Transcript Modeling of Electrical Circuits

Modeling Methods of Electric Circuits
There are two main methods to model electrical systems.
In the first method, Kirchhoff's point rule is applied. At any node in an
electrical circuit, the sum of currents flowing into that node equals to the
sum of currents flowing out of that node.
Or equivalently, the algebraic sum of currents in a circuit meeting at a
point is zero. This law is also called Kirchhoff's first law
The sum of the electrical potential differences (voltage) around any
closed network is zero. This law is also called Kirchhoff's second law or
Kirchhoff's loop rule.
1. METHOD
Kirchoff’s current law
Kirchoff’s voltage law
In the second method, Kirchhoff's point rule is applied. By Kirchhoff's
first law, the algebraic sum of currents in a circuit meeting at a point will
be zero.
The expressions of total magnetic energy, electrical energy, and virtual
work are written for the system. Then, Lagrange equation is applied. In
this course, the second method is used.
2. METHOD
Current law
Magnetic energy, electric energy, virtual work. Lagrange equation
In this course, 2th Method will be applied.
BASIC ELECTRIC CIRCUIT ELEMENTS
Passive Elements
In electrical systems, there are three passive elements, which are inductor
L, capacitor C, resistor R.
L
C
R
Active Elements
In this course, op-amp will be introduced here as an active element.
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Op-amp’s properties will be explained in the future examples.
The mathematical models of passive elements can be given with an analogy
between mechanics and electrics. The elements and their definitions in
mechanics and electrics are given below.
There is an analogy among the quantities in the columns. Mass and inductor
have the same equations and the elements and their equations are given below,
respectively. Kinetic and magnetic energy equations are written for mass and
inductor.
Mechanical-Electrical Analogy
x
Disp.
(m)
.
x
Vel.
(m/s)
..
x
Acc.
(m/s2)
m
Mass
(kg)
k
Spring const.
(N/m)
c
Damp. const.
(Ns/m)
f
Force
(N)
R
Resistance
(Ohm)
V
Voltage
(Volt)
f  mx
1
E1  mx 2
2
q
Charge
(Coulomb)
i
Current
(Amper)
di
dt
L
1/C
Inductance C:Capacitance
(Henry)
(Farad)
VL
di
dt
1
E1  Lq 2
2
Spring and capacitor have the same equations. The equations for these elements
are given below, respectively.
Potenatial and electric energy equations are written for spring and capacitor
elements.
x
Disp.
(m)
.
x
Vel.
(m/s)
Mechanical-Electrical Analogy
..
x
Acc.
(m/s2)
m
Mass
(kg)
k
Spring const.
(N/m)
c
Damp. const.
(Ns/m)
f
Force
(N)
R
Resistance
(Ohm)
V
Voltage
(Volt)
f  kx
f  mx
1
1
E1  mx 2 E 2  kx2
2
2
q
Charge
(Coulomb)
i
Current
(Amper)
di
dt
L
1/C
Inductance C:Capacitance
(Henry)
(Farad)
VL
di
dt
1
E1  Lq 2
2
1
q
C
1 2
E2 
q
2C
V
Damper and resistor have the same equations. The equations for these elements
are given below, respectively.
The virtual work equations are written for damper and resistor elements.
x
Disp.
(m)
.
x
Vel.
(m/s)
Mechanical-Electrical Analogy
..
x
Acc.
(m/s2)
m
Mass
(kg)
k
Spring const.
(N/m)
f  kx
f  mx
1
1
E1  mx 2 E 2  kx2
2
2
q
Charge
(Coulomb)
i
Current
(Amper)
di
dt
L
1/C
Inductance C:Capacitance
(Henry)
(Farad)
VL
di
dt
1
E1  Lq 2
2
c
Damp. const.
(Ns/m)
f  cx
W  cx x
R
Resistance
(Ohm)
f
Force
(N)
W  fx
V
Voltage
(Volt)
1
V  Ri
q
C
W  Vq
1 2 W   Rq q
E2 
q
2C
V
Analogy in Mechanical (Translational and Rotational) and Electrical Systems
The table shows an analogy among the mechanical and electrical systems. There
is also an analogy between the translational and rotational mechanical quantities.
There are general coordinates in mechanical systems, whereas there are general
charges in electrical systems. Lagrange equations are applied to both of
mechanical and electrical systems.
x
Disp.
(m)
.
..
x
Vel.
(m/s)
x
Acc.
(m/s2)
.
..
θ
Angular
pos.
(rad)
θ
Angular
vel.
(rad/s)
q
Charge
(Coulomb)
i
Current
(Amper)
θ
Ang.
acc.
(rad/s2)
m
Mass
(kg)
k
Spring cons.
(N/m)
c
Damp. Cons.
(Ns/m)
f
Force
(N)
IG
mass m.
ineratia
(kg-m2)
L
Induc.
(Henry)
Kr
Rot. Spring
const.
(Nm/rad)
Cr
Rot. damp.
const.
Nm/(rad/s)
M
Moment
1/C
C:Capac.
(Farad)
R
Resistance
(Ohm)
(Nm)
V
Voltage
(Volt)
E1
There is an
analogy
among
the
quantities in
the columns.
Definitions
for fluid and
thermal
systems will
be given in
next
courses.
E2
m
Mass
(kg)
k
Spring const.
(N/m)
c
Damp. const.
(Ns/m)
f
Force
(N)
IG
Mass moment
of inertia
(kg-m2)
Kr
Rot. spring
const.
(Nm/rad)
Cr
Rot. damp.
const.
Nm/(rad/s)
M
Moment
L
Inductance
(Henry)
1/C
C:Capacitance
(Farad)
R
Resistor
(Ohm)
V
Voltage
(Volt)
If
Fluid inertia
(kg/m4)
1/Cf
Cf: Fluid
capacitance
Rf
Fluid
resistance
P
Pressure
(N/m2)
1/Ct
Ct:Heat
capacitance
Rt
Heat resistor
T
Temperature
(oC)
(Nm)
Difference among the forces which act on a system makes a body translate.
Difference among the moments makes the body rotate.
Potential difference generates current in electrical circuits.
E1, the first type of energy in Lagrange equation can be stored in moving
elements in mechanical systems and inductors in electrical systems.
E2, the second type of energy in Lagrange equation can be stored in springs in
mechanical systems and capacitors in electrical systems.