Unit 9:
SeriesParallel Circuits
In Units 7 and 8 you studied series
circuits and parallel circuits. Many reallife
circuits are more complicated, with combinations of both seriesconnected
and parallelconnected components. In this unit we'll study such seriesparallel
circuits. You'll find that the basic rules
of the game are the ones that you've already learned for series circuits
and parallel circuits, but the job of applying those rules can get pretty
tricky.
Unit 8 Review
 This unit will build on material that you studied in Unit
8. So let's
begin by taking this selftest to review what you learned in that unit.

Identifying SeriesParallel Relationships
 A seriesparallel circuit is a circuit that contains combinations
of seriesconnected and parallelconnected components, which are in
turn connected in series and/or in parallel with other such combinations.
Here's an example:
 Recall our definitions of series connections and parallel connections
from previous units:
 Two components are connected in series if they are connected
to each other at exactly one point and no other component is connected
to that point.
 Two components are connected in parallel if
they are connected to each other at two points.
 In the example circuit shown above, R2 is in series with R3, which
is in series with R4. So R2, R3, and R4 form a series path. Also, this
entire series path, taken as a whole, is in parallel with R1.
Notation for SeriesParallel Relationships
 As a shorthand way of describing seriesparallel relationships in
a circuit, we'll use two symbols:
 we use the symbol + to indicate
a series connection between components
 we use the symbol  to
indicate a parallel connection

So for the example circuit shown above, we would write
R2 + R3 + R4
to indicate that R2, R3, and R4 form a series path. Then, to show that R1
is in parallel with this series path, we would write
R1  (R2 + R3 + R4)
Analyzing SeriesParallel Circuits
 Units 2 and 3 gave stepbystep rules for analyzing series circuits
and parallel circuits. It was possible to do this because one basic
approach works on all series circuits, and another basic approach works
on all parallel circuits.
 But it's not possible to give a stepbystep procedure that will
work on all seriesparallel circuits. There is
too much variety among seriesparallel circuits, and an approach
that works for one circuit may not work for other circuits.
 The key is practice, practice, and more practice. To develop your
skills, you need to study as many examples and work out as many problems
as you can.
 Usually, a good first step is to find the circuit's total equivalent
resistance by combining pairs of seriesconnected or parallelconnected
resistors until you've combined all the resistors into a single resistance.
 For example, in the circuit shown below, we would add R2, R3, and R4
together (since they're in series), then use the reciprocal formula
to combine that value with R1, since R1 is in parallel
with the series path containing R2, R3, and R4. The result would be
the circuit's total equivalent resistance, R_{T}.

 After you have the total equivalent resistance, you can use Ohm's
law to find the circuit's total current:
I_{T} = V_{S} ÷ R_{T}
 After that, you can apply any of the rules and laws
from previous units, in any sequence that leads to the
solution for a particular voltage or current. The most useful of these
rules and laws are:
 Ohm's law
 voltage/current rules for series or parallel combinations
 Kirchhoff's Voltage Law
 Kirchhoff's Current Law
 the voltagedivider rule
 the currentdivider rule
 Let's review these rules and laws, and discuss their application to seriesparallel
circuits.
Ohm's Law in SeriesParallel Circuits
Voltage/Current Rules for Series or Parallel Combinations
 Here are two more simplebutpowerful rules. Recall
that:
 Components connected in series must have the same
current (but usually
they won't have the same voltage).
 Components connected in parallel must have the same
voltage (but usually
they won't have the same current).
 These two rules apply not only to individual components but also to portions
of circuits.
 In our example circuit, for instance, resistor R1 is
connected in parallel with the series path containing R2, R3, and
R4. Therefore, we know that the voltage across R1 must be equal
to the voltage across the series path. In symbols:
V_{1} = V_{2+3+4}
 Also, in the same circuit, since R2, R3, and
R4 form a series path, we know that the three resistors must
carry the same current. In symbols:
I_{2} = I_{3} = I_{4}
Kirchhoff's Voltage Law in SeriesParallel Circuits
 Recall that KVL says that the sum of the voltage drops around
any closed loop in a circuit equals the sum of the voltage rises
around that loop.
 We originally studied KVL in the context of series circuits, but
KVL holds for all circuits.
Kirchhoff's Current Law in SeriesParallel Circuits
 Recall that KCL says that the sum of all currents entering a point
is equal to the sum of all currents leaving that point.
 We originally studied KCL in the context of parallel circuits, but
KCL holds for all circuits.
 In our example circuit, for instance, consider the point where the voltage
source meets R1 and R2. KCL tells us that the current flowing into
this point from the voltage source must be equal to the sum of the
currents flowing out of this point through R1 and R2. In symbols:
I_{T} = I_{1} + I_{2}
The VoltageDivider Rule in SeriesParallel
Circuits
The CurrentDivider Rule in SeriesParallel
Circuits
Examples
 We have now reviewed the use of Ohm's law, Kirchhoff's Laws, and
the divider rules in seriesparallel circuits. Analysis of any particular
seriesparallel circuit will involve the use of one or more of these
rules and laws.
 Usually there will be more than one way to approach a particular
problem or circuit. In finding a particular voltage, for instance,
one student might use Ohm's law and KVL, while another student
uses KVL and the voltagedivider rule, and a third student uses the
currentdivider rule and Ohm's law. All three approaches will give
the same answer, as long as the students apply the rules correctly.
 As mentioned earlier, a good first step is usually to find the circuit's
total equivalent resistance by combining pairs of seriesconnected
or parallelconnected resistors until you've combined all the resistors
into a single resistance.
Then find the circuit's total current by using Ohm's Law in the form:
I_{T} = V_{S} ÷ R_{T}
Power in a SeriesParallel Circuit
 To find the power dissipated in a resistor in a seriesparallel circuit,
use any of the same formulas that you used for series circuits and
parallel circuits:
P = V × I
P = I^{2} × R
P = V^{2} ÷ R
 Recall that in each of these equations, R is the resistor's
resistance, V is the voltage across the resistor, and I is
the current through the resistor.
Total Circuit Power
 Just as with series circuits and parallel circuits, there are two
ways to compute total power dissipated in a seriesparallel circuit.
You'll get the same answer either way:
 Either find the power for each resistor, and
then add these powers:
P_{T} = P_{1} + P_{2} + P_{3} +
... + P_{n}
 Or apply any one of the power formulas to the
entire circuit:
P_{T} = V_{S} × I_{T}
P_{T} = I_{T}^{2} × R_{T}
P_{T} = V_{S}^{2} ÷ R_{T}
These are the same power formulas from above, except that now we're
applying them to the entire circuit, instead of to a single resistor.

Ground (or Common) Symbol
 Recall from Unit 4 that instead of drawing lines to show
connections in a schematic diagram, we often use a ground symbol,
also called a common
symbol,
to designate terminals that are connected together. The symbol looks
like this:
 The main reason for using this symbol is that it let you eliminate
lines that clutter a schematic diagram and make it difficult to visualize
current flow between components.
 Example: Take a look again at this circuit diagram:
Notice that the voltage source, R1, and R4 all come together at a common
point. Using the ground symbol to show that these three components
are connected together, we could redraw the diagram as shown below.
 This is not a very good example, because the redrawn diagram with
the ground symbol is no easier to read than the original diagram. But
in a complicated schematic diagram with many resistors, the ground
symbol can eliminate a lot of lines and therefore make the diagram
much easier to read. The important point in this example is to see
that the two diagrams are equivalent.
Bubble Symbol for Voltage Sources
 In schematic diagrams where the common symbol is used, we often omit
that symbol from the side of any voltage source that is connected to
common.
 The terminal that is not connected to common is shown
by a small circle (or "bubble") and is labeled + or  the
voltage of the voltage source.
 Example: Continuing the same example from above, we could redraw
the schematic diagram once more using the bubble symbol to arrive at
the following diagram:
Circuit Geometry
 The geometry of a circuit (the way the component connections
are shown in a schematic diagram) can often make it hard to see voltage
and current relations in the circuit.
 You should carefully redraw a complicated schematic diagram in as
many ways as necessary to make it easier to read.
Unit 9 Review
 This eLesson has covered several important topics, including:
 identifying series and parallel relationships
 analyzing seriesparallel circuits
 power in seriesparallel circuits
 ground symbol and bubble symbol.
 To finish the eLesson, take this selftest to check your understanding
of these topics.

Congratulations! You've completed the eLesson for this unit.