1. Electronics for Physicists
Lecture 13
Combinatorial Logic

2. Electronics for physicists
Combinatorial logic
Combinatorial logic is the combination of many elementary logic gates to more general and complex logical modules.
The output state (= result) depends only on the current input states. It does not depend on the history of the input states or on previous output states.
The number of input and output states may vary wildly.
input states
output states
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Electronics for physicists

3. Electronics for physicists
Multiplexer (Mux)
Multiplexer with 3 inputs (E1, E2, E3), control line S, and output A.
Mux truth table
Multiplexer with 2 AND-gates, 1 inverter, and 1 OR-gate
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4. Multiplexing data streams / serializer
circuit with 2 inputs (E1, E2), clock and output (A)
If you need to transmit the full information of all inputs (here E1 and E2), you must increase the output clock and data rate.
Input frequency (40 MHz) 1 2 3 4 5 6 7 8 9 10 E1 E2
Output frequency (80 MHz) 12 34 56 78 90 A 01 00 11
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5. Electronics for physicists
Serializer
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6. Multiplexer as a “memory”
4 input lines set to “0” or “1” (here this values are kept constant)
2 address select lines a, b (was used as control line before!)
1 output line A
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7. Electronics for physicists
Decoder
truth table for E = 1
2 address lines a, b
4 output lines A0 to A3
1 enable line E. For E = “0”, all Ai are “0”.
A decoder sets the selected output line to “1”.
All other outputs are “0”.
Purpose: chip select, row select, etc.
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8. Decoder Think of “a, b” as an address which
2-to-4 decoder with enable input
Circuit diagram of a 2 – to – 4 decoder
Think of “a, b” as an address which
asserts a single corresponding
address line Ai.
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9. Electronics for physicists
Encoder
An encoder is the reverse of a decoder.
4-to-2 Encoder
The encoder output a, b corresponds to the input line set to “1”. Only one input line must be set to “1”.
(A parity encoder relaxes this condition and selects the address with the highest input at to “1”.)
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10. Comparator A comparator checks if two input numbers are equal.
4-bit-comparator
Truth table of 1-bit-comparator
Algebraic expressions for 1-bit comparator.
“a = b” = ab + ab
“a > b” = a b
“a < b” = a b
This digital comparator is not to be confused with the op-amp comparator.
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11. Electronics for physicists
Half adder
The half adder is used for addition of binary numbers.
s = sum c = carry
For addition of numbers with more than 1 bit, need full adder.
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12. Adding arbitrarily long binary numbers
Full adder
Full adder
Half adder
LSB requires one half adder.
For all other bits one additional full adder is needed.
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13. Electronics for physicists
Full adder
Full adder considers carry from previous stage.
Half adder
Half adder
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Electronics for physicists