EE 4271 VLSI Design, Fall 2013
Combinational Circuits
Overview
• Combinational Circuit
• Chip Design styles
– Full-custom design
– Cell library based design
– Programmable Logic Array
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Combinational Circuits
• A combinational circuit consists of logic gates
whose outputs, at any time, are determined
by combining the values of the inputs.
• For n input variables, there are 2n possible
binary input combinations.
• For each binary combination of the input
variables, there is one possible output.
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Combinational Circuits (cont.)
•
Hence, a combinational circuit can be described by:
1. A truth table that lists the output values for each
combination of the input variables, or
2. m Boolean functions, one for each output variable.
n-inputs
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••
•
Combinational
Circuit
Combinational Logic
••
•
m-outputs
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Combinational vs. Sequential Circuits


Combinational circuits are memory-less.
Thus, the output value depends ONLY on the
current input values.
Sequential circuits consist of combinational
logic as well as memory elements (used to
store certain circuit states). Outputs
depend on BOTH current input values and
previous input values (kept in the storage
elements).
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Combinational vs. Sequential Circuits
n-inputs
Combinational
Circuit
m-outputs
(Depend only on inputs)
Combinational Circuit
n-inputs
Combinational
Circuit
m-outputs
Next
state
Storage
Elements
Present
state
Sequential Circuit
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Important Design Concepts
• Modern digital design deals with various methods
and tools that are used to design and verify complex
circuits and systems.
• Concepts:
– Design Hierarchy
– Computer-Aided-Design (CAD) tools
– Hardware Description Languages (HDLs)
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Design Hierarchy
• “Divide-and-Conquer” approach used to cope
with the challenges of designing complex
circuits and systems (many times in the order of
millions of gates).
• Circuit is broken into blocks, repetitively.
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Design Hierarchy
Example: 9-input odd function (for counting # of 1 in inputs)
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Why is Hierarchy useful?
• Reduces the complexity required to design
and represent the overall schematic of the
circuit.
• Reuse of blocks is possible. Identical blocks
can be used in various places in a design, or in
different designs.
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Reusable Functions and CAD
• Whenever possible, we try to decompose a complex design into
common, reusable function blocks
• These blocks are
– verified and well-documented
– placed in libraries for future use
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Integrated Circuits
• Integrated circuit (a chip) is a semiconductor crystal (most
often silicon) containing the electronic components for the
digital gates and storage elements which are interconnected on
the chip.
• Terminology - Levels of chip integration
–
–
–
–
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SSI (small-scale integrated) - fewer than 10 gates
MSI (medium-scale integrated) - 10 to 100 gates
LSI (large-scale integrated) - 100 to thousands of gates
VLSI (very large-scale integrated) - thousands to 100s of millions of
gates
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Technology Parameters
• Specific gate implementation technologies are characterized by the
following parameters:
– Fan-in – the number of inputs available on a gate
– Fan-out – the number of standard loads driven by a gate output
– Cost for a gate - a measure of the contribution by the gate to the cost of the
integrated circuit
– Propagation Delay – The time required for a change in the value of a signal to
propagate from an input to an output
– Power Dissipation – the amount of power drawn from the power supply and
consumed by the gate
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Propagation Delay
• Propagation delay is the time for a change on an input of a gate to
propagate to the output.
• Delay is usually measured at the 50% point with respect to the H and L
output voltage levels.
• High-to-low falling and low-to-high rising delays.
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OUT (volts)
IN (volts)
Propagation Delay Example
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1.0 ns per division
Combinational Logic
t (ns)
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Chip Design Styles
•
Full custom - the entire design of the chip down to the smallest detail of the layout
is performed
– Expensive, its timing and power is hard to analyze
– only for dense, fast chips with high sales volume
•
Standard cell - blocks have been design ahead of time or as part of previous
designs
– Intermediate cost
– Less density and speed compared to full custom
•
Gate array - regular patterns of gate transistors that can be used in many designs
built into chip - only the interconnections between gates are specific to a design
–
–
–
–
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Lowest cost
Less density compared to full custom and standard cell
Prototype design
The base of FPGA
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Cell Libraries
• Cell - a pre-designed primitive block
• Cell library - a collection of cells available for design using
a particular implementation technology
• Cell characterization - a detailed specification of a cell for
use by a designer
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Cell Library Based Design Procedure
1.
Specification
–
2.
Write a specification for the circuit if one is not already available
Formulation
–
3.
Derive a truth table or initial Boolean equations that define the
required relationships between the inputs and outputs, if not in the
specification
Optimization
–
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Draw a logic diagram or provide a netlist for the resulting circuit
using ANDs, ORs, and inverters
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Cell Library Based Design Procedure
4. Technology Mapping
– Map the logic diagram to the implementation
technology selected
– Map to CMOS
5. Evaluation
– Evaluate the timing and power
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Design Example
1. Specification
– BCD to Excess-3 code converter
– Transforms BCD code for the decimal digits to Excess-3
code for the decimal digits
– BCD code words for digits 0 through 9: 4-bit patterns 0000
to 1001, respectively
– Excess-3 code words for digits 0 through 9: 4-bit patterns
consisting of 3 (binary 0011) added to each BCD code word
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Design Example (continued)
2.
Formulation
–
–
–
–
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Conversion of 4-bit codes can be most easily formulated by a
truth table
Variables
- BCD:
Input BCD
Output Excess-3
A,B,C,D
ABCD
WXYZ
Variables
0000
0011
- Excess-3
W,X,Y,Z
0001
0100
0010
0101
Don’t Cares
- BCD 1010
0011
0110
to 1111
0100
0111
0101
0110
0111
1000
1001
Combinational Logic
1000
1001
1010
1011
1011
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Design Example (continued)
3. Optimization
z
1
1
3
4
5
7
X
X
12
13
8
9
1
B
1
4
5
A
X
CD
X
13
8
9
1
X
10
C
1
2
0
4
5
7
6
4
1
1
1
X
13
1
B
14
11
w
3
8
X
X
1
A
6
15
1
0
12
7
X
12
10
C
X
2
D
x
1
3
1
X
14
11
0
D
X
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X
X
1
1
6
15
1
C
2
1
X
A
y
1
0
1
a. K-maps
W = A + BC + BD
X = BC + DB+ B
Y = CD + CD
Z= D
C
X
15
X
9
11
Combinational Logic
D
B
X
14
10
1
X
1
1
8
7
X
13
6
X
15
X
9
2
1
5
12
A
X
3
11
14
X
10
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B
Design Example (continued)
3. Optimization (continued)
Multiple-level using transformations
W = A + BC + BD
X = BC + DB+ B
Y = CD + C D
Z=
–
CD
D
Perform extraction, finding factor:
T1 = C + D
W = A + BT1
X = B T1 + B C D
Y = CD + CD
Z=
D
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Design Example (continued)
4. Technology Mapping
Map with a library containing inverters and 2-input NAND, and
then map it to a CMOS based circuit
A
W
B
X
C
D
Y
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Z
Combinational Logic
Z
Technology Mapping Example
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Timing Analysis
• Use static timing analysis which has been
covered.
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Transmission Gate Based Design Multiplexer
• “Selects” binary information from one of many
input lines and directs it to a single output line.
• Also know as the “selector” circuit,
• Selection is controlled by a particular set of inputs
lines whose # depends on the # of the data input
lines.
• For a 2n-to-1 multiplexer, there are 2n data input lines
and n selection lines whose bit combination
determines which input is selected.
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Multiplexer (cont.)
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2-to-1-Line Multiplexer
• Since 2 = 21, n = 1
• The single selection variable S has two values:
– S = 0 selects input I0
– S = 1 selects input I1
• The equation:
Y = S’ I0 + SI1
• The circuit:
Enabling
Circuits
Decoder
I0
Y
S
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I1
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Example: 4-to-1 MUX - Cell Library Based Design
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4–to–1-Line Multiplexer
using Transmission Gates
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MUX as a Universal Gate
• We can construct AND and NOT gates using 2-to-1
MUXs. Thus, 2-to-1 MUX is a universal gate.
z = 0x + 1x’ = x’
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z = x1x0 + 0x0’ = x1x0
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Programmable Logic Array
• The set of functions to be implemented is first transformed to product
terms
• Since output inversion is available, terms can implement either a function
or its complement
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Programmable Logic Array Example
• To implement
– F1= A’B’C+A’BC’+AB’C’=(AB+AC+BC+A’B’C’)’
– F2=AB+AC+BC
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Programmable Logic Array Example
A
B
C
X
X
X
1
X
X
X
2
X
X
X Fuse intact
Fuse blown
X
X
X
3
X
X
4
C C B B A A
X
X
X
X
0
X
1
F1
F2
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Summary
• Three design styles
– Full custom design
– Gate library based design
– PLA based design
• Transmission gate based design
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