Equivalent Resistance of 5×n-laddered Network

The basic structure of 5×n -step network was studied by matrix transform method. Thus, the resistances of both the infinite and finite network were obtained. In addition, the resistances of the finite 5×n -laddered network were measured experimentally by NI Multisim 10 when n equals to positive integer such as one,two,three ... ,and so on. It is found that the measured values of the equivalent resistances of finite 5×n-laddered networks are consistent with the calculated ones.


Introduction
Equivalent Resistance of n×n-laddered network is a general problem in introductory physics.With the development of communication technology, network technologies have become more and more important in many fields such as resistance network and self-control systems.Lots of actual network technological problems can also be simulated by resistance network.This stimulates the research of network's resistance.Up to now, studies have been devoted to the equivalent resistance of different resistance networks, [1][2][3][4][5][6][7][8][9] but those results have no experimental foundations.Unfortunately, some results are even wrong. 3Therefore, it is necessary to develop an effective method (matrix transform) to calculate the equivalent resistance of resistance network.
Matrix transform is commonly used in mathematics and theoretical physics, especially in quantum physics.However, students trained in the algebra-based physics course are used to matrix transform only in linear equations, but not in mechanics,electromagnetism,and difference equations.In this paper, we will construct difference equations of closed circuits and find relation of matrix transform and difference equations to calculate the equivalent resistance of 5×n-laddered network.Furthermore, we measured its resistance by using NI Multisim 10 simulation software.Agreement is achieved between the measured values and the theoretical ones.

The development of the differential equation on current
Figure 1 is a schematic diagram of 5×n-laddered resistance network (n→∞).The equivalent resistance of between a and f nodes was assumed to be af R .
Suppose each resistor has the same value r.The direction of the steady current is from nodes f to a as shown in Fig 1 .Each branch current and its flowing direction were marked in Fig. 2, which is a part of 5×n-laddered resistance network (hereafter denoted as 5×n-laddered resistance sub-networks).Assume the currents of the Kirchhoff's current law 10 is used for the all nodes in the second rank.Differential current equations were obtained as follows: In terms of the symmetry, we obtain the following equations Trace all the meshes in the (k-1)-th rank and apply Kirchhoff's voltage law (KVL). 10 In the same manner, trace the all meshes in the k-th rank and apply KVL.
Multiply both sides of the matrix ( 21) by three order undetermined matrix on the left, we have Assuming the existence of constants 1 t , 2 t , 3 t , and let the following matrix equation be correct.
Expand the matrix ( 23) and simplify, we achieve The solutions of the above equations are 23), ( 24), (25), and (24) into Eq.( 16), hence the matrix equation can be written as x , Simplify the matrix (23), we hold where 1 t is the known constant, based on the definition of differential equation, Eq. ( 28) is obviously second order linear differential equation with fixed coefficients.Let Suppose the roots of the equation with respect to x are α and β , the roots of the equation with respect to y are γ and δ , and those of the equation with respect to z are μ and ν , the solutions of characteristic Eq. ( 33) are as follows: ) ( From the matrix equation we obviously see that α ≠ β , and then subtracting Eq. ( 38) from Eq. (37) gives In the same manner, we get Eqs. ( 39), (40), and (41) can be written in matrix form where 42) gives the current properties of vertical resistance r in any sub-network.

The Properties of Boundary Current
When the current flows toward node a and flows away from node b in Fig. 1, applying the current continuity equation, we have If we consider Eq. ( 43), summation of Eq. ( 42 Up to now, we obtain differential equations model of current parameters under boundary conditions by analyzing 5×n-laddered resistance network.Based on Kirchhoff's current law and mesh analysis, 11 we also find the following formula from Fig. 3, Due to the symmetry, we have  47), ( 48), (49), and (50) into Eqs.( 51), ( 52), and (53), and then simplify them, we obtain Eq. ( 54) can be written in matrix form,consequently the differential equations model of 5×n-laddered resistance network under boundary conditions are obtained.
, substitute Eq. ( 54) into it and simplify, we achieve 2 In the same manner for 1 , and β α + = 4-3 , Eq. ( 56) can be expressed as 2 Eqs. ( 57), ( 58), ( 59) can be written in matrix form Substitute Eq. (60) into Eqs.( 44), ( 45), ( 46), and simplify them, we get , therefore, these matrix equations give the current properties of 5×n-laddered resistance network under boundary conditions.65) is a general expression which denotes the equivalent resistance af R of 5×n-laddered resistance network between nodes a and f, and af R has a limited value in this case.

Measurements of af R (n) by Simulation Experiments
The equivalent resistance af R of 5×n-laddered resistance network is measured by NI Multisim 10 when n is a series of positive integer.In the meantime,the equivalent resistance af R of 5×n-laddered resistance network is calculated from Eq. ( 63).The relationship curves of the equivalent resistance af R versus n are plotted in Fig. 4. It is found that agreement is achieved between the experimental values and the theoretical ones.These results indicate that the equivalent resistances of n×n-laddered resistance network were not only calculated by matrix transform and but also the calculated results are reliable.achieved by the matrix transform method to solve a set of differential equations.In addition, the equivalent resistance af R of 5×n-laddered resistance network is calculated from Eq. ( 63).Moreover,the equivalent resistance af R is measured by NI Multisim 10 when n is a series of positive integer.The result exhibits that the theoretical values are consistent with the experimental ones for the equivalent resistance af R of 5×n-laddered resistance network.This study also reveals that the matrix transform method may be extended to calculate the equivalent resistances of n×n-step resistance network.

Conclusion
k k x x = , substituting it into Eq. (28)yields the following characteristic equation = Eq.(30), we also obtain the following characteristic equation.characteristic equation of differential equation is equivalent resistance afR (∞) in infinite networkAs n tends to infinity, Fig.1becomes a 5×n-laddered resistance network.From Eqs. (29), (30), and (31 laddered resistance network in infinite and finite networks are

Figure 1 .
Figure 1.Schematic diagram of 5×n-laddered resistance network horizontal resistances in the second rank are ak