Download Circuit Theory

Dr inż. Agnieszka Wardzińska
Room: 105 Polanka
agnieszka.wardzinska@put.poznan.pl
cygnus.et.put.poznan.pl/~award
Advisor hours:
Monday: 9.30-10.15
Wednesday: 10.15-11.00
Basic bibliography:
1. Introductory Circuit Analysis, Robert L. Boylestad, Prentice Hall PTR, 2000, 2003,
2007, 2010;
2. AC and DC Network Theory, A. J. Pointon, H. M. Howarth, Springer Netherlands,
1991;
3. Electrical Circuit Theory and Technology, Bird, John, Elsevier Newnes, 2003;
Additional bibliography:
1. Circuits Systems with Matlab and PSpice, Won Y. Yang, Seung C. Lee, Wiley, Asia,
2007.
2. Linear and Nonlinear Circuits, L.O. Chua, C.A. Desoer, E.S. Kuh , McGraw-Hill
Inc., 1987.
3. Analog and digital filters: design and realization, H. Y.,-F. Lam , Prentice_Hall, Inc.,
Englewood Cliffs, New Jersey, 1979.
4. Classical Circuit Theory, Omar Wing, Springer US, 2009
Credits for the course:
• written final exam
Course description
1.
2.
3.
4.
5.
6.
Basic laws in circuit theory: voltage and current Kirchoff's laws,
Tellegen’s theorem. Real circuit and its mathematical model, Thevenin
and Norton theorem.
Linear and non-linear passive components and active elements of analog
circuits. The basic principles, theorems and methods in the analysis of
resistive circuits.
Circuits with harmonic currents in steady state - Method of complex
numbers, phasor diagrams. Coupled and resonant circuits.
Transients, analysis in time and frequency domain (Laplace transform).
Two-ports and their description using the matrices: Z, Y, H, A, etc., and S.
The concept of transfer function, amplitude and phase characteristics.
Basic concepts of circuits stability.
Electric Circuit
• An electric circuit is an interconnection of electrical
elements.
• A circuit consists of a mesh of loops
 Represented as branches and
nodes in an undirected graph.
 Circuit components reside in
the branches
 Connectivity resides in the
nodes
 Nodes represent wires
 Wires represent
equipotentials
Charge
 Basic SI unit, measured in Coulombs (C)
 Counts the number of electrons (or positive charges)




present.
Charge of single electron is 1.602*10-19 C
One Coulomb = 6.24*1018 electrons.
Charge is always multiple of electron charge
Charge cannot be created or destroyed, only
transferred.
Current
 The movement of charge.
 We always note the direction of the equivalent positive
charges, even if the moving charges are negative.
 It is the time derivative of charge passing through a
circuit branch
dq
i
dt
 Unit is Ampere (A), is one Coulomb/second
 Customarily represented by i (AC) or I (DC).
Voltage
 a difference in electric potential
 always taken between two points.
 It is a line integral of the force exerted by an electric
field on a unit charge.
 Customarily represented by u (AC) or U (DC) or v and
V alternativelly.
 The SI unit is the Volt [V].
Power
 Power is the product of voltage by current.
 It is the time derivative of energy delivered to or
extracted from a circuit branch.
 Customarily represented by P or S or W.
 The SI unit is the Watt [W].
AC vs. DC circuits
• Direct Current (DC) is a
current that remains constant
with time is called
• A common source of DC is a
battery.
• A current that varies sinusoidally
with time is called Alternating
Current (AC)
• Mains power is an example of AC
AC analysis
 AC signal eg. voltage switches polarity over time. The
signal graphed over time takes on sine wave. The
reason for such "shape" of the signal derives from the
Faraday’s Law of Electromagnetic Induction. We can
describe the sine function both by sinus or cosinus
function:
Imax is the amplitude of the signal,
ω is angular frequency and is related to the physical frequency ω= 2πf,
φ is the phase angle in degrees or radians that the waveform has shifted either left
or right from the reference point.
The difference between sine or cosine wave is in the assumption of the time when
the phase is equal zero.
AC analysis
We can write the relationships between this two types of waveforms:
If in the circuit there are two or more sources and all of them generates the same
sinus or cosinus function we do need make the above conversions. If some sources
are described by sine function, and some by cosine function wee need to convert
them all have the same function. Because of power analysis the AC voltage is often
expressed as a root mean square (RMS) value, written as VRMS. For the
sinusoidal waveform:
AC analysis
 Considering two AC signals in the circuit eg two voltages or
currents or voltage and current, we can define the phase shift. It
mean that two or more waveforms are out of step with each
other. The amount of phase shift can be expressed in terms of
degreesor radians on the horizontal axis of the waveform the
trigonometric sine function. A leading waveform is defined as
one waveform that is ahead of another. The waveform which is
behind is a lagging waveform.
 AC circuit analysis must take into consideration both amplitude
and phase shift of voltage and current waveforms. This requires
the use of a mathematical system called complex numbers.
 Basic theories and network analysis for AC circuit is analogous
for AC and DC circuits but using complex numbers to describe
the singals. After considering the voltages and currents in
complex domain, we rewrite them to time domain.
Complex Numbers in AC analysis
Phasor Diagram of a Sinusoidal
Waveform
 Analysing the voltages and currents in the circuit, we
describe them “frozen” at some point in time. Then we
can represent magnitude and direction by the the
vector (Phasor) as a scaled line.
 The positive rotation of the phase in phasor diagram is
anti-clockwise rotation.
 Then in the phasor diagram we can show the relation
between currents and voltages in phase and
magnitude. Below there are some examples of the
phasor diagram.
Phasor Diagram of a Sinusoidal
Waveform
Basic circuit elements - resistor
Resistors are circuit elements that resist the flow of current. When this
is done a voltage appears across the resistor's two wires.
A pure resistor turns electrical energy into heat. Devices similar to
resistors turn this energy into light, motion, heat, and other forms of
energy.
Resistors don't care which leg is connected to positive or negative. We note
the current flow opposite to the voltage. This is called the an "positive charge”
sign convention. Some circuit theory books assume "electron flow" flow sign
convention.
Basic circuit elements - resistor
Resistance is measured in terms of units called "Ohms" (volts per ampere), which is
commonly abbreviated with the Greek letter Ω ("Omega"). Ohms are also used to
measure the quantities of impedance and reactance. The variable most commonly used
to represent resistance is "r" or "R". Resistance is defined as:
where ρ is the resistivity of the material, L is the length of the resistor, and A is the
cross-sectional area of the resistor.
Conductance is the inverse of resistance. Conductance has units of "Siemens" (S). The
associated variable is "G":
G
1
r
Basic circuit elements - inductor
Inductance is the property whereby an inductor exhibits opposition to the
change of current flowing through it, measured in henrys (H).
Basic circuit elements – inductor (2)
 The dependence between the current and the voltage
of the inductor is described by the equations:
 The power stored by an inductor:
An inductor acts like a short circuit to dc (di/dt = 0) and its
current cannot change abruptly.
Basic circuit elements - capacitor
A capacitor is a passive element designed to store energy in its electric field.
A capacitor consists of two conducting
plates separated by an insulator (or
dielectric).
 Capacitance C is the raBo of the charge q on one plate of a capacitor
to the voltage difference v between the two plates, measured in farads
(F).
Where ε is the permittivity of the dielectric material between the plates, A is the
surface area of each plate, d is the distance between the plates.
Basic circuit elements – capacitor (2)
 The dependence between the charge and voltage is:
 Then current –voltage relationship of the capacitor is
described by the equations:
 The power stored by an inductor:
A capacitor is an open circuit to dc (dv/dt = 0). And its voltage cannot change
abruptly (depends on integral of i).
Impedance of AC components
 When talking about the AC circuits we talk about the impedance
rather than resistance. It is denoted as Z and compose of resistance R
(the part from resistors) and reactance X (the part from inductors and
capacitors). The SI unit of impedance is the Ohm (symbol ).
 The inverse of impedance is admitance Y .
 Similar to the impedance we can distinguish the real part conductance
G and imaginary part susceptance B. The SI unit of admittance is
the siemens (symbol S).
 We must remember that in the general case the inverse of resistance is
different from the conductance and the inverse of reactance is different
from the susceptance.
AC resistor circuits
AC capacitor circuits
AC capacitor circuits
AC inductor circuits
AC inductor circuits
AC inductor circuits
Circuit Elements Ideal
Independent Voltage Source
 provides a specified voltage or current that is completely independent
of other circuit variables
 The voltage at the nodes is strictly defined by voltage of the source, the
current flow depends on the other elements in the circuit
The ideal voltage source is only a mathematical model.
Generally we can divide the voltage sources into three groups:
• Batteries
• Generators
• Supplies
Circuit Elements Ideal
independent current source
 The current flow in the branch is strictly defined by current
of the source, the voltage at the nodes of the source
depends on the other elements in the circuit
 The symbols used for AC current sources (similarly as for
voltage sources) are the same as for the DC current sources,
but described with noncapital letters (e.g. j(t)).
The ideal current source similarly to ideal
voltage source is only a mathematical model.
Circuit Elements – dependent
sources
 Ideal dependent
voltage source
 Ideal dependent
current source
The voltage defined by the
source depends on the
voltage or current
determined in this or other
circuit
The current defined by the
source depends on the
voltage or current
determined in this or other
circuit
The real voltage sources
The real AC voltage source is represented
The real voltage source can be
by ideal voltage source in series with
represented by ideal voltage source
impedance Zs. That impedance can have
in series with resistance Rs.
resistive, inductive or capacitive character.
The real current sources
The real current source can be
represented by ideal current source
in parallel with resistance Rs.
The real AC current source is represented
by ideal current source with finite
impedance Zs placed across an ideal current
source. That impedance can have resistive,
inductive or capacitive character.
Ideal Wires
 we will assume that an ideal wire has zero total resistance,
no capacitance, and no inductance. A consequence of these
assumptions is that these ideal wires have infinite
bandwidth, are immune to interference, and are — in
essence — completely uncomplicated.
 This is not the case in real wires, because all wires have at
least some amount of associated resistance. Also, placing
multiple real wires together, or bending real wires in
certain patterns will produce small amounts of capacitance
and inductance, which can play a role in circuit design and
analysis.
Ideal Junctions or Nodes
 Nodes are also called "junctions”
 A junction is a group of wires that share the same electromotive force
(not voltage). Wires ideally have no resistance, thus all wires that touch
wire to wire somewhere are part of the same node.
 Sometimes a node is described as where two or more wires touch. This
only works on simple circuits.
One node has to be labeled ground in
any circuit drawn before voltage can be
computed or the circuit simulated.
Typically this is the node having the
most components connected to it.
Logically it is normally placed at the
bottom of the circuit logic diagram.
Ground is not always needed physically.
Some circuits are floated on purpose.
Ohms Law
The potential difference (voltage) across an ideal conductor is proportional
to the current through it.
For the DC
For the AC
where: U - voltage I - current R – resistance, Z- inductance
Kirchhoff’s Circuit Laws
 Kirchhoff’s circuit laws were first described in 1845 by
Gustav Kirchhoff. They consist from two equalities for
the lumped element model of electrical circuits. They
describe the current and voltage behaviour in the
circuit.
Kirchhoff’s First Law - Kirchhoff’s
Current Law (KCL)
 The algebraic sum of currents in a network of conductors meeting at a node is
zero.
It can be described by the equation:
The currents flowing into the node (I1, I6) we describe as positive, the
currents flowing out the node (I2, I3, I4, I5) we describe as negative.
Kirchhoff’s Second Law Kirchhoff’s Voltage Law (KVL)
 The algebraic sum of the potential rises and drops
around a closed loop or path is zero.
where Ui describes both the potential
drops at the elements and the
voltages generated by sources.
To use the KVL one need to set up a rotation in the circuit. Potentials with direction
of the circuit have a positive sign, voltage opposite to the direction of circulation of
the circuit have a negative sign.
Series Connection
 All components are connected end-to-end.
 Voltage drops add to total voltage.
 Due to all components goes the same (equal) current.
 Impedance (or simply resistance in DC) add to total impedance
(resistance).
Parallel Connection
 All components are conected between the same two sets of electrically
common points.
 Currents add to total current.
 Voltage drop on the components are the same.
 Conductances (inverse of resistance) add to total conductance.
or
Series-Parallel Connection
 Typical circuits have some series
connected components in some parts
of the circuit and parallel in others.
Then it is impossible to apply a single
set of rules to the all circuit. Instead, it
is possible to identify which parts of
that circuit are series and which parts
are parallel, then selectively apply
series and parallel rules.
Delta-Y conversions
 In many applications circuits or fragments of circuits form
a three-terminal network. Components are then connected
together in one of two ways: the “Delta,” or (also known as
the “Pi,” or ) configuration, and the “Y” (also known as the
“T”) configuration.
 The Delta (left side) and (right side) configuration take the
form presented below:
Delta-Y conversions
 The Y (left side) and T (right side) configuration take
the form presented below:
Delta-Y conversions