ELECTRONICS

BASIC ELECTRONICS

S/No	Descriptions	Link
	Basic Electronics
1.	DC Theory - Relationship between Voltage, Current and Resistance	1-1 , 1-2 , 1-3
2.	Ohm’s Law	2
3.	Resistor	3
4.	The Inductor	4
5.	Introduction to Capacitors	5
6.	Semi-conductor (Diode)	6
7.	Bipolar Transistor Basics	7
8.	The AC Theory	8
	Digital Electronics
9.	Binary Numbers	9
10.	Hexadecimal Numbers	10
11.	Octal Numbers	11
12.	Boolean Algebra	12
	Radio and Data Communications
13.	Introduction to Signals	13
14.	Modulation	14
15.	Transmission Channels	15
16.	Types of Transmission	16
17.	Transmission Synchronization Types	17
18.	FDM Vs TDM Multiplexing methods	18

18. FDM vs TDM Multiplexing methods

FDM vs. TDM

TDM (Time Division Multiplexing) and FDM (Frequency Division Multiplexing) are two methods of multiplexing multiple signals into a single carrier. Multiplexing is the process of combining multiple signals into one, in such a manner that each individual signal can be retrieved at the destination. Since multiple signals are occupying the channel, they need to share the resource in some manner. The primary difference between FDM and TDM is how they divide the channel. FDM divides the channel into two or more frequency ranges that do not overlap, while TDM divides and allocates certain time periods to each channel in an alternating manner. Due to this fact, we can say that for TDM, each signal uses all of the bandwidth some of the time, while for FDM, each signal uses a small portion of the bandwidth all of the time.

TDM provides greater flexibility and efficiency, by dynamically allocating more time periods to the signals that need more of the bandwidth, while reducing the time periods to those signals that do not need it. FDM lacks this type of flexibility, as it cannot dynamically change the width of the allocated frequency.

The advantage of FDM over TDM is in latency. Latency is the time it takes for the data to reach its destination. As TDM allocates time periods, only one channel can transmit at a given time, and some data would often be delayed, though it’s often only in milliseconds. Since channels in FDM can transmit at any time, their latencies would be much lower compared to TDM. FDM is often used in applications where latency is of utmost priority, such as those that require real-time information.

17. Transmission synchronization types

a. Asynchronous Systems

In asynchronous systems, a separate timing channel is not used. The transmitter and receiver must be preset in advance to an agreed-upon baud rate. A very accurate local oscillator within the receiver will then generate an internal clock signal that is equal to the transmitters within a fraction of a percent. For the most common serial protocol, data is sent in small packets of 10 or 11 bits, eight of which constitute message information. When the channel is idle, the signal voltage corresponds to a continuous logic '1'. A data packet always begins with a logic '0' (the start bit) to signal the receiver that a transmission is starting. The start bit triggers an internal timer in the receiver that generates the needed clock pulses. Following the start bit, eight bits of message data are sent bit by bit at the agreed upon baud rate. The packet is concluded with a parity bit and stop bit.

Parity Bit

Noise and momentary electrical disturbances may cause data to be changed as it passes through a communications channel. If the receiver fails to detect this, the received message will be incorrect, resulting in possibly serious consequences. As a first line of defense against data errors, they must be detected. If an error can be flagged, it might be possible to request that the faulty packet be resent, or to at least prevent the flawed data from being taken as correct. If sufficient redundant information is sent, one- or two-bit errors may be corrected by hardware within the receiver before the corrupted data ever reaches its destination.

A parity bit is added to a data packet for the purpose of error detection. In the even-parity convention, the value of the parity bit is chosen so that the total number of '1' digits in the combined data plus parity packet is an even number. Upon receipt of the packet, the parity needed for the data is recomputed by local hardware and compared to the parity bit received with the data. If any bit has changed state, the parity will not match, and an error will have been detected. In fact, if an odd number of bits (not just one) have been altered, the parity will not match. If an even number of bits has been reversed, the parity will match even though an error has occurred. However, a statistical analysis of data communication errors has shown that a single-bit error is much more probable than a multibit error in the presence of random noise. Thus, parity is a reliable method of error detection.

b. Synchronous Systems

Although synchronous transmission is relatively simple to implement, it is efficient because of the start and stop bits required. For example, eleven or more bits is needed to transmit an 8 bit character.

In synchronous transmission, the start and stop bits are eliminated. The data is usually transmitted in one continuous block, rather than one character at a time. There are no pauses between characters. The blocks of data are separated by special characters called SYN character. The SYNC character usually consists of a 8 bit data code, although more can be used.

It should also be noted that transmission efficiency is higher for systems that sends out large packets. As proof, we shall consider the example below:

Amount of data bits to be sent: 1024 bits

With asynchronous transmission, 8 bits will be sent at a time, and therefore the efficiency is fixed at 8/11 ~ 72 % (with 1 start bit, 1 stop bit and 1 parity bit). However, with synchronous transmission, the efficiency is calculated to be 1024/(1024+8) assuming a 8-bit synchronization character which gives 99.4 % efficiency.

For small data packets however, it is entirely possible asynchronous transmission will be preferred.

Next topic will be on Multiplexing methods

16. Types of transmission

Most digital messages are vastly longer than just a few bits. Because it is neither practical nor economic to transfer all bits of a long message simultaneously, the message is broken into smaller parts and transmitted sequentially. Bit-serial transmission conveys a message one bit at a time through a channel. Each bit represents a part of the message. The individual bits are then reassembled at the destination to compose the message. In general, one channel will pass only one bit at a time. Thus, bit-serial transmission is necessary in data communications if only a single channel is available. Bit-serial transmission is normally just called serial transmission and is the chosen communications method in many computer peripherals.

Parallel transmission with 8 bits conveys eight bits at a time through eight parallel channels. Although the raw transfer rate is eight times faster than in bit-serial transmission, eight channels are needed, and the cost may be as much as eight times higher to transmit the message. When distances are short, it may nonetheless be both feasible and economic to use parallel channels in return for high data rates. The popular Centronics printer interface is a case where byte-serial transmission is used. As another example, it is common practice to use a 16-bit-wide data bus to transfer data between a microprocessor and memory chips; this provides the equivalent of 16 parallel channels. On the other hand, when communicating with a timesharing system over a modem, only a single channel is available, and bit-serial transmission is required. This figure illustrates these ideas:

The baud rate refers to the signalling rate at which data is sent through a channel and is measured in electrical transitions per second. In the EIA232 serial interface standard, one signal transition, at most, occurs per bit, and the baud rate and bit rate are identical. In this case, a rate of 9600 baud corresponds to a transfer of 9,600 data bits per second with a bit period of 104 microseconds (1/9600 sec.). If two electrical transitions were required for each bit, as is the case in non-return-to-zero coding, then at a rate of 9600 baud, only 4800 bits per second could be conveyed. The channel efficiency is the number of bits of useful information passed through the channel per second. It does not include framing, formatting, and error detecting bits that may be added to the information bits before a message is transmitted, and will always be less than one.

The data rate of a channel is often specified by its bit rate (often thought erroneously to be the same as baud rate). However, an equivalent measure channel capacity is bandwidth. In general, the maximum data rate a channel can support is directly proportional to the channel's bandwidth and inversely proportional to the channel's noise level.

A communications protocol is an agreed-upon convention that defines the order and meaning of bits in a serial transmission. It may also specify a procedure for exchanging messages. A protocol will define how many data bits compose a message unit, the framing and formatting bits, any error-detecting bits that may be added, and other information that governs control of the communications hardware. Channel efficiency is determined by the protocol design rather than by digital hardware considerations. Note that there is a trade-off between channel efficiency and reliability - protocols that provide greater immunity to noise by adding error-detecting and -correcting codes must necessarily become less efficient.

Next topic will be Transmission synchronization Types

15. Transmission channels

A communications channel is a pathway over which information can be conveyed. It may be defined by a physical wire that connects communicating devices, or by a radio, laser, or other radiated energy source that has no obvious physical presence. Information sent through a communications channel has a source from which the information originates, and a destination to which the information is delivered. Although information originates from a single source, there may be more than one destination, depending upon how many receive stations are linked to the channel and how much energy the transmitted signal possesses.

In a digital communications channel, the information is represented by individual data bits, which may be encapsulated into multibit message units. A byte, which consists of eight bits, is an example of a message unit that may be conveyed through a digital communications channel. A collection of bytes may itself be grouped into a frame or other higher-level message unit. Such multiple levels of encapsulation facilitate the handling of messages in a complex data communications network.

Any communications channel has a direction associated with it:

The message source is the transmitter, and the destination is the receiver. A channel whose direction of transmission is unchanging is referred to as a simplex channel. For example, a radio station is a simplex channel because it always transmits the signal to its listeners and never allows them to transmit back.

A half-duplex channel is a single physical channel in which the direction may be reversed. Messages may flow in two directions, but never at the same time, in a half-duplex system. In a telephone call, one party speaks while the other listens. After a pause, the other party speaks and the first party listens. Speaking simultaneously results in garbled sound that cannot be understood.

A full-duplex channel allows simultaneous message exchange in both directions. It really consists of two simplex channels, a forward channel and a reverse channel, linking the same points. The transmission rate of the reverse channel may be slower if it is used only for flow control of the forward channel.

Next topic will be on Types of Transmission

14. Modulation

Modulation is the process of shifting a frequency signal to a higher frequency band suitable for transmission. The modulating (original) signal is modulated onto the carrier signal. The following diagram is an illustration of the basic modulation process. Note that the actual process may vary based on the modulation technique used.

Basic diagram workflow for Amplitude Modulation

The original signal is recovered at the receiving end through the demodulation process. The demodulation technique often mirrors the modulation technique in reverse; if the modulation technique involves mapping various voltage values to its corresponding frequency values (e.g in the case of frequency modulation where 2V could be represented by 5kHz and 3V to be presented by 10kHz), then the demodulation process will involve deriving the values of the original signal based on the different frequency values of the signal. (10 kHz -> 3V and 5 kHz -> 2V).

a. Modulation techniques

While there are various techniques for modulation, these can be largely classified into two categories:

• Analog Modulation
• Digital Modulation

Digital modulation techniques are employed to transmit digital data across an analog channel. Aside from shifting the signal frequency band to a suitable transmission band, it also converts the digital signal to a suitable analog signal that is to be decoded at the receiving end.

Analog Modulation Techniques

Types of Modulation

In analog modulation, the modulation is applied continuously in response to the analog information signal. Common analog modulation techniques are:

Amplitude modulation (AM) (here the amplitude of the carrier signal is varied in accordance to the instantaneous amplitude of the modulating signal)

Frequency modulation (FM) – the frequency of the carrier signal is varied in accordance to the instantaneous amplitude of the modulating signal.

Phase modulation (PM) – the phase of the carrier signal is varied in accordance to the instantaneous amplitude of the modulating signal.

Comparison between the modulation techniques

	Advantages	Disadvantages
Amplitude Modulation	1) Simple and cheap to implement One advantage of adopting AM systems is that AM Systems is relatively simple to construct – fewer components are needed to actually build such a system. As such, it is actually cheaper to implement compared to FM or PM. 2) Less bandwidth needed As only one frequency for transmission is used for AM, the bandwidth being used is considerably less than FM. This translates into cheaper costs for AM.	1) High power requirement. As the extraction of the AM signal on the receiving side is dependent on the amplitude level of the receiving signal, it is important to amplify the voltage signal so that the effects of attenuation (a phenomenon where a signal gets smaller with distance) will be minimized. It is important to keep the signal at a significantly high voltage level compared to the noise(interfering signals) so that the noise can be negligible compared to the signal (i.e adding 1V noise to a 20V signal would be less pronounced than adding 1V to a 20V signal). 2) Poor noise immunity As the demodulation process is based on processing the amplitude level of the receiving signal to retrieve the original signal, AM is very susceptible to noise. Therefore, it is not very ideal for it to be transmitted over large distances or components with a known high noise factor. 3) Limited bandwidth range AM cannot be deployed at high frequencies. This is a huge limitation as the antenna for such systems has to be longer due to longer wavelength needed.
Frequency Modulation	1) Less susceptible to noise (See note on Poor Noise Immunity under Disadvantages of AM). With FM, the FM modulator converts the voltage variation in the modulating signal to a frequency variation. At the receiver, this frequency variation is converted back to a voltage variation. If the frequency variation is large, then the output voltage will also be large. The demodulation process being not dependent on amplitude level on signal essentially means that noise will not have so much of a large effect on FM systems. 2) No need to transmit at high power Since the demodulated output level is not dependent on the received FM level, there is no need to ensure a high level FM signal at the receiver. Hence, transmission power can be lower than the AM to cover the same area. Together with the advantage of constant power, this makes FM suitable for battery operated transmitted like cordless phones, wireless microphone and Walkie-Talkie. 3) Able to transmit at high frequencies FM is known for its application in VHF and UHF systems, and being able to transmit at high frequency lessens the need for a long antenna, as required length of antenna is directly correlated to the wavelength of the signal.	1) More bandwidth needed With FM, the carrier frequency is varied in response to the voltage level of the original signal. Therefore, a larger range of frequencies is actually sent across the transmission channel. This translates into higher costs. 2) Complexity of circuit Designing a FM system is considerably more complex compared to AM systems.
Phase Modulation	1) Not very susceptible to noise distortion As the modulation process is based on reading the different phases of the signals, it is not as susceptible to noise distortion as AM, which relies completely on reading the voltage level. However, noise can still affect the accuracy of demodulation, although to a lesser extent. 2) Less bandwidth needed As PM does not incur additional bandwidth in the same way FM does, it retains the AM advantage of lesser bandwidth costs.	1) Extremely complex to design The complexity of a phase modulator is extremely expensive. This is because the circuit needs to be sensitive enough to detect phase changes which can be very minute. 2) Usually associated with high error rate As it is difficult to design a perfect system to be able to detect phase variations accurately, PM systems often run into error rate problems.

b. Digital Modulation Techniques

Digital modulation is similar to analog modulation, but rather than being able to continuously change the amplitude, frequency, or phase of the carrier, there are only discrete values of these attributes that correspond to digital codes. In data communications, these discrete values generally take on the binary values of ‘0’ and ‘1’s.

It is important to note that since electrical signals are analog, conversion from the digital values to its analog counterpart has to be done to be able to transmit it across the channel.

Types of digital modulation techniques

As can be observed from the figure above, the digital bit 1 maps to a significantly higher sinusoidal waveform while the digital bit 0 maps to a smaller sinusoidal waveform. This is analogous to its counterpart of AM, with the main difference being that this signal can only take on a finite set of values. (as opposed to a continuous analog range; in this case, only 2 values can be used).

It is important to note that the digital bit stream is not restricted to mapping one bit at a time, it is possible to design a ASK system that takes in 2 bits at a time; in which case there could be four different values being interpreted (2² = 4 values).

In this illustrated example of FSK above, the digital bit ‘1’ maps to a higher frequency waveform and ‘0’ maps to a lower frequency waveform. FSK shares the same advantages as FM: lower amount of power needed for transmission, better noise immunity and ability to transmit at higher frequencies.

Phase-shift keying (PSK) is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal (the carrier wave).

Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases; each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data.

Quadrature Amplitude Modulation

QAM combines ASK and PSK techniques to increase the number of bits evaluated at a time. Quadrature amplitude modulation (QAM) requires changing the phase and amplitude of a carrier sine wave. An example of how the mapping of the values to the analog waveforms can be found in the table below:

2nd bit	1st bit	Result
0	0	2V sin wave with phase angle of 0 degrees
0	1	2V sin wave with phase angle of 180 degrees
1	0	5V sin wave with phase angle of 0 degrees
1	1	5V sin wave with phase angle of 180 degrees.

QAM is usually used with 8-bit, 16-bit, 32-bit and 64-bit requirements.

Summary of Digital Modulation advantages and disadvantages

Next Topic will be on Transmission channels

13. Introduction to Signals

a. What is a signal?

A signal is the transmission of data. We deal with signals constantly during the span of our lives. We interact with signals from music, power lines, telephones, and cellular devices. This means the use of antennas, satellites, and of course wires. In "computer land" signals are very important. Anyone that uses a computer should know how the machine transforms data into signals that other computers and devices can understand. In many cases, knowing how signals work will help you solve some kind of technical problem over the span of your life.

b. How is data transmitted?

Data is transmitted through a transmission medium. Transmission media come in two forms: wired and wireless.

Wired transmission media include: twisted pairs, coaxial cables and optical fibres. They involve the use of a physical connection from point to point and are generally tedious to adopt in systems which require many communication terminals to be set up.

Wireless transmission media offer the advantage of mobility at the cost of compromising speed and security. Examples of wireless media are: RF waves, microwaves and infrared.

c. Analog vs Digital Signals

The two main types of electrical signals are analog and digital signals. An analog signal generally takes on a continuous waveform, taking a possible infinite set of values across its range. A digital signal on the other hand, is discrete, with the amount of possible values being finite.

The following diagram depicts the differences between an analog and digital waveform:

d. What is RF?

Radio frequency (RF) is a type of transmission which involves pushing a oscillation of analog signals across a medium which corresponds to the frequency of radio waves. While RF waves take the form of analog signals, digital signals can be transmitted through RF as long as it is first properly modulated onto an analog carrier. It is subsequently demodulated at the receiving end to extract the original signal.

The following two tables outline the various nomenclatures for the frequency bands. The third table outlines some of the applications at each of the various frequency bands.

 

Table 1: Frequency Band Designations

Table 1 shows a relationship between frequency (f) and wavelength (λ). A wave or sinusoid can be completely described by either its frequency or its wavelength. They are inversely proportional to each other and related to the speed of light through a particular medium. The relationship in a vacuum is shown in the following equation:

where c is the speed of light. As frequency increases, wavelength decreases. For reference, a 1 GHz wave has a wavelength of roughly 1 foot, and a 100 MHz wave has a wavelength of roughly 10 feet.

 

Table 2: Microwave Letter Band Designations

 

e. Why Operate at Higher Frequencies?

Reasons accounting for this push into higher frequencies include efficiency in propagation, immunity to some forms of noise and impairments as well as the size of the antenna required. The antenna size is typically related to the wavelength of the signal and in practice is usually ¼ wavelength.

Next topic will be Modulation

12. BOOLEAN ALGEBRA

a. Introduction

In 1854, George Boole performed an investigation into the "laws of thought" which were based on a simplified version of the "group" or "set" theory, and from this Boolean or "Switching" algebra was developed. Boolean Algebra deals mainly with the theory that both logic and set operations are either "TRUE" or "FALSE" but not both at the same time. For example, A + A = A and not 2A as it would be in normal algebra. Boolean algebra is a simple and effective way of representing the switching action of standard Logic Gates and the basic logic statements which concern us here are given by the logic gate operations of the AND, the OR and the NOT gate functions.

b. The logic AND Function

The Logic AND Function states that two or more events must occur together and at the same time for an output action to occur. But the order at which they occur is unimportant as it does not affect the final result. For example, A & B = B & A. In Boolean algebra the Logic AND Function follows the Commutative Law which allows a change in position of either variable.

The AND function is represented in electronics by the dot or full stop symbol ( . ) Thus a 2-input (A B) AND Gate has an output term represented by the Boolean expression A.B or just AB.

Switch Representation of the AND Function

Here the two switches A and B are connected in series and both Switch A AND Switch B must be closed (Logic "1") in order to put the light on. Then this type of logic gate only produces and output when "ALL" of its inputs are present and in Boolean Algebra terms the output will be TRUE only when all of its inputs are TRUE. In electrical terms, the logic AND function is equal to a series circuit.

As there are only two Switches, each with two possible states "open" or "closed", there are then 4 different ways or combinations of arranging the two switches as shown.

Truth Table

Switch A Switch B Output Description 0 0 0 A and B are both open, lamp OFF 0 1 0 A is open and B is closed, lamp OFF 1 0 0 A is closed and B is open, lamp OFF 1 1 1 A is closed and B is closed, lamp ON Boolean Expression (A AND B) A . B

AND Gates are available as standard i.c. packages such as the common TTL 74LS08 Quadruple 2-input Positive AND Gates, (or the 4081 CMOS equivalent) the TTL 74LS11 Triple 3-input Positive AND Gates or the 74LS21 Dual 4-input Positive AND Gates. AND Gates can also be "cascaded" together to produce circuits with more than just 4 inputs.

c. The Logic NOT Function

The Logic NOT Function is simply a single input inverter that changes the input of a logic level "1" to an output of logic level "0" and vice versa. The logic NOT function is so called because its output state is NOT the same as its input state. It is generally denoted by a bar or overline ( ¯ ) over its input symbol which denotes the inversion operation. As NOT gates perform the logic INVERT or COMPLEMENTATION function they are more commonly known as Inverters because they invert the signal. In logic circuits this negation can be represented by a normally closed switch.

Switch Representation of the NOT Function

If A means that the switch is closed, then NOT A or simply A says that the switch is NOT closed or in other words, it is open. The logic NOT function has a single input and a single output as shown.

Truth Table

Switch Output 1 0 0 1 Boolean Expression Ā

The inversion indicator for a logic NOT function is a "bubble", ( O ) symbol on the output (or input) of the logic elements symbol. In Boolean algebra the Logic NOT Function follows the Complementation Law producing inversion.

NOT gates or Inverters can be used with standard AND and OR gates to produce NAND and NOR gates. Inverters can also be used to produce "Complementary" signals in more complex decoder/logic circuits for example, the complement of logic A is A and a double inversion will give the original value of A.

When designing logic circuits and you need only one or two inverters, but do not have the space or the money for a dedicated Inverter chip such as the 74LS04, you can easily make inverter functions using any spare NAND or NOR gates by simply connecting their inputs together as shown below.

d. The Logic OR Function

The Logic OR Function states that an output action will occur or become TRUE if either one "OR" more events are TRUE, but the order at which they occur is unimportant as it does not affect the final result. For example, A + B = B + A. In Boolean algebra the Logic OR Function follows the Commutative Law the same as for the logic AND function, allowing a change in position of either variable.

The OR function is sometimes called by its full name of "Inclusive OR" in contrast to the Exclusive-OR function we will look at later in tutorial six.

The logic or Boolean expression given for a logic OR gate is that for Logical Addition which is denoted by a plus sign, (+). Thus a 2-input (A B) Logic OR Gate has an output term represented by the Boolean expression of:  A+B = Q.

Switch Representation of the OR Function

Here the two switches A and B are connected in parallel and either Switch A OR Switch B can be closed in order to put the light on. Then this type of logic gate only produces and output when "ANY" of its inputs are present and in Boolean Algebra terms the output will be TRUE when any of its inputs are TRUE. In electrical terms, the logic OR function is equal to a parallel circuit.

Again as with the AND function there are two switches, each with two possible positions open or closed so therefore there will be 4 different ways of arranging the switches.

Truth Table

Switch A Switch B Output Description 0 0 0 Both A and B are open, lamp OFF 0 1 1 A is open and B is closed, lamp ON 1 0 1 A is closed and B is open, lamp ON 1 1 1 A is closed and B is closed, lamp ON Boolean Expression (A OR B) A + B

OR Gates are available as standard i.c. packages such as the common TTL 74LS32 Quadruple 2-input Positive OR Gates. As with the previous AND Gate, OR can also be "cascaded" together to produce circuits with more inputs such as in security alarm systems (Zone A or Zone B or Zone C,etc).

e. The NAND Function

The NAND or Not AND function is a combination of the two separate logical functions, the AND function and the NOT function connected together in series. The logic NAND function can be expressed by the Boolean expression of, A.B

The Logic NAND Function only produces and output when "ANY" of its inputs are not present and in Boolean Algebra terms the output will be TRUE only when any of its inputs are FALSE.

Switch Representation of the NAND Function

The truth table for the NAND function is the opposite of that for the previous AND function because the NAND function performs the reverse function of the AND gate. Then the NAND gate is the complement of the AND gate.

Truth Table

Switch A	Switch B	Output	Description
0	0	1	A and B are both open, lamp ON
0	1	1	A is open and B is closed, lamp ON
1	0	1	A is closed and B is open, lamp ON
1	1	0	A is closed and B is closed, lamp OFF
Boolean Expression (A NAND B)			_____ A . B

The NAND Function is sometimes known as the Sheffer Stroke Function and is denoted by a vertical bar or upwards arrow operator, for example, A NAND B = A|B or A↑B.

NAND Gates are used as the basic "building blocks" to construct other logic gate functions and are available in standard i.c. packages such as the very common TTL 74LS00 Quadruple 2-input NAND Gates, the TTL 74LS10 Triple 3-input NAND Gates or the 74LS20 Dual 4-input NAND Gates. There is even a single chip 74LS30 8-input NAND Gate.

f. The NOR Function

Like the NAND Gate above, the NOR or Not OR Gate is also a combination of two separate functions, the OR function and the NOT function connected together in series and is expressed by the Boolean expression as, A + B

The Logic NOR Function only produces and output when "ALL" of its inputs are not present and in Boolean Algebra terms the output will be TRUE only when all of its inputs are FALSE.

Switch Representation of the NOR Function

The truth table for the NOR function is the opposite of that for the previous OR function because the NOR function performs the reverse function of the OR gate. Then the NOR gate is the complement of the OR gate.

Truth Table

Switch A	Switch B	Output	Description
0	0	1	Both A and B are open, lamp ON
0	1	0	A is open and B is closed, lamp OFF
1	0	0	A is closed and B is open, lamp OFF
1	1	0	A is closed and B is closed, lamp OFF
Boolean Expression (A NOR B)			______ A + B

The NOR Function is sometimes known as the Pierce Function and is denoted by a downwards arrow operator as shown, A NOR B = A↓B.

NOR Gates are available as standard i.c. packages such as the TTL 74LS02 Quadruple 2-input NOR Gate, the TTL 74LS27 Triple 3-input NOR Gate or the 74LS260 Dual 5-input NOR Gate.

g. The Laws of Boolean

As well as the logic symbols "0" and "1" being used to represent a digital input or output, we can also use them as constants for a permanently "Open" or "Closed" circuit or contact respectively. Laws or rules for Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean.

Boolean Algebra uses these "Laws of Boolean" to both reduce and simplify a Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules which are used to define and reduce Boolean expressions. The variables used in Boolean Algebra only have one of two possible values, a "0" and a "1" but an expression can have an infinite number of variables all labeled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.

Examples of these individual Boolean laws, rules and theorems for Boolean Algebra are given in the following table.

Truth Tables for the Laws of Boolean

Boolean Expression	Description	Equivalent Switching Circuit	Boolean Algebra Law or Rule
A + 1 = 1	A in parallel with closed = CLOSED		Annulment
A + 0 = A	A in parallel with open = A		Identity
A . 1 = A	A in series with closed = A		Identity
A . 0 = 0	A in series with open = OPEN		Annulment
A + A = A	A in parallel with A = A		Indempotent
A . A = A	A in series with A = A		Indempotent
NOT Ā = A	NOT NOT A (double negative) = A		Double Negation
A + Ā = 1	A in parallel with not A = CLOSED		Complement
A . Ā = 0	A in series with not A = OPEN		Complement
A+B = B+A	A in parallel with B = B in parallel with A		Commutative
A.B = B.A	A in series with B = B in series with A		Commutative
____       _ A+B = Ā.B	invert and replace OR with AND		de Morgan's Theorem
___         _ A.B = Ā +B	invert and replace AND with OR		de Morgan's Theorem

The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the distributive Law allowing the factoring of an expression, are the same as in ordinary algebra. Each of the laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs to the expression. The above laws can be used to prove any given Boolean expression and for simplifying complicated digital circuits. A brief description of the Laws of Boolean is given below.

h. Description of the Laws and Theorems

• Annulment Law - A term AND´ed with a "0" equals 0 or OR´ed with a "1" will equal 1.   
• A . 0 = 0,    A variable AND'ed with 0 is always equal to 0.
• A + 1 = 1,    A variable OR'ed with 1 is always equal to 1.

• Identity Law - A term OR´ed with a "0" or AND´ed with a "1" will always equal that term.   
• A + 0 = A,   A variable OR'ed with 0 is always equal to the variable.
• A . 1 = A,    A variable AND'ed with 1 is always equal to the variable.

• Indempotent Law - An input AND´ed with itself or OR´ed with itself is equal to that input.
• A + A = A,    A variable OR'ed with itself is always equal to the variable.
• A . A = A,    A variable AND'ed with itself is always equal to the variable.

•  Complement Law - A term AND´ed with its complement equals "0" and a term OR´ed with its complement equals "1".
•  A . Ā = 0,    A variable AND'ed with its complement is always equal to 0.
• A + Ā = 1,    A variable OR'ed with its complement is always equal to 1.

• Commutative Law - The order of application of two separate terms is not important.   
• A . B = B . A,    The order in which two variables are AND'ed makes no difference.
• A + B = B + A,    The order in which two variables are OR'ed makes no difference.

• Double Negation Law - A term that is inverted twice is equal to the original term    
•  _        
• Ā = A,     A double complement of a variable is always equal to the variable.

• de Morgan´s Theorem - There are two "de Morgan´s" rules or theorems,
• (1) Two separate terms NOR´ed together is the same as the two terms inverted 
• (Complement) and AND´ed                                                                                           for example, A+B = Ā. B.
• (2) Two separate terms NAND´ed together is the same as the two terms inverted
•  (Complement) and OR´ed                                                                                                  for example, A.B = Ā +B.

Other algebraic laws not detailed above include:

• Distributive Law - This law permits the multiplying or factoring out of an expression.
• Absorptive Law - This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.
• Associative Law - This law allows the removal of brackets from an expression and regrouping of the variables.

Boolean Algebra Functions

Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table.

Function	Description	Expression
1.	NULL	0
2.	IDENTITY	1
3.	Input A	A
4.	Input B	B
5.	NOT A	_ A
6.	NOT B	_ B
7.	A AND B (AND)	A . B
8.	A AND NOT B	_ A . B
9.	NOT A AND B	Ā . B
10.	NOT A AND NOT B (NAND)	_____ A . B
11.	A OR B (OR)	A + B
12.	A OR NOT B	_ A + B
13.	NOT A OR B	Ā + B
14.	NOT OR (NOR)	_____ A + B
15.	Exclusive-OR	A.B ⊕ A.B
16.	Exclusive-NOR	________ A.B ⊕ A.B

Example No1

Using the above laws, simplify the following expression:  (A + B)(A + C)

Q =	(A + B)(A + C)
	AA + AC + AB + BC	- Distributive law
	A + AC + AB + BC	- Identity AND law (A.A = A)
	A(1 + C) + AB + BC	- Distributive law
	A.1 + AB + BC	- Identity OR law (1 + C = 1)
	A(1 + B) + BC	- Distributive law
	A.1 + BC	- Identity OR law (1 + B = 1)
Q =	A + BC	- Identity AND law (A.1 = A)

Then the expression:  (A + B)(A + C) can be simplified to A + BC

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