physics module

7/29/2019 physics module

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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!

physics module

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