MCC EEE202 Chapter 1, G. Formicone

1
Circuit Elements Models Chapter 1
MCC - EEE202 - Chapter 1, G. Formicone
2
OUTLINE Objective Introduction Network/Circuit
Elements Sources Models Homework 1
MCC - EEE202 - Chapter 1, G. Formicone
3

• Objective
• Understand current-voltage (v vs. i, or v-i)
  relationship for various electrical elements or
  components
• Learn to apply Kirchhoffs current and voltage
  laws for any circuit topology.
• Understand the limitations of a mathematical
  model that describes the behavior of an
  electrical element.

For most practical applications, an electrical
circuit or network accepts input signals, such as
the output of a sensor or an antenna, and convert
them into output signals, such as a computer or
instrument display or cell phone speaker, etc. A
circuit usually responds differently to different
types of input signals.
MCC - EEE202 - Chapter 1, G. Formicone
4
Introduction Network or circuit analysis deals
with the measurement and calculation of voltages
and currents. An electrical network or circuit
is simply a connection of different elements or
components that will affect the way in which
voltage and current vary throughout that
connection. Circuit elements divide into two
categories passive and active. Examples of
passive elements are resistors, capacitors,
inductors. Examples of active elements are
voltage and current sources, whether dependent or
independent, transistors, operational amplifiers
(op-amps). Each element is characterized by a
mathematical model that describes the
current-voltage relations, i.e. the equation v
v(i). Each circuit is solved by two fundamental
laws, Kirchhoffs laws, that applied to a circuit
yield a set of equations that solved provide
voltage and current throughout the circuit.
MCC - EEE202 - Chapter 1, G. Formicone
5
UNITS
multiple
prefix
symbol
1024 yotta Y 1021 zetta Z 1018 exa
E 1015 peta P 1012 tera T 109 giga
G 106 mega M 103 kilo k 1 10-3 milli
m 10-6 micro m 10-9 nano n 10-12 pico
p 10-15 femto f 10-18 atto a 10-21 zepto z
10-24 yocto y
MCC - EEE202 - Chapter 1, G. Formicone
6
Electric Circuit
An electric circuit is an interconnection of
electrical components, each of which will be
described by a mathematical model.
Examples of electrical components resistor,
capacitor, inductor voltage source
(batteries) current source (generators) diode,
transistor
Passive elements
Source elements
Active elements
An active element is capable of generating
energy, and a passive element can dissipate
and/or store energy, but NOT generate it.
MCC - EEE202 - Chapter 1, G. Formicone
7
Electric Charge and Current
The most elementary electrical quantity is the
electric charge q(t). The time rate of change
of the electric charge is the electric current
i(t).
The electric charge is measured in coulomb C.
The electric current is measured in ampere
A. 1 A 1 C/s.
MCC - EEE202 - Chapter 1, G. Formicone
8
Voltage or Potential Difference
The voltage or potential difference of a unit
charge between two points in a circuit is the
energy per unit charge (i.e. potential
difference) it takes to move that charge from one
point to the other point.
Energy is measured in Joule J. It is indicated
as U(t). Potential is measured in Volts V. It
is indicated as V(t). Voltage or potential
difference v(t) is also measured in Volts V.
1 V 1 J/C 1J 1N m 1V 1 N m / C
The voltage associated with a two-terminal
element is the energy required to move a unit
charge form its negative terminal (-) to its
positive terminal ().
MCC - EEE202 - Chapter 1, G. Formicone
9
Energy and Power
The power p(t) is the time rate of change of
energy U(t). Power is measured in watts W, or
J/s, i.e. 1 W 1 J/s.
MCC - EEE202 - Chapter 1, G. Formicone
10
Passive Sign Convention
If a current i(t) flows from a point at a higher
potential () to a point at a lower potential (-)
in a circuit element, v(t) being the voltage or
potential difference between the two points, a
power p(t) i(t)v(t) is absorbed by or
dissipated in such element.
MCC - EEE202 - Chapter 1, G. Formicone
11
Passive Sign Convention
If a current i(t) moves from a point at a lower
potential (-) to a point at a higher potential
() in a circuit element, -v(t) being the voltage
or potential difference between the two points, a
power p(t) - i(t)v(t) is generated in or
supplied by such element.
MCC - EEE202 - Chapter 1, G. Formicone
12
Passive Sign Convention
When the element is absorbing / dissipating
energy, a positive current enters the positive
terminal and leaves via the negative terminal.
When the element is supplying / delivering
energy, a positive current enters the negative
terminal and leaves via the positive terminal.
MCC - EEE202 - Chapter 1, G. Formicone
13
Passive Sign Convention
Given a circuit element, the product of voltage
v(t) across its input and output terminals and
current i(t) through it, with their appropriate
signs, will determine the magnitude and sign of
the power. If the power is positive, power is
absorbed by the element if the power is
negative, power is supplied by the element.
MCC - EEE202 - Chapter 1, G. Formicone
14
Resistors and Ohms Law
The voltage vR(t) across an ideal resistor is
directly proportional to the current iR(t)
flowing through it.
or
Resistance (R) is measured in Ohm (W). 1 W 1
V / A. Conductance (G) is measured in Siemens
(S). 1 S 1 A / V, or 1 S 1 / W. R
0 circuit is a short vR(t) 0 G 0 circuit is
a open iR(t) 0
In general, the voltage across the terminals of a
short circuit is always zero regardless of the
current flowing through it the current through
an open circuit is always zero regardless of the
voltage between its terminals.
MCC - EEE202 - Chapter 1, G. Formicone
15
Resistors and Ohms Law
In an ideal resistor, the current iR(t) always
flows from the higher potential () to the lower
potential (-). Therefore, the power pR(t) will
always be absorbed power.
The power absorbed or dissipated is
or, in terms of conductance
A linear behavior defines the v - i relationship
of an ideal resistor
MCC - EEE202 - Chapter 1, G. Formicone
16
Inductors
The voltage vL(t) across an ideal inductor is
directly proportional to the time derivative of
the current iL(t) flowing through it.
Conversely, the current iL(t) flowing through an
ideal inductor is directly proportional to the
time integration of the voltage vL(t) across it.
The parameter L is called inductance and is
measured in Henry (H). 1 H 1 Vs / A. At
DC an inductor is a short circuit!
MCC - EEE202 - Chapter 1, G. Formicone
17
Inductors
The power stored in an inductor is
Therefore, the energy stored in the magnetic
field of an inductor is
and the total energy stored (obtained for
iL(t0)0) is
MCC - EEE202 - Chapter 1, G. Formicone
18
Capacitors
The current iC(t) flowing through an ideal
capacitor is directly proportional to the time
derivative of the voltage vC(t) across it.
Conversely, the voltage vC(t) across an ideal
capacitor is directly proportional to the time
integration of the current iC(t) flowing through
it.
The parameter C is called capacitance and is
measured in Farad (F). 1 F 1 As / V
C/V. At DC a capacitor is an open circuit!
MCC - EEE202 - Chapter 1, G. Formicone
19
Capacitors
The power flowing into or out of a capacitor is
Therefore, the energy stored in the electric
field of a capacitor, is
and the total energy stored (obtained for
vC(t0)0) is
since
MCC - EEE202 - Chapter 1, G. Formicone
20
Independent Voltage Sources
An ideal voltage source is a two-terminal element
that maintains a given voltage vS(t) across its
terminals regardless of the current flowing
through them.
An off voltage source, with vS(t) 0, is a
short circuit!
A horizontal line defines the v - i relationship
of an ideal voltage source
MCC - EEE202 - Chapter 1, G. Formicone
21
Independent Current Sources
An ideal current source is a two-terminal element
that maintains a given current iS(t) flowing
through its terminals regardless of the voltage
across them.
An off current source, with iS(t) 0, is an
open circuit!
A vertical line defines the v - i relationship
of an ideal current source
MCC - EEE202 - Chapter 1, G. Formicone
22
Dependent Voltage Sources (VCVS)
An ideal voltage controlled voltage source
(vcvs) is a three or more terminal element that
maintains a given voltage vS(t) across two of its
terminals dependent on the control voltage vc(t)
across two other terminals.
MCC - EEE202 - Chapter 1, G. Formicone
23
Dependent Voltage Sources (CCVS)
An ideal current controlled voltage source
(ccvs) is a three or more terminal element that
maintains a given voltage vS(t) across two of its
terminals dependent on the control current ic(t)
flowing through two other terminals.
MCC - EEE202 - Chapter 1, G. Formicone
24
Dependent Current Sources (VCCS)
An ideal voltage controlled current source
(vccs) is a three or more terminal element that
maintains a given current iS(t) flowing through
two of its terminals dependent on the control
voltage vc(t) across two other terminals.
MCC - EEE202 - Chapter 1, G. Formicone
25
Dependent Current Sources (CCCS)
An ideal current controlled current source
(cccs) is a three or more terminal element that
maintains a given current iS(t) flowing through
two of its terminals dependent on the control
current ic(t) flowing through two other terminals.
MCC - EEE202 - Chapter 1, G. Formicone
26
Examples of dependent sources
Operation Amplifier (Op-Amp) as a VCVS
MCC - EEE202 - Chapter 1, G. Formicone
27
Examples of dependent sources
Field Effect Transistor (FET) as a VCCS
MCC - EEE202 - Chapter 1, G. Formicone
28
Examples of dependent sources
npn Bipolar Junction Transistor (BJT) as a CCCS
MCC - EEE202 - Chapter 1, G. Formicone
29
Kirchhoffs Laws
A network or circuit is made of any arrangement
of elements and sources connected at their
terminals. For current to flow, at least one
closed path must exist in such a network,
otherwise you have an open circuit!
The points at which two or more elements and/or
sources are connected to each other are called
nodes. At each node a potential is defined,
called the node potential. Any closed path made
of such elements and sources is called a
loop. Any loop that does not contain another loop
inside of it is called mesh. Within each mesh a
current is defined, called a mesh current.
Each element and source is described by the v-i
relationship characteristic of such a device,
independent of the network/circuit
topology. Kirchhoffs laws are two constraints
that must be satisfied any time a network is
formed, and they are the following

• Kirchhoffs Current Law (KCL) imposes a
  constraint for the total
• current at each node (or through a closed
  surface) to be zero, therefore
• satisfying the principle of conservation of
  charge.

2) Kirchhoffs Voltage Law (KVL) imposes a
constraint for the total voltage around each
loop or mesh to be zero, therefore satisfying the
principle of conservation of energy.
MCC - EEE202 - Chapter 1, G. Formicone
30
Kirchhoffs Current Law (KCL)
KCL (Kirchhoff Current Law) (current ?
node) The algebraic sum of the current entering
(or leaving) any node (or closed surface ) is
zero. (...also stated as) The sum of the
currents entering a node (or closed surface) is
equal to the sum of the currents leaving the same
node (or closed surface).
(...also stated as)
where Ne is the total number of elements and
sources connected to such a node, whereas Nin is
the subset of currents entering the node and Nout
is the subset of currents leaving the same node.
In doing the algebraic sum, currents going into a
node take a positive sign, whereas currents
leaving from a node take a negative sign.
MCC - EEE202 - Chapter 1, G. Formicone
31
Kirchhoffs Current Law (KCL)
By applying KCL to a circuit with N nodes, we get
a system of N equations. This system of equations
is NOT linearly independent, i.e. one of the
equations can be obtained as a linear combination
of the other N-1 equations. Therefore, it is
sufficient to write KCL equations at N-1 nodes
only.
Recalling what stated earlier, at each node a
potential is defined, called the node potential.
A circuit with N nodes will have N node
potentials. The voltage across an element or
source is the potential difference between
its node potentials. By selecting a common
reference for all nodes in the circuit and
measuring the potential difference between N-1
nodes with respect to the common potential, we
get N-1 node voltages.
Therefore, we select one node and we arbitrarily
set its potential equal to zero. This node is
called the GROUND, or reference node or reference
potential.
Now, the N-1 KCL equations translate into N-1
equations in N-1 unknowns, which are the N-1 node
voltages.
We are now left with a system of N-1 linearly
independent equations in N-1 unknowns.
MCC - EEE202 - Chapter 1, G. Formicone
32
Kirchhoffs Voltage Law (KVL)
KVL (Kirchhoff Voltage Law) (voltage ?
loop/mesh) The algebraic sum of the element and
source voltages around any loop or mesh is
zero. (...also stated as) The sum of the
voltage rises around any loop/mesh is equal to
the sum of the voltage drops around the same loop.
(...also stated as)
where L is the total number of elements and
sources connected in such a loop, whereas Lup is
the subset of voltage rises along the loop and
Ldown is the subset of voltage drops around the
same loop.
In doing the algebraic sum, voltage drops around
a loop take a positive sign, whereas voltage
rises around the loop take a negative sign.
MCC - EEE202 - Chapter 1, G. Formicone
33
Kirchhoffs Voltage Law (KVL)
By applying KVL to a circuit with L loops, we get
a system of L equations. This system of equations
is NOT linearly independent, i.e. some of the
equations can be obtained as a linear combination
of other equations. How to select a minimal set
of loops that leads to a set of independent KVL
equations is not a simple process. However, for
planar networks we can obtain a minimal set of M
independent KVL equations by writing one KVL
equation for each mesh in the network. Lets
recall that a mesh in a circuit is a loop that
does not contain any other loop inside
itself! For each mesh we can define a mesh
current, that is the current that would flow
along the mesh if it were the only mesh in the
network. Therefore, in a circuit with M meshes we
have M unknown mesh currents.
By applying KVL to a circuit with M meshes, we
get a system of M equations in M unknowns.
MCC - EEE202 - Chapter 1, G. Formicone
34
Examples of Circuit Topologies
using element voltages and currents
MCC - EEE202 - Chapter 1, G. Formicone
35
Examples of Circuit Topologies
using element voltages and currents
MCC - EEE202 - Chapter 1, G. Formicone
36
Examples of Circuit Topologies
using node voltages and mesh currents
MCC - EEE202 - Chapter 1, G. Formicone
37
Examples of Circuit Topologies
using node voltages and mesh currents
MCC - EEE202 - Chapter 1, G. Formicone
38
Examples of Circuit Topologies
using node voltages and mesh currents
MCC - EEE202 - Chapter 1, G. Formicone
39
Solution of a Circuit / Network
The complete solution of a circuit is a set of
time-varying voltages and currents such that the
v-i relationships of all elements and sources are
satisfied, KCL is satisfied at each node and KVL
is satisfied along each loop or mesh.

• Lets enounce the exhaustive method for the
  complete solution of a network
• To find the complete solution for a planar
  network it is sufficient to write
• a v-i relation for each element
• a KCL equation at each independent node
• a KVL equation for each mesh

The approach leads to a number of equations equal
to the number of unknowns that solved has a
unique solution.
MCC - EEE202 - Chapter 1, G. Formicone
40
Solution of a Circuit / Network
For circuits containing resistors only, the
equations are algebraic equations.
For circuits containing resistors, capacitors and
inductors, the equations are ordinary
differential equations (ODE).
Using the Laplace transform, these equations can
be converted into algebraic equations for
transient problems.
On the other hand, using the phasor notation and
the concept of impedance, these differential
equations can be converted into complex algebraic
equations for AC problems.
MCC - EEE202 - Chapter 1, G. Formicone
41
Series Parallel Connection
When a node connects exactly two elements, those
elements are said to be connected in series.
Three or more elements connected to the same
node does not make a series connection!
Applying KCL to two elements connected in series
leads to the following
The currents flowing through two elements
connected in series are the same.
When a mesh connects exactly two elements, those
elements are said to be connected in parallel.
Three or more elements connected by the same
mesh does not make a parallel connection!
Applying KVL to two elements connected in
parallel leads to the following
The voltages across two elements connected in
parallel are the same.
The above properties of series and parallel
connection can be used to reduce the number of
variables when solving for the complete solution
of a network by eliminating redundancies.
MCC - EEE202 - Chapter 1, G. Formicone
42
Series Connection
When a node connects exactly two elements, those
elements are said to be connected in series.
Three or more elements connected to the same
node does not make a series connection!
Applying KCL to two elements connected in series
leads to the following
The currents flowing through two elements
connected in series are the same.
ie1
ie1
ie2
ie2
ie1 ie2
The above properties of series and parallel
connection can be used to reduce the number of
variables when solving for the complete solution
of a network by eliminating redundancies.
MCC - EEE202 - Chapter 1, G. Formicone
43
Parallel Connection
When a mesh connects exactly two elements, those
elements are said to be connected in parallel.
Three or more elements connected by the same
mesh does not make a parallel connection!
Applying KVL to two elements connected in
parallel leads to the following
The voltages across two elements connected in
parallel are the same.
- Ve1 Ve2 0
-
-
i.e.
Ve1
Ve2
Ve1 Ve2
The above properties of series and parallel
connection can be used to reduce the number of
variables when solving for the complete solution
of a network by eliminating redundancies.
MCC - EEE202 - Chapter 1, G. Formicone
44
Models
What described so far for capacitors, inductors,
resistors, dependent and independent sources are
ideal mathematical models providing a simplified
description of the underlying physics.
Ideal mathematical models ignore the time it
takes charge to move through the element itself,
and the electromagnetic interactions between its
parts that depend on the specific design,
material, geometry and more in general on the
details of the physical realization the element
itself is made of.
Depending on the specific application, it may
become necessary to extend the simplified model
to include features that are otherwise
negligible.
The development of physical models for real
components is a field of expertise of its own.
model of a real resistor incorporating some of
the parasitic effects !
MCC - EEE202 - Chapter 1, G. Formicone
45
Examples of Circuit Equations
using the exhaustive method
4 elements
A 4-element circuit has 4 element voltages and 4
element currents, for a total of 8 unknowns. 8
equations are needed to analyze the circuit.
4 equations are given by the Ve - ie relations. 3
equations are given by the (4-1) KCLs. 1
equation is given by the 1-loop KVL.
4-13 node equations (KCL)
ie1 ie2
ie4 ie1
but this is redundant, since it can be obtained
by adding the three equations on the left!
ie2 ie3
ie3 ie4
1 loop equation (KVL)
- Ve1 Ve2 Ve3 Ve4 0
MCC - EEE202 - Chapter 1, G. Formicone
46
Examples of Circuit Equations
using the exhaustive method
A 7-element circuit has 7 element voltages and 7
element currents, for a total of 14
unknowns. Therefore, 14 equations are needed to
analyze this circuit.
7 equations are given by the Ve - ie relations. 5
equations are given by the (6-1) KCLs. 2
equations are given by the 2 KVLs.
6-15 node equations (KCL)
2 loop equations (KVL)
ie1 ie2
ie5 ie6
- Ve1 Ve2 Ve4 Ve7 0
ie2 ie3 ie4
ie7 ie4 ie6
- Ve4 Ve3 Ve5 Ve6 0
ie3 ie5
MCC - EEE202 - Chapter 1, G. Formicone
47
Examples of Circuit Equations
using the exhaustive method
A 10-element circuit has 10 element voltages and
10 element currents, for a total of 20 unknowns.
Therefore, 20 equations are needed to analyze
this circuit.
10 equations are given by the Ve - ie
relations. 7 equations are given by the (8-1)
KCLs. 3 equations are given by the 3 KVLs.
MCC - EEE202 - Chapter 1, G. Formicone
48
Examples of Circuit Equations
using the exhaustive method
8-17 node equations (KCL)
3 loop equations (KVL)
ie1 ie2
ie7 ie8
- Ve1 Ve2 Ve5 Ve10 0
ie2 ie3 ie5
ie9 ie8 ie6
- Ve5 Ve3 Ve6 Ve9 0
ie3 ie4 ie6
ie10 ie9 ie5
- Ve6 Ve4 Ve7 Ve8 0
ie4 ie7
MCC - EEE202 - Chapter 1, G. Formicone
49
Examples of Circuit Equations
using the exhaustive method
A 15-element circuit has 15 element voltages and
15 element currents, for a total of 30 unknowns.
Therefore, 30 equations are needed to analyze
this circuit.
15 equations are given by the Ve - ie
relations. 10 equations are given by the (11-1)
KCLs. 5 equations are given by the 5 KVLs.
MCC - EEE202 - Chapter 1, G. Formicone
50
Examples of Circuit Equations
using the exhaustive method
11-110 node equations (KCL)
ie1 ie2 ie11
ie2 ie14 ie3 ie5
ie3 ie15 ie4 ie6
ie4 ie7
ie7 ie8
ie9 ie8 ie6
ie10 ie9 ie5
ie11 ie12
ie12 ie13 ie14
ie13 ie15
5 loop equations (KVL)
- Ve1 Ve2 Ve5 Ve10 0
- Ve11 Ve12 Ve14 - Ve2 0
- Ve5 Ve3 Ve6 Ve9 0
- Ve14 Ve13 Ve15 - Ve3 0
- Ve6 Ve4 Ve7 Ve8 0
MCC - EEE202 - Chapter 1, G. Formicone
51
Homework 1
Problem 1 The charge entering the circuit below
is given by the expression q(t) 4t2 .
Determine the current entering the circuit at t
2 sec.
Problem 2 Find the magnitude and direction of
the voltage across the following elements
(b)
(a)
MCC - EEE202 - Chapter 1, G. Formicone
52
Homework 1
Problem 3 Find the power absorbed or supplied
by the elements in the following circuit
Problem 4 Find the power absorbed or supplied
by the elements in the network below
MCC - EEE202 - Chapter 1, G. Formicone
53
Homework 1
Problem 5 Is the voltage source VS in the
following network absorbing or supplying power?
And how much?
Problem 6 Given the following circuit, find I
and V0.
MCC - EEE202 - Chapter 1, G. Formicone
54
Homework 1
Problem 7 If, in the following circuit, VR2 is
known to be 6 V, find VR1 and VA.
Problem 8 Find I and V0 in the network below
MCC - EEE202 - Chapter 1, G. Formicone
55
Homework 1
Problem 9 Find the voltage V0 in the following
network
Problem 10 Determine the current I in the
circuit below
MCC - EEE202 - Chapter 1, G. Formicone