1) whilethe other end is grounded. Resistances, each with value R , are connected to each sitein all four directions. Our objective is to find out the effective resistance ( R eff ) for thegeometry and how R eff depends on the dimensions L x and L y . To find out the effective resistance in the lattice, for a moment, we assume there areno resistive connections in the y (vertical) direction. Thus for a lattice of size L x × L y ,points are connected only in the x -direction (see Fig. 2). There are total L y branchesof parallel resistances with each branch consisting of a set of resistances in series. Nowin each set of resistances in series, we can notice that the equivalent resistance betweentwo adjacent sites is R + R = 2 R (resistance R on the left of one site and on the rightof the other site). Thus we find L x − R between firstand last lattice points and two resistances of value R on the left and right ends of thelattice. Thus in total, we have L x number of resistances of value 2 R in series on eachbranch. Thus equivalent resistance of each branch is 2 RL x . Since there exist L y suchbranches in parallel, the overall effective resistance of this simplified circuit: R simpeff = (cid:2) / (2 RL x ) + · · · ( L y times) (cid:3) − = 2 RL x /L y . (2) ffective resistances of two dimensional resistor networks RV RRR R RRRRV RR R RRRV RR R RR L y L x Fig. 1.
A resistor network on a rectangular lattice geometry. R R (1,1) 2 R R ( L x , R R (1,2) 2 R R ( L x , R R (1 , L y ) 2 R R ( L x , L y ) V Fig. 2.
A network of resistors connected only in the x -direction. We know that when two resistances ( R and R ) are connected in series, as shownin Fig. 3(a), the potential drop after the first resistance R (i.e. in the middle of R and R ) will be given by V − V = V R R + R , (3)which gives the potential at the middle of R and R : V = V (cid:18) R R + R (cid:19) . (4) ffective resistances of two dimensional resistor networks R R V (a) R R V R R R (b) Fig. 3. (a) Two resistances in series. (b) A simple Wheatstone bridge, which is arealization of the case L x = 1, L y = 2 in the rectangular network. Extending this argument to our case, the potential at a point ( i, j ) will be V i,j = V (cid:18) iR RL x (cid:19) = V (cid:18) iL x (cid:19) . (5)since there are equivalent resistance of value ( i − . R + R = 2 iR [( i −
1) resistanceswith value 2 R plus a single resistance with value R ] resistances to the left of point ( i, j ).Eq. (5) shows that the potential at any branch is independent of y -coordinate.Thus, even if we were to connect the points in the y -direction using resistances ofthe same value (which was our original network to begin with), no current would flow inthe y -direction for the same x -coordinate. This means that the original network, withall the lattice points joined, is equivalent to the network with lattice points joined onlyin the x -direction. Since the two networks are equivalent, the effective resistances of theoriginal rectangular network will be the same as the one in Eq. (2): R recteff = 2 R L x L y = R z L x L y (6)where we attempt to write the formula in a more generic form by looking at thecoordination number z (number of nearest neighbor sites, z = 4 for a rectangularlattice). We can easily notice that a balanced Wheatstone bridge [2] with resistance R on each of its branches is the L x = 1 and L y = 2 case of the rectangular network (seeFig. 3(b)). There, by applying Eq. (6), we get R eff = 2 R. / R which is supposed tobe the desired result for the bridge network. In our rectangular lattice of size L x × L y , we can mark out distinct 9 kinds of latticepoints: • Left bottom corner point ( i = 1, j = 1) • Left top corner point ( i = 1, j = L y ) ffective resistances of two dimensional resistor networks • Right bottom corner point ( i = L x , j = 1) • Right top corner point ( i = L x , j = L y ) • Left Non-corner edge points ( i = 1, j ∈ [2 , L y − • Bottom non-corner edge points ( i ∈ [2 , L x − j = 1) • Right non-corner edge points ( i = L x , j ∈ [2 , L y − • Top non-corner edge points ( j = L y , i ∈ [2 , L x − • Non-border inner points ( i ∈ [2 , L x − j ∈ [2 , L y − Left bottom corner point → i = 1, j = 1: V V , R + V , V , R + V , V , R = 0 . (7)(ii) Left top corner point → i = 1, j = L y : V V ,Ly R + V ,Ly V ,Ly R + V ,Ly V ,Ly R = 0 . (8)(iii) Right bottom corner point → i = L x , j = 1: V Lx , V Lx, R − V Lx, R + V Lx, V Lx, R = 0 . (9)(iv) Right top corner point → i = L x , j = L y : V Lx ,Ly V Lx,Ly R − V Lx,Ly R + V Lx,Ly V Lx,Ly R = 0 . (10)(v) Left non-corner edge point → i = 1, j = 2 to L y V V ,j R + V ,j V ,j R + V ,j V ,j R + V ,j +1 V ,j R = 0 . (11)(vi) Right non-corner edge point → i = L x , j = 2 to L y V Lx ,j V Lx,j R − V Lx,j R + V Lx,j V Lx,j R + V Lx,j +1 V Lx,j R = 0 . (12)(vii) Bottom non-corner edge point → i = 2 to L x , j = 1: V i , V i, R + V i +1 , V i, R + V i, V i, R = 0 . (13)(viii) Top non-corner edge point → i = 2 to L x , j = L y : V i ,Ly V i,Ly R + V i +1 , V i,Ly R + V i,Ly V i,Ly R = 0 . (14)(ix) Non-border inner point → i = 2 to L x j = 2 to L y V i ,j V i,j R + V i +1 ,j V i,j R + V i,j V i,j R + V i,j +1 V i,j R = 0 . (15) ffective resistances of two dimensional resistor networks V ij :(i) Left bottom corner point → i = 1, j = 1: (cid:20) R + 12 R + 12 R (cid:21) V , − R V , − R V , = VR . (16)(ii)
Left top corner point → i = 1, j = L y : (cid:20) R + 12 R + 12 R (cid:21) V ,L y − R V ,L y − R V ,L y − = VR . (17)(iii)
Right bottom corner point → i = L x , j = 1: (cid:20) R + 1 R + 12 R (cid:21) V L x , − R V L x − , − R V L x , = 0 . (18)(iv) Right top corner point → i = L x , j = L y : (cid:20) R + 1 R + 12 R (cid:21) V L x ,L y − R V L x − ,L y − R V L x ,L y − = 0 . (19)(v) Left non-corner edge point → i = 1, j = 2 to L y − (cid:20) R + 12 R + 12 R + 12 R (cid:21) V ,j − R V ,j − R V ,j − − R V ,j +1 = VR . (20)(vi)
Right non-corner edge point → i = L x , j = 2 to L y − (cid:20) R + 1 R + 12 R + 12 R (cid:21) V L x ,j − R V L x − ,j − R V L x ,j − − R V L x ,j +1 = 0 . (21)(vii) Bottom non-corner edge point → i = 2 to L x , j = 1: (cid:20) R + 12 R + 12 R (cid:21) V i, − R V i − , − R V i +1 , − R V i, = 0 . (22)(viii) Top non-corner edge point → i = 2 to L x , j = L y : (cid:20) R + 12 R + 12 R (cid:21) V i,L y − R V i − ,L y − R V i +1 , − R V i,L y − = 0 . (23)(ix) Non-border inner point → i = 2 to L x − j = 2 to L y − (cid:20) R + 12 R + 12 R + 12 R (cid:21) V i,j − R V i − ,j − R V i +1 ,j − R V i,j − − R V i,j +1 = 0 . (24)Now V ij ’s constitute a L x × L y matrix, but if we linearize (see Appendix for details) itas a column vector V of length L x L y , The above equations can be represented in matrixnotation as GV = I (25) ffective resistances of two dimensional resistor networks I is a column vector whose values are given by the right hand side of equations (7)to (15) and G is a matrix consisting of the coefficients of the variables in the equations.Since G has units 1 /R , we are calling it the conductance matrix . Since the potential atany lattice point ( i, j ) depends only on its neighboring points, the matrix G is generallysparse, and can be solved using a sparse matrix solver numerically. The method is verysimilar to typical transfer matrix method used in circuit analysis [22] and also similarto the method used in the context of disordered resistor network [23].Once the above matrix system is solved and we know the potential at all latticepoints, the effective resistance can be determined by dividing the total applied voltageby net current flowing through the lattice in the direction of the applied voltage. It canbe observed that the net current can be determined using the potential of the end-pointsof the lattice. The net current, in this case, would be given by I net = Σ L y j =1 V Lx,j
R . (26)Effective Resistance can then be determined as R eff = VI net . (27)In practice, we choose R = 1 and V = 1. We first plot R eff against L x for several fixed values of L y . As expected from theanalytical solution expressed in Eq. (6), R eff grows linearly as L x increases and the slopeof the linear curve drops at a larger value of L y (see Fig. 4(a)). L x R e ff Rectangular L y = 10L y = 20L y = 50L y = 100L y = 200 (a) L y (b) Fig. 4.
A plot of R eff as (a) L x is varied for different values of L y and (b) L y is variedfor different values of L x for a rectangular lattice. When L x is kept constant, R eff decreases as L y increases and R eff vs 1 /L y plots showlinear, establishing that R eff ∝ L x /L y . Now to find out the proportionality constant, we ffective resistances of two dimensional resistor networks r ≡ R eff L y RL x (28)which according to Eq. (6) should be equal to z/ r is a constant when L x and L y are varied respectively, keeping the otherdimension as a fixed parameter. The value of the constant is 2 and hence we see thatthe numerical results very well agree with our analytical formula. (a) (b) Fig. 5.
A plot of r = R eff L y / ( RL x ) as (a) L x and (b) L y is varied while otherparameters are kept fixed. Both show R ratio = 2 and it is independent of dimensions L x and L y .
4. Hexagonal Network Model
Now we consider the hexagonal or the graphene [24] type honeycomb lattice network.Out of two possible orientations, we select a hexagonal lattice which has armchair edgesin the x -direction and zigzag edges in the y -direction (see Fig. 6) and dub this armchairhexagonal lattice. Here for our convenience, we break the sites into two categories – (i) M -type sites, sittingat the middle corners of a hexagon and such sites connect to the bias and grounding,and (ii) S -type sites, sitting on the top or bottom sides of a hexagon. We add extraindices 0 and 1 to specify M and S sites respectively. Now we can see there must bealways equal and even numbers of M and S sites in the x -direction in a lattice withcomplete hexagons. The number of M and S sites ( L M or L S ) sets the measurement ofthe length L x : L x = L Mx = L Sx . On the other hand, the number of voltage connectiondetermines the length L y : L y = L My , L Sy = L My + 1 = L y + 1. Total number of sites can ffective resistances of two dimensional resistor networks V R
M M M MS S S SS S S S R R R R R R R R R R R R R R R R RV R R R RV R R R R R R R R R L y L x Fig. 6.
A resistor network on a hexagonal lattice geometry. be determined as N site = L Mx L My + L Sx L Sy = L x L y + L x ( L y + 1) = L x (2 L y + 1). We candistinguish 6 kinds of sites in this system: • Left Border Points ( i = 1, j = 1 to L y , k = 0) • Right Border Points ( i = L x , j = 1 to L y , k = 0) • Top border points ( i = 1 to L x , j = L y + 1 , k = 1) • Bottom Border Points ( i = 1 to L x , j = 1, k = 1) • M -type inner points ( i = 2 to L x − j = 1 to L y , k = 0) • S -type inner points ( i = 1 to L x , j = 2 to L y , k = 1)The KCL for the above 6 kinds of points would be:(i) Left Border Points → i = 1, j = 1 to L y , k = 0: V − V ,j, R + V ,j, − V ,j, R + V ,j +1 , − V ,j, R = 0 . (29)(ii) Right Border Points → i = L x , j = 1 to L y , k = 0: − V Lx,j, R + V Lx,j, − V Lx,j, R + V Lx,j +1 , − V Lx,j, R = 0 . (30)(iii) Top Border Points → i = 1 to L x , j = L y + 1, k = 1:if i=odd, V i,Ly, − V i,Ly +1 , R + V i +1 ,Ly +1 , − V i,Ly +1 , R = 0 . (31) ffective resistances of two dimensional resistor networks V i,Ly, − V i,Ly +1 , R + V i − ,Ly +1 , − V i,Ly +1 , R = 0 . (32)(iv) Bottom Border Points → i = 1 to L x , j = 1, k = 1:if i=odd, V i, , − V i, , R + V i +1 , , − V i, , R = 0 . (33)if i=even, V i, , − V i, , R + V i − , , − V i, , R = 0 . (34)(v) M -type inner points → i = 2 to L x − j = 1 to L y , k = 0:if i=odd, V i − ,j, − V i,j, R + V i,j, − V i,j, R + V i,j +1 , − V i,j, R = 0 . (35)if i=even, V i +1 ,j, − V i,j, R + V i,j, − V i,j, R + V i,j +1 , − V i,j, R = 0 . (36)(vi) S -type inner points → i = 1 to L x , j = 2 to L y , k = 1:if i=odd, V i +1 ,j, − V i,j, R + V i,j, − V i,j, R + V i,j − , − V i,j, R = 0 . (37)if i=even, V i − ,j, − V i,j, R + V i,j, − V i,j, R + V i,j − , − V i,j, R = 0 . (38)Note that here we have two types of sites, namely M and S , and that though we use( i, j ) to denote the two-dimensional location different types of sites, a particular kindof site belongs to a particular type and hence one type’s i or j should not coincide withanother type’s i or j and this distinction is taken care by the index k . As before, theabove equations can be represented in matrix notation and are solved using a sparsematrix solver. Like in the previous case, we first plot R eff as L x is varied, keeping L y fixed at differentvalues. Even for a hexagonal lattice, R eff seems to increase linearly with L x , as seenin Fig. 7(a). We then plot R eff as L y is varied, keeping L x fixed at different values.The result is shown in Fig. 7(b). Though R eff decreases with increasing L y like in therectangular lattice case, R eff vs 1 /L y plots are not exactly linear (see Fig. 4.2). ffective resistances of two dimensional resistor networks (a) L y (b) Fig. 7.
Plot of R eff as (a) L x is varied for different values of L y and (b) L y is variedfor different values of L x for an armchair hexagonal lattice based network. To see what is the actual dependence, we again plot the ratio r = R eff L y / ( RL x )against L x and L y keeping other parameters fixed. We found a few interestingobservations: (i) When L y is fixed, the ratio r becomes independent of L x ; (ii) When L x is fixed, r becomes universal and it approaches a constant at the thermodynamic limit( L y → ∞ ). These two observations, as shown in Fig. 4.2, let us arrive at a conclusionthat the ratio is a sole function of L y : r = α ( L y ) . (39)This leads to an empirical formula for the effective resistance: R hexeff = α ( L y ) L x L y . (40) ffective resistances of two dimensional resistor networks (a)(b)(c) Fig. 8.
Plot of r = R eff L y / ( R (cid:32)L x ) as (a) L y is varied for different values of L x and(b) as L x is varied for different values of L y for an armchair hexagonal lattice. (c) ln r plotted against 1 /L y to verify the formula given in Eq. (41) for α where r = α ( z, L y ). Now we further notice that α ( z, L y ) approaches z as L y → ∞ where z = 3 ffective resistances of two dimensional resistor networks V R R R R R RV R R R R R R R R R R R R R RV R R R R RV R R R R R R R R L y L x Fig. 9.
A resistor network on a triangular lattice geometry. is the lattice coordination number for a hexagonal lattice. Since α ( z, L y ) has to bedimensionless to keep Eq. (39) physically consistent, a convenient guess could be α ( L y ) = z e − c/L y (41)which implies ln α ( L y ) = ln z − cL y − . (42)where c is a constant. Now Fig. 8(c) plots ln α against L y − for various L x and we cansee when L y − approaches zero (thermodynamic limit), ln α approaches ln z = ln 3 (cid:39) . α in Eq. (41).
5. Triangular Network Model
We now move to the case where the lattice is triangular. The network consideredis shown in Fig. 9. Clearly this lattice is similar to a rectangular lattice, with theexception that diagonal points in one particular direction are also connected via anequivalent resistance 2 R . For a triangular lattice, we have 9 kinds of lattice points: • Left bottom corner point ( i = 1 , j = 1) • Left top corner point ( i = 1 , j = L y ) • Right bottom corner point ( j = 1 , i = L x ) • Right top corner point ( i = L x , j = L y ) • Left non-corner edge points ( i = 1 , j ∈ [2 , L y − • Right non-corner edge points ( i = L x , j ∈ [2 , L y − ffective resistances of two dimensional resistor networks • Bottom non-corner edge points ( j = 1 , in ∈ [2 , L x − • Top non-corner edge points ( j = L y , i ∈ [2 , L x − • Non-border inner points ( i ∈ [2 , L x − , j ∈ [2 , L y − i, j ) with its neighboring point, forthe above 9 kinds of points would be as follows:(i) Left bottom corner point → i = 1, j = 1: V V , R + V , V , R + V , V , R = 0 . (43)(ii) Left top corner point → i = 1, j = L y : V V ,Ly R + V ,Ly V ,Ly R + V ,Ly V ,Ly R + V ,Ly V ,Ly R = 0 . (44)(iii) Right bottom corner point → i = L x , j = 1: V Lx , V Lx, R − V Lx, R + V Lx, V Lx, R + V Lx − , V Lx, R = 0 . (45)(iv) Right top corner point → i = L x , j = L y : V Lx ,Ly V Lx,Ly R − V Lx,Ly R + V Lx,Ly V Lx,Ly R = 0 . (46)(v) Left non-corner edge point → i = 1, j = 2 to L y V V ,j R + V ,j V ,j R + V ,j V ,j R + V ,j +1 V ,j R + V ,j − − V ,j R = 0 . (47)(vi) Right non-corner edge point → i = L x , j = 2 to L y V Lx ,j V Lx,j R − V Lx,j R + V Lx,j V Lx,j R + V Lx,j +1 V Lx,j R + V Lx − ,j +1 V Lx,j R = 0 . (48)(vii) Bottom non-corner edge point → i = 2 to L x , j = 1: V i , V i, R + V i +1 , V i, R + V i, V i, R + V i − , V i, R = 0 . (49)(viii) Top non-corner edge point → i = 2 to L x , j = L y : V i ,Ly V i,Ly R + V i +1 , V i,Ly R + V i,Ly V i,Ly R + V i +1 ,Ly V i,Ly R = 0 . (50)(ix) Non-border inner point → i = 2 to L x j = 2 to L y V i ,j V i,j R + V i +1 ,j V i,j R + V i,j V i,j R + V i,j +1 V i,j R + V i − ,j +1 V i,j R + V i +1 ,j − V i,j R = 0 . (51) ffective resistances of two dimensional resistor networks (a) L y (b)(c) (d) Fig. 10.
Plot of R eff against (a) L x for different values of L x and (b) 1 /L y (inset:against L y ) for different values of L x for a triangular lattice. (c) R eff /L x vs L x plotsshow that R eff depends linearly on L x only at large L x or L y . (d) R eff /L y vs L y plotsshow that R eff does not strictly depends linearly on 1 /L y . After solving the above equations in the matrix form, we plot R eff against L x keeping L y fixed at various values. We again notice that R eff increases with L x and 1 /L y (see Fig. 10(a)). However, unlike the rectangular or hexagonal lattice network, thedependence of R eff is not strictly linear, rather it only becomes linear in L x at largevalue of L x or L y (see Fig. 10(c)).Now we look at the ratio r = R eff L y / ( RL x ) and find that it depends non-triviallyon both L x and L y . As can be noticed from Fig. 11(a) and Fig. 11(b), r approaches aconstant value only when L x /L y (when L x varied, L y fixed) or L y /L x (when L y varied, L x fixed) is significantly large (i.e. L x or L y approaches the thermodynamic limit comparedto the other dimension). However, unlike the earlier two lattice cases, we could nottrivially figure out any empirical function or formula for the R eff ’s dependence on L x and L y for the triangular lattice network. We presume that this non-triviality arisesbecause of the diagonal resistance dependence of the circuit current which is absentin hexagonal and rectangular lattice networks. Also, one should note that hexagonal ffective resistances of two dimensional resistor networks L x and L y havebeen reflected in all three different lattice networks discussed in our paper. (a) (b) Fig. 11.
Plot of r = R eff L y / ( RL x ) vs L y as L y is varied for different values of L x fora triangular lattice.
6. Summary
Now in Table 1, we briefly summarize R eff ’s dependence on the dimensions for variousnetwork geometries that we discussed already in the previous sections.Network geometry L x ( L y fixed) L y ( L x fixed) FormulaRectangular ∝ L x ∝ /L y R z L x /L y Hexagonal (armchair) ∝ L x ∝ /L y at L x (cid:28) L y R α ( z, L y ) L x /L y Triangular ∝ L x at L x (cid:28) L y not strictly ∝ /L y R α ( z, L x , L y ) L x /L y Table 1.
Table for dependence of R eff on various 2D lattice geometries. Our numerical codes (in Python) are freely available to the public on the Githubrepository: https://github.com/hbaromega/2D-Resistor-Network .
7. Experiment: Determining the effective resistance of a resistor network
Our theoretical findings can be easily verified by setting up simple circuits made upof resistors of equal magnitudes. We first constructed a 2 × × R eff should be 2 R = 200 Ω and 2 . R = 271 . ffective resistances of two dimensional resistor networks
18Ω respectively (see Section 3 and Section 4). Our multimeter readings show 200 Ωfor case A and 271 Ω for case B respectively, showing consistent agreement with ourtheoretical predictions (Fig. 12(c) and Fig. 12(d)). (a) (b)(c) (d)
Fig. 12. (a) Setup of 2 × × R eff for (c) the square and (d) hexagonalnetworks. Each resistor in the networks has resistance R = 100 Ω. We then connect the networks to a DC power supply (manufactured by Keltronix,India) and determined the effective resistance by measuring the voltage and currentacross the circuit (figure 13(a)). The following are the readings obtained for the 2 × ffective resistances of two dimensional resistor networks Table 1: Rectangular/square case V (Volts) I (Amperes)3.01 0.0173.48 0.0194.01 0.0224.49 0.0245.01 0.0275.63 0.036.01 0.0326.51 0.0357.06 0.0377.47 0.047.95 0.0428.47 0.0459.16 0.0489.49 0.0510.13 0.05310.53 0.05511.06 0.05811.49 0.0612.3 0.06413.02 0.06813.5 0.0714.06 0.07314.71 0.07615.1 0.078 ffective resistances of two dimensional resistor networks Table 2: Hexagonal (armchair) case V (Volts) I (Amperes)2.54 0.0133.52 0.0154 0.0174.55 0.0195 0.0215.52 0.0236.15 0.0256.52 0.0267.08 0.0297.5 0.038.06 0.0328.48 0.0349.02 0.0369.5 0.03810.06 0.0410.45 0.04111.05 0.04311.59 0.04512.09 0.04712.46 0.04913.02 0.05113.49 0.05314.03 0.05514.74 0.05715.1 0.059From the above two tables, we plot the I - V (current vs voltage) curves and fit eachof them with linear regression lines using the least square method [25]. The slopesof the regression lines estimate the values of conductance, G eff = 1 /R eff . We find G eff = 0 . − and G eff = 0 . − , implying R eff = 196 .
08 Ω and R eff = 270 .
27 Ωfor case A (square) and B (hexagonal) respectively. The values are slightly off thetheoretical values: 1.96% below for case A and 0.43% below for case B. In case A,the root mean square error (RMSE) and coefficient of determination ( R score) [25]of the regression line are 5.76 × − and 0.999831 respectively. The same for caseB are 1.6 × − and 0.999139 respectively. The values reflect that regression lineshave reasonably high accuracy. The lines, however, show very small finite interceptsof values 0 . . ffective resistances of two dimensional resistor networks (a)(b) Fig. 13. (a) Setup for determining internal resistance using voltage regulator. Theresistance is obtained by dividing the value of voltage by the value of current. (b)Current vs voltage plots for square and hexagonal lattices. The slopes s of the curvesestimate the values of R eff ( R eff = 1 /G eff ). Here we find R eff = 196 .
08 Ω for the squarenetwork and R eff = 270 .
27 Ω for the hexagonal network. power supply) since we have already checked that the networks accurately producethe theoretical result when measured separately with a multimeter. Thus as inference,we must say that our experiment validates the theory within very low error bars.The Python codes of our experimental plots and regression analysis can be found at https://github.com/hbaromega/2D-Resistor-Network/tree/master/EXPT .
8. Outlook
The detailed but simple derivations of finite size lattice networks of three distinctgeometries and the discussed simple experiment on a breadboard setup offer veryeasy and effective way to teach network analysis to students or even adults since ffective resistances of two dimensional resistor networks
Acknowledgments
HB and RCM owe to the NIUS Camp 2019, HBCSE, Mumbai, which made the projectto be worked out and successful. They also thank to Dr. Rajesh Khaprade and Dr.Praveen Pathak for providing the necessary hospitality and experimental facilities.
Appendix A. Linearization of V-matrix
The KCL equations contain two-dimensional V i,j elements. For the rectangular ortriangular lattice, when we linearize it to a one-dimensional vector or column matrix,we take either of these two mappings: Mapping 1: V , , · · · , V L x , → V , · · · , V L x .V , , · · · , V L x , → V L x +1 , · · · , V L x . ... V ,L y , · · · , V L x ,L y → V ( L y − L x +1 , · · · , V L y L x . Generically, ( i, j ) → ( j − L x + i . (A.1) Mapping 2: V , , · · · , V ,L y → V , · · · , V L y .V , , · · · , V ,L y → V L y +1 , · · · , V L y . ... V L x , , · · · , V L x ,L y → V ( L x − L y +1 , · · · , V L x L y . Generically, ( i, j ) → ( i − L y + j . (A.2)Now a typical equation such as Eq. (16) looks like αV + βV + · · · + γV L x = I . (A.3)which can be recast as G V + G V + · · · + G L x V L x = I . (A.4)Generically this can be written as G i V + G i V + · · · + G iL x V L x = I i . (A.5) ffective resistances of two dimensional resistor networks G G · · · G N L G G · · · G N L ... ... . . . ... G N L G N L · · · G N L N L V V ... V N L = I I ... I N L . (A.6)where N L ≡ L x L y . Finding corresponding row and column of G , given row of I : Since each lattice point follows a particular KCL depending on its neighborhood andvoltage connection, the rank of that lattice (reflected by the row or index i of currentvector I in Eq. (A.5)) in the mapped 1D array will denote the row of G and the indexof V (which is a vector or column matrix) will yield the column of G . Mapping in the hexagonal lattice case:
Since we introduce another index k in the armchair hexagonal lattice, we extend thelinear mapping as ( i, j, k ) → i + ( j − L x + kL x L y . (A.7)One can check the mapping conserves the total number of points N site = L x (2 L y +1): k = 0 case: V , , , · · · , V L x , , → V , · · · , V L x .V , , , · · · , V L x , , → V L x +1 , · · · , V L x . ... V ,L y , , · · · , V L x ,L y , → V ( L y − L x +1 , · · · , V L y L x . (A.8) k = 1 case: V , , , · · · , V L x , , → V L x L y +1 , · · · , V ( L y +1) L x .V , , , · · · , V L x , , → V ( L y +1) L x +1 , · · · , V ( L y +2) L x . ... V ,L y , , · · · , V L x ,L y , → V (2 L y − L x +1 , · · · , V L y L x .V ,L y +1 , , · · · , V L x ,L y +1 , → V (2 L y L x +1 , · · · , V (2 L y +1) L x . (A.9) ffective resistances of two dimensional resistor networks References [1] I.E. Irodov.
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