Introductory Circuit Analysis Illustrations
By way of a preliminary familiarization with circuit analysis we consider several relatively simple
circuits. The circuits actually are more significant that may appear at first, but the emphasis here is on
familiarization.
SERIES Circuit Illustration
The circuit drawn to the right will serve to illustrate several basic
circuit concepts; assume E, R1, and R2 are known constant
values. The circuit itself consists of two resistors connected in
'series', meaning the two resistors carry the same (the same, not
just equal) current. KCL justifies the use of a single arrow to
serve all circuit elements, since it is the same current that flows
through all the elements. Drawing the current arrow in the form
of a partial loop suggests this commonality. Analytically what
we have done is to use KCL to reduce the number of 'unknown' currents we need to calculate. A single
current variable I supported by KCL provides the current through all the circuit elements.
KVL is the basis for writing E = V1 + V2, and the terminal volt-ampere relations for the resistors are
V1 = IR1 and V2 = IR2 respectively. Note that the resistor voltage and current polarity conventions
used mean R1 and R2 implicitly are positive numbers. Substituting the Ohm's Law relations in the
KVL expression requires
Back-substitution into the Ohm's Law expressions obtains
(A symmetry argument indicates that the expression for V2 may be obtained simply by exchanging
subscripts 1 and 2 in the equation for V1. The electrical properties of the circuit would not depend on
the particular subscripts used, i.e., on whether R1 is above or below R2 in the circuit diagram.)
This simple example illustrates the basic character of much of electric circuit analysis in general; apply
KVL to relate circuit voltages, apply KCL to relate circuit currents, and apply the circuit element
volt–ampere relations that specify the relationships between currents and voltages to obtain a set of
solvable equations. For consistent linear circuits it can be shown that a solution always can be obtained
this way, and moreover that the solution obtained is unique. Much, although certainly not all, of basic
circuit analysis consists of a study of efficient methods of applying the basic equations.
It is instructive to examine the physical aspects of the currents and voltages in the illustrative circuit.
The series combination or two resistors turns out to be a very commonly used circuit, so much so that it
generally is given the special name 'voltage divider'. The name reflects the fact that the voltage across
the series combination is divided between the resistors in proportion to the resistance values, i.e.
.
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This conclusion is equivalent to observing both resistors carry the same current.
The battery voltage E is in effect a statement that every unit of charge associated with current I flowing
from - to + through the battery is endowed with a work-doing capability E. This ability is created by the
physical processes 'inside' the battery which do work to separate + and - electricity in the battery.
Conservation of Energy (KVL) requires this energy to be dissipated in the two resistors (where else?);
this is the physical content of E = V1 + V2 since V1 and V2 are the work per unit charge corresponding
to the (same) current through R1 and R2 respectively. The energy balancing and the resistor voltampere relations limit the current to E/(R1+ R2).
The series resistor circuit configuration represents a fundamental means of controlling electrical workdoing ability, albeit in this elementary form it is neither a very sophisticated nor a particularly efficient
means for doing so. Suppose R2 represents a device doing useful work, perhaps an incandescent light
or a radiant heater or a loudspeaker or whatever. Then if R1 is made a variable (adjustable) resistor the
proportion of the total work done by each unit of charge, which is expended in R2, can be varied, with
the unused portion dissipated in R1. For low power applications, where low energy efficiency is not
critical, this unsophisticated but inexpensive control is not uncommon, e.g., a dimmer switch for
automobile dashboard lights.
PARALLEL Circuit Illustration
A different circuit, drawn to the right, illustrates another
basic approach to control of electrical work-doing ability
and provides another illustration of an elementary circuit
analysis. In this case two resistors are placed in parallel,
i.e., there is the same (the same, not just equal) voltage
drop across each resistor. The icon on the left is that of
a 'current source', the current analog of the voltage source. A current source supplies a fixed current I
whatever the voltage across the source happens to be. (Although the current source is an important
circuit element conceptually it has practical disadvantages compared to a voltage source. Voltage
sources have the advantage of not dissipating energy in a circuit in a standby 'no current' state. On the
other hand KCL requires the current from a current source to flow in some closed path at all times, and
as a practical matter there inevitably is dissipation of one sort or another from this current flow.)
Note that a single voltage-polarity assignment used for all the circuit elements, a commonality justified
by KVL. (Contrast this with the use of a single current polarity assignment for the series circuit.) The
resistor volt-ampere relation requires V = I1R1 and V = I2R2. KCL requires I = I1 + I2. From these
equations one finds the current division ratio with a symmetrical expression (replace subscript 1 by 2,
and vice versa) for I1
This circuit illustrates (in an unsophisticated form) power control of a different sort than the series
illustration. Here a unit of charge does the same amount of work whichever resistor branch it passes
through; the voltage drop (work per unit charge) across both branches is the same. Overall work-doing
is controlled by the division of the fixed total (source) current between the resistors, i.e., how much of
the charge from the current source passes through each branch. The more current through a resistor the
greater the power dissipated in that resistor. By varying one of the resistors the current division ratio
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can be varied. This control mechanism, like the other one, has limited practical use in the simple form
illustrated.
SERIES-PARALLEL Circuit Illustration
As a somewhat more involved illustration consider
the circuit drawn to the right; the circuit consists of a
series connection of one resistor to a parallel
connection of two other resistors. Note the 'ground'
symbol at the node just below the I2 current arrow.
At one time the symbol literally meant a physical
connection to the ground, i.e. to earth; such a
connection is important in certain contexts.
However its use has become less specific so that as
often used now it simply indicates a node used as a common reference in describing voltages. Thus
when a reference is made such as '…the voltage e at the upper end of resistor R2…', which if taken
literally is meaningless, it really means that e is the voltage difference between the designated terminal
and the reference terminal. More precisely here we regard e as the voltage drop from the specified node
to the ground reference. We make more use of this convention later.
Another use for this particular circuit illustration is to carry the idea of 'lumping' a bit further. Thus if we
can lump physical processes inside a single package called a resistor we should be able to lump several
such physical processes corresponding to several resistors into a single package. The 'lump'
corresponding to this combined packaging is called an 'equivalent circuit'. Expressed more formally
two circuits are 'equivalent' if their terminal volt-ampere relations are identical, for then there is no way
to distinguish the two circuits from a terminal measurement. The concept of an equivalent circuit is very
useful in that it enables a lot of detailed information to be packaged in a form that makes it easier to use.
For example the parallel combination of resistors R2 and R3 can be replaced by a single 'equivalent'
resistor R whose terminal volt-ampere relation is exactly the same as that of the parallel resistor
combination. Since the 'insides' of the equivalent element are not accessible (by definition) there is no
way to distinguish a single lumped element from a combination of such elements inside a 'box' if the
terminal volt-ampere relations are the same in the two cases. For the parallel resistor combination
KCL tells us that I1 = I2+I3, and KVL tells us that e=I2R2 and e=I3R3. Hence it follows that
Suppose the two paralleled resistors are
replaced by a single resistor whose terminal
volt-ampere relation (the relationship
between e and I1) is the same as that given
above. The rest of the circuit would not be
aware of the substitution and so its
behavior would be no different. The
'equivalent' resistor would have a resistance
equal to the coefficient of I1 as indicated in
the figure to the right.
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Replacing the parallel resistor combination by the single equivalent resistor brings us back to the series
circuit already analyzed. We just substitute R for R2 (in the series circuit analysis). That means we can
calculate e and I1 as before. Now we replace the equivalent resistor by the equivalent two-resistor
combination; the equivalency is reciprocal of course. Then the calculation of the currents in the two
resistors is straightforward.
We shall see in due course that this idea of equivalent circuits is much more powerful than the relatively
simple use illustrated.
This circuit allows for both series and parallel control mechanisms. Thus R1 can be varied to control
how much of the total work-doing ability E each unit of charge retains after passing through R1, while
R2 (say) can be used to control what fraction of the charge transport passes through R3. However this is
not a particularly elegant arrangement; for one thing the adjustments of R1 and R2 interact, and for
another it's not very efficient. But it will 'work', and serves as a counterpoint to more elegant
arrangements studied later
The equations listed below summarize the application of KVL, KCL, and the circuit element voltampere relation to the calculation of the equivalent resistor.
In a similar manner one can show that the single resistor equivalent to two resistors in series has a
resistance equal to the sum of the series resistances; doing so is left as an exercise.
Voltage Divider and Current Divider (again)
The preceding illustrations have involved two ubiquitous resistor combinations which, precisely because
they are found so often in electrical circuits, should be committed to memory. The series combination
of two resistors is drawn to the right, and is characterized by the fact that
the same current flows through each resistor. It is important to get this
straight; current flows through one resistor directly into the other,
without any diversion.
From Ohm's Law and the series connection (same current) it follows that
(Use a symmetry argument to
obtain the expression for V1/V.
The series combination of
resistors often is called a 'voltage divider' and used to obtain a
voltage that is a fixed fraction of another voltage.)
Another property of a series combination of resistors that should be committed to memory is the formula
for calculating the resistance of a single resistor equivalent to a series connection of resistors. The
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previous illustration calculated this for two resistors in series; the general expression for any number of
series-connected resistors (apply perfect induction) is
The dual of the voltage divider is the current divider formed by
two resistors in parallel (or 'shunt'). Two resistors are
connected in parallel if they have a common voltage drop, as
illustrated to the right. Because they have a common voltage
drop it follows that
A symmetry argument can be used to obtain the division expression for I1/I. These expressions should
be memorized. Also well worth committing to memory is the generalized expression for the resistance
of a single resistor equivalent to a parallel connection of resistors:
One other thing, almost an afterthought. Writing 1/R occurs so often for one reason or another that a
special unit is used to simplify doing so. The 'conductance' of a resistor is defined as G = 1/R, so that
V=RI could be rewritten (and often is) as I =GV. The unit for conductance is the mho, which is 'ohm'
spelled backwards. The Greek capital letter Ω (omega) often is used to mean 'ohm'; if you draw the
omega upside down it means 'mho'. A relatively new replacement name for the unit of conductance is
'Sieman'.
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