Controlling The Real World With Computers
::. Control And Embedded Systems .::

How to Read a Schematic And Basic Electronics

We will begin at the beginning by discussing a little basic electronics. Don't worry. I've taught 10-year-old girls and boys the same material as part of a church ministry, and they know pretty much everything you will read in this section. I hope there are other adults who will teach kids this material. Work through an organization where you can get a background check and go for it. It's worth the effort, even if you are a 72-year-old like me. In my case it's because my kids need to know that education is the surest way out of their poverty. Maybe you know children in the same situation.

This web site covers the basic concepts of embedded and control systems. With that knowledge, a person can open the door to a career that represents the occupational future of this country. Run a search for "embedded and control systems salary survey" to see how well the effort pays off monetarily. Just as important is the pure joy of working with robots and other grown up toys.

So let's get started. In general terms, a circuit can be described as any group of electrical or electronic devices connected together by conductors. Conductors are most often metallic, and wires were the conductor of choice in the past. Old radios and other electronic equipment were often a rat's nest of wires, a little like the old Silvertone guitar amp I've been refurbishing:

Today, it's more common to find metallic pathways, often called
traces, on a board constructed of a mixture of fiberglass and epoxy, such as the board for one of my products, The Listening Car:

The terms
board and card are synonymous in this context. You will often hear professionals use the initials P.C.B. (Printed Circuit Board).

A schematic in electronics is a drawing representing a circuit. It uses symbols to represent real-world objects. The most basic symbol is a simple conductor, shown simply as a line. If wires connect in a diagram, they are often shown with a dot at the intersection:

Conductors that do not connect are shown without a dot, or with a bridge formed by one wire over the other:

Among the connections are power and ground, the high and low system voltages respectfully. The 5 or 3 volt system power in the schematic of a computer is shown simply as 5V or 3V. There is also a +12V supply and a -12V supply. Ground, or 0 volts, has its own symbol:

A switch is a device that is capable of allowing the user to make or break the circuit as if the wire had been connected or broken. Its symbol reflects this characteristic:

A resistor is a device that resists the flow of charge. Its symbol reflects this characteristic by making the line jagged:

Just in case you have seen or heard "flow of current" elsewhere rather than "flow of charge", see the definition of current below.

The unit of resistance is the ohm, pronounced om with a long o. The K you will find in a lot of schematics stands for kilohm or thousands of ohms. 10K means the same as 10,000. Meg and sometimes M mean megohm or million ohms. 4.7Meg or 4.7M is the same as 4,700,000.

You will sometimes see two variations on resistors in some schematics. One is the resistor array or network. It is commonly found in a Single In-line Package (SIP) containing several resistors connected together. They can be found in many configurations. One arrangement simply connects one end of the resistors to each other and brings them out to a common connection. The other end of each resistor is left free:

Another, more common variation is the variable resistor. It has a third contact that can move along the resistor element to permit the values at that point to be variable. The moveable part is called the wiper and is shown as an arrow in a schematic:

There is a relationship between voltage, current and resistance that is expressed by Ohm's Law, which says that Voltage is equal to Current times Resistance, or:

V = I * R

V is voltage (often referred to as Electromotive Force where E rather than V is used), I is current and R is resistance. Current is expressed in Amperes, or amps for short. Very little current is used in typical electronic circuits, so milliamps, which means 1/1000 amp, is used. One milliamp = .001 amp. It's abbreviated ma, or sometimes MA.

To paraphrase a definition of charge from whatis.com :

"The coulomb (symbolized C) is the standard unit of electric charge in the International System of Units (SI). It is a dimensionless quantity. A quantity of 1 C is equal to approximately 6.24 x 1018, or 6.24 quintillion."

"In terms of SI base units, the coulomb is the equivalent of one ampere-second. Conversely, an electric current of 1 amp represents 1 C of unit electric charge carriers flowing past a specific point in 1 second. The unit electric charge is the amount of charge contained in a single electron. Thus, 6.24 x 1018 electrons have 1 C of charge. This is also true of 6.24 x 1018 positrons or 6.24 x 1018 protons, although these two types of particles carry charge of opposite polarity to that of the electron."

Since we deal mostly with electrons in electronics, 1 amp represents the effect of 6,240,000,000,000,000,000 electrons flowing past a point per second. Thus, since current is already defined as something flowing, to say "current flow" would be to say "..... flowing flow" which is incorrect because it is redundant.

Now let's say we have a 10K resistor and 2 milliamps of current. The voltage across the resistor will be:

V = I * R = .002 * 10,000 = 20 volts

We can use the original Ohm's Law equation and a little simple algebra to generate an equation for each of the three variables. It would be a good idea to learn the following well, as knowing at least some simple algebra will be necessary in electronics. It requires remembering just two things:

1. It's OK to do something to one side of an equation as long as the same thing is done to the other side. The two sides will remain equal.

2. Anything divided by itself is equal to 1.

V = I * R
Now divide both sides by R. Since R/R = 1, the right side now becomes I * 1 which is simply I, giving us V/R = I. If we switch sides and put the I on the left we end up with:
I = V/R

V = I * R
Now divide both sides by I. Since I/I = 1, the right side now becomes R * 1 which is simply R, giving us V/I = R. If we switch sides and put the R on the left we end up with: R = V/I

Thus, all three equations are:
V = I * R
I = V/R
R = V/I

Always work a problem by writing down the equation first. For example,
What is I if V = 10 and R = 5?

Write down the equation:
I = V/R

Now plug in the values, so here's what you get:
I = V/R = 10/5 = 2 Amps

One way to remember the three equations is to say, "The Vulture looks down and sees the Iguana and the Rabbit side by side (V = I * R), the Iguana sees the Vulture over the Rabbit (I = V/R) and the Rabbit sees the Vulture over the Iguana (R = V/I)." But please learn the algebra. It will serve you well in the future.

A very common circuit is a voltage divider. An example might be like the following:

Two components connected end-to-end, such as the resistors above, are said to be connected
in series. Here, the total resistance is simply the sum of the two. In this case, it would be 22000 + 33 = 22033 ohms. If 1 volt is applied to the open end of the 22K resistor, the current through the whole circuit would be I = V/R = 1/22033 or .00004538646576 amps, or about .05 milliamps.

The voltage across the 33 ohm resistor is then
V = I * R = .00004538646576 * 33 = .00149775337 volts, or about 1.5
millivolts (1/1000 volt).

Resistors are also often connected in parallel , such as below:

The value of the above parallel network is:
R = 1/(1/R1 + 1/R2 + 1/R3)
The equation is good for any number of resistors. There's more on parallel resistors below.

The following shows the symbols and units for voltage, resistance and current. Power is also shown since it can be found from the other quantities: P = VI:
 Quantity Symbol Unit Voltage or Electromotive Force V or E Volts Resistance R Ohms Current I Ampere or Amp (often milliamp in electronics) Power P Watts

A color code is used to determine the value of a resistor:

 Number Color Memory Word 0 Black Bad 1 Brown Boys 2 Red Rob 3 Orange Our 4 Yellow Young 5 Green Girls 6 Blue But 7 Violet Violet 8 Gray Gives 9 White Willingly 5% or .1 multiplier Gold 10.00% Silver

To determine the value, first write down the first 2 numbers from the first two colors. Then put the number of 0s indicated by the third color after the first 2 numbers.

Consider these examples:

The above is yellow, violet, brown. Yellow=4, Violet=7 and Brown=1, so the first two digits are 47 and we put 1 zero on the end to make 470 ohms.

Or this:

This one is gray, red and black. The last color indicates the temperature coefficient. Gray=8, Red=2 and Black=0, so the first two numbers are 82 and there are no zeros on the end, so it's 82 ohms. You can see the gold tolerance band a little better here, which means the resistor is 82 ohms +/- 5%.

Download my free ColorCode program HERE to help you learn the code. There are some other applications there that might be useful as well.

Capacitors are devices which have metal plates separated by an insulator. They are used to temporarily store an electrical charge. Their symbol reflects their construction:

The unit of capacitance is the Farad, but it's so large that the
microfarad is used in practice. Microfarad means millionths of a Farad. It's often abbreviated mf, MF or some variation, although the correct abbreviation is µF. A value without a designator is assumed to be in microfarads. For example, in a schematic you might see several capacitors simply designated .1, but they are actually .1µF capacitors.

Some capacitors must have their leads connected to the positive or negative side of a circuit. They are polarized capacitors. When that is the case, one side will be shown with a + sign where the positive side must be, or a - sign where the negative side must be, or both.

It's also very common to see picofarads abbreviated pf in some schematics. A picofarad is 10-12 Farad, and is sometimes called a micro microfarad.

A diode permits the flow of charge in only one direction. Its symbol reflects this characteristic, but with a slight problem:

Anode Cathode

The slight problem comes from the fact that flow of charge, at least in a wire, is from where there are a greater number of electrons to where there are fewer. Electrons are negatively charged. Thus, electrical flow of charge is from negative to positive in a wire. The problem with the symbol is that the cathode, not the anode, is the negative side. Electrical flow of charge is from the cathode to the anode, against the direction of the arrow. It's backwards because Benjamin Franklin thought the flow of charge was from positive to negative. A special kind of diode is the Light Emitting Diode (LED), shown as a diode with lines representing light waves:

Integrated Circuits contain many individual components. They, in turn, usually form several functional blocks. For example, the following page shows a pinout for the 74LS08 Quad 2 Input AND gate, along with its truth table. VCC is the 5 volt supply, and GND is ground. Sometimes ground is shown as VSS. The gate inputs are the As and Bs, and the outputs are the Ys. Thus, the inputs to gate 1 are 1A and 1B, and the output is 1Y. You will see variations on these conventions, but they hold true in many cases.

An Operational Amplifier also contains many individual components, but is not a digital circuit. It looks a little like a buffer which can be found in the Boolean Logic section, but has 2 inputs:

You can find a more detailed treatment of operational amplifiers on the Internet. For a simplified coverage of the subject, look at the circuit below:

An Op-Amp has many important characteristics. One of them is that the above circuit, called an inverting amplifier, attempts to prevent any current through the inverting (-) input. In this circuit, Rin connects to the inverting input. Rfb also connects to the inverting input, with its other end connected to the output. Rfb is called the feedback resistor. Let's attempt to drive a current through the inverting input by placing 1V on the unconnected end of Rin and assume that the right end has 0 volts on it. If Rin == 1K, the current will be Iin = V/Rin = 1/1K = 1ma

The output will try to counter this by driving a current of the opposite polarity through the feedback resistor into the inverting input. With a 10K feedback resistor, the required voltage to do that will be: V = -(I * Rfb) = -(1ma * 10K) = -10V.

Thus, we get a voltage to current conversion, a current to voltage conversion, a polarity inversion and, most importantly, amplification. Amplification or gain is commonly labeled G. In the case of the inverting amplifier,
G = -(Feedback Resistor / Input Resistor)
In this case, it's G = -(Rfb/Rin) = -10.

Since the feedback cancels out the input, there is no voltage at the inverting input. It is said to be at virtual ground .

Now look at the circuit below taken out of a schematic from one of my projects:

The gain is a little over -1000 in order to provide enough amplification for the low-level output of a microphone. The signal is not only amplified but inverted because we are going into the inverting input. The inversion however, is not quite the same as it is in a digital device. Here, we are talking about an audio analog signal that, once transformed into an electrical signal by the microphone, moves smoothly and continuously in the negative and positive voltage directions. Inversion here means that when the input moves in the positive direction, the output moves in the negative direction. When the input goes toward negative, the output goes toward positive. C1 prevents DC voltages from even getting into the circuit. This blocking action will be discussed in a future section.

The non-inverting side is designated by the +. It is there that a positive offset voltage is applied. If R1 were not connected to C1 but rather to ground, the non-inverting side would exhibit a gain of (R2/R1)+1 for the bias voltage. With C1 however, there is no DC gain for the non-inverting side, and AC is effectively shorted to ground by C2. The result is a gain of 1 on the non-inverting side for DC voltages.

The following is a self-test over this section. You will find other, shorter tests in various sections below. It would be a very good idea to make sure you know the answers to all of the questions in this document since the sections that follow will build on this one.

1. _____ is a drawing that represents a circuit.

A) Switch
B) Schematic
C) Ground
D) Diagram

2. A _____ is a device that allows the user to break the circuit.

A) Scissors
B) Schematic
C) Resistor
D) Switch

3. A _____ is a device that resists the flow of charge.

A) Resistor
B) Buffer
C) Diode

4. The unit of resistance is the ____ . The relationship between voltage, current, and resistance is expressed by ____ .

A) Buffer B) Ohm's Law C) Amplifier D) Capacitors
E) Ohm F) Diode G) Circuits H) Switch

5. The ____ is the unit of current. If there is very little current, it is expressed as ____, which means 1/1000th.

A) Amperes (or Amps) B) Volts C) Millivolts

D) Picofarads (or pf) E) Milliamps (or Ma or ma)

F) Microfarads (or µF;) G) Amplifier, Circuits

6. _____ are devices which have metal plates separated by an insulator. They temporarily store an electrical charge.

A) In Series
B) Cathode
C) Capacitors

7. What permits the flow of charge in only one direction?

A) Anode
B) Diode
C) Cathode
D) Schematic

8. _____ contain many individual components and usually form several functional blocks.

A) Schematics
B) Diodes
C) Amplifiers
D) Integrated Circuits

9. The _____ also contains many components, but is not a digital device.

A) Inverting Amplifier
B) Operational Amplifier
C) Volt
D) Electron

10. This is _________________________________________

11. This is ___________________________________

12. This is __________________________________________

13. This is _______________________________________

14. This is ________________________________________

15. This is _________________________________________

16. This is _________________________________________

17. Ohm's Law: ________________________________________

18. I = 4, R = 10 so V = ________________ = ________________

19. V = 12, R = 6 so I = ________________ = ________________

20. I = 75, V = 150K Volts so R = ________________ = ________________

21. What's the power for 20?: P = ________________ = ________________

22. The value of the following is ______________

Resistors that are connected end-to-end are in series with each other. They are not a series circuit since the ends are not connected. Remember the phrase “series circle” to get an idea of what a series circuit looks like. The following resistors are connected in series, but are not a series circuit since there is no “circle”:

Remember the two rules about a series circuit:

1. The current at all points of a series circuit is the same.

2. The voltage drops of the parts of a series circuit add up to the source voltage (the battery voltage in this case).

To start with an easy circuit, look at the following. To find the total resistance of resistors connected in series, just add the values:

So, the total resistance in the above circuit = ___________ ____________ (there are two blanks because both the value and units should be included)

Now, remember that the current in a circuit = I = V/R.

The voltage is ____________ ____________

So, remembering to write the equation first, plug in the values to get the current:

I = ____________ = ____________ = ____________ ____________

To measure the voltage drop across a resistor, just put your meter wires on each end of the resistor:

You can get a meter for about \$6 at Harbor Freight and probably at other stores. I got one for each of the kids in my class:

Since all three resistors are 1000 ohms in the above circuit, there is a total of 3000 ohms, and I = 3/3000 = 1ma. Remember, ma means milliamps or thousandths of an amp, so 1ma is the same as 1/1000th of an amp. Now, to find the voltage across each 1K resistor, use V = IR.

So, the calculated voltage across each resistor =

V = IR = ____________ = ____________ ____________

This one is a little more complicated:

To find the total resistance of resistors connected in series, just add the values. So the total resistance is 1000 + 2200 + 470 = 3670 ohms.

Then the current for the whole circuit is I = V/R = 3/3670 = about .8ma.

In this experiment, you will use a breadboard:

The holes outlined in green are connected to each other. That's true of all of the vertical rows of holes in the center section, and the horizontal rows of holes on the outside edges. Horizontal holes in the center section and vertical holes in the outside sections are not connected to each other.

So, if three resistors are plugged like this, they are connected end-to-end:

If you connect a 3 volt battery pack to the left and right end of the whole circuit, you will have the series circuit above with the three resistors in series with the 3 volt battery:

Now do this experiment. Connect three resistors in series with each other and a battery to make a series circuit like the one above. Pick the resistors so that they are not the same values and add up to a total resistance between 1000 and 10000 ohms.

Would including a 10K resistor work? ________. Why? ________________________

R1 = ________ ________

R2 = ________ ________

R3 = ________ ________

Calculate (don't measure, calculate) the total of the series resistance.

Total Series Resistance: ______________ Ohms

Now, measure (don't just read it off of the battery) your source voltage.

Source Voltage (Battery): ____________ Volts

Again, remember the two rules for a series circuit:

1. The current at all points of a series circuit is the same.

2. The voltage drops of the parts of a series circuit add up to the source voltage (the battery voltage in this case).

Calculate the current in the circuit. Remember, I = V/R, where V is the battery voltage and R is the total series resistance. Remember, enter the equation first, then plug in the numbers, then the answer.

Calculated Current: I = _____________ = _____________ = _____________

Measure (don't calculate) the voltage drops across each resistor:

R1 voltage drop: _____________ Volts

R2 voltage drop: _____________ Volts

R3 voltage drop: _____________ Volts

Sum of Voltage Drops: ____________ Volts

Now that you know the voltage across each resistor and the value of each resistor, calculate the current for each. Remember, enter the equation first, then plug in the numbers, then the answer:

R1 Current: I = ____________ = ____________ = ____________ ____________

R2 Current: I = ____________ = ____________ = ____________ ____________

R3 Current: I = ____________ = ____________ = ____________ ____________

Now look at this:

Most LEDs need 2 volts at 20ma (.02 amps) as shown in the following:

What is your power supply voltage? ____________

What is the required voltage drop across R? ___________

So what is the value needed for R?

R = ____________ = ____________ = ____________

For younger learners and those who might be a little rusty on the subject, we will be talking here about raising a number to a power. It's not hard. A number raised to a power is shown with two parts. The base is the big number (5 in the example below) and the exponent is the little number (7 in the example below). The exponent is the power to which a number is raised. To raise a number to a power, just write it down the exponent number of times and multiply. In the following example, we would say that 5 is raised to the power of 7:

57

One way to recognize an exponent is by the fact that it is raised when written:
52 = 5 * 5 = 25
23 = 2 * 2 * 2 = 8
44 = 4 * 4 * 4 * 4 = 256

The number raised, such as the 5, 2 and 4 above, is called the base. Humans have used a base of 10 for thousands of years, probably because we have 10 fingers.

Each position has a weight. For all numbering systems I am aware of, the number on the right end of a whole number (a number without a decimal fraction) is known as the 1's place. There, the weight is equal to the base raised to the power of 0.
Any number raised to the power of 0 is equal to 1.

The exponent is increased by 1 with each move to the left. Thus, the second place from the right in a whole number has a weight equal to the base raised to the power of 1.
Any number raised to the power of 1 is equal to itself.

We were taught that the second place from the right is the 10's place. That's because we were using a base of 10 and we were raising it to the power of 1.

We can also raise our base to a negative power. This gives us decimal fractions when using the base of 10.

So we have this:
 10 Raised to a Power 103 102 101 100 10-1 10-2 10-3 Weight (place) 1000 100 10 1 .1 .01 .001 Weight as Fraction 1000/1 1000/1 10/1 1/1 1/10 1/100 1/1000 Place in words Thousands Hundreds Tens Ones Tenths Hundredths Thousandths

For example, many times you will have a lot less current than 1 amp. In fact, most of the time in electronics you have only milliamps, abbreviated ma. A milliamp is one thousandth of an amp. For example, one of the circuits above was found to have a current of about .8ma. You might see 8.174386921-4 on one calculator, or 817.4386921 x 10-6 on another (it might show e-6 rather than x 10-6). To find the number of amps, just move the decimal the number of places shown by the power number. In the above cases, that's four or six places to the left, since the power is negative. Both give you a little over .0008 amps, or about .8ma. If the power number is positive, move the decimal to the right. Remember that a whole number has an imaginary decimal on the right end. So 54, 54.0 and 54. are the same thing.

Study the chart above a little and you will also notice that the weight in a position is the weight of the place to its right times the base.

Try these for practice:

1,385,247 0 = ________________

1 X 103 = _____________________

.1ma = ______________ amps

101 = _______________

10-4 = _______________

For the number 432.178,

the 4 is in the ____________s place,

the 7 is in the ____________s place, and

the 8 is in the ____________s place.

And for a bit of a head scratcher:
xy0 = x raised to the power of y which is raised to the power of 0 = _______________
Just take it one step at a time from right to left.

When you connect polarized devices, such as batteries, diodes, etc. in series, the voltages add. You connect them positive to negative. So, two 3 volt battery packs would be connected in series like the following. Since they are 3 volts each, the total series voltage is 6 volts:

In a problem above, you were supposed to figure out what size of resistor you needed to power a LED. You learned that the LED needs 2 volts at 20ma. You also know that the current is the same everywhere in s series circuit, and that the voltage drops add up to the source voltage. So this is what happens if you use your two battery packs in series:

You need to find the value of the resistor you need to do the job, which is easy. Just remember that the equation for resistance is R = V/I (or the Rabbit sees the Vulture over the Iguana). Now plug in the numbers, and you get R = 4/.02 (.02 is the same as 20ma, and 4 is the voltage drop you need). So, R = 4/.02 = 350 ohms.

Now, figure out how to use your 6 volt source with no resistor but with more LEDs. Hint: the sentence that's talking about polarized devices above will help a lot.

Another way to connect things is to wire them in parallel. It might help to understand what a parallel circuit looks like by thinking of a railroad track. A railroad track has two steel beams running along side (in parallel with) each other. The wheels of a train roll on the beams, called rails, which are kept the correct distance apart by railroad ties. The rails are held down on the ties with metal plates and spikes (large nails). The whole thing sits on a bed of gravel or other material called a ballast:

A parallel circuit looks a lot like the railroad track's rails and ties. Components (parts) of the circuit are connected across two conductors like the ties from rail to rail. You will even find the term rail-to-rail in some electronics documents. A simple parallel circuit using just resistors follows.

Remember the two rules about a series circuit:

1. The current at all points of a series circuit is the same.

2. The voltage drops of the parts of a series circuit add up to the source voltage (usually the battery voltage in our experiments).

The parallel circuit also has two rules:

1. The current through each component can be different, but the sum of the current through all components add up to the source current.

2. The voltage across each component is the same as the source voltage.

The usual way to calculate the combined resistance in a circuit like the above is to use this equation:
R
total = 1/(1/R1 + 1/R2 + 1/R3) = 1/(1/30K + 1/300 + 1/3K).

But since you know the voltage, you can use the rules for a parallel circuit and Ohm's law for resistance and current. So calculate the following. Remember, the voltage is 3 volts across each resistor. Also remember to write Ohm's law for current first, which is ___________ :

Current through R1: _________ = __________ = __________ __________

Current through R2: _________ = __________ = __________ __________

Current through R3: _________ = __________ = __________ __________

Total current through all 3 = __________ __________

Since you know the source voltage is 3 volts and you know the total current, you can calculate the combined resistance using Ohm's law for resistance, which is _____________ :

Rtotal = __________ = ____________ = ____________ __________

Now look at this circuit:

Notice the three LEDs. They are in series with each other, but the three of them together are in parallel with the resistors and light bulb. The group of parallel components are in series with the power source and switch. This makes a parallel-series or series-parallel circuit. You can say it either way. Both are correct.

Such circuits are very common. Any room in a house that has more than one light controlled by a single switch is usually a parallel-series circuit.

If you are working on this material in a classroom setting, then draw a schematic diagram of the series-parallel lighting system in a room as a team project. You could also work on it alone, but the point is that a lot, maybe even most, engineering is done in teams. You will use the power that is supplied by the electric company, which is about 120VAC. VAC means volts, alternating current, which will be discussed shortly. You will use 4 lighting units, each of which needs 120VAC to power it. Finally, you will use a switch to turn on all 4 lights at once. Use engineering grid paper for rough drawings and a white board for the final drawing if available. Use any symbols you like for the different parts. Some ideas are below:

When you use a battery, you are using something called DC, or direct current. The positive stays positive and the negative stays negative. But that's not the same with AC (alternating current). With AC, the voltage moves from 0 to a maximum positive value, back to 0, then down to a maximum negative value. If you could see it, it might look like a sine wave:

You can see the above wave form with an oscilloscope:

The AC wave form shown above is an example of an analog signal. Not all analog signals look like a sine wave. Some, such as the human voice, are full of different widths and heights all mixed together. The oscilloscope display of a voice pattern can be seen by amplifying the signal from a microphone, then connecting the signal to an oscilloscope. What's being seen is not the actual sound wave. A sound wave can be visualized as bursts of compressed air particles. The darker areas below represent the compressed parts of a sound wave:

What is seen on the oscilloscope is the sound wave converted to an electrical signal. Most computers, however, work with signals that are either on of off.

The next section will begin the programming part of the tutorial. We will eventually work with on/off signals, which are called digital signals. When we reach the point where we are attempting to look at analog signals we will need to convert them to digital patterns that the computer can work with.

But for now, let's just talk about basic programming in the next section.