We then add controlled sources and coupling devices to the IPE. In Section 3, we discretize the matrix equation so that it can be used for a time-domain analysis of the lumped parameter circuit, which includes capacitors and inductors. In Section 4, the matrix equations are combined with the discretized equations for the MTL system.
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This provides a boundary condition for the connection of the lumped parameter circuit to the MTL system. We shall begin with construction of coupled equations for the lumped parameter circuit in order to connect smoothly with the MTL equations at the boundary. The potential and current are the independent variables in the MTL equations 9 , 15 , and it is better to use node potentials rather than element voltages in the lumped parameter circuit.
We consider first a lumped parameter circuit with independent power sources without the coupled elements as dependent power sources. We introduce the incidence matrix A T , where the column numbers describe element currents associated with each node point numbered as a row number in the circuit. We thus have the relation,. The sparse-tableau type form 22 is obtained from these formulae by bringing the unknown quantities U and I in the left hand side, while the current sources J and the voltage sources E are known and brought to the right hand side:.
This is the fundamental equation to be used for the boundary equation between the lumped parameter circuit and the MTL system, and we name this Eq. We formulate next the dependent power supplies as shown in Fig. The treatment of the dependent power sources can be applied to other coupling devices such as mutual inductances and the transistors as the non-linear devices. The KCL equation is written as,. The element numbers and the node potentials are indicated. Since the third and fourth terms of Eq. As shown in Fig. This allows us to write the following equation using the potential vector U :.
We can then write the CCCS term as.
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The rewritten E expresses independent voltage sources. Here, H and R are diagonal matrices that include the voltage transfer ratios and transresistances, respectively. We can now use all the above expressions containing U and I to rewrite the IPE 3 to include the controlled sources. We base on this Eq. We note here that this Eq. In this case, by calculating the inverse of the matrix on the right-hand side of Eq.
We want to write the IPE 10 in the time domain for the analysis of noise. For the time-domain analysis, the time-domain impedances can be defined as elements of the impedance matrix Z From Eq. Using the above expressions 11 , we obtain a discretize IPE from Eq. That is,. Since the origin of the dependent voltage sources D CCVS and D VCVS are the same as the independent voltage sources E , we ought to introduce the minus sign in the right hand side of the above equation.
The sign changes in Eq. Since this Eq.
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In our approach, we prefer to treat these quantities explicitly as the consequence of the discretization of the time dependent equations as shown in the time domain IPE This is because MTL equations of distributed circuit are given in the coupled differential equations for potentials and currents in the standard FDTD method 9 , and the expression in time dependent IPE 12 can naturally express the boundary condition between the lumped parameter circuit and the MTL system as discussed in the next section.
We write here the MTL equations including the retardation effect 15 , We consider the case of a MTL system and introduce n thin wires. We introduce the charge per length by integrating the charge density q by the cross section assuming that the charge density is unchanged within the cross section of area S :. Using the same procedure we arrive at the expression:. Here, we consider the case that the wire is thin and the direction of the current and the vector potential is the x -direction and drop the suffix x in the above expression.
We need two more relations for each wire in order to complete the MTL equation. One is the continuity equation for the relation between the current and the charge:. We drop the x suffix and express the electric field in terms of the scalar and vector potentials:. We can express the four relations in terms of two relations by taking the partial derivative of the potentials with respect to time. We get one relation for the scalar potential by using the continuity Eq. Here, we have dropped the y , z coordinates in the scalar potential and write only the x coordinate. In the above expression, we have neglected the effect of the size of the wires and the distances between wires in the retardation time.
When these sizes and distances become large, we ought to formulate the MTL system as a two three dimensional problem. The retardation time is calculated at the place of the charge. In this case we define them as.
We have a similar expression for L i , j. We note here that these local coefficients of potential and inductance are to be contrasted from the global coefficients of potential P , which are inverse of coefficients of capacitance C and the global coefficients of inductance L of the Heaviside telegraphic equations.
These global coefficients are obtained by integrating over the full length of the wires, and use only the currents at the position of the potentials in the telegraphic equations. T , representing all the transmission lines.
The suffix d denotes the potentials and currents of the MTL system. R is a diagonal matrix that indicates the resistance per unit length of the MTL. The time difference of charge Q can be expressed by the spatial difference of the current I using the continuity equation. Without the nonlocal terms, the above MTL Eqs 30 and 31 correspond the telegraphic equations, which are the most common form of transmission equations for the MTL system 9. The discretized Eqs 12 , 30 and 31 can be combined to obtain the boundary condition. Using this relation we are able to combine the discretized Eq.
The elements of matrix Z are the time-domain impedance of the lumped parameter circuit Z L and the characteristic impedances Z d of the MTL 9. Here, the impedance matrix Z d for the MTL part has both the diagonal and non-diagonal elements. The current in the right hand side should be defined as. Here, they are. For completeness, the boundary equations between the lumped parameter circuit and the MTL would be expressed as. The time dependent IPE 44 expresses the boundary conditions of the lumped parameter circuit and the MTL equations, which include terms treating radiation.
We are now able to perform numerical calculations for various problems. In particular, it is important to analyze three line circuit for the discussion of the electromagnetic noise. Hence, there are three modes to be expressed using the normal mode, and common mode, and additionally the antenna mode We should first calculate the potentials and currents in the present formalism, and take linear combinations to get the normal, common and antenna modes:.
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Here, the potentials and currents with the suffix n denote those of the normal mode, suffix c the common mode, and suffix a the antenna mode 15 , Control Theory and Technology , , 12 1 : 13— Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. Akar, K. On the existence of common quadratic Lyapunov functions for second-order linear timeinvariant discrete-time systems. Hu, Z. Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions.
Kapila, W. A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach. Haddad, V. Robust stabilization for discrete-time systems with slowly time-varying uncertainty. Lazar, W. Maurice, M. Heemels, et al.
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Lyapunov functions, stability and input-to-state stability subtleties for discrete-time discontinuous systems. Zhai, B. Hu, K. Yasuda, et al. Stability and 2-gain analysis of discrete-time switched systems.