Download Document

Computer Architecture I: Digital Design
Dr. Robert D. Kent
Logic Design
Sequential Circuits
Part I
Review
• We have studied logic design in the contexts of Medium Scale Integration (MSI)
of gate devices and programmable logic devices (PLD).
• We have studied the design of a number of specific, practical functional circuits
with a view to re-using those circuits as components in MSI design.
Adders
Subtractors
Comparator
Decoders
Multiplexers
• We note the differing design approaches, or emphases, effected by differential
layering of abstraction. (The same design issue arises in the context of software
engineering as well.)
SSI:
MSI:
Boolean algebra / Simplification / Logic gates
Interconnection networks / Iterative re-use / Components
Goals
• Previously, we studied Combinational circuits, or networks.
– These are time independent because the inputs, once provided,
immediately establish what the outputs will be.
Goals
• Previously, we studied Combinational circuits, or networks.
– These are time independent because the inputs, once provided,
immediately establish what the outputs will be.
• We now continue to consider Sequential Networks
– These are time dependent in that the initial values of the circuit
outputs are used to provide input to the same circuit.
Goals
• Previously, we studied Combinational circuits, or networks.
– These are time independent because the inputs, once provided,
immediately establish what the outputs will be.
• We now continue to consider Sequential Networks
– These are time dependent in that the initial values of the circuit
outputs are used to provide input to the same circuit.
– This is called feedback.
Goals
• The properties of sequential networks yield the capability to design
memory circuits
– characterized by internal states and secondary states that describe the
behaviour and values in a circuit before and after inputs are applied.
Goals
• The properties of sequential networks yield the capability to design
memory circuits
– characterized by internal states and secondary states that describe the
behaviour and values in a circuit before and after inputs are applied.
• There are two kinds of sequential networks
Goals
• The properties of sequential networks yield the capability to design
memory circuits
– characterized by internal states and secondary states that describe the
behaviour and values in a circuit before and after inputs are applied.
• There are two kinds of sequential networks
– Synchronous - behaviour is governed by the inputs only during
specific discrete time intervals
Goals
• The properties of sequential networks yield the capability to design
memory circuits
– characterized by internal states and secondary states that describe the
behaviour and values in a circuit before and after inputs are applied.
• There are two kinds of sequential networks
– Synchronous - behaviour is governed by the inputs only during
specific discrete time intervals
– Asynchronous - behaviour is governed by the inputs immediately as
they are applied
Goals
• The basic logic element is called the Flip-Flop circuit.
Goals
• The basic logic element is called the Flip-Flop circuit.
• We will study first a primitive element - the basic bi-stable
element.
Goals
• The basic logic element is called the Flip-Flop circuit.
• We will study first a primitive element - the basic bi-stable
element.
– ... then study Latches.
Goals
• The basic logic element is called the Flip-Flop circuit.
• We will study first a primitive element - the basic bi-stable
element.
– ... then study Latches.
– ... then proceed to Flip-Flops and Gated Latches/Flip-Flops.
Goals
• The basic logic element is called the Flip-Flop circuit.
• We will study first a primitive element - the basic bi-stable
element.
– ... then study Latches.
– ... then proceed to Flip-Flops and Gated Latches/Flip-Flops.
• Finally, we will establish an MSI based model of a register and
discuss how to construct load, read, shift and count capabilities into
the register designs.
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
X
X’
Q
Q’
Y
Y’
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
Trace: Starting from the top gate
1. If X = 0 then Q = X’ = 1
X
X’
Q
Q’
Y
Y’
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
X
X’
Q
Q’
Y
Trace: Starting from the top gate
1. If X = 0 then Q = X’ = 1
2. Thus, Y = X’ = Q = 1 implies Q’ = Y’ = 0
Y’
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
X
Q
Q’
Y
Trace: Starting from the top gate
1. If X = 0 then Q = X’ = 1
2. Thus, Y = X’ = Q = 1 implies Q’ = Y’ = 0
This is self-consistent, since X = Y’ = Q’.
X’
Y’
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
X
X’
Q
Q’
Y
Trace: Starting from the top gate
1. If X = 0 then Q = X’ = 1
2. Thus, Y = X’ = Q = 1 implies Q’ = Y’ = 0
This is self-consistent, since X = Y’ = Q’.
The same self-consistency applies when X = 1 (Y = 0).
Therefore, we say the state is stable.
Y’
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
X
– no inputs !!!
– Two outputs.
X’
Q
• This circuit has the representation:
Q’
Y
Y’
• The term bi-stable implies that there are two possible states
Q = 0 , Q’ = 1
and
Q = 1 , Q’ = 0
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no
Q inputs !!!
– Two outputs.
X
X’
Q
Transition Voltage
Smooth Signal Profile
• This circuit has the representation:
Q’
Q’
Transition VoltageY
Y’
• The term bi-stable implies that there are two possible states
Q = 0 , Q’ = 1
and
Q = 1 , Q’ = 0
– There is a third state that is technically possible, called the metastable state. This applies when the voltage signal values of X and Y
(hence, Q and Q’) are precisely half way between their HI and LO
values; however, these in-between states are typically short lived.
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
X
– no
Q inputs !!!
– Two outputs.
X’
Q
Transition Voltage
Noisy Signal Profile
• This circuit has the representation:
Q’
Q’
Transition VoltageY
Y’
• The term bi-stable implies that there are two possible states
Q = 0 , Q’ = 1
and
Q = 1 , Q’ = 0
– There is a third state that is technically possible, called the metastable state. This applies when the voltage signal values of X and Y
(hence, Q and Q’) are precisely half way between their HI and LO
values; however, these in-between states are typically short lived.
Basic Bi-stable Element
• The basic bi-stable element is a simple device characterized by
– no inputs !!!
– Two outputs.
• This circuit has the representation:
X
X’
Q
Q’
Y
Y’
• Although the bi-stable element is worth studying for its simple
properties, it is relatively useless as a computer circuit because
– its value cannot be changed from the “outside” - once power is
applied its value is set (after a brief time period to achieve stability)
and does not change henceforth.
Latches
Latches
• A Flip-Flop is a bistable device that permits both probing of its
current state (value) and modification of the state.
Latches
• A Flip-Flop is a bistable device that permits both probing of its
current state (value) and modification of the state.
– Set the state - store a value 1 in the circuit; also called pre-setting the
state.
Latches
• A Flip-Flop is a bistable device that permits both probing of its
current state (value) and modification of the state.
– Set the state - store a value 1 in the circuit; also called pre-setting the
state.
– Reset the state - store a value 0 in the circuit; also called clearing the
state.
Latches
• A Flip-Flop is a bistable device that permits both probing of its
current state (value) and modification of the state.
– Set the state - store a value 1 in the circuit; also called pre-setting the
state.
– Reset the state - store a value 0 in the circuit; also called clearing the
state.
• We will consider next a class of flip-flops called Latches.
Latches
• A Flip-Flop is a bistable device that permits both probing of its
current state (value) and modification of the state.
– Set the state - store a value 1 in the circuit; also called pre-setting the
state.
– Reset the state - store a value 0 in the circuit; also called clearing the
state.
• We will consider next a class of flip-flops called Latches.
– Characterized by the fact that the timing of the output changes is not
controlled (except possibly by an Enable, or Clock, signal).
SR Latch
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
R
Q
Q’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
R
Q
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
Q’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
Q and Q ’ are the output signal
0
0
– two
inputs,
S and
to as set and reset inputs
values when
theR,Sreferred
and R inputs
are applied - they are also applied
inputs
the
– two as
outputs,
Q to
and
Q’nor gates.
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
R
Q0 ’
Q
Q0
Q’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
Once the nor gates
have stabilized the
– two inputs,
S and R, referred to as set and reset inputs
outputs, Q1 and Q1’
are then fed back
as inputs.
– two outputs,
Q and Q’
R
Q
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
Q’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
The nor gates must stabilize to a
final output , Q2 and Q2’.
– two outputs, Q and Q’
R
Q
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
Q’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
0
Q
1
• Truth table:
S
R
Q0
Q0’
0
0
0
1
Q1
Q1’
Q2
Q2’
0
0
Q’
1
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
0
Q
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
0
0
0
1
0
1
Q2
Q2’
0
0
Q’
1
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
0
Q
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
1
0
1
0
1
0
0
Q’
1
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
0
Q
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
1
1
0
0
1
0
1
1
0
Q’
0
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
0
Q
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
1
1
0
0
1
1
0
0
1
1
0
Q’
0
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
0
Q
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
Q’
0
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0 > 0
1
Q
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
Q’
1 > 1
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
1
Q
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
0
1
1
0
1
0
1
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
1
1
0
Q’
0
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
Q2’
0
0
0
0
0
0
1
1
0
1
0
1
1
0
1
0
0
1
0
0
1
0
1
0
0
1
0
1
0
1
1 > 0
0 > 0
Q
1 > 0
Q’
0
0 > 0
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
0
0
1
1
0
1
0
1
1
0
1
0
0
1
0
0
1
0
1
0
0
1
0
0
Q2’
1
0
1
1 *
1 > 0 > 0
0 > 0
Q
1 > 0
Q’
0
0 > 0 > 1
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
0
Q2’
1
0
1
1 *
1
1
Q
0
Q’
1
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
0
0
1
0
0
Q2’
1
0
1
1 *
0 > 0
1 > 0
Q
0 > 0
Q’
1
1 > 0
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
0
0
1
0
0
1
Q2’
1
0
1
1 *
0 *
0 > 0 > 1
1 > 0
Q
0 > 0
Q’
1
1 > 0 > 0
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
0
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
1
1
0
0
1
1
0
0
0
1
0
1
0
1
1
0
1
0
1
0
0
1
0
0
0
1
1
0
1
0
0
0
0
1
0
0
1
1
Q2’
1
0
1
1 *
0 *
0
1 > 1
0 > 0
Q
1 > 1
Q’
1
0 > 0
Stable!
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
1
• Truth table:
S
R
Q0
Q0’
Q1
Q1’
Q2
0
0
0
0
1
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
x
1
0
1
0
1
0
x
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
0
1
1
0
Q2’
1
0
1
1 *
0 *
0
0
x > 0
x > 0
Q
x > 0
Q’
1
x > 0
Stable!
But not
complementary!
( Q = Q’ )
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
R
Q+
• Truth table: (Adapted)
S
R
Q0
Q0’
Q2
Q2’
Q+
Q+’
0
0
0
0
1
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
x
1
0
1
0
1
0
x
0
1
0
0
1
1
0
1
0
1
1
0
0
0
Q
Q’
Q
Q’
0
1
0
1 *
1
0 *
1
0
Forbidden
Q +’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
R
Q+
• Truth table: (Adapted - Simplified)
S
R
0
0
1
1
0
1
0
1
Q+
Q
0
1
(0) Forbidden
Q +’
S
SR Latch
• This circuit consists of two cross-coupled nor gates with
– two inputs, S and R, referred to as set and reset inputs
– two outputs, Q and Q’
R
Q+
• Truth table: (Adapted - Simplified)
S
R
0
0
1
1
0
1
0
1
Q +’
Q+
S
Q
Two commonly used MSI symbols.
0
S Q
S Q
1
(0) Forbidden
R
Q’
R
Q
S’R’ Latch
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs,
Q+
and Q
+’
S’
R’
Q+
Q +’
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs, Q+ and Q +’
0
Q+ = 1
• Truth table:
0
S’
R’
0
0
Q+
(1) Forbidden
Q +’ = 1
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs, Q+ and Q +’
• Truth table:
S’
R’
0
0
0
1
Q+ = 1
S’ = 0
R’ = 1
Q+
(1) Forbidden
1
Q +’ = 0
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs, Q+ and Q +’
• Truth table:
S’
R’
0
0
1
0
1
0
Q+ = 0
S’ = 1
R’ = 0
Q+
(1) Forbidden
1
0
Q +’ = 1
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs, Q+ and Q +’
• Truth table:
S’
R’
0
0
1
1
0
1
0
1
Q+ = Q
S’ = 1
R’ = 1
Q+
(1) Forbidden
1
0
Q
Q +’ = Q’
S’R’ Latch
• This circuit consists of two cross-coupled nand gates with
– two complemented inputs, S’ and R’, referred to as set and reset
inputs
– two outputs,
Q+
and Q
S’
+’
Q+
• Truth table:
Q +’
R’
S’
0
0
1
1
R’
0
1
0
1
Q+
(1**)
1
0
Q
Two commonly used MSI symbols.
S
Q
S
Q
R
Q’
R
Q
D Flip-Flop
D Flip-Flop
• The D flip-flop overcomes the problem of the SR (S’R’) latch
having forbidden states. It features a single input, D, and a control
input, Clk, provided by the system clock, and produces outputs Q
and Q’.
(Enable)
Clk
D
Q
Q’
D Flip-Flop
• The D flip-flop overcomes the problem of the SR (S’R’) latch
having forbidden states. It features a single input, D, and a control
input, Clk, provided by the system clock, and produces outputs Q
and Q’.
– Note the SR latch
sub-circuit element
(Enable)
Clk
D
Q
Q’
D Flip-Flop
• The D flip-flop overcomes the problem of the SR (S’R’) latch
having forbidden states. It features a single input, D, and a control
input, Clk, provided by the system clock, and produces outputs Q
and Q’.
– Note the SR latch
sub-circuit element
– The control input, Clk,
controls a sub-circuit
called a “gate”.
(Enable)
Clk
D
Q
Q’
D Flip-Flop
• The D flip-flop overcomes the problem of the SR (S’R’) latch
having forbidden states. It features a single input, D, and a control
input, Clk, provided by the system clock, and produces outputs Q
and Q’.
– Note the SR latch
sub-circuit element
– The control input, Clk,
controls a sub-circuit
called a “gate”.
(Enable)
Clk
D
• Since D is the only input, the forbidden values, S = R = 1, never
occur.
Q
Q’
D Flip-Flop
• The DWhen
flip-flop
the problem of the SR (S’R’) latch
Clk =overcomes
0, it
havingfollows
forbidden
states. It features a single input, D, and a control
that:
input, Clk,
by the system clock, and produces outputs Q
S =provided
R=0
and Q’.
This implies no
0
change of state:
– Note the
latch
Q+SR
=Q
sub-circuit element
– The control input, Clk,
controls a sub-circuit
called a “gate”.
(Enable)
Clk
D
Q
0
Q’
0
• Since D is the only input, the forbidden values, S = R = 1, never
occur.
D Flip-Flop
• The D flip-flop overcomes the problem of the SR (S’R’) latch
When Clk = 1, it
havingfollows
forbidden
states. It features a single input, D, and a control
that:
input,SClk,
provided by the system clock, and produces outputs Q
= D and R = D’
and Q’.
Thus, the behaviour
D’
is:
– DNote
the1 SR latch
=S=
Q+ = 1
sub-circuit element
D = 0 (R = 1)
Q+ = 0
– The control input, Clk,
controls a sub-circuit
called a “gate”.
(Enable)
Clk
D
Q
1
Q’
D
• Since D is the only input, the forbidden values, S = R = 1, never
occur.
JK Flip-Flop
JK Flip-Flop
• The basic JK flip-flop provides a solution to the problem of the SR
latch with S=R=1 that produces forbidden output states.
JK Flip-Flop
• The basic JK flip-flop provides a solution to the problem of the SR
latch with S=R=1 that produces forbidden output states.
• The JK flip-flop can be constructed from the gated SR latch by
coupling additional feedback from the (Q, Q’) outputs into the J
and K inputs.
J
S
Q
Clk
Q
C
Q
K
R
Q’
JK Flip-Flop
• The basic JK flip-flop provides a solution to the problem of the SR
latch with S=R=1 that produces forbidden output states.
• The JK flip-flop can be constructed from the gated SR latch by
coupling additional feedback from the (Q, Q’) outputs into the J
and K inputs.
• With the clock disabled
(C = 0) the SR latch
retains the state (Q, Q’).
S
J
Clk
0
Q
Q
C
Q
K
R
Q’
JK Flip-Flop
• The basic JK flip-flop provides a solution to the problem of the SR
latch with S=R=1 that produces forbidden output states.
• The JK flip-flop can be constructed from the gated SR latch by
coupling additional feedback from the (Q, Q’) outputs into the J
and K inputs.
• With the clock enabled
(C = 1) the SR latch
produces outputs that
depend on the J and K
inputs.
S
J
Clk
1
Q
Q
C
Q
K
R
Q’
JK Flip-Flop TRACE
• The JK flip-flop can be constructed from the gated SR latch by
coupling additional feedback from the (Q, Q’) outputs into the J
and K inputs. We denote the final output as Q+.
J K Qin Qout
META-STABLE !
Actions:
J
J=K=0
Do nothing
Q
J=1, K=0 Set Q=1
K
J=0, K=1 Reset Q=0
J=K=1
Complement Q = Q’
Q’
0 0 0
0 0 1
0
1
1 0 0
1 0 1
1
1
0 1 0
0 1 1
0
0
1 1 0
1 1 1
1
0
This leads to the algebraic expression for the final output,
labeled Q+, in terms of J, K and initial input Q:
Q+ = JQ’ + K’Q
JK Flip-Flops – edge triggering
• The previous implementation of a JK flip-flop is considered
unstable under certain circumstances. Utilizing an edge-triggered
master-slave latch is used to produce a stable circuit.
– Below is given a more typical JK using a master-slave approach
Q
Q
– WARNING: Tracing the logic may prove confusing as the actual
circuits employ both leading-edge and trailing-edge gate elements in
order to avoid forbidden states.
JK, D and T Flip-Flops
• JK flip-flops can be used to produce
– D flip-flops
• Connect K to J using an inverter so they have different values
– T flip-flops
• Sometimes called a complementer
• Connect J and K so they have the same values
– If J = K = 0, nothing happens (Q stays the same)
– If J = K = 1, the complement of Q is outputted
• This illustrates, once again, the principle that common components
can be used to achieve design goals in different ways.
Timing Considerations
Timing Considerations
• Real (physical) circuit elements take a finite time to respond to the
stimulus of changing state (voltage).
• The behaviour of a logic device is characterized by the following
times:
Timing Considerations
• Real (physical) circuit elements take a finite time to respond to the
stimulus of changing state (voltage).
• The behaviour of a logic device is characterized by the following
times:
– Propagation Delay
S
The time it takes to
produce a change in an
output signal based on
the input signals.
R
Tp,LH
Q
Q’
Tp,HL
Timing Considerations
As the value of S begins to
change, it is only when it has
• Real (physical) circuit elements
takeaacertain
finite voltage
time tolevel
respond to the
reached
stimulus of changing state (voltage).
that the value of Q (Q’) begins
to change.
S must be maintained at a
• The behaviour of a logic device
is characterized
by the following
certain
level for a minimum
time period before Q can
times:
stabilize.
– Propagation Delay
S
The time it takes to
produce a change in an
output signal based on
the input signals.
R
Tp,LH
Q
Q’
Tp,HL
Timing Considerations
• Real (physical) circuit elements take a finite time to respond to the
stimulus of changing state (voltage).
• The behaviour of a logic device is characterized by the following
times:
– Propagation Delay
S
– Minimum Pulse Width
R
The minimum amount of
time an input signal must
be applied in order to
produce a change in the
output.
Q
Timing Considerations
• Real (physical) circuit elements take a finite time to respond to the
stimulus of changing state (voltage).
• The behaviour of a logic device is characterized by the following
times:
– Propagation Delay
– Minimum Pulse Width
– Setup Time - the minimum time the input signals must be held fixed
before the latching action begins
Timing Considerations
• Real (physical) circuit elements take a finite time to respond to the
stimulus of changing state (voltage).
• The behaviour of a logic device is characterized by the following
times:
– Propagation Delay
– Minimum Pulse Width
– Setup Time - the minimum time the input signals must be held fixed
before the latching action begins
– Hold Time - the minimum time the input signals must be held fixed
until the latching action is completed
State Tables and Diagrams
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State tables are a form of truth tables where current values of flip-flop
outputs are used as inputs, along with other specified inputs, to determine
outputs after a clock pulse.
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State diagrams are graphical representations of all possible transitions that
are described by a state table.
• Example: JK flip-flop
Present
State
Inputs
Next
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State diagrams are graphical representations of all possible transitions that
are described by a state table.
• Example: JK flip-flop
0
Draw possible Q output states
in circles (or ellipses)
1
Present
State
Inputs
Next
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State diagrams are graphical representations of all possible transitions that
are described by a state table.
• Example: JK flip-flop
00,01
0
1
Present
State
Inputs
Next
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State diagrams are graphical representations of all possible transitions that
are described by a state table.
• Example: JK flip-flop
00,01
0
10,11
1
Present
State
Inputs
Next
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
State Tables and Diagrams
• Complex circuits are difficult to represent simply in a compact notation.
– State diagrams are graphical representations of all possible transitions that
are described by a state table.
• Example: JK flip-flop
00,01
0
10,11
01,11
1
00,10
Present
State
Inputs
Next
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
State Tables and Diagrams
•
Draw transitions between output states. Allow
Complex circuits are difficult
to change
represent
inwell
a compact
notation.
for no
of simply
value, as
as changes
in
value.representations
Label each transition
by thetransitions
(list) of allthat
JK
– State diagrams are graphical
of all possible
inputs that effect the transition.
are described by a state table.
• Example: JK flip-flop
00,01
0
10,11
01,11
1
00,10
If the inputs are themselves changed in
transition,
list the initial
and final input Next
values
Present
Inputs
Stateseparated by a slash ‘/’.
state
Q(t)
J
K
Q(t+1)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
Characteristic Equations
Characteristic Equations
• Before proceeding, we stop briefly to recapitulate the various basic
flip-flop circuits derived so far.
• Each circuit has an associated set of expressions that describe the
outputs in terms of the inputs and the internal state at the time the
circuit is enabled. These expressions are called the characteristic
equations.
Characteristic Equations
SR flip-flop
S
Q
C
R
Q’
Q+ = S + R’Q : (SR) = 0
Q+ refers to the output value
after the next clock interval,
or Q(t+1).
Characteristic Equations
SR flip-flop
S
Q
C
R
D flip-flop
D
Q
C
Q’
Q+ = S + R’Q : (SR) = 0
Q’
Q+ = D
Characteristic Equations
SR flip-flop
S
Q
C
R
Q’
JK flip-flop
Q
C
K
D
Q
C
Q+ = S + R’Q : (SR) = 0
J
D flip-flop
Q’
Q+ = JQ’ + K’Q
Q’
Q+ = D
Characteristic Equations
SR flip-flop
S
Q
D flip-flop
C
R
D
Q
C
Q’
Q’
Q+ = S + R’Q : (SR) = 0
JK flip-flop
Q+ = D
T flip-flop
J
Q
C
K
T
Q
C
Q’
Q+ = JQ’ + K’Q
Q’
Q+ = TQ’ + T’Q
= T xor Q
Gated Latches
Gated Latches
• The concept of a gate, or a controlling element, is important in
computer circuits.
• During the execution of a program only specific circuit elements
should be active at a given time. These are often controlled using a
strobe signal that provides a regular sequence of alternating
voltage-HI (1) and voltage-LO (0) signals.
• Because of the regular nature of the signal sequence the strobe is
called a “clock”.
• Thus, gate control is often achieved using a clock. Another type of
control signal is called an “enable” signal.
Gated SR Latch
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
S
Q
(Enable) Clk
Q’
R
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 0
S
0
Q
(Enable) Clk
Q’
R
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 0
S
0
(Enable) Clk
R
1
Q’
Q
1
Q
Q’
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 0
S
0
(Enable) Clk
R
1
Q
Q
Q’
Q
1
Q’
Q’
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 0
The outputs (Q, Q’)
remain unchanged when
the circuit is disabled.
The (Q,Q’) are stored in
a stable manner.
S
0
(Enable) Clk
R
1
Q
Q
Q’
Q
1
Q’
Q’
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 1
S
1
Q
(Enable) Clk
Q’
R
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 1
The first stage nand
gates are activated by the
Clk signal to produce the
outputs (S’, R’).
S
S’
Q
1
(Enable) Clk
R
R’
Q’
Gated SR Latch
• The SR latch is modified to a gated SR latch by applying a clock
signal to the latch input.
• This gives rise to the behaviour:
Clk = 1
The first stage nand
gates are activated by the
Clk signal to produce the
outputs (S’, R’).
S
S’
Q
1
(Enable) Clk
R
The remaining sub-circuit
is just an S’R’ latch whose
properties were discussed previously.
R’
Q’
Gated SR Latch
Theiseffect
of the
input
• The SR latch
modified
to aClk
gated
SR latch by applying a clock
is to control
signal to the latch
input. the latch
circuit. Changes to (Q,Q’)
may only occur when Clk=1
• This gives riseenables
to the behaviour:
the circuit.
Clk = 1
The first stage nand
gates are activated by the
Clk signal to produce the
outputs (S’, R’).
S
S’
Q
1
(Enable) Clk
R
The remaining sub-circuit
is just an S’R’ latch whose
properties were discussed previously.
R’
Q’
Gated D Latch
Gated D Latch
• We previously considered this circuit. See earlier notes.
(Enable)
Clk
D
Q
Q’
Gated D Latch
• We previously considered this circuit. See earlier notes.
• The MSI representation
may be given in two
forms:
(Enable)
Clk
(A)
D
C
Q
D
Q’
(B)
D
C
Q
Q
Q
Q’
Master-Slave Flip-Flops
Master-Slave Flip-Flops
• We have just considered the category of flip-flops called latches.
– Changes on the information input lines produce immediate responses
on the output lines.
– This is called transparency.
• Now we consider the category of Master-Slave (pulse-triggered)
flip-flop circuits.
– These circuits feature a control signal that enables one stage of a
circuit while disabling a second stage, then the second stage is
enabled while the first stage is disabled.
– This is called cascading of circuits.
Master-Slave SR Flip-Flop
Master-Slave SR Flip-Flop
• The SR flip-flop is adapted to the Master-Slave flip-flop by
cascading two SR circuits with sequential enabling of each circuit
by a Clk pulse.
Master-Slave SR Flip-Flop
• The SR flip-flop is adapted to the Master-Slave flip-flop by
cascading two SR circuits with sequential enabling of each circuit
by a Clk pulse.
S
S
C
C
R
R
Q
Q
QM
QM’
S
Q
C
R
Q
QS
Q
QS’
Q’
Master-Slave SR Flip-Flop
• The SR flip-flop is adapted to the Master-Slave flip-flop by
cascading two SR circuits with sequential enabling of each circuit
by a Clk pulse.
– When the clock is set, C = 1, the first stage gated SR latch is enabled,
but the second stage is disabled.
S
C
R
S
1
Q
C
R
Q
QM
QM’
S
0
Q
C
R
Q
QS
Q
QS’
Q’
Master-Slave SR Flip-Flop
• The SR flip-flop is adapted to the Master-Slave flip-flop by
cascading two SR circuits with sequential enabling of each circuit
by a Clk pulse.
– When the clock is set, C = 1, the first stage gated SR latch is enabled,
but the second stage is disabled.
– When the clock signal returns to C = 0, the first stage is disabled and
the second stage is enabled.
S
C
R
S
0
Q
C
R
Q
QM
QM’
S
1
Q
C
R
Q
QS
Q
QS’
Q’
Registers
Registers
• A register is a collection of flip-flops taken as a single entity.
• Since flip-flops are memory units for single bits, then registers are
the equivalent, multi-bit storage units.
– Since registers are comprised of a finite number, N, of flip-flops, the
total number of 0 and 1 combinations is 2N.
– Each of these combinations is known as the content or state of the
register.
• In addition to storage alone, registers may also have other
capabilities associated with them.
– Clear, Load, Shift, Count
Registers
• A simple storage register based on the Master-Slave D flip-flop is
constructed by chaining n of them as shown. The entire memory
unit is controlled by the Clock (C) pulse.
D0
D
Q
Q0
C
Q’
D1
D
Q
Q0
Q0’
Q1
Q1’
D1
Q0 ’
.
.
.
Q1
Dn-1
C
C
D0
Q’
Q1 ’
C
Qn-1
Q’n-1
Registers
• In a similar fashion, the Master-Slave T flip-flop is constructed by
chaining n of them as shown, controlled by the clock (C) pulse.
T0
T
Q
Q0
C
Q’
T1
T
Q
Q0
Q0’
Q1
Q1’
T1
Q0 ’
.
.
.
Q1
Tn-1
C
C
T0
Q’
Q1 ’
C
Qn-1
Q’n-1
Registers
• Thus, a register is a special multi-bit storage unit that is used to
store data in a collective representation (eg. signed binary, BCD,
and so on).
Input Data
N-bit Register
I0
D0
I1
D1
In-1
Q0
Q0’
Q1
Q1’
.
.
.
Dn-1
Enable
Clock Enable
Q0
Q1
Stored Values
(Potential output)
Qn-1
Q’n-1
Qn-1
Registers
• We will discuss registers in more detail due to their importance in
CPU design and in other places in a computer
• CPU registers used in the textbook (Mano):
–
–
–
–
–
–
–
–
PC :: Program counter
IR
:: Instruction register
AR :: Address register
DR :: Data register
AC :: Accumulator
INR :: Input buffer register
OUTR :: Output buffer register
SCR :: Sequence counter register (or just SC)
Summary
• We considered details and MSI views of:
– Latches:
SR , S’R’ , D
– Gated Latches: SR , D
– Master-Slave: SR , JK, D, T
• We also discussed the issue of timing and response as important
behaviours that characterize and typify logic devices.
– Including propagation delay, minimum pulse width, set-up and hold
times.
• We concluded by considering registers as conceptual extensions
of the basic flip-flops.