THE TRANSFER FUNCTION

Linear, time-invariant differential equation:

Example:

Find the transfer function represented by

Solution:

Examples:1. Find the transfer function corresponding to

the differential equation

2. Find the differential equation correspondingto the transfer function

Answers:

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Electric Network Transferfunctions We now combine electrical components into

circuits, decide on the input and output, andfind the transfer function. Our guidingprinciples are Kirchhoff’s laws.

Table 2.3Voltage-current, voltage-charge, andimpedance relationships for capacitors,resistors, and inductors

Example:Find the transfer function relating the capacitorvoltage to the input voltage.

Example:Given the network,find the transferfunction I2(s) / V(s)

Operational Amplifiers

Is an electronic amplifier used as a basicbuilding block to implement transferfunctions

Differential input High input impedance Low output impedance High constant gain amplification

Inverting operationalamplifier

Example:Find the transfer function Vo(s) / Vi(s)

Noninverting Operational Amplifier

Example:Find the transferfunction Vo(s) /Vi(s) for the givencircuit

Translational MechanicalSystem Transfer Functions Mechanical systems, like electrical networks,

have three passive, linear components. Twoof them, the spring and the mass, are energy-storage elements; one of them the viscousdamper, dissipates energy.

Force-velocity, force-displacement, andimpedance translational relationshipsfor springs, viscous dampers, and mass

Example:Find the transfer function X(s) / F(s) for thesystem.

Example:Write the equations of motion for the mechanicalnetwork

a. Forces on M1 due only to motion of M1b. forces on M1 due only to motion of M2c. all forces on M1

a. Forces on M2 due only to motion of M2;b. forces on M2 due only to motionof M1;c. all forces on M2

Example:Write, but do not solve, the equations of motion forthe mechanical network

Example:Find the transfer function G(s) = X2(s) / F(s)for the translational mechanical system