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**Goal**• To understand how digital a computer can work, at the lowest level. • To understand what is possible and the limitations of what is possible for a digital computer.**Logic Gates**• Digital circuits are hardware components that manipulate binary information. • Logic gates are implemented using transistors and integrated circuits. • Each basic circuit is referred to as a logic gate.**Logic Gates**• All basic logic gates have the ability to accept either one or two input signals (depending upon the type of gate) and generate one output signal.**Logic Gates Symbols**Inputs and outputs • Gates have two or more inputs, except a NOT gate which has only one input. All gates have only one output. Usually the letters A, B, C and so on are used to label inputs, and Q is used to label the output. On this page the inputs are shown on the left and the output on the right. The inverting circle (o) • Some gate symbols have a circle on their output which means that their function includes inverting of the output. It is equivalent to feeding the output through a NOT gate. For example the NAND (Not AND) gate symbol shown on the right is the same as an AND gate symbol but with the addition of an inverting circle on the output.**Basic logic gates**• NOT • AND • OR • NAND • NOR**Truth Table**• A truth table is a good way to show the function of a logic gate. • It shows the output states for every possible combination of input states. • The symbols 0 (false) and 1 (true) are usually used in truth tables.**Logic Gate: NOT**• The NOT gate is also known as an inverter, simply because it changes the input to its opposite (inverts it). • The NOT gate accepts only one input and the output is the opposite of the input. • A common way of using the NOT gate is to simply attach the circle to the front of another gate. This simplifies the circuit drawing and simply says: "Invert the output from this gate."**Logic Gate: AND**• The AND gate requires two inputs and has one output. • The AND gate only produces an output of 1 when BOTH the inputs are a 1, otherwise the output is 0.**Logic Gate: OR**• The AND gate requires two inputs and has one output. • The AND gate only produces an output of 1 when BOTH the inputs are a 1, otherwise the output is 0.**Logic Gate: NAND**• This is an AND gate with the output inverted, as shown by the 'o' on the output. • The output is true if input A AND input B are NOT both true: Q = NOT (A AND B) • A NAND gate can have two or more inputs, its output is true if NOT all inputs are true.**Logic Gate: NOR**• This is an OR gate with the output inverted, as shown by the 'o' on the output. • The output Q is true if NOT inputs A OR B are true: Q = NOT (A OR B) • A NOR gate can have two or more inputs, its output is true if no inputs are true.**Logic Circuit DesignExample 1**Find the Boolean expressions output of the following circuit: Answer: (x+y)y**___**_ _ Logic Circuit DesignExample 2 Find the Boolean expressions output of the following circuit: Answer: xy**x+y**x Logic Circuit DesignExample 3 Draw the circuits for the following Boolean algebraic expressions: x+y __**(x+y)x**(x+y)x x+y Logic Circuit DesignExample 4 Draw the circuits for the following Boolean algebraic expressions: x+y**A**AND AND B Y OR NOT C A B C Y 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Circuit-to-Truth Table Example 2# ofInputs = # of Combinations 2 3 = 8**A**A B A AND AND B Y = A B + A C OR } NOT } C A C A B C Y 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 Circuit-to-Truth Table Example**Half-Adder**• To understand what is a half adder you need to know what is an adder first. • Adder circuit is a combinational digital circuit that is used for adding two numbers. A typical adder circuit produces a sum bit (denoted by S) and a carry bit (denoted by C) as the output. • Besides addition, adder circuits can be used for a lot of other applications in digital electronics like address decoding, table index calculation etc. • Adder circuits are of two types: Half adder and Full adder.**Half-Adder**• Half adder is a combinational arithmetic circuit that adds two numbers and produces a sum bit (S) and carry bit (C) as the output. • The arithmetic operation, addition of two binary digits has four possible elementary operations, namely: • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0 with 1 carry • If A and B are the input bits, then sum bit (S) is the X-OR of A and B and the carry bit (C) will be the AND of A and B. • From this it is clear that a half adder circuit can be easily constructed using one X-OR gate and one AND gate.**Half-Adder**• Half adder is a combinational arithmetic circuit that adds two numbers and produces a sum bit (S) and carry bit (C) as the output. • The arithmetic operation, addition of two binary digits has four possible elementary operations, namely: • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0 with 1 carry • If A and B are the input bits, then sum bit (S) is the X-OR of A and B and the carry bit (C) will be the AND of A and B. • From this it is clear that a half adder circuit can be easily constructed using one X-OR gate and one AND gate.**Half-Adder**• Inputs: A + B • Outputs: Sum (S) , Carry (C) Boolean Expressions: Sum (S) = A’B + AB’ Carry (C) = AB Logic Circuit Truth Table**Full-Adder**• This type of adder is a little more difficult to implement than a half-adder. • The main difference between a half-adder and a full-adder is that the full-adder has three inputsand two outputs. • The first two inputs are A and B and the third input is an input carry designated as CIN. • The output carry is designated as COUT and the normal output is designated as S. Take a look at the truth-table.**Full-Adder**• From the truth-table, the full adder logic can be implemented. We can see that the output S is an EXOR between the input A and the half-adder SUM output with B and CI inputs. We must also note that the C will only be true if any of the two inputs out of the three are HIGH. • Thus, we can implement a full adder circuit with the help of two half adder circuits. • The first will half adder will be used to add A and B to produce a partial Sum. • The second half adder logic can be used to add CI to the Sum produced by the first half adder to get the final S output. If any of the half adder logic produces a carry, there will be an output carry. • Thus, C will be an OR function of the half-adder Carry outputs. Take a look at the implementation of the full adder circuit shown in the previous slide.