Grade 9

Intermediate

10 min

Lesson 12 of 12

Building a CPU: Simple Logic Circuits

Not Started

See how multiple gates combine to perform calculations, like the Half-Adder circuit.

Every time you use a calculator, millions of tiny switches are flipping inside your phone. But how do you turn a simple 'on/off' switch into a machine that can actually do math?

Learning Objectives

1.Analyze a multi-gate circuit diagram to determine the final output
2.Explain how a Half-Adder circuit uses XOR and AND gates to add two bits
3.Create a logic circuit diagram to solve a specific real-world safety problem

The Power of Combination

In isolation, a single logic gate is simple: an AND gate only outputs $1$ if both inputs are $1$ . But when we chain gates together, we create combinational logic. This is the foundation of a CPU. By connecting the output of one gate to the input of another, we can process complex instructions. In binary math, we only have two digits: $0$ and $1$ . When we add $1 + 1$ , the result is $10_2$ (which is $2$ in decimal). To represent this, we need two separate output wires: one for the Sum and one for the Carry bit.

Quick Check

If we add $1 + 1$ in binary, what is the 'Carry' bit value?

The result is

10_2

, where the left digit is the carry.

Answer

The Half-Adder Circuit

The Half-Adder is the simplest circuit that performs addition. It takes two inputs (

A

and

B

) and produces two outputs. To find the Sum, we use an XOR (Exclusive OR) gate. The XOR gate is perfect because it returns

1

only when the inputs are different (

0+1

1+0

). To find the Carry, we use an AND gate. The AND gate only returns

1

when both inputs are

1

(

1+1

). Together, they solve the problem:

Sum = A \oplus B

Carry = A \cdot B

Let's trace the signals when $A=1$ and $B=1$ : 1. The inputs reach the XOR gate. Since both are $1$ (the same), the XOR output (Sum) is $0$ . 2. The inputs also reach the AND gate. Since both are $1$ , the AND output (Carry) is $1$ . 3. The final result is $Carry=1, Sum=0$ , which reads as $10_2$ .

Quick Check

Which logic gate is used to calculate the 'Sum' in a Half-Adder?

Think about which gate outputs 0 when both inputs are 1.

Answer

The XOR gate

Real-World Logic: The Safety Interlock

Logic circuits aren't just for math; they control the world. Imagine a heavy industrial paper cutter. For safety, the blade should only drop if the operator has both hands on two separate buttons (to keep hands away from the blade) AND the safety plastic guard is down. This requires a multi-gate circuit. We can use an AND gate for the two buttons, and then take that result and feed it into another AND gate with the guard sensor.

Design a circuit for an automated greenhouse. The water should turn on if: 1. The soil is Dry ( $A=1$ ) 2. AND (It is Morning ( $B=1$ ) OR the Manual Override ( $C=1$ ) is pressed).

This circuit requires an OR gate for $B$ and $C$ , and the output of that must be fed into an AND gate with $A$ .

A Half-Adder is limited because it cannot accept a 'Carry' input from a previous calculation. To add larger numbers, we use a Full-Adder. 1. A Full-Adder uses two Half-Adders and an OR gate. 2. It handles three inputs: $A$ , $B$ , and $Carry_{in}$ . 3. This allows computers to chain 64 of these together to process 64-bit numbers instantly!

Key Takeaways

1A Half-Adder uses an XOR gate for the Sum and an AND gate for the Carry.
2Combinational logic allows multiple gates to solve complex problems by chaining outputs to inputs.
3Binary addition follows the rule: $1 + 1 = 10_2$ , requiring two output bits.
4Logic gates can be used to enforce real-world safety conditions through boolean logic.

Quick Check

Multiple Choice

In a Half-Adder, if Input $A=0$ and Input $B=1$ , what are the Sum and Carry?

Multiple Choice

Why can't a Half-Adder be used to add large multi-digit binary numbers alone?

True/False

An AND gate is used for the Carry bit because it only outputs 1 when both inputs are 1.

What's Next?

Review Tomorrow

In 24 hours, try to sketch the Half-Adder circuit from memory and write out its truth table for all four input combinations.

Practice Activity

Try to design a logic circuit for a home alarm: The alarm sounds if the 'System is Armed' AND ('Window is Open' OR 'Door is Open').

Browse

Browse

Building a CPU: Simple Logic Circuits

Learning Objectives

The Power of Combination

The Half-Adder Circuit

Real-World Logic: The Safety Interlock

Key Takeaways

Quick Check

What's Next?

Building a CPU: Simple Logic Circuits

Learning Objectives

The Power of Combination

The Half-Adder Circuit

Real-World Logic: The Safety Interlock

Key Takeaways

Quick Check

What's Next?