3.6.1 Basic Concepts

A sequential circuit defines its output as a function of both its current inputs and its previous inputs. Therefore, the output depends on past inputs. To remember previous inputs, sequential circuits must have some sort of storage element. We typically refer to this storage element as a flip-flop. The state of this flip-flop is a function of the previous inputs to the circuit. Therefore, pending output depends on both the current inputs and the current state of the circuit. In the same way that combinational circuits are generalizations of gates, sequential circuits are generalizations of flip-flops.

3.6.2 Clocks

Before we discuss sequential logic, we must first introduce a way to order events. (The fact that a sequential circuit uses past inputs to determine present outputs indicates we must have event ordering.) Some sequential circuits are asynchronous, which means they become active the moment any input value changes. Synchronous sequential circuits use clocks to order events. A clock is a circuit that emits a series of pulses with a precise pulse width and a precise interval between consecutive pulses. This interval is called the clock cycle time. Clock speed is generally measured in megahertz (MHz), or millions of pulses per second. Common cycle times are from one to several hundred MHz.

A clock is used by a sequential circuit to decide when to update the state of the circuit (when do "present" inputs become "past" inputs?). This means that inputs to the circuit can only affect the storage element at given, discrete instances of time. In this chapter we examine synchronous sequential circuits because they are easier to understand than their asynchronous counterparts. From this point, when we refer to "sequential circuit", we are implying "synchronous sequential circuit."

Most sequential circuits are edge-triggered (as opposed to being level-triggered). This means they are allowed to change their states on either the rising or falling edge of the clock signal, as seen in Figure 3.17.

3.6.3 Flip-Flops

A level-triggered circuit is allowed to change state whenever the clock signal is either high or low. Many people use the terms latch and flip-flop interchangeably. Technically, a latch is level triggered, whereas a flip-flop is edge triggered. In this book, we use the term flip-flop.

In order to "remember" a past state, sequential circuits rely on a concept called feedback. This simply means the output of a circuit is fed back as an input to the same circuit. A very simple feedback circuit uses two NOT gates, as shown in Figure 3.18.

In this figure, if Q is 0, it will always be 0. If Q is 1, it will always be 1. This is not a very interesting or useful circuit, but it allows you to see how feedback works.

A more useful feedback circuit is composed of two NOR gates resulting in the most basic memory unit called an SR flip-flop. SR stands for "set/reset." The logic diagram for the SR flip-flop is given in Figure 3.19.

We can describe any flip-flop by using a characteristic table, which indicates what the next state should be based on the inputs and the current state, Q. The notation Q(t) represents the current state, and Q(t + 1) indicates the next state, or the state the flip-flop should enter after the clock has been pulsed. Figure 3.20 shows the actual implementation of the SR sequential circuit and its characteristic table.

An SR flip-flop exhibits interesting behavior. There are three inputs: S, R, and the current output Q(t). We create the truth table shown in Table 3.13 to illustrate how this circuit works.

Table 3.13: Truth Table for SR Flip-Flop

		Present State	Next State
S	R	Q(t)	Q(t+1)
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	0
1	0	0	1
1	0	1	1
1	1	0	undefined
1	1	1	undefined

For example, if S is 0 and R is 0, and the current state, Q(t), is 0, then the next state, Q(t + 1), is also 0. If S is 0 and R is 0, and Q(t) is 1, then Q(t+1) is 1. Actual inputs of (0,0) for (S,R) result in no change when the clock is pulsed. Following a similar argument, we can see that inputs (S,R) = (0,1) force the next state, Q(t + 1), to 0 regardless of the current state (thus forcing a reset on the circuit output). When (S,R) = (1,0), the circuit output is set to 1.

There is one oddity with this particular flip-flop. What happens if both S and R are set to 1 at the same time? This forces both Q and to 1, but how can Q = 1 = ? This results in an unstable circuit. Therefore, this combination of inputs is not allowed in an SR flip-flop.

We can add some conditioning logic to our SR flip-flop to ensure that the illegal state never arises-we simply modify the SR flip-flop as shown in Figure 3.21. This results in a JK flip-flop. JK flip-flops were named after the Texas Instruments engineer, Jack Kilby, who invented the integrated circuit in 1958.

Another variant of the SR flip-flop is the D (data) flip-flop. A D flip-flop is a true representation of physical computer memory. This sequential circuit stores one bit of information. If a 1 is asserted on the input line D, and the clock is pulsed, the output line Q becomes a 1. If a 0 is asserted on the input line and the clock is pulsed, the output becomes 0. Remember that output Q represents the current state of the circuit. Therefore, an output value of 1 means the circuit is currently "storing" a value of 1. Figure 3.22 illustrates the D flip-flop, lists its characteristic table, and reveals that the D flip-flop is actually a modified SR flip-flop.

3.6.4 Examples of Sequential Circuits

Latches and flip-flops are used to implement more complex sequential circuits. Registers, counters, memories, and shift registers all require the use of storage, and are therefore implemented using sequential logic.

Our first example of a sequential circuit is a simple 4-bit register implemented using four D flip-flops. (To implement registers for larger words, we would simply need to add flip-flops.) There are four input lines, four output lines, and a clock signal line. The clock is very important from a timing standpoint; the registers must all accept their new input values and change their storage elements at the same time. Remember that a synchronous sequential circuit cannot change state unless the clock pulses. The same clock signal is tied into all four D flip-flops, so they change in unison. Figure 3.23 depicts the logic diagram for our 4-bit register, as well as a block diagram for the register. In reality, physical components have additional lines for power and for ground, as well as a clear line (which gives the ability to reset the entire register to all zeros). However, in this text, we are willing to leave those concepts to the computer engineers and focus on the actual digital logic present in these circuits.

Another useful sequential circuit is a binary counter, which goes through a predetermined sequence of states as the clock pulses. In a straight binary counter, these states reflect the binary number sequence. If we begin counting in binary: 0000, 0001, 0010, 0011, . . . , we can see that as the numbers increase, the low-order bit is complemented each time. Whenever it changes state from 1 to 0, the bit to the left is then complemented. Each of the other bits changes state from 0 to 1 when all bits to the right are equal to 1. Because of this concept of complementing states, our binary counter is best implemented using a JK flip-flop (recall that when J and K are both equal to 1, the flip-flop complements the present state). Instead of independent inputs to each flip-flop, there is a count enable line that runs to each flip-flop. The circuit counts only when the clock pulses and this count enable line is set to 1. If count enable is set to 0 and the clock pulses, the circuit does not change state. You should examine Figure 3.24 very carefully, tracing the circuit with various inputs to make sure you understand how this circuit outputs the binary numbers from 0000 to 1111. You should also check to see which state the circuit enters if the current state is 1111 and the clock is pulsed.

We have looked at a simple register and a binary counter. We are now ready to examine a very simple memory circuit.

The memory depicted in Figure 3.25 holds four 3-bit words (this is typically denoted as a 4 x 3 memory). Each column in the circuit represents one 3-bit word. Notice that the flip-flops storing the bits for each word are synchronized via the clock signal, so a read or write operation always reads or writes a complete word. The inputs In₀, In₁, and In₂ are the lines used to store, or write, a 3-bit word to memory. The lines S₀ and S₁ are the address lines used to select which word in memory is being referenced. (Notice that S₀ and S₁ are the input lines to a 2-to-4 decoder that is responsible for selecting the correct memory word.) The three output lines (Out₁,Out₂, and Out₃) are used when reading words from memory.

You should notice another control line as well. The write enable control line indicates whether we are reading or writing. Note that in this chip, we have separated the input and output lines for ease of understanding. In practice, the input lines and output lines are the same lines.

To summarize our discussion of this memory circuit, here are the steps necessary to write a word to memory:

1. An address is asserted on S₀ and S₁.

2. WE (write enable) is set to high.

3. The decoder using S₀ and S₁ enables only one AND gate, selecting a given word in memory.

4. The line selected in Step 3 combined with the clock and WE select only one word.

5. The write gate enabled in Step 4 drives the clock for the selected word.

6. When the clock pulses, the word on the input lines is loaded into the D flip-flops.

We leave it as an exercise to create a similar list of the steps necessary to read a word from this memory. Another interesting exercise is to analyze this circuit and determine what additional components would be necessary to extend the memory from, say, a 4 x 3 memory to an 8 x 3 memory or a 4 x 8 memory.