Circuit Theory
Goals and Underlining Assumptions
1
Circuit Theory's Goals
• Being able to predict the behavior of complex electrical
systems and design better ones
• To analyze any complex system engineers typically
describe the system in terms of simplified
(approximated) models (abstractions) of its components
2
Engineering discipline
F
If I tell you that I'm applying a certain
force on a given object, and I want to
predict how much is the object going to
be accelerated, all I need to ask for is
the mass of the object !!
• What about the object's size, shape, density, temperature,
…?
• The material-point abstraction (approximation) enables
us to disregard most of the object's properties and it
makes very simple to deal with complex physical
systems
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A quick review of Physics
• Any electrical phenomenon can be fully described in terms
of the four Maxwell's equations + the continuity equation
(a.k.a. the conservation of charge equation)
•
Maxwell's equations:
– Gauss's Law for Electricity
– Gauss's Law for Magnetism
– Faraday's Law
– Ampere-Maxwell's Law
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Gauss's law for electric fields
• Electric charge produces an electric field, and the flux of
that field passing through any closed surface is
proportional to the total charge contained within that
surface
5
Gauss's law for magnetic fields
• The net magnetic flux through any closed surface must
always be zero
• Magnetic Fields do not originate and terminate on charges;
they form closed loops.
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Gauss's law for magnetic fields
• All static magnetic fields are produced by moving electric
charge. The contribution dB to the magnetic field at a
specified point p from a small element of electric current
is given by Biot-Savart law:
• Biot-Savart law is equivalent to Coulomb's
law for static electric fields produced by fixed electric
charges
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Faraday's law
• If the magnetic flux through a surface changes, an electric
field is induced along the boundary of the surface. If a
conducting material is present along that boundary, the
induced electric field provides an emf that drives a
current through the material
The integral in this expression is over any surface S,
whereas the integral in Gauss's law is over a closed
surface. The magnetic flux through an open surface
may be non zero – it is only when the surface is closed
that the number of magnetic field lines passing through
the surface in one direction must equal the number
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passing through in the other direction.
Faraday's law
• The E in Faraday's law represent the induced electric field
at each point along path C (the boundary of the surface
through which the magnetic flux is changing over time).
The path may be through empty space or through a
physical material – the induced electric field exists in
either case.
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Faraday's law
• The induced electric fields produced by changing magnetic
flux are very different from the electric fields produced by
electric charge.
• The field lines of induced electric fields form closed loops,
so this fields are capable of keep “driving” charged
particles around continuous “circuits”
• Charge moving through a circuit is the very definition of
electric current, so the induced electric field may act as a
generator of electric current
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Faraday's law
• The circulation of the induced electric field around a path is
the work that the electric field does when moving a “test”
charge around the path.
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Faraday's law
• It is important to understand that changing magnetic flux
induces an electric field whether or not a conducting
path exists in which a current may flow.
• If you find the idea disturbing think of an unconnected
battery. The battery emf is there whether we attach the
battery to something or not. The emf is produced by the
chemical work that happens inside the battery.
VB
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Faraday's law
• The emf produced by a changing magnetic flux is due to
the work spent to make the magnetic flux change (e.g.
moving a magnet to change B, or moving a loop within
a magnet to change dA).
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Ampere-Maxwell's law
• A magnetic field is produced along a a path if any current is
enclosed by the path or if the electric flux through any
surface bounded by the path changes over time
• It is important to understand that the path may be real or
purely imaginary – the magnetic field is produced
whether the path exists or not
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Ampere-Maxwell's law
• The magnetic fields induced by changing electric flux are
extremely weak
• A changing electric field produces a magnetic field even
when no charges are present and no physical current
(i.e. no physical movement of charges) flows.
There is neither charge or physical current
on the surface between the plates.
A changing electric field can be obtained
either moving the plates or applying a time
varying voltage to the plates
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Electromagnetic waves
• Since changing magnetic
fields induce electric fields,
and changing electric fields
induce magnetic fields
electromagnetic waves
can propagate even
through a perfect vacuum
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Charge conservation law
• Electric charge is never lost or created. Electric charges
can move from place to place but never appear from
nowhere (we say that charge is conserved).
• If there is a net current out of a closed surface, the amount
of charge inside must decrease by the corresponding
amount
Total current flowing out of a volume V is
equal to the current density J through the
surface S, which in turn is equal to the 17
rate of decrease of the charge enclosed in V.
From Physics to EE
VB
What is the current through the light bulb ?
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How do we go from Physics to EE?
• So far we do not know much about EE, so let's start from
what we know (Maxwell's equations + charge-current
continuity equation) and when we find it reasonable let's
try to simplify things
• Key Assumption: (Lumped elements assumption)
– All elements in the system have negligible physical
dimension
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Lumped Elements Assumption
• To say that all elements in the systems have negligible
physical dimension it means that all electrical effects
happen simultaneously throughout the system.
• If the size of a system is small enough we can think of the
system as “lumped”
• But how small is small enough ?
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Lumped Elements Assumption
• If we want every point of the system to be reached
simultaneously the wavelength ! of the signals traveling
through the system must be much bigger than the
dimension d of the circuit
!"d (rule of thumb: at least 10 times)
!#
c
speed of light
f max
highest frequency of operation
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Lumped Elements Assumption
• !#
c
f max
"d
means that at the timescales of interest the
propagation delay of the electromagnetic
waves across the circuit is negligible
signal timescale: T #
1
f max
d
c
• To assume that in a lumped system there is no e.m. wave
propagation means that:
propagation delay through the circuit:
td #
– the rate of change w.r.t. time of any e.m. quantity in
the system is zero
d
$0
dt
!
!
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Back to our EE problem
VB
• From an EE perspective
we'd like to treat any
lumped element in the
system as a black-box
accessible through its
terminals.
light bulb's filament
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Back to our EE problem
• Let's apply Faraday's law by taking a closed path C that
include the terminals x and y of the filament:
x
magnetic flux which
passes through
surface outlined
by the closed path
%d & B
dt
!!d !l
E
!!d !l
E
y
%d & B
(
(
' E)d l # dt *
C
y
x
%d & B
(
(
(
(
+ E)d l ,+ E)d l # dt
x
y
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x
Back to our EE problem
• For a lumped-element:
y
x
%d &B
(
(
(
(
+ E)d l ,+ E)d l # dt
x
y
y
=0
y
x
x
y
( (l #%+ (
E)d (l
+ E)d
V yx
%V xy
• Which in plain english means that a unique voltage V can
be defined across the terminal x and y of our lumpedelement (the light bulb's filament), and the voltage does
not depend on the path we use to go from x to y
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Back to our EE problem
• In summary when applying Faraday's law under the
lumped-parameters assumption we must remember that:
under LPA
at all time through any
closed path outside the
element:
d &B
#0
dt
• There is no magnetic coupling between the lumped
elements forming a system (magnetic coupling can
occur within a component, but not outside)
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Back to our EE problem
• Let's now apply the charge-current continuity equation by
taking a gaussian surface S enclosing the filament:
(J
Ix
x
Sx
y
Iy
d(
S
S
d(
S
Sy
The only parts of S that
give a contribute are Sx and Sy
(J
total amount of charge
enclosed by S
current density out of S
' (J)d (S#%
S
dqenc
*
dt
dq enc
(
(
(
(
+ J)d S,+ J )d S #% dt
S
S
x
y
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!J
Back to our EE problem
d !S
d !S
• For a lumped element:
!J
=0
dq enc
(
(
(
(
J
)d
S
,
J
)d
S#%
+
+
dt
S
S
x
y
%+ (J)d (
S #+ (J)d (S
Sx
%I x
Sy
Iy
• Which in plain english means that the current I flowing
through the lumped element is uniquely defined, or in
other words the current entering into terminal x is the
same as the current exiting from terminal y (i.e. there is
no net charge accumulated inside the limped element)
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Back to our EE problem
• In summary when applying the current-charge continuity
law under the lumped-parameters assumption we must
remember that:
under LPA
at all time inside any
closed surface
enclosing the element:
dqenc
#0
dt
• There is no net charge inside the lumped elements forming
a system.
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We are getting closer ...
• Thanks to the LPA, Faraday's law and the charge
conservation's law allows us to characterize any
electrical system as the interconnection of a bunch of
black boxes. The structural behavior of the system is
completely captured by the V-I relationship at each black
box's terminals
Ix
IB
+
VB
+
Vxy
-
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and closer ...
• So far everything we have found does not depend on the
internal nature of the circuit (i.e. we have found only
topological constraints).
• To solve the problem we also need to approximate
(model) the internal behavior of the black box with its
primary physical property
F
a?
In our mechanics analogy the primary
property of the object is the mass. If we
do not know how to model the object
with a certain mass there is no way to
solve the problem.
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The filament's resistance
• Physicist have found that for most substances over a wide
range of electric strengths of the electric field the current
density is proportional to the electric field
(J #- E
(
Conductivity of the material
(a very large number for metallic
conductors, extremely small for
poor conductors)
dI # (
J )d (A
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The filament's resistance
• To keep things simple let's assume the filament's is
cylindrical shaped with cross section A
Area A
(J
I
!
E
L
+
Vxy
-
Resistivity
It depends on the geometry
! E()d (l
V xy L
EL
EL
L
1 L
#
#
#
#
#.
I
! (J)d (l J A - E A - A A
A
/R
Resistance
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Lumped abstraction
for the light bulb
• The only property of interest for the bulb is its resistance
x
VB
I
VB
y
I#
V
with V #V B
R
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