To do this, evaluate the expression, following proper mathematical order of operations multiplication before addition, operations inside parentheses before anything else , and draw gates for each step. Obviously, this circuit is much simpler than the original, having only two logic gates instead of five. Such component reduction results in higher operating speed less delay time from input signal transition to output signal transition , less power consumption, less cost, and greater reliability.

Electromechanical relay circuits, typically being slower, consuming more electrical power to operate, costing more, and having a shorter average life than their semiconductor counterparts, benefit dramatically from Boolean simplification. As before, our first step in reducing this circuit to its simplest form must be to develop a Boolean expression from the schematic. For example, examine the following resistor network with its resistors arranged in the same connection pattern as the relay contacts in the former circuit, and corresponding total resistance formula:.

Remember that parallel contacts are equivalent to Boolean addition, while series contacts are equivalent to Boolean multiplication.

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Write a Boolean expression for this relay contact circuit, following the same order of precedence that you would follow in reducing a series-parallel resistor network to a total resistance. Now that we have a Boolean expression to work with, we need to apply the rules of Boolean algebra to reduce the expression to its simplest form simplest defined as requiring the fewest relay contacts to implement :.

As you can see, the reduced circuit is much simpler than the original, yet performs the same logical function:. Load More Articles. Published under the terms and conditions of the Design Science License. Home Textbook Vol. Digital signals can be easily stored in an electronic memory and can be easily understood by digital microprocessors.

Examples of engineering problems in digital electronics are: how to efficiently perform arithmetic operations with digital signals on a microprocessor, how to communicate data without losing information, and how to design a reusable reconfigurable digital processor. Career choices are abundant in locations around the world. Commonly advertised positions include: digital electronics engineer, digital circuit design engineer and digital integrated circuit design engineer.

After that, engineers always used the cheapest available electronic switches that could still fulfill the requirements. The earliest integrated circuits were a happy accident. They were constructed not to save money, but to save weight, and permit the Apollo Guidance Computer to control an inertial guidance system for a spacecraft. To everyone's surprise, by the time the circuits were mass-produced, they had become the least-expensive method of constructing digital logic.

Improvements in this technology have driven all subsequent improvements in cost. With the rise of integrated circuits , reducing the absolute number of chips used represented another way to save costs. The goal of a designer is not just to make the simplest circuit, but to keep the component count down.

## Basic Gates and Functions

Sometimes this results in slightly more complicated designs with respect to the underlying digital logic but nevertheless reduces the number of components, board size, and even power consumption. For example, in some logic families, NAND gates are the simplest digital gate to build. All other logical operations can be implemented by NAND gates.

If a circuit already required a single NAND gate, and a single chip normally carried four NAND gates, then the remaining gates could be used to implement other logical operations like logical and. This could eliminate the need for a separate chip containing those different types of gates. The "reliability" of a logic gate describes its mean time between failure MTBF. Digital machines often have millions of logic gates. Also, most digital machines are "optimized" to reduce their cost. The result is that often, the failure of a single logic gate will cause a digital machine to stop working.

## Science for Kids: Digital Electronics

Digital machines first became useful when the MTBF for a switch got above a few hundred hours. Even so, many of these machines had complex, well-rehearsed repair procedures, and would be nonfunctional for hours because a tube burned-out, or a moth got stuck in a relay. The "fan out" describes how many logic inputs can be controlled by a single logic output.

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The minimum practical fan out is about five. Modern electronic logic using CMOS transistors for switches have fanouts near fifty, and can sometimes go much higher. The "switching speed" describes how many times per second an inverter an electronic representation of a "logical not" function can change from true to false and back. Faster logic can accomplish more operations in less time.

Digital logic first became useful when switching speeds got above fifty hertz , because that was faster than a team of humans operating mechanical calculators. Modern electronic digital logic routinely switches at five gigahertz 5x10 9 hertz , and some laboratory systems switch at more than a terahertz 1x10 12 hertz.

It is possible to construct non-electronic digital mechanisms. In principle, any technology capable of representing discrete states and representing logic operations could be used to build mechanical logic.

### Digital Logic Fundamentals

Danny Hillis , co-author of The Connection Machine , once built a working computer from Tinker Toys , string, a brick, and a sharpened pencil, which is supposed to be in the Houston Museum of Science. Hydraulic, pneumatic and mechanical versions of logic gates exist and are used in situations where electricity cannot be used. The first two types are considered under the heading of fluidics. One application of fluidic logic is in military hardware that is likely to be exposed to a nuclear electromagnetic pulse nuclear EMP, or NEMP that would destroy any electrical circuits.

Mechanical logic is frequently used in inexpensive controllers, such as those in washing machines. Famously, the first computer design, by Charles Babbage , was designed to use mechanical logic. Mechanical logic might also be used in very small computers that could be built by nanotechnology. Another example is that if two particular enzymes are required to prevent the construction of a particular protein, this is the equivalent of a biological "NAND" gate.

The term digital signal is used to refer to more than one concept. It can refer to discrete-time signals that are digitized, or to the waveform signals in a digital system. Template:Main Digital signals are digital representations of discrete-time signals , which are often derived from analog signals. An analog signal is a datum that changes over time—say, the temperature at a given location; the depth of a certain point in a pond; or the amplitude of the voltage at some node in a circuit—that can be represented as a mathematical function, with time as the free variable abscissa and the signal itself as the dependent variable ordinate.

A discrete-time signal is a sampled version of an analog signal: the value of the datum is noted at fixed intervals for example, every microsecond rather than continuously.

## The Beginner’s Guide to Digital Electronics

If individual time values of the discrete-time signal, instead of being measured precisely which would require an infinite number of digits , are approximated to a certain precision—which, therefore, only requires a specific number of digits—then the resultant data stream is termed a digital signal.

The process of approximating the precise value within a fixed number of digits, or bits, is called quantization. In conceptual summary, a digital signal is a quantized discrete-time signal; a discrete-time signal is a sampled analog signal. In the Digital Revolution , the usage of digital signals has increased significantly. Many modern media devices, especially the ones that connect with computers use digital signals to represent signals that were traditionally represented as continuous-time signals ; cell phones, music and video players, personal video recorders, and digital cameras are examples.

In most applications, digital signals are represented as binary numbers , so their precision of quantization is measured in bits. Suppose, for example, that we wish to measure a signal to two significant decimal digits. Since seven bits, or binary digits, can record discrete values viz. Digital signal: 1 Low level, 2 High level, 3 Rising edge, and 4 Falling edge. In computer architecture and other digital systems, a waveform that switches between two voltage levels representing the two states of a Boolean value 0 and 1 is referred to as a digital signal , even though it is an analog voltage waveform, since it is interpreted in terms of only two levels.

The clock signal is a special digital signal that is used to synchronize digital circuits.

The image shown can be considered the waveform of a clock signal. Logic changes are triggered either by the rising edge or the falling edge. The two states of a wire are usually represented by some measurement of electric current : Voltage is the most common, but current is used in some logic families. A threshold is designed for each logic family.

When below that threshold, the wire is "low," when above "high. The circuits avoid that area, in order to avoid indeterminate results.