Lab 11: Sequential Circuits

As mentioned in Lab 10, sequential logic circuits are a type of logic circuit where the output of the circuit depends not only on the input, as in combinational logic circuits, but also on the sequence of past inputs that are used to determine the state of the circuit.

As we observed in Lab 10, flip-flops can be used to hold (remember) the state of the system. Each flip-flop holds one bit of data and can be used to represent one state variable within a system. A system that has N states will require log₂N state variables and hence log₂N flip-flops. If N is not a power of two, then you have to round log₂N up to the nearest integer.

So, if you need 4 states, you will need two state variables. Let's assume that the names of the state variables are A and B. Then the four states are provided by the following values of A and B: AB = 00, 01, 10, and 11. If you need 8 states, you will need three state variables, for example, A, B, and C, where (ABC = 000, 001, 010, 011, 100, 101, 110, and 111). And if you need 6 states, you still need three state variables since log₂6 = 2.58 which will be 3 after rounding up (since you can’t have a fractional number of state variables). In this case, you could use any 6 of the 8 states, leaving the other 2 unused.

For example, if you had an elevator controller for a building that had floors 1 through 6, the states that indicate which floor the elevator is at would likely be 001 (1), 010 (2), 011 (3), 100 (4), 101 (5), and 110 (6), with states 000 (0) and 111 (7) unused.

Now, we are designing a two-bit saturating up/down counter. There are two output variables, A and B, and one control input X. When X = 1, the counter will count up with each clock trigger until it reaches state AB = 11 where it will remain until X is changed to 0 (i.e. the counter is saturated). Similarly, when X = 0, the counter will count down until it reaches the lowest count AB = 00, remaining until X is changed to 1. If X is changed whenever the upper or lower limit state is reached, the system will oscillate up and down from 00, 01, 10 to 11 and back.

A state diagram is used to represent the behavior of a sequential system where the state is represented by a circle, and the transitions between states are represented by an arrow (also called an edge or arc). The state diagram below represents a 2-bit saturating up/down counter.

The block diagram below shows the overall design of a sequential circuit. The state register is built by flip-flops that are used to hold the state of the system. The flip-flop Q outputs represent the current state of the system or the present state. The state will change or transition on the triggering edge of the clock (in Lab 10, we learned that triggering can be a positive edge or a negative edge).

Figure 11.2: Sequential Circuit Architecture

In the saturating-counter example, there are four states (S_0, S_1, S_2, and S_3), so we need two bits to represent the state values. If the current state is S_1, the Q outputs for the current state will be 01, which also can be used to represent the current counter value (AB).

The combinational circuit in the diagram combines the current state-register outputs (representing the current state) with any external inputs to generate the state-register inputs that will determine the next state of the circuit at the next clock trigger. In this experiment, we use D flip-flops to store the current state. Since the next state of a D flip-flop follows the D input, the flip-flop inputs will be the same as the following state (this is not true for the other types of flip-flops, SR, JK, and T). In the saturating-counter example, if the current state is S_1 (output is AB = 01) and the input is X = 0, the next state will be S_0 (output will be AB = 00). Therefore, the following state values of inputs D_AD_B must also equal 00. (Note: you can use any flip-flop in a sequential logic design, but the flip-flop inputs will differ depending on the type of flip-flop used.)

The saturating-counter example has one external input U but no output other than the flip-flop outputs AB (i.e. the state of the counter). The elevator example of Lab 10 is more complex: the state of the system would be the current floor that the elevator is on, the inputs would be the buttons on each floor pushed to call the elevator, and the buttons inside that select the destination floor, and the output would be used to control the motors to make the elevator go up and down. As we will see in this experiment, where the output depends on both the current state and the inputs, the circuit is called a Mealy machine. Where it depends only on the current state, it is called a Moore machine. (The elevator would be an example of the Mealy machine.)

A state table is used to design the combinational circuit within a sequential circuit. A state table differs from a truth table in that, in addition to inputs and outputs, it also represents both the current state of the system (as an input) and the next state of the system (as an output). Below is the state table for the saturating up/down counter:

Notice in the state table that the State Variables are shown twice as the current state value (before the clock trigger) and as the next state value (after the clock trigger). To design the combinational logic circuit, we need to follow these steps.

1. Derive the state table based on the desired behavior of the sequential circuit (represented by the state diagram in Figure 11.1).
2. Given the current and following states, based on the type of flip-flops used, determine the flip-flop inputs (since we are using D flip-flops, notice in the above table how the flip-flop inputs are the same as the following state values for states A and B).
3. Derive K-maps from the state table and determine the combinational logic equations for flip-flop inputs An and Bn as a function of external input U and the present state of flip-flop outputs A and B.
4. If the sequential circuit has an explicit output(s), use K-maps to determine the combinational logic equations for the outputs as a function of external input(s) and present state (for Mealy machines) or only as a function of the present state (for Moore machines).
5. Draw the logic diagram with one flip-flop per state variable, the combinational logic circuits for the flip-flop inputs, and, if used, the combinational logic circuits for the circuit outputs (we do not have one in our saturating counter example).

Notice that AX appears in both equations, so it is only necessary to create it once and then use it as an input to the circuit for A_n and for B_n.

Question

1. In your lab report, copy the state table and the above equations. Next, draw the circuit diagram, including two D flip-flops (7474) and the logic circuits for A_n and B_n. Do not forget to include the clock input.

In this section, you will design (but not build) a sequential circuit, or state machine, using two D-flip-flops (7474) and some gates. This will be a "Mealy" machine; i.e. a circuit whose output is a function of both current state and inputs. The circuit has an external input X, an output Y, and two state-variable D-flip-flops A and B.

The state diagram is shown below with X/Y values along with the arrows. Because this is a Mealy machine, the output Y is determined by input X as well as by the present state AB.

Figure 11.3: Mealy State Machine Architecture

For example: alongside the arrow from state S_0 to state S_1, we see X=0 / Y=1. This means that with X=0, there will be a transition to state S_1 when the clock pulse comes. It also means that until the clock pulse comes and while the circuit is still in state S_0, the output Y will be 1 as long as X=0. Therefore, X is the current input and Y is the current output for any given current state.

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From the above state diagram, finish the state table (on your own paper). Notice in the table that each pair of rows represents the same state of AB (i.e. the same circle in the diagram).

Next, design (but do not build) the circuit. Do this with just three chips: one 7474 and 2 NAND chips. (If you need to complement A and/or B, just use inverted flip-flop outputs — there is no need to use gates as inverters. But you will need a gate to produce X .)

Questions

For this experiment:

2. From the state table, derive maps for A_n, B_n, and Y from the maps, and derive their equations. Each map is labeled as shown. Since A's column in the state table is at the left, it appears at the bottom row of the map.
   

   BX A	00	01	11	10
   0
   1

   A_n =

   BX A	00	01	11	10
   0
   1

   B_n =

   BX A	00	01	11	10
   0
   1

   Y =

3. Draw the circuits using only a 7474 (dual D flip-flops) and NANDs (7400's and/or 7410's, but no 7408's or 7432's). Use DeMorgan symbols for NANDs but only where doing so preserves the original AND/OR structure. A diagram with all normal or all DM gate symbols is not acceptable. (Remember: connecting wires must either have bubbles at both ends or no bubbles at all.)

This section is similar to Exp#11.2 except that the sequential circuit here is a "Moore" machine — a state machine whose output depends only on the current state, not on inputs. The circuit has two D-flip-flops, A, B, an input X (from a switch), and an output Y (connected to an LED). Its state diagram is shown below.

Figure 11.5: More State Machine Architecture

You can think of this circuit as a primitive entry-code detector. You enter a sequence of values for input X and move the circuit from state to state. (This would be like pressing a sequence of keys in a special order to gain entry to a room, a car, or whatever.)

Here, each step involves setting input X to 0 or 1 (using a switch) and then pressing a pulser. You do this at least 3 times. If you set X to the right value each time, the state machine reaches state 3 (AB = 11). At this time, Y goes high and turns on its LED, indicating that you have gained entry. If at any point, you clock in a wrong value for X, the state machine returns to 0 where you have to try a new sequence for X.

Of course, with a real entry-code detector, if you did not know the code there would be too many possible sequences to try — here there are only 8. Also, unlike here, just pressing a key enters it; you do not have to clock it in with a pulser.

Design steps:

1. Choose a sequence of values for X. Make X2 the same as X0 (X0 = 0, X2 = 0 or X0 = 1, X2 = 1) and make X1 the opposite. Example: X0 = 0, X2 = 0, so X1 = 1, or the opposite. (This may simplify the circuit you will build.)
   X0 = X1 = X2 =
2. Derive the state table from Figure 11.6 state diagram. The table has the same format as that in Exp# 11.2.

   Current State			Input	Output	Next State
   State	A	B	X	Y	State	An	Bn
   S_0	0	0	0
   	0	0	1
   S_1	0	1	0
   	0	1	1
   S_2	1	0	0
   	1	0	1
   S_3	1	1	0
   	1	1	1

3. From the table, draw K-maps for A_n, B_n, and Y, and from the maps, and derive equations in their simplest form.

   BX A	00	01	11	10
   0
   1

   A_n =

   BX A	00	01	11	10
   0
   1

   B_n =

   BX A	00	01	11	10
   0
   1

   Y =

4. Use KiCad to draw the circuit. Only three chips are needed: a 7474 (dual D-flip-flops), a 7410 (NAND), and a 7402 (NOR). Do not use a 7400. Remember: a 7410 NAND gate can also be drawn as an invert-OR and a 7402 NOR gate can also be drawn as an invert-AND. Use DeMorgan symbols where appropriate. Inverters are not needed for A and B since they are available as flip-flop outputs.
5. Build the circuit. Bring state variables A and B as well as Y to LEDs so you can monitor the sequence of states and the output.
   
   

Start by momentarily grounding the clear inputs of both flip-flops to put the circuit into state 0. Then,

• Set X to X0 and clock the circuit with a pulser. The state should remain at S_0.
• Repeat with X = X0. The state should move to S_1.

Repeat this test method, moving to the next state by entering the correct subsequence to get there. Then enter X for that state and return to S_0. Continue until you have checked all the states. (Of course, from state S_3 you can only go back to S_0 — there is no correct or incorrect value for X.)

• For the on-campus lab, have your instructor check your results.
• For the online lab, take pictures for each input step, and post them in the report.

Mealy example: Table 2 is similar to but not the same as the one in Exp #11.3. The values of Y depend on state variables A and B and on input X, not just on A and B (see the first two rows as an example). So this represents a Mealy machine.

Since the values of Y in each row exist at the same time as the corresponding values of A, B, and X (i.e. all in the present state), then the first 4 columns (shaded) can be treated as a simple truth table, and we can design a circuit for Y using gates but no flip-flops. (You would just fill in an 8-square K-map.)

Here we omit Y in Table 3 to concentrate on A_n and B_n. Using D flip-flops simplifies the next-state design. The D values are the same as the n.s. values since at the triggering edge of the clock, the D values are stored into the flip-flops and emerge as the new A and B. Since the D's exist at the same time as A, B (current state), and X, we can design circuits for them using K-maps, just like for Y.

Figure 11.7: K-Map for Next State

From the K maps of Figure 11.7, we derive expressions for the flipflop inputs:

A_n = B

B_n = B X + A X + A B X

Now, suppose you build circuits for A_n and B_n and connect them to their flip-flops' inputs. You then supply pulses to the flipflop CLK inputs and also input a series of values for X, synchronized with the clock.

The diagrams in Figure 11.8 below demonstrate the behavior of the circuit as it responds to the clock pulses. The result is a sequence of states over time representing the dynamic behavior of the circuit for a particular series of input X values.

Figure 11.8: Timing Diagram of the Circuit

At the start of each present state, the circuit uses values of flip-flops A and B and an input X to generate A_n and B_n. This is indicated by the down arrows in the diagram. Thus, A, B, X, A_n, and B_n all exist at the same time. Then, when the clock pulse comes, A_n and B_n values are moved into their respective flip-flops and become the next present state of A and B as indicated by the up arrows. At the same time, a new value is supplied to X.

The waveforms shown below indicate how A and B change from state to state as input X assumes various values over time (assigned here at random). This illustrates the dynamics of state machine behavior.

Question

4. Based on Table 2, use the online diagram website (https://app.diagrams.net/) to draw the state diagram, and post it in your report.
5. Assume that X will equal 0 after the final clock pulse. What will be the new values of A_n and B_n? (Include the answer in Exp# 11.1.)

Finally, suppose output Y was included in the circuit. In Figure 11.8 it would appear at the bottom of each column along with outputs A_n and B_n. Now consider the cases where A and B are the same but X is not; e.g. columns 3 and 6 (or 4 and 8). In a Moore circuit, Y depends only on A and B and not on X, so the value of Y in both columns would be the same. But in a Mealy circuit, Y depends as well on X, so its value could change.

6. Draw the state of the signal Y on the timing diagram in Figure 11.8.