MODELING IN THE FREQUENCY DOMAIN

Laplace Transform:

System represented by DE is difficult to model as block diagram. Thus, LT is used. By LT input, output and system can be represented separately.

The LT is defined as

we can use Laplace Transform theorems to assist in transforming between f(t) to F(s) and F(s) to f(t) or we called as Inverse Laplace Transform.

The Transfer Function:

The transfer function is the ratio of the Laplace transform of the output of a system to the Laplace transform of the input. As a example, the transfer function, G(s) for a system representation is C(s)/R(s)

Electric Network Transfer Function:
-Apply transfer function to mathematical modeling of electrical circuit including passive Network and Op-Amp circuit.
-Equivalent circuits for the electric networks that we work with first consist of three passive linear components: resistors, capacitors and inductors.
-summarizes the components and the relationships between voltage and current and between current and charge under zero initial conditions.
-From these relationships, we can write the differential equations for the circuit using Khirchhoff’s laws.
-Then we can take the Laplace transforms of the differential equations and finally solve for the transfer function.

Or we can use transform methods:
1. loop or mesh analysis – Kirchhoff’s voltage law
2. nodal analysis – Kirchhoff’s current law

TRANSLATIONAL MECHANICAL SYSTEM TRANSFER FUNCTIONS:

Mechanical systems, like electrical network, can be have 3 passive linear components. Two of them, the spring and the mass, are energy-storage elements; and one of them, the viscous damper, dissipates energy.

where K, fv and M are called spring constant, coefficient of viscous friction and mass, respectively.

ROTATIONAL MECHANICAL SYSTEM TRANSFER FUNCTIONS:

Rotational mechanical systems are handled the same way as translational mechanical systems, except that torque replaces force and angular replaces translational displacement.
The components along with the relationships between torque and angular velocity, as well as angular displacemanet.
Notice that the symbols for the components look the same as translational symbols, but they are undergoing rotation and no translation.
The values of K, D and J are called spring constant, coefficient of viscous friction and moment of inertia, respectively.

ELectric circuit analogs:

The commonality of systems from the various disciplines by demonstrating that the mechanical systems can be represented by equivalent electric circuits.
We have pointed out the similarity between the equations resulting from Kirchhoff’s laws for electrical systems and the equations of motion of mechanical systems.
An electric circuit that is analogous to a system from another discipline is called an electric circuit analog. Analogs can be obtained by comparing the describing equations, such as the equations of motion of a mechanical system, with either electrical mesh or nodal equations.
When compared with mesh equations, the resulting electric circuit is called a series analog.
When compared with nodal equations, the resulting electric circuit is called a parallel analog.