Circuit network theory of n-horizontal bridge structure

This research investigates a complex n order cascading circuit network with embedded horizontal bridge circuits with the N-RT method. The contents of the study include equivalent resistance analytical formula and complex impedance characteristics of the circuit network. The research idea is as follows. Firstly the equivalent model of n-order resistance network is established, and a fractional difference equation model is derived using Kirchhoff’s law. Secondly, the equivalent transformation method is employed to transform the fractional equation into a simple linear difference equation, and its particular solution is computed. Then the solution to the difference equation is used to derive the effective resistance of the resistance network of the embedded horizontal bridge circuit, and various special cases of equivalent resistance formula are analyzed and the correctness of the analysis model gets verified. Finally, as an expanded application, the equivalent complex impedance of LC network is studied, and Matlab drawing tool is employed to offer the equivalent impedance with various variables of the graph. Our results provide new research ideas and theoretical basis for relevant scientific researches and practical applications.


Results
Considering the complex network of Fig. 1, the positional arrangement of the node A k and B k in the network model are shown in Fig. 1. The N-RT technique 14,27 is used to study the resistive network and a simple analytical formula of resistance formula is found as follows, where R 0 is the any resistance of the right edge, as well as and, along with that is with Methods Modeling recursive equation. We use the N-RT technique to study the circuit network according to Fig. 1, assuming the effective resistance between the left edge A n and the B n is R n , and the equivalent resistance between the two nodes at the left edge of the n − 1 network is R n−1 . Therefore, the network in Fig. 1 can be equivalently simplified to the model in Fig. 2. It is the key to solving the problem how to build the recursive relations between A n and B n . Here, we use the circuit theory to establish the relationship between R n and R n-1 . The equivalent model of bridge circuit network with current parameters is shown in Fig. 2.
a = (2r 1 + r 3 + r)r 2 + r 1 r 3 , b = (2r 2 + r 3 + r)r 1 + r 2 r 3 .  www.nature.com/scientificreports/ In Fig. 2, we also obtain the node current equation through the node current law, Through the multiple substitution simplification of the above current equations, we can eliminate the parameters of I 1 − I 8 in Eqs. (7)- (14), and finally obtain the relationship between I 0 ∝ I and R n−1 (solving the above eight equations is complex, and the derivation process is omitted here), then use Ohm's law R n = U A n B n /I = I 0 r 0 /I , the resistance relation equation is as follows where c, d are constants determined by the resistance in Eqs. (7)-(10), which are given by Eqs. (5) and (6). Thus, the analytical formula of equivalent resistance can be studied based on this recurrence equation.
Solving the equation by substitution of variables. How to solve Eq. (15) is the key to solving the problem. Fortunately, Ref. 14 established a good method for solving the complex difference equations, and we solve the Eq. (15) by variable substitution method, assuming that there is a sequence {x n } to be determined, and we make the following transformation: We suggest that the initial term be x 0 = 1 , from Eq. (16), we can determine x 1 , so we have Replace Eq. (16) and its recurrence formula R n−1 into Eq. (15) and we can obtain it through algebraic calculation In order to solve Eq. (18), we first need to solve its characteristic root, by Ref. 14 we know its characteristic equation is obtained from Eq. (18) So, let α and β be the two roots of the characteristic Eq. (19), and (18) can be transformed into (9) I 3 r 3 + I 7 r 2 + I 5 r − I 2 r 2 = 0, (10) I 6 r 1 + I 7 r 2 − I 8 R n−1 = 0.

Discussion
Analysis of special cases. Figure 1 is a general circuit network model, and formula (1) is a general equivalent resistance result. Since the network in Fig. 1 contains six independent parameters, it is a multi-functional and multi-purpose network, including a variety of network model topologies, which has important application values. As several interesting applications, a series of special results of formula (1) are given below.
Special case-6 n → ∞. When n → ∞ , from Eq. (4), we know 0 < β/α < 1 , so lim Research on n-LC impedance network of bridge circuit. The research results of the resistor network in this paper can also be applied to the LC complex impedance network shown in Fig. 5 through the following variable substitution techniques. Assume that the element frequency of AC is ω, and we can carry out the transformation of resistance and complex impedance as follows Substituting (48) into Eqs. (2), (4) and (5), we can obtain (45) R A n B n (n → ∞) = − β. .
According to the eigenvalue expression (50), we need to discuss and analyze many cases in which the eigenvalues are real numbers and imaginary numbers respectively.
The case of ω 2 LC ≥ 1. When ω 2 LC = x ≥ 1 , and its characteristic roots δ and η are real numbers, the complex impedance expression is the structure of Eq. (52). Next, we draw a 3D characteristic curve with MATLAB to reveal the variation law of complex impedance when the characteristic root is real numbers, the change curve is shown in Fig. 6. Figure 6 shows that when 1 ≤ x ≤ 2 and q = 1 , the complex impedance Z r 0 in Eq. (52) changes irregularly with the increase of ω 2 LC and n . It should be noted that there is no oscillation in this case. Next, let's explore the transformation law of the complex impedance when n is the given value and q is the variable. Figure 7 shows that when 1 ≤ x ≤ 2 and n = 30 , the complex impedance Z r 0 increases with the increase of ω 2 LC based on the Eq. (52). In the interval ω 2 LC ≥ 1 , when ω 2 LC → 1 , the complex impedance Z r 0 increases rapidly.
The case of ω 2 LC = 4/3 . When ω 2 LC = 4/3 , we can get Z n → ∞ . It is known from Eq. (52) that impedance resonance occurs in the equivalent complex impedance Z n . Obviously, ω 2 LC = 4/3 is a very interesting number.
The case of 0 < ω 2 LC < 1 . When 0 < ω 2 LC < 1 , its characteristic roots δ and η are imaginary numbers. So complex analysis is needed to carry out on the characteristic roots. We can get from Eq. (50) where θ = arc cos[(1 − 2x)/(8x 2 − 8x + 1)] . Substituting (55) into (50), we can obtain (55) 1 − 2x + 2 x(x − 1) = cos θ + i sin θ,  www.nature.com/scientificreports/ where θ = arc cos(1 − 2x) , and φ = 2(1−x) 4−3x and µ = 2 4−3x are defined. Next, we draw the characteristic curve of equivalent complex impedance at 0 < ω 2 LC < 1 by MATLAB to reveal the variation law of equivalent complex impedance when the eigenvalue is complex in many cases. Figure 9 shows that when 0 < ω 2 LC < 1 and q = 1 , the complex impedance Z r 0 in Eq. (57) changes irregularly with the increase of ω 2 LC and n . When 0 < ω 2 LC < 1 and 0 < n < 50 , the complex impedance Z r 0 has irregular oscillation characteristics and resonance characteristics, the oscillation is dense, and the distribution of oscillation amplitude is not obviously regular. Figure 10 shows that when 0 < ω 2 LC < 1 and n = 30 , the complex impedance Z r 0 in Eq. (57) changes irregularly with the increase of ω 2 LC and q , where, when ω 2 LC is a definite value and q is a variable, the complex impedance Z r 0 has regular oscillation characteristics, and the distribution of oscillation amplitude is obviously regular.
A summary. In this paper, a general n-order resistance network model with horizontal bridge circuit ( Fig. 1) is proposed, which has not been studied before. The n-order recursive transformation method (N-RT method) is used to evaluate the equivalent resistance of n-order resistance network with horizontal bridge circuit, and the analytical expression of the equivalent resistance between A n and B n nodes, Eq. (27), is given by using the solution of the fractional difference equation. Using Eq. (27) can help us continue to analyze and study the equivalent resistance in a variety of special cases. Since the network in Fig. 1 contains five independent parameters, some interesting conclusions are obtained by discussing some cases of equal or limit parameters, for instance, when r 3 = r 1 , r 1 = r , r 3 = 0 etc. And compared with relevant literature, the results of the present research obtained are correct, which is also due to the self-consistency of our research. As the derivation process and method are rigorous and accurate, the results can be applied to other engineering or scientific aspects in the future. In addition, the research ideas and techniques in this paper can also be employed to study complex impedance, such as the LC network shown in Fig. 6. In this paper, the basic characteristics of LC network are studied and analyzed in detail, and the characteristic curves of equivalent complex impedance varying with different parameters are drawn. Besides, the theoretical results of this paper have potential application value. For example, the research of Metagratings and super surface material is an important scientific and engineering problem. The research of Metagratings and super surface material may need to be simulated by equivalent circuit model 33,34 .