MIO400S Mechatronic Design and Simulation (Mathematical Modelling)

Session 6 - Sensor Modelling

What is a Sensor?

It is a device that receives and responds to a signal or stimulus.

Examples

Position sensorTemperature sensorSpeed sensorVoltmeterMicrophone

Sensor Components

A sensor usually consists of

• The Sensor• The conditioning circuit

Both need to be modelled

Inaccuracies need to be modelled

• Random errors• Systematic errors (bias)• Dynamic response

Sensors Contd

Other characteristics of Sensors may need to be modelled

• Input Range• Sensitivity• Linearity• Hysterisis• Sample rate• Slew rate• Noise

Sensors Contd

Dynamic response

Response to a changing signal is different to that of a steady state input

The response “lags” - this is due to energy storing components in the sensor.

Examples• Intertia• Capacitance• Inductance

Sensors Contd

Types of Dynamic behavior of sensorsWeb resource http://www.science.smith.edu/~jcardell/Courses/EGR326/Modeling_1st2nd_order_systems.pdf

1. Zero order – responds instantaneously

2. First order – Example a Voltage sensor with an RC circuit as a filter

How do we models this – we will use the RC circuit example

The differential equation for this is DV/dT = (1/RC)[Vin - Vc]

Sensors ContdThis equation is modelled in Xcos/Simulink as follows

Gain 1/RC Integrator

Feedback

Step Input V

dV/dt V

If this is plotted we see the RC curve

Sensors Contd3. Second order – Example a mass spring damper

How do we models this

The differential equation for this is

Where Xddot is the acceleration A of mass mAnd Xdot is the Velocity V of the massAnd x is the displacement

Sensors ContdThis is modelled in the following way

Produces a plot as follows

DC Motor Simulation ExampleThe parameters of a measured DC motor as [resented inhttp://www.engin.umich.edu/group/ctm/examples/motor/motor.htmlAndhttp://vlabs.iitkgp.ernet.in/RCSLab/exp1.htmlwere Where Input V = Source voltageOutput I = current w = angular velocityParameters R = Resistance (1 Ohm) L = Inductance (0.1H) J = Inertia (0.01kg*m2/s2) K = Ke = Kt = EMF Force constant (0.05Nm/Amp) b = damping ratio (or viscous friction)0.1Nms

DC Motor Contd 1

The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back emf, e, is related to the rotational velocity by the following equations:

T = Ki

Back emf e = K w

ThereforeJ dw/dt + b w = KiL di/dt + Ri = V - Kw

DC Motor Contd 2

DC Motor Contd 3