Download EE61: Introduction to Electric Circuits Chapter 1: Basic Principles

EE61: Introduction to Electric Circuits
Chapter 1: Basic Principles
Copyright 1994 { 1999, Dean E. McCumber
January 20, 1999
Contents
1 Basic Electrical Quantities
1.1
1.2
1.3
1.4
1.5
Electric Charge . . . . . . . . . . . . . . . . .
Electric Current: The Time Flow of Charge .
Energy . . . . . . . . . . . . . . . . . . . . . .
Voltage: Potential Energy per Unit of Charge
Power: The Time Flow of Energy . . . . . . .
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1
1
2
3
4
4
2 A Water-Flow Analogy
5
3 Sign Conventions
6
4 Linear Circuit Elements (Ohm's Law)
8
4.1 The Linear Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 The Linear Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 The Linear Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Open and Short Circuits
13
6 Switches
15
5.1 Open Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Short Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
i
EE61 Electric Circuits, Ch1: Basic Principles
ii
6.1 Ideal Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 Practical Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7 Sources
7.1 Ideal Independent Sources . . . .
7.2 Ideal Dependent Sources . . . . .
7.3 Practical Sources . . . . . . . . .
7.3.1 Practical Voltage Sources .
7.3.2 Practical Current Sources
7.4 Independent Sources in SPICE . .
16
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8 Kirchho's Laws (1847)
8.1 Kirchho's Current Law (KCL) . . . . . . . . .
8.2 Kirchho's Voltage Law (KVL) . . . . . . . . .
8.3 Maxwell's Equations . . . . . . . . . . . . . . .
8.3.1 Kirchho's Current Law . . . . . . . . .
8.3.2 Kirchho's Voltage Law . . . . . . . . .
8.3.3 Maxwell's Equations, Special Relativity .
9 Ideal Ammeters and Voltmeters
17
18
19
19
21
22
24
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26
27
28
28
29
30
30
9.1 Ideal Ammeters and Voltmeters . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.2 Wire \Equivalents" to Ideal Meters . . . . . . . . . . . . . . . . . . . . . . . 31
10 Equivalent Resistance
32
10.1 Current Division, Parallel Resistors . . . . . . . . . . . . . . . . . . . . . . . 33
10.2 Voltage Division, Series Resistors . . . . . . . . . . . . . . . . . . . . . . . . 34
11 Source Equivalents
37
11.1 Thevenin and Norton Equivalents . . . . . . . . . . . . . . . . . . . . . . . . 37
11.2 Sources in Series and Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . 38
11.2.1 Additive Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
EE61 Electric Circuits, Ch1: Basic Principles
iii
11.2.2 Dominating Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
11.2.3 Trouble! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
12 Circuit Reduction
39
Introduction to Electric Circuits
Chapter 1: Basic Principles
1 Basic Electrical Quantities
This course deals with the analysis, design and evaluation of linear circuits. A circuit models
an interconnection of electric devices that processes energy or information. Understanding
circuits is important because energy and information are the underlying technological commodities in electrical engineering. The study of circuits provides a foundation for other
areas of electrical engineering such as electronics, power systems, communication systems,
computers, and control systems.
1.1 Electric Charge
All electricity stems from the fundamental physical quality of electric charge. Charge is a
fundamental physical property of matter.
Total charge is conserved { that is, in any physical (or chemical) reaction the total charge
out equals the total charge in.
Electric charge is quantized. At all energies of relevance to chemistry and life on this planet
as we know it, the quantum of charge is the charge on the electron, and the three important
fundamental particles of matter are
Particle Quantity of Charge
Proton
Neutron
Electron
q
q
q
Rest Mass
= 1 = 1 602
(coulombs)
1 67 10;24g
= 0, uncharged
1 67 10;24g
= ;1 = ;1 602 10;19C (coulombs) 9 11 10;28g
e
10;19C
:
:
:
e
:
:
Table 1: Table of Elementary Particles
In the International System of Units (SI), charge is measured in coulombs. The symbol for
coulombs is capital C. Rule on Unit Symbols: Symbols for units that derive from proper
names are capitalized, others are not periods are not used after symbols (except to end a
sentence!) and the symbols do not take on plural forms.
Protons and neutrons are of comparable mass (1 67 10;24g= 1 67 10;27kg) and each is
about 2000 times heavier than the electron (9 11 10;28g= 9 11 10;31kg).
The mechanisms of \electricity" relate to the accumulation and ow of electric charge.
:
:
:
:
EE61 Electric Circuits, Ch1: Basic Principles
2
In this course we are concerned with the ow of electricity through electric circuits modeling the interconnection of various circuit elements by metallic conductors. In metallic
conductors, electricity ows through the motion of electrons.
In ionized gasses and in aqueous electrolytes, electricity ows through the motion of heavy
charged conglomerates called ions. In both cases, charged particles move, carrying charge
from one location to another.
To understand the physics of particular circuit elements and electronic devices, it is usually
important to know the nature of the charge carriers. In circuit analysis knowledge of the
charge carriers is usually irrelevant. The important thing is that charges are induced to move
by the action of external potentials.
1.2 Electric Current: The Time Flow of Charge
The motion of electric charges give rise to what are called electric currents. Current
is the time rate of ow of electric charge past a given point . . . or, better, through a
surface. Charge and current are related by the equations:
( ) = ( )
i t
dq t
dt
Zt
( ) = (0) +
1 ( 1)
0
where ( ) is the net ow of charge through the surface. Examples, exercises and problems
are given in the text exploring these relations. Work some of them out to become comfortable
with the relationship between charge and current.
Table 2 lists some comparative velocities relevant to currents generated by electron ow.
The very low mean velocity of electrons in a good conductor like copper is noteworthy and
a direct consequence of the high electron density in metals.
q t
q
dt
i t
q t
Situation
Automobile @ 67 mph
Electrons in 14-gauge Cu wire to 100 W lamp @ 120 V
Free electron rms velocity at 300 K
Free electron with 1 eV energy
Free electron with 1,000 eV energy
Velocity of light in vacuum
Velocity
30 m/s
5 0 10;5 m/s
1 2 105 m/s
5 9 105 m/s
1 9 107 m/s
3 0 108 m/s
:
:
:
:
:
Table 2: Electron velocities for important situations.
In the International System of Units (SI), current is measured in coulombs per second, or
amperes. The symbol for amperes, singular and plural, is capital A.
EE61 Electric Circuits, Ch1: Basic Principles
3
Current is a through variable. It measures the ow of charge through a conductor.
1.3 Energy
Energy is a second fundamental physical quantity. As is the case with charge, energy is
conserved in all physical reactions.
In general, it takes energy to move electric charges. That energy can be provided chemically
(as in a battery), mechanically (as in a generator, where conductors are moved through a
magnetic eld), or \electrically" (by the application of a \voltage" . . . itself provided ultimately by some other prime mover).
For the moment, let's not ask whence the energy came. Let's simply acknowledge that it is
required, and that work is done by the movement of electric charges.
Energy can ow into and out of a circuit element. Consider the following circuit element,
which is representative of the lumped circuit elements we shall consider in this course.1
c
+
(a)
c
-
() ;
()
i t
(b)
i t
Figure 1: Circuit element with standard passive-element labeling conventions.
Charge ow is indicated by the current ( ). The current ( ) that ows into the circuit
element at the (+) terminal equals the current that ows out of the circuit element at the
(;) terminal. The arrow is does not necessarily indicate the direction of the actual physical
charge ow (which might actually reverse many times a second) but it does indicate the ow
direction when the variable ( ) is positive { that is, it establishes a sign convention for the
current variable ( ).
Let us assume that the charge that has owed through the element up to time is ( )
and that the work done (energy expended) to cause that charge to ow is ( ). In the
International System of Units (SI), energy is measured in joules. The symbol for joules,
singular and plural, is capital J.
i t
i t
i t
i t
t
q t
w t
Lumped circuit elements are appropriate when circuit distances are small compared to the velocity
of light (c = 3 1010 cm/s times the time scales of relevance). When distances are not small, one has
distributed circuit elements. Distributed circuit elements are the stu of interest in EE170.
1
EE61 Electric Circuits, Ch1: Basic Principles
1.4 Voltage: Potential Energy per Unit of Charge
4
The energy required at time to move one unit of charge through the above circuit element
is
( ) = ( ) = ab( ) = a( ) ; b ( )
t
v t
dw t
v
dq
t
v
t
v
t where a( ) and b ( ) are measured relative to some common reference. That is, ( ) is
the work required to move a positive charge of one coulomb from the (+) terminal through
the circuit element to the (;) terminal. We say that ( ) is the voltage across the circuit
element. Or, using a term that reminds us of energy, we say that ( ) is the potential
dierence across the circuit element.
Voltage is not measured at a point, but rather between two points or across an element.
Sometimes one of the points is a reference point. Such a reference point is commonly called
ground. It may, but usually does not, designate a physical connection to the earth. Common
reference (ground) symbols are shown in Figure 2.
v
t
v
t
v t
v t
v t
s
s
s
@; ;;;
Figure 2: Standard reference (ground) symbols.
In the International System of Units (SI), voltage is measured in volts. The symbol for volts,
singular and plural, is capital V.
Notice that current is a through variable and voltage is an across variable, because they
respectively ow through and appear across the circuit element.
1.5 Power: The Time Flow of Energy
Power is the time rate of change of energy:
()=
p t
()
dw t
dt
:
In the International System of Units (SI), power is measured in watts. The symbol for watts,
singular and plural, is capital W.
Power measures the rate at which energy is transformed. If we view the energy ( ) as a
function of the transferred charge ( ), then it follows from the chain rule of dierentiation
that
( ) = ( )] ( ) = ( ) ( ) = a ( ) ; b( )] ( )
()
w t
q t
p t
dw
dw q t
dt
dq t
dq t
dt
v t
i t
v
t
v
t
i t :
EE61 Electric Circuits, Ch1: Basic Principles
5
For the sign conventions shown in Figure 1, ( ) is the work done on the circuit element. If
( ) 0, energy ows into the circuit element: electrical energy is absorbed by the circuit
element.
w t
p t
>
2 A Water-Flow Analogy
The ow of charges in wires is su!ciently removed from everyday experience that it is useful
to seek a more familiar physical analog, even if imperfect. The analogy we introduce here is
the ow of water in pipes.
Consider a system of pipes connecting two water reservoirs at dierent vertical levels { that
is, at two dierent potential-energy levels{, so that water tends to ow through the pipes
from the higher reservoir to the lower one. The quantity of water owing past a given point
in a pipe is the water analog of the quantity of charge owing past a given point in a metalic
wire. The rate of ow (gallons per minute, for example) of water through a pipe is analogous
to rate of ow of charge { that is, to electric current. The pressure dierence across a
section of pipe is analogous to a voltage dierence.
Real pipes provide resistance to water ow, just as real conductors provide resistance to
the ow of charge. If water is to ow through a real pipe, a pressure dierence must be
developed across the pipe: the greater the desired ow, the greater the required pressure
dierence. In the simplest mathematical model, the rate of water ow would be proportional
to the pressure dierence. Using electrical notation, the governing equation would be
()=
i t
()
(1)
Gv t where is bigger for big pipes and smaller for small pipes.2 That is, the resistance = 1
is less for big pipes and larger for small pipes, where size here means cross-sectional area.
The ideas translate directly into correct ideas for electric ow in wires, with equation (1)
being a form of Ohm's Law.
Because a pressure dierence is required to ow water through a pipe, the system does work
on the water-plus-pipe section. That is, the system causes energy to ow into the resistive
section of pipe. This energy is dissipated as heat, which we could detect by very careful
measurement. The rate at which energy is expended causing water to ow through the pipe
is proportional to the rate of water ow and to the pressure dierence { that is, in electrical
notation, the power ( ) being expended to sustain the ow through the section of pipe under
scrutiny is
()= () ()
(2)
where ( ) is the pressure dierence across the pipe section. The total energy ( ) expended
G
R
=G
p t
p t
v t
v t
i t w t
Formula (1) is qualitatively correct in that larger pressure dierences cause larger ows, but it is incorrect
for small ow velocities. Careful analysis shows that for uids like water the pressure dierence goes as i2(t)
at low velocities. This is a deciency in the water analogy.
2
EE61 Electric Circuits, Ch1: Basic Principles
6
to force a quantity ( ) of water through the pipe section is
q t
()=
w t
Zt
;1
( ) ( 1) with ( ) =
dt1 v t1
i t
Zt
q t
;1
( )
(3)
dt1 i t1 :
The energy ( ) required can depend upon the ow history as well as upon the total charge
( ) transferred, even when equation (1) is exact.
w t
q t
3 Sign Conventions
In Section 1 we were very careful to choose our direction of positive current ow (indicated
by the arrow) to be through the circuit element from the plus-labeled terminal to the minuslabeled terminal. This is consistent with what we would expect from the water analog:
water ows through pipes from high pressures to lower pressures. If ( ) 0 is the pressure
dierence across a pipe section, with the pressure at the (+) end greater than that at the
(;) end, one expects the current ( ) to ow in the direction indicated. Because a pipe, like a
resistor in a circuit, is a passive dissipative element, these polarity and directional denitions
are said to conform to the passive-element sign convention.
v t
>
i t
High Potential
or \Pressure"
c
+
(
)
(a)
Low Potential
or \Pressure"
- ;c
()
i t
i t
(b)
( ) = a( ) ; b( )
v t
v
t
v
t
Figure 3: Circuit element with standard passive-element sign conventions, in which the
current is \pushed through" the passive element by the potential or pressure dierence. The
power requirred to push the charge through the circuit is absorbed by the circuit element.
The power absorbed is abs ( ) = ( ) ( ) with ( ) and ( ) dened as shown.
p
t
v t i t
v t
i t
With this sign convention, the power ( ) owing into the circuit element will be positive
if the voltage ( ) and current ( ) have the same sign. The circuit element in this case is
said to absorb energy. If all or some of the energy can subsequently be extracted { that is,
the circuit element can deliver electrical energy {, the circuit element is said to be energy
storing. If energy cannot subsequently be extracted but is instead converted into heat,
the circuit element is said to be dissipative. Resistors are always fully dissipative. Ideal
capacitors and inductors are pure energy storing devices.
It is important to remember that the voltage polarity and current directional arrow are
denitions, to give meaning to the mathematical variables ( ) and ( ). The physical voltage
p t
v t
i t
v t
i t
EE61 Electric Circuits, Ch1: Basic Principles
7
polarity and current direction in a real circuit may or may not correspond to the signs shown.
It is su!cient that the mathematical quantities ( ( ) ( )) take on both positive and negative
values.
The quantity ( ) will be positive if charge ows in the direction of the arrow it will be
negative if charge ows in the opposite direction. The arrow is simply a label dening the
positive direction of the mathematical function ( ). The real physical ow of charge may
be in the direction of the arrow, in which case ( ) will be a positive number, or it may be
in the opposite direction, in which case ( ) will be a negative number. While it would be
nice to choose our arrows so that we were always dealing with positive quantities, that is not
usually possible . . . for two reasons: we cannot correctly guess ow directions in complicated
circuits, and sometimes ow directions change, like the tide.
Similar statements apply to the signs representing the polarity of the voltage ( ). For
the situation shown in Figure 3, the voltage
( ) = a( ) ; b( )
(4)
That is, the voltage ( ) is the electric potential of the (+)-labeled terminal relative to
the (;)-labeled terminal. If the (+)-labeled terminal actually has greater potential than
the (;)-labeled terminal, the mathematical function ( ) will be a positive number. If the
opposite conditions prevail, it will be a negative number. A particular choice of signs xes the
meaning of the mathematical function ( ), but it tell us nothing about the actual physical
situation.
The sign conventions shown in Figure 3 are such that, if the current ows in the direction
of the arrow and if the voltage across the element is positive when measured at the (+)
terminal relative to the (;) terminal, then energy will ow into the circuit element at the
rate ( ) = ( ) ( ).
What if the labeling convention has the current arrow reversed, as in Figure 4? In that case,
v t i t
i t
i t
i t
i t
v t
v t
v
t
v
t :
v t
v t
v t
p t
v t i t
High Potential
or \Pressure"
c
+ ( )
(a)
Low Potential
or \Pressure"
()
i t
i t
c
;
(b)
( ) = a( ) ; b( )
v t
v
t
v
t
Figure 4: Circuit element with energy-source labeling convention, in which the circuit element delivers power by \pushing current" from the lower to the higher potential or pressure.
The power delivered by the circuit element is del = ( ) ( ) with ( ) and ( ) dened as
shown.
if the voltage is positive when measured in the same fashion, energy will be delivered by
p
v t i t
v t
i t
EE61 Electric Circuits, Ch1: Basic Principles
8
the circuit element at a rate ( ) = ( ) ( ) { that is, the circuit element will be a source of
energy.
Because we cannot always know in advance whether a particular circuit element is absorbing
or delivering energy . . . and at dierent times some circuit elements may do dierent things,
we shall always use the passive-element sign convention and let the sign of the computed
voltages and currents sort out the actual physical situation. Under the passive-element sign
convention, positive current ows into the terminal labeled with the (+) sign. Under this
convention, if the current and the voltage are both positive numbers, then energy ows into
the element. If the product of the current and voltage are negative, energy will ow out of
the element, and we shall say that the element delivers energy or power. By using a single
consistent sign convention, we automatically keep track of whether elements are energy
sinks or energy sources.
p t
v t i t
The passive-element sign convention is the convention used in SPICE, the important computer program for circuit analysis. It is used for all circuit elements,
independent of whether they actually absorb or deliver energy.
4 Linear Circuit Elements (Ohm's Law)
An important subclass of circuit elements are those which are linear. Briey stated, circuit
elements are linear if the voltage across the element is directly proportional to the magnitude
of current through the element or, equivalently, the current through the element is directly
proportional to the magnitude of voltage across the element.
Somewhat more abstractly, if the total current through a linear circuit element is
( ) = 1( ) + 2( )
i t
i
t
i
t then the total voltage ( ) across the element equals the sum of the two voltages 1 ( ) and
2 ( ) that would obtain if 1 ( ) and 2 ( ) owed separately. That is,
v t
v
t
i
v
t
i
t
t
( ) = 1( ) + 2( )
v t
v
t
v
t
for any two currents 1 ( ) and 2 ( ). This is the important property of superposition.3
Linear circuit elements are an idealization, but for very many important applications, many
real devices are approximately linear over the relevant voltage and current values. For
network and systems engineering analyses, circuit devices are often modeled as ideal linear
elements.4
3 There is a related property of homogeneity that, in eect, says that this relationship obtains contini
t
i
t
uously if the currents vary continuously. For the present physical situation, this latter property is implicit
in the superposition property.
4 Deviations from linearity can be important in certain applications. In those cases it is not sucient to
use a linear model. If the eects of nonlinearities are small, they sometimes are only investigated after the
linear analysis is completed.
EE61 Electric Circuits, Ch1: Basic Principles
9
Don't be confused. Many important devices and circuit elements are nonlinear, and their
importance depends upon their nonlinearity. Diodes, for example, are highly nonlinear,
permitting current to ow in only one direction, so not all the world is linear. Moreover,
devices which are nominally linear, usually become nonlinear if driven hard enough.
Let me give you a very simple mechanical example. Consider a gong. When struck softly
with a padded hammer, it gives an eerie and distinctive sound. But, if struck hard with an
iron hammer, it will complain dissonantly and be dented.
Some electronic circuit elements are also \dented" when driven hard: stereo speakers are
blown out if driven too hard. Other circuit components may not be permanently destroyed
or damaged, if the driving signal is not too large, but the response can still be far from linear.
For example, some unusual eects are obtained by rock groups by \clipping" the amplied
music . . . or by driving ampliers into saturation. We'll look at such nonlinearities from a
dierent perspective later in the course, but for the moment, let's come back to the ideal
linear circuit element.
There are three simple linear circuit elements of great importance. These fundmental circuit
elements are the resistor, the capacitor and the inductor. They are the basic building
blocks of passive electric circuits.
4.1 The Linear Resistor
For the ideal linear resistor, voltage and current are directly proportional:
()
v t
()
Ri t (5)
where is a constant of proportionality with the dimension in SI units of volts/ampere or
ohms (#). This constant of proportionality is called resistance, and apt but arbitrary
term. Think of it simply as a number with appropriate units which describes the linear
relationship between voltage and current for a particular kind of circuit element.
A plot of voltage versus current for an ideal resistor is a straight line, as shown in Figure 5.
The slope of the line,
= , the resistance.
This particular kind of circuit element is su!ciently common and su!ciently important that
it has its own name (resistor) and circuit symbol. The circuit symbol is shown in Figure 6.
The linear relationship (5) between voltage and current is called Ohm's law (1827).
If the voltage is directly proportional to the current, then the current is directly proportional
to the voltage:
(6)
()= 1 () ()
R
R
dv=di
R
i t
where
5
G
R
v t
Gv t is called the conductance. In SI units it is measured in siemens (S). 5
Formerly the unit of conductance was the mho (0), ohms spelled backwards (if not upside down!).
EE61 Electric Circuits, Ch1: Basic Principles
10
V vs I for Linear Resistor (R=2)
4
3
2
1
00
-1
1
2
3
i
-1
Figure 5: Plot of voltage versus current for an ideal resistor.
c
+
(a)
-
()
i t
c
B B B B B
R
;
(b)
Figure 6: Circuit symbol for a resistor.
EE61 Electric Circuits, Ch1: Basic Principles
11
Water analogy, continued: If the ow of water in a system of pipes is anal-
ogous to the ow of electricity in a circuit, we would expect that the pressure
drop across a length of pipe would increase with the rate at which water is forced
through the pipe. Moreover, we would expect that this rate of increase would be
larger for smaller pipes than for larger ones. We can say that the smaller pipes
have greater resistance to water ow than do large pipes.
Is the ideal resistor a linear circuit element? Yes. The condition for linearity is formally the
following: If the current j ( ) produces the voltage j ( ), then the current ( ) = 1( ) + 2 ( )
produces the voltage ( ) = 1( ) + 2 ( ). This linearity condition can be easily veried for
the resistor:
i
t
v t
()=
v t
v
v
t
v
( ) = 1 ( ) + 2 ( )] =
Ri t
R i
t
i t
i
t
i
t
t
t
i
t
( )+
R i1 t
( ) = 1( ) + 2( )
R i2 t
v
t
v
(7)
t :
What can we say about power and energy if Ohm's law obtains? With the passive-circuitelement sign conventions of Figure 3, the power owing into the resistor is
()= () ()=
pinto t
v t
i t
2
Ri
()=
t
Gv
2
()
t
(8)
positive. The ideal resistor always absorbs, and dissipates,
power, independent of the sign of the current or voltage. The resistor is not an
energy storage element.
with
R
and
G
=1
=R
4.2 The Linear Capacitor
Voltage is proportional to the integral of the current:
Zt
( )= ()
() 1
v t
C
;1
dt1 i t1
q t
C
(9)
where is a constant of proportionality with the dimension in SI units of coulombs/volt or
farads (F).
The circuit element with this property is called a capacitor, with its own parallel-plate
circuit symbol.
It is easy to verify that the ideal capacitor is a linear circuit element:
Zt
Zt
Zt
Zt
1
1
1
1
( )=
( )+ ( )] =
( )+
( ) = ( )+ ( )
()=
C
v t
C
dt1 i t1
C
dt1 i1 t1
i2 t1
C
dt1 i1 t1
C
dt1 i2 t1
v1 t
v2 t :
(10)
If we substitute the capacitor's voltage expression (in terms of charge ( )) into the power
expression and recognize that =
, we obtain
2
()
(11)
( ) = ( ) ( ) = ( ) ( ) = 21
q t
i
p t
dq=dt
v t i t
q t
C
dq t
dt
dq
C
dt
t
:
EE61 Electric Circuits, Ch1: Basic Principles
c
+
(a)
12
c
-
()
;
C
i t
(b)
Figure 7: Circuit symbol for a capacitor.
The energy stored in the capacitor is, to within a reference energy,
Z
(12)
()=
( ) = 21 2( ) = 12 2( )
For an ideal capacitor, without resistance in its leads or leakage across its dielectric, this
energy is stored reversibly, without loss.
w t
dt p t
C
q
t
v
t
C:
4.3 The Linear Inductor
Voltage is proportional to the derivative of the current:
()
()
v t
L
di t
dt
(13)
where is a constant of proportionality with the dimension in SI units of volt-seconds/ampere
or henries (H).
The circuit element with this property is called an inductor, with its own curly circuit
symbol.
L
c
+
(a)
-
()
i t
L
c
;
(b)
Figure 8: Circuit symbol for an inductor.
EE61 Electric Circuits, Ch1: Basic Principles
13
It is easy to verify that the ideal inductor is a linear circuit element:
()= () =
1( ) + 2( )] = 1( ) + 2 ( ) = 1( ) + 2( )
v t
L
di t
L
dt
d
dt
i
t
i
t
L
di
t
dt
L
di
t
v
dt
t
v
t :
(14)
If we substitute the inductor's voltage expression into the power expression, we obtain
( ) ( ) = 12
It follows from this expression and the power denition
()= ()
()= () ()=
p t
v t
i t
Li t
di t
dt
dw t
p t
dt
2
L
di
()
dt
t
:
(15)
(16)
that the energy stored in the inductor is, to within a reference energy,
( ) = 21 2( )
(17)
For an ideal inductor, without resistance in its windings, this energy is stored reversibly,
without loss.
w t
Li
t :
Resistors will be our principal concern for about the rst third of this course. We'll consider
inductors and capacitors when we take up Energy Storage elements and circuit dynamics.
5 Open and Short Circuits
Two limiting special cases of an ideal linear circuit element deserve separate discussion: the
open circuit and the short circuit.
5.1 Open Circuit
Think of an open circuit as two disconnected pieces of wire. Properties of an ideal open
circuit:
1. Zero current ow independent of voltage across the element.
2. Ability to withstand an arbitrary voltage, no matter how high, without breakdown.
3. - curve that is a vertical line with = 0 (innite slope).
V
I
i
Such a circuit element resembles a resistor with innite resistance or zero conductance.
EE61 Electric Circuits, Ch1: Basic Principles
c
+
(a)
-
()
i t
14
c
;
(b)
Figure 9: Representation of an open circuit.
c
+
(a)
-
()
i t
c
;
(b)
Figure 10: Representation of a short circuit.
EE61 Electric Circuits, Ch1: Basic Principles
5.2 Short Circuit
15
Think of a short circuit as a single piece of ideal wire. Properties of an ideal short circuit:
1. Ability to support an arbitrary current, no matter how high, without any voltage
drop.
2. Zero voltage dierence across the ends of the element, independent of the current
through the element.
3. - curve that is a horizontal line with = 0 (zero slope).
V
I
v
Such a circuit element resembles a resistor with zero resistance or innite conductance.
6 Switches
6.1 Ideal Switch
A switch is a device that makes or breaks a connection between two parts of a circuit, as
determined by some external actuator.
An ideal switch provides an ideal short circuit when the switch is closed, and an ideal open
circuit when the switch is open. The circuit symbol representing the (open) switch is shown
in Figure 11. This is a single-pole, single-throw switch. There are all sorts of complicated
Sw
z
z
Figure 11: Circuit symbol for a SPST switch.
ways to gang together switches, and to admit multiple contact points . . . limited only by
the ingenuity of mechanical designers. The simple SPST switch is su!cient for most of our
applications.
6.2 Practical Switches
Practical switches are subject to constraints of cost, size and materials. They dier from
ideal switches primarily in four ways: nite (instead of zero) resistance when closed, nite
EE61 Electric Circuits, Ch1: Basic Principles
16
(instead of innite) resistance when open, nite (instead of innite) breakdown voltage, and
nite (instead of zero) time to open or close.
Example: Household light switch.
This usually involves a pair of metallic contacts. When the switch is open, the contacts are
separated by an air gap. While there may be some electrical leakage along the surface of
the supporting material (especially if the surfaces are contaminated with ionic salts and the
humidity is high), the practical open-circuit resistance is high, approximately innite.
Likewise, there is good metallic contact when the switch is closed, so that the closed resistance
is usually much less than 1#. There is some resistance, however, so there is a rated current
that should not be exceeded if resistive temperature rise is not to be signicant. In a typical
wall light switch, that rated current is 15 A.
When the contact is open, stray inductance (more about that later) will cause an arc to form.
To avoid excessive arcing, and to ensure that the switch will perform well under anticipated
conditions of temperature and humidity, the switch has a maximum voltage rating as well
as a maximum current rating. For a typical home wall switch that rating is 250V(rms).6
Example: Semiconductor switch.
Semiconductor switches usually have a relatively low breakdown voltage (from a few volts to
approximately one hundred volts), a relatively low open resistance (hundreds of megohms),
a relatively high on resistance (one to one hundred ohms), and a nite switching time (microseconds to picoseconds). Compared to a wall light switch, the semiconductor switches
do not look very good, but that is an unfair comparison! They are designed for dierent
applications, and for their designed applications they are very eective and economical.
This is an example of an engineering tradeo: choosing designs that balance the requirements
of performance, size and cost.
7 Sources
Resistors, inductors and capacitors are passive circuit elements. Inductors and capacitors
temporarily store energy. Resistors (always) dissipate energy.7 None generates energy.
In order to have interesting electrical circuits, we need sources of electrical energy. We need
sources to represent external signals and to power useful loads.
For circuit analysis there are voltage sources and current sources, and there are two
kinds of each of these: independent sources and dependent sources. We restrict ourRatings for DC applications are sometimes dierent from those for AC applications. DC systems are
more prone to sustained arcing than AC systems.
7 To the extent that capacitors and inductors have resistive conductors and imperfect insulation, they also
dissipate energy. That is, their \round trip storage eciency" is always less than 100%.
6
EE61 Electric Circuits, Ch1: Basic Principles
17
selves initially to independent sources { that is, to sources whose characteristics are xed
by independent external factors. Later, we shall introduce dependent sources whose properties are determined by internal circuit variables. Dependent sources are useful for modeling
various kinds of active devices.
7.1 Ideal Independent Sources
There are two basic kinds of ideal independent sources: independent voltage sources and
independent current sources.
The circuit symbol for an independent voltage source is a circle containing polarity
indicators. This source is ideal in that the voltage source always has its specied terminal
c
+
(a)
-
()
i t
+ ;
v (t)
c
;
(b)
Figure 12: Circuit symbol for an independent voltage source.
voltage independent of the current owing through the source.
A special case of an independent voltage source is a battery. A battery is so common, and
so important, that (like the resistor, inductor and capacitor) it has its own unique circuit
symbol. It is an alternative form of ideal voltage source for xed (DC) voltages.
c
+
(a)
c
-
()
i t
B
V
;
(b)
Figure 13: Circuit symbol for a battery.
The circuit symbol for an ideal independent current source is a circle containing a
directional arrow. This source is ideal in that the current source always drives its specied
EE61 Electric Circuits, Ch1: Basic Principles
c
+
(a)
-
()
i t
-
i(t)
18
c
;
(b)
Figure 14: Circuit symbol for an independent curent source.
current independent of the voltage across the source.
What is the response of ideal sources to short and open circuits? For a voltage
source: Open circuit is OK the source maintains its prescribed voltage. Short circuit is a
disaster! The source, trying to maintain its output voltage, will increase the current through
the short to innity . . . in the expectation that, as in an ordinary resistor, the voltage will
rise to the dened value.
For a current source the situation is just the reverse: A short circuit is OK the source
maintains its prescribed current. Open circuit is a disaster! The source, trying to maintain
its output current, will increase the voltage across the open circuit to innity . . . in the
expectation that, as in an ordinary resistor, the current will rise to the dened value.
Water analogy, continued: In our water analogy, sources would be reservoirs. Voltage sources would correspond to reservoirs that always supplied the
same water pressure, no matter how much water was requested. Picture a water
source at the bottom of a very, very large reservoir. Provided the water level at
the top of the reservoir remains the same, the reservoir would supply the same
pressure no matter how large the tapping hole.
Current sources would correspond to water sources that always supplied the same
amount of water, no matter what the pressure, as from a xed-feed-rate pump.
If you wanted to receive the water in a large hose, the pressure might be quite
low, but if you tried to shut o the valve feeding the hose or if you shifted to a
smaller hose, the pressure would increase to whatever was required to keep the
same amount of water owing! (In real systems there is usually a relief valve
which shunts uid from the output back to the input, to ensure that the output
pressure does not become destructively high.)
7.2 Ideal Dependent Sources
There are four basic kinds of ideal dependent sources. We list them here for notational
completeness, but we postpone a detailed discussion to a later chapter.
EE61 Electric Circuits, Ch1: Basic Principles
19
In dependent sources the strength of the source is controlled by some other circuit voltage
or current. Thus, we can distinguish:
Voltage-controlled voltage sources: VCVS.
Current-controlled voltage sources: CCVS.
Voltage-controlled current sources: VCCS.
Current-controlled current sources: CCCS.
Ideal dependent sources appear on circuit diagrams as a diamond. The symbols for a voltagecontrolled voltage source (VCVS) and a voltage-controlled current source (VCCS) are shown
in Figure 15. The nature of the source (voltage or current) is indicated inside the diamond.
c
+
(a)
-
()
i t
;@@
;
+
@ ;;
@;
=
v
c
v
c
;
(b)
c
+
(a)
-
()
i t
;@;-@@;
@;
i
=
c
gv
a) Dependent voltage source: VCVS.
b) Dependent current source: VCCS.
Figure 15: Voltage-controlled dependent sources.
The control equation is indicated adjacent to the diamond symbol.
7.3 Practical Sources
Water analogy, continued: In certain circumstances, a real water source might
be best represented as a constant-pressure source or as a constant-volume source.
More likely, however, the real water source does not maintain its constant pressure
at very large volumes or constant volume through very small constrictions. The
reason is that water from real sources must ow through pipes of nite size and
that the available pressure is also limited. The same kind of limitations apply to
practical electric sources.
7.3.1 Practical Voltage Sources
To get insight, return again to the water analogy.
c
;
(b)
EE61 Electric Circuits, Ch1: Basic Principles
20
Water analogy, continued: In our water analogy, a constant-voltage source is
a constant pressure reservoir, supplying its water through a pipe. As the amount
of water requested increases, there will be a drop in perceived pressure due to
resistance in the ow of water through the pipes.
This suggests the equivalent circuit for practical voltage sources shown in Figure 16. The
s
s
cv
s
?
i
V
-
i
B B B B B
R
+
;
c
Figure 16: Equivalent circuit for a practical voltage source.
resistor s is said to be the source resistance.
What is the response of this practical source to short and open circuits?
If there is a short across the output { that is, a direct connection across the output {
the output voltage will be zero. All of the source voltage s will appear across the series
resistance s. By Ohm's law, the current through that resistor will be the short-circuit
current ss = s s.
If the output is open, then the open-circuit output current oc will be zero, the voltage
drop across the series resistance will be s oc = 0, and the voltage s will appear across the
terminals.
By measuring the open-circuit voltage and the short-circuit current, we can completely
characterize the practical voltage-source equivalent.
Example: 9 V Alkaline Electronics Battery
Open circuit voltage is 9 V, so s = 9 V. Short circuit current is about 3 A, so
R
V
R
i
V =R
i
R i
V
V
s=
R
s
isc
V
= 3#
:
This is a pretty good practical model of a that battery.
Notice that when the battery is short-circuited, the internal resistance will dissipate
a signicant amount of heat.
2
s
Rs
V
= 3W
EE61 Electric Circuits, Ch1: Basic Principles
21
7.3.2 Practical Current Sources
Again, let's start with our water analogy, which is here a little less clear.
Water analogy, continued: In our water analogy, a current source provides a
constant amount of water independent of the size of the receiving vessel. If the
receiver is a small pipe, the pressure across the pipe will increase, in principle to
innity. In practice, as the pressure at the source increases, water will begin to
ow through other pipes (or through leaks), limiting the pressure at the source,
and diverting water elsewhere.
This suggests the equivalent circuit for practical current sources shown in Figure 17. The
-
i
s
(;
)s
s
I
6
s
R
(+)s
cv
PPPP
PP
PPPP
s
c
Figure 17: Equivalent circuit for a practical current source.
resistor s is here a shunt resistance, rather than a series resistance as it was in the practical
voltage source. For reasons which may become clear later, it is also called the source
resistance.
What is the response of this practical source to short and open circuits?
If there is a short across the output { that is, a direct connection across the output { the
output voltage will be zero. The voltage across the shunt resistor s is also zero, so the
current owing in the shunt is zero. All of the current s ows through the short: ss = s.
If the output is open, then the output current will be zero: oc = 0. All of the source current
will ow through the shunt. By Ohm's law, the voltage across the shunt will be s s, which
will also be the open-circuit output voltage: oc = s s.
Again, by measuring the open-circuit voltage and the short-circuit current, we can completely
characterize this practical current-source equivalent.
R
R
I
i
i
R I
v
R I
I
EE61 Electric Circuits, Ch1: Basic Principles
7.4 Independent Sources in SPICE
22
Let's pause here and look at the following SPICE source code:8
First EE61 Spice Exercise, <your name here> <== First line, title (always)
R1 1 0 20
V1 1 0 DC 30
.OP
.END
<== Last line, always
This consists of ve lines.
In SPICE, the rst line is always a title line. It is used by the SPICE program as a label
and is otherwise ignored.
The last line is always the .END statement. Modern versions of SPICE are somewhat more
tolerant of a .END omission, but it is always a good idea to include it.
The lines in between describe the circuit and control the method of analysis. In this simple
circuit there is one resistor and one independent voltage source.
The simplied SPICE syntax for resistors and independent DC (or constant) sources is:
Rname
(node)
Vname
Iname
(+ node)
(+ node)
(node)
value
(- node) DC value
(- node) DC value
There is a separate line for each circuit element. Resistor lines begin with R, independent
voltage sources with V, and independent current sources with I. This initial symbol is immediately followed (without any space) by a suitable \name" indicator, to distinguish among
the various components of the same type.
In our simple circuit, there is only one resistor and one voltage source.
Following the name, after a space, are two numbers indicating how the circuit element is
connected. Circuit elements are connected at nodes. Nodes are regions of equipotential
connected to other nodes through circuit elements dierent from ideal conductors.9 Nodes
do not need to include \solder blobs", although they often do, but they also do include
the ideal conductors connecting elements to each other or to solder blobs. Every ideal wire
belongs to one, and only one, node.
In SPICE, nodes are always labeled by nonnegative integers. In this circuit there are only
two nodes, a reference node (0) and another node (1).
See Muhammad H. Rashid, SPICE for Circuits and Electronics Using PSpiece, 2nd.ed. (Prentice Hall,
1995).
9 Recall that voltage is related to the change of energy associated with the movement of charge. Charges
can be moved on an equipotential without a concomitant change in energy.
8
EE61 Electric Circuits, Ch1: Basic Principles
23
By convention, node zero (0) is always reserved for the reference node. All circuit voltages are measured relative to the reference node. We indicate the reference node in circuit
diagrams by the \ground" symbol.
Because the resistor is a bilateral (symmetric) element, it does not make any dierence
whether we use
R1 0 1 20
or
R1 1 0 20
to represent the resistor in our circuit.
By contrast, because each source has a particular polarity and/or current direction, it does
make a dierence how the node numbers are assigned to the source lines. All voltage and
current sources are represented in SPICE by a single consistent sign convention, the passiveelement sign convention we saw earlier, and repeated in Figure 18. Current enters the
c
+
(a)
-
()
i t
c
- ;
()
i t
(b)
Figure 18: Passive-element sign convention used in SPICE.
element at the positive (+) voltage terminal. The positive-voltage (+) terminal is always the
rst node listed, the (;) node the second tt DC tells PSpice which of several models to use
for the source, and for DC (or constant-value) sources the \value" can be either positive or
negative, as required by the circuit.
SPICE command lines begin with a period (.). The .OP command is described by Rashid
(ibid.). If it is omitted, PSpice outputs only the list of the node voltages. If .OP is present,
PSpice outputs the currents through each voltage source (with the standard passive sign
conventions of Figure 1) and the total power delivered by all voltage sources (only!). In a
circuit containing only voltage sources and linear resistors, that total power delivered by
the voltage sources will be a positive number equal to the total power absorbed by the
resistors. It is not necessary to separately output the voltage across current sources, since
those voltages can be inferred from the node voltages.
EE61 Electric Circuits, Ch1: Basic Principles
24
8 Kirchho's Laws (1847)
Our emphasis heretofore has been upon the individual elements from which an electrical
circuit is constructed. We have a collection of passive linear elements (resistors, inductors,
and capacitors) and a collection of ideal independent and dependent sources. We also know
the rules and sign conventions for energy and power ow into individual elements. In order
to analyze circuits, we need rules governing the ow of electricity among circuit elements.
These are provided by Kirchho's two laws: Kirchho's Current Law (KCL) and Kirchho's Voltage Law (KVL). These two laws, rst enunciated by the German scientist Gustav
Kirchho about 1847, follow from the conservation laws for electric charge and energy.
Before describing Kirchho's laws, we need to dene two circuit terms: nodes and loops.
Briey, a node is the electrical juncture of two or more circuit elements. To rst order, it is
the \solder blob" on the circuit diagram where the leads connect, but more generally it is the
whole equipotential net of ideal wires and ideal connections (\solder blobs") providing the
\connection" of circuit element ends.10 Three examples of a node are shown in Figures 19 21.
c
+
(a)
- BBB
()
i t
(c)
BBB
c
;
(b)
Figure 19: A node (c) connecting two resistors, without any explicit \solder blobs".
The concept of a loop is somewhat easier, at least in rst order. Briey, a loop is any closed
path in a circuit, where the path consists of circuit elements and their interconnection and
where the path does not cross any intermediate node more than once.
A mesh is a special case of a loop. A mesh is a loop that does not contain any other loops
within it.
We now have the denitions necessary to state Kirchho's two laws. The two laws are:
Kirchho's Current Law (KCL): The algebraic sum of the currents into a node at any
instant is zero. Equivalently: The sum of the currents owing into a node equals the sum of
the currents owing out. Equivalently: The algebraic sum of the currents owing into any
Recall that voltage is related to the potential energy dierence associated with the movement of charge.
On an eqipotential, charges can move without a change in potential energy.
10
EE61 Electric Circuits, Ch1: Basic Principles
c
+
(a)
(c)
- B B B PPs
PPPP
()
B B B
i t
25
c
;
(b)
Figure 20: A node (c) connecting three resistors.
c
+
(a)
(c)
s
s
- B B B PP PP
PPPP PPPP
()
i t
B B B ;c
(b)
Figure 21: A node (c) connecting four resistors.
EE61 Electric Circuits, Ch1: Basic Principles
26
section of a circuit is zero. Equivalently: The current owing in any leg of a circuit can be
represented as the algebraic sum of currents owing in closed loops.
Kirchho's Voltage Law (KVL): The algebraic sum of the voltages around any closed
path (loop) in a circuit is identically zero at all times. Equivalently: The voltage at any
point in a circuit is a single-valued function. Equivalently: Each node has a unique potential
(voltage) measured relative to a xed reference node.
These two laws, together with Ohm's law and its inductor and capacitor syblings, are the
basic laws of circuit theory. These laws are the basis for the general methods of circuit
analysis to be described in Chapter 3.
8.1 Kirchho's Current Law (KCL)
Kirchho's current law is a statement of charge conservation.
For a typical two-terminal circuit element, like that shown guratively in Figure 18, Kirchho's current law is that at each instant the current coming out one lead equals that going
in the other. This was explicit in Figure 18 where we showed the same current ( ) in both
input and output leads.
More generally, if the \pig tails" at one end of each of several circuit elements are electrically
connected together, as shown in Figure 22, the currents into the common connection sum
algebraically to zero. Equivalently, the sum of the currents into the connection equals the
i t
-s
i1
s
?
i2
i3
?
-
i4
Figure 22: Kirchho's current law:
i1
= 2 + 3 + 4.
i
i
i
sum of the currents coming out of the connection.
Notice that the \connection" in Figure 22 consists of two \solder blobs" (the solid circles)
plus ve ideal conductors. Both blobs and all of the wires are at the same electrcal potential
{ that is, they are at the same voltage. A node is an equipotential net of \perfect
conductors" and \solder blobs". It is NOT just the \solder blobs".
There are many topologically equivalent ways to redraw this node, making connections
EE61 Electric Circuits, Ch1: Basic Principles
27
through dierent numbers of \solder blobs", but all still represent only one \node". From
the point of view of circuit analysis, the important consideration is the identication and
numbering of nodes. Solder blobs have no relevance, except in so far as they indicate on a
circuit diagram that two wires connect.
Kirchho's current law applies to each and every node separately: the sum of the currents
into the node equal the sum of the currents out of that node. Charge does not
accumulate at any node.
More generally, if we draw an envelope around any part of a circuit, with the envelope
intersecting only conductors (and not cutting through circuit elements), KCL implies that
the sum of the currents into that circuit part equals the sum of the currents out of that circuit
part { that is, charge cannot accumulate in any part of a circuit.
As an exercise: Write the KCL equation for node (c) in each of Figures 19 - 21. Show
current labels and directional arrows on each circuit diagram.
8.2 Kirchho's Voltage Law (KVL)
Kirchho's voltage law is a statement of energy conservation. More precisely, it is a
statement that the electrostatic potential is a single-valued function.
Consider the subcircuit shown in Figure 23(a), with the four nodes numbered by the integers
1 to 4. These four circuit elements are connected together in such a way that they form a
2s
3s
(+)
s
v2
(;)
s
2
(+)
v1
(+)
1
3
(;)
1s
4s
(;)
s
(+)
a) KVL:
v12
v3
4
v4
s
(;)
b) KVL: ; 1 + 2 + 3 ; 4 = 0.
+ 23 + 34 + 41 = 0.
Figure 23: Alternate representations of Kirchho's voltage law.
v
v
v
v
v
v
v
\loop" or \mesh", with connections to other circuit elements suggested but not shown.
The important fact captured as Kichho's Voltage Law is that as one progresses around any
circuit loop, one returns to the same potential.
EE61 Electric Circuits, Ch1: Basic Principles
28
The four nodes at the corners of the loop in Figure 23 have voltages ( 1 2 3 4 ) relative
to some reference. In terms of these node voltages, the voltage dierences across each
element of the loop are
v v v v
=
23 =
34 =
41 =
;
2;
3;
4;
v12
v1
v2
v
v
v3
v
v
v
v
(18)
(19)
(20)
(21)
v4
v1
It follows trivially from this notation that the algebraic sum of the voltages around the closed
loop is zero:
v12
+
v23
+
v34
+
v41
= ( 1 ; 2) + ( 2 ; 3) + ( 3 ; 4) + ( 4 ; 1) = 0
v
v
v
v
v
v
v
v
:
(22)
KVL is more di!cult to use in practice than KCL because of sign problems. If we use the
following simple rules, KVL sign problems are minimized:
1. Assign voltage variables, with associated signs, to each circuit element { using the
passive sign convention, say, if there are also element currents assigned.
2. Start from one node on the loop, move around the loop in a single clockwise, or counterclockwise direction, adding or subtracting element voltages according to whether
the (+) or (;) sign of that element is rst encountered.
Consider the situation illustrated in Figure 23(b), which is a simple relabeling of Figure 23(a).
Proceeding from the node previously labeled 1 and moving clockwise around the loop, one
writes down KVL in the form
; 1+ 2+ 3; 4=0
v
v
v
v
:
(23)
Each voltage appears in the r.h.s. of equation (23) with the sign rst encountered in clockwise
progression around the loop.
Kirchho's voltage law applies to any closed loop in a circuit, even to loops that encircle
many smaller loops.
8.3 Maxwell's Equations
8.3.1 Kirchho's Current Law
Kirchho's circuit equations are the foundation of circuit analysis, and will serve us well in
this course, but they are not strictly true in all circumstances. To understand statement,
consider the the circuit symbol for a capacitor. As indicated in Figure 24, this is actually a
simplied physical representation of a capacitor: a pair of plates separated by a dielectric.
EE61 Electric Circuits, Ch1: Basic Principles
29
plane 1 plane 3 plane 2
c
+
(a)
-
()
i t
c
C
;
(b)
Figure 24: Physical representation of a capacitor.
By Kirchho's current law (KCL) the current out the right side of the element equals the
current into the left side. Equivalently, if we draw a plane through, and perpendicular to,
the left-hand lead and if we draw a similar plane through the right-hand lead, KCL implies
that the currents through the two planes will be equal.
So far so good. But, what if we draw one of the planes inside the capacitor. We know
that charge carriers do not actually cross the gap separating the two plates. Instead, charge
accumulates on one plate, and an equal and opposite charge accumulates on the other plate.
The current in the two leads will be the same everywhere, but clearly there is no \current
ow" between the plates.
Kirchho's current law states that current (charge) cannot accumulate. While this is true
for the device as a whole, it is not true \inside" the device itself.
To resolve this dilemma, Maxwell proposed in the mid 19th century that in addition to the
current carried by physical carriers, there was a displacement current proportional to the
rate of change of electric eld between the plates in a capacitor. If the \total current" was
the sum of the charge current plus this displacement current, than, Maxwell asserted, the
current crossing a plane would be a constant everywhere in a circular loop, including \inside"
devices like the capacitor.
This assertion immediately raised another question: Is the displacement current \real"? In
particular, we know from Ampere's law of magnetism that moving charges produce magnetic
elds. Do the displacement current produce magnetic elds?
It took many years to resolve this issue, but by the turn of the century the answer was
known: Yes! The displacement currents do produce magnetic elds, and their associated
magnetic elds are essential for radio waves.
8.3.2 Kirchho's Voltage Law
So much for Kirchho's current law. There is a somewhat similar deciency in Kirchho's
voltage law (KVL). Implicit in KVL is the requirement that the potential be a single valued
EE61 Electric Circuits, Ch1: Basic Principles
30
function. This is true in electrostatics, but it is not true in the presence of time-varying
magnetic elds. The so-called Faraday law of magnetic induction, which is also essential for
radio waves and which underlies electric transformer technology, induces electric eects that
cannot be described by a single-valued electrostatic potential.
8.3.3 Maxwell's Equations, Special Relativity
All of these eects were captured late in the 19th century by Maxwell in his famous four
equations of electromagnetics. When Maxwell proposed his equations, they seemed very
mysterious, and their validity was doubted. All doubts were laid to rest in the early 20th
century when Einstein showed that they followed from, and were consistent with, the special
theory of relativity. This agreement served to conrm both Maxwell's and Einstein's results.
Maxwell's equations were a triumph of 19th century science. Formulated well before the
discovery of relativity and quantum mechanics, they have been shown over the course of the
intervening century to be fully consistent with modern knowledge of electromagnetics, fully
relativistic, and fully consistent with quantum mechanics.
Maxwell's equations are studied in depth in EE170.
What about Kirchho's laws? Are they still valid? Yes, they are, for the conditions that
prevail in circuits involving lumped circuit elements, provided one does not look too
closely at the insides of those elements and provided that distances inside circuits are not
so large (or times so short) that one needs to worry about the velocity of light. In short,
Kirchho's laws are accurate and applicable to the vast majority of electric circuits.
9 Ideal Ammeters and Voltmeters
Kirchho's current and voltage laws are fundamental to circuit analysis. This knowledge
suggests that, if we wish to measure currents and voltages within circuits without disturbing
those circuits, we should strive to use instruments (ammeters for measuring currents and
voltmeters for measuring potential dierences) that do not signicantly change the currents
owing into nodes (KCL) or the voltage dierences around loops (KVL). Ideal meters
measure without changing or disturbing anything about the system being measured.
9.1 Ideal Ammeters and Voltmeters
Current and voltage are, respectively, through variables and across variables. In order to
measure current, we must ow the current through the measuring device { that is, we
must put the meter in series with other elements in the circuit \leg" whose current we wish
to measure. If the current measuring device is ideal, the meter will not change the current
ow nor change any voltage dierences around any loop containing that leg { that as, there
EE61 Electric Circuits, Ch1: Basic Principles
31
will be no voltage drop across an ideal ammeter. A concomitant is that the ideal ammeter
will not dissipate energy.11 Real ammeters have nite, but very small voltage drops and do
dissipate energy.
In order to measure voltage, we must place the measuring device across the circuit
element. If the voltage measuring device is ideal, circuit voltages will not be perturbed
and current will not be diverted to ow through the meter. Current ows into the nodes at
the two ends of the voltmeter will be undisturbed. No current will ow into or out of any
node through an ideal voltmeter. No power will be dissipated in an ideal voltmeter. Real
voltmeters have nite, but very small current ows and do dissipate energy.
Care must be taken to ensure that the properties of real meters are suited to the circuit
being measured. Also, care must be taken to operate meters within their design range, so
that the readings will be accurate . . . and, equally important, the meter will not be damaged
by voltage breakdown or overheating.
9.2 Wire \Equivalents" to Ideal Meters
Current ows through an ideal ammeter without any potential dierence across the meter.
What is the simplest \circuit element" that would permit current ow without voltage drop?
A \perfect wire" without resistance. When connected between two points, this wire would
c
+
(a)
c
-
;
()
(b)
i t
Figure 25: Circuit equivalent to an ideal ammeter:
v
ab (t) = 0
for all ( ).
i t
be a short circuit.
An ideal voltmeter measures a potential dierence (or voltage dierence) with zero current
ow through the meter. What is the simplest \circuit element" that would sustain voltage
without current ow? A wire with a gap along its length. When connected between to
points, this circuit element would be an open circuit.
The power dissipated in the meter equals the product of the current through the meter times the voltage
across the meter. If the voltage across an ideal ammeter is zero, than that power will be zero. Similarly, if
the current through an ideal voltmeter is zero, then the power dissipated by an ideal voltmeter will also be
zero.
11
EE61 Electric Circuits, Ch1: Basic Principles
c
+
(a)
32
c
-
;
()
(b)
i t
Figure 26: Circuit equivalent to an ideal voltmeter: ( ) = 0 for all
i t
v
ab (t).
An open circuit is not usually a disaster, but it certainly would change the circuit if it were
substituted in series with the circuit element under investigation.
Laboratory Caution: One of your laboratory exercises will be to measure current and
voltage. The lab manual has been somewhat confusing in the past, causing some students
to connect ammeters across circuit elements, with reworks. Fortunately, the lab ammeters
are protected with fuses, so the results are not truly disastrous.
10 Equivalent Resistance
The analysis, design and manufacture of a circuit can sometimes be simplied by replacing
part of a circuit with one that is equivalent but simpler. This is more often feasible with
resistor circuits than with circuits containing resistors, inductors and capacitors. Let's see
how it goes with resistor circuits.
Two circuits are said to be equivalent if they have identical - characteristics at a specied
pair of terminals. Given a pair of terminals, what we seek is a black box with two terminals
which cannot be distinguished through voltage and current measurements made only at the
two pairs of terminals.
That equivalent circuits exist is perhaps surprising, but what we shall nd is the even more
startling fact that for any specied pair of terminals in a resistor network there always exists
an equivalent circuit of a very simple form. There are two such forms, called Thevenin equivalent and Norton equivalent circuits. We shall study them in a later session.
For the moment, consider two particular examples: a set of parallel-connected resistors and a
set of series-connected resistors. They relate respectively to what is called current division
and voltage division.
i v
EE61 Electric Circuits, Ch1: Basic Principles
10.1 Current Division, Parallel Resistors
33
Current division follows as an application of KCL. Consider the following simple circuit of
a current source driving a set of parallel-connected resistors. How many nodes are there in
PPPP?
PP
1 PPPP
PP ?
P
2 PP
i1
?s
?
i
I
s s
(+)
R
i2
R
i
R
s s
(;)
N
PP ?
NP
PP
Figure 27: Current source driving parallel-connected resistors.
this circuit? (Two.)
Let us choose the lower node to be the reference node (ground symbol), with the voltage
( ) of the upper node measured relative to the reference node.
Kirchho's current law (KCL) for this circuit becomes (with s = )
0 = + R1 + R2 + + RN
(24)
However, if we assume that the voltage ( ) is known, the individual resistor currents are
given by Ohm's law:
()
(25)
j=
v t
i
I
i
i
i
I
:
v t
v t
i
j
R
:
Thus, we can rewrite KCL in the form
1
(
)
(
)
(
)
1
1
0= +
+
++
= +
+ ++
()
I
v t
v t
R1
R2
v t
I
N
R
R1
R2
N
R
v t :
(26)
Let us ask the question whether all of these parallel resistors can, from the perspective of
the voltage source, be replaced by an equivalent single resistor. For this simple circuit KCL
and Ohm's law give
(27)
0= + 1 ()
eq
It follows that the \equivalent resistance" is dened by the equation
1 = 1 + 1 ++ 1
(28)
eq
1
2
N
For the special case of only two resistors this gives
I
R
R
eq =
R
v t :
R
R
R
R1 R2
R1
+
R2
:
:
(29)
EE61 Electric Circuits, Ch1: Basic Principles
(+)
R
PPPP?
PPPP
eq PP
i
?s
?
i
I
34
(+)
6
()
R
s
(;)
v t
(;)
Figure 28: Equivalent circuit.
I recommend that you memorize this equation. It is simple, and it is very useful.
For the special case of two identical resistors in parallel, this expression gives
1
eq = + = 2
RR
R
R
R:
R
(30)
The equivalent resistance produces the same voltage ( ), for the specied source current
, so it follows that the power supplied by the source is equal in the actual and equivalent
circuits. You can show that the total power dissipated in the two circuits is also the same.
As an exercise, you might show this for identical resistances in parallel.
v t
I
N
R
10.2 Voltage Division, Series Resistors
Voltage division follows as an application of Kirchho's voltage law (KVL). Consider the
following simple circuit of a voltage source driving a set of series-connected resistors. What
are the nodes in this circuit? (Remember that the nodes are the equipotential interconnections between elements.) There are nodes plus the reference node, which we again take
to be at the bottom of the diagram (ground symbol),
Kirchho's current law for this circuit is equivalent to the statement that the same current
( ) ows clockwise around the whole loop (or closed path). (Remember that our choice of a
clockwise ( ) is arbitrary. If ( ) should prove to be a positive number, then the real current
does indeed ow clockwise. But, if ( ) should prove to be negative, the real current ows
counterclockwise.) For this denition of ( ) it follows that
N
i t
i t
i t
i t
i t
R (t) = i(t)
i
s (t) = ;i(t):
and
(31)
i
If we dene the voltage at the topmost node to be ( ), then our ( ) and ( ) choices
correspond to the standard energy absorbing convention for the stack of resistors: current
into the stack at the positive-voltage terminal. For this denition of ( ) we also have
v t
i t
v t
v t
()=
v t
V
(32)
EE61 Electric Circuits, Ch1: Basic Principles
(+)
()
i t
?s
-
PPPP R
PP ?
PPPP
PP
i
R1
i
V
35
+
;
R2
N
R
(;)
PPPP
PP
s
Figure 29: Voltage source driving series-connected resistors.
the source voltage.
Now, what is the voltage across each resistor? . . . what is the voltage (relative to the reference) of each node in the resistor stack? . . . and what is the equivalent resistance for the
series-connected stack of resistors?
If Rn is the voltage across resistor , with each resistor voltage measured according
to the standard convention, with (+) sign at the top where the current ( ) enters
the resistor,12 then it follows from Kirchho's Voltage Law (KVL) that
v
n
i t
0=
R
v 1
+
R
v 2
++
RN
v
;
V
(33)
or, equivalently,
= R1 + R2 + + RN
(34)
As a memory device to help us get the sign right (always a problem with KVL!), we used
in deriving the rst form of this equation the standard convention: assign to each element
voltage the voltage sign on the terminal rst encountered as we circulate around the closed
path. If voltages are not labeled, but the current is, we implicitly use the standard passiveelement sign convention (current enters at the positive voltage terminal), so that Ohm's law
takes the familiar form.13
Using Ohm's law, we write the resistor voltages
V
v
v
Rn (t) = i(t) Rn :
v
v
:
(35)
It is not important for a resistor how we dene the current i(t). But, once the current is dened, it is
important to identify resistor voltages consistent with the passive-element sign convention.
13 To derive our result (33), we started at the node at the top of the circuit of Figure 29 and progressed
clockwise, encountering rst the positive terminal of R1 and last the negative terminal of the voltage source.
12
EE61 Electric Circuits, Ch1: Basic Principles
36
Substituting these expressions into the KVL expression, we have
V
= ( ) 1 + ( ) 2 ++ ( )
i t R
i t R
and
i t R
()=
= ( )( 1 +
N
i t
R
R2
++
N)
R
(36)
(37)
+ 2 ++ N
From this and Ohm's law it follows that the voltages across the individual resistors are
i t
v
Rn (t) =
V
R1
R1
R
:
R
n
R
+
R2
++
N
R
(38)
V:
If we let j ( ) be the voltage of the node immediately above the resistor
from this result and KVL that
v
t
() =
2( ) =
v1 t
v
t
=
then it follows
V
R
N (t)
()=
2 + 3 ++ N
1 + 2 + 3 ++
v t
j,
R
R
R
R
R
R
R
N
R
N
V
+ 2 + 3 ++ N
so that the stack of series-connected resistors acts as a voltage divider.14
Finally, what is the equivalent resistance for the series-connected resistors? It follows immev
(+)
s
?
+
;
i
V
R1
R
()
i t
R
R
-
R
PPPP?
PPPP
eq PP
i
V
(+)
6
()
R
s
(;)
v t
(;)
Figure 30: Equivalent circuit.
diately from the expression for the loop current ( ) that
i t
R
14
eq =
R1
+
R2
++
N:
R
The parallel-connected resistors are sometimes called a current divider.
(39)
EE61 Electric Circuits, Ch1: Basic Principles
s
-
i
B B B B B
cv
s
?
i
s
(;
)s
R
+
s
;
V
s
I
6
R
s
(+)s
c
-
i
37
cv
PP
PP
P
s
c
a) Practical voltage source.
b) Practical current source.
Figure 31: Series-connected and parallel-connected soruces.
11 Source Equivalents
11.1 Thevenin and Norton Equivalents
In the preceding section we derived a procedure by which we might replace parallel-connected
and series-connected resistors by equivalent resistors. In a later section we shall use these
resistor equivalents to simplify the analysis of resistor networks.
We can use these techniques to greater advantage if we have a procedure by which we might
replace a series-connected voltage source by an equivalent parallel-connected current source,
and vice versa. Again, we use the principle of terminal equivalence enunciated earlier:
two circuits are equivalent at a pair of specied terminals if the - characteristics at the
specied terminals are identical.
There is no requirement that circuits equivalent at a specied pair of terminals be equivalent
at other terminals . . . or even that they have the same number of terminals.
Consider the two circuits shown in Figure 31. You will recognize them as the circuit models
of \practical" voltage and current sources proposed earlier.
Assume that these two circuits are each enclosed in a \black box" with only their two
terminals exposed. Is it possible to choose the component values such that the terminal
characteristics of these two sources are indistinguishable? By terminal characteristics, we
mean for our resistor-circuit applications the current-voltage relationship of the the two
circuits.
When we considered these circuits before, as models of practical sources, we looked at the
open-circuit voltage and the short-circuit current. Let's do that again. The results are shown
in Table 3. We also consider the eect of an arbitrary \load" resistor hung across the
source terminals. Comparing the two columns of this table, we see that the two sources will
be equivalent, within the meaning of our terminal-equivalence convention, if s is the same
in both circuits and s = s s.
It follows that we may, for whatever useful reason, use either of the two source forms in our
i v
R
R
V
I R
EE61 Electric Circuits, Ch1: Basic Principles
Open-Circuit Voltage
Short-Circuit Current
Load- Voltage
Load- Current
38
Series Connected Circuit Parallel Connected Circuit
s
V
s
s
V =R
s ( + s)
s ( + s)
R
RV = R
R
V = R
s s
Is
R I
R
R
s
s
( + s)
( + s)
I RR = R
s s
I R = R
R
R
Table 3: Terminal characteristics of the source circuits of Figure 31.
circuits. In particular, when analyzing circuit containing one of the forms of Figure 31, we
may substitute the other form if doing so simplies our analysis. (We give an example in the
next section.) In some cases we may need to switch back later to determine voltages and
currents in the devices in the original circuit. For example, if we are given the series form
of Figure 31(a) and we wish to know the current through the voltage source, we will need
to convert any answers based upon the equivalent source conguration 31(b) to the required
form. This is usually not a di!cult step.
11.2 Sources in Series and Parallel
11.2.1 Additive Sources
This includes voltage sources in series and current sources in parallel. These are
equivalent, respectively, to voltage sources and current sources with values equal to the sum
of the separate source values.
11.2.2 Dominating Sources
These include a current source in series with a voltage source and a voltage source in parallel
with a current source. They also include certain resistor/source combinations: a resistor in
series with a current source and a resistor in parallel with a voltage source.
When a current source is in series with either a voltage source or a resistor, or both, the
current source dominates. The combination is equivalent to a single current source, with the
source value equal to the original current value. Any voltage drops across resistors or
voltage sources is compensated by voltage drops across the current source, because an ideal
current source can provide whatever voltage is needed to ensure that the specied source
current ows.15
When a voltage source is in parallel with either a current source or a resistor, or both, the
volteage source dominates. The combination is equivalent to a single voltage source, with
The voltage across the original current source diers from that across the equivalent current source by
the voltage drop across the original-circuit resistor and voltage source.
15
EE61 Electric Circuits, Ch1: Basic Principles
39
the source value equal to the original voltage value. Any currents demanded by the
resistor or the current source is compensated by current provided by the voltage source,
because an ideal voltage source can provide whatever current is needed to ensure that the
specied voltage is maintained.16
11.2.3 Trouble!
When two ideal current sources of unequal value are in series, there is trouble. Each tries
to establish its own current ow (by varying its voltage), but because the two target values
are unequal, the competition is doomed to failure.
Similarly, two ideal voltage sources of unequal value in parallel spell trouble. Each tries to
establish its own voltage value (by varying its current), but because the two target values
are unequal, the competition is doomed to failure.
In principle, current sources in series and voltage sources in parallel can succeed when the
target values are equal, but stability is illusory because small changes in any real system
will cause the target values to become unequal. Stabilizing systems like this is a practical
engineering problem, since series and parallel systems can be necessary when a single system
is incapable of delivering su!cient power by itself.
To stabilize such systems, the designer must usually include shunt and series impedances
{ in series-current and parallel-voltage source sytems, respectively { so that the subcomponent sources are no longer strictly ideal. Equivalently, the control systems can actively
monitor power output from each subsystem and provide adjustments to bring the systems
into balance. How this is done is beyond the scope of this course.
12 Circuit Reduction
For certain not-too-complex circuits for which manual methods of analysis are su!cient,
one can sometimes use the methods of equivalent resistors and equivalent sources to
simplify the circuit analysis. For complex, multicomponent circuits SPICE-based computer
methods are required, and there is little to be gained from prior circuit reduction.
Consider the circuit shown in Figure 32. Here one replaces each of the two sets of parallel
resistors with its equivalent resistance. The result is a simple loop circuit with two series
resistors and one voltage source. (Do in class for 1 = = 4 = 2#.)
Consider the circuit shown in Figure 33. Here one can convert one of the sources to the
equivalent other form, and then combine resistors and sources easily.
One can use reduction methods on parts of networks. Consider the resistor network shown
in Figure 34. This falls readily to circuit-reduction simplication. The key is to use a neat,
R
16
R
The current through the original voltage source is dierent from that through the equivalent voltage
EE61 Electric Circuits, Ch1: Basic Principles
a
s
?
P
P
P
P
P
P
1 P
P
P
P
PP ?
P
2 PP
i1
i2
vs
b
s
+ ;
-
PP ?
P
3 PP
s
i
R
R
PPPP?
PP
4 PPPP
i3
i4
R
R
s
s
40
s
Figure 32: Circuit with four resistors and a voltage source.
1
B B B B B i
v
s
PPPP?
PP
2 PPPP
i2
R1
+
V1
;
R
6
I2
s
Figure 33: Circuit with two resistors, a voltage source and a current cource.
c
c
1#
B B B
s
1#
B B B
P
P
P
P
P
P
2#
s
s
PPPP
PP 2#
s
1#
B B B
s
PPPP
PP 2#
s
Figure 34: Array of resistors.
PPPP
PP 2#
EE61 Electric Circuits, Ch1: Basic Principles
41
systematic method to keep track of successive reductions.
Here, start from the right-hand side of the circuit. Draw successive vertical brackets to mark
and digest successive \bites".
Finally, consider a repetitive network that is innitely long. Consider the circuit shown in
Figure 35. Here the key is to recognize that, if one makes a vertical cut through the network
c
1#
1#
s
B B B
P
P
P
2#
c
1#
s
B B B
s
B B B
PP
P 2#
s
:::
PP
P 2#
s
s
:::
Figure 35: Repetitive innite array of resistors.
at any stage, the network to the right of the cut is identical to the whole network . . . and
that equivalent network can be replaced by the (unknown, as yet) equivalent resistance eq .
Using this resistance, one can proceed to set up an equations that, when solved, will yield
the unknown eq . The equations for the example of Figure 35 are:
2 eq
eq = 1 +
2 + eq
R
R
R
R
2
or
Req
+
2
Req
R
=2+
2
Req
Using the quadratic formula, we obtain
;
Req
Req
+2
Req
;2=0
=2+3
Req
:
p
1 1 + 8 = 1 3 = 2 or ; 1
eq =
2
2
Because eq must be non-negative, it follows that eq = 2. The plausibility of this result
can be established by solving the system by successive approximation.
It is easy to generalize these results into an innite geometric progression of resistors, as
illustrated in Figure 36. The starting equation governing the analysis of this array becomes
2 eq
eq = 1 +
2 + eq
The positive solution is
p4 ; 4 + 9 2
;
2
+
3
+
eq =
2
R
R
R
R
R
R
R
:
EE61 Electric Circuits, Ch1: Basic Principles
c
1#
s
B B B
1 #
B B B
P
P
P
2#
c
1 2#
s
s
B B B
PP
P 2 #
:::
PP 2
P 2 #
s
42
s
s
:::
Figure 36: Repetitive innite geometric progression of resistors,
>
0.
Equivalent Resistance, Geometric Array
3
2.5
2
1.5
1
0.5
00
2
4
6
8
10
gamma
Figure 37: Equivalent resistance for geometric array as function of .
EE61 Electric Circuits, Ch1: Basic Principles
43
This solution is plotted as a function of in Figure 37. When = 0, the rst 2-# resistor
is eectively shorted by the subsequent array, and the equivalent resistance is eq = 1 #.
When = 1, the array after the rst 2-# resistor is eectively an open circuit, and the
equivalent resistance is eq = 3 #. These limits are evident in the gure.
R
R
WARNING: These circuits were selected to be amenable to circuit reduction. Not all
circuits are signicantly simplied by circuit reduction. Sometimes it is useful. Sometimes
it is more trouble than it is worth. As you gain experience, you will be able to make that
judgement call. As you begin to tackle more di!cult problems, you will often punt and go
straight to SPICE!
/ee61/ch1.tex
source by the current ow through the resistor and current source.