SLIDE 1 Slides for Lecture 6
ENEL 353: Digital Circuits — Fall 2013 Term Steve Norman, PhD, PEng
Electrical & Computer Engineering Schulich School of Engineering University of Calgary
20 September, 2013
SLIDE 2 ENEL 353 F13 Section 02 Slides for Lecture 6
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Previous Lecture
◮ ranges for two’s-complement systems ◮ overflow in two’s-complement addition ◮ BCD (binary coded decimal) codes for decimal digits ◮ Gray codes
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Today’s Lecture
A little more about Gray codes. Introduction to logic gates. Related reading in Harris & Harris: Section 1.5
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Gray code to unsigned binary code conversion, unsigned binary code to Gray code conversion
... ... n-bit Gray code n-bit binary code extra 0 not part
The Gray code bit is 0 if the adjacent binary code bits match each other. Otherwise the Gray code bit is 1. Let’s convert 4-bit binary 0110 to Gray code, and 6-bit Gray code 101110 to binary.
SLIDE 5 A 4-bit shaft encoder: Optical sensors make a digital measurement
- f the rotation angle of a spinning metal disk. This could be used
for a digital weathervane, indicating one of 16 wind directions N, NNE, NE, ENE, . . . , NNW. This version uses unsigned binary encoding for positions 0 to 15. Why is this encoding a poor design choice?
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Practical shaft encoders use Gray codes
If the light and dark markings on the spinning disk are
- rganized using a Gray code, that eliminates the problem of
having two or more bits change at almost but not exactly the same time. Very precise shaft encoders can measure very small changes in angles in industrial equipment. For example, a 12-bit shaft encoder measures rotation angle to within 360◦/212 = 0.088◦.
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Combinational versus Sequential Logic
It’s IMPORTANT to know what these words mean. The outputs of a combinational logic circuit depend only the current values of its inputs. The outputs of a sequential logic circuit depend on the history of its input values.
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Example combinational circuit: 1-bit full adder
Cin Cout A B S full adder The values of Cout and S depend on what the values of A, B, and Cin are RIGHT NOW, not what they were a few minutes ago, or even a few nanoseconds ago.
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What does the term logic gate mean?
A logic gate is a combinational logic circuit that has
◮ one or more inputs, each of which is a bit; ◮ exactly one output, which is a bit.
According to this definition, is the full adder a logic gate?
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Graphical and algebraic notations for some common 1- and 2-input logic gates
A A B A B B B A Y = A + B Y = A ⊕ B Y = A A B A A A B Y = AB Y = A Y = A ⊕ B Y = A + B Y = AB
We’ll now examine these gates one or two at a time . . .
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1-input gates
A NOT gate, also called an inverter:
Y = A A
In A is the input and Y is the output. The “bubble” in the symbol indicates inversion: 0 converted to 1, and 1 converted to 0. The equation Y = A is read, “Y equals NOT A.” A buffer:
A Y = A
The output Y is just a copy of the input A. Logically this is no different from a wire, but electrically, a buffer can help with circuit timing, making sure a device gets enough input current, and so on.
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An alternate notation for the NOT operation
The “overline” is probably the most common notation for the NOT operation, but there is another symbol in wide use. Many textbooks and other documents use the ′ symbol (“prime”) instead. This includes the textbook used last year in ENEL 353. This year we will use an “overline” for the NOT operation, to be consistent with this year’s textbook. If you see A′ in course material from past years, translate that in your head to A.
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Truth tables
A truth table describes a combinational logic element by making a list of the values of the output for all possible combinations of input bits. (We’ve already seen the truth table for the 1-bit full adder.) Let’s write out the very simple truth tables for a NOT gate and a buffer.
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When writing in English about logic gates, use ALL CAPS for the names of gates
Confusing: “For this design you may use or or and gates but not not gates.” Better: “For this design you may use OR or AND gates but not NOT gates.”
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The AND gate
Here is the symbol and the algebraic notation:
A B Y = AB
One way to remember the symbol is to remember that the word AND contains a D, and the symbol looks like a D. An alternate algebraic notation for AND uses the · symbol, as in Y = A · B. Here is how the AND operation is defined: AB = 1 if both A and B are 1
Let’s write the truth table for AND.
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The OR gate
Symbol and algebraic notation . . .
Y = A + B A B
Here is how the OR operation is defined: A + B = if both A and B are 0 1
In other words, in digital logic, OR means “one or the other or both are true.” Let’s write the truth table for OR.
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In discussion of digital logic, + does not mean arithmetic addition
◮ In decimal arithmetic, what is 1 + 1? ◮ In binary arithmetic, what is 1 + 1? ◮ In digital logic, what is 1 + 1?
This can be confusing, especially when discussing digital logic circuits that are designed to do arithmetic!
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The XOR gate
XOR is short for exclusive OR.
A B Y = A ⊕ B
“Exclusive or” means “one or the other but not both.” A ⊕ B = 1 if one of A and B is 1 and the other is 0
Let’s write the truth table for XOR.
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Convention for ordering of truth table input combinations
◮ Start with the row for all inputs = 0. ◮ Alternate 0’s and 1’s most slowly on the left. ◮ Alternate 0’s and 1’s more quickly as you move left to
right.
◮ Alternate 0’s and 1’s most quickly—every row—on the
right.
◮ In other words, write the input bit patterns in unsigned
binary ordering.
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Convention for ordering of truth table input combinations: Examples
This follows the convention . . . A B C Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 These two tables convey the same information, but should be AVOIDED! A B C Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A B C Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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Next Lecture
More 2-input gates: NAND, NOR and XNOR. Examples of gates with 3 or more inputs. Correspondences between bit values and voltage levels in digital circuits. Related reading in Harris & Harris: Sections 1.5 and 1.6.
