Steps in Obtaining a State Model
The following four steps will create a set of state equations (a state-space model) from a linear graph:
1. Choose the state variables (forces and velocities for a mechanical system; currents and voltages would be chosen for an electrical system).
2. Write the constitutive equations (characteristic relationships) for the independent inertia and stiffness elements in the system (independent capacitors and inductors in an electrical system).
3. Do the same for the remaining elements (dependent stiffness and inertia elements, and damping elements; dependent inductors and capacitors, and resistors).
4. Develop the state equation (i.e., retain the state and input variables) by eliminating all other variables, using continuity (node) equations and compatibility (loop) equations.
If a linear graph has s sources (forcing functions) and a number of branches equal to b, then the total number of unknown variables in the system is
Electrical Systems
Thus far we have primarily considered the modeling of mechanical systems—systems with inertia, flexibility, and mechanical energy dissipation.
In view of the analogies that exist between mechanical, electrical, fluid, and thermal components and associated variables, there is an “analytical” similarity between these four types of physical systems.
Accordingly, once we have developed procedures for modeling and analysis of one type of systems (say, mechanical systems) the same procedures may be extended (in an “analogous” manner) to the other three types of systems.
First we will make use of these analogies to model electrical systems, by making use of the same procedures that have been used for mechanical systems.
Next we will specifically consider fluid systems and thermal systems. These procedures can be extended to mix systems—systems that use a combination of two or more types of physical components (mechanical, electrical, fluid, and thermal) in an integrated manner.
Source Elements
An electrical system has two types of source elements:
1. Voltage source
2. Current source.
A voltage source is able to provide a specified voltage without being affected by the current (loading). Hence it has a low output impedance. A current source is able to provide a specified current without being affected by
the load voltage. Hence it has a high output impedance. These are idealizations of actual elements, because in practice, the source output changes due to loading.
Circuit Equations
1. Node equations for currents: The sum of currents into a circuit node is zero. This is the well-known Kirchhoff’s current law.
2. Loop equations for voltages: The sum of voltages around a circuit loop is zero. This is the celebrated Kirchhoff’s voltage law.
Finally, we eliminate the unwanted (auxiliary) variables from the three types of equations (constitutive, node, loop) to obtain the analytical model (say, state equations). Linear graphs can be used for this purpose as usual.
