1. Basic circuit analysis and design
I don’t care!
You don’t always need all 2n input combinations in an n-variable function.
If you can guarantee that certain input combinations never occur.
If some outputs aren’t used in the rest of the circuit.
We mark don’t-care outputs in truth tables and K-maps with Xs.
Within a K-map, each X can be considered as either 0 or 1. You should pick the interpretation that allows for the most simplification.
Basic circuit analysis and design
4/25/2019

2. Example: Seven Segment DisplayB C D e 1 2 3 4 5 6 7 8 9 X
Input: digit encoded as 4 bits: ABCD a f g c
Table for e b
Assumption: Input represents a legal digit (0-9) e d CD AB 00 01 11 10 1 X
CD’ + B’D’
Basic circuit analysis and design
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3. Example: Seven Segment DisplayB C D a 1 2 3 4 5 6 7 8 9 X a
Table for a f b g e c d CD AB 00 01 11 10 1 X
A + C + BD + B’D’
The expression in book (p 110) is different because it assumes “0” for “illegal” inputs: A’C+A’BD+B’C’D’+AB’C’
Basic circuit analysis and design
4/25/2019

4. Practice K-map 3 Find a MSP for
f(w,x,y,z) = m(0,2,4,5,8,14,15), d(w,x,y,z) = m(7,10,13)
This notation means that input combinations wxyz = 0111, 1010 and 1101 (corresponding to minterms m7, m10 and m13) are unused.
Basic circuit analysis and design
4/25/2019

5. Solutions for practice K-map 3
Find a MSP for:
f(w,x,y,z) = m(0,2,4,5,8,14,15), d(w,x,y,z) = m(7,10,13)
All prime implicants are circled. We can treat X’s as 1s if we want, so the red group includes two X’s, and the light blue group includes one X.
The only essential prime implicant is x’z’. The red group is not essential because the minterms in it also appear in other groups.
The MSP is x’z’ + wxy + w’xy’. It turns out the red group is redundant; we can cover all of the minterms in the map without it.
Basic circuit analysis and design
4/25/2019

6. Basic circuit analysis and design
K-map Summary
K-maps are an alternative to algebra for simplifying expressions.
The result is a minimal sum of products, which leads to a minimal two-level circuit.
It’s easy to handle don’t-care conditions.
K-maps are really only good for manual simplification of small expressions... but that’s good enough for CS231!
Things to keep in mind:
Remember the correct order of minterms on the K-map.
When grouping, you can wrap around all sides of the K-map, and your groups can overlap.
Make as few rectangles as possible, but make each of them as large as possible. This leads to fewer, but simpler, product terms.
There may be more than one valid solution.
Basic circuit analysis and design
4/25/2019

7. Basic circuit analysis and design
We have learned all the prerequisite material:
Truth tables and Boolean expressions describe functions.
Expressions can be converted into hardware circuits.
Boolean algebra and K-maps help simplify expressions and circuits.
Now, let us put all of these foundations to good use, to analyze and design some larger circuits.
Basic circuit analysis and design
4/25/2019

8. Basic circuit analysis and design
Circuit analysis involves figuring out what some circuit does.
Every circuit computes some function, which can be described with Boolean expressions or truth tables.
So, the goal is to find an expression or truth table for the circuit.
The first thing to do is figure out what the inputs and outputs of the overall circuit are.
This step is often overlooked!
The example circuit here has three inputs x, y, z and one output f.
Basic circuit analysis and design
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9. Write algebraic expressions...
Next, write expressions for the outputs of each individual gate, based on that gate’s inputs.
Start from the inputs and work towards the outputs.
It might help to do some algebraic simplification along the way.
Here is the example again.
We did a little simplification for the top AND gate.
You can see the circuit computes f(x,y,z) = xz + y’z + x’yz’
Basic circuit analysis and design
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10. Basic circuit analysis and design
...or make a truth table
It’s also possible to find a truth table directly from the circuit.
Once you know the number of inputs and outputs, list all the possible input combinations in your truth table.
A circuit with n inputs should have a truth table with 2n rows.
Our example has three inputs, so the truth table will have 23 = 8 rows. All the possible input combinations are shown.
Basic circuit analysis and design
4/25/2019

11. Simulating the circuit
Then you can simulate the circuit, either by hand or with a program like LogicWorks, to find the output for each possible combination of inputs.
For example, when xyz = 101, the gate outputs would be as shown below.
Use truth tables for AND, OR and NOT to find the gate outputs.
For the final output, we find that f(1,0,1) = 1. 1
Basic circuit analysis and design
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12. Finishing the truth table
Doing the same thing for all the other input combinations yields the complete truth table.
This is simple, but tedious.
Basic circuit analysis and design
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13. Expressions and truth tables
Remember that if you already have a Boolean expression, you can use that to easily make a truth table.
For example, since we already found that the circuit computes the function f(x,y,z) = xz + y’z + x’yz’, we can use that to fill in a table:
We show intermediate columns for the terms xz, y’z and x’yz’.
Then, f is obtained by just OR’ing the intermediate columns.
Basic circuit analysis and design
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14. Truth tables and expressions
The opposite is also true: it’s easy to come up with an expression if you already have a truth table.
We saw that you can quickly convert a truth table into a sum of minterms expression. The minterms correspond to the truth table rows where the output is 1.
You can then simplify this sum of minterms if desired—using a K-map, for example.
f(x,y,z) = x’y’z + x’yz’ + xy’z + xyz
= m1 + m2 + m5 + m7
Basic circuit analysis and design
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15. Circuit analysis summary
After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does.
You can easily convert between expressions and truth tables.
Find the circuit’s
inputs and outputs
Find a Boolean
expression
for the circuit
Find a truth table
for the circuit
Basic circuit analysis and design
4/25/2019

16. Basic circuit analysis and design
Basic circuit design
The goal of circuit design is to build hardware that computes some given function.
The basic idea is to write the function as a Boolean expression, and then convert that to a circuit.
Step 1:
Figure out how many inputs and outputs you have.
Step 2:
Make sure you have a description of the function, either as a
truth table or a Boolean expression.
Step 3:
Convert this into a simplified Boolean expression. (For this course,
we’ll expect you to find MSPs, unless otherwise stated.)
Step 4:
Build the circuit based on your simplified expression.
Basic circuit analysis and design
4/25/2019

17. Design example: Comparing 2-bit numbers
Let’s design a circuit that compares two 2-bit numbers, A and B. The circuit should have three outputs:
G (“Greater”) should be 1 only when A > B.
E (“Equal”) should be 1 only when A = B.
L (“Lesser”) should be 1 only when A < B.
Make sure you understand the problem.
Inputs A and B will be 00, 01, 10, or 11 (0, 1, 2 or 3 in decimal).
For any inputs A and B, exactly one of the three outputs will be 1.
Basic circuit analysis and design
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18. Step 1: How many inputs and outputs?
Two 2-bit numbers means a total of four inputs.
We should name each of them.
Let’s say the first number consists of digits A1 and A0 from left to right, and the second number is B1 and B0.
The problem specifies three outputs: G, E and L.
Here is a block diagram that shows the inputs and outputs explicitly.
Now we just have to design the circuitry that goes into the box.
Basic circuit analysis and design
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19. Step 2: Functional specification
For this problem, it’s probably easiest to start with a truth table. This way, we can explicitly show the relationship (>, =, <) between inputs.
A four-input function has a sixteen-row truth table.
It’s usually clearest to put the truth table rows in binary numeric order; in this case, from 0000 to 1111 for A1, A0, B1 and B0.
Example: 01 < 10, so the sixth row of the truth table (corresponding to inputs A=01 and B=10) shows that output L=1, while G and E are both 0.
Basic circuit analysis and design
4/25/2019

20. Step 2: Functional specification
For this problem, it’s probably easiest to start with a truth table. This way, we can explicitly show the relationship (>, =, <) between inputs.
A four-input function has a sixteen-row truth table.
It’s usually clearest to put the truth table rows in binary numeric order; in this case, from 0000 to 1111 for A1, A0, B1 and B0.
Example: 01 < 10, so the sixth row of the truth table (corresponding to inputs A=01 and B=10) shows that output L=1, while G and E are both 0.
Basic circuit analysis and design
4/25/2019

21. Step 3: Simplified Boolean expressions
Let’s use K-maps. There are three functions (each with the same inputs A1 A0 B1 B0), so we need three K-maps.
G(A1,A0,B1,B0) =
A1 A0 B0’ +
A0 B1’ B0’ +
A1 B1’
E(A1,A0,B1,B0) =
A1’ A0’ B1’ B0’ +
A1’ A0 B1’ B0 +
A1 A0 B1 B0 +
A1 A0’ B1 B0’
L(A1,A0,B1,B0) =
A1’ A0’ B0 +
A0’ B1 B0 +
A1’ B1
Basic circuit analysis and design
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22. Step 4: Drawing the circuits
G = A1 A0 B0’ + A0 B1’ B0’ + A1 B1’
E = A1’ A0’ B1’ B0’ + A1’ A0 B1’ B0 + A1 A0 B1 B0 + A1 A0’ B1 B0’
L = A1’ A0’ B0 + A0’ B1 B0 + A1’ B1
LogicWorks has gates with NOTs attached (small bubbles) for clearer diagrams.
Basic circuit analysis and design
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23. Testing this in LogicWorks
Where do the inputs come from? Binary switches, in LogicWorks
How do you view outputs? Use binary probes.
probe
switches
Basic circuit analysis and design
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24. Basic circuit analysis and design
Example wrap-up
Data representations.
We used three outputs, one for each possible scenario of the numbers being greater, equal or less than each other.
This is sometimes called a “one out of three” code.
K-map advantages and limitations.
Our circuits are two-level implementations, which are relatively easy to draw and follow.
But, E(A1,A0,B1,B0) couldn’t be simplified at all via K-maps. Can you do better using Boolean algebra?
Extensibility.
We used a brute-force approach, listing all possible inputs and outputs. This makes it difficult to extend our circuit to compare three-bit numbers, for instance.
We’ll have a better solution after we talk about computer arithmetic.
Basic circuit analysis and design
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25. Basic circuit analysis and design
Summary
Functions can be represented with expressions, truth tables or circuits. These are all equivalent, and we can arbitrarily transform between them.
Circuit analysis involves finding an expression or truth table from a given logic diagram.
Designing a circuit requires you to first find a (simplified) Boolean expression for the function you want to compute. You can then convert the expression into a circuit.
Next time we’ll talk about some building blocks for making larger combinational circuits, and the role of abstraction in designing large systems.
Basic circuit analysis and design
4/25/2019