EE365
Adv. Digital Circuit Design
Clarkson University
Lecture #11
State Machines
Topics
• State Machine Types
• State machine Design
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Clocked synchronous seq. circuits
• A.K.A. “state machines”
• Use edge-triggered flip-flops
• All flip-flops are triggered from the same
master clock signal, and therefore all change
state together
• Feedback sequential circuits
– No explicit flip-flops; state stored in feedback
loops
– Example: edge-triggered D flip-flop itself (4 states)
– Sections 7.9, 7.10 (advanced courses)
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State-machine structure
depends on
(Mealy) output
state and input
typically edge-triggered
D flip-flops
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State-machine structure
output depends
(Moore)
on state only
typically edge-triggered
D flip-flops
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Remember
Mealy  state and input
Moore  state
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Notation, characteristic
equations
• Q means “the next value of Q.”
• “Excitation” is the input applied to a
device that determines the next state.
• “Characteristic equation” specifies the
next state of a device as a function of its
excitation.
• S-R latch: Q = S + R´ · Q
• Edge-triggered D flip-flop: Q = D
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State-machine analysis steps
• Assumption: Starting point is a logic diagram.
1. Determine next-state function F and output
function G.
2a. Construct state table
– For each state/input combination, determine the
excitation value.
– Using the characteristic equation, determine the
corresponding next-state values (trivial with D ff’s).
2b. Construct output table
– For each state/input combination, determine the
output value. (Can be combined with state table.)
3. (Optional) Draw state diagram
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Example state machine
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Excitation equations
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Transition equations
• Excitation equations
• Characteristic equations (D flip-flop is easy)
• Substitute excitation equations into
characteristic equations
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Transition and state tables
(transition
equations)
(output equation)
transition
table
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state table
state/output
table
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State diagram
• Circles for states
• Arrows for transitions (note output info)
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Modified state machine
MAXS
MAXS = Q0  Q1
• Moore machine (output depends only on
current state)
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Updated
state/output
table, state
diagram
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Timing diagram for state machine
• Not a complete description of machine
behavior
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State-machine design and synthesis
The creative part, like
writing a program
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Turning the crank, like
a compiler does
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A Design example (from text)
• Design a machine inputs A and B with output
Z that is 1 if:
– A had the same value at the two previous ticks
– B has been 1 since the last time the above was
true
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State Assignment
• Can minimize number of states (see text), but
hardly anyone bothers anymore.
• Need to assign state-variable combinations to
states.
– Minimum number of variables for n states is log2
n
– Using more than minimum number may be
advantageous in some situations, e.g., one
variable per state (“one-hot”) (see text).
– Example -- 4 states, 2 state variables (Q1,Q2):
A ==> 00
B ==> 01
C ==> 10
D ==> 11
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Up to this point is “art”, the
rest is just “turning the crank.”
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State assignment contd
.
• There are 6,720 different state
assignments of 5 states to 3 variables.
– And there are even more using 4 or more
variables
• Here are a few “obvious” or “interesting”
ones:
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Transition equations; circuit
• Transition table specifies each state
variable (Q1, Q2) as a combinational
logic function of Q1, Q2, X1, X2.
– Find a realization of each function by your
favorite means -- ad hoc, minimal sum-ofproducts, etc.
• Build the circuit.
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Transition/output table (decomposed
assignment)
D1 D2 D3
• Simple textual substitution
• With D flip-flops, excitation table is identical to
transition table (now with D1 D2 D3)
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Develop
excitation
equations
• Assume unused states have next-state = 000
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Option:
Minimize
with “Don’t
Cares”
D=1
D2=Q1 • Q3´ • A´ + Q3 • A + Q2 • B
D3= A
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Circuit using minimized equations
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Unused States
‘0’ versus ‘don’t-care’
• Minimal-Risk versus Minimal-Cost
• Don’t-care uses less gates, but can give
unpredictable output if the system goes into a
state that it was not meant to
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In Class Practice Problem
1s-counting machine.
Design a clocked synchronous state machine with two
inputs, X and Y, and one output, Z. The output should be 1 if
the number of 1 inputs on X and Y since reset is a multiple of
4, and 0 otherwise. Use D flip-flops.
• Write State and Output table
• Write Transition/Excitation and Output table
• Write K-maps
• Find Excitation and Output Equations
• Draw AND-OR and D flip-flop circuit
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In Class Practice Problem
State and Output Table
I started the table for you. Essentially, in this problem, the
output indicates the number of inputs received modulo 4.
Here the state represents the current count and the inputs X
and Y represent the new ‘1s’ that can be added to the count.
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In Class Practice Problem
State and Output Table
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In Class Practice Problem
Transition/Excitation and Output Table
Here’s the layout of the Transition/Excitation
and Output Table:
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In Class Practice Problem
Transition/Excitation and Output Table
Next:
• Write K-maps
• Find Excitation and Output Equations
• Draw AND-OR and D flip-flop circuit
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In Class Practice Problem
K-maps, Excitation Eqns. and Output Eqns.
D1 = Q2•X’•Y + Q1’•X•Y + Q1•X’•Y’ + Q2•X•Y’
D2 = Q1’•X’•Y + Q1’•X•Y’ + Q2•X’•Y’ + Q2’•X•Y
Z = Q1’•Q2’
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In Class Practice Problem
Circuit (outline)
X
Y
Combinational
AND-OR
D1
(flip flop)
D2
(flip flop)
Comb.
(AND)
Z
D1 = Q2•X’•Y + Q1’•X•Y + Q1•X’•Y’ + Q2•X•Y’
D2 = Q1’•X’•Y + Q1’•X•Y’ + Q2•X’•Y’ + Q2’•X•Y
Z = Q1’•Q2’
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Next time
• Registers
• Counters
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