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

4.2 Complex Impedance Z(s)

• Complex Impedance (under zero state/initial condition)

4.1 Circuit Transformation from Time to Complex Frequency

Chapter 4 Circuit Analysis with Laplace Transform

• Resistive Network Transformation

• Inductive Network Transformation:

• Capacitive Network Transformation

3.4 Alternate Method of Partial Fraction Expansion

• Partial Fraction Expansion can also be performed with the method of clearing the fractions

• Method:

3.3 Improper Rational Function

• Expression (where N(s)/D(s) is a proper rational function):

3.2 Partial Fraction Expansion

• Laplace transform in most case appear in a rational form of s:

• The coefficient ak and bk are real numbers for k = 1, 2, ..., n
• m < n : F(s) is a proper rational function
• m >= n : F(s) is an improper rational function
• In proper rational function, roots of N(s) (found by setting N(s) = 0) are called zeros of F(s) ; root of D(s) (found by setting D(s) = 0) are called poles of F(s)
• Distinct poles:
• If all the poles (p1, p2, p3, ..., pn) are distinct, F(s):

 

• Using partial fraction expansion method, F(s):

• Residue rk :

• Complex poles:

• Complex poles occur in complex conjugate pairs, the number of complex poles is even

• Multiple (Repeated) poles:

• Partial Fraction Expansion:

• Residue for repeated poles:

3.1 Inverse Laplace Transform Integral

•  Inverse Laplace Transform Integral:

• The integral is difficult to evaluate because it requires contour integration using complex variable theory
• For most engineering problems, we can refer to common Laplace transform pairs to lookup the inverse Laplace transform 

2.4 Laplace Transform of Common Waveforms

• Pulse

• Linear

• Triangular

• Rectangular 

• Half Rectified Sine Wave

2.3 Laplace Transform of Common Functions of Time

2.2 Properties and Theorems of Laplace Transform

2.1 Laplace Transformation

• Two-side or bilateral Laplace Transform:

• One-side or unilateral Laplace Transform:

• Convergence:

Re {s} is the real part of complex variable s

1.5 Higher Order Delta Function

• nth-order delta function:

• sampling property:

• sifting property:

1.4 Unit Impulse / Delta Function

• Unit Impulse function is the derivative of unit step u0(t):

• Two useful delta function properties are sampling & sifting property 
• Sampling property:

• Multiplication of any function f(t) by the delta function results in sampling the function at the time instants where the delta function is not zero. (foundation of discrete-time systems)
• Sifting property:

1.3 Unit Ramp Function

• Unit Ramp function u1(t) is defined as 

• u0(t) is the derivative of u1(t)

1.2 Unit Step Function

• A function is said to be discontinuous if it exhibits point of discontinuity
• A well known discontinuous function is called unit step function:

  

•