Finite State Machines
School of Electrical and Computer Engineering
Georgia Tech1
Contents:
Background
Prelab
Experiment
Page 1
Page 6
Page 8
Background:
Combinational logic circuits can perform many complex tasks, but they have no memory of past actions
or events. They generate outputs based only on the current input values. A Finite State Machine, a type
of sequential circuit, combines a combinational logic circuit with registers in order to include memory in
the logic decisions. The “state” is a set of system variables that contain information about the sequence of
inputs that led from an initial state to the current state. The current state is stored as a value in one or
more registers. This circuit is easily built using a decoder and OR gates.
A Finite State Machine can be represented by a transition diagram that shows what states are available
and what input values cause a transformation from one state to another. The following transition diagram
shows the states named as A, B, and C. The states are defined by the values stored in the registers; for
example, State B corresponds to S1S0 = 01 where S1 is the value of Register 1 and S0 is the value of
Register 0. The external input is X.
X=1
A
(00)
X=0
X=1
X=1
C
(10)
X=0
X=0
B
(01)
Figure 1: State Transition Diagram.
The value of X on each arrow indicates what input value will cause the state to transition to the next state.
For example, to get from state A to state B, the input X must be 0. If the input X is 1, then state A will
stay in state A. The actual transition occurs after the next clock pulse in the circuit. This diagram can be
summarized in a truth table where NS1 represents the next value of S1 , that is, the value after the state
transition:
1
For comments, please contact Professor Bonnie Ferri.
1
Finite State Machines
State S1 S0 X New State NS1 NS0
A
0 0 0
B
0
1
A
0
0
1
A
0
0
B
0
1
0
C
1
0
B
0
1
1
A
0
0
C
1
0
0
C
1
0
C
1
0
1
A
0
0
--
1
1
0
--
--
--
--
1
1
1
--
--
--
Because there are three states, two registers are needed to be able to count up to three. The last two rows
are empty, because there is no fourth state. If the state machine is built correctly, it should never end up
in the fourth state (which would be represented as S1=1, S0 =1), and these rows can be ignored.
The right side of the table shows the values in the next state. The top row of the table shows that State A
(S1=0, S0 =0) transitions to State B (S1=0, S0=1) when X=0. The second line shows that the system stays
in State A (00) when X=1. The third line shows that the system transitions from State B (01) to State C
(10) when X = 0; and so on.
A finite state machine can be built from a decoder, OR gates, and registers (or flip-flops).
Decoders: A decoder has n inputs and 2n outputs. Below is the truth table for a 3-to-8 decoder—that
is, 3 inputs and 8 outputs where A0 – A2 represent the inputs and Y0-Y7 represent the outputs. The circuit
converts the binary representation of a number into an output pattern which has a “1” on the output pin
associated with the decimal representation of the input with all other outputs are “0”. For example, the
binary number 011 is equal to the decimal value 3, so Y3 = 1 with all other outputs equal to 0.
A2 A1 A0 Y7 Y6 Y5 Y4 Y3 Y2 Y1 Y0
0 0 0 0 0 0 0 0 0 0 1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
The icon for a 3 to 8 decoder is
A0
A1
A2
3 to 8
decoder
2
Y0
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Finite State Machines
A decoder can implement a combinational circuit that has multiple inputs and outputs. For example, look
at the output NS0 in the state transition table. NS0 = 1 when S1S0 X = 000. Let the decoder inputs be
defined as A2A1 A0 = S1S0 X, then NS0 = Y0. Similarly, NS1 = 1 when S1 S0X = 010 or when S1S0 X = 100.
So let NS1 = Y2 OR Y4.
The NS1 and NS0 signals are connected to the registers, so that the registers will be updated with the new
state when the clock signal goes from low to high.
Registers: Registers are the basic units of memory in a digital circuit. They can store a 0 or 1 and
output it for other circuits to use. To change the value in the register, the new value is put on the
incoming pin and the clock signal is changed from 0 to 1 (positive edge) to store the new value. Registers
can be made from D flip flops.
IN
The value of IN (0 or 1) is only
stored in the register when
CLOCK goes from 0 to 1. OUT
is always equal to whatever is
currently stored in the register.
OUT
CLOCK
State Machine Circuit Diagram: A state machine can be implemented with registers to hold the
memory and a decoder and OR gates to perform the combinational logic. The state transition diagram in
Figure 1 can be implemented using the circuit diagram shown below.
X
NS0
NS1
Decoder
S1
CLOCK
S0
CLOCK
Figure 2: Schematic for the state machine with state transition diagram shown in Figure 1.
3
Finite State Machines
Protoboards: Protoboards (also known as breadboards) make many connections under the board in
order to reduce the number of wires that you have to connect. Typically, groups of 5 holes are connected.
There are also lines of groups of 5 that are all connected. See the example figure.
Terminals
Every hole in
this vertical line
is connected.
This is true of
every vertical
line.
Each horizontal
group of 5
holes is
connected.
Figure 3: Protoboard.
Integrated Chip: An integrated chip (IC) can have several gates (circuits) on it. For example, the
74HC32 IC has 4 2-input OR gates. It has 14 wire pins coming out the sides with the connections from
the pins to the logic gates shown in Figure 4.
pin 1 near
notch
Figure 4b: Logic gates for the
74HC32 IC.
Figure 4a: Pin Diagram to the
74HC32 Quad 2-input OR gate.
4
Finite State Machines
Pin Connection Diagram: A pin connection diagram shows the physical layout of the connections
between components. Each component is shown along with the corresponding pin numbers for that
component. The pin diagram for the logic expression F = (A OR B) OR (C OR D) is shown below. The
pin numbers correspond to the pins as labeled above for the 74HC32 2-input OR chip.
A
1
B
2
OR Gate 74HC32
3
9
OR Gate 74HC32
8
10
C
4
D
5
F
OR Gate 74HC32
6
To build this circuit, you would need to connect a wire from pin 3 of the IC to pin 9, another wire from
pin 6 to pin 10. Pins 1, 2, 4, and 5 are inputs and connect to outputs from other components while pin 8 is
connected to an LED or an input to another component.
The pin diagram for the state machine with schematic shown in Figure 2 is
To LED 0
Input from dip
switch
1
2
3
2
D
Q
A0
Y0
15
A1
Y1
14
74HC74A
13
D flip flop
A2
74HC238
Decoder
Y2
Y3
Y4
Y5
Y6
Y7
5
12
11
10
9
7
1
OR Gate
74HC32
To LED 1
3
2
B
12
D
Q
9
74HC74A
D flip flop
Figure 5: Pin diagram for state machine in Figures 1 and 2. For simplicity, some of the connections, such
as ground, Vcc, clock and reset are not shown.
5
Finite State Machines
Prelab:
X=0
A
(00)
X=1
B
(01)
X=0
X=1
X=1
X=0
X=0
D
(11)
X=1
C
(10)
1) For the state transition diagram shown, complete the truth table below.
STATE S1 S0 X NEW STATE NS1 NS0
A
0 0 0
A
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
6
Finite State Machines
2) Draw the circuit diagram for the state machine. Your circuit should include a decoder, two
registers, and 2- or 3-input OR gates.
3) Draw the pin connection diagram for the state machine. Start with the basics of the pin diagram
shown below, and complete it using Figure 5 as a guideline. Use the IC descriptions in the
appendix to determine the proper pins to use.
To LED 0
Input from dip
switch
1
A0
A1
A2
74HC238
2
D
Q
Y0
15
Y1
14
74HC74A
13
D flip flop
Y2
Y3
Y4
Y5
Y6
Y7
5
12
11
To LED 1
10
9
12
7
D
Q
74HC74A
D flip flop
Prelab sign off, TA or Instructor ______________
7
9
Finite State Machines
Experiment:
Part A: Build the Finite State Machine defined in the introduction of this lab with the pin
connection diagram shown in Figure 5..
Two registers, two pushbuttons (RESET and CLOCK), DIP Switches, a decoder, registers, and OR gates
are already placed on the breadboard. The IC diagrams and pin layouts are given in the appendix.
Terminals
VCC
GND
DIP Switches
3- input OR gates
LED for S0
LED for S1
L0
L1
Decoder Chip
2 input OR gate
Register Chip
Debouncer
Clock Button
Reset Button
Figure 6: Layout of the protoboard.
The circuit is partially wired as shown in Figure 6:
• The battery pack supplies high voltage (Vcc) from the red lead and the ground (GND) from the
black lead. The power routed through the uppermost dip switch, which must be turned on to
power the circuits.
• Vcc and GND are connected to all of the ICs.
• The ICs are all oriented so that the upper left pin is PIN 1 (when the board is oriented as shown
above).
• The light emitting diodes (LEDs) are connected to S0 and to S1. The LEDs light when the signal
is “1”.
• The reset button resets the state to State A. It is already connected to the registers.
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Finite State Machines
•
•
•
•
The clock button advances the clock. The register stores the values of its input and passes the
value to its output after each time that the clock button is pushed. It is already connected to the
registers.
The resistors (small cylindrical components with stripes) are used to limit the current in some of
the lines. Otherwise, the components many burn out too quickly.
A “debounce” circuit, consisting of an additional IC and a capacitor, is connected to the clock
button. A bounce is when a switch does not produce a clean signal. There may be some ripple in
the signal (or perhaps a small spike) that might be interpreted by the digital logic as a second
input signal, in this case, an extra clock pulse. The debounce circuit filters the signal to remove
the ripple or spike.
The decoder has the enable pins already wired.
Steps:
1) Make sure that the power is turned off of the board prior to constructing the circuit, that is, turn
off the uppermost dip switch.
2) Complete the circuit according to the PIN Connection Diagram shown in Figure 5. Use the loose
lead wires that are supplied with the kit. The color of wire is insignificant. Refer to the
protoboard description to see which holes are connected within the protoboard, and refer to the
appendix to see the pin numbers of the ICs.
3) Turn on the power to the circuit.
4) Now, test the State Machine to make sure it is working properly. First reset the state machine to
put it in state A. Set the DIP switch to 0 and then press the clock button. Determine the state
from the LEDs and mark it in the table below. Change the input to 1 and press the clock button
to see the next state value. Move the DIP switch to either the on or off position according to the
table, and then press the clock button to change the state. Complete the table below to show the
states corresponding to the given input sequence. Verify that this sequence follows the
transitions as dictated by the state transition table in Figure 1.
State
Input
A
0
State
Input
State
1
Input
State
1
Input
State
Input
0
State
0
Input
State
0
Input
State
1
Instructor or TA Initials: ___________
Part B: Build the Finite State Machine designed for the Prelab.
1) Turn off the power to the circuit before making any changes.
2) Remove only the wires that you added to the protoboard for Part A (do not remove the wires that
were already there) .
3) Build the circuit using the pin connection diagram that was completed for the Prelab.
4) When the state machine is complete, turn on the power. Reset the state machine to put it in state
A, and input the following sequence into the circuit. Write down the state after each step:
State
Input
A
0
State
Input
1
State
Input
1
State
Input
1
Instructor or TA Initials: ___________
9
State
Input
0
State
Input
1
State
Input
1
State
Finite State Machines
Appendix
Decoder IC (74HC238):
Register (D-FlipFlop) IC (74HC74):
10
Finite State Machines
2 Input OR Gate IC (74HC32):
3 Input OR Gate IC (74HC4075B1R)
11