In the last chapter the binary values zero and one are represented by two different voltages. Binary zero is a voltage close to 0 V and binary one by a voltage close to +5 V (some logic circuits use other voltage levels but this is a popular value and will serve as an example). A gate is a simple electronic circuit that has a single output voltage that corresponds to one of the two binary values. These gates are often referred to as ‘logic gates’ and the output voltages as ‘logic 0’ or ‘logic 1’ instead of binary 0 and 1. The distinction is just in the name. If you were to ask a mathematician or a computer programmer, they will refer to the outputs as binary values but an electronics engineer will call them logic levels. It really doesn’t matter.
We connect one or more voltages to the input of the gate. These input voltages are either logic 0 or logic 1 levels. The logic gate looks at the input voltages and ‘decides’, depending on its design, what voltage to produce at the output of the circuit.
There are only four basic designs of gate. They are called the NOT gate, the AND gate, the OR gate and the XOR gate. Notice how we use capital letters for the names of the gates otherwise we can finish up with some almost indecipherable sentences. Not not or and not and or not…
A little reminder before we start. Logic gates are clever little chaps but they are not magic. Just like any other electronic circuit, they need power supplies to make them work. Now, because all gates and microprocessors need power supplies, we tend to assume that everyone knows that. You will notice that power supplies are not shown in any of the diagrams in this chapter but that doesn’t mean that they are not there!
We will explore these gates now, starting from the simplest.
It has only one input and performs a very simple function. It simply reverses the binary value. If we put a logic 1 into it, we get a logic 0 at the output. Similarly, a 0 at the input gives a 1 at the output. On a diagram, we represent a NOT gate by a symbol as shown in Figure 5.1.
Figure 5.1 Symbols for a NOT gate
This is an alternative to the wordy description of how a gate works. It simply lists all the possible inputs to the gate together with the corresponding outputs. The truth table for a NOT gate is really easy. There are only two possible inputs: 0 and 1 as we can see in Figure 5.2.
Figure 5.2 The truth table for a NOT gate
So, how is it used in the microprocessor?
The truth table only shows what happens to a single bit but in the microprocessor we may want to use a NOT gate to invert a hex number like A4H. In this case the hex number is first converted to an 8-bit binary number. This process is not performed by the microprocessor but by other external circuits. By the time it reaches the microprocessor it has been converted to the binary equivalent of 101001002.
The NOT gate has only one input so, to handle an 8-bit binary word, we will need eight NOT gates. Now it becomes much easier. Each NOT gate inverts just one of the bits and all the outputs are grouped together to form a new hex number. See how it works in Figure 5.3. The result was the hex number 5BH. This is curious. If we add A4H to this result of 5BH we get FFH or all ‘ones’ in binary, 111111112. There was nothing special about the number A4H. It happens with any pair of numbers generated by NOT gates. Why is this? Figure 5.3 gives a clue.
Figure 5.3 Inverting a hex number
A little extra bit
We can show an inversion by drawing a line over the top. In Figure 5.2 the input was given the letter A and the output was shown as X. We could say: X=.
Unlike the NOT gate, an AND gate has more than one input. In fact we can have as many inputs as we like but the good news is that in microprocessors only two inputs are used. This simplifies the symbols and the truth tables considerably.
An AND gate is any circuit that gives a logic (or binary) 1 if (and only if) every input to the circuit is at logic 1. So in microprocessors, with only two inputs, it is easier to say that it gives a 1 out if both of the inputs is at logic 1. The symbols for the AND gate are shown in Figure 5.4.
Figure 5.4 Symbols for an AND gate
The truth table in Figure 5.5 has four rows to cover all the possible combinations of inputs.
Figure 5.5 The truth table for an AND gate
What is the point of an AND gate?
We often meet an AND gate without realizing it. When we climb into an elevator the door must be closed AND the floor button pressed before the motor will start. This is an AND gate in action. In a microprocessor, groups of AND gates are used to handle pairs of inputs at the same time just like we did with the NOT gates. For convenience we usually use hex numbers to describe groups of inputs and outputs but remember it’s all really happening in binary.
Unless we appreciate that the hex numbers are really just groups of ones and zeros, it may seem odd to talk about putting numbers through AND gates. Figure 5.6 may make it (a little) clearer. In this figure, we have used hex inputs of 37H and 5BH giving a result of 13H. Note that the AND gate does not add numbers.
Figure 5.6 An AND gate in action
A little extra bit
In data books and manuals, the AND function is abbreviated to a dot like a period (full-stop). So, if an AND gate had two inputs called A and B and an output called X, then we could write X = A·B
Sometimes it is further simplified to X = AB
Very occasionally we come across a symbol ∧ so we can write X=A∧B.
The word NAND is just a fancy contraction of NOT and AND. The NAND gate is just an AND gate followed by a NOT gate so all the outputs shown in the AND truth table are just inverted by the NOT gate.
On a diagram, this combination is indicated by putting a small circle on the end of the symbol. The symbol and truth table are shown in Figure 5.7.
Figure 5.7 The truth table for a NAND gate
It is better not to make a big effort to learn the NAND gate. Just remember that it’s the same as the AND gate except the output has been inverted by the NOT gate that has been added internally. The symbol has a line over the top to indicate the added NOT function. A two input NAND could be written as X= or X= or very occasionally X=.
This follows on nicely from the AND gate. The OR gate gives a logic one at its output if either (or both) of the inputs is at a logic one. Just like the AND gate, the OR gate can have as many inputs as we wish but in a microprocessor, we only use two input versions. Its symbols and truth table are shown in Figures 5.8 and 5.9.
Figure 5.8 Symbols for an OR gate
Figure 5.9 The truth table for an OR gate
Another extra bit
The OR function can be written as + or sometimes ∨. So, if an OR gate had two inputs called A and B and an output called X, then we could write X=A+B or X=A∨B. Don’t mistake this + sign as ‘plus’ as in addition 3+4=7.
As we would expect, this is just the same as the OR gate except for the NOT gate added to the output. The symbol has the inversion line over it to give X= or (see Figure 5.10).
Figure 5.10 The NOR gate
This is called the Exclusive-OR gate, which is abbreviated to XOR or EOR. Here are two examples of everyday English, both using the word ‘or’.
We get into an elevator and the operator says to us ‘Do you want to go up or down?’ We have a choice, we can decide to go up or we can go down. We have two possible answers – ‘up’ or ‘down’.
On the way home we buy a burger. We are asked ‘Do you want ketchup or mustard?’ This time we could answer ‘ketchup’ or ‘mustard’ or we could say ‘both’.
Figure 5.11 The XOR or difference gate
This is a fine example of how we can make something which is really easy appear difficult. The word ‘or’ in English has two different meanings which can be referred to as exclusive and inclusive. The first example used the exclusive ‘or’ because we could have one or the other but not both. The second situation uses the inclusive ‘or’ because we could have one or the other or both. We use both meanings everyday and understand them so well that we know automatically which one is meant.
In the XOR gate the output is a logic one if either input is a logic 1 but not both. Only two-input XOR gates are manufactured, whether for use in a microprocessor or not (see Figure 5.11).
To look at it in another way, the output from an XOR is a logic one if the two inputs are different. For this reason, it is often called a ‘Difference’ gate.
The electronic padlock
When we want to take some money out of our bank account we can go to the money machine and pop our card in. It then asks us to key in our code number which is usually a four-digit number. If the number is correct, we have access to our account.
In the background is a microprocessor with two useful attributes:
1 Microprocessors are very good at spotting whether a number is zero or not.
2 They contain a series of XOR gates.
So how can it check my number? One easy method is to compare our keyed-in number with the number read from the magnetic strip on the back of the card. Each of our code numbers is treated as a hex number and converted to a four-bit binary number. This results in one 16-bit binary number from the keyboard and another from the magnetic strip on the card, which now have to be compared.
Inside the microprocessor are 16 XOR gates each with two inputs – one from the card and one from the keyboard. Every time the bits coincide, whether they are both zeros or both ones, the output will be a zero.
The 16 results are quickly scanned looking for any output which is not zero. This would indicate an incorrect number. The process is shown in Figure 5.12 but for clarity, only four of the XOR gates have been shown. In real life, there will be 16 of them, of course.
Figure 5.12 Using an XOR gate to compare numbers
The extra bit
The XOR function can be written as ⊕. So, if an XOR gate had inputs called A and B and an output called X, then it would be abbreviated as X=A⊕B.
This is the inverted version of the XOR gate. These result in the output being at logic 1 only when the two inputs have the same value or are equivalent. For this reason, it is often referred to as the Equivalence gate (see Figure 5.13).
Figure 5.13 The truth table for an XNOR or equivalence gate
The tri-state buffer
This looks like a logic gate but behaves more like a switch. In Figure 5.14 we can see that it is quite simple having only an input, an output and another connection called an ‘enable’. The purpose of the enable line is to switch the buffer on or off. When the buffer is switched on, any signal applied to the input appears at the output and when it is switched off, the buffer is disconnected so that there is no output signal present.
Figure 5.14 A tri-state buffer is like a switch
So, why not just use a switch?
The problem with a switch is that, once closed, the input and output are physically joined so that input and output circuits are connected together. The buffer is a one-way device for signals so that the output is isolated from the input to prevent any changes in the next circuit from interfering with the input circuits.
Look for the circles
In the NOT, NAND NOR and XNOR gates a small circle was shown at the output to indicate that the output has been inverted. The same thing occurs with the buffer. If a circle is shown, the output is an inverted copy of the input.
We apply a similar convention to the enable input. If the input has to be a logic 1 level to switch it on, then it is as shown in Figure 5.15. If, however, the enable input has to be a logic 0 to enable it a small circle is shown at the point where the enable line connects with the buffer. When the buffer is switched off, it is said to be disabled. This would give us four possible buffers as in Figure 5.16.
Figure 5.15 An active-low tri-state buffer
Figure 5.16 The four types of tristate buffer
In each case, choose the best option.
1 Which of the gates shown in Figure 5.17 would have an output of logic zero?
(a) A, B, C.
(b) A and D.
(c) B and D.
(d) None of them.
Figure 5.17 Which outputs are at logic 0?
2 If the hex number 4C is applied to eight NOT gates, the output would have the value of:
(a) B3.
(b) 4C.
(c) FF.
(d) CD.
3 Which two-input gate has an output of logic 0 when both inputs are at logic 1:
(a) OR.
(b) AND.
(c) NOT.
(d) XOR.
4 A gate is:
(a) necessary to keep the cattle in the field. (b) always operated from a supply of +5 V.
(c) an electronic circuit with three connecting wires. (d) an electronic switching circuit whose output voltage depends on the inputs.
5 Adding the input and output binary values of a NOT gate:
(a) can be 0, 1 or 2 depending on the inputs.
(b) will give a voltage around 2.5 V.
(c) always results in 1.
(d) is not possible.