22.2.1 Hexadecimal Number System

Hexadecimal represents a number system whose base is 16 (made out of the Greek word hex for 6 and decimal denoting association with 10). Initially, therefore, 16 symbols are necessary to represent all numbers. Also, in writing hexadecimal numbers it is necessary to somehow indicate that its base is 16 and not 10.

The 16 numbers for the base of the hexadecimal system are numbers 0 to 15. For numbers 0 to 9 the same symbols used in the decimal system are employed. For numbers 10 to 15 the letters A to F are utilized, without any name. The base for hexadecimal system is then

and their corresponding decimal values are

The following number after F in hexadecimal system is 10, in the same way that in decimal numbers after 9 we have 10. To show that a number is written in the hexadecimal system (or its base is 16), we can put the base as a subscript for the number, like 543₁₆. A more common way is to put an X before the number (e.g., X543) or add an H after the number, like 543H. Thus,

The following shows examples of hexadecimal numbers (without the preceding X):

To have a better feeling for the values of these numbers, we calculate their decimal equivalents. You notice that if we do not use the preceding X, it is not clear if numbers such as 48 and 312 are decimal or hexadecimal, and these are quite different. The decimal magnitudes of hexadecimal numbers are calculated in the same way shown earlier for 543 in both cases of having 10 or 16 as for base. Because 16⁰ = 1, we refrain from its repetition and use 1 for the multiplier of the rightmost (least significant) digit.

Also, notice the following

Note that equivalent to 99, 999, and similar decimal numbers (i.e., those figures before the number of digits increases) in hexadecimal system we have FF, FFF, and so on. Thus,

(Note that it is not necessary to write X1 instead of 1 because there is no ambiguity.)

The reverse process of finding the hexadecimal expression of decimal numbers can be done by successive divisions by 16. The remainder of a division determines the digit for the number. Consider, for example, the small figure 23. Dividing this figure by 16, we can write

The arrows on the left show the procedure, and the arrows on the right show the answer. The answer is 17, as defined by the numbers 7 and 1 denoted by arrows. The oblique arrow starting from 1 on the first row toward the second row denotes the transfer of the quotient to a new line for continuation of the same process of dividing by 16. The vertical arrow shows that when the quotient becomes 0 the process should stop.

For this small number, only one division is sufficient (the last one is not really necessary). But for larger numbers the dividing process must be repeated a number of times until no more division is possible (or no extra results are obtained). The following examples show the same procedure for larger numbers. It is not necessary to show the arrows. Here they help understanding.

Solution

We carry out dividing by 16 showing the process in the same format as above.

The corresponding hexadecimal number, therefore, is X7D2 (D is for the decimal 13).

22.2.2 Binary Number System

The binary number system has only two base numbers, 0 and 1. The reason for developing such a number system is its applicability to many physical processes that have two states, and thus a binary number is sufficient to describe them. These physical states can be in the form of

• On and off
• True and false
• High and low
• Yes and no
• Positive and negative

depending on the application. For example, for switches and switching devices in an electrical circuit the on-off defines the state of one switch. This can be the basis of storing data, which can consist of the many switches or devices where each one assumes one of the two states at a time.

A number in the binary system, thus, can have only 0’s and 1’s because there is no other symbol. Note that in the same way that in the decimal system the number after 9 (the last number element) is 10, in the binary system the number after 1 is 10. The following lines show a few examples of some decimal numbers when written in the binary system. A subscript b or 2 is used to indicate that a number is binary.

Before we learn how to find the binary equivalent of a number, note that in the same way that 10 + 1 = 11, in the given binary numbers the same thing can be verified:

and after 1111_b (the last four digit binary number) is 10000 (the first five-digit number), in the same way that after 9999 in decimal numbers we have 10000.

The decimal equivalent of any binary number can be found by the same procedure we have used earlier for the number 543. We use the above figures to verify that they really are equivalent to each other:

Solution

The sequence of divisions by 2 can be arranged in the following format:

The binary representation of 86, thus, is 1010110.

22.2.3 Octal Number System

Another number system that is more utilized in industry is the octal numbers, having number 8 as the base. The rules for conversion from octal to decimal and vice versa are the same as the other number systems. Numbers 0 to 7 are used for the eight main symbols. Thus, the positive numbers start from 0 and are

Again, each octal number must be followed by a subscript 8 to indicate its base. The following examples show calculation of the decimal equivalent of an octal number and the octal equivalent of a decimal number.

22.2.4 Conversion between Binary and Hexadecimal Numbers

The fact that hexadecimal numbers are based on 16, and 16 is the fourth power of 2 makes the conversion between hexadecimal and binary numbers easy. Consider a single digit hexadecimal number. The values expressed by this digit are between 0 and 15. Now consider the numbers that may be represented by four digits in a binary number. Again, they are between 0 and 15. All these numbers are compared together below (the prefix X for hexadecimal numbers is omitted for simplicity).

The above equivalence can be directly carried to all other digits of a hexadecimal number and the set of four digits of a binary number. The following are some examples.

Note that in the last number the leftmost 0 is not shown in the hexadecimal number, whereas in a binary number it is very common to have 0’s on the left, as you will see in the next section.

22.2.5 Direct Conversion between Binary and Octal Numbers

On the basis of the discussion in Section 22.2.4, a direct conversion between octal and binary numbers is possible. A binary number can be divided in groups of three, each of which corresponds to one digit of its associated octal number. Only the first 8 out of 16 in the preceding list correspond to octal numbers, and the 0 on the left must be omitted so that there are only three digits for each binary number. Some examples are as follows.

As said before, the 0’s on the left side of a binary number do not need to be included in the converted number.

22.3.1 Bits and Bytes

In dealing with binary numbers, the terms bit and byte come into picture. A bit is the unit for the digits that a binary number can have. Thus, the smallest binary number has 1 bit, and its value can be 0 or 1. A byte is a group of 8 bits together. The memory and the digital processor for many devices may contain 8 bits (one byte), but computers have larger number of bits for their number processing. Today’s laptops are made with 32-bit or 64-bit processors and the larger computers can be based on 128 bits. This number of bits is for representing or storing one digital number, only. A computer can handle millions of numbers. Thus, it has many units of the 128-bit device. This defines the computer memory, which is normally measured in bytes; for example, 32 megabytes or 4 gigabytes, etc. Associated with bit and byte in computer terminology a word is a group of 16 bits or two bytes.

Figure 22.2 Representation of bits, bit numbers, and their values.

Figure 22.2 shows the representation of the electronic circuits that hold binary numbers. We refer to this electronic device as a register. At this time we are only concerned about the bits and bytes, without caring about what circuits constitute the register. When there are a number of bits, like in a byte, the rightmost bit is called the least significant bit (LSB), and the leftmost bit is called the most significant bit (MSB). Each bit is numbered for ease of reference. The rightmost bit is bit 0, followed by bit 1, bit 2 and so forth. The last bit in a byte, therefore, is bit 7. There is a value associated with each bit. This value is obtained by raising “2” to the power defined by the bit number. For instance, the first one from the right has a value 2⁰ = 1, and the most significant figure in a byte has a value of 2⁷ = 128. These numbers are also shown in Figure 22.2 for a byte.

As can be understood, with 4 bits only, the smallest number that can be addressed or stored is zero, when all bits are 0, and the largest number is 15, when all bits are 1. Altogether, 16 values can be recorded. Similarly, with 8 bits, the smallest number is zero and the largest number is 255. This implies that only 256 values are possible. These are 255 positive integer values plus zero.

You may ask where such a large number is used. Remember that negative and fractional numbers need to be handled, too. Computers use 64 bits, but for industrial use sometimes even 4 bits may suffice for very simple applications.

In practice, we often have negative values, particularly in measurements. We continue this discussion to see how negative binary numbers can be stored. For dealing with negative binary numbers it is crucial to adopt a fixed number of bits. If not, numbers cannot be interpreted correctly. Normally, a commercial device comes with a fixed bit size, and the greater the number of bits, the more expensive the device is.

22.3.2 Representation of Negative Binary Numbers

For simplicity and better clarity, in what follows only 8 bits are used. With 8 bits the total numbers that can be addressed is limited. Dealing with larger numbers requires larger bit size. Nevertheless, in many small applications, 8 bits is sufficient.

When an 8-bit device is in use we need to fill all the places in the register with 0’s and 1’s. In other words, for each bit a value must be defined and it cannot be left without a value. For instance, for storing the decimal number 86 in a binary form it was found that (see Section 22.2) 86 = 1010110_b. This number has only 7 digits and occupies 7 bits in a register. To represent this number with 8 bits the remaining bit on the left is filled with a 0 and thus 86 is represented by 0101,0110_b. For any other number in the same way all remaining bits from left are filled with 0’s.

In computer terminology, positive numbers are referred to as unsigned numbers, and when we have both positive and negative numbers, then we need to use signed numbers. With 8 bits there are 256 possible numbers. For any application an initial decision must be made if these 256 possibilities are to be used for the range of 0 to 255 unsigned numbers, or half are negative and half positive and, therefore, the 256 positions must accommodate 256 signed numbers.

Figure 22.3 Signed and unsigned 8-bit numbers.

In unsigned numbers (for 8 bits) the most significant bit has a value of 2⁷ = 128, whereas in signed numbers this bit represents the negative sign. Figure 22.3 shows this fact. The unsigned numbers start from 0 and end with 255, and are as follows:

and the signed numbers are from −128 to +127, as follows:

Below we see some examples of signed numbers. Pay attention to the fact that if a positive number and a negative number of the same quantity are added, the resultant is zero. Note and compare the numbers having similar bits except for the most significant bit.

Although a way to find the value of a negative number is shown above (in the brackets), this is not the best way. Also, we want to find the negative of a given number. There is a rule for finding the negative of a binary number. This is discussed under one’s and two’s complement in Section 22.4.

22.3.3 Precision in Analog-to-Digital Conversion

Today, in many industries, microprocessors are used for control of processes and plants. Examples of that are plenty. Today’s cars are equipped with a microprocessor that controls the performance of the engine by measuring a number of parameters, such as air temperature, car speed, position of the gas pedal, and other pertinent parameters. Likewise, the operation of a wind turbine, steam turbine, and gas turbine is supervised and controlled by a microprocessor based on many parameters that need to be measured.

A microprocessor is the brain of any computer and any equipment with digital control. There are many different microprocessors with different bit size and different capabilities. In all microprocessor-based control systems, when an analog quantity is measured, it must be converted to its digital equivalent, so that it can be read or stored digitally or better, say, by a binary number.

The output of most measurement devices is in the form of a voltage. For some variables like wind speed this value is always positive (we do not have negative wind speed) and for some other ones, like temperature this can be positive or negative. According to what was said before, a signed or unsigned number can be decided on based on the requirement.

Suppose that the output of a measurement device is a voltage that can vary between 0 and 10 V. Considering an 8-bit converter, there are 256 possibilities for the values that can be recorded, processed, and displayed. In this case the 256 numbers correspond to 10 V range. Because there are 255 intervals (between 0 and 255), each digital number corresponds to 10/255 V, which is approximately 0.039 V. This implies that within the converted digital values the difference between any number and the number next to it is 0.039 V. This value of 0.039 V is the resolution of the representation of 10 V by 8 bits. It determines the precision of the operation. Samples of the binary values and their corresponding voltages for this example are shown below:

The above numbers show that the difference between any two consecutive numbers that can be displayed through a digital display is 0.039 V. For instance, considering the two first numbers, a measured value can be 0 or 0.039 V, and nothing in between. With 8 bits it is not possible to have a better precision. To have a higher precision, either the voltage range must be smaller or the number of bits (bit size) must be increased, as shown in the following examples.

22.4.1 Addition

The addition of two binary numbers is performed in the same way as we add two decimal numbers with the exception that the only nonzero element in the set of basic elements of the binary number system is the number 1. Thus, in the same way that in decimal numbers if we add 1 to 9, the answer is 10, in binary numbers if we add 1 to 1, the answer is 10. That is, the addition of 1 and 1 is a 0 with a 1 carried over to the next digit on the left. Also, practically, which is the way this operation is performed in microprocessors, only two numbers are added together at a time. We consider the following three examples. Assume unsigned numbers.

Because any binary numbers is practically the contents of a register, only a limited number of bits are available. This is because of the hardware, which determines the bit size to be 8, 10, 12, or some other. In the result of adding two numbers you see that the very last digit on the left is shown in red, when it happens to exist. This digit (bit) is ignored. There is physically no place for this bit to go to. It is either not significant (in a subtraction operation, as you will see next), or it is an indication of overflow (as shown above). Overflow is a situation where a number is larger than the capacity of the machine. When there is an overflow situation, the last bit is ignored, but it signals out a flag. A flag is an indicator of an undesirable or fault situation in digital systems. A message will be issued when a flag is signaled. An example is when you have a division by zero in your calculator.

22.4.2 Subtraction

To subtract a number B from number A the common method is to add A to the negative of B.

The subtraction, then, changes to an addition operation. Then, in the same way as discussed above the two numbers are added.

Consider for example subtraction of 44 (0010,1100_b) from 108 (0110,1100_b) in binary operation. Instead, we add −44 (1101,0100_b) to 108

You can easily check that 108 − 44 = 64, and the resultant binary number (0100,0000_b) is 64. Also, you can verify the number for −44 to be correct. If it is added to 44, the resultant is 0.

Note that in both cases the very last digit, shown in red, does not play any role and is ignored.

22.4.3 One’s Complement of a Binary Number

The one’s complement of a binary number is used to find the two’s complement of a number, which is used to find the negative of that number. Note that negatives are valid with signed numbers. The one’s complement of a number is obtained by switching each bit to its opposite value, 1’s to 0’s and 0’s to 1’s. For example, the one’s complement of 0110,1010 (106) is 1001,0101.

22.4.4 Two’s Complement of a Binary Number

Two’s complement of a binary number is obtained by adding 1 to the one’s complement of that number. For instance,

The two’s complement defines the negative of a number. Therefore, the negative of 0110,1010_b is 1001,0110_b. We can check this by adding the two numbers together.

The given numbers was −25. You cannot say that easily just from the number directly by looking at the digits, except that one has a look-up table showing all the numbers in the range. Figure 22.4 shows some numbers between −128 and −1. You may notice a pattern to determine the numbers for 8 bits as shown in the table. On the basis of this pattern, find the decimal magnitude of the binary number leaving out the leftmost bit, and subtract 128 from that number. For other bit numbers the value to subtract (128 in this case) must be changed accordingly.

Figure 22.4 Negative binary numbers.

22.4.5 Multiplication and Division of Unsigned Binary Numbers

Multiplication of binary numbers is very easy. This is the way multiplication is performed in a microprocessor. First, consider multiplication by 10_b and 100_b, which is, in fact, multiplication by 2 and 4. Consider 0011,1011_b (always assume a limited number of bits, here 8). Assume an unsigned number.

As in the case of decimal numbers 1 or 2, 0’s are added to the right side of the number for multiplication by 10_b and 100_b, respectively. Because the number of bits must remain the same, this operation is like bringing the 0’s from the left hand to the right hand. Figure 22.5 depicts this. This operation of moving 0’s to the right is called rotation.

Rotation to the right is performed for multiplication, 1 bit for multiplication by 10_b, 2 bits for multiplication by 100_b, and so on. Note that here if multiplication is continued, then eventually overflow happens when the result becomes larger than a machine can handle (here because there are only 8 bits, not many numbers can be handled, but in a machine with 128 bits the operation can go on much further).

For multiplication by numbers like 3 (11_b) and 6 (110_b) and so forth, these numbers are broken into their components. For example,

Figure 22.5 Multiplication of a number by 10_b and 100_b.

Thus, multiplication of a binary number by 11_b includes multiplication of that number by 10_b, as above, and multiplication of the number by 1(multiplication by 1 is trivial and does not change the number), and adding the results. Similarly, multiplication by 110_b includes multiplication by 100_b plus multiplication by 10_b,

both of which we have already seen. The results of the two multiplications then must be added together. Multiplication by all other numbers can be treated the same way.

You may want to check the results with decimal numbers. The initial number was 103, and the final answer is 721, which is 103 × 7.

In conjunction with multiplication is division. For division, numbers are shifted right. Shift right is shown in Figure 22.6. In dividing integer numbers, round-off errors can occur, as can be seen in the figure.

Figure 22.6 Division in binary unsigned numbers.

We can verify the result. The original number is 109 and the multiplier is 11. The result of multiplication must be 109 × 11 = 1199, which is true.

22.6.1 Sampling

In a continuous process any measurement must be frequently repeated for updated values. Therefore, for microprocessor control of a device another factor comes into picture: the rate at which the data are updated.

This updating of measured values is called sampling and how often a variable is updated is determined by the sampling rate. The sampling rate can also be called sampling frequency, which implies how many times in one second the variable is measured. The time interval between each two measurements is called the sampling time.

For example, if in each second a variable is measured 1000 times, the sampling rate is 1000 per second, and the sampling time is 0.001 second. Depending on how important a role a parameter has in a process, this updating of values must be performed at a higher sampling rate.

Figure 22.7 Sampling a variable parameter.

In Figure 22.7 a variable is sampled between times A and B. The variable is assumed to remain constant at the measured value for the total period of the sampling time until the next measurement is performed. In reality, it is more likely that the value of a variable changes even if the time interval is very small. For example, the value of the parameter at M is changing to that at N, whereas in the microcontroller it is kept constant until the new reading comes in, which implies a jump from N′ to N. It is thus obvious that more precision in measurement and control can be achieved by having a smaller sampling time. The measured value of any variable is stored by a circuit called sample and hold.

Figure 22.8 Sampled data and their binary conversion.

Nowadays, fast sampling, for instance, one million samples per second (a sampling time of 1 microsecond) is possible for many measurements. This implies that the necessary time for measuring the variable under question, converting its value to digital and storing the result can be less than 1 millionth of a second. It must be noted that, in general, the higher the sampling rate is, the higher the price of the sampling equipment.

Increasing the sampling rate is not always necessarily essential, useful, or worth the cost even if possible. For example, consider the pitch angle of a wind turbine. If it takes 1 sec for the pitch angle to change (through its mechanism), any measurement faster than 1 per second does not have any useful effect. Also, if the measurement itself takes a long time, or the process is slow, such as heating a liquid, there is no point in sampling at a high rate. It can only add to the cost.

After sampling is performed at any instant, the measured value is converted to its binary form. This is repeated after each sampling time interval. Figure 22.8 shows a typical example of sampled data and its conversion to binary data (the binary conversion is only for demonstration).

22.6.2 Clock Signal

Each microprocessor has a clock to regulate its operation. Each and every single operation, such as adding two bits of data, is performed with the clock pulse. Each clock pulse is a step for synchronized work of all components (for reading new values, conversions, mathematical operations, etc.) and all operations progress step by step. A clock signal is a square pulse train with a fixed frequency. The clock rate can be decreased, and slower pulse trains be made from it for slower processes.

Figure 22.9 shows the pulse train of the clock signal and two other signals with half and one quarter of the clock frequency. Each pulse has a rising edge, a falling edge, and a level part. A device can be sensitive to the rising edge or the falling edge, and, consequently, its function takes place with that part of a pulse. Some devices are level sensitive.

Figure 22.9 Representation of fixed frequency pulse signals.

Figure 22.10 Representation of clock and other signals in a microprocessor.

A rising edge is also called a leading edge or positive going edge. A falling edge is also called a trailing edge or a negative going edge. A pulse train continuously switches between zero and a positive value. We refer to these values as low and high.

Although microprocessors are very fast and their clock frequencies are in the range of MHz (million hertz), the change from a low level to a high level and vice versa is not instant, and, thus the time for switching (duration of the rising edge and falling edge) is not zero. In this sense, the actual clock signal and other signals are represented as shown in Figure 22.10.