physics module
TRANSCRIPT
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PHYSICS II
ELECTRICITY, MAGNETISM, SOUND, AND LIGHT
SERIES-PARALLEL CIRCUIT
In module 1 and 2 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 tricky.
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 fromprevious modules:
o Two components are connected in series if they are connected toeach other at exactly one point and no other component is
connected to that point.
o Two components are connected in parallel if they are connected toeach other at two points.
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In the example circuit shown above, R2 is in series with R3, which is inseries 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.
The only way to learn series-parallel circuits is by studying lots ofexamples.
Notation for Series-Parallel Relationships
As a short-hand way of describing series-parallel relationships in a circuit,we'll use two symbols:
o we use the symbol + to indicate a series connection betweencomponents
o 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
Modules 1 and 2 gave step-by-step rules for analyzing series circuits andparallel circuits. It was possible to do this because one basic approachworks 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 allseries-parallel circuits. There is too much variety among series-parallel
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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, youneed 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 resistanceby combining pairs of series-connected or parallel-connected resistors
until you've combined all the resistors into a single resistance.
o For example, in the circuit shown below, we would add R2, R3, andR4 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 tofind the circuit's total current:
IT = VS RT
After that, you can apply any of the rules and laws from previous units, inany sequence that leads to the solution for a particular voltage or current.
The most useful of these rules and laws are:
o Ohm's law
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o voltage/current rules for series or parallel combinationso Kirchhoff's Voltage Lawo Kirchhoff's Current Lawo the voltage-divider ruleo the current-divider rule
Let's review these rules and laws, and discuss their application to series-parallel circuits.
Ohm's Law in Series-Parallel Circuits
Recall that Ohm's law relates resistance, current, and voltage. As youknow, the three forms of Ohm's law are:
I = V R
V= I R
R = V I
Ohm's law can be applied to a single resistor, or to an entire resistivecircuit, or to any resistive portion of a circuit.
o For instance, in the example circuit that we've been discussing,Ohm's law can be applied to any one of the resistors by itself .
Applying it to resistor R3, we can write:
I3 = V3 R3
o Or we can apply Ohm's law (in any of its forms) to the entire circuit,letting us write:
IT = VS RT
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Remember that IT means the total current passing through the
voltage source.
o Or we can apply Ohm's law to a portion of the circuit. For instance,recall that our example circuit contains a series path made up of R2,
R3, and R4. Applying Ohm's law just to this series path, we can say
that the current through the series path is equal to the voltage
across the series path divided by the resistance of the series path.
In symbols:
I2+3+4
= V2+3+4
R2+3+4
To avoid being confused by this notation, remember that here
we're using + to stand for series connection, not mathematical
addition. So we're using the subscript 2+3+4 to denote the series
combination of R2, R3, and R4. So R2+3+4 is the resistance of this
series path, and V2+3+4 is the voltage across this series path, and
I2+3+4 is the current through this series path.
So although Ohm's law is one of the simplest laws in electronics, it is alsovery powerful and very flexible. Many complicated-looking problems in
electronics can be solved using nothing more than Ohm's law.
Voltage/Current Rules for Series or Parallel Combinations
Here are two more simple-but-powerful rules. Recall that:o Components connected in series must have the same current (but
usually they won't have the same voltage).
o Components connected in parallel must have the same voltage (but usually they won't have the same current).
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These two rules apply not only to individual components but also toportions of circuits.
o In our example circuit, for instance, resistor R1 is connected inparallel 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
o Also, in the same circuit, since R2, R3, and R4 form a series path, weknow 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 closedloop 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 forall circuits.
o In our example circuit, for instance, one closed loop contains justthe voltage source and R1. Therefore KVL tells us that the voltage
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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.
o Another closed loop in our example circuit contains the voltagesource 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 + V4
Kirchhoff's Current Law in Series-Parallel Circuits
Recall that KCL says that the sum of all currents entering a point is equalto the sum of all currents leaving that point.
We originally studied KCL in the context of parallel circuits, but KCL holdsfor all circuits.
o In our example circuit, for instance, consider the point where thevoltage source meets R1 and R2. KCL tells us that the current
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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 Circuits
The rules listed above (Ohm's law, KVL, and KCL) are your most importanttools for analyzing circuits. Most series-parallel circuits can be analyzed
using just those rules. In some cases, the voltage-divider rule and the
current-divider rule are also useful. However, it's easy to make mistakes
when you apply the voltage-divider rule and current-divider rule to series-
parallel circuits. Therefore, although we'll now discuss these two rules, I
advise you to rely primarily on Ohm's law, KVL, and KCL.
In discussing series circuits, we stated the voltage-divider rule as follows:the voltage across any resistance in a series circuit is equal to the ratio of
that resistance to the circuit's total resistance, multiplied by the source
voltage. In equation form:
Vx= (Rx RT) VS
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The same basic rule applies to any series path in a series-parallel circuit(but not to an entire series-parallel circuit, and not to any portion of a
series-parallel circuit that is not a series path). Since the original wording
of the rule applies only to series circuits, we need to change the wording
slightly, as follows: the voltage across any resistance in a series path is
equal to the ratio of that resistance to the path's total resistance,
multiplied by the voltage across the series path. In equation form:
Vx= (Rx Rpath) Vpath
Here I'm using two symbols whose meanings hopefully are clear:
Rpathmeans the total resistance of the series path, and Vpath means the
voltage across the entire series path. As in the original voltage-divider rule,
x is a variable representing the number of the resistor that you're
interested in.
o In our example circuit, for instance, we have a series pathcontaining resistors R2, R3, R4. If we know the voltage across the
entire series path, then the voltage-divider rule lets us find the
voltage across any one resistor in that path. For instance, the
voltage across R2 is given by:
V2 = (R2 Rpath) Vpath
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o It's very important to remember that this rule applies only toresistors in a series path. For instance, since R1 is not part of a
series path in this circuit, we cannot use the voltage-divider rule to
find its voltage:
V1 = (R1 RT) VS Incorrect!
Because students often apply the voltage-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 voltage-divider rule.
The Current-Divider Rule in Series-Parallel Circuits
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 equation form:
Ix= (RT Rx) IT
The same rule applies to parallel branches in a series-parallel circuit. Butyou 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.
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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 current.
o In our example circuit, for instance, we have two parallel branches.One branch contains just R1, and the other branch is a series path
containing R2, R3, R4. If we know the total current entering both
branches, then the current-divider rule lets us find the current into
either branch. For instance, the current into the series path is given
by:
Ipath = (RT Rpath) IT
o But the following would not be a proper application of the current-divider rule, since R2 does not by itself form a complete branch:
I2 = (RT R2) IT Incorrect!
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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 rule.
Examples
We have now reviewed the use of Ohm's law, Kirchhoff's Laws, and thedivider 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 problemor 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 totalequivalent 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, useany of the same formulas that you used for series circuits and parallel
circuits:
P = V I
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P = I2
R
P = V2
R
Recall that in each of these equations, R is the resistor's resistance, Vis thevoltage 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 tocompute total power dissipated in a series-parallel circuit. You'll get the
same answer either way:
1. Either find the power for each resistor, and then add these powers:PT = P1 + P2 + P3 + ... + Pn
2. Or apply any one of the power formulas to the entire circuit: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.
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Ground (or Common) Symbol
Instead of drawing lines to show connections in a schematic diagram, weoften 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 thatclutter 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
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connected together, we could redraw the diagram as shown below.
This is not a very good example, because the redrawn diagram with theground 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 omitthat symbol from the side of any voltage source that is connected to
common.
The terminal that isnot 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 theschematic diagram once more using the bubble symbol to arrive at the
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following diagram:
Circuit Geometry
The geometry of a circuit (the way the component connections are shownin 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 manyways as necessary to make it easier to read.
Review
This Lesson has covered several important topics, including:o identifying series and parallel relationshipso analyzing series-parallel circuitso power in series-parallel circuitso ground symbol and bubble symbol.
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To finish the Lesson, take this self-test to check your understanding ofthese topics.
Congratulations! You've completed the e-Lesson for this unit. What's next?
What will happen next is up to you!