5.4 Computation of State Transition Matrix e^AT

• Evaluation of state transition matrix is based on the Cayley-Hamilton Theorem:

• The eigenvalues of A are the roots of the nth order polynomial:

where A is n x n matrix, I is n x n identity matrix

• Distinct eigenvalues:

• 
  

5.3 State Transition Matrix

5.2 Solution of Single State Equations

• If a circuit contains only one energy-storing device, the state equations are written as:

• Solution of the first state equation:

5.1 Expressing Differential Equations in State Equation Form

Chapter 5 State Variables and State Equations

• Circuits that contain energy-storing devices (capacitor & inductor) result in integro-differential equations via KCL & KVL
• One energy-storing device = first order; Two energy-storing device = second order; so on
• A first order linear time-invariant circuit can be described by a differential equation of the form:

• An nth-order differential equation can be resolved to n first-order simultaneous differential equation with a set of auxiliary variables called state variables
• The resulting first order differential equations are called state-space equations or simply state equations
• State variables method can also be used on non-linear, time-varying devices
• State equations can also be solved with numerical methods such as Taylor series and Runge-Kutta methods

4.5 Using the Simulink Transfer Fcn Block

• The Simulink Transfer Fcn block implements a transfer function where the input Vin (s) and the output Vout (s) can be expressed in transfer function form as G(s) = Vout(s) / Vin(s)

4.4 Transfer Function

• In an s-domain circuit, the ratio of the output voltage Vout(s) to the input voltage Vin(s) under zero state conditions, is of great interest in network analysis
• The ratio is referred as the voltage transfer function Gv(s):

• Current transfer function Gi(s) - rarely used :

4.3 Complex Admittance Y(s)

• Y(s) is the ratio of the current excitation Is(s) to the voltage response V(s) under zero state (zero initial condition)
• Complex Admittance