Category: Education
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9.8 Using Multiple Combinational Circuits
As an example of using several circuits together, we are going to make a device that will have 16 inputs, representing a four digit number, to a four digit 7-segment display but using just one binary-to-7-segment encoder. First, the overall architecture of our circuit provides what looks like our the description provided. Follow this circuit…
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9.7 Multiplexers
A multiplexer, abbreviated mux, is a device that has multiple inputs and one output. The schematic symbol for multiplexers is The truth table for a 2-to-1 multiplexer is Using a 1-to-2 decoder as part of the circuit, we can express this circuit easily. Multiplexers can also be expanded with the same naming conventions as demultiplexers.…
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9.6 Demultiplexers
A demultiplexer, sometimes abbreviated dmux, is a circuit that has one input and more than one output. It is used when a circuit wishes to send a signal to one of many devices. This description sounds similar to the description given for a decoder, but a decoder is used to select among many devices while…
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9.5 Encoder
What is an Encoder? An encoder is a circuit that changes a set of signals into a code. Let’s begin making a 2-to-1 line encoder truth table by reversing the 1-to-2 decoder truth table. This truth table is a little short. A complete truth table would be One question we need to answer is what…
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9.4 Decoder
A decoder is a circuit that changes a code into a set of signals. It is called a decoder because it does the reverse of encoding, but we will begin our study of encoders and decoders with decoders because they are simpler to design. Types of Decoders Line Decoder A common type of decoder is…
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9.3 Full-Adder
The half-adder is extremely useful until you want to add more that one binary digit quantities. The slow way to develop a two binary digit adders would be to make a truth table and reduce it. Then when you decide to make a three binary digit adder, do it again. Then when you decide to…
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9.2 Half-Adder
As a first example of useful combinational logic, let’s build a device that can add two binary digits together. We can quickly calculate what the answers should be: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 102 So we well need two inputs (a…
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9.1 Introduction to Combinational Logic Functions
The term “combinational” comes to us from mathematics. In mathematics a combination is an unordered set, which is a formal way to say that nobody cares which order the items came in. Most games work this way, if you rolled dice one at a time and get a 2 followed by a 3 it is…
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8.11 Larger 5 & 6-variable Karnaugh Maps
Larger Karnaugh maps reduce larger logic designs. How large is large enough? That depends on the number of inputs, fan-ins, to the logic circuit under consideration. One of the large programmable logic companies has an answer. Altera’s own data, extracted from its library of customer designs, supports the value of heterogeneity. By examining logic cones,…
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8.10 Don’t Care Cells in the Karnaugh Map
Up to this point we have considered logic reduction problems where the input conditions were completely specified. That is, a 3-variable truth table or Karnaugh map had 2n = 23 or 8-entries, a full table or map. It is not always necessary to fill in the complete truth table for some real-world problems. We may…
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8.9 Sum and Product Notation
For reference, this section introduces the terminology used in some texts to describe the minterms and maxterms assigned to a Karnaugh map. Otherwise, there is no new material here. Σ (sigma) indicates sum and lower case “m” indicates minterms. Σm indicates sum of minterms. The following example is revisited to illustrate our point. Instead of…
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8.8 Minterm vs Maxterm Solution
So far we have been finding Sum-Of-Product (SOP) solutions to logic reduction problems. For each of these SOP solutions, there is also a Product-Of-Sums solution (POS), which could be more useful, depending on the application. Before working a Product-Of-Sums solution, we need to introduce some new terminology. The procedure below for mapping product terms is…
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8.7 Larger 4-variable Karnaugh Maps
Knowing how to generate Gray code should allow us to build larger maps. Actually, all we need to do is look at the left to right sequence across the top of the 3-variable map, and copy it down the left side of the 4-variable map. See below. Reductions of 4 Variable K Maps The following…
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8.6 Logic Simplification With Karnaugh Maps
The logic simplification examples that we have done so far could have been performed with Boolean algebra about as quickly. Real world logic simplification problems call for larger Karnaugh maps so that we may do serious work. We will work some contrived examples in this section, leaving most of the real world applications for the…
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8.5 Karnaugh Maps, Truth Tables, and Boolean Expressions
Who Developed the Karnaugh Map? Maurice Karnaugh, a telecommunications engineer, developed the Karnaugh map at Bell Labs in 1953 while designing digital logic based telephone switching circuits. The Use of Karnaugh Map Now that we have developed the Karnaugh map with the aid of Venn diagrams, let’s put it to use. Karnaugh maps reduce logic…