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

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