Question

Another way to construct an or-gate is as a compound digital logic device, built from and-gates and inverters. Define a procedure or-gate that accomplishes this. What is the delay time of the or-gate in terms of and-gate- delay and inverter-delay?

Short answer

The delay of the constructed OR gate is \(2 \times \text{inverter-delay} + \text{and-gate-delay}\).

Step-by-step solution

Understanding an OR Gate

An OR gate is a digital logic gate that outputs true or 1 when at least one of its inputs is true or 1. Your goal is to construct an OR gate using AND gates and NOT gates (inverters).

Using De Morgan's Theorem

According to De Morgan's theorem, the expression \( A + B \) (which represents an OR operation) is equivalent to \( \overline{\overline{A} \cdot \overline{B}} \). This means that an OR operation can be created using AND gates and inverters.

Constructing the Circuit

To implement \( \overline{\overline{A} \cdot \overline{B}} \), first pass the inputs A and B through inverters to get \( \overline{A} \) and \( \overline{B} \). Next, feed these outputs into an AND gate to get \( \overline{A} \cdot \overline{B} \). Finally, pass the output through another inverter to get \( \overline{\overline{A} \cdot \overline{B}} \), which is equivalent to \( A + B \).

Calculating the Delay Time

Given the delays for individual gates, the delay time for the constructed OR gate is calculated as follows: The inputs go through two inverters and one AND gate. Therefore, the total delay is \[\text{total delay} = 2 \times \text{inverter-delay} + \text{and-gate-delay}\] This delay accounts for the time taken for inputs to pass through the two inverters and one AND gate in sequence.

Key concepts

Digital Logic and The OR Gate

In the realm of digital logic, an OR gate is vital for operations where output is true if at least one input is true. Essentially, if you imagine flipping switches, the OR gate light turns on if any switch is flipped up. This principle is fundamental in digital electronics, constructing circuits that perform logical operations.

Constructing an OR Gate with AND-Gates and Inverter-Gates

Creating an OR gate using just AND-gates and inverter-gates is a clever application of De Morgan's Theorem. Using these gates, you can build complex logic from simpler components. Begin by inverting inputs A and B, obtaining \( \overline{A} \) and \( \overline{B} \).
Feed these inverted inputs into an AND-gate. The output is then inverted again, which effectively results in \( A + B \). This step showcases how different gates interact to form the desired logic function.

• Input Inversion: Use inverters to get \( \overline{A} \) and \( \overline{B} \)
• AND Operation: Compute \( \overline{A} \cdot \overline{B} \)
• Final Inversion: Obtain final OR output by inverting \( \overline{A} \cdot \overline{B} \)

Understanding De Morgan's Theorem

De Morgan's Theorem is a cornerstone of digital logic, especially when manipulating expressions. It helps in expressing one type of logic operation in terms of others. According to De Morgan's, an expression like \( A + B \) (OR) is equivalent to \( \overline{\overline{A} \cdot \overline{B}} \). This is crucial in situations where converting OR-gate outputs using inverters becomes necessary.
It is instrumental in minimizing and transforming complex circuits.

Measuring Circuit Delay

Circuit delay is an important factor in digital circuits, representing the time taken for input changes to propogate through the circuit to produce a final output. In our constructed OR gate, delay plays a key role. It's computed as the sum of delays from each gate it passes through. To determine it for our artificial OR gate:
The delay involves times for:

• Two pass-throughs of inverter-gates
• One pass-through of the AND-gate

The formula for this delay is:
\[ \text{total delay} = 2 \times \text{inverter-delay} + \text{and-gate-delay} \] This guides in predicting the processing time of logic operations.

Role and Use of Inverter-Gates

Inverter gates, also known as NOT gates, are fundamental in digital logic circuits. They operate simply, producing the opposite of the input logic level. If the input is true (1), the output will be false (0), and vice versa.
This simple inversion capability is vital in creating complex circuits, like the composite OR gate discussed earlier. Inverters allow us to switch between logic states, enabling intricate logical expressions through elementary means.

• Used to modify input logic states
• Essential in creating certain logic functions via De Morgan's applications

Understanding and leveraging inverters is key in building effective and efficient digital logic systems.