# Transmission Lines And Lumped Circuits (Electro...

An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, current sources, resistances, inductances, capacitances). An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources (voltage or current), linear lumped elements (resistors, capacitors, inductors), and linear distributed elements (transmission lines), have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

## Transmission Lines and Lumped Circuits (Electro...

Discrete passive components (resistors, capacitors and inductors) are called lumped elements because all of their, respectively, resistance, capacitance and inductance is assumed to be located ("lumped") at one place. This design philosophy is called the lumped-element model and networks so designed are called lumped-element circuits. This is the conventional approach to circuit design. At high enough frequencies, or for long enough circuits (such as power transmission lines), the lumped assumption no longer holds because there is a significant fraction of a wavelength across the component dimensions. A new design model is needed for such cases called the distributed-element model. Networks designed to this model are called distributed-element circuits.

Since it is difficult to find causes of invisible EM noise, various measures based on experience and know-how of trained engineers have been taken to reduce it. To describe EM noise, the researchers used a three-line (multi-conductor transmission line (MTL)) circuit, to which lumped-parameter circuits were connected. In addition to a conventional two-line circuit configuration, another conductor line was connected on the source side as the ground.

Their method has enabled theoretical calculations of electric circuits with various configurations and electrical connections, confirming that a symmetrical configuration of three transmission lines together with lumped circuits was the only solution to eliminate EM noise.

In order to find the origins of electromagnetic noise in the time domain, we formulate a system of lumped parameter circuits and multiconductor transmission lines (MTL). We present a discretized approach to treat any lumped parameter circuits and MTL systems, and the boundary conditions between these systems, where the lumped parameter circuits are described by coupled differential equations, and the MTL systems by coupled partial-differential equations. The introduction of the time-domain impedance and the element matrices enables us to perform a time-domain analysis that includes dependent sources and the coupling devices in the framework of the circuit theory. For three-line systems, we are able to calculate the coupling of the normal, common, and antenna modes, and to find out methods to reduce the noise.

Electromagnetic noise is troublesome; it not only affects the transmission of signals, but also causes a malfunction. An integrated circuit (IC) has electromagnetic noise that interferes with the signal, and an important area of research for IC is electromagnetic compatibility (EMC)1. Many scientists have studied this phenomenon by taking a symptomatic approach to systems from the IC level to power lines2,3,4,5,6. In the attempt to locate the sources of electromagnetic noise, many authors have considered environments in terms of conductors in the ground and power plane and/or the signal lines. The electromagnetic noise was studied also experimentally using printed circuit boards (PCB) and the results were analyzed by introducing a phenomenological model7,8. In principle, we should be able to calculate the origin of the electromagnetic noise using the multi-conductor transmission line (MTL) theory based on the Heaviside transport equations9. There are many studies in this line of thoughts in various research fields as the performance of multi-wire systems and motor-stater winding-coils with high frequency signals10,11. However, it is not easy to identify the sources of the noise in the manipulation of the MTL equations.

In order to understand the role of the symmetrization for the reduction of noise, there was an attempt to describe theoretically the three-conductor transmission lines14. In an effort to introduce the variables in the normal and common modes in the MTL equations, it was realized that the use of the concept of capacitor made the manipulation of the MTL equations very difficult and unclear. Instead, if we rewrite the MTL equations in terms of the coefficients of potential, we were able to express the MTL equations straightforwardly and found the condition to decouple the normal mode from the common mode14. The decoupling condition states that the two main circuit lines have to be symmetrically arranged around the third ground line, and all the electric components have to be arranged symmetrically around the third line as found in the accelerator science.

For the purpose of simulating the mechanism of noise generation and its reduction, we have to facilitate these findings by introducing lumped parameter circuits connected with the MTL system. In this formalism, we have to make the concept of the common mode clear, and to define the noise, which is not included in the standard MTL theory proposed by Heaviside a century ago. To this end, we should derive the MTL equations from the Maxwell equations, and identify the noise terms by deriving the Heaviside telegraphic equations by several approximations. Those terms which are neglected in the process of reducing the full equations down to the Heaviside equations are to be identified as noise terms.

We have to derive boundary equations for both the MTL equations and the lumped parameter circuits, which are shown in Fig. 1. The boundary conditions play an important role for two kinds of differential equations to be solved simultaneously: ordinary coupled differential equations for the lumped parameter circuit and partial differential equations for the MTL system15. Recently, we have introduced the fundamental concept for the boundary conditions of lumped parameter circuits and the MTL system16,17. There were, however, several ingredients, whose details were not described explicitly. One is the coupling devices such as dependent power sources in the lumped parameter circuit and another is the explicit algorithm for the treatment of the radiation terms.

A coupled system of lumped parameter circuits and multi-conductor transmission lines. Model of discretization at the boundary of lumped parameter circuit with any number of MTLs, where lumped parameter circuits are connected at both ends. The finite-difference time-domain (FDTD) method is used to solve for the transmission of signals and emission of noise in the MTLs. Lumped circuits consist of passive elements and independent and dependent sources and coupled elements. Here, U and I are the potential and current vectors and the subscript d represents the MTL, and m and n denote discretized time and space, respectively.

We base on this Eq. (10) for further discussion to express the boundary condition between lumped parameter circuits with a MTL system. We note here that this Eq. (10) can also be used to solve for the steady state AC and DC of a lumped parameter circuit. In the AC steady state, R, 1/(jωC), and jωL can be used as elements in the Z matrix. In this case, by calculating the inverse of the matrix on the right-hand side of Eq. (10), we obtain the node potentials U and the element currents I. A mutual inductor M can be regarded as two CCVSs that have a combined transresistance of jωM in the AC steady state.

FIGURE 1. Lossless transmission line and its three equivalent lumped-circuits. (A) Lossless transmission line. (B) Lumped-circuits consisted of M Π-type circuits. (C) Lumped-circuits consisted of M inverse Γ-type circuits. (D) Lumped-circuits consisted of M T-type circuits.

FIGURE 3. Example analysis. (A) Step response model of the lossless transmission line. (B) Calculation results of the implicit Euler method. (C) Calculation results of the implicit trapezoidal method. (D) Calculation results of the four-step method. (E) Unit step response of the T-type lumped-circuits.

FIGURE 4. Chained number and numerical algorithm analysis of lumped-circuits numerical simulation. (A) M = 4 (the terminal resistance is 30 Ω). (B) M = 12 (the terminal resistance is 30 Ω). (C) M = 12 (the terminal resistance is 29.65 Ω). (D) Amplitude-frequency response error curve of the TR algorithm. (E) Amplitude-frequency response error curve of algorithm IE. (F) Amplitude-frequency response error curve of algorithms TR and IE.

In order to design noiseless electromagnetic (EM) devices, it is necessary to clarify the mechanism behind EM noise and theoretical calculations and computer simulations are performed for prediction assessment of devices. Two researchers at Osaka University developed an algorithm for numerical calculation of EM noise (interference) in electric circuits. googletag.cmd.push(function() googletag.display('div-gpt-ad-1449240174198-2'); ); EM noise is a problem that has proven to be difficult to solve. Caused by interference from transmission lines and connecting parts, various approaches have been taken to reduce it, such as adding filters and/or passive devices to circuits or using the symmetry of the configuration. 041b061a72