Chapter 4. Digital Logic
Embedded Systems - Shape The World

Jonathan Valvano and Ramesh Yerraballi

This chapter introduces digital logic. We will first define what it means to be digital, and then introduce logic, voltages, gates, flip flops, registers, adders and memory. This chapter is foundational, laying the ground work for the remainder of the class.

 Learning Objectives: Understand N-channel and P-channel MOS transistors. Learn digital logic as implemented on a computer. Know how to build simple logic from transistors. Learn how to construct the basic components of a computer from the logic gates. Know the terms: flip flop, register, binary adder and memory. Video 4.0. Digital Logic

## 4.1. Binary Information Implemented with MOS transistors

Information is stored on the computer in binary form. A binary bit can exist in one of two possible states. In positive logic, the presence of a voltage is called the ‘1’, true, asserted, or high state. The absence of a voltage is called the ‘0’, false,  not asserted, or low state. Figure 4.1 shows the output of a typical complementary metal oxide semiconductor (CMOS) circuit. The left side shows the condition with a true bit, and the right side shows a false. The output of each digital circuit consists of a p-type transistor “on top of” an n-type transistor. In digital circuits, each transistor is essentially on or off. If the transistor is on, it is equivalent to a short circuit between its two output pins. Conversely, if the transistor is off, it is equivalent to an open circuit between its outputs pins. Figure 4.1. A binary bit is true if a voltage is present and false if the voltage is 0.

Every family of digital logic is a little different, but on a Stellaris® microcontroller powered with 3.3 V supply, a voltage between 2 and 5 V is considered high, and a voltage between 0 and 1.3 V is considered low, as drawn in Figure 4.2. Separating the two regions by 0.7 V allows digital logic to operate reliably at very high speeds. The design of transistor-level digital circuits is beyond the scope of this class. However, it is important to know that digital data exist as binary bits and encoded as high and low voltages.

. What does binary mean? Figure 4.2. Mapping between analog voltage and the corresponding digital meaning on the TM4C123.

The maximum allowable voltage that the input will consider as low is called VIL. For the TM4C123, VIL is 1.3V. The minimum allowable voltage that the input will consider as high is called VIH. For the TM4C123, VIH is 2.0V.

The output pin also has a range of normal operating voltages. When the output is low, the maximum possible voltage that an output can be is called VOL. For the TM4C123, VOL is 0.4V. This means the output of the TM4C123 will be between 0 and 0.4V when the microcontroller is sending a low. When the output is high, the minimum possible voltage that an output can be is called VOH. For the TM4C123, VOH is 2.4V. This means the output of the TM4C123 will be between 2.4 and 3.3V when the microcontroller is sending a high. This information can be found in Section 24.2 of TM4C123 data sheet.

If the information we wish to store exists in more than two states, we use multiple bits. A collection of 2 bits has 4 possible states (00, 01, 10, and 11). A collection of 3 bits has 8 possible states (000, 001, 010, 011, 100, 101, 110, and 111). In general, a collection of n bits has 2n states. For example, a byte contains eight bits, and is built by grouping eight binary bits into one object, as shown in Figure 4.3. Another name for a collection of eight bits is octet (octo is Latin and Greek meaning 8.) Information can take many forms, e.g., numbers, logical states, text, instructions, sounds, or images. What the bits mean depends on how the information is organized and more importantly how it is used. This figure shows one byte in the state representing the binary number 01100111. Again, the output voltage 3.3V means true or 1, and the output voltage of 0V means false or 0. Figure 4.3. A byte is comprised of 8 bits, in this case representing the binary number 01100111.

What these 8 bits mean depends on how the computer software chooses to interpret them. Possibilities include but are not limited to an unsigned integer, a signed integer, a part of a machine code, and a character.

. Assume the circuit in Figure 4.3 contains an unsigned integer. What is the smallest unsigned integer that can be represented? What is the largest unsigned integer that can be represented?

. Assume the circuit in Figure 4.3 contains a signed 2’s complement integer. What is the smallest unsigned integer that can be represented? What is the largest unsigned integer that can be represented?

. If the data stored in Figure 4.3 represent characters, how many characters could it represent?

## 4.2. Digital Logic

In this section, we will give just a little taste of how the computer digital logic in the computer works. Transistors made with metal oxide semiconductors are called MOS. In the digital world MOS transistors can be thought of as voltage controlled switches. Circuits made with both p-type and n-type MOS transistors are called complementary metal oxide semiconductors or CMOS. The 74HC04 is a high-speed CMOS NOT gate, as shown in Figure 4.4. Figure 4.4. CMOS implementation of a NOT gate.

Video 4.0. Design of a NOT gate in digital logic

There are just a few rules one needs to know for understanding how CMOS transistor-level circuits work. Each transistor acts like a switch between its source and drain pins. In general, current can flow from source to drain across an active p-type transistor, and no current will flow if the switch is open. From a first approximation, we can assume no current flows into or out of the gate. For a p-type transistor, the switch will be closed (transistor active) if its gate is low. A p-type transistor will be off (its switch is open) if its gate is high.

The gate on the n-type works in a complementary fashion, hence the name complementary metal oxide semiconductor. For an n-type transistor, the switch will be closed (transistor active) if its gate is high. An n-type transistor will be off (its switch is open) if its gate is low. Therefore, consider the two possibilities for the circuit in Figure 4.4. If the input A is high (+3.3V), then the p-type transistor is off and the n-type transistor is active. The closed switch across the source-drain of the n-type transistor will make the output low (0V). Conversely, if A is low (0V), then p-type transistor is active and the n-type transistor is off. The closed switch across the source-drain of the p-type transistor will make the output high (+3.3V).

The AND, OR, EOR digital logic takes two inputs and produces one output; see Figure 4.5 and Table 4.1. We can understand the operation of the AND gate by observing the behavior of its six transistors. If both inputs A and B are high, both T3 and T4 will be active. Furthermore, if A and B are both high, T1 and T2 will be off. In this case, the signal labeled ~(A&B) will be low because the T3–T4 switch combination will short this signal to ground. If A is low, T1 will be active and T3 off. Similarly, if B is low, T2 will be active and T4 off. Therefore if either A is low or if B is low, the signal labeled ~(A&B) will be high because one or both of the T1, T2 switches will short this signal to +3.3V. Transistors T5 and T6 create a logical complement, converting the signal ~(A&B) into the desired result of A&B. We can use the and operation to extract, or mask, individual bits from a value. Figure 4.5. Logical operations can be implemented with discrete transistors or digital gates.

 A B AND NAND OR NOR EOR Ex NOR 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 Symbol A&B ~(A&B) A|B ~(A|B) A^B ~(A^B)

Table 4.1. Two-input one-output logical operations.

We can understand the operation of the OR gate by observing the behavior of its six transistors. If both inputs A and B are low, both T1 and T2 will be active. Furthermore, if A and B are both low, T3 and T4 will be off. In this case, the signal labeled ~(A|B) will be high because the T1–T2 switch combination will short this signal to +3.3V. If A is high, T3 will be active and T1 off. Similarly, if B is high, T4 will be active and T2 off. Therefore if either A is high or if B is high, the signal labeled ~(A|B) will be low because one or both of the T3, T4 switches will short this signal to ground. Transistors T5 and T6 create a logical complement, converting the signal ~(A|B) into the desired result of A|B. We use the OR operation to set individual bits.

When writing software we will have two kinds of logic operations. When operating on numbers (collection of bits) we will perform logic operations bit by bit. In other words, the operation is applied independently on each bit. In C, the logic operator for AND is &. For example, if number A is 01100111 and number B is 11110000 then

A =      01100111

B  =     11110000

A&B          01100000

The other type of logic operation occurs when operating on Boolean values. In C, the condition false is represented by the value 0, and true is any nonzero value. In this case, if the Boolean A is 01100111 and B is 11110000 then both A and B are true. The standard value for true is the value 1. In C, the Boolean operator for AND is &&. Performing Boolean operation yields

A =      01100111

B  =     11110000

A&&B                       1

In C, the logic operator for OR is |. The logic operation is applied independently on each bit. E.g.,

A =      01100111

B  =     11110000

A|B          11110111

In C, the Boolean operator for OR is ||. Performing Boolean operation of true OR true yields true. Although 1 is the standard value for a true, any nonzero value is considered as true.

A =      01100111

B  =     11110000

A||B                       1

Other convenient logical operators are shown as digital gates in Figure 4.6. The NAND operation is defined by an AND followed by a NOT. If you compare the transistor-level circuits in Figures 4.5 and 4.6, it would be more precise to say AND is defined as a NAND followed by a NOT. Similarly, the OR operation is a NOR followed by a NOT. The exclusive NOR operation implements the bit-wise equals operation. Figure 4.6. Other logical operations can also be implemented with MOS transistors.

Video 4.1. Building a NAND gate

Boolean Algebra is the mathematical framework for digital logic. Some fundamental laws of Boolean Algebra are listed in Table 4.2. With these laws, we consider A, B, C either as Booleans or as individual bits of a logic operation.

 A & B = B & A A | B = B | A (A & B) & C = A & (B & C) (A | B) | C = A | (B | C) (A | B) & C = (A & C) | (B & C) (A & B) | C = (A | C) & (B | C) A & 0 = 0 A | 0 = A A & 1 = A A | 1 = 1 A | A = A A | (~A) = 1 A & A = A A & (~A) = 0 ~(~A) = A ~(A | B) = (~A) & (~B) ~(A & B) = (~A) | (~B) Commutative Law Commutative Law Associative Law Associative Law Distributive Law Distributive Law Identity of 0 Identity of 0 Identity of 1 Identity of 1 Property of OR Property of OR Property of AND Property of AND Inverse De Morgan’s Theorem De Morgan’s Theorem

Table 4.2. Fundamental laws of Boolean Algebra.

. Let A bit an 8-bit number, and consider the operation B=A&0x20, where A&0x20 is performed bit by bit. Now, if we consider B as a Boolean value, what is the relationship between A and B?

. Let C be an 8-bit number and consider the operation C=C&0xDF. How does this operation affect C?

. Let D bit an 8-bit number, and consider the operation D=D|0x20. How does this operation affect D?

When multiple operations occur in a single expression, precedence is used to determine the order of operation. Usually NOT is evaluated first, then AND, and then OR. This order can be altered using parentheses.

There are multiple ways to symbolically represent the digital logic functions. For example,  ~A  A !A and ⌐A are five ways to represent NOT(A). One can use the pipe symbol (|) or the plus sign to represent logical OR: A|B  A+B. In this class we will not use the plus sign to represent OR to avoid confusion with arithmetic addition. One can use the ampersand symbol (&) or a multiplication sign (* • × ) to represent logical AND: A&B  AB. In this class we will not use the multiplication sign to represent AND to avoid confusion with arithmetic multiplication. Another symbolic rule is adding a special character (* n \) to a name to signify the signal is negative logic (0 means true and 1 means false). These symbols do not signify an operation, but rather are part of the name used to clarify its meaning. E.g., Enable* is a signal than means enable when the signal is zero.

. Let C bit an 8-bit number. Are these two operations the same or different?  C=C&0xDF   C=C&(~0x20)

## 4.3. Flip-flops are used for storage

While we’re introducing digital circuits, we need digital storage devices, which are essential components used to make registers and memory. The simplest storage device is the set-reset latch. One way to build a set-reset latch is shown on the left side of Figure 4.7. If the inputs are S*=0 and R*=1, then the Q output will be 1. Conversely, if the inputs are S*=1 and R*=0, then the Q output will be 0. Normally, we leave both the S* and R* inputs high. We make the signal S* go low, then back high to set the latch, making Q=1. Conversely, we make the signal R* go low, then back high to reset the latch, making Q=0. If both S* and R* are 1, the value on Q will be remembered or stored. This latch enters an unpredictable mode when S* and R* are simultaneously low.

The gated D latch is also shown in Figure 4.7. The front-end circuits take a data input, D, and a control signal, W, and produce the S* and R* commands for the set-reset latch. For example, if W=0, then the latch is in its quiescent state, remembering the value on Q that was previously written. However, if W=1, then the data input is stored into the latch. In particular, if D=1 and W=1, then S*=0 and R*=1, making Q=1. Furthermore, if D=0 and W=1, then S*=1 and R*=0, making Q=0. So, to use the gated latch, we first put the data on the D input, next we make W go high, and then we make W go low. This causes the data value to be stored at Q. After W goes low, the data does not need to exist at the D input anymore. If the D input changes while W is high, then the Q output will change correspondingly. However, the last value on the D input is remembered or latched when the W falls, as shown in Table 4.3.

The D flip-flop, shown on the right of Figure 4.7, can also be used to store information. D flip-flops are the basic building block of RAM and registers on the computer. To save information, we first place the digital value we wish to remember on the D input, and then give a rising edge to the clock input. After the rising edge of the clock, the value is available at the Q output, and the D input is free to change. The operation of the clocked D flip-flop is defined on the right side of Table 4.3. The 74HC374 is an 8-bit D flip-flop, such that all 8 bits are stored on the rising edge of a single clock. The 74HC374 is similar in structure and operation to a register, which is high-speed memory inside the processor. If the gate (G) input on the 74HC374 is high, its outputs will be HiZ (floating), and if the gate is low, the outputs will be high or low depending on the stored values on the flip-flop. The D flip-flops are edge-triggered, meaning that changes in the output occur at the rising edge of the input clock. Figure 4.7. Digital storage elements.

 D W Q D clock Q 0 0 Qold 0 0 Qold 1 0 Qold 0 1 Qold 0 1 0 1 0 Qold 1 1 1 1 1 Qold 0 ↓ 0 0 ↑ 0 1 ↓ 1 1 ↑ 1

Table 4.3. D flip-flop operation. Qold is the value of the D input at the time of fall of W or rise of clock.

The tristate driver, shown in Figure 4.8, can be used dynamically control signals within the computer. It is called tristate because there are three possible outputs: high, low, and HiZ. The tristate driver is an essential component from which computers are built. To activate the driver, we make its gate (G*) low. When the driver is active, its output (Y) equals its input (A). To deactivate the driver, we make its G* high. When the driver is not active, its output Y floats independent of A. We will also see this floating state with the open collector logic, and it is also called HiZ or high impedance. The HiZ output means the output is neither driven high nor low. The operation of a tristate driver is defined in Table 4.4. The 74HC244 is an 8-bit tristate driver, such that all 8 bits are active or not active controlled by a single gate. The 74HC374 8-bit D flip-flop includes tristate drivers on its outputs. Normally, we can’t connect two digital outputs together. The tristate driver provides a way to connect multiple outputs to the same signal, as long as at most one of the gates is active at a time. Figure 4.8. A 1-bit tristate driver and an 8-bit tristate driver (if G* is low, then Y equals A, if G* is high, then Y is HiZ). The signal G* is negative logic.

Table 4.4 describes how the tristate driver in Figure 4.8 works. Transistors T1 and T2 create the logical complement of G*. Similarly, transistors T3 and T4 create the complement of A. An input of G*=0 causes the driver to be active. In this case, both T5 and T8 will be on. With T5 and T8 on, the circuit behaves like a cascade of two NOT gates, so the output Y equals the input A. However, if the input G*=1, both T5 and T8 will be off. Since T5 is in series with the +3.3V, and T8 in series with the ground, the output Y will be neither high nor low. I.e., it will float.

 A G* T1 T2 T3 T4 T5 T6 T7 T8 Y 0 0 on off on off on off on on 0 1 0 on off off on on on off on 1 0 1 off on on off off off on off HiZ 1 1 off on off on off on off off HiZ

Table 4.4. Tristate driver operation. HiZ is the floating state, such that the output is not high or low.

The output of an open collector gate, drawn with the ‘×’, has two states low (0V) and HiZ (floating) as shown in Figure 4.9. Consider the operation of the transistor-level circuit for the 74HC05. If A is high (+3.3V), the transistor is active, and the output is low (0V). If A is low (0V), the transistor is off, and the output is neither high nor low. In general, we can use an open collector NOT gate to switch current on and off to a device, such as a relay, an LED, a solenoid, or a small motor. The 74HC05, the 74LS05, the 7405, and the 7406 are all open collector NOT gates. 74HC04 is high-speed CMOS and can only sink up to 4 mA when its output is low. Since the 7405 and 7406 are transistor-transistor-logic (TTL) they can sink more current. In particular, the 7405 has a maximum output low current (IOL) of 16 mA, whereas the 7406 has a maximum IOL of 40 mA. Figure 4.9. Two transistor implementations of an open collector NOT gate.

The computer performs many arithmetic and logic operations. We will show one of them to illustrate some of the computation possible in the computer. We begin the design of an adder circuit with a simple subcircuit called a binary full adder, as shown in Figure 4.10. There are two binary data inputs A, B, and a carry input, Cin. There is one data output, Sout, and one carry output, Cout. As shown in Table 4.5, Cin, A, and B are three independent binary inputs each of which could be 0 or 1. These three inputs are added together (the sum could be 0, 1, 2, or 3), and the result is encoded in the two-bit binary result with Cout as the most significant bit and Sout as the least significant bit. Cout is true if the sum is 2 or 3, and Sout is true if the sum is 1 or 3. Figure 4.10. A binary full adder.

 A B Cin A+B+Cin Cout Sout 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 2 1 0 1 0 0 1 0 1 1 0 1 2 1 0 1 1 0 2 1 0 1 1 1 3 1 1

Table 4.5. Input/output response of a binary full adder.

Figure 4.11 shows an 8-bit adder formed by cascading eight binary full adders. Similarly, we build a 32-bit adder by cascading 32 binary full adders together. The carry into the 32-bit adder is zero, and the carry out will be saved in the carry bit. For an 8-bit unsigned number, there are only 256 possible values, which are 0 to 255. When we add two 8-bit numbers the sum can be any number from 0 to 510, which is a 9-bit number. The 9-bit result in Figure 4.11 exists as the 8 bits R7–R0 plus carry.

We can think of 8-bit unsigned numbers as positions along a circle, like a clock. There is a discontinuity in the clock at the 0|255 interface; everywhere else adjacent numbers differ by ±1. If we add two unsigned numbers, we start at the position of the first number a move in a clockwise direction the number of steps equal to the second number. If  96+64 is performed in 8-bit unsigned precision, the correct result of 160 is obtained. In this case, the carry bit will be 0 signifying the answer is correct. On the other hand, if 224+64 is performed in 8-bit unsigned precision, the incorrect result of 32 is obtained. In this case, the carry bit will be 1, signifying the answer is wrong.

. If A has the value 100 (0x64) and B has the value 50 (0x32), what will be the value of the output (R7-R0) of the circuit in Figure 4.11? Also what will the carry signal be?

. If A has the value 255 (0xFF) and B has the value 2 (0x02), what will be the value of the output (R7-R0) of the circuit in Figure 4.11? Also what will the carry signal be?

## 4.5. Digital Information stored in Memory Figure 4.12. Memory is a sequential collection of data storage elements.

In the computer, we can build an 8-bit storage element, shown logically as Figure 4.12, by combining 8 flip-flops. This basic storage element is called a register, as shown in Figure 4.13. A bus is a collection of wires used to pass data from one place to another. In this circuit, the signals D7–D0 represent the data bus. Registers on the Stellaris® microcontrollers are 32-bits wide, but in this example we show an 8-bit register. We call it storage because as long the circuit remains powered, the digital information represented by the eight voltages Q7–Q0 will be remembered. There are two operations one performs on a register: write and read. To perform a write, one first puts the desired information on the 8 data bus wires (D7–D0). As you can see from Figure 4.13, these data bus signals are present on the D inputs of the 8 flip-flops. Next, the system pulses the Write signal high then low. This Write pulse will latch or store the desired data into the 8 flip-flops. The read operation will place a copy of the register information onto the data bus. Notice the gate signals of the tristate drivers are negative logic. This means if the Read* signal is high, the tristate drivers are off, and this register does not affect signals on the bus. However, the read operation occurs by setting the Read* signal low, which will place the register data onto the bus.

. What does negative logic mean? Figure 4.13. Digital logic implementation of a register.

A great deal of confusion exists over the abbreviations we use for large numbers. In 1998 the International Electrotechnical Commission (IEC) defined a new set of abbreviations for the powers of 2, as shown in Table 4.6. These new terms are endorsed by the Institute of Electrical and Electronics Engineers (IEEE) and International Committee for Weights and Measures (CIPM) in situations where the use of a binary prefix is appropriate. The confusion arises over the fact that the mainstream computer industry, such as Microsoft, Apple, and Dell, continues to use the old terminology. According to the companies that market to consumers, a 1 GHz is 1,000,000,000 Hz but 1 Gbyte of memory is 1,073,741,824 bytes. The correct terminology is to use the SI-decimal abbreviations to represent powers of 10, and the IEC-binary abbreviations to represent powers of 2. The scientific meaning of 2 kilovolts is 2000 volts, but 2 kibibytes is the proper way to specify 2048 bytes. The term kibibyte is a contraction of kilo binary byte and is a unit of information or computer storage, abbreviated KiB.

1 KiB = 210 bytes = 1024 bytes

1 MiB = 220 bytes = 1,048,576 bytes

1 GiB = 230 bytes = 1,073,741,824 bytes

These abbreviations can also be used to specify the number of binary bits. The term kibibit is a contraction of kilo binary bit, and is a unit of information or computer storage, abbreviated Kibit.

A mebibyte (1 MiB is 1,048,576 bytes) is approximately equal to a megabyte (1 MB is 1,000,000 bytes), but mistaking the two has nonetheless led to confusion and even legal disputes. In the engineering community, it is appropriate to use terms that have a clear and unambiguous meaning.

 Value SI          Decimal SI          Decimal Value IEC          Binary IEC          Binary 10001 k kilo- 10241 Ki kibi- 10002 M mega- 10242 Mi mebi- 10003 G giga- 10243 Gi gibi- 10004 T tera- 10244 Ti tebi- 10005 P peta- 10245 Pi pebi- 10006 E exa- 10246 Ei exbi- 10007 Z zetta- 10247 Zi zebi- 10008 Y yotta- 10248 Yi yobi-

Table 4.6. Common abbreviations for large numbers.

## 4.6. Chapter 4 Quiz

4.1 Which of the following is true. Positive logic is defined as

a) The configuration where the "true" state has a higher voltage than the "false" state.

b) The state where the signal is "high".

c) The state where the transistor is "on".

d) The state where the transistor is "off".

e) None of the above

4.2 If a voltage on an input of the TM4C123 is between 0 and 1.3 V, how is that input considered?

a) Low or logic "0"

b) Unknown or illegal

c) High or logic "1"

4.3 If a voltage on an input of the TM4C123 is between 1.3 and 2 V, how is that input considered?

a) Low or logic "0"

b) Unknown or illegal

c) High or logic "1"

4.4 If a voltage on an input of the TM4C123 is between 2 and 5 V, how is that input considered?

a) Low or logic "0"

b) Unknown or illegal

c) High or logic "1"

4.5 Calculate the logic expression for each set of inputs A, B, C

 A B C A & (B | C) 01002 = 4 01012 = 5 01102 = 6 01112 = 7 10102 = 10 00012 = 1 11102 = 14 10012 = 9 01112 = 7

4.6 Calculate the Boolean expression for each set of inputs A, B, C

 A B C A && (B || C) True False False True False True False True False True True False

Reprinted with approval from Embedded Systems: Introduction to ARM Cortex-M Microcontrollers, 2014, ISBN: 978-1477508992, http://users.ece.utexas.edu/~valvano/arm/outline1.htm 