- February 15, 2021

ACS133 – Physical Systems Lecture 8: Electrical and Analogous Systems Simon Pope [email protected] Autumn semester – ACS133 1 Introduction to modelling and analysis of dynamic systems 2 Explore different types of physical systems 2.1 Mechanical Systems 2.2 Electrical Systems 2.3 Thermal Systems 2.4 Flow Systems 3 System simulation using Matlab and Simulink 4 Practical laboratory sessions Electrical Systems Electrical Systems Variables Element Laws Interconnection Laws Developing a system model Analogous Systems Case study: Lung Mechanics – Electrical Analogous System Electrical Systems 1. Electrical Systems 2. Analogous Systems 3. System Linearisation Electrical Systems 1. Variables 2. Element Laws 3. Interconnection Laws 4. Developing a system model Variables Electrical Systems I Electricity is created when electrons travel around a circuit I Each electron carries energy with it and has a negative charge Charge Electric charge (denoted by Q) is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. I The amount of electrical charge that moves in a circuit depends on the current flow and how long it flows for I Electrical charge is measured in coulomb (C) Variables Electrical Systems Current Current (denoted by i) is the flow of charge. I Because charge is measured in C, its flow is in coulombs/second which are also called Amperes (A) Voltage Voltage (denoted by v ) can be thought of as the “force” that pushes the flow of charge. I For the operation of circuit elements it is the voltage difference across those elements that matters I The standard unit is the Volt (V) Element Laws Electrical Systems 1. Resistor Definition An ideal resistor (denoted by R) is a passive two-terminal electrical component that implements electrical resistance as a circuit element. I The standard unit is the Ohm (Ω) Ohm’s law The time domain expression relating voltage and current for the resistor is given by Ohm’s law: vR(t) = iR(t)R (1) Element Laws Electrical Systems 2. Capacitor Definition A capacitor (denoted by C) is a passive two-terminal electrical component that stores electrical energy in an electric field. I The standard unit is the farad (F) I The time domain expression relating voltage and current for the Capacitor is given as: vC(t) = 1 C ∫ iC(t)dt (2) Element Laws Electrical Systems 3. Inductor Definition An inductor (denoted by L) is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. I The standard unit is the henry (H) I The time domain expression relating voltage and current for the inductor is given as vL(t) = L diL(t) dt (3) Element Laws Electrical Systems Element Laws Electrical Systems 4. Voltage sources Definition A voltage source (denoted by ei ) is a two-terminal device which can maintain a fixed voltage. I An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current 5. Ground Definition Ground is the reference point in an electrical circuit from which voltages are measured. Interconnection Laws Electrical Systems Kirchhoff’s current law The sum of current flowing into a junction of conductors is zero. ∑ j ij = 0 (4) Kirchhoff’s voltage law The directed sum of voltages around a loop is zero. ∑ j vj = 0 (5) Developing a system model Electrical Systems 1. Express currents/voltages using the element laws 2. Apply Kirchhoff’s laws 3. Derive ODEs Electrical Systems Example: RC circuit Consider a two-port electric circuit as shown in the figure. The input voltage is denoted by vi(t) and the output voltage is denoted by vo(t). Find the transfer function Vo(s)Vi (s) of the circuit. Electrical Systems Example: RC circuit The voltage across the resistor is given by vR(t) = iR(t)R. The voltage across the capacitor is given by vC(t) = 1C ∫ iC(t)dt . Note that the current through the resistor and through the capacitor is the same: iC(t) = iR(t) = i(t). Applying Kirchhoff’s voltage law for the left loop gives vi(t) = vR(t) + vC(t) = i(t)R + 1 C ∫ i(t)dt Applying Kirchhoff’s voltage law for the right loop gives vo(t) = vC(t) = 1 C ∫ i(t)dt Electrical Systems Example: RC circuit vi(t) = i(t)R + 1 C ∫ i(t)dt vo(t) = 1 C ∫ i(t)dt Using the Laplace transform assuming zero initial conditions gives Vi(s) = I(s)R + 1 Cs I(s) Vo(s) = 1 Cs I(s) Substituting I(s) from the second equation into the first one gives Vo(s) Vi(s) = 1 1 + RCs Electrical Systems 1. Electrical Systems 2. Analogous Systems 3. System Linearisation Analogous Systems I We can relate the behaviour of our system’s parameters to an equivalent (analogous) known system that is easier to understand and analyse. I Examples: I The Phillips Hydraulic Computer MONIAC used the flow of water to model economic systems (https: //www.youtube.com/watch?v=rAZavOcEnLg) I Electronic circuits can be used to represent both physiological and ecological systems I A mechanical device can be used to represent mathematical calculations Analogous Systems I We can relate the behaviour of our system’s parameters to an equivalent (analogous) known system that is easier to understand and analyse. I Examples: I The Phillips Hydraulic Computer MONIAC used the flow of water to model economic systems (https: //www.youtube.com/watch?v=rAZavOcEnLg) I Electronic circuits can be used to represent both physiological and ecological systems I A mechanical device can be used to represent mathematical calculations Analogous Systems Lung Mechanics – Electrical Analogous System Analogous Systems Lung Mechanics – Electrical Analogous System Analogous Systems Lung Mechanics – Electrical Analogous System Lung Mechanics – Electrical Analogous System Mathematical model I We can now apply Kirchhoff’s laws to derive the system’s dynamic model I However, the real behaviour of the lung volume/pressure is not linear! I In the analogue electric circuit this corresponds to a Non-linear resistor I dVolume dPressure ≈ dCurrentdVoltage I A good approximation for low-medium lung volume is io = 1 7 e3o Lung Mechanics – Electrical Analogous System Mathematical model I Applying Kirchhoff’s laws gives Lung Mechanics – Electrical Analogous System Mathematical model I Applying Kirchhoff’s laws gives 1 2 e˙o + ( eo − ei(t) ) + 1 7 e3o = 0 I This is equivalent to 1 2 e˙o + 1 7 e3o + eo = ei(t) Lung Mechanics – Electrical Analogous System Mathematical model I Applying Kirchhoff’s laws gives 1 2 e˙o + ( eo − ei(t) ) + 1 7 e3o = 0 I This is equivalent to 1 2 e˙o + 1 7 e3o + eo = ei(t) Non-linear! Lecture 8: Take-home points I Electrical systems I Variables I Elements laws (ideal resistor, capacitor, inductor, voltage source) I Connecting laws – Kirchoff’s current and voltage law I Modelling steps I Analogous systems 欢迎咨询51作业君