SWITCHES ~ Lessons In Electric Circuits -- Volume IV Chapter 4

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Switch types

An electrical switch is any device used to interrupt the flow of electrons in a circuit. Switches are essentially binary devices: they are either completely on ("closed") or completely off ("open"). There are many different types of switches, and we will explore some of these types in this chapter.
Though it may seem strange to cover this elementary electrical topic at such a late stage in this book series, I do so because the chapters that follow explore an older realm of digital technology based on mechanical switch contacts rather than solid-state gate circuits, and a thorough understanding of switch types is necessary for the undertaking. Learning the function of switch-based circuits at the same time that you learn about solid-state logic gates makes both topics easier to grasp, and sets the stage for an enhanced learning experience in Boolean algebra, the mathematics behind digital logic circuits.
The simplest type of switch is one where two electrical conductors are brought in contact with each other by the motion of an actuating mechanism. Other switches are more complex, containing electronic circuits able to turn on or off depending on some physical stimulus (such as light or magnetic field) sensed. In any case, the final output of any switch will be (at least) a pair of wire-connection terminals that will either be connected together by the switch's internal contact mechanism ("closed"), or not connected together ("open").
Any switch designed to be operated by a person is generally called a hand switch, and they are manufactured in several varieties:


ELECTROMECHANICAL RELAYS ~ Lessons In Electric Circuits -- Volume IV Chapter 5

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ELECTROMECHANICAL RELAYS

Relay construction

An electric current through a conductor will produce a magnetic field at right angles to the direction of electron flow. If that conductor is wrapped into a coil shape, the magnetic field produced will be oriented along the length of the coil. The greater the current, the greater the strength of the magnetic field, all other factors being equal:

Inductors react against changes in current because of the energy stored in this magnetic field. When we construct a transformer from two inductor coils around a common iron core, we use this field to transfer energy from one coil to the other. However, there are simpler and more direct uses for electromagnetic fields than the applications we've seen with inductors and transformers. The magnetic field produced by a coil of current-carrying wire can be used to exert a mechanical force on any magnetic object, just as we can use a permanent magnet to attract magnetic objects, except that this magnet (formed by the coil) can be turned on or off by switching the current on or off through the coil.

BOOLEAN ALGEBRA ~ Lessons In Electric Circuits -- Volume IV Chapter 7

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BOOLEAN ALGEBRA

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
Rules of addition for Boolean quantities

Introduction

Mathematical rules are based on the defining limits we place on the particular numerical quantities dealt with. When we say that 1 + 1 = 2 or 3 + 4 = 7, we are implying the use of integer quantities: the same types of numbers we all learned to count in elementary education. What most people assume to be self-evident rules of arithmetic -- valid at all times and for all purposes -- actually depend on what we define a number to be.
For instance, when calculating quantities in AC circuits, we find that the "real" number quantities which served us so well in DC circuit analysis are inadequate for the task of representing AC quantities. We know that voltages add when connected in series, but we also know that it is possible to connect a 3-volt AC source in series with a 4-volt AC source and end up with 5 volts total voltage (3 + 4 = 5)! Does this mean the inviolable and self-evident rules of arithmetic have been violated? No, it just means that the rules of "real" numbers do not apply to the kinds of quantities encountered in AC circuits, where every variable has both a magnitude and a phase. Consequently, we must use a different kind of numerical quantity, or object, for AC circuits (complex numbers, rather than real numbers), and along with this different system of numbers comes a different set of rules telling us how they relate to one another.

Lessons In Electric Circuits -- Volume IV Chapter 8

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KARNA UGH MAPPING

Original author: Dennis Crunkilton




Introduction

Why learn about Karnaugh maps? The Karnaugh map, like Boolean algebra, is a simplification tool applicable to digital logic. See the "Toxic waste incinerator" in the Boolean algebra chapter for an example of Boolean simplification of digital logic. The Karnaugh Map will simplify logic faster and more easily in most cases.
Boolean simplification is actually faster than the Karnaugh map for a task involving two or fewer Boolean variables. It is still quite usable at three variables, but a bit slower. At four input variables, Boolean algebra becomes tedious. Karnaugh maps are both faster and easier. Karnaugh maps work well for up to six input variables, are usable for up to eight variables. For more than six to eight variables, simplification should be by CAD (computer automated design).

Lessons In Electric Circuits -- Volume IV Chapter 9

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COMBINATION LOGIC FUNCTIONS

Introduction

The term "combinational" comes to us from mathematics. In mathematics a combination is an unordered set, which is a formal way to say that nobody cares which order the items came in. Most games work this way, if you rolled dice one at a time and get a 2 followed by a 3 it is the same as if you had rolled a 3 followed by a 2. With combinational logic, the circuit produces the same output regardless of the order the inputs are changed.
There are circuits which depend on the when the inputs change, these circuits are called sequential logic. Even though you will not find the term "sequential logic" in the chapter titles, the next several chapters will discuss sequential logic.
Practical circuits will have a mix of combinational and sequential logic, with sequential logic making sure everything happens in order and combinational logic performing functions like arithmetic, logic, or conversion.
You have already used combinational circuits. Each logic gate discussed previously is a combinational logic function. Let's follow how two NAND gate works if we provide them inputs in different orders.
We begin with both inputs being 0.

We then set one input high.

We then set the other input high.

So NAND gates do not care about the order of the inputs, and you will find the same true of all the other gates covered up to this point (AND, XOR, OR, NOR, XNOR, and NOT).

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Lessons In Electric Circuits -- Volume IV Chapter 10

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MULTI VIBRATORS

Digital logic with feedback

With simple gate and combinational logic circuits, there is a definite output state for any given input state. Take the truth table of an OR gate, for instance:

For each of the four possible combinations of input states (0-0, 0-1, 1-0, and 1-1), there is one, definite, unambiguous output state. Whether we're dealing with a multitude of cascaded gates or a single gate, that output state is determined by the truth table(s) for the gate(s) in the circuit, and nothing else.
However, if we alter this gate circuit so as to give signal feedback from the output to one of the inputs, strange things begin to happen:

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Lessons In Electric Circuits -- Volume IV Chapter 11

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SEQUENTIAL CIRCUITS COUNTERS

Binary count sequence

If we examine a four-bit binary count sequence from 0000 to 1111, a definite pattern will be evident in the "oscillations" of the bits between 0 and 1:

Note how the least significant bit (LSB) toggles between 0 and 1 for every step in the count sequence, while each succeeding bit toggles at one-half the frequency of the one before it. The most significant bit (MSB) only toggles once during the entire sixteen-step count sequence: at the transition between 7 (0111) and 8 (1000).
If we wanted to design a digital circuit to "count" in four-bit binary, all we would have to do is design a series of frequency divider circuits, each circuit dividing the frequency of a square-wave pulse by a factor of 2:

J-K flip-flops are ideally suited for this task, because they have the ability to "toggle" their output state at the command of a clock pulse when both J and K inputs are made "high" (1):

If we consider the two signals (A and B) in this circuit to represent two bits of a binary number, signal A being the LSB and signal B being the MSB, we see that the count sequence is backward: from 11 to 10 to 01 to 00 and back again to 11. Although it might not be counting in the direction we might have assumed, at least it counts!
The following sections explore different types of counter circuits, all made with J-K flip-flops, and all based on the exploitation of that flip-flop's toggle mode of operation.
  • REVIEW:
  • Binary count sequences follow a pattern of octave frequency division: the frequency of oscillation for each bit, from LSB to MSB, follows a divide-by-two pattern. In other words, the LSB will oscillate at the highest frequency, followed by the next bit at one-half the LSB's frequency, and the next bit at one-half the frequency of the bit before it, etc.
  • Circuits may be built that "count" in a binary sequence, using J-K flip-flops set up in the "toggle" mode.

 

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