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ELE2120 Digital Circuits and Systems
Tutorial Note 5
Outline
1. Complex PLD
2. Field - Programmable Gate Array (FPGA)
3. Lookup Table (LUT)
4. Multiplexer
5. Shannon’s Expansion
Complex PLD(CPLD)
•Combined by PAL-like blocks
•With interconnection wires to connect all the PAL-like blocks
Complex PLD(CPLD)
•XOR with 1 programmable input:
If input = 1, complement the OR gate output.
If input = 0, retain the OR gate output.
•D-type flip flop: Store value of the previous state
•Multiplexer: Select between value of current state or previous state
•Tri-state buffer: Control the pin to be input or output
Close: output. Open: input.
Field Programmable Gate Array(FPGA)
•
FPGAs are a distinct from SPLDs and CPLDs and typically offer the highest logic
capacity. An FPGA consists of an array of logic blocks, surrounded by
programmable I/O blocks, and connected with programmable interconnect. A
typical FPGA contains from 64 to tens of thousands of logic blocks and an even
greater number of flip-flops. Most FPGAs do not provide 100% interconnect
between logic blocks (to do so would be prohibitively expensive). Instead,
sophisticated software places and routes the logic on the device much like a PCB
auto-router would place and route components.
Field – Programmable Gate Array (FPGA)
CPLD
FPGA
Structure
AND and OR planes
Constructed by many
logic blocks
Compatible
Combinational circuits Sequential circuits
Capability
10,000 equ. Gates
>1000,000 equ. Gates
Power consumption
Large
Reasonable small
Lookup Table (LUT)
Steps for generate Lookup Table:
1.Write down the truth table
2. Transfer the output column to lookup table.
For the function (X1+X2)’
2 input multiplexer
Exercise: LUT for FPGA
Q1
For the logic function F1=X1X3+X2X3+X1X2,
X1 X2 X3 F
1.Write down the truth table for F1.
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
2. Draw the LUT for F1.
3. What would be the result if Select = 0?
Q2
Q3
Exercise: FPGA
Q1
For the logic function F1=X1X3+X2X3+X1X2,
X1 X2 X3 F
1.Write down the truth table for F1.
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
2. Draw the LUT for F1.
3. What would be the result if Select = 0?
Q2
0
0
0
1
0
1
1
1
Step 2
fill the lookup table
Q3
Step 1
fill the truth table
Multiplexer
Multiplexer circuit has a number of data inputs, one or more select inputs,
and one output.
Multiplexer
Transmission Gate
Symbol
Transmission Gate
Equivalent circuit
Truth Table
Why we use both PMOS and NMOS transistors to build transmission gate?
Answer:
1.NMOS transistors passes 0 well and 1 poorly while PMOS transistor passes 1 well and 0 poorly. It is possible to
combine an NMOS and a PMOS transistor into a single switch that is capable of driving its output terminal either to
a low or high voltage equally well.
2.Small impedance(~100ohm) between x and f when closed. However, high impedance (~10 9ohm) when opened.
3. Very short time delay between x and f.
Exercise(1): Multiplexer
What is the truth table for Load, Data, A, B and output?
Load
Data
Output
Exercise(1): Multiplexer
Analysis:
(1) Load = 0: TG1 closed,TG2 opened; A=Data, B=A’, Output=Data; State Transmission
(2) Load = 1:TG1 opened, TG2 closed; A=Output, B=A’, Output=A ; State Retained
Exercise(1): Multiplexer
Data
Q
Load
_
Q
D-Latch
Load
Data
Output
Multiplexer
4-to-1 multiplexer
3 2-to-1 multiplexer
Larger multiplexer can be built based on small multiplexers such as 4-to-1
or 2-to-1 multiplexer, e.g., a 4-to-1 multiplexer can be built with 3 2-to1multiplexers,a 8-to-1 multiplexer can be composed of 2 4-to-1 and a 2-to1 multiplexers, and a 16-to-1 multiplexer can be set up using 5 4-to-1
multiplexers.
Exercise(2): Multiplexer
Implement the circuit for the truth table shown below by using a 4-to-1
multiplexer:
W1 W2 W3 F
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
1
Shannon’s Expansion
Multiplexer implementations of logic functions require that a given function be
decomposed in terms of the variables that are used as the select inputs. This can
be accomplished by means of a theorem proposed by Claude Shannon’s
Expansion Theorem
Any Boolean function
can be written in the form
This expansion can be done in terms of any of the n variables.
Example: f(w1,w2,w3)=w1w2+w1w3+w2w3
Expanding this function in terms of w1 gives which is the expression that we derived above
Method I
f(w1,w2,w3)=w1(w2+w3)+(w1+w1’)w2w3= w1(w2+w3+ w2w3)+w1’w2w3=w1(w2+w3)+w1’w2w3
Method II
f(0,w2,w3)=w2w3
f(1,w2,w3)=w2+w3+w2w3 =w2+w3
f(w1,w2,w3)=w1’ f(0,w2,w3)+w1 f(1,w2,w3)= w1’w2w3+ w1(w2+w3)
Shannon’s Expansion
Steps for solving Shannon’s Expansion
Method I
1. Expand(or split) the function into two part (with respect to w1 and w1’).
2. Combine the terms into two groups of w1 and w1’.
Method II
1. For f(w1,w2,….wn), let w1=0, the remain minimized function is the factor for w1’
2. For f(w1,w2,….wn), let w1=1, the remain minimized function is the factor for w1
Exercise: Shannon’s Expansion
Use the Shannon’s Expansion to expand the following function:
w1+w2’w3+w2w3’ with respect of w1, w2: