Unit 5.
Ohm's Law
The main topic in this unit is the most fundamental equation in electronics:
Ohm's Law. This is an equation that relates voltage, current, and resistance.
We'll start to discuss a topic that's not covered in the book, but
that's important for a technician: how to estimate answers to
mathematical problems without using a calculator. Finally, we'll look
beyond series circuits and parallel circuits to more complicated circuits
called seriesparallel circuits.
Unit 4 Review
 This unit will build on material that you studied in Unit
4. So let's
begin by taking this selftest to review what you learned in that unit.

A Simple Circuit
 Here is a very simple circuit that contains just a voltage source
(V_{S}) and a resistor (R):
 Note: In drawings like this, sometimes we'll use the label V_{S} for
the source voltage (as we have done here), and sometimes we'll use
the label V for the source voltage. The two labels mean
the same thing, so don't think that there's an important difference
between using V_{S} and using V.
 Some textbooks use the label E for a source voltage,
so you may see this label from time to time.
 How will the current in this circuit change if we change either the
source voltage or the resistance? (Read on for the answer.)
Current Depends on Voltage
 Voltage is a measure of a source's ability to produce current.
The greater the voltage, the greater the current that the source can
produce in a fixed resistance.
 In other words, current produced in a resistance is directly proportional to
the voltage of the source.
 So, for instance, in the circuit shown above, if we double the
source voltage V_{S} while we keep the resistance R constant,
then the current through the circuit will double.
 On the other hand, if we reduce the source voltage V_{S} by
half while we keep the resistance R constant, then
the current through the circuit will be reduced by half.

Current Depends on Resistance
 Resistance reduces the flow of current. The greater the resistance,
the smaller the current produced through it by a fixed source voltage.
 In other words, current is inversely proportional to resistance.
 So, for instance, in the circuit shown above, if we double the
resistance R while we keep the source voltage V_{S} constant,
then the current through the circuit will be reduced by half.
 On the other hand, if we reduce the resistance R by half while
we keep the source voltage V_{S} constant, then the current
through the circuit will double.


Ohm's Law
 The points made above can be combined in a single mathematical expression
called Ohm's law:
I = V ÷ R
 In words, the current in a circuit is equal to the source voltage
divided by the circuit's resistance.

Other Forms of Ohm's Law
 Using basic algebra, we can rewrite Ohm's law in two different
forms. One is:
V =
I × R
 In words, this says that voltage equals current times resistance.
 The other form of Ohm's law is:
R = V ÷ I
 In words, resistance equals voltage divided by current.
Ohm's Law Triangle
 Many people use the following memory aid to
help them remember the three different forms of Ohm's law:
 Draw
the triangle shown below.
 Use your finger to cover up the
letter that you're solving for.
 The two remaining letters will then
be in the correct position:
 If the two letters are sidebyside,
then you have to multiply them.
 If one letter is positioned over the
other one, then you have to divde.
 For example, if you cover up the I, then you're left with
a V over
an R, which tells you that I = V ÷ R.
 Of course,
when you draw the triangle you have to put the letters in the correct
positions. To help you remember that the V goes on top and
the I and R go below, remember this little
saying: "The Vulture flies over the Indian,
who walks by the River."
 Try it out on the following
questions!

Voltage Drops
 So far we've been discussing a circuit with just one resistor connected
across a voltage source. In such a case, we say that the entire source
voltage is dropped across the resistor. This means
that if we use a voltmeter to measure the voltage across the resistor,
we'll get the same value that we would get if we measured across the
voltage source
itself.
 But most circuits contain more than one resistor.
In this case, each resistor will have what's called a voltage
drop,
but usually this is not
equal to the source voltage.
 Example: The circuit shown below has two resistors, R_{1} and
R_{2}, connected to a 12volt source. In this circuit, R_{1} will
have a voltage drop of 4 volts, and R_{2} will have
a voltage drop of 8 volts. (Later you'll learn
the equations that let you find these values.)
 We express these facts by writing V_{1} = 4 V, and V_{2} =
8 V. As you can probably guess, V_{1}means the voltage drop
across
R_{1}, and V_{2}means the voltage
drop across R_{2}.
Ohm's Law Again
 Ohm's law applies to all circuits containing resistors, but you have
to be careful how you use it.
 When you apply it to the entire circuit, Ohm's law says
that the circuit's current is equal to the source voltage divided
by the circuit's total resistance.
 When you apply it to a single resistor, Ohm's law says
that a resistor's current is equal to the resistor's voltage drop
divided by the resistor's resistance.
 In symbols, when we're talking about the entire circuit, we'll write
I_{T} = V_{S} ÷ R_{T
}
where I_{T} is
the circuit's total current (that's the current coming out of the battery),
and V_{S} is the source voltage, and R_{T} is the circuit's total resistance.
 But when we're talking about a single resistor, we'll write
I_{1} = V_{1} ÷ R_{1
}
where I_{1} is
the current through resistor R1, and V_{1} is
the voltage drop across resistor R1, and R_{1} is the resistance
of resistor R1.
 This relation holds for every resistor. So we can also write I_{2} = V_{2} ÷ R_{2},
and so on. (And we can also rearrange these equations to solve for
voltage or resistance.)

Linearity
 An electrical device is said to be linear if
its voltagecurrent relationship possesses two mathematical properties named
homogeneity and additivity.
 Homogeneity: If a linear element's voltage is v when
its current is i, then for any constant k, its voltage
is kv when its current is ki.
 Additivity: If a linear element's voltage is v_{1}
when its current is i_{1}, and its voltage is v_{2} when its current is
i_{2}, then its voltage is v_{1} + v_{2} when its current is i_{1} + i_{2}.
 A resistor is an example of a linear device. Other examples are capacitors
and inductors, which you will study later. If you drew a graph of voltage
versus current for a 3.3 kΩ resistor, the graph would be
a straight line, as shown below.
 A diode is an example of a device that is not linear. If you
drew a graph of voltage versus current for a diode, the graph would
look something like the one shown below. We will not study diodes or
any other nonlinear devices in this course, but you will study them
in later courses.
 An circuit is linear if it contains only linear devices,
independent sources, and linear dependent sources.
 Some circuitanalysis techniques that we'll study apply only to
linear circuits.
 All of the circuits we will study in this course are
linear circuits.

Estimation
k and m Are "Opposites"
 When I say that kilo and milli are opposites, what I mean mathematically
is that kilo stands for 10^{3} and milli stands for 10^{3}.
Note that the number part of the exponents are the same, but one exponent
is positive while the other exponent is negative.
 Here are a few situations in which it's useful to know that kilo
and milli are the "opposite" of each other:
 When you multiply kilo times milli,
they cancel each other, leaving you with no metric prefix:
 Example: 6 mA × 2 kΩ =
12 V.
 When you're doing a division, if you have kilo in
the denominator and no metric prefix in the numerator, your
answer's metric prefix will be milli:
 Example: 10 V ÷ 5 kΩ =
2 mA.
 When you're doing a division, if you have milli in
the denominator with no metric prefix in the numerator, your
answer's metric prefix will be kilo :
 Example: 15 V ÷ 3 mA
= 5 kΩ.

M and μ Are "Opposites"
 In the same way that kilo and milli are the opposite of each other,
mega and micro are also the opposite of each other, since mega stands
for 10^{6} and micro stands for 10^{6}.
 Here are a few situations in which it's useful to know that mega
and micro are the "opposite" of each other:
 When you multiply mega times micro,
they cancel each other, leaving you with no metric prefix:
 Example: 5 µA × 3 MΩ =
15 V.
 Also, when you're doing a division, if you have mega in
the denominator and no metric prefix in the numerator, your
answer's metric prefix will be micro :
 Example: 12 V ÷ 4 MΩ =
3 µA.
 And if you have micro in
the denominator with no metric prefix in the numerator, your
answer's metric prefix will be mega :
 Example: 16 V ÷ 8 µA
= 2 MΩ.

SeriesParallel Circuit
 A seriesparallel circuit is one that contains combinations
of series and parallelconnected components, which are in turn connected
in series and/or parallel with other such combinations.
 Example: The circuit shown below, which appeared in some of
the selftest questions for Unit 4, is a seriesparallel circuit.
Seeing Series Connections and Parallel Connections
 In working with seriesparallel circuits, it's very important that
you be able to see which components are connected in series with each
other, and which are connected in parallel with each other.
 Example: In the circuit above, R2 is connected in parallel
with R3. Also, V_{S} is connected in series with R1, and V_{S} is connected
in series with R4. (If you don't understand why, go back and review
the meanings of "connected in series" and "connected
in parallel" in Unit 4.)

Breadboarding SeriesParallel Circuits
 Once you've spent some time studying how the components in a particular
circuit are connected with each other, you should be ready to build
the circuit on a breadboard.
 Example: Repeated below is the seriesparallel circuit from
above. Beneath the schematic diagram is a breadboard diagram showing one
way of building this circuit. (Of course, there are many other
correct ways of building it.)
 Study this
example until you are satisfied that the schematic diagram and the
breadboard diagram represent the same circuit.
 Now here's a photograph of the same circuit built on an actual breadboard.
 Be
sure to try these SelfTest questions!
Measurements in SeriesParallel Circuits
 Once you've built a seriesparallel circuit, you'll want to be able
to make measure voltage drops and currents in the circuit.
 Measuring voltage drops is easy. To measure the voltage drop
across a resistor, you set the meter to measure voltage, and then you
touch the meter's two leads to the two legs of the resistor.
 Another way of saying this is that the voltmeter must be connected
in parallel with the resistor whose voltage you wish to measure.
 Measuring current is usually trickier. To measure the current
through a resistor, you must first lift one leg of that resistor out
of the breadboard. (It does not matter which of the resistor's legs
you lift.) Then set the meter to measure current, and connect one meter
lead to the detached leg of the resistor, and connect the other meter
lead to whatever the resistor leg was originally attached to.
 A simpler way of saying this is that the ammeter must be connected
in series with the resistor whose current you wish to measure.
 Example: In the circuit that we've been discussing, suppose
you wish to measure R2's current. Shown below is one way of doing this.
Notice that one leg of R2 has been detached from what it was originally
attached to (namely, R1 and R3), and then the meter was inserted between
R2 and those things to which R2 was originally attached (R1 and R3).

Unit 5 Review
 This eLesson has covered several important topics, including:
 Ohm's law
 estimation
 seriesparallel circuits.
 To finish the eLesson, take this selftest to check your understanding
of these topics.

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