In this unit you’ll begin learning how to analyze circuits. **Circuit
analysis** means looking at a schematic diagram for a circuit
and computing the voltage, current, or power for any component in that
circuit. Closely related to the task of circuit analysis is the task
of **troubleshooting**, which means figuring out what
is wrong in a circuit that is not working correctly. Analyzing and
troubleshooting go hand in hand; when a circuit is not working correctly,
the easiest way to figure out what’s wrong is usually to measure voltages
in the circuit and compare those measured values to the values that
the voltages should have (which you compute by analyzing the circuit).

The rest of this course will concentrate on analyzing and troubleshooting **resistive
circuits** (circuits that contain only resistors in addition
to power supplies). In later courses you’ll learn how to analyze and
troubleshoot circuits that contain other components, such as capacitors,
inductors, diodes, and transistors.

This unit covers just about everything you could ever want to know about
the simplest type of circuit, which is called a **series resistive circuit**. Some
of the things you’ll learn here apply only to series resistive circuits,
but other things apply to any kind of circuit. For instance, the rule
called **Kirchhoff’s Voltage Law ** applies to all circuits,
and is therefore a very important general rule of circuit analysis.

- This unit will build on material that you studied in Unit 6. So let’s begin by taking this self-test to review what you learned in that unit.

- First, we’ll define some terms:
**series connection**,**series path**, and**series circuit**. And even before we define those terms, we’ll define what we mean by saying that two components are**connected**to each other. - Defining these terms carefully will be a big help later when we get to more complicated series-parallel circuits.

- Two components are said to be
**connected**to each other when there is a path of zero resistance joining a terminal of one component to a terminal of the other component. - Basically, this means that either the components are connected directly to each other, or there’s a conductor (such as a wire, or a trace on a circuit board) that connects them together.
- Examples: In the circuit shown below,
*V*_{S}is connected to R1 at one point;- R2 and R3 are connected to each other at two points;
*V*_{S}and R2 are not connected to each other.

- Now that we’ve defined what we mean by saying that two components are connected to each other, we can define what we mean by a series connection.
- 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. - Notice that there are two halves to this definition, both of which
must be met in order to have a series connection.
- First, the components must be directly or indirectly connected
**at exactly one point**, no more and no less. - Second,
**no other component can be connected to that point**where the other two components meet.

- First, the components must be directly or indirectly connected
- The most
**common mistake**that students make here is to remember the first half of the definition but forget about the second half. For example, in the circuit shown below, R1 and R2 are connected to each other at exactly one point, but the voltage source is also connected to that same point. Therefore, R1 and R2 are**not**connected in series.

- The most important property of series connections is that
**the current is the same in every series-connected component**. - Example: In the circuit shown below,
*V*_{S}and R1 are connected in series, so we know that the current through*V*_{S}must be the same as the current through R1. But R1 and R3 are not connected in series, so we cannot assume that the current through R1 is equal to the current through R3.

- Now that we’ve defined what we mean by a series connection, we can define what we mean by a series path.
- A
**series path**is a group of connected components in which each connection is a series connection. - As noted above, components connected in series have the same current.
Therefore,
**all of the components in a series path must have the same current as each other**

- Now that we’ve defined what we mean by a series path, we can define what we mean by a series circuit.
- A
**series circuit**is a (complete) circuit that consists**exclusively**of one series path. In other words, it’s a complete circuit in which every connection is a series connection. - As noted above, components connected in series have the same current as each other. Therefore, all of the components in a series circuit must have the same current as each other.
- In fact, this feature of having the same current is how some textbooks define
series circuits: "
**A series circuit provides only one path for current between two points so that the current is the same through each series component.**"

- We’ve already said this once or twice, but it’s so important that
we’ll repeat it:
**the current is the same everywhere in a series circuit**. - So, for example, in the series circuit shown below, if you measure or compute the current through any one of the components, then you immediately know the current through each of the other components. All of the components have the same current.
- By the way, this is true for
**all**series circuits, not just series resistive circuits. In the circuit shown below, if we replaced R2 with a diode, inductor, or other component, we would still be able to say that that component has the same current as R1, R3, and the voltage source.

- The
**total resistance**of resistors connected in series is simply equal to the sum of the individual resistance values. - Example: if a 1.2 kΩ resistor is connected in series with a 4.7 kΩ resistor, then the total resistance is 5.9 kΩ.
- We’ll use the symbol
to stand for total resistance. So for the case above, we could write*R*_{T}*R*_{T}= 5.9 kΩ. - The general formula for finding the total resistance of resistors
connected in series is
*R*_{T}=*R*_{1}+*R*_{2}+*R*_{3}+ … +*R*_{n}

- Instructors in the Wisconsin Technical College System have created a library of short online animations and quizzes to help students learn electronics. I’ll include links to some of these "learning objects." Whenever you see the icon below, click it to see a learning object on the material you’re studying. The Wisconin learning object will open in a new window; close the winow when you’re finished and want to return to this lesson.
- This first one will give you more practice calculating total series resistance.

- We now know enough to be able to find currents and voltage drops
in a series resistive circuit. There are four basic steps.
- Find the total resistance by adding all of the individual resistances:
*R*_{T}=*R*_{1}+*R*_{2}+*R*_{3}+ … +*R*_{n} - Apply Ohm’s law in the form
*I*=*V*÷*R*to the entire circuit. In words, the total current produced by the voltage source is equal to the source voltage divided by the total resistance. In symbols,*I*_{T}=*V*_{S}÷*R*_{T} - Recall that in a series circuit, every component has the same
current. Therefore, each resistor’s current is equal to the total
current. In symbols,
*I*_{T}=*I*_{1}=*I*_{2}=*I*_{3}= … =*I*_{n} - Use Ohm’s law in the form
*V*=*I*×*R*to find the voltage drop across each resistance. In symbols,*V*_{1}=*I*_{1}×*R*_{1}

*V*_{2}=*I*_{2}×*R*_{2}

*V*_{3}=*I*_{3}×*R*_{3}

and so on for each of the resistors.

- Find the total resistance by adding all of the individual resistances:
- These four steps
**do not apply to all circuits**. In particular, Steps 1 and 3 do not work for circuits that aren’t series circuits. (But Steps 2 and 4 do work for all resistive circuits.)

- Consider the series circuit shown below:

- We want to determine the current through each resistor and the voltage across each resistor.
- Our analysis of this circuit has four steps:
- Add the resistor values together to find the total resistance
*R*_{T}.

*R*_{T}= *R*_{1}+*R*_{2}= 100 Ω + 200 Ω = 300 Ω - Use Ohm’s law on the entire circuit to find the circuit’s total
current
*I*_{T}:

*I*_{T}= *V*_{S}÷*R*_{T}= 12 V ÷ 300 Ω = 40 mA - Recall that since this is a series circuit, current is the same
everywhere. Therefore, each resistor receives the total current:

*I*_{1}= *I*_{T}= 40 mA *I*_{2}= *I*_{T}= 40 mA - Use Ohm’s law on each resistor to find the voltage drops
*V*_{1}and*V*_{2}across the resistors:

*V*_{1}= *I*_{1}×*R*_{1}= 40 mA × 100 Ω = 4 V *V*_{2}= *I*_{2}×*R*_{2}= 40 mA × 200 Ω = 8 V

- Add the resistor values together to find the total resistance

- When we analyze a circuit to figure out the current, we’re interested
not only in the current’s
**magnitude**(or size), but also in its**direction**. - For example, in the series circuit shown below, the current has a
**magnitude**of 1.93 mA. What is the current’s**direction**? The current’s direction is clockwise around the circuit, because current comes out of the voltage source’s positive terminal (represented by the longer line in the symbol for a voltage source) and goes into the voltage source’s negative terminal.

- Also, when we analyze a circuit to figure out a particular voltage,
we’re interested not only in the voltage’s
**magnitude**(or size), but also in its**polarity**. Polarity means which end of the voltage is positive and which end is negative. Here’s how to figure out the polarity of the voltage across a resistor:- The end of a resistor into which current enters is the positive end;
- The end of a resistor from which current leaves is the negative end.

- For example, in the circuit shown above, we’ve determined that current flows clockwise around the circuit, which means that it flows into the left-hand end of R1 and out of the right-hand end of R1. Therefore, the polarity of R1’s voltage is: positive on the left-hand end, and negative on the right-hand end.
- The diagram below uses + and – signs to show the polarity of each
voltage in the circuit we’ve been discussing:

- Notice an important difference between voltage sources and resistors:
- Current flows
**out of a voltage source’s positive end**and into its negative end. - Current flows
**into a resistor’s positive end**and out of its negative end.

- Current flows

- Some circuit-analysis techniques (including one called Kirchhoff’s Voltage Law that we’ll introduce soon) require you to take an imaginary journey around a circuit, keeping track of the voltage changes as you travel.
- As you mentally move through the circuit, if you pass through a component
from its − end to its + end, we’ll call that a
**voltage rise**. - On the other hand, if you mentally pass through a component from
its + end to its − end, we’ll call that a
**voltage drop**. - Example: In the circuit shown below, suppose you decide to "travel" around
the circuit in a clockwise direction. Then you’ll encounter voltage
**drops**as you pass through R1, R2, and R3, and you’ll encounter a voltage**rise**as you pass through the voltage source.

- But, continuing the same example, suppose you now decide to "travel" around
the circuit in a counter-clockwise direction. Then you’ll encounter
voltage
**rises**as you pass through R1, R2, and R3, and you’ll encounter a voltage**drop**as you pass through the voltage source. - So a particular voltage can be considered as either a voltage drop or a voltage rise, depending on the direction of your imaginary trip around the circuit.

- Sometimes you’ll encounter a circuit with two or more voltage sources connected in series.
- A common example is a flashlight. As you probably know, most flashlights require two batteries. Each battery is a voltage source, and when you load the batteries into the flashlight, they go in end-to-end, so they’re connected in series.
- It’s pretty easy to analyze voltage sources connected in series.
You just have to be careful to notice whether the voltage sources are
trying to drive current in the same direction or in opposite directions.
We call these two cases
**series-aiding**voltage sources and**series-opposing**voltage sources.

- If two series-connected voltage sources are connected so as to produce
current in the same direction, they are said to be
**series-aiding**. - The net effect on the circuit is the same as that of a single source whose voltage equals the sum of the two voltage sources.
- Example: in the circuit shown below,
*V*_{S1}and*V*_{S2}are both connected so as to push current in a clockwise direction around the circuit. Therefore, they are series-aiding, and they combine to have the same effect as a 22-V source.

- When two series-connected sources are connected so as to produce
current in opposite directions, they are said to be
**series-opposing**. - The net effect on the circuit is the same as that of a single source
equal in magnitude to the
**difference**between the source voltages and having the same polarity as the larger of the two. - Example: in the circuit shown below,
*V*_{S1}tries to push current in a clockwise direction around the circuit, but*V*_{S2}tries to push current in a counter-clockwise direction. Therefore, they are series-opposing, and they combine to have the same effect as a 2-V source with the polarity of*V*_{S1}(pushing current clockwise around the circuit).

- You’re familiar with the concept of the voltage across
a resistor or other component. When we wish to refer to the voltage
across a particular resistor, we usually use a notation such as
*V*_{1}, which means the voltage across resistor R1. In this notation, we write a capital*V*with a single number as a subscript: this number is the number of the resistor whose voltage we’re talking about. - Sometimes, when we want to talk about a voltage between two points
in a circuit, we’ll use a different notation, which has a capital
*V*with two letters as a subscript, such as*V*._{AB} - In such cases, the points
*A*and*B*would be labeled in the circuit ‘s schematic diagram to identify them, as in the diagram below.

*V*means the voltage between points_{AB}*A*and*B*, with point*A*regarded as + and point*B*regarded as −. In other words, it’s the voltage that you would measure with a voltmeter if you placed the meter’s positive (red) lead at point*A*and the meter’s negative (black) lead at point*B*.- On the other hand,
*V*means the voltage between points_{BA}*A*and*B*, but with*B*regarded as + and*A*regarded as −. In other words, it’s the voltage that you would measure with a voltmeter if you placed the meter’s positive (red) lead at point*B*and the meter’s negative (black) lead at point*A*. - So in any given circuit,
*V*and_{AB}*V*will have the same magnitude, but one of them will have a negative sign and the other will not. For example, in the circuit shown above,_{BA}*V*= −12 V and_{AB}*V*= 12 V._{BA}

- Kirchhoff’s Voltage Law 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 use the abbreviation
**KVL**as a shorthand way of referring to Kirchhoff’s Voltage Law.

- When applied to a complete series resistive circuit with a single voltage source, KVL says that if you add the voltages across all of the resistors, the sum must be equal to the value of the source voltage.
- For example, consider the circuit shown below, which shows the polarities
of the voltages across the source and across the resistors.

- If we "travel" clockwise around the circuit, then voltages
*V*_{1},*V*_{2}, and*V*_{3}are voltage drops, and voltage*V*_{S}is a voltage rise. Since KVL says that the sum of the voltage drops must equal the sum of the voltage rises, we know that*V*_{1}+*V*_{2}+*V*_{3}=*V*_{S} - On the other hand, if we "travel" counter-clockwise
around the circuit, then voltages
*V*_{1},*V*_{2}, and*V*_{3}are voltage rises, and voltage*V*_{S}is a voltage drop. Since KVL says that the sum of the voltage drops must equal the sum of the voltage rises, we know that*V*_{S}*= V*_{1}+*V*_{2}+*V*_{3} - Either way, we reach the same conclusion: the sum of the resistor voltages is equal to the source voltage. So in the circuit shown above, we know that if we add together the voltages across the three resistors, we’ll get a sum of 10 V.

- If we "travel" clockwise around the circuit, then voltages

- KVL is a general rule that
**applies in all circuits**, not just series circuits and not just circuits containing resistors. In more complicated circuits, it can get tricky to apply KVL correctly, but when applied correctly it is a powerful tool. We’ll see this in later units.

- A string of series resistors is often called a
**voltage divider**because the total voltage across the entire string is divided among the various resistors in direct proportion to the resistance of each one.- For example, if you have two resistors in series and one resistor
is
**twice**as large as the other one (for example, suppose that one is 20 kΩ and the other is 10 kΩ), then there will be**twice**as much voltage across the larger resistor as there is across the smaller one. - On the other hand, if one of the series resistors is
**three times**as large as the other one (say, 30 kΩ and 10 kΩ), then there will be**three times**as much voltage across the larger resistor as there is across the smaller one.

- For example, if you have two resistors in series and one resistor
is

- The voltage-divider rule is a shortcut rule that you can use to find the voltage drop across a resistor in a series circuit.
- The rule says that
**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, this rule is expressed as:
*V*= (_{x}*R*÷_{x}*R*_{T})*× V*_{S} - Here
*x*is a variable representing the number of the resistor that you’re interested in.- For instance, if you’re trying to find the voltage across resistor
R1, you would replace
*x*with 1 to get:*V*_{1}= (*R*_{1}÷*R*_{T})*×**V*_{S} - On the other hand, applying the rule to resistor R4 in a series
circuit gives us:
*V*_{4}= (*R*_{4}÷*R*_{T})*×**V*_{S}

- For instance, if you’re trying to find the voltage across resistor
R1, you would replace
- Of course, you can also find these voltage drops using the procedure we used earlier: first find total resistance, then use Ohm’s law to find the current, and then use Ohm’s law to find the voltage drop that you’re interested in. Doing it this way will give you the same answer that you get by using the voltage-divider rule.

- In Unit
3 you learned that a
**potentiometer**is a type of variable resistor with three terminals, represented by the following schematic symbol:

- Recall that when you adjust a potentiometer you are moving the middle
terminal (called the
**wiper terminal**) toward one end or the other of the resistor. The resistance between the two end terminals stays constant, but the resistance between the wiper and either end terminal will change as you adjust the potentiometer. - In effect, what you have here is an adjustable voltage divider. In other words, it’s like having two resistors in series whose total resistance is constant, but you can adjust the relative sizes of the individual resistors by moving the wiper in one direction or the other.

- In Unit 1 of this course you learned three formulas for computing
the power dissipated in a resistor:
*P*=*I*^{2}×*R**P = V × I**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. - These formulas let you find a resistor’s power in any kind of circuit,
including a series resistive circuit. But you need to be a little careful. The
most common mistake that students make when using these formulas
is to use the wrong value for
*V*. In particular, students often mistakenly use the value of the source voltage when they should use the value of a single resistor’s voltage. - In the circuit shown below, for example, the source voltage (
*V*_{S}) equals 12 V, and R1’s voltage (*V*_{1}) equals 4 V. Here’s one correct way to find the power dissipated in R1:*P*_{1}=*V*_{1}^{2}÷*R*_{1}= (4 V)^{2}÷ 100 Ω = 160 mW**Correct answer!**Many students would incorrectly use the source voltage instead, which gives the wrong answer:

*P*_{1}=*V*_{S}^{2}÷*R*_{1}= (12 V)^{2}÷ 100 Ω = 1.44 W Incorrect answer!

- We’ve been talking about the power dissipated in a single resistor.
Not surprisingly, we use the symbol
*P*_{1}for the power dissipated in resistor R1,and the symbol*P*_{2}for the power dissipated in resistor R2, and so on. We can also talk about the**total power**dissipated in an entire circuit, for which we use the symbol*P*_{T}. - Here are two ways to compute total power in a resistive circuit.
You’ll get the same answer either way:
- You can find the power for each resistor, and then add these
powers:
*P*_{T}=*P*_{1}+*P*_{2}+*P*_{3}+ … +*P*_{n} **Or**you can apply any one of the power formulas to the entire circuit:*P*_{T}=*I*_{T}^{2}×*R*_{T}*P*_{T}=*V*_{S}×*I*_{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.*P*_{T}=*V*_{S}^{2}÷*R*_{T}

- You can find the power for each resistor, and then add these
powers:

- Earlier in this unit you learned about the double-subscript notation
that we use to talk about a voltage between two points in a circuit.
For instance, if we have points labeled
*A*and*B*in a circuit, we use the symbol*V*to refer to the voltage between those two points. As mentioned earlier, this is the voltage that you would measure with a voltmeter by placing the meter’s positive (red) lead at point_{AB}*A*and the meter’s negative (black) lead at point*B*. - In many cases, we’re interested in knowing the voltage at a point
in a circuit relative to the circuit’s ground. In such cases we use
a notation that has a capital
*V*with one letter as a subscript, such as*V*. This is the voltage at point_{A}*A***relative to ground**, which simply means the voltage that you would measure with a voltmeter by placing the meter’s positive (red) lead at point*A*and the meter’s negative (black) lead at the circuit’s ground point.

**Troubleshooting**a non-working circuit means finding the problem that is preventing the circuit from working correctly.- The two most common types of problems are open circuits and short circuits.

- An
**open circuit**, or "open," is a break in a circuit path. - For example, when a light bulb burns out, it causes an open. A resistor or other component can also fail by becoming open. This can happen,for instance, if too much current passes through a resistor, causing it to "burn out."
- A circuit containing an open is said to be an
**open circuit**, or to be**open-circuited**.

- The most important thing to remember about opens is that
**no current can flow through an open**. - Therefore, no current can flow anywhere in a series circuit containing an open.
- Since no current flows through an open, you can think of the open
as having infinite resistance (
*R*= ∞).

- A
**common mistake**is to believe that since the current through an open is zero, the voltage across the open must also be zero. - Usually, an open will
**not**have a voltage drop of 0 V. In fact, in a series circuit that contains an open,**the entire source voltage will appear across the open, and no voltage will appear across any of the other resistors**. - So if you measure the voltage between any two points in a series
circuit containing an open, you’ll measure 0 V if the two points
are on the same side of the open, but you’ll measure the entire source
voltage if the points are on opposite sides of the open.
- For example, suppose R3 is open in the circuit shown below. Then
there will be 0 V across R1, across R2, and across R4. Also,
*V*= 0 V. But there will be 9 V across R3. Also,_{ab}*V*= 9 V, and_{ac}*V*= 9 V._{bc}

- For example, suppose R3 is open in the circuit shown below. Then
there will be 0 V across R1, across R2, and across R4. Also,

- A
**short circuit**, or "short," is a path of zero resistance connecting two points in a circuit that are not supposed to be connected. - For example, a wire clipping or a loose lump of solder can accidentally touch the leads of two resistors, thereby connecting those resistors to each other.

- Since a short has zero resistance, the voltage across it must be
zero. This follows from Ohm’s law,
*V*=*I*×*R*.

- A component is said to be short-circuited, or "shorted out," when
there is a short circuit connected across it. No current flows
through a short-circuited component. Instead, current is diverted
through the short itself.
- For example, suppose that in the circuit shown below there is a
short between points
*a*and*b*, perhaps caused by a loose wire clipping that connects these two points. Then R2 is short-circuited. No current will flow through R2; instead, current will follow the path of zero resistance through the short itself (the wire clipping).

- For example, suppose that in the circuit shown below there is a
short between points
- A short in a series circuit reduces the circuit’s total resistance,
causing more current to flow out of the voltage source.
- For example, in the circuit shown above, if R2 is short-circuited
by a wire clipping that connects points
*a*and*b*, then R2’s resistance disappears from the circuit, and the circuit’s total resistance is equal to*R*_{1}+*R*_{3}+*R*_{4}.

- For example, in the circuit shown above, if R2 is short-circuited
by a wire clipping that connects points

- The following animated lesson shows some of the real-world conditions that typically cause shorts or opens in circuits. It’s got some good practical examples, so be sure no to skip it.

- This e-Lesson has covered several important topics, including:
- series connections and series paths
- series circuits
- voltage drops and voltage rises
- voltage sources in series
- Kirchhoff’s Voltage Law (KVL)
- voltage dividers and the voltage-divider rule
- power in series circuits
- open circuits and short circuits.

- 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.