Implementing Boolean Equations Using PAL Four Inputs And Outputs
Introduction to Programmable Array Logic (PAL)
Programmable Array Logic (PAL) devices are essential components in digital logic design, offering a flexible and efficient way to implement complex Boolean functions. A PAL device consists of a programmable AND array and a fixed OR array. This architecture allows designers to customize the AND array to generate specific product terms, which are then combined in the OR array to produce the desired output functions. This article delves into implementing Boolean equations using a PAL with four inputs, four outputs, and a three-wired AND-OR structure, focusing on the equations:
- w(A, B, C, D) = Σm(2, 12, 13)
- x(A, B, C, D) = Σm(7, 8, 9, 10, 11, 12, 13, 14, 15)
- y(A, B, C, D) = Σm(0, 2, 3, 4, 5, 6, 7, 8, 10, 11)
Understanding the structure and functionality of a PAL is crucial for optimizing digital circuits and achieving efficient implementations. The programmable nature of the AND array provides versatility in creating various logic functions, making PAL devices a valuable tool in digital design.
Understanding Boolean Equations and Sum of Minterms
Before implementing these equations in a PAL, it is essential to understand the notations and concepts involved. The equations are expressed in the sum of minterms form, which is a standard way to represent Boolean functions. Each minterm corresponds to a specific combination of inputs that make the function true (logic 1). The notation Σm(…) indicates the sum of minterms, where the numbers inside the parentheses represent the decimal equivalents of the binary input combinations. For example, Σm(2, 12, 13) means the function w(A, B, C, D) is true for minterms 2, 12, and 13.
To elaborate, let's break down each minterm:
- Minterm 2: Binary representation is 0010, which means A'B'CD'
- Minterm 12: Binary representation is 1100, which means ABC'D'
- Minterm 13: Binary representation is 1101, which means ABC'D
Similarly, the equation x(A, B, C, D) = Σm(7, 8, 9, 10, 11, 12, 13, 14, 15) covers a broader range of input combinations:
- Minterm 7: Binary representation is 0111, which means A'BCD
- Minterm 8: Binary representation is 1000, which means AB'C'D'
- Minterm 9: Binary representation is 1001, which means AB'C'D
- Minterm 10: Binary representation is 1010, which means AB'CD'
- Minterm 11: Binary representation is 1011, which means AB'CD
- Minterm 12: Binary representation is 1100, which means ABC'D'
- Minterm 13: Binary representation is 1101, which means ABC'D
- Minterm 14: Binary representation is 1110, which means ABCD'
- Minterm 15: Binary representation is 1111, which means ABCD
The equation y(A, B, C, D) = Σm(0, 2, 3, 4, 5, 6, 7, 8, 10, 11) includes:
- Minterm 0: Binary representation is 0000, which means A'B'C'D'
- Minterm 2: Binary representation is 0010, which means A'B'CD'
- Minterm 3: Binary representation is 0011, which means A'B'CD
- Minterm 4: Binary representation is 0100, which means A'BC'D'
- Minterm 5: Binary representation is 0101, which means A'BC'D
- Minterm 6: Binary representation is 0110, which means A'BCD'
- Minterm 7: Binary representation is 0111, which means A'BCD
- Minterm 8: Binary representation is 1000, which means AB'C'D'
- Minterm 10: Binary representation is 1010, which means AB'CD'
- Minterm 11: Binary representation is 1011, which means AB'CD
Understanding these minterms is crucial for programming the PAL effectively. Each minterm represents a unique product term that needs to be generated by the AND array and then summed up by the OR array.
PAL Architecture: Four Inputs, Four Outputs, Three Wired AND-OR Structure
The PAL device in question is designed with specific characteristics that dictate how Boolean functions can be implemented. It features four inputs (A, B, C, D), four outputs (w, x, y, and an unused output), and a three-wired AND-OR structure. This structure means that each output is the result of an OR operation on up to three AND terms. The architecture limits the complexity of the Boolean functions that can be directly implemented, as each output can only combine a maximum of three product terms. Understanding these architectural constraints is vital for efficient PAL programming.
Four Inputs (A, B, C, D)
The four inputs (A, B, C, D) represent the input variables for the Boolean functions. These inputs can be used in their true or complemented forms, providing a total of eight possible input signals (A, A', B, B', C, C', D, D'). These signals are fed into the programmable AND array, where they are combined to generate the necessary product terms.
Four Outputs (w, x, y)
The PAL has four outputs, labeled w, x, and y, with one output remaining unused in this specific implementation. Each output corresponds to a Boolean function that needs to be realized. The outputs are generated by the fixed OR array, which combines the product terms generated by the AND array. Each output is limited to a maximum of three product terms due to the three-wired AND-OR structure.
Three-Wired AND-OR Structure
The three-wired AND-OR structure is a critical aspect of this PAL architecture. It means that each OR gate, which generates the final output, can have a maximum of three inputs. These inputs come from the AND array, where the product terms are generated. This limitation affects the complexity of the Boolean functions that can be implemented directly. For functions requiring more than three product terms, additional logic or simplification techniques may be necessary.
Implementing Boolean Equations in PAL
To implement the given Boolean equations, we need to express each function as a sum of products (SOP) and then map these product terms to the PAL architecture. Given the three-wired AND-OR structure, each output can have a maximum of three product terms. We will now detail the implementation for each output (w, x, and y).
Implementing w(A, B, C, D) = Σm(2, 12, 13)
The function w(A, B, C, D) = Σm(2, 12, 13) can be expressed as the sum of the minterms:
w(A, B, C, D) = A'B'CD' + ABC'D' + ABC'D
This equation already has three product terms, which is the maximum allowed by the PAL architecture. Therefore, we can directly implement this function by programming the AND array to generate these three product terms and connecting them to the OR gate for output w:
- Product Term 1: A'B'CD'
- Product Term 2: ABC'D'
- Product Term 3: ABC'D
Each of these product terms will be generated by configuring the AND array to perform the logical AND operation on the corresponding input signals. The outputs of these AND gates are then connected to the OR gate that generates the output w. This direct mapping simplifies the implementation process and efficiently utilizes the PAL resources.
Implementing x(A, B, C, D) = Σm(7, 8, 9, 10, 11, 12, 13, 14, 15)
The function x(A, B, C, D) = Σm(7, 8, 9, 10, 11, 12, 13, 14, 15) has nine minterms. Given the three-wired AND-OR structure, we can only implement a maximum of three product terms directly. Therefore, we need to simplify the equation or use alternative implementation strategies. One approach is to use Boolean algebra to simplify the expression or to implement the complement of the function and then invert the output. However, for this specific PAL structure, simplifying using a Karnaugh map (K-map) is beneficial.
Using a K-map for x(A, B, C, D), we can identify the following simplification:
x(A, B, C, D) = A'BCD + AB'C' + ABC'
Even after simplification, we still have more than three product terms. However, recognizing that x(A, B, C, D) covers most of the minterms from 7 to 15, it might be more efficient to consider the complement of x, denoted as x'.
x'(A, B, C, D) = Σm(0, 1, 2, 3, 4, 5, 6)
Using a K-map for x'(A, B, C, D), we can derive a simplified expression:
x'(A, B, C, D) = A'B' + A'C' + A'D'
This simplified expression for x' has three product terms, which can be implemented directly in the PAL. The final output x can be obtained by inverting x':
x(A, B, C, D) = (A'B' + A'C' + A'D')'
However, since our PAL does not have an inverter at the output, we need to rethink our strategy. The three-wired AND-OR structure limits us, and we must find three minterms or combine minterms to fit this architecture.
Alternatively, we can express x(A, B, C, D) using the original minterms, selecting three terms that cover a significant portion of the function. For example:
x(A, B, C, D) ≈ A'BCD + AB'C'D' + AB'C'D
This approximation uses only three product terms, making it implementable within the PAL's constraints. However, it's essential to acknowledge that this is an approximation and will not produce the correct output for all input combinations. A more accurate implementation might require a different PAL architecture or additional logic gates.
Implementing y(A, B, C, D) = Σm(0, 2, 3, 4, 5, 6, 7, 8, 10, 11)
The function y(A, B, C, D) = Σm(0, 2, 3, 4, 5, 6, 7, 8, 10, 11) has ten minterms. Similar to the implementation of x, we need to reduce the number of product terms to fit within the three-wired AND-OR structure. Using a K-map to simplify this expression is essential.
By analyzing the K-map, we can identify the following simplified expression:
y(A, B, C, D) = B'C'D' + B'CD' + B'CD + A'B
This expression still has four terms, exceeding the PAL's capacity. We need to further simplify or approximate the function. Let's try grouping the minterms differently to fit within the three-term limit. We can choose three product terms that cover a significant portion of the minterms:
y(A, B, C, D) ≈ A'B'C'D' + A'B'CD' + A'B'CD
This approximation uses three product terms:
- Product Term 1: A'B'C'D'
- Product Term 2: A'B'CD'
- Product Term 3: A'B'CD
These terms can be directly implemented in the PAL. However, this is an approximation, and the output will not be accurate for all input combinations. A more precise implementation would require a PAL with more product terms or additional logic gates.
Programming the PAL Device
Programming a PAL device involves configuring the AND array to generate the required product terms. This is typically done using a PAL programmer, which applies the appropriate voltages to the programmable links within the device. The programming process essentially involves blowing fuses in the AND array to disconnect unwanted inputs, leaving only the desired inputs connected to the AND gates.
Configuring the AND Array for w
For the function w(A, B, C, D) = A'B'CD' + ABC'D' + ABC'D, we need to program the AND array to generate the following product terms:
- A'B'CD': Connect A', B', C, and D' to the first AND gate.
- ABC'D': Connect A, B, C', and D' to the second AND gate.
- ABC'D: Connect A, B, C', and D to the third AND gate.
The outputs of these three AND gates are then connected to the OR gate for output w.
Configuring the AND Array for x
For the approximated function x(A, B, C, D) ≈ A'BCD + AB'C'D' + AB'C'D, we program the AND array as follows:
- A'BCD: Connect A', B, C, and D to the first AND gate.
- AB'C'D': Connect A, B', C', and D' to the second AND gate.
- AB'C'D: Connect A, B', C', and D to the third AND gate.
The outputs of these three AND gates are connected to the OR gate for output x. It is crucial to remember that this implementation is an approximation.
Configuring the AND Array for y
For the approximated function y(A, B, C, D) ≈ A'B'C'D' + A'B'CD' + A'B'CD, the AND array is programmed as:
- A'B'C'D': Connect A', B', C', and D' to the first AND gate.
- A'B'CD': Connect A', B', C, and D' to the second AND gate.
- A'B'CD: Connect A', B', C, and D to the third AND gate.
The outputs of these three AND gates are then connected to the OR gate for output y. Again, this is an approximation due to the PAL's limitations.
Limitations and Considerations
The primary limitation in this implementation is the three-wired AND-OR structure of the PAL. This restricts the complexity of the Boolean functions that can be directly implemented. For functions with more than three product terms, approximations or alternative implementation strategies are necessary. These approximations can lead to inaccuracies in the output, as seen in the implementations of x and y.
Alternative PAL Architectures
For more complex Boolean functions, PAL devices with more product terms per output or other programmable logic devices (PLDs) such as Complex Programmable Logic Devices (CPLDs) or Field-Programmable Gate Arrays (FPGAs) may be more suitable. These devices offer greater flexibility and capacity, allowing for more complex logic functions to be implemented.
Karnaugh Maps (K-maps)
Karnaugh Maps (K-maps) are invaluable tools for simplifying Boolean expressions. They provide a visual method for minimizing the number of terms in a Boolean function, which is crucial when implementing logic functions in devices with limited resources, such as the PAL discussed in this article.
Boolean Algebra Simplification
Boolean algebra provides a set of rules and theorems that can be used to simplify Boolean expressions. Techniques such as DeMorgan's laws, distribution, association, and absorption can help reduce the complexity of Boolean functions, making them easier to implement in hardware.
Conclusion
Implementing Boolean equations using a PAL with four inputs, four outputs, and a three-wired AND-OR structure requires careful consideration of the device's limitations. While direct implementation is possible for functions with three or fewer product terms, more complex functions may require simplification, approximation, or alternative implementation strategies. The functions w(A, B, C, D), x(A, B, C, D), and y(A, B, C, D) illustrate these challenges and demonstrate the importance of understanding PAL architecture and Boolean algebra techniques. For more complex designs, exploring alternative programmable logic devices with greater capacity and flexibility is often necessary. Understanding these limitations and trade-offs is essential for efficient digital logic design.