Selected Linear Electrical Circuit Elements
Formalities (review)
KVL and KCL are physical laws generally applicable to the analysis of an electrical circuit without
regard to the specific components with which the circuit is assembled. Further KVL, representing an
expression of Conservation of Energy, is a relationship between circuit voltages independent of the
circuit currents. Similarly KCL, representing an expression of Conservation of Charge provides
relationships between circuit currents independent of the circuit voltages. The missing link between the
two is a means of relating currents to voltages, and this is provided by the properties of the specific
circuit elements constituting the circuit.
What distinguishes one electrical circuit with a specific topology from another with the same topology,
what gives each circuit a special character of its own, are the specific electrical elements which are
connected together to form the circuit. Electrical circuits of the sort considered in this course involve
what are described colloquially as 'lumped' circuit elements. 'Lumping' refers to the manner in which
the electrical properties of the element are described, rather than a property of a circuit element. The
details of the internal physical phenomena of an electrical circuit component are suppressed in favor of
an overall terminal description. The component may be imagined to be totally enclosed in a container
(shape and dimensions are otherwise not pertinent considerations) with electrical terminals brought out
as the only available means of access to the device. The component circuit properties then are described
entirely in terms of quantities that can be measured at or between terminals, i.e., without need to go
inside the box directly or indirectly.
There are just two basic electrical terminal parameters for the circuit description of the behavior of a
circuit element; the voltage difference between the terminals (work involved in the transport of a unit of
charge between the terminals independent of the path between terminals), and the current into one
terminal and (as required by KCL) an equal current out of the other terminal. The electrical properties
of the device, the collective consequences of whatever phenomena are occurring inside the device
container, are expressed through the relationship they impose between the terminal voltage and current
parameters. However, before actually specifying the terminal volt-ampere relation for various circuit
components it is useful first to confirm the conventions to be used to express that specification; we have
to be sure we understand the language to be used. Actually this protocol consists largely of what we
have already discussed. But it is all too often misunderstood and misused, and so fundamental a
misunderstanding clouds subsequent discussion quite unnecessarily. Hence there is some value to
repeating at least part of material discussed previously.
The voltage difference between two terminals, for example, has two
attributes; a magnitude (how much work is involved) and a polarity (work
either is done by or it is done by the circuit element (i.e., on the current
through the element during the charge transport). The figure to the left
represents a circuit element; the amorphous geometry is intended to
emphasize the unimportance of internal details insofar as a lumped element
description is involved. Indeed for the immediate purpose the exact nature
of the circuit element is not germane. This is a two-terminal element; two is
the minimum number of terminals needed to enable KCL to be satisfied.
Multi-terminal elements will be introduced in due course. The character V is used as an algebraic
symbol representing the voltage difference between the terminals. The distinctive symbols + and - are
used to differentiate the two terminals, and may be assigned arbitrarily without qualification.
This arbitrariness of assignment is a matter of some importance that is misused all too often. While in
specific instances which terminal is assigned which symbol might affect convenience, esthetics, or
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psyche, it has no formal analytical consequence at all. The sole formal purpose for the assignment is to
be able to distinguish in simple terms the process of moving charge from the + end to the - end, from the
opposite transport from - to +. It is the fact that the symbols are different that is what is essentially
important. And, of course, it is obviously important that all parties collectively considering a particular
circuit agree on a common assignment is so they can communicate with each other. In practice the
polarity assignment often is made by a device manufacturer, e.g., this is done for a car battery and
coordinated with terminal polarity assignments for battery chargers. What this means in a practical
sense are that the manufacturer's specifications describing the properties of the battery use that
assignment. If you really want to be an iconoclast reverse the assignment; just make sure that your
assignment is known to whoever reads your specifications! But aside from tangible benefits of adhering
to common conventions and industry standards, the specification of + and – generally is formally
arbitrary.
It is conventional (but with occasional exception) to reserve the ± symbols for use only for voltage
descriptions. Conventional jargon defines a voltage ‘drop’ as the voltage difference in moving from +
to -, and the voltage ‘rise’ the difference going the other way. Note these terms have nothing to do
directly with the actual voltage difference between the terminals, and it is not necessary even to know
what the actual voltages are to apply them. They simply are colloquial substitutes for "…from + to …" or conversely for "…from - to +…".
Some more jargon to consider. Arithmetic ± signs often are used as prefixes for a voltage specification.
(The arithmetic ± signs while of course identical in appearance to the ± polarity markings but have quite
different meaning. The meaning of the latter lies entirely in the fact that the symbols are distinguishably
different, while the former refers to arithmetic properties.) Suppose for example the voltage
specification for the circuit element whose polarity conventions are understood to be those shown in the
figure above is a voltage drop of +10 volts. This means that the magnitude of the voltage difference is
10 volts, and that the difference is to be taken from the '+' terminal to the '-' terminal of the element. An
explicit '+' sign usually is omitted; if the sign is not explicitly '-' then it is to be presumed to be '+'. On
the other hand suppose the specification were, say, a voltage drop of -12 volts. The magnitude of the
voltage difference again is 12 volts. The '-' sign means effectively to take the difference using the
reverse of the direction indicated by the use of the word 'drop', i.e., go from '-' to '+'. Of course this
could have been expressed directly as a voltage rise of +12 volts.
These peculiarities of notation arise not infrequently spontaneously in electrical circuit analysis where
the actual (physical) polarity is not known initially. The important thing is not to confuse what you are
describing with the terms in which you describe it. The name of a thing is different from the thing itself.
The physical voltage difference between two terminals is not the same thing as the conventions with
which you describe that voltage difference.
Now consider specifying the current, represented by the algebraic symbol I, flowing through the circuit
element. The current also has two attributes; a magnitude (how much current is involved) and a polarity
(the direction of charge transport between the two terminals).
First I is assigned a magnitude equal to the amount of current, for example 10 amperes. What is needed
in addition is a convention for specifying the polarity of the current flow. There are only two
possibilities, and to distinguish them it is common (but not universal) to use the arrow symbol. The
direction in which the arrow points can be chosen arbitrarily, independently of the polarity choice made
for the voltage difference. After all, the arrow does not indicate the actual direction of current flow. It
only serves to distinguish one possible direction of flow between terminals from the other.
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To indicate direction in the current specification use is made of the arithmetic ± prefix convention
similar to the use with voltage. Thus a '+' prefix (which conventionally is omitted) means the magnitude
of the flow applies to a movement in the same direction as the arrow. The '-' sign means the magnitude
of the flow is applied to a movement in the direction opposite to the arrow. It may be intellectually tidy
and aesthetically pleasing to have the actual (physical) current flow in the direction of the arrow, and in
fact if the current direction is known beforehand most people probably would make just that choice to
avoid the '-' sign. But the description convention has to allow for situations in which the current
direction is not known initially, and an algebraic variable whose value is unknown initially is assigned
for a circuit analysis.
Never assume the arrow points the direction of actual current flow before you know the sign of the
current specification.
Formalities (extended)
We have reviewed conventions for describing voltage, and conventions for describing current. Now we
return to the original objective of describing the electrical properties of circuit elements. What circuit
elements 'do' that is different from KVL (which relates voltages to each other) and KCL (which relates
currents to each other) is to relate the voltage across to the current through the element, i.e., a voltage to
a current. The overall effect of the physical processes 'inside' the circuit element is summed up in a
'terminal volt-ampere relation' (or 'constitutive' relation), i.e., an explicit relationship between the current
through and the voltage across the element.
As has been noted several times the ± voltage assignment and the current arrow assignment formally can
be made independently of one another. However because the terminal voltage and the terminal current
for an element are related, i.e., the physical properties of the element, are described by that relationship,
it should not be too surprising that certain advantages can accrue by using freedom
in making assignments to make convenient assignments. Since the ± and arrow
polarity assignments separately are arbitrary any convenience must lie in the relative
choice of assignment. Thus in describing the terminal volt-ampere relation for a
two-terminal circuit element the current arrow by convention will be drawn from
+ to -, as indicated in the figure to the right. The advantage of this convention is
that the product of the voltage and current indicates directly whether power is
consumed (positive product) or generated (product negative). It is not necessary to
repeatedly review the details of the physics for each device considered to renew this conclusion
Resistor
We have not actually described the terminal properties of any circuit element as yet, just the manner in
which the description will be made. We rectify the omission now; that is we specify the terminal voltampere relation of a ‘resistor’. More precisely we define an idealized (theoretical) resistor, and not any
of the several commercial devices called 'resistors'. The latter are commercial approximations to the
properties of the idealized resistor, but in certain circumstances in which they may be used they can have
significantly different characteristics. A more detailed description is needed in the latter
case, consisting of a combination of idealized elements. This 'modeling' of real devices
as combinations of idealized elements is an important consideration in more advanced
work. First we deal with the abstract idealized elements.
The idealized resistor is an abstraction of the property of energy dissipation. In addition
to non-distinctive labeled shapes sometimes used in circuit diagrams ( e.g., a rectangle
enclosing the word 'resistor') the idealized resistor has been a special distinguishing icon
as illustrated to the left. The icon is symbolic of a geometrical arrangement which fits a long length of
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wire in a small distance, an arrangement still used in the construction of resistors for certain
applications.
Note particularly the assignment of the polarity indicators, because they have not been made entirely
arbitrarily here. Instead the relative assignment as described before is used here, i.e., the current arrow
is chosen specifically to point in the direction from '+' to '-'. This convention is assumed more or less
universally in describing the properties of a resistor.
The volt-ampere relation for the idealized resistor, using the polarity conventions shown in the figure, is
V = RI
where V is the voltage drop across the resistor, I is the current through the resistor, and R is a constant of
proportionality called the ‘resistance’ (and is a property of the resistor composition and geometry).
Because of the polarity convention chosen R is a positive constant (with units of ohms if the voltage is
expressed in volts and current in amperes). If for some reason you reverse one or the other of the
polarity assignments, i.e., change the relative relationship; R must be made negative to correctly
describe the physical properties of the resistor. Incidentally it doesn't matter whether the resistor
drawing is turned upside down or not, i.e., whether the arrow points up or down (or sideways for that
matter). The properties of the resistor don't depend on which way you hold the paper. What is
important for the validity of V = RI with R a positive constant is the relative relationship between the
± assignment and the arrow direction.
The V = RI relationship (together with the polarity convention) is called Ohm's Law. It differs from
KVL or KCL in that these latter laws apply generally, without regard to the specific electrical devices
involved. KVL describes a relationship among voltages and KCL describes a relationship among
currents. Ohm's Law, on the other hand, describes a relationship between a voltage and a current
imposed by specific electrical phenomena lumped within a package accessible at a pair of terminals.
Suppose, for example, the current were 10 amperes and actually to flow in the direction opposite to the
arrow. Then I would have the value -10 amperes. The resistor property then requires the voltage drop
across the resistor to be -10 volts, meaning that the physical phenomena inside the resistor will conspire
to make this so. (Keep in mind the implicit assumption about the relative polarity assignments; -10
amperes and - 10 volts mean absolutely nothing without the polarity references.)
The power associated with a current I flowing through the resistor is, by definition of power, IV = I(IR)
= I2R. The square of the current is positive whether current actually flows in the direction of the arrow
or not, i.e. whether or not I is positive. Thus resistor power consumed always is positive; the current
flowing through a resistor always does work, so energy is continually consumed.
Incidentally, in a way resistors have a quite undeserved tarnished reputation; energy is not necessarily
wasted in a resistor, only consumed. The value judgment as to whether the energy expenditure is wasted
or not is a separate consideration. A resistor can represent the expenditure of energy by a motor under
load, or a speaker playing music, or other things sometimes judged desirable.
Commercial circuit elements called resistors are available in a variety of forms and varying
specifications. The reason for the diversity is that commercial resistors are approximations to the
idealized circuit element and no single device is a good approximation in all respects. The theoretical
usefulness of the idealized element is that in combination with other idealized elements it can be used to
approximate the characteristics of real devices. An analogy might be the use of frictionless pulleys as an
idealization with which to examine basic characteristics of pulley use, and then to incorporate important
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additional considerations such as friction forces and material strength, which modify rather than change
the basic idealized conclusions.
Ohm's Law itself actually is an approximation to a nonlinear characteristic valid for 'small' voltages and
currents. The resistance also is temperature sensitive, and there are secondary phenomena occurring
(capacitive and inductive effects) which can be significant under certain conditions. Energy is
dissipated as heat in a resistor, and with enough dissipation physical changes can occur up to and
including destruction.
Probably one of the most familiar resistor forms is the 'axial lead composite resistor' whose general
appearance is illustrated below. Depending on power rating the drawing is roughly full-scale (1 watt) to
about 4X scale for a 1/4 watt resistor.
Resistors come with resistance specified in well-defined tolerance ranges. (The manufacturing process
itself is not held to costly high tolerance constraints; product is measured and sorted to suit.) The figure
below describes the industry standard color codes for the axial lead composition resistors used in the lab
experiments.
The ±5% and ±10% standard values are listed (±10% values, a subset of the ±5% values, are √’ed).
Each value is a constant multiple of its predecessor, starting from 10. For the 5% values the multiplier is
(approx.) the 24th root of 10 = 1.1, and for the 10% values it is (approximately) the 12th root of
10 = 1.2.
Two special cases of a resistor are of some interest. A 'short-circuit' is an idealized resistor having zero
resistance. It carries current without consuming energy. A wire often is modeled as a short-circuit with
adequate accuracy. Similarly an ‘open-circuit’ is (so-to-speak) a resistor that carries no current
whatever the voltage difference applied.
Independent Sources
Another familiar electrical circuit element is the (idealized) battery, represented by the
icon drawn to the right; the icon symbolizes the cell structure of a wet cell battery.
Note that while ± voltage symbols are assigned there is no current arrow assignment.
This is because the terminal-volt ampere relation for the battery does not involve
current. The volt-ampere relation simply states that the voltage drop across the battery
is a constant value E whatever current flows through the battery. Of course this specification is for an
idealized device and ignores ancillary phenomena occurring in a real battery. If current flows from - to
+, and whether it does so or not depends on the circuit in which the battery is embedded (e.g., depending
on the circuit a battery might be either charging or discharging) then the battery is doing work. (A
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chemical battery converts chemical energy into the potential energy stored in the separation of
electricity. By providing an external path for the separated electricities to flow together and recombine
the stored energy is recovered.)
On the other hand if current flows + to - the work is done by the current. By implication there is another
source elsewhere in the circuit forcing that current flow direction.
Here is some advice with respect to idealized circuit elements. The terminal volt-ampere relation is a
definition of the properties of that element. You must use that definition as is, even if a conclusion
based on the definition is clearly unrealistic. A definition need not be realistic, only consistency is
required. Of course you can argue that the definition is an inadequate approximation to a real world
device, at least for certain circumstances. Well then, don't use it for that
approximation; a more appropriate model must be defined.
The icon drawn to the right represents a more general 'voltage source'; the voltage
drop is V whatever current flows. The difference between this source and the battery
is that there is no requirement that V be a constant. Instead the specific character of V
must be specified separately., e.g. V = 10sin(2t).
The dual of a voltage source is a 'current' source, which supplies a specified current, whatever voltage is
required to do so. There is no 'DC' current source icon dual to the battery; such a device is in general
impractical. A battery delivers power only if there is a current flow; a 'standby' state is
available in which no current flows and no energy is expended. (In practice there will be
various losses but the 'shelf life' of a battery can be relatively long.) A current source by
definition maintains a current and in practice this will almost inevitably cause significant
continual losses. There is a general-purpose current source icon, shown to the left, which
has a number of theoretical as well as practical uses; the current for such a source is not
necessarily constant.
The current source, as for the other circuit elements is a theoretical abstraction. Thus, for example, there
are apparent absurdities such as an unconnected current source, which then requires an infinite voltage
drop across its terminals. Obviously this circumstance is not a reflection of a practical condition, and
the approximation of a 'real source by the idealized source is as inappropriate as where a voltage source
is short-circuited.
Controlled (Dependent) Sources
Controlled sources are an abstraction of certain twoterminal pair elements where the voltage or current
source strength at one place in a circuit is
proportional to either a current or voltage at another
place in the circuit. The four possible controlledsources are represented by the icons drawn below;
the nature of the control strength - source strength
relationship is self-evident from the icon. All the
relationships are linear, i.e., the proportionality
factor is a constant. Although the input terminal pair
is drawn proximate to the output terminal pair below
this is only a convenience; no particular topological
relationship in a circuit is implied. The input
terminals merely indicate, respectively, a voltage
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across a branch or a current through a branch; the specific nature or location of a branch is not implied
by the icons. As with the other circuit elements described controlled (or 'dependent') sources are
idealized abstractions.
Controlled sources add enormous variety to the performance of electrical circuits, but that is a matter to
consider in due course. Incidentally note that the coefficients µ and A are dimensionless, while r m has
dimensions of ohms and gm has dimensions of siemans.
Consistency, Redundancy, and Indeterminacy
Idealized elements in particular impose certain conditions that seem to contradict ordinary experience.
But an idealized circuit element is a definition, and not a real device. Whether or not an idealized
element can adequately model a real element depends on the circumstances in which it is to be used, and
this is not always an easy matter to resolve. In any event it is not a matter considered in any detail as
yet. For now we note simply that a definition requires only consistency, whether or not it conflicts with
intuition. The figure to the left uses six possible
combinations of voltage and current sources to illustrate
some incompatibilities and redundancies.
Thus in item a) KVL requires that V1 = V2 = V. Except
for consistency with KCL there is no direct basis for
determining how the input current divides between the
sources. A symmetry argument suggests the current
divides equally. However it is unlikely that any benefit
accrues from the use of two sources; replace the
combination with a single voltage source.
Connecting two voltage sources in series, as shown in b),
offers no problem with consistency or indeterminacy.
Ordinarily however absent some special reason for the use
of two sources (as for example in this illustration) the
circuit would likely be simplified to a single equivalent
voltage source with a combined strength V1 + V2.
Similarly item c) is equivalent to a single current source
with strength I1 + I2, and could be so replaced as a
simplification. Note that there is no indeterminacy with
respect to the voltage across the current sources; KVL requires the same voltage difference.
The fourth item d) is required by KCL to have I1 = I2 = I, and could be replaced by a single source of
strength I. In this case there is indeterminacy in that the voltage division across the current sources is
arbitrary, subject only to the condition that the sum of the voltages across the source equal the terminal
voltage difference. Of course ‘symmetry’ can be invoked to argue the voltage across each source is the
same.
Item e) offers no contradictions but it is redundant. The parallel combination of sources can be replaced
by the voltage source alone, since that source can carry whatever terminal current is required. (The
current source alone would not fix the terminal voltage.) Note also that the current source does not
place any constraint on the terminal current. Even with the current source removed the voltage source
carries whatever terminal current is required.
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And finally item f offers no contradictions but it is redundant. The terminal current necessarily is I, and
the current source alone, by definition, can support whatever voltage difference is required.
Matters become rather more involved, and concurrently more interesting, when dependent sources are
involved. For example the current- controlled voltage source drawn to the left has a reflexive
dependency, i.e. The source voltage depends on the current through the source itself, and
unlike an independent source relates both the source voltage and the source current. If the
circuit is configured so that K is a positive constant, e.g., K = R then the terminal voltampere relationship is v = Ri; the source behaves as does a resistor following Ohm’s Law.
However this is somewhat deluding; the arrangement is not a resistor but simply mimics
one. Moreover it is possible to arrange for K = -R, so that the terminal relationship is
v = –Ri. All that this means is that the source behaves as does would a fictional ‘negative’ resistance.
Contrary to an ohmic resistance the dependent source generates rather than dissipates power.
It is possible also to construct a circuit that, at least over a limited range of voltage and
current, is represented closely by the idealized arrangement drawn to the right. The
voltage drop between the terminals is zero, whatever the current! What this means is
simply that the power generated by the dependent source equals the power consumed
by the resistor. No additional power need be (or can be) provided by external sources.
Aside from the interesting possibilities such as ‘negative’ resistance afforded by dependent sources it is
possible to obtain indeterminacies such as that exhibited by the circuit fragment
drawn to the right. Here the voltage across the dependent source is controlled by
the current through the resistor. The circuit is legitimate except that there is no
way to determine the division of an input current between the source and the
resistor. Any current division proposed will satisfy KVL and KCL!
Fortunately instances of indeterminacy and redundancy outside of academic circumstances are unlikely;
we simply ignore such possibilities.
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