Circuits consisting of just one battery and one load resistance are
very simple to analyze, but they are not often found in practical
applications. Usually, we find circuits where more than two components
are connected together.
There are two basic ways in which to connect more than two circuit components:
series and
parallel. First, an example of a series circuit:
Here, we have three resistors (labeled R
1, R
2, and R
3),
connected in a long chain from one terminal of the battery to the
other. (It should be noted that the subscript labeling -- those little
numbers to the lower-right of the letter "R" -- are unrelated to the
resistor values in ohms. They serve only to identify one resistor from
another.) The defining characteristic of a series circuit is that there
is only one path for electrons to flow. In this circuit the electrons
flow in a counter-clockwise direction, from point 4 to point 3 to point 2
to point 1 and back around to 4.
Now, let's look at the other type of circuit, a parallel configuration:
Again,
we have three resistors, but this time they form more than one
continuous path for electrons to flow. There's one path from 8 to 7 to 2
to 1 and back to 8 again. There's another from 8 to 7 to 6 to 3 to 2 to
1 and back to 8 again. And then there's a third path from 8 to 7 to 6
to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path
(through R
1, R
2, and R
3) is called a
branch.
The defining characteristic of a parallel circuit is that all
components are connected between the same set of electrically common
points. Looking at the schematic diagram, we see that points 1, 2, 3,
and 4 are all electrically common. So are points 8, 7, 6, and 5. Note
that all resistors as well as the battery are connected between these
two sets of points.
And, of course, the complexity doesn't
stop at simple series and parallel either! We can have circuits that are
a combination of series and parallel, too:
In
this circuit, we have two loops for electrons to flow through: one from
6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3
to 2 to 1 and back to 6 again. Notice how both current paths go through
R
1 (from point 2 to point 1). In this configuration, we'd say that R
2 and R
3 are in parallel with each other, while R
1 is in series with the parallel combination of R
2 and R
3.
This is just a preview of things to come. Don't worry! We'll explore
all these circuit configurations in detail, one at a time!
The
basic idea of a "series" connection is that components are connected
end-to-end in a line to form a single path for electrons to flow:
The
basic idea of a "parallel" connection, on the other hand, is that all
components are connected across each other's leads. In a purely parallel
circuit, there are never more than two sets of electrically common
points, no matter how many components are connected. There are many
paths for electrons to flow, but only one voltage across all components:
Series
and parallel resistor configurations have very different electrical
properties. We'll explore the properties of each configuration in the
sections to come.
- REVIEW:
- In a series circuit, all components are connected end-to-end, forming a single path for electrons to flow.
- In a parallel circuit, all components are connected across each other, forming exactly two sets of electrically common points.
- A "branch" in a parallel circuit is a path for electric current formed by one of the load components (such as a resistor).
Let's start with a series circuit consisting of three resistors and a single battery:
The
first principle to understand about series circuits is that the amount
of current is the same through any component in the circuit. This is
because there is only one path for electrons to flow in a series
circuit, and because free electrons flow through conductors like marbles
in a tube, the rate of flow (marble speed) at any point in the circuit
(tube) at any specific point in time must be equal.
From the
way that the 9 volt battery is arranged, we can tell that the electrons
in this circuit will flow in a counter-clockwise direction, from point 4
to 3 to 2 to 1 and back to 4. However, we have one source of voltage
and three resistances. How do we use Ohm's Law here?
An
important caveat to Ohm's Law is that all quantities (voltage, current,
resistance, and power) must relate to each other in terms of the same
two points in a circuit. For instance, with a single-battery,
single-resistor circuit, we could easily calculate any quantity because
they all applied to the same two points in the circuit:
Since
points 1 and 2 are connected together with wire of negligible
resistance, as are points 3 and 4, we can say that point 1 is
electrically common to point 2, and that point 3 is electrically common
to point 4. Since we know we have 9 volts of electromotive force between
points 1 and 4 (directly across the battery), and since point 2 is
common to point 1 and point 3 common to point 4, we must also have 9
volts between points 2 and 3 (directly across the resistor). Therefore,
we can apply Ohm's Law (I = E/R) to the current through the resistor,
because we know the voltage (E) across the resistor and the resistance
(R) of that resistor. All terms (E, I, R) apply to the same two points
in the circuit, to that same resistor, so we can use the Ohm's Law
formula with no reservation.
However, in circuits containing
more than one resistor, we must be careful in how we apply Ohm's Law. In
the three-resistor example circuit below, we know that we have 9 volts
between points 1 and 4, which is the amount of electromotive force
trying to push electrons through the series combination of R1, R2, and R3.
However, we cannot take the value of 9 volts and divide it by 3k, 10k
or 5k Ω to try to find a current value, because we don't know how much
voltage is across any one of those resistors, individually.
The figure of 9 volts is a total quantity for the whole circuit, whereas the figures of 3k, 10k, and 5k Ω are individual
quantities for individual resistors. If we were to plug a figure for
total voltage into an Ohm's Law equation with a figure for individual
resistance, the result would not relate accurately to any quantity in
the real circuit.
For R1, Ohm's Law will relate the amount of voltage across R1 with the current through R1, given R1's resistance, 3kΩ:
But, since we don't know the voltage across R1
(only the total voltage supplied by the battery across the
three-resistor series combination) and we don't know the current through
R1, we can't do any calculations with either formula. The same goes for R2 and R3:
we can apply the Ohm's Law equations if and only if all terms are
representative of their respective quantities between the same two
points in the circuit.
So what can we do? We know the voltage of the source (9 volts) applied across the series combination of R1, R2, and R3,
and we know the resistances of each resistor, but since those
quantities aren't in the same context, we can't use Ohm's Law to
determine the circuit current. If only we knew what the total resistance was for the circuit: then we could calculate total current with our figure for total voltage (I=E/R).
This brings us to the second principle of series circuits: the total
resistance of any series circuit is equal to the sum of the individual
resistances. This should make intuitive sense: the more resistors in
series that the electrons must flow through, the more difficult it will
be for those electrons to flow. In the example problem, we had a 3 kΩ,
10 kΩ, and 5 kΩ resistor in series, giving us a total resistance of 18
kΩ:
In essence, we've calculated the equivalent resistance of R1, R2, and R3 combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of R1, R2, and R3:
Now we have all the necessary information to calculate circuit current,
because we have the voltage between points 1 and 4 (9 volts) and the
resistance between points 1 and 4 (18 kΩ):
Knowing
that current is equal through all components of a series circuit (and
we just determined the current through the battery), we can go back to
our original circuit schematic and note the current through each
component:
Now
that we know the amount of current through each resistor, we can use
Ohm's Law to determine the voltage drop across each one (applying Ohm's
Law in its proper context):
Notice
the voltage drops across each resistor, and how the sum of the voltage
drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts.
This is the third principle of series circuits: that the supply voltage
is equal to the sum of the individual voltage drops.
However,
the method we just used to analyze this simple series circuit can be
streamlined for better understanding. By using a table to list all
voltages, currents, and resistances in the circuit, it becomes very easy
to see which of those quantities can be properly related in any Ohm's
Law equation:
The rule with such a table is to apply Ohm's Law only to the values within each vertical column. For instance, ER1 only with IR1 and R1; ER2 only with IR2 and R2; etc. You begin your analysis by filling in those elements of the table that are given to you from the beginning:
As you can see from the arrangement of the data, we can't apply the 9 volts of ET (total voltage) to any of the resistances (R1, R2, or R3) in any Ohm's Law formula because they're in different columns. The 9 volts of battery voltage is not applied directly across R1, R2, or R3.
However, we can use our "rules" of series circuits to fill in blank
spots on a horizontal row. In this case, we can use the series rule of
resistances to determine a total resistance from the sum of individual resistances:
Now, with a value for total resistance inserted into the rightmost
("Total") column, we can apply Ohm's Law of I=E/R to total voltage and
total resistance to arrive at a total current of 500 µA:
Then,
knowing that the current is shared equally by all components of a
series circuit (another "rule" of series circuits), we can fill in the
currents for each resistor from the current figure just calculated:
Finally, we can use Ohm's Law to determine the voltage drop across each resistor, one column at a time:
Just
for fun, we can use a computer to analyze this very same circuit
automatically. It will be a good way to verify our calculations and also
become more familiar with computer analysis. First, we have to describe
the circuit to the computer in a format recognizable by the software.
The SPICE program we'll be using requires that all electrically unique
points in a circuit be numbered, and component placement is understood
by which of those numbered points, or "nodes," they share. For clarity, I
numbered the four corners of our example circuit 1 through 4. SPICE,
however, demands that there be a node zero somewhere in the circuit, so
I'll re-draw the circuit, changing the numbering scheme slightly:
All
I've done here is re-numbered the lower-left corner of the circuit 0
instead of 4. Now, I can enter several lines of text into a computer
file describing the circuit in terms SPICE will understand, complete
with a couple of extra lines of code directing the program to display
voltage and current data for our viewing pleasure. This computer file is
known as the netlist in SPICE terminology:
series circuit
v1 1 0
r1 1 2 3k
r2 2 3 10k
r3 3 0 5k
.dc v1 9 9 1
.print dc v(1,2) v(2,3) v(3,0)
.end
Now, all I have to do is run the SPICE program to process the netlist and output the results:
v1 v(1,2) v(2,3) v(3) i(v1)
9.000E+00 1.500E+00 5.000E+00 2.500E+00 -5.000E-04
This printout is telling us the battery voltage is 9 volts, and the voltage drops across R1, R2, and R3
are 1.5 volts, 5 volts, and 2.5 volts, respectively. Voltage drops
across any component in SPICE are referenced by the node numbers the
component lies between, so v(1,2) is referencing the voltage between
nodes 1 and 2 in the circuit, which are the points between which R1
is located. The order of node numbers is important: when SPICE outputs a
figure for v(1,2), it regards the polarity the same way as if we were
holding a voltmeter with the red test lead on node 1 and the black test
lead on node 2.
We also have a display showing current (albeit
with a negative value) at 0.5 milliamps, or 500 microamps. So our
mathematical analysis has been vindicated by the computer. This figure
appears as a negative number in the SPICE analysis, due to a quirk in
the way SPICE handles current calculations.
In
summary, a series circuit is defined as having only one path for
electrons to flow. From this definition, three rules of series circuits
follow: all components share the same current; resistances add to equal a
larger, total resistance; and voltage drops add to equal a larger,
total voltage. All of these rules find root in the definition of a
series circuit. If you understand that definition fully, then the rules
are nothing more than footnotes to the definition.
- REVIEW:
- Components in a series circuit share the same current: ITotal = I1 = I2 = . . . In
- Total resistance in a series circuit is equal to the sum of the individual resistances: RTotal = R1 + R2 + . . . Rn
- Total voltage in a series circuit is equal to the sum of the individual voltage drops: ETotal = E1 + E2 + . . . En
Let's start with a parallel circuit consisting of three resistors and a single battery:
The
first principle to understand about parallel circuits is that the
voltage is equal across all components in the circuit. This is because
there are only two sets of electrically common points in a parallel
circuit, and voltage measured between sets of common points must always
be the same at any given time. Therefore, in the above circuit, the
voltage across R1 is equal to the voltage across R2 which is equal to the voltage across R3
which is equal to the voltage across the battery. This equality of
voltages can be represented in another table for our starting values:
Just
as in the case of series circuits, the same caveat for Ohm's Law
applies: values for voltage, current, and resistance must be in the same
context in order for the calculations to work correctly. However, in
the above example circuit, we can immediately apply Ohm's Law to each
resistor to find its current because we know the voltage across each
resistor (9 volts) and the resistance of each resistor:
At
this point we still don't know what the total current or total
resistance for this parallel circuit is, so we can't apply Ohm's Law to
the rightmost ("Total") column. However, if we think carefully about
what is happening it should become apparent that the total current must
equal the sum of all individual resistor ("branch") currents:
As
the total current exits the negative (-) battery terminal at point 8
and travels through the circuit, some of the flow splits off at point 7
to go up through R1, some more splits off at point 6 to go up through R2, and the remainder goes up through R3.
Like a river branching into several smaller streams, the combined flow
rates of all streams must equal the flow rate of the whole river. The
same thing is encountered where the currents through R1, R2, and R3
join to flow back to the positive terminal of the battery (+) toward
point 1: the flow of electrons from point 2 to point 1 must equal the
sum of the (branch) currents through R1, R2, and R3.
This is the second principle of parallel circuits: the total circuit
current is equal to the sum of the individual branch currents. Using
this principle, we can fill in the IT spot on our table with the sum of IR1, IR2, and IR3:
Finally, applying Ohm's Law to the rightmost ("Total") column, we can calculate the total circuit resistance:
Please note something very important here. The total circuit resistance is only 625 Ω: less
than any one of the individual resistors. In the series circuit, where
the total resistance was the sum of the individual resistances, the
total was bound to be greater than any one of the resistors
individually. Here in the parallel circuit, however, the opposite is
true: we say that the individual resistances diminish rather than add
to make the total. This principle completes our triad of "rules" for
parallel circuits, just as series circuits were found to have three
rules for voltage, current, and resistance. Mathematically, the
relationship between total resistance and individual resistances in a
parallel circuit looks like this:
The same basic form of equation works for any
number of resistors connected together in parallel, just add as many
1/R terms on the denominator of the fraction as needed to accommodate
all parallel resistors in the circuit.
Just as with the
series circuit, we can use computer analysis to double-check our
calculations. First, of course, we have to describe our example circuit
to the computer in terms it can understand. I'll start by re-drawing the
circuit:
Once
again we find that the original numbering scheme used to identify
points in the circuit will have to be altered for the benefit of SPICE.
In SPICE, all electrically common points must share identical node
numbers. This is how SPICE knows what's connected to what, and how. In a
simple parallel circuit, all points are electrically common in one of
two sets of points. For our example circuit, the wire connecting the
tops of all the components will have one node number and the wire
connecting the bottoms of the components will have the other. Staying
true to the convention of including zero as a node number, I choose the
numbers 0 and 1:
An
example like this makes the rationale of node numbers in SPICE fairly
clear to understand. By having all components share common sets of
numbers, the computer "knows" they're all connected in parallel with
each other.
In order to display branch currents in SPICE, we
need to insert zero-voltage sources in line (in series) with each
resistor, and then reference our current measurements to those sources.
For whatever reason, the creators of the SPICE program made it so that
current could only be calculated through a voltage source. This
is a somewhat annoying demand of the SPICE simulation program. With each
of these "dummy" voltage sources added, some new node numbers must be
created to connect them to their respective branch resistors:
The
dummy voltage sources are all set at 0 volts so as to have no impact on
the operation of the circuit. The circuit description file, or netlist, looks like this:
Parallel circuit
v1 1 0
r1 2 0 10k
r2 3 0 2k
r3 4 0 1k
vr1 1 2 dc 0
vr2 1 3 dc 0
vr3 1 4 dc 0
.dc v1 9 9 1
.print dc v(2,0) v(3,0) v(4,0)
.print dc i(vr1) i(vr2) i(vr3)
.end
Running the computer analysis, we get these results (I've annotated the printout with descriptive labels):
v1 v(2) v(3) v(4)
9.000E+00 9.000E+00 9.000E+00 9.000E+00
battery R1 voltage R2 voltage R3 voltage
voltage
v1 i(vr1) i(vr2) i(vr3)
9.000E+00 9.000E-04 4.500E-03 9.000E-03
battery R1 current R2 current R3 current
voltage
These values do indeed match those calculated through Ohm's Law earlier: 0.9 mA for IR1, 4.5 mA for IR2, and 9 mA for IR3.
Being connected in parallel, of course, all resistors have the same
voltage dropped across them (9 volts, same as the battery).
In
summary, a parallel circuit is defined as one where all components are
connected between the same set of electrically common points. Another
way of saying this is that all components are connected across each
other's terminals. From this definition, three rules of parallel
circuits follow: all components share the same voltage; resistances
diminish to equal a smaller, total resistance; and branch currents add
to equal a larger, total current. Just as in the case of series
circuits, all of these rules find root in the definition of a parallel
circuit. If you understand that definition fully, then the rules are
nothing more than footnotes to the definition.
- REVIEW:
- Components in a parallel circuit share the same voltage: ETotal = E1 = E2 = . . . En
- Total resistance in a parallel circuit is less than any of the individual resistances: RTotal = 1 / (1/R1 + 1/R2 + . . . 1/Rn)
- Total current in a parallel circuit is equal to the sum of the individual branch currents: ITotal = I1 + I2 + . . . In.
When students first see the parallel resistance equation, the natural question to ask is, "Where did that thing come from?" It is truly an odd piece of arithmetic, and its origin deserves a good explanation.
Resistance, by definition, is the measure of friction
a component presents to the flow of electrons through it. Resistance is
symbolized by the capital letter "R" and is measured in the unit of
"ohm." However, we can also think of this electrical property in terms
of its inverse: how easy it is for electrons to flow through a component, rather than how difficult. If resistance
is the word we use to symbolize the measure of how difficult it is for
electrons to flow, then a good word to express how easy it is for
electrons to flow would be conductance.
Mathematically, conductance is the reciprocal, or inverse, of resistance:
The greater the resistance, the less the conductance, and vice versa.
This should make intuitive sense, resistance and conductance being
opposite ways to denote the same essential electrical property. If two
components' resistances are compared and it is found that component "A"
has one-half the resistance of component "B," then we could
alternatively express this relationship by saying that component "A" is twice as conductive as component "B." If component "A" has but one-third the resistance of component "B," then we could say it is three times more conductive than component "B," and so on.
Carrying this idea further, a symbol and unit were created to represent
conductance. The symbol is the capital letter "G" and the unit is the mho,
which is "ohm" spelled backwards (and you didn't think electronics
engineers had any sense of humor!). Despite its appropriateness, the
unit of the mho was replaced in later years by the unit of siemens
(abbreviated by the capital letter "S"). This decision to change unit
names is reminiscent of the change from the temperature unit of degrees Centigrade to degrees Celsius, or the change from the unit of frequency c.p.s. (cycles per second) to Hertz.
If you're looking for a pattern here, Siemens, Celsius, and Hertz are
all surnames of famous scientists, the names of which, sadly, tell us
less about the nature of the units than the units' original
designations.
As a footnote, the unit of siemens is never
expressed without the last letter "s." In other words, there is no such
thing as a unit of "siemen" as there is in the case of the "ohm" or the
"mho." The reason for this is the proper spelling of the respective
scientists' surnames. The unit for electrical resistance was named after
someone named "Ohm," whereas the unit for electrical conductance was
named after someone named "Siemens," therefore it would be improper to
"singularize" the latter unit as its final "s" does not denote
plurality.
Back to our parallel circuit example, we should be
able to see that multiple paths (branches) for current reduces total
resistance for the whole circuit, as electrons are able to flow easier
through the whole network of multiple branches than through any one of
those branch resistances alone. In terms of resistance, additional branches result in a lesser total (current meets with less opposition). In terms of conductance, however, additional branches results in a greater total (electrons flow with greater conductance):
Total parallel resistance is less than any one of the individual branch resistances because parallel resistors resist less together than they would separately:
Total parallel conductance is greater than any of the individual branch conductances because parallel resistors conduct better together than they would separately:
To be more precise, the total conductance in a parallel circuit is equal to the sum of the individual conductances:
If we know that conductance is nothing more than the mathematical
reciprocal (1/x) of resistance, we can translate each term of the above
formula into resistance by substituting the reciprocal of each
respective conductance:
Solving
the above equation for total resistance (instead of the reciprocal of
total resistance), we can invert (reciprocate) both sides of the
equation:
So,
we arrive at our cryptic resistance formula at last! Conductance (G) is
seldom used as a practical measurement, and so the above formula is a
common one to see in the analysis of parallel circuits.
- REVIEW:
- Conductance is the opposite of resistance: the measure of how easy is it for electrons to flow through something.
- Conductance is symbolized with the letter "G" and is measured in units of mhos or Siemens.
- Mathematically, conductance equals the reciprocal of resistance: G = 1/R
When
calculating the power dissipation of resistive components, use any one
of the three power equations to derive the answer from values of
voltage, current, and/or resistance pertaining to each component:
This is easily managed by adding another row to our familiar table of voltages, currents, and resistances:
Power for any particular table column can be found by the appropriate Ohm's Law equation (appropriate based on what figures are present for E, I, and R in that column).
An interesting rule for total power versus individual power is that it is additive for any
configuration of circuit: series, parallel, series/parallel, or
otherwise. Power is a measure of rate of work, and since power
dissipated must equal the total power applied by the source(s)
(as per the Law of Conservation of Energy in physics), circuit
configuration has no effect on the mathematics.
- REVIEW:
- Power is additive in any configuration of resistive circuit: PTotal = P1 + P2 + . . . Pn
One
of the most common mistakes made by beginning electronics students in
their application of Ohm's Laws is mixing the contexts of voltage,
current, and resistance. In other words, a student might mistakenly use a
value for I through one resistor and the value for E across a set of
interconnected resistors, thinking that they'll arrive at the resistance
of that one resistor. Not so! Remember this important rule: The
variables used in Ohm's Law equations must be common to the same
two points in the circuit under consideration. I cannot overemphasize
this rule. This is especially important in series-parallel combination
circuits where nearby components may have different values for both
voltage drop and current.
When using Ohm's Law to
calculate a variable pertaining to a single component, be sure the
voltage you're referencing is solely across that single component and
the current you're referencing is solely through that single component
and the resistance you're referencing is solely for that single
component. Likewise, when calculating a variable pertaining to a set of
components in a circuit, be sure that the voltage, current, and
resistance values are specific to that complete set of components only! A
good way to remember this is to pay close attention to the two points
terminating the component or set of components being analyzed, making
sure that the voltage in question is across those two points, that the
current in question is the electron flow from one of those points all
the way to the other point, that the resistance in question is the
equivalent of a single resistor between those two points, and that the
power in question is the total power dissipated by all components
between those two points.
The "table" method presented for
both series and parallel circuits in this chapter is a good way to keep
the context of Ohm's Law correct for any kind of circuit configuration.
In a table like the one shown below, you are only allowed to apply an
Ohm's Law equation for the values of a single vertical column at a time:
Deriving values horizontally across columns is allowable as per the principles of series and parallel circuits:
Not only does the "table" method simplify the management of all
relevant quantities, it also facilitates cross-checking of answers by
making it easy to solve for the original unknown variables through other
methods, or by working backwards to solve for the initially given
values from your solutions. For example, if you have just solved for all
unknown voltages, currents, and resistances in a circuit, you can check
your work by adding a row at the bottom for power calculations on each
resistor, seeing whether or not all the individual power values add up
to the total power. If not, then you must have made a mistake somewhere!
While this technique of "cross-checking" your work is nothing new,
using the table to arrange all the data for the cross-check(s) results
in a minimum of confusion.
- REVIEW:
- Apply Ohm's Law to vertical columns in the table.
- Apply rules of series/parallel to horizontal rows in the table.
- Check
your calculations by working "backwards" to try to arrive at originally
given values (from your first calculated answers), or by solving for a
quantity using more than one method (from different given values).
The job of a technician frequently entails "troubleshooting" (locating
and correcting a problem) in malfunctioning circuits. Good
troubleshooting is a demanding and rewarding effort, requiring a
thorough understanding of the basic concepts, the ability to formulate
hypotheses (proposed explanations of an effect), the ability to judge
the value of different hypotheses based on their probability (how likely
one particular cause may be over another), and a sense of creativity in
applying a solution to rectify the problem. While it is possible to
distill these skills into a scientific methodology, most practiced
troubleshooters would agree that troubleshooting involves a touch of
art, and that it can take years of experience to fully develop this art.
An essential skill to have is a ready and intuitive
understanding of how component faults affect circuits in different
configurations. We will explore some of the effects of component faults
in both series and parallel circuits here, then to a greater degree at
the end of the "Series-Parallel Combination Circuits" chapter.
Let's start with a simple series circuit:
With
all components in this circuit functioning at their proper values, we
can mathematically determine all currents and voltage drops:
Now let us suppose that R2 fails shorted. Shorted
means that the resistor now acts like a straight piece of wire, with
little or no resistance. The circuit will behave as though a "jumper"
wire were connected across R2 (in case you were wondering,
"jumper wire" is a common term for a temporary wire connection in a
circuit). What causes the shorted condition of R2 is no matter to us in this example; we only care about its effect upon the circuit:
With R2 shorted, either by a jumper wire or by an internal resistor failure, the total circuit resistance will decrease.
Since the voltage output by the battery is a constant (at least in our
ideal simulation here), a decrease in total circuit resistance means
that total circuit current must increase:
As the circuit current increases from 20 milliamps to 60 milliamps, the voltage drops across R1 and R3 (which haven't changed resistances) increase as well, so that the two resistors are dropping the whole 9 volts. R2,
being bypassed by the very low resistance of the jumper wire, is
effectively eliminated from the circuit, the resistance from one lead to
the other having been reduced to zero. Thus, the voltage drop across R2, even with the increased total current, is zero volts.
On the other hand, if R2
were to fail "open" -- resistance increasing to nearly infinite levels
-- it would also create wide-reaching effects in the rest of the
circuit:
With R2
at infinite resistance and total resistance being the sum of all
individual resistances in a series circuit, the total current decreases
to zero. With zero circuit current, there is no electron flow to produce
voltage drops across R1 or R3. R2, on the other hand, will manifest the full supply voltage across its terminals.
We can apply the same before/after analysis technique to parallel
circuits as well. First, we determine what a "healthy" parallel circuit
should behave like.
Supposing that R2 opens in this parallel circuit, here's what the effects will be:
Notice
that in this parallel circuit, an open branch only affects the current
through that branch and the circuit's total current. Total voltage --
being shared equally across all components in a parallel circuit, will
be the same for all resistors. Due to the fact that the voltage source's
tendency is to hold voltage constant, its voltage will not
change, and being in parallel with all the resistors, it will hold all
the resistors' voltages the same as they were before: 9 volts. Being
that voltage is the only common parameter in a parallel circuit, and the
other resistors haven't changed resistance value, their respective
branch currents remain unchanged.
This is what happens in a
household lamp circuit: all lamps get their operating voltage from power
wiring arranged in a parallel fashion. Turning one lamp on and off (one
branch in that parallel circuit closing and opening) doesn't affect the
operation of other lamps in the room, only the current in that one lamp
(branch circuit) and the total current powering all the lamps in the
room:
In
an ideal case (with perfect voltage sources and zero-resistance
connecting wire), shorted resistors in a simple parallel circuit will
also have no effect on what's happening in other branches of the
circuit. In real life, the effect is not quite the same, and we'll see
why in the following example:
A
shorted resistor (resistance of 0 Ω) would theoretically draw infinite
current from any finite source of voltage (I=E/0). In this case, the
zero resistance of R2 decreases the circuit total resistance
to zero Ω as well, increasing total current to a value of infinity. As
long as the voltage source holds steady at 9 volts, however, the other
branch currents (IR1 and IR3) will remain unchanged.
The critical assumption in this "perfect" scheme, however, is that the
voltage supply will hold steady at its rated voltage while supplying an
infinite amount of current to a short-circuit load. This is simply not
realistic. Even if the short has a small amount of resistance (as
opposed to absolutely zero resistance), no real voltage source
could arbitrarily supply a huge overload current and maintain steady
voltage at the same time. This is primarily due to the internal
resistance intrinsic to all electrical power sources, stemming from the
inescapable physical properties of the materials they're constructed of:
These
internal resistances, small as they may be, turn our simple parallel
circuit into a series-parallel combination circuit. Usually, the
internal resistances of voltage sources are low enough that they can be
safely ignored, but when high currents resulting from shorted components
are encountered, their effects become very noticeable. In this case, a
shorted R2 would result in almost all the voltage being
dropped across the internal resistance of the battery, with almost no
voltage left over for resistors R1, R2, and R3:
Suffice it to say, intentional direct short-circuits across the
terminals of any voltage source is a bad idea. Even if the resulting
high current (heat, flashes, sparks) causes no harm to people nearby,
the voltage source will likely sustain damage, unless it has been
specifically designed to handle short-circuits, which most voltage
sources are not.
Eventually in this book I will lead you through the analysis of circuits without the use of any numbers,
that is, analyzing the effects of component failure in a circuit
without knowing exactly how many volts the battery produces, how many
ohms of resistance is in each resistor, etc. This section serves as an
introductory step to that kind of analysis.
Whereas
the normal application of Ohm's Law and the rules of series and
parallel circuits is performed with numerical quantities ("quantitative"), this new kind of analysis without precise numerical figures is something I like to call qualitative analysis. In other words, we will be analyzing the qualities of the effects in a circuit rather than the precise quantities. The result, for you, will be a much deeper intuitive understanding of electric circuit operation.
- REVIEW:
- To
determine what would happen in a circuit if a component fails, re-draw
that circuit with the equivalent resistance of the failed component in
place and re-calculate all values.
- The ability to intuitively determine what will happen to a circuit with any given component fault is a crucial
skill for any electronics troubleshooter to develop. The best way to
learn is to experiment with circuit calculations and real-life circuits,
paying close attention to what changes with a fault, what remains the
same, and why!
- A shorted component is one whose resistance has dramatically decreased.
- An open
component is one whose resistance has dramatically increased. For the
record, resistors tend to fail open more often than fail shorted, and
they almost never fail unless physically or electrically overstressed
(physically abused or overheated).
In the course of learning about electricity, you will want to construct
your own circuits using resistors and batteries. Some options are
available in this matter of circuit assembly, some easier than others.
In this section, I will explore a couple of fabrication techniques that
will not only help you build the circuits shown in this chapter, but
also more advanced circuits.
If all we wish to construct is a simple single-battery, single-resistor circuit, we may easily use alligator clip jumper wires like this:
Jumper
wires with "alligator" style spring clips at each end provide a safe
and convenient method of electrically joining components together.
If
we wanted to build a simple series circuit with one battery and three
resistors, the same "point-to-point" construction technique using jumper
wires could be applied:
This
technique, however, proves impractical for circuits much more complex
than this, due to the awkwardness of the jumper wires and the physical
fragility of their connections. A more common method of temporary
construction for the hobbyist is the solderless breadboard, a
device made of plastic with hundreds of spring-loaded connection sockets
joining the inserted ends of components and/or 22-gauge solid wire
pieces. A photograph of a real breadboard is shown here, followed by an
illustration showing a simple series circuit constructed on one:
Underneath
each hole in the breadboard face is a metal spring clip, designed to
grasp any inserted wire or component lead. These metal spring clips are
joined underneath the breadboard face, making connections between
inserted leads. The connection pattern joins every five holes along a
vertical column (as shown with the long axis of the breadboard situated
horizontally):
Thus,
when a wire or component lead is inserted into a hole on the
breadboard, there are four more holes in that column providing potential
connection points to other wires and/or component leads. The result is
an extremely flexible platform for constructing temporary circuits. For
example, the three-resistor circuit just shown could also be built on a
breadboard like this:
A parallel circuit is also easy to construct on a solderless breadboard:
Breadboards have their limitations, though. First and foremost, they are intended for temporary
construction only. If you pick up a breadboard, turn it upside-down,
and shake it, any components plugged into it are sure to loosen, and may
fall out of their respective holes. Also, breadboards are limited to
fairly low-current (less than 1 amp) circuits. Those spring clips have a
small contact area, and thus cannot support high currents without
excessive heating.
For
greater permanence, one might wish to choose soldering or
wire-wrapping. These techniques involve fastening the components and
wires to some structure providing a secure mechanical location (such as a
phenolic or fiberglass board with holes drilled in it, much like a
breadboard without the intrinsic spring-clip connections), and then
attaching wires to the secured component leads. Soldering is a form of
low-temperature welding, using a tin/lead or tin/silver alloy that melts
to and electrically bonds copper objects. Wire ends soldered to
component leads or to small, copper ring "pads" bonded on the surface of
the circuit board serve to connect the components together. In wire
wrapping, a small-gauge wire is tightly wrapped around component leads
rather than soldered to leads or copper pads, the tension of the wrapped
wire providing a sound mechanical and electrical junction to connect
components together.
An example of a printed circuit board, or PCB, intended for hobbyist use is shown in this photograph:
This
board appears copper-side-up: the side where all the soldering is done.
Each hole is ringed with a small layer of copper metal for bonding to
the solder. All holes are independent of each other on this particular
board, unlike the holes on a solderless breadboard which are connected
together in groups of five. Printed circuit boards with the same 5-hole
connection pattern as breadboards can be purchased and used for hobby
circuit construction, though.
Production printed circuit boards have traces
of copper laid down on the phenolic or fiberglass substrate material to
form pre-engineered connection pathways which function as wires in a
circuit. An example of such a board is shown here, this unit actually a
"power supply" circuit designed to take 120 volt alternating current
(AC) power from a household wall socket and transform it into
low-voltage direct current (DC). A resistor appears on this board, the
fifth component counting up from the bottom, located in the middle-right
area of the board.
A
view of this board's underside reveals the copper "traces" connecting
components together, as well as the silver-colored deposits of solder
bonding the component leads to those traces:
A
soldered or wire-wrapped circuit is considered permanent: that is, it
is unlikely to fall apart accidently. However, these construction
techniques are sometimes considered too permanent. If anyone
wishes to replace a component or change the circuit in any substantial
way, they must invest a fair amount of time undoing the connections.
Also, both soldering and wire-wrapping require specialized tools which
may not be immediately available.
An alternative construction technique used throughout the industrial world is that of the terminal strip. Terminal strips, alternatively called barrier strips or terminal blocks,
are comprised of a length of nonconducting material with several small
bars of metal embedded within. Each metal bar has at least one machine
screw or other fastener under which a wire or component lead may be
secured. Multiple wires fastened by one screw are made electrically
common to each other, as are wires fastened to multiple screws on the
same bar. The following photograph shows one style of terminal strip,
with a few wires attached.
Another, smaller terminal strip is shown in this next photograph. This
type, sometimes referred to as a "European" style, has recessed screws
to help prevent accidental shorting between terminals by a screwdriver
or other metal object:
In the following illustration, a single-battery, three-resistor circuit is shown constructed on a terminal strip:
If
the terminal strip uses machine screws to hold the component and wire
ends, nothing but a screwdriver is needed to secure new connections or
break old connections. Some terminal strips use spring-loaded clips --
similar to a breadboard's except for increased ruggedness -- engaged and
disengaged using a screwdriver as a push tool (no twisting involved).
The electrical connections established by a terminal strip are quite
robust, and are considered suitable for both permanent and temporary
construction.
One of the essential skills for anyone interested
in electricity and electronics is to be able to "translate" a schematic
diagram to a real circuit layout where the components may not be
oriented the same way. Schematic diagrams are usually drawn for maximum
readability (excepting those few noteworthy examples sketched to create
maximum confusion!), but practical circuit construction often demands a
different component orientation. Building simple circuits on terminal
strips is one way to develop the spatial-reasoning skill of "stretching"
wires to make the same connection paths. Consider the case of a
single-battery, three-resistor parallel circuit constructed on a
terminal strip:
Progressing
from a nice, neat, schematic diagram to the real circuit -- especially
when the resistors to be connected are physically arranged in a linear
fashion on the terminal strip -- is not obvious to many, so I'll
outline the process step-by-step. First, start with the clean schematic
diagram and all components secured to the terminal strip, with no
connecting wires:
Next,
trace the wire connection from one side of the battery to the first
component in the schematic, securing a connecting wire between the same
two points on the real circuit. I find it helpful to over-draw the
schematic's wire with another line to indicate what connections I've
made in real life:
Continue
this process, wire by wire, until all connections in the schematic
diagram have been accounted for. It might be helpful to regard common
wires in a SPICE-like fashion: make all connections to a common wire in
the circuit as one step, making sure each and every component with a
connection to that wire actually has a connection to that wire before
proceeding to the next. For the next step, I'll show how the top sides
of the remaining two resistors are connected together, being common with
the wire secured in the previous step:
With
the top sides of all resistors (as shown in the schematic) connected
together, and to the battery's positive (+) terminal, all we have to do
now is connect the bottom sides together and to the other side of the
battery:
Typically
in industry, all wires are labeled with number tags, and electrically
common wires bear the same tag number, just as they do in a SPICE
simulation. In this case, we could label the wires 1 and 2:
Another
industrial convention is to modify the schematic diagram slightly so as
to indicate actual wire connection points on the terminal strip. This
demands a labeling system for the strip itself: a "TB" number (terminal
block number) for the strip, followed by another number representing
each metal bar on the strip.
This
way, the schematic may be used as a "map" to locate points in a real
circuit, regardless of how tangled and complex the connecting wiring may
appear to the eyes. This may seem excessive for the simple,
three-resistor circuit shown here, but such detail is absolutely
necessary for construction and maintenance of large circuits, especially
when those circuits may span a great physical distance, using more than
one terminal strip located in more than one panel or box.
- REVIEW:
- A solderless breadboard
is a device used to quickly assemble temporary circuits by plugging
wires and components into electrically common spring-clips arranged
underneath rows of holes in a plastic board.
- Soldering is
a low-temperature welding process utilizing a lead/tin or tin/silver
alloy to bond wires and component leads together, usually with the
components secured to a fiberglass board.
- Wire-wrapping
is an alternative to soldering, involving small-gauge wire tightly
wrapped around component leads rather than a welded joint to connect
components together.
- A terminal strip, also known as a barrier strip or terminal block
is another device used to mount components and wires to build circuits.
Screw terminals or heavy spring clips attached to metal bars provide
connection points for the wire ends and component leads, these metal
bars mounted separately to a piece of nonconducting material such as
plastic, bakelite, or ceramic.
Contributors to this chapter are listed in chronological order of their
contributions, from most recent to first. See Appendix 2 (Contributor
List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting, which led to a much better-looking second edition.
Ron LaPlante (October 1998): helped create "table" method of series and parallel circuit analysis