CHAPTER THREE, logic gates. Now that we've had a look at binary number systems, let's put them to work in a practical application. Gates. Gates are used in everyday intelligent electrical devices or IDs, such as computers, microprocessors, smart house devices, metering and relays just to mention a few. We looked at binary numbers because they are the route to binary quantities binary variables in binary systems. A binary quantity is one that can take only two states.
This is a binary arrangement with a switch in series With a power source and a light, the switch can be open or closed, the light can either be on or off, we can now assign a binary number to the switch and to the light. For the light on would be one off would be zero. For the switch closed would be the equivalent of one open would be the equivalent of zero. And in order to analyze these binary arrange arrangements, we devised a truth table. And the truth table has an output for every possible input. So the switch can either be open or closed can be one or zero.
And the light would either be on or off significant signified by a one or a zero. This is a binary arrangement that has two switches in series with a power source and a light. Either switch can be opened or closed the light will either be on or off and as before we can now assign our binary number to the light and the switch for the light on is one off is zero for the switches closed as one open is zero. And in order to analyze the binary arrangement, we have devised a truth table. The two switches form what isn't known in logic terms or in gate terms as an AND gate. We state that the light will be on or that l is equal to s one and s two.
In other words, the light is on with both when both switch one and switch To our on and a tooth table hold displeases where you have zeros if either of the switches are off, but if both switches, s one and s two are on, then you have a one for the light. Here is another binary arrangement with two switches this time in parallel, which generates this truth table. When either of the switches are closed or a one than the light will be on or one it requires both switches to be open or zero in order for the light to be off. The two switches form what is known in logic terms as an OR gate. We state that the light in order for the light to be on s one or s two have to be one or closed. Here is yet another binary arrangement, this time with three switches in series, which generates this truth table.
The three switches form what is known in logic terms as a NAND gate and we state that l in order for LTV one, s one and s two and s three have to be closed. So, all three switches have to be a one. If any of these switches are zero, the output is zero or the light is off of course, we have to look at the Parallel arrangement of the switches as well. And this generates the following truth table. The three switches form what is known in logic terms as an OR gate. And we state that the light in order for it to be on s one or s two or s three have to be on or closed.
Looking at the truth table, it is stating the same thing. If any of the switches are closed, you will get a one. In order for the light not to be on. All of the switches have to be open. One more example before pressing on to other matters. We have a three switch arrangement this time A combination of a series parallel arrangement of the switches this circuit would generate this truth table.
The logic of the switch three switches can only be described by the equation L and L is one or the light is on if switch S one and either s two or s three are closed. And the truth table demonstrates this as well. If all of the switches are one of course you get one if any one of s one or s two is closed and then s one has to be closed as well. You would get one if s one is open or zero, then you're going to get a zero. And if both s one are sorry if both s two and s three are open, then you also get a zero. Mechanical switches are really good for demonstrating no logic arrangement of switches and lights.
However in most solid state systems computers etc The logic is set up in terms of voltage levels. A zero volt or a low logic level is a 05 volts is equal to a high or a logic level of one in this transistor circuit arrangement here The transistor is in a state of saturation by virtue of the applied input voltage to five volts through a two position switch, because it is saturated the transistor drops very little voltage between the collector and the emitter resulting in an output voltage of practically zero volts. If we are using this circuit to represent binary bits we would say that the input signal is a binary one and the output signal is a binary zero. Any voltage close to full supply voltage measured in reference to ground of course, is considered a one and a lack of voltage is considered a zero.
Alternative terms for these voltage levels are a high the same as a binary one and a low to save a binary zero is a general term for the representation of a binary bit by a circuit voltage is logic level. Moving a switch to the other position we apply a binary 02 the input and receive a binary one at the output putting the transistor into cutoff. What we've created here with a single transistor is a circuit generally known as a solid state logic gate or simply a gate. A gate is a special type of amplifier circuit designed to accept and generate voltage signals corresponding to binary ones and zeros. As such gates are not intended to be used for amplifying analog signals used together multiple gates made be applied to the task of binary numbers storage memory circuits or the manipulation computing circuits, each gate, each gates output representing one bit of a multi bit binary number.
Just how this is done is the subject of further study. Right now it's important to focus on the operation of these individual gates. The gates shown here with a single transistor is known as an inverter or a NOT gate because the output is the exact opposite from what is on the input. For convenience, gate circuits are generally represented by their own symbols rather than by their circuit diagrams of transistors and resistors. logic gates are the building blocks used to create digital circuits. There are three elementary logic gates and a range of other simple gates you'll see these shortly.
Each gate has its own logic symbol, which allows complex function functions to be represented by a logic diagram. The function of each gate can be represented by a truth table and that truth table is unique. Looking back very quickly at the single transistor switching circuit that provided us with the logic of what is called an inverter or a NOT gate. Rather than drawing out a network of switches and or transistors, gates symbols are used in conjunction with their logic truth tables to symbol belies logic functions. This is the symbol for an inverter or a NOT gate. The logic is described by its truth table.
It has one input and one output. The output is THE COMPLETE REVERSE OF THE input. In other words, a zero on the input provides a one on the output, a one on the input provides a zero on the output. Remember our series switched circuit. We can now represent this system by the standard two input AND gate symbol with its truth table, which is identical to the two series switch version. Reading the truth table.
If a and b are zero, you have a zero Output if one of the inputs is zero, you have a zero output, you need both inputs to be a one to have a one on the output. Then we can remember our three switches in series. We can now represent this system by this by the standard three input AND gate symbol with a truth table, which describes how that AND gate works. And how it works is you need all of the inputs to be a one. Otherwise you would have a zero on the output that agrees with the same truth table we used in our mechanical switches. What this truth table means in practical terms is shown in the following sequence of illustrations with to input AND gate subjected to all possibilities of input logic levels, an LED light emitting diode provides the visual indication of the output logic level.
So with zeros on the input, the output is zero. In other words, the LED is not lit. If you switch a to have a high input, we still don't have high on the output. In other words, the light emitting diode is not lit. If we switch the B input to a high level or a one, we still don't get an output. However, if we switch both A and B high, then the end gate gives us the logical level one output which will light the light emitting diode.
So let's have a look at that. Just to repeat it so we can see how it works and cemented into our memory. If we change the input a from zero to one, nothing happens. If we change the input B to one, nothing happens. However, if we change both inputs to a one p output, he gives us a logic level one and lights the light emitting diode. A variation on the idea of an end gate is called a NAND gate.
The word NAND is a verbal contraction of the words, not an end. Essentially, a NAND gate behaves the same as an end gate with a knot or an inverter gate connected to the output terminal. to symbolize this output signal inversion, the NAND gate symbol has an bubble on the output, which differentiates it from a simple AND gate. The truth table for a NAND gate is as one might expect exactly the inverse, as that of an end gate. As with an end gate, NAND gates are made with more than one inputs sometimes, in such cases, the same general principle applies. The output output will be low or zero if and only if all the inputs are high, a one.
If any of the inputs are low as zero, the output will go high. It'll be a one. Our next gate to investigate is the OR gate that they can have any number of inputs. For example, This one has two inputs, an A and a B input. And your data symbolized like this. It can also have multiple inputs and this one is indicating three inputs A, B and C. And the symbol is the same, it just has more inputs.
The truth table for each are depicted here. So called an OR gate because the output will be high a one if any of the inputs are high or a one, the output will be low zero, if and only if all of the inputs are low, or zero. So let's hook up our OR gate now to our lit light emitting diode and the following sequence of illustrations demonstrates the OR gates function with the two inputs experiencing all possible logic levels. an LED or a light emitting diode provides the visual indication of the gates output logic level. So if we switch one, from a low to a high or a from a low to a high, the light emitting diode will light. In other words, one on the A input will give us a one on the output.
If we switch B to A one and switch a to zero, we still get the light emitting diode to light. And if we switch both of them to a one, again we get the same result the output is a logic level one and the light emitting diode will light So let's repeat that just have a second look at it. If any of the inputs are one, you get an output that will light the light emitting diode. As you might have suspected, there exists an inverted OR gate known as a NOR gate, which is an OR gate with its output inverted. You will notice that there's a differentiation between the OR gate and the NOR gate, by the placement of a little bubble on the output of an OR gate, which symbolizes it to be a NOR gate. NOR gates, like all other multiple input gates seen thus far can be manufactured with With more than just two inputs, still, the same logic applies.
The output goes low zero if any of the inputs are made high one, the output is high one, only when all of the inputs are low, zero. Another gated function is the negative and gate. The negative and gate functions the same as an AND gate with all of its inputs inverted, connected through not gates. In keeping with standard gates symbolization convention, these inverted inputs are signal thought signified by little bubbles on the input. Contrary to most people's first instinct, the logic behavior of a negative and GUID is not the same as a NAND gate. Its truth table.
Actually, is identical to a NOR gate. The output goes low, zero if any of the inputs are made high one, the output is high one only when all of the inputs are low, zero. Following the same pattern, and negative or gate functions the same as an OR gate with all of its inputs inverted. In keeping with standard gates symbol convention, these inverted inputs are signified by bubbles. This function can also be written as these equivalent gate circuits, to behavior and truth table. Have a negative or gate is the same as an NAND gate.
The Exclusive OR gate. The last six gate types are fairly direct variations of three basic functions AND, OR, and NOT. The Exclusive OR gate however, is something quite different. The Exclusive OR gates, output is a high one logic level. If the inputs are at a different logic level, they're either going to be 01 or one and zero on the input. Conversely, the output is a low a zero logic level if the inputs are at the same logic level.
The Exclusive OR is sometimes called an XOR gate has both a symbol and a truth table pattern that is unique There are equivalent circuits for an exclusive OR gate made up of an OR and NOT gates, just as there were for NAND nor and negative input gates. A rather direct approach to simulating an exclusive OR gate is to start with a regular OR gate, then add additional gates that in combinations work like this. If a and b are the same, both ones or both zeros, the both end gates output will be zero making the output of the OR gate zero. If a and b are different, one and 01 of the two AND gates output will be one making the output of the OR gate one. Thus, the truth table holds true for this combination of gates meaning that we can replace them with The Exclusive OR gate, Exclusive OR gates are very useful circuits, where two or more binary numbers are to be compared bit for bit, and also for error detection or parody checks and code conversions.
Finally, our last gate for analysis is the Exclusive NOR gate, otherwise known as a next NOR gate. It is equivalent to an exclusive OR gate with an inverted output. The truth table for this gate is exactly opposite to that of an exclusive OR gate. As indicated by the truth table, the purpose of an Exclusive NOR gate is to have the output high. One logic level whenever both inputs are at the same logic level, either 00 or one one. It should have Have the output low zero logic level if the inputs are different.
So, in review, we have considering two input gates rather than multi input gates but the two inputs logic can be extrapolated to multi level as well. An aim gate, the output is high only of both inputs A and B are high. For a NOR gate the output is high if the input a or input B are high NAND gate the output is not high of both a input and B inputs are high, the NOR gate, the output is not high if either a input or beat input are high. negative end gate behaves like an OR gate. And negative or gate behaves like a NAND gate. An exclusive OR gate the output is high if the input logic levels are different.
And an Exclusive NOR gate the output is high if the input logic levels are the same. Let's look at something called gate universality. NAND gate and NOR gates possess a special property that is they can be considered universal. Given enough gates either type of gate is able to mimic the operation of any other gate type. The ability for a single gate type to be able to mimic any other gate type is one enjoyed only by the NAND gate and or the NOR gate. In fact, digital control systems have been designed around nothing but either NAND or an orange gates, all the necessary logic functions being delivered from the collection of interconnected Nan's or Norse.
Let's have a look at this universality and see how all the basic gates types can be formed using only Nan's or only Norse. As you can see, there are two ways to use a NAND gate as an inverter. And two ways to use a NOR gate as an inverter. constructing the and function to make the and function from NAND gates, all it is needed is an inverter stage on the output of the NAND gate, this extra inversion cancels out the first n inand leaving the N function Takes a little bit more work to wrestle the same functionality out of a NOR gate. But it can be done by running all of the inputs of the NOR gate through, not gates, which are NOR gates made to function as not gates. We can construct the NAND function using NOR gates.
To make a NOR gate perform the NAND function we must invert all the inputs of the NOR gate, as well as the NOR gates output. constructing the or function inverting the output of a NOR gate with another NOR gate connected as an inverter results in the or function. The NAND gate on the other hand requires inversion of all inputs to mimic The or function just as we needed to invert all the inputs of a NOR gate to obtain the and function. Remember that inversion of all inputs to a gate results in changing that gates essential function from and to or, or vice versa, plus an inverted output. That's with all the inputs inverted. A Nan behaves as an or a nor behaves as an and, and an and behaves as a nor an or behaves as an end.
And the last function constructing a nor function much the same as the procedure for making a NOR gate behave as a NAND gate we must invert all the inputs and the outputs to make a NAND gate function as a NOR gate and this ends chapter three