Laplace operator ‘s’ Reminder from ENGR201 Know how components behave in an instant of time Consider differential response Combine and simplify into standard forms Integration by lookup table determines long- term response t domain s domain

Signals Components are connected together by signals. Signals have many different forms, Must also have direction & name Signals continue until interrupted! Signals and components are considered ideal. We add other signals and components to alter the properties VoltageV Electrical Force N Mechanical FlowF Fluid TemperatureK Thermal

We wish to know how the output signal varies with an input signal for a fixed (invariant) system Consider a spring (constant k), with force F applied to stretch it a distance x We may plot two signals against each other invariant of time (system relationship) Signals Distance x ForceF Distance x ForceF

System constants are time invariant for the given system We now consider a different system as the spring has been changed. However, the analysis stays the same Constants Distance x ForceF Spring constant k = 1 Distance x ForceF Now k = 2

Systems In order to analyse a system: We identify an input signal [a variable] Using block diagram components [Basic block Summing junction Take-off point] We combine internal signals [modified variables] To produce the output signal [another variable]. The Input-Output relationship may then be determined

FlowF 1 F 1 + F 2 Fluid F 2 Components Summing junction + and/or – the system signals VoltageV 1 V 1 – V 2 Electrical V 2 + _ + + Up to three inputs and only one output

Components Block or / or function of the system signal VariablesConstants InputDirectionOutput Only one input and only one output ForceF Distance x Spring CurrentI V R 1 / k

Components Take-off point The system signal can be used elsewhere, but is not affected by the split VoltageV 1 V 1 Electrical V 1 Only one input and many outputs

Laplace operator ‘s’ Reminder from ENGR201 Know how components behave in an instant of time Consider differential response Combine and simplify into standard forms Integration by lookup table determines long- term response t domain s domain

Mechanical Components We know that distance (x) is related to velocity (v) is related to acceleration (a) through differentiation. It gets messy writingall the time! Therefore, we will write ‘s’ instead. x = x v = sx a = sv = s 2 x NB, Both with respect to the variable

Mechanical Components Spring Damper Mass Source Nise 2004 Force - Impedance Distance

Capacitor Resistor Inductor Source Nise 2004 Electrical Components Voltage - Impedance Current

Assume armature inductance is negligible. Armature resistor: Back emf of motor: Torque proportional to armature current: Torque is opposed by the inertia torque: Hint: apply Kirchhoff’s voltage law to the armature circuit We need to form a relationship between input voltage and output velocity: R2-D2 Motor System

Combine components Components of Motor System R + _

Reduce block diagram: Block Diagram of Motor System + _ /R + _

Output linked to input: Can be expressed much more simply!: Where: Time Constant: Gain: Transfer Function of System

Past Assignment question Determine the block diagram for each system? Source Nise 2004

Notes EN315_2010T1/CourseOutlinehttp://ecs.victoria.ac.nz/Courses/EC EN315_2010T1/CourseOutline PIO - plane HPv0qt03aAhttp:// HPv0qt03aA PIO - microlight 963Tc8R5o0&feature=relatedhttp:// 963Tc8R5o0&feature=related Millennium wobbly bridge AXVa__XWZ8http:// AXVa__XWZ8 Mass-spring system ZNnwQ8HJHUhttp:// ZNnwQ8HJHU