State Variables: Concept and Formulation 7.1 Introduction 7. 2 State Equations in Normal Form 7.3 The Concept of State and State Variables and Normal Tree 7.4 Systematic Procedure in Writing State Equations 7.5 State Equations for Networks Described by Scalar Wai-Kai Chen Differential Equations University of Illinois, chica 7.6 Extension to Time-Varying and Nonlinear Networks 7.1 Introduction An electrical network is describable by a system of algebraic and differential equations known as the primary system of equations obtained by applying the Kirchhoffs current and voltage laws and the element v-i relations In the case of linear networks, these equations can be transformed into a system of linear algebraic equations by means of the Laplace transformation, which is relatively simple to manipulate. The main drawback is that it contains a large number equations. To reduce this number, three secondary systems of equations are available the nodal system, the cutset system, and the loop system. If a network has n nodes, branches, and ccompone there are n- linearly independent equations in nodal or cutset analysis and b-n+ c linearly independent equations in loop analysis. These equations can then be solved to yield the Laplace transformed solution. To obtain the final time-domain solution, we must take the inverse Laplace transformation. For most practical networks, the procedure is usually long and complicated and requires an excessive amount of computer time. As an alternative we can formulate the network equations in the time domain as a system of first-order differential equations, which describe the dynamic behavior of the network. Some advantages of representing the network equations in this form are the following. First, such a system has been widely studied in mathe matics, and its solution, both analytic and numerical, is known and readily available. Second, the representation can easily and naturally be extended to time-varying and nonlinear networks. In fact, computer-aided solution of time-varying, nonlinear network problems is almost always accomplished using the state-variable approach Finally, the first-order differential equations can easily be programmed for a digital computer or simulated on an analog computer. Even if it were not for the above reasons, the approach provides an alternative view of the hysical behavior of th The term state is an abstract concept that may be represented in many ways. If we call the set of inst values of all the branch currents and voltages as the state of the network, then the knowledge of the inst values of all these variables determines this instantaneous state. not all of these instantaneous values ar in order to determine the instantaneous state. however because some can be calculated from the others a set of data qualifies to be called the state of a system if it fulfills the following two requirements 1. The state of any time, say, to, and the input to the system from to on determine uniquely the state at any time t> tr c 2000 by CRC Press LLC© 2000 by CRC Press LLC 7 State Variables: Concept and Formulation 7.1 Introduction 7.2 State Equations in Normal Form 7.3 The Concept of State and State Variables and Normal Tree 7.4 Systematic Procedure in Writing State Equations 7.5 State Equations for Networks Described by Scalar Differential Equations 7.6 Extension to Time-Varying and Nonlinear Networks 7.1 Introduction An electrical network is describable by a system of algebraic and differential equations known as the primary system of equations obtained by applying the Kirchhoff’s current and voltage laws and the element v-i relations. In the case of linear networks, these equations can be transformed into a system of linear algebraic equations by means of the Laplace transformation, which is relatively simple to manipulate. The main drawback is that it contains a large number equations. To reduce this number, three secondary systems of equations are available: the nodal system, the cutset system, and the loop system. If a network has n nodes, b branches, and c components, there are n – c linearly independent equations in nodal or cutset analysis and b – n + c linearly independent equations in loop analysis. These equations can then be solved to yield the Laplace transformed solution. To obtain the final time-domain solution, we must take the inverse Laplace transformation. For most practical networks, the procedure is usually long and complicated and requires an excessive amount of computer time. As an alternative we can formulate the network equations in the time domain as a system of first-order differential equations, which describe the dynamic behavior of the network. Some advantages of representing the network equations in this form are the following. First, such a system has been widely studied in mathe￾matics, and its solution, both analytic and numerical, is known and readily available. Second, the representation can easily and naturally be extended to time-varying and nonlinear networks. In fact, computer-aided solution of time-varying, nonlinear network problems is almost always accomplished using the state-variable approach. Finally, the first-order differential equations can easily be programmed for a digital computer or simulated on an analog computer. Even if it were not for the above reasons, the approach provides an alternative view of the physical behavior of the network. The term state is an abstract concept that may be represented in many ways. If we call the set of instantaneous values of all the branch currents and voltages as the state of the network, then the knowledge of the instantaneous values of all these variables determines this instantaneous state. Not all of these instantaneous values are required in order to determine the instantaneous state, however, because some can be calculated from the others. A set of data qualifies to be called the state of a system if it fulfills the following two requirements: 1. The state of any time, say, t0, and the input to the system from t0 on determine uniquely the state at any time t > t0. Wai-Kai Chen University of Illinois, Chicago