In Units 7 and 8 you studied series
circuits and parallel circuits. Many real-life
circuits are more complicated, with combinations of both series-connected
and parallel-connected components. In this unit we'll study such series-parallel
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
Unit 8 Review
- This unit will build on material that you studied in Unit
8. So let's
begin by taking this self-test to review what you learned in that unit.
Identifying Series-Parallel Relationships
- A series-parallel circuit is a circuit that contains combinations
of series-connected and parallel-connected 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 Series-Parallel Relationships
- As a short-hand way of describing series-parallel 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 Series-Parallel Circuits
- Units 2 and 3 gave step-by-step 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 step-by-step procedure that will
work on all series-parallel circuits. There is
too much variety among series-parallel 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 series-connected or parallel-connected
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, RT.
- After you have the total equivalent resistance, you can use Ohm's
law to find the circuit's total current:
IT = VS ÷ RT
- 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 voltage-divider rule
- the current-divider rule
- Let's review these rules and laws, and discuss their application to series-parallel
Ohm's Law in Series-Parallel Circuits
Voltage/Current Rules for Series or Parallel Combinations
- Here are two more simple-but-powerful rules. Recall
- 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
- 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:
V1 = V2+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:
I2 = I3 = I4
Kirchhoff's Voltage Law in Series-Parallel 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.
- In our example circuit, for instance, one closed loop contains just the
source and R1. Therefore KVL tells us that the voltage rise across
the source must be equal to the voltage drop across R1. In symbols:
VS = V1
Of course, we could also have seen this from the fact that the source and
R1 are connected in parallel. In many cases, there will be more than
one way to arrive at the same conclusion.
- Another closed loop in our example circuit contains the voltage
source and resistors R2, R3, R4. Therefore KVL tells us that the
voltage rise across the source must be equal to the sum of the voltage
drops across those three resistors. In symbols:
VS = V2 + V3
Kirchhoff's Current Law in Series-Parallel 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:
IT = I1 + I2
The Voltage-Divider Rule in Series-Parallel
The Current-Divider Rule in Series-Parallel
- In discussing parallel circuits, we stated the current-divider rule
as follows: for branches in parallel, the current Ix through
any branch equals the ratio of the total parallel resistance RT to
the branch's resistance Rx, multiplied by the total
current IT entering the parallel combination. In
Ix = (RT ÷ Rx) × IT
- The same rule applies to parallel branches in a series-parallel
circuit. But you have to be careful to apply it correctly. In particular,
note that Rx is the total resistance of the branch
whose current you are trying to find. Depending on the circuit you're
analyzing, Rx may be a single resistor's
resistance, or it may be the combined resistance of several resistors
in series and/or parallel. Also, note that in this equation RT means
the total resistance of the parallel branches, which may or may not
be the same as the entire circut's total resistance. Similarly, IT in
this equation means the total current entering the parallel
branches, which may or may not be the same as the entire circut's total
- Because students often apply the current-divider rule incorrectly in series-parallel
circuits, I recommend that you rely more heavily on Ohm's law, KVL,
and KCL, and try to avoid using the current-divider
- We have now reviewed the use of Ohm's law, Kirchhoff's Laws, and
the divider rules in series-parallel circuits. Analysis of any particular
series-parallel 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 voltage-divider rule, and a third student uses the
current-divider 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 series-connected
or parallel-connected 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:
IT = VS ÷ RT
Power in a Series-Parallel Circuit
- To find the power dissipated in a resistor in a series-parallel circuit,
use any of the same formulas that you used for series circuits and
P = V × I
P = I2 × R
P = V2 ÷ 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 series-parallel circuit.
You'll get the same answer either way:
- Either find the power for each resistor, and
then add these powers:
PT = P1 + P2 + P3 +
... + Pn
- Or apply any one of the power formulas to the
PT = VS × IT
PT = IT2 × RT
PT = VS2 ÷ RT
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
to designate terminals that are connected together. The symbol looks
- 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
- 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:
- 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 e-Lesson has covered several important topics, including:
- identifying series and parallel relationships
- analyzing series-parallel circuits
- power in series-parallel circuits
- ground symbol and bubble symbol.
- To finish the e-Lesson, take this self-test to check your understanding
of these topics.
Congratulations! You've completed the e-Lesson for this unit.