1 Signals and systems, A. V. Oppenhaim, A. S. Willsky, Prentice Hall, 2 nd edition, FUNDAMENTALS. Electrical Engineering. 2.

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1 1 Signals and systems, A. V. Oppenhaim, A. S. Willsky, Prentice Hall, 2 nd edition, FUNDAMENTALS Electrical Engineering 2.Processing - Analog data An analog signal is a signal that varies continuously. The name derives from the fact that such a signal is analogous to the physical signal that it represents. The magnitude of an analog signal can take on any value; that is, the amplitude of an analog signal exhibits a continuous variation over its range of activity. The vast majority of signals in the world around us are analog. From a conceptual point of view the simplest signal-processing task is that of signal amplification [3]. The need for amplification arises because transducers provide signals that are said to be weak, that is, in the microvolt or millivolt range and possessing little energy. Such signals are too small for reliable processing, and processing is much easier if the signal magnitude is made larger. The functional block that accomplishes this task is the signal amplifier. The band of frequencies over which the gain of the amplifier is almost constant, is called the amplifier bandwidth. Normally the amplifier is designed so that its bandwidth coincides, or it is wider, than the spectrum of the signals it is required to amplify. If this were not the case, the amplifier would distort the frequency spectrum of the input signal, with different components of the input signal being amplified by different amounts. Amplification of signals is possible thanks to electronic active devices known as transistors. There are two basic transistor devices: The bipolar junction Transistor (BJT) and the Field-Effect Transistor (FET). Although the two devices are physically quite different, the basic principle involved in both devices is the use of the voltage between two terminals to control the current flowing in the third terminal. In this way a three-terminal device can be used to realize a controlled source, which is the basis for amplifier design.

2 2 Bipolar Junction Transistor Field-Effect Transistor For a BJT, the voltage across the base-emitter (B-E) junction controls the current flow from collector (C) to emitter (E). In particular, when the B-E voltage is below 0.6 V, the BJT is turned off, so that no current can flow from C to E. On the contrary, when the B-E voltage reaches 0.6V, the BJT turns on, so that a high current, with no voltage drop, can flow from C to E. In a FET, the voltage drop between the gate terminal and the source terminal controls the resistance between drain and source: when the gate voltage differs from the source voltage from a quantity below a well-defined threshold (typically 1 V), the resistance between drain and source contacts is very high, so that no current flows between these two terminals even if the drain is at a higher voltage tan the source. Vice versa, when the voltage difference between gate ands source raises above the FET threshold, the resistance between drain and source is low so that current can easily flow through the FET. Typically, only one transistor is not enough to produce an adequate amplification of a signal. This is also because the amplifier circuit must exhibit an input and output resistance that is well suited with respect to other parts of a circuit. For these reasons, in most cases a quite complex circuit, featuring different transistors, is employed for signal amplification. A notable example is constituted by the operation amplifier (op amp). The op amp is a DC-coupled high-gain electronic voltage amplifier with differential inputs and, usually, a single output. Note that the term DCcoupled means that even the DC (direct) component of the input signal is amplified. This is in contrast with AC-coupled circuits, in which only the AC component of the input signal is amplified, whereas its DC part is filtered out. In its ordinary usage, the output of the op-amp is controlled by negative feedback which, because of the amplifier's high gain, almost completely determines the output voltage for any given

3 3 input. In other words, the response of an op amp with feedback is typically determined entirely by its feedback network. This is of very importance with respect to reliability issues of the circuit. Operation Amplifier Although the op amp internal circuit is complex, typically incorporating 20 or more transistors, its almost ideal terminal behaviour makes it possible to treat the op amp as a circuit element and to use it in the design of powerful circuits without any knowledge of its internal construction. In this context, an ideal behaviour means a very high gain, a very high input resistance, and a very low output resistance. Since the appearance of the first integrated-circuit (IC) op amp in the mid-1960s, electronic engineers started using op amps in large quantities. One of the reasons for the popularity of the op amp is its versatility: With only a handful of external components forming the feedback network, op amps can be made to perform a wide variety of analog signal processing tasks. Equally important is the fact that the IC op amp has characteristics that closely approach the assumed ideal. This implies that it is quite easy to design circuits using the IC op amp. The latter are widely available as DIP-8 or DIP-16 integrated circuits, in which two of the IC pins are dedicated to power supply (a dual supply is typically needed for op amp operation). ICs typically contain more than one op amps, (dual as well as quad configurations are easily found), making it easier to realize complex circuits involving more than one op amp. - Amplifiers - Circuit models for Amplifiers - Frequency response of Amplifiers - The Bipolar Junction Transistor (BJT)

4 4 - The Field-Effect Transistor (FET) - The transistor as an amplifier - Basic Single-Stage BJT and FET Amplifier Configurations - Operational Amplifiers - The Op-Amp terminals - The Ideal Op Amp - Feedback principles: application to Operational Amplifiers The use of feedback is essential and extensively employed in electronics circuits. Schematics of feedback The feedback is achieved by sampling the output y of the system A, and reporting the processed output By to the input of the same system A. The basic idea behind feedback is to trade-off gain (which can be very large, especially for operation amplifiers) with some other desired characteristics: for example feedback can result in changes in input/output resistance, better stability, increase of bandwidth, and reduction of nonlinearity. Typically, these benefits are achieved by negative feedback: In negative feedback, the output of a circuit (for example the output of an operation amplifier) is sampled, and part of it is subtracted from the input. In this way, the original circuit (i.e. the open-loop network) will see a reduced input with respect to the external one. - Negative Feedback and Positive Feedback - Amount of feedback

5 5 - Benefits of Feedback in Electronic Applications - Case study: Feedback in Operation Amplifiers Operational amplifier are usually operated with feedback: as already explained, the large gain of the op amp in open loop configuration (i.e., without feedback), is traded off with other benefits, such as stability, bandwidth and noire rejection. There are two basic closed-loop configurations: the inverting configuration and the noninverting configuration. Let us consider the inverting configuration. The basic circuit topology is shown in the figure below. Op amp inverting configuration Starting from the assumption that the input impedance of the op amp is infinite, and that there exists a sort of virtual short circuit between the two inputs of the op amp, it is easy to show that the transfer function of the above configuration is given by: V R = 2 out V in R1 Note that the gain of the op amp is entirely defined by the resistance values, so that it can be controlled very precisely by choosing high-precision resistances. Note also that the output is inverted with respect to the input signal, that s why we talk about inverting configuration. Obviously, the above relation will be valid as long as the output voltage does not exceed (in absolute value) the saturation voltage of the amplifier. The latter is typically very close to the supply voltage. Another basic assumption behind the above relation is that the input voltage signal has a bandwidth

6 6 that is lower tan the bandwidth of the amplifier. Note the latter depends on the gain in an inverse fashion, so that higher gains can be achieved at the expense of a reduced bandwidth. The inverting configuration can be used to implement a very large number of functions, with only slight modifications with respect to the basic configuration shown above. For instance, it is possible to realize a summing amplifier by simply applying the input signals to the inverting input of the op amp, by means of opportune resistances. Summing amplifier In this case, it is easy to show that the output will be given by: V out R0 R0 = V1 + V2 R1 R2 Hence, by choosing opportunely the various resistances, it is possible to realize a weighted sum of the various input signals. The basic noninverting configuration is shown below: Op amp noninverting configuration

7 7 In this case, it is easy to show that the output voltage is related to the input voltage by the relation: V R R 1 2 out = + V in R1 Note that in this case the output voltage is in phase with the input voltage, yet the output voltage is subtracted from the input one, so that negative feedback is achieved also in this case. - Analysis of Circuits containing Ideal Op Amps The inverting Configuration The Inverting Configuration with General Impedances The Inverting Integrator The Inverting Differentiator The Weighed Summer - The Noninverting Configuration - Effect of Finite Open-Loop Gain and Bandwidth on Circuit Performance - ADC and DAC Converters An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic circuit that converts continuous signals to discrete digital numbers [4]. The reverse operation is performed by a digital-to-analog converter (DAC). Such operations are necessary whenever we want to transmit a signal in its digital form, or we want to process it by using digital circuits. Typically, an ADC is an electronic device that converts an input analog voltage to a sequence of digital numbers. The digital output may be using different coding schemes, such as binary and two's complement binary. However, some non-

8 8 electronic or only partially electronic devices, such as rotary encoders, can also be considered ADCs. Analog-to-digital converter Conversion from analog to digital is always accompanied by some form of noise, or error. Quantization error (assuming the ADC is intended to be linear) intrinsic to any analog-to-digital conversion and is related to the fact that we represent an analog signal, which potentially can assume any value within an opportune range, with a discrete quantity belonging to a finite set of values. These errors are measured in a unit called the LSB, which is an abbreviation for least significant bit. In case of an eight-bit ADC, an error of one LSB is 1/256 of the full signal range, or about 0.4%. These are the most common ways of implementing an electronic ADC: a) A direct conversion ADC or flash ADC has a comparator that fires for each decoded voltage range. In other words, the input signal to be converted is simultaneously compared to different, fixed voltage values. At the output of each comparator, there will be a low or high signal depending whether the input signal is higher or lower than the corresponding reference signal. The comparator bank feeds a logic circuit that, on the basis of the results of any comparison, generates a code for each voltage range. The following illustration shows a 3-bit flash ADC circuit:

9 9 Flash ADC converter Vref is a stable reference voltage provided by a precision voltage regulator as part of the converter circuit, not shown in the schematic. As the analog input voltage exceeds the reference voltage at each comparator, the comparator outputs will sequentially saturate to a high state. The priority encoder generates a binary number based on the highest-order active input, ignoring all other active inputs. When operated, the flash ADC produces an output that looks something like this: Operation of a flash ADC converter

10 10 Direct conversion is very fast, but usually has only 8 bits of resolution (256 comparators) or fewer, as it needs a large, expensive circuit. ADCs of this type have a large die size, a high input capacitance, and are prone to produce glitches on the output. They are often used for video or other fast signals. b) A successive-approximation ADC features a very special counter circuit known as a successive-approximation register. Instead of counting up in binary sequence, this register counts by trying all values of bits starting with the most-significant bit and finishing at the least-significant bit. Throughout the count process, the register monitors the comparator's output to see if the binary count is less than or greater than the analog signal input, adjusting the bit values accordingly. The advantage to this counting strategy is much faster results: the DAC output converges on the analog signal input in much larger steps than with the 0-to-full count sequence of a regular counter. Without showing the inner workings of the successive-approximation register (SAR), the circuit looks like this: SAR A/D converter It should be noted that the SAR is generally capable of outputting the binary number in serial (one bit at a time) format, thus eliminating the need for a shift register. Plotted over time, the operation of a successive-approximation ADC looks like this:

11 11 Opearation of a SAR ADC converter Note how the updates for this ADC occur at regular intervals, unlike the digital ramp ADC circuit. c) In a tracking ADC, Instead of a regular "up" counter driving the DAC, this circuit uses an up/down counter. The counter is continuously clocked, and the up/down control line is driven by the output of the comparator. So, when the analog input signal exceeds the DAC output, the counter goes into the "count up" mode. When the DAC output exceeds the analog input, the counter switches into the "count down" mode. Either way, the DAC output always counts in the proper direction to track the input signal. Tracking ADC Notice how no shift register is needed to buffer the binary count at the end of a cycle. Since the counter's output continuously tracks the input (rather than counting to meet

12 12 the input and then resetting back to zero), the binary output is legitimately updated with every clock pulse. An advantage of this converter circuit is speed, since the counter never has to reset. Note the behavior of this circuit: Operation of a tracking ADC Note the much faster update time than any of the other "counting" ADC circuits. Also note how at the very beginning of the plot where the counter had to "catch up" with the analog signal, the rate of change for the output was identical to that a digital ramp ADC. Also, with no shift register in this circuit, the binary output would actually ramp up rather than jump from zero to an accurate count as it did with the counter and successive approximation ADC circuits. Perhaps the greatest drawback to this ADC design is the fact that the binary output is never stable: it always switches between counts with every clock pulse, even with a perfectly stable analog input signal. This phenomenon is informally known as bit bobble, and it can be problematic in some digital systems. - Resolution - Accuracy - Sampling rate - ADC basic sctructures - Digital Logic Elements Digital logic elements are the basic circuits that make up all digital equipment. These are the circuits that are used to process the binary data. The logic element has one or more binary data inputs that are to be processed. The logic element processes

13 13 and manipulates the binary input signals in a fixed way and generates an appropriate output signal. The output is a function of the binary states of inputs and the unique processing capability of the logic element. The basilar digital logic element is the inverter, which provides an output digital signal that is inverted with respect to the input signal [5]. Although the inverter is a very simple digital circuit, it is generally possible to realize more complex circuits (such as AND gates and OR gates), starting from the inverter configuration with very slight modifications. The implantation of an inverter circuit is achieved by using one or more transistors, where the latter act as switches, which can be open or close depending on the control signals. A very popular technology is the CMOS (Complementary MOS) technology, in which the inverter is realized by using two FET transistors, and the input signal is applied simultaneously to the gates of both transistors. The two transistors are complementary, that is one of the two is an n-channel FET (NMOS) while the other one is a p-channel FET (PMOS). The complementary nature of the two FETs implies that the input signal is able to turn on only one of the transistors, while the other one is switched off. CMOS Inverter In particular, following the scheme of the above figure, when the input signal is low (0 V), the NMOS (the lower transistor) is switched off, while the PMOS is on. In this way, the output capacitance (always present in CMOS technology) is connected to the VCC power supply via the PMOS. Eventually, the output capacitance will be completely charged, i.e. the output voltage will be VCC (logic 1). Vice versa, when applying a VCC voltage on the input gates, the NMOS will be on while the PMOS will be off. In this way, the output capacitance will be connected to ground via the NMOS

14 14 and will completely discharged. Hence, the output voltage will reach 0 V (logic 0). Note that the circuit is designated so that no dc current flows from power supply to ground, as one of the transistors is always off. This is very important as regards power dissipation issues. The major advantages of CMOS technology are the high level of integration, the very low power dissipation, and immunity to noise. - Inverter - Buffer - AND gate - OR gate - NAND and NOR gates - Multivibrators A multivibrator is an electronic circuit used to implement a variety of simple two-state systems such as oscillators, timers and flip-flops. The most common form is the astable or oscillating type, which generates a square wave - the high level of harmonics in its output is what gives the multivibrator its common name. There are three types of multivibrator circuit [5]: - astable, in which the circuit is not stable in either state - it continuously oscillates from one state to the other. In its simplest form the multivibrator circuit consists of two cross-coupled transistors. Using resistor-capacitor networks within the circuit, it is possible to define the time periods of the unstable states.

15 15 Astable multivibrator The circuit keeps one transistor switched on and the other switched off. Suppose that initially, Q1 is switched on and Q2 is switched off. Q1 holds the bottom of R1 (and the left side of C1) near ground (0V). The right side of C1 (and the base of Q2) is being charged by R2 from below ground to 0.6V. R3 is pulling the base of Q1 up, but its base-emitter diode prevents the voltage from rising above 0.6V. R4 is charging the right side of C2 up to the power supply voltage (+V). Because R4 is less than R2, C2 charges faster than C1. When the base of Q2 reaches 0.6V, Q2 turns on, and the following positive feedback loop occurs: Q2 abruptly pulls the right side of C2 down to near 0V. Because the voltage across a capacitor cannot suddenly change, this causes the left side of C2 to suddenly fall to almost -V, well below 0V. Q1 switches off due to the sudden disappearance of its base voltage. R1 and R2 work to pull both ends of C1 toward +V, completing Q2's turn on. The process is stopped by the B-E diode of Q2, which will not let the right side of C1 rise very far. This now takes us to State 2, the mirror image of the initial state, where Q1 is switched off and Q2 is switched on. Then R1 rapidly pulls C1's left side toward +V, while R3 more slowly pulls C2's left side toward +0.6V. When C2's left side reaches 0.6V, the cycle repeats. - monostable, in which one of the states is stable, but the other is not - the circuit will flip into the unstable state for a determined period, but will eventually return to the

16 16 stable state. Such a circuit is useful for creating a timing period of fixed duration in response to some external event. Monostable multivibrator In the circuit shown above, only the state corresponding to Q1 turned off and Q2 turned on is stable: in this case, the collector of Q2 is at 0 V, so that Q1 remains off. When applying a negative pulse to the base of Q2, Q1 turns on so that the left side of C1 is brought to 0 V. As voltage on C1 can not change abruptly, the voltage on the right side of C1 is also brought to 0 V, so that Q2 remains off. However, the right side of C2 is charged to +V through R2, so that, after some time (i.e. when the voltage on Q2 s base reaches 0.6 V), Q2 turns on again, eventually resulitng in Q1 s turn off. - bistable, in which the circuit will remain in either state indefinitely. The circuit can be flipped from one state to the other by an external event or trigger. Such a circuit is important as the fundamental building block of a register or memory device. This circuit is also known as a flip-flop.

17 17 Bistable multivibrator This circuit is similar to astable multivibrator, except that there is no charge or discharge time, due to the absence of capacitors. Hence, when the circuit is switched on, if Q1 is on, its collector is at 0 V. As a result, Q2 gets switched off. This results in nearly +V volts being applied to base of Q1, thus keeping it on. Thus, the circuit remains stable in a single state continuously. Similarly, Q2 remains on continuously, if it happens to get switched on first. However, in practice, it is preferable to determine the switching of the transistor manually. For this, the Set and Reset terminals are used. For example, if when Q2 is on, Set is grounded, this switches Q2 off, and as described above, makes Q1 on. Thus, Set is used to 'set' Q1 on, and Reset is used to 'reset' it to off state. Multivibrators find applications in a variety of systems where square waves or timed intervals are required, but the simple circuits tend to be fairly inaccurate, so are rarely used where precision is required. - RS Latch - Clocked RS Latch - D Latch - JK Master Slave Flip Flop - JK Edge-triggered Flip-Flop - Monostable oscillators - Applications of monostable oscillators

18 18 Bibliography 1) Communication systems Author: Marcelo S. Alencar Springer, ) Microelectronic circuits Authors: Adel S. Sedra, Kenneth C. Smith, Oxford University Press, 4 th edition, ) Signals and systems Authors: A. V. Oppenhaim, A. S. Willsky Prentice Hall, 2 nd edition, ) Electrical Engineering Authors: Ralf Kories, Heinz Schmidt-Walter Springer, ) Elettronica digitale Author: Paolo Spirito McGraw-Hill, 2006.

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