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The Design and Implementation of the Ripple-Carry Adder: a Review of the Fundamentals of Digital Electronics

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The Design and Implementation of the Ripple-Carry Adder: a Review of the Fundamentals of Digital Electronics The Design and Implementation of the Ripple-Carry Adder: a Review of the Fundamentals of Digital Electronics Michael Stirchak June 1, 2016 Abstract To gain an understanding of digital electronics, I designed and implemented a 4-bit, ripple-carry adder using discrete n-type MOSFET's. To accomplish this, I first studied the design of bipolar junction and field-effect transistors to understand the physics of electronic switching. Next, I reviewed the principles of Boolean algebra in order to develop logic gates. Using these gates, I was able to design and build a digital circuit capable of binary addition under the 2's complement encoding scheme. 1 Contents 1 Introduction 3 2 Transistor Design 4 2.1 The Bipolar Junction Transistor . .5 2.2 The Field-Effect Transistor . .6 2.3 Transistor Comparison for Digital Electronics . .9 3 Boolean Algebra 10 3.1 Logical Operations and Truth Tables . 10 3.1.1 Example: The XOR Gate . 12 3.1.2 Example: Implementation of the NAND gate using MOSFET's . 14 3.2 Binary Encoding: the 2's Complement Method . 15 3.2.1 Limitations of 2's Complement . 17 4 The Ripple-Carry Adder 17 4.1 Design of the Adder . 18 4.1.1 Binary Addition . 18 4.1.2 The 1-Bit Adder . 19 4.1.3 The 4-Bit Adder . 21 4.1.4 Overflow Detection . 22 4.2 Implementation of the Adder . 24 5 Conclusion 26 6 Acknowledgments 26 Appendices 27 A Appendix I: Transistor-Level Implementations of Logic Gates 27 B Appendix II: Laws of Boolean Algebra 29 C Appendix III: Transistor-Level Diagrams of the Adder 30 D Appendix IV: Technologies and Materials Used 31 Bibliography 33 2 1 Introduction The 20th century has been called the Quantum Century for the breathtaking advances made in the field of physics during that time. The quality and breadth of these break- throughs, particularly in the latter half of the 20th century, were due in no small part to the advent of the digital computer. Computers allowed physicists to tackle problems that had previously been deemed unsolvable, and also gave these scientists an invaluable tool with which to attack future problems. Today, there is no single tool more indispensable to physics research than the digital computer. The theory and construction of computers, then, should be of great interest to physicists. Computers have been around in some form or another for thousands of years. Ancient computers like the Greek Antikythera were intricate, but purely mechanical; more recent computers employed bulky vacuum tubes to regulate the flow of electric signals in a produc- tive manner; finally, almost all modern computers contain vast arrays of transistors capable of performing billions of calculations every second. A common theme in most of these devices is the digital nature of their inputs and outputs: they work with signals that are either high (on) or low (off). Transistors in particular have proven themselves to be indispensable to the construction of digital circuits, as they function as highly-efficient electronic switches. Indeed, the overall goal of this paper is to explore how transistors can be used to perform some logical and arithmetic operations on electronic signals. Although transistors are the basis of almost all components in a modern computer, in- cluding memory, storage, and processors, this paper will focus solely on the construction of a simple arithmetic logic unit capable of binary addition. To that end, I will first study the design of bipolar junction and field-effect transistors to understand the physics behind electronic switching. Next, to explore the mathematical basis for logical and arithmetic oper- ations, an overview of Boolean algebra will be given. Finally, with this theoretical foundation I will design and implement a 4-bit, ripple-carry adder. 3 Figure 1: Diagram of a discrete npn-type bipolar junction transistor. As current flows into the base, the positively-charged holes are attracted to the emitter region, causing electrons to pass to the collector. This results in a steady current from collector to emitter. 2 Transistor Design Prior to 1947, the dominant switching component in electronics was the vacuum tube. These devices performed their job adequately, but they were by nature large, cumbersome to use, prone to failure, and inefficient in terms of power consumption. Physicists and engineers, recognizing the growing importance of these devices, sought to improve on them by making a smaller, safer, and more efficient electronic switch. The result of these efforts were realized in December 1947, when Bell Laboratories created the first bipolar junction transistor [1]. The BJT was compact, efficient, and most of all easy to manufacture, and as a result they quickly replaced vacuum tubes in electronics. Research on transistors did not stop there, however, and after a few more years the first field-effect transistors were being manufactured as well. Today, most digital circuits use FET's instead of BJT's, for reasons I shall discuss. Nevertheless, both types of transistors can be used in digital circuits and are worth discussing. 4 2.1 The Bipolar Junction Transistor The bipolar junction transistor was the first transistor to be widely used in industry. Its general structure, as seen in Figure 1, consists of the collector, base, and emitter regions. There are two distinct varieties of BJT's: npn and pnp. npn transistors consist of a collector doped with electrons, an emitter heavily doped with electrons so that it is more negatively charged than the collector, and a positively-charged base region sandwiched between them. pnp transistors are just the opposite, with a negatively-doped base in between the positively- doped collector and emitter regions. In either case, the BJT effectively consists of two pn- junctions which allows it to act as a current-controlled diode [2]. If, for example, there is no current IB running into the base of the transistor, there is effectively a pn-junction between the collector and emitter that prevents current from flowing between them. However, if IB is increased, the positive charge-carrying holes flowing into the base will be attracted to the emitter, causing the emitter electrons to flow into the base and emitter. This results in a controllable current flow from collector to emitter. From Figure 1 it is clear that the collector current is dependent on the base current. Exactly what relationship the two currents have depends on the specific type of the BJT; in general however, IC is directly proportional to IB and a unit-less variable β, which is usually around 50-100 [2]. In other words, IC = β · IB: This relationship makes BJT's phenomenally useful as current amplifiers, as a small base current can be used to drive a much larger collector current [2]. However, the exact value of β depends on the base current, the transistor's temperature, and a few other factors, meaning that IC may be inconsistent during continuous operation. Usually though, a BJT is used an an amplifier when the exact output current does not matter, so long as it is sufficiently large. There are limits on the amplifying behavior of BJT's. To understand these limits, it 5 is useful to study BJT's not only as current-driven driven devices, but also as voltage- driven devices. If we define VC , VE, and VB to be the voltages at the three terminals of the transistor, we can define the emitter current in terms of the voltage difference at the base-emitter junction: VBE V IE = IS(e T − 1); where VT = kT=e; k being Boltzmann's Constant, T being the transistor's temperature, and e being the charge of an electron. IS is the reverse-saturation current across the base- emitter junction (leakage current), and is usually in the pA-fA range. This is known as the Ebers-Moll equation, and it is the mathematical foundation for the behavior of BJT's [3]. From it, we can see that the emitter current will increase as VBE increases, but only until the transistor becomes saturated. Also, an increase in temperature could cause significant change in emitter current. These will be important quantities to keep in mind as other types of transistors are introduced. These two relationships, IC = β · IB and the Ebers-Moll equation, represent two separate and equally valid ways of analyzing the behavior of bipolar junction transistors. For digital electronics, it is usually preferable to take the current-driven view of BJT's because the relationship between IB and IC is relatively linear. In the voltage-driven view, current depends exponentially on VBE, which is needlessly complex for a circuit whose signals need only be \high" or \low." Having established that the BJT is a current-driven device for our purposes, it will be interesting to begin an analysis of the design of the field-effect transistor. 2.2 The Field-Effect Transistor The design of the field-effect transistor is in many ways less complex than the bipolar junction transistor, but also more subtle. Similar to the BJT, the FET consists of three terminals, now called the gate, source, and drain (see Figure 2). Also like the BJT, FET's come in two flavors, the n-type FET and p-type FET. In both cases, the drain and source are 6 Figure 2: Diagram of an n-type Metal Oxide Semiconductor Field-Effect Transistor (MOS- FET). The main body is a positively-doped silicon substrate, while the drain and source are negatively-doped. As a voltage is applied across the gate, positive charge accumulates on the terminal's plate, attracting electrons from the body and forming the conductive channel.
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