Ebook - Radio frequency integrated circuit design

Ebook - Radio frequency integrated circuit design

John Rogers Calvin Plett

Artech House Boston • London www.artechhouse.com

p cm — (Artech House microwave library)

Includes bibliographical references and index.

ISBN 1-58053-502-x (alk paper)

1 Radio frequency integrated circuits—Design and construction 2 Very high speed integrated circuits I Plett, Calvin II Title III Series.

TK7874.78.R64 2003

British Library Cataloguing in Publication Data

Rogers, John

Radio frequency integrated circuit design — (Artech House microwave library)

1 Radio circuits—Design and construction 2 Linear integrated circuits—Design and construction 3 Microwave integrated circuits—Design and construction

4 Bipolar integrated circuits—Design and construction I Title II Plett, Calvin 621.3’812

ISBN 1-58053-502-x

Cover design by Igor Valdman

2003 ARTECH HOUSE, INC.

685 Canton Street

Norwood, MA 02062

All rights reserved Printed and bound in the United States of America No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission

in writing from the publisher.

All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized Artech House cannot attest to the accuracy of this information Use

of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

International Standard Book Number: 1-58053-502-x

Library of Congress Catalog Card Number: 2003041891

10 9 8 7 6 5 4 3 2 1

Foreword xv

1 Introduction to Communications Circuits 1

Design Versus Radio Frequency Integrated

2.2 Noise 9

3.8 Base Shot Noise Discussion 55

5.3 Sheet Resistance and the Skin Effect 97

Passives and Some Common De-Embedding

5.25 Packaging 135

7.10.3 Image Rejection with Amplitude and Phase

7.11.3 Mixer with Simultaneous Noise and Power

7.12 General Design Comments 231

8.11 A Modified Common-Collector Colpitts

9.5 Some Simple Image Rejection Formulas 333

10.8.2 Variation on Class F: Quarter-Wave

10.11 Summary of Amplifier Classes for RF Integrated

I enjoyed reading this book for a number of reasons One reason is that itaddresses high-speed analog design in the context of microwave issues This is

an advanced-level book, which should follow courses in basic circuits andtransmission lines Most analog integrated circuit designers in the past worked

on applications at low enough frequency that microwave issues did not arise

As a consequence, they were adept at lumped parameter circuits and often notcomfortable with circuits where waves travel in space However, in order todesign radio frequency (RF) communications integrated circuits (IC) in thegigahertz range, one must deal with transmission lines at chip interfaces andwhere interconnections on chip are far apart Also, impedance matching isaddressed, which is a topic that arises most often in microwave circuits In mycareer, there has been a gap in comprehension between analog low-frequencydesigners and microwave designers Often, similar issues were dealt with in twodifferent languages Although this book is more firmly based in lumped-elementanalog circuit design, it is nice to see that microwave knowledge is brought inwhere necessary

Too many analog circuit books in the past have concentrated first on thecircuit side rather than on basic theory behind their application in communica-tions The circuits usually used have evolved through experience, without asatisfying intellectual theme in describing them Why a given circuit works bestcan be subtle, and often these circuits are chosen only through experience Forthis reason, I am happy that the book begins first with topics that require anintellectual approach—noise, linearity and filtering, and technology issues I

am particularly happy with how linearity is introduced (power series) In therest of the book it is then shown, with specific circuits and numerical examples,how linearity and noise issues arise

xv

In the latter part of the book, the RF circuits analyzed are ones thatexperience has shown to be good ones Concentration is on bipolar circuits,not metal oxide semiconductors (MOS) Bipolar still has many advantages athigh frequency The depth with which design issues are addressed would not

be possible if similar MOS coverage was attempted However, there might beroom for a similar book, which concentrates on MOS

In this book there is a lot of detailed academic exploration of someimportant high-frequency RF bipolar ICs One might ask if this is important

in design for application, and the answer is yes To understand why, one mustappreciate the central role of analog circuit simulators in the design of suchcircuits At the beginning of my career (around 1955–1960) discrete circuitswere large enough that good circuit topologies could be picked out by bread-boarding with the actual parts themselves This worked fairly well with someanalog circuits at audio frequencies, but failed completely in the progression tointegrated circuits

In high-speed IC design nowadays, the computer-based circuit simulator

is crucial Such simulation is important at four levels The first level is the use

of simplified models of the circuit elements (idealized transistors, capacitors,and inductors) The use of such models allows one to pick out good topologiesand eliminate bad ones This is not done well with just paper analysis because

it will miss key factors, such as the complexities of the transistor, particularlynonlinearity and bias and signal interaction effects Exploration of topologieswith the aid of a circuit simulator is necessary The simulator is useful for quickiteration of proposed circuits, with simplified models to show any fundamentalproblems with a proposed circuit This brings out the influence of modelparameters on circuit performance This first level of simulation may be avoided

if the best topology, known through experience, is picked at the start.The second level of simulation is where the models are representative ofthe type of fabrication technology being used However, we do not yet usespecific numbers from the specific fabrication process and make an educatedapproximation to likely parasitic capacitances Simulation at this level can beused to home in on good values for circuit parameters for a given topologybefore the final fabrication process is available Before the simulation begins,detailed preliminary analysis at the level of this book is possible, and manyparameters can be wisely chosen before simulation begins, greatly shorteningthe design process and the required number of iterations Thus, the analysisshould focus on topics that arise, given a typical fabrication process I believethis has been done well here, and the authors, through scholarly work and realdesign experience, have chosen key circuits and topics

The third level of design is where a link with a proprietary industrialprocess has been made, and good simulator models are supplied for the process.The circuit is laid out in the proprietary process and simulation is done, including

estimates of parasitic capacitances from interconnections and detailed models

of the elements used

The incorporation of the proprietary models in the simulation of thecircuit is necessary because when the IC is laid out in the actual process,fabrication of the result must be successful to the highest possible degree This

is because fabrication and testing is extremely expensive, and any failure canresult in the necessity to change the design, requiring further fabrication andretesting, causing delay in getting the product to market

The fourth design level is the comparison of the circuit behavior predictedfrom simulation with that of measurements of the actual circuit Discrepanciesmust be explained These may be from design errors or from inadequacies inthe models, which are uncovered by the experimental result These modelinadequacies, when corrected, may result in further simulation, which causesthe circuit design and layout to be refined with further fabrication

This discussion has served to bring attention to the central role thatcomputer simulation has in the design of integrated RF circuits, and the accompa-nying importance of circuit analysis such as presented in this book Such detailedanalysis may save money by facilitating the early success of applications Thisbook can be beneficial to designers, or by those less focused on specific design,for recognizing key constraints in the area, with faith justified, I believe, thatthe book is a correct picture of the reality of high-speed RF communicationscircuit design

Miles A Copeland Fellow IEEE Professor Emeritus Carleton University Department of Electronics

Ottawa, Ontario, Canada

April 2003

This book has evolved out of a number of documents including technical papers,course notes, and various theses We decided that we would organize some ofthe research we and many others had been doing and turn it into a manuscriptthat would serve as a comprehensive text for engineers interested in learningabout radio frequency integrated circuits (RFIC) We have focused mainly onbipolar technology in the text, but since many techniques in RFICs are indepen-dent of technology, we hope that designers working with other technologieswill also find much of the text useful We have tried very hard to identify andexterminate bugs and errors from the text Undoubtedly there are still manyremaining, so we ask you, the reader, for your understanding Please feel free

to contact us with your comments We hope that these pages add to yourunderstanding of the subject

Nobody undertakes a project like this without support on a number oflevels, and there are many people that we need to thank Professors MilesCopeland and Garry Tarr provided technical guidance and editing We wouldlike to thank David Moore for his input and consultation on many aspects ofRFIC design David, we have tried to add some of your wisdom to these pages.Thanks also go to Dave Rahn and Steve Kovacic, who have both contributed

to our research efforts in a variety of ways We would like to thank Sandi Plettwho tirelessly edited chapters, provided formatting, and helped beat the wordprocessor into submission She did more than anybody except the authors tomake this project happen We would also like to thank a number of graduatestudents, alumni, and colleagues who have helped us with our understanding

of RFICs over the years This list includes but is not limited to Neric Fong,

xix

Bill Toole, Jose´ Macedo, Sundus Kubba, Leonard Dauphinee, Rony Amaya,John J Nisbet, Sorin Voinegescu, John Long, Tom Smy, Walt Bax, BrianRobar, Richard Griffith, Hugues Lafontaine, Ash Swaminathan, Jugnu Ojha,George Khoury, Mark Cloutier, John Peirce, Bill Bereza, and Martin Snelgrove.

Introduction to Communications

Circuits

1.1 Introduction

Radio frequency integrated circuit (RFIC) design is an exciting area for research

or product development Technologies are constantly being improved, and asthey are, circuits formerly implemented as discrete solutions can now be inte-grated onto a single chip In addition to widely used applications such as cordlessphones and cell phones, new applications continue to emerge Examples of new

products requiring RFICs are wireless local-area networks (WLAN), keyless entry for cars, wireless toll collection, Global Positioning System (GPS) navigation,

remote tags, asset tracking, remote sensing, and tuners in cable modems Thus,the market is expanding, and with each new application there are uniquechallenges for the designers to overcome As a result, the field of RFIC designshould have an abundance of products to keep designers entertained for years

to come

This huge increase in interest in radio frequency (RF) communications has resulted in an effort to provide components and complete systems on an integrated

circuit (IC) In academia, there has been much research aimed at putting a

complete radio on one chip Since complementary metal oxide semiconductor (CMOS) is required for the digital signal processing (DSP) in the back end,

much of this effort has been devoted to designing radios using CMOS gies [1–3] However, bipolar design continues to be the industry standardbecause it is a more developed technology and, in many cases, is better modeled.Major research is being done in this area as well CMOS traditionally had theadvantage of lower production cost, but as technology dimensions become

technolo-1

smaller, this is becoming less true Which will win? Who is to say? Ultimately,both will probably be replaced by radically different technologies In any case,

as long as people want to communicate, engineers will still be building radios

In this book we will focus on bipolar RF circuits, although CMOS circuits willalso be discussed Contrary to popular belief, most of the design concepts inRFIC design are applicable regardless of what technology is used to implementthem

The objective of a radio is to transmit or receive a signal between sourceand destination with acceptable quality and without incurring a high cost Fromthe user’s point of view, quality can be perceived as information being passedfrom source to destination without the addition of noticeable noise or distortion.From a more technical point of view, quality is often measured in terms of biterror rate, and acceptable quality might be to experience less than one error inevery million bits Cost can be seen as the price of the communications equipment

or the need to replace or recharge batteries Low cost implies simple circuits tominimize circuit area, but also low power dissipation to maximize battery life

1.2 Lower Frequency Analog Design and Microwave Design Versus Radio Frequency Integrated Circuit Design

RFIC design has borrowed from both analog design techniques, used at lowerfrequencies [4, 5], and high-frequency design techniques, making use of micro-wave theory [6, 7] The most fundamental difference between low-frequencyanalog and microwave design is that in microwave design, transmission lineconcepts are important, while in low-frequency analog design, they are not.This will have implications for the choice of impedance levels, as well as howsignal size, noise, and distortion are described

On-chip dimensions are small, so even at RF frequencies (0.1–5 GHz),transistors and other devices may not need to be connected by transmissionlines (i.e., the lengths of the interconnects may not be a significant fraction of

a wavelength) However, at the chip boundaries, or when traversing a significantfraction of a wavelength on chip, transmission line theory becomes veryimportant Thus, on chip we can usually make use of analog design concepts,although, in practice, microwave design concepts are often used At the chipinterfaces with the outside world, we must treat it like a microwave circuit

1.2.1 Impedance Levels for Microwave and Low-Frequency Analog Design

In low-frequency analog design, input impedance is usually very high (ideallyinfinity), while output impedance is low (ideally zero) For example, an opera-tional amplifier can be used as a buffer because its high input impedance doesnot affect the circuit to which it is connected, and its low output impedance

can drive a measurement device efficiently The freedom to choose arbitraryimpedance levels provides advantages in that circuits can drive or be driven by

an impedance that best suits them On the other hand, if circuits are connectedusing transmission lines, then these circuits are usually designed to have aninput and output impedance that match the characteristic impedance of thetransmission line

1.2.2 Units for Microwave and Low-Frequency Analog Design

Signal, noise, and distortion levels are also described differently in low frequencyanalog versus microwave design In microwave circuits, power is usually used

to describe signals, noise, or distortion with the typical unit of measure beingdecibels above 1 milliwatt (dBm) However, in analog circuits, since infinite orzero impedance is allowed, power levels are meaningless, so voltages and currentare usually chosen to describe the signal levels Voltage and current are expressed

voltages are assumed to be across 50⍀

vrms = vpp

Similarly, noise in analog signals is often defined in terms of volts oramperes, while in microwave it will be in terms of dBm Noise is usuallyrepresented as noise density per hertz of bandwidth In analog circuits, noise

is specified as squared volts per hertz, or volts per square root of hertz Inmicrowave circuits, the usual measure of noise is dBm/Hz or noise figure, which

is defined as the reduction in signal-to-noise ratio caused by the addition ofthe noise

In both analog and microwave circuits, an effect of nonlinearity is theappearance of harmonic distortion or intermodulation distortion, often at newfrequencies In low-frequency analog circuits, this is often described by the ratio

of the distortion components compared to the fundamental components Inmicrowave circuits, the tendency is to describe distortion by gain compression(power level where the gain is reduced due to nonlinearity) or third-orderintercept point (IP3)

Noise and linearity are discussed in detail in Chapter 2 A summary oflow-frequency analog and microwave design is shown in Table 1.2

1.3 Radio Frequency Integrated Circuits Used in a

Communications Transceiver

A typical block diagram of most of the major circuit blocks that make up atypical superheterodyne communications transceiver is shown in Figure 1.1.Many aspects of this transceiver are common to all transceivers

Table 1.2

Comparison of Analog and Microwave Design

Parameter Analog Design Microwave Design

(most often used on chip) (most often used at chip

boundaries and pins)

Signals Voltage, current, often peak Power, often dBm

or peak-to-peak Noise nV/√Hz Noise factor F, noise figure NF

Nonlinearity Harmonic distortion, Third-order intercept point IP3

intermodulation, clipping 1-dB compression

Figure 1.1 Typical transceiver block diagram.

This transceiver has a transmit side (Tx) and a receive side (Rx), whichare connected to the antenna through a duplexer that can be realized as a switch

or a filter, depending on the communications standard being followed Theinput preselection filter takes the broad spectrum of signals coming from theantenna and removes the signals not in the band of interest This may be

required to prevent overloading of the low-noise amplifier (LNA) by

out-of-band signals The LNA amplifies the input signal without adding much noise.The input signal can be very weak, so the first thing to do is strengthen thesignal without corrupting it As a result, noise added in later stages will be ofless importance The image filter that follows the LNA removes out-of-bandsignals and noise (which will be discussed in detail in Chapter 2) before thesignal enters the mixer The mixer translates the input RF signal down to theintermediate frequency, since filtering, as well as circuit design, becomes mucheasier at lower frequencies for a multitude of reasons The other input to the

mixer is the local oscillator (LO) signal provided by a voltage-controlled oscillator

inside a frequency synthesizer The desired output of the mixer will be thedifference between the LO frequency and the RF frequency

At the input of the radio there may be many different channels or frequencybands The LO frequency is adjusted so that the desired RF channel or frequency

band is mixed down to the same intermediate frequency (IF) in all cases The

IF stage then provides channel filtering at this one frequency to remove the

unwanted channels The IF stage provides further amplification and automatic

gain control (AGC) to bring the signal to a specific amplitude level before the

signal is passed on to the back end of the receiver It will ultimately be convertedinto bits (most modern communications systems use digital modulation schemes)that could represent, for example, voice, video, or data through the use of ananalog-to-digital converter

On the transmit side, the back-end digital signal is used to modulate thecarrier in the IF stage In the IF stage, there may be some filtering to removeunwanted signals generated by the baseband, and the signal may or may not

be converted into an analog waveform before it is modulated onto the IF carrier

A mixer converts the modulated signal and IF carrier up to the desired RFfrequency A frequency synthesizer provides the other mixer input Since the

RF carrier and associated modulated data may have to be transmitted over large

distances through lossy media (e.g., air, cable, and fiber), a power amplifier (PA)

must be used to increase the signal power Typically, the power level is increasedfrom the milliwatt range to a level in the range of hundreds of milliwatts towatts, depending on the particular application A lowpass filter after the PAremoves any harmonics produced by the PA to prevent them from also beingtransmitted

1.4 Overview

We will spend the rest of this book trying to convey the various design constraints

of all the RF building blocks mentioned in the previous sections Componentsare designed with the main concerns being frequency response, gain, stability,noise, distortion (nonlinearity), impedance matching, and power dissipation.Dealing with design constraints is what keeps the RFIC designer employed.The focus of this book will be how to design and build the major circuitblocks that make up the RF portion of a radio using an IC technology Tothat end, block level performance specifications are described in Chapter 2 Abrief overview of IC technologies and transistor performance is given in Chapter

3 Various methods of matching impedances, which are very important at chipboundaries and for some interconnections of circuits on-chip, will be discussed

in Chapter 4 The realization and limitations of passive circuit components in

an IC technology will be discussed in Chapter 5 Chapters 6 through 10 will

be devoted to individual circuit blocks such as LNAs, mixers, voltage-controlled

oscillators (VCOs), filters, and power amplifiers However, the design of complete

synthesizers is beyond the scope of this book The interested reader is referred

to [8–10]

References

[1] Lee, T H., The Design of CMOS Radio Frequency Integrated Circuits, Cambridge, England:

Cambridge University Press, 1998.

[2] Razavi, B., RF Microelectronics, Upper Saddle River, NJ: Prentice Hall, 1998.

[3] Crols, J., and M Steyaert, CMOS Wireless Transceiver Design, Dordrecht, the Netherlands:

Kluwer Academic Publishers, 1997.

[4] Gray, P R., et al., Analysis and Design of Analog Integrated Circuits, 4th ed., New York:

John Wiley & Sons, 2001.

[5] Johns, D A., and K Martin, Analog Integrated Circuit Design, New York: John Wiley &

Sons, 1997.

[6] Gonzalez, G., Microwave Transistor Amplifiers Analysis and Design, 2nd ed., Upper Saddle

River, NJ: Prentice Hall, 1997.

[7] Pozar, D M., Microwave Engineering, 2nd ed., New York: John Wiley & Sons, 1998.

[8] Crawford, J A., Frequency Synthesizer Design Handbook, Norwood, MA: Artech House,

a perfect copy of the input signal at the output In a real circuit, the amplifierwill introduce both noise and distortion to that waveform Noise, which ispresent in all resistors and active devices, limits the minimum detectable signal

in a radio At the other amplitude extreme, nonlinearities in the circuit blockswill cause the output signal to become distorted, limiting the maximum signalamplitude

At the system level, specifications for linearity and noise as well as manyother parameters must be determined before the circuit can be designed Inthis chapter, before we look at circuit details, we will look at some of thesesystem issues in more detail In order to design radio frequency integratedcircuits with realistic specifications, we need to understand the impact of noise

on minimum detectable signals and the effect of nonlinearity on distortion.Knowledge of noise floors and distortion will be used to understand the require-ments for circuit parameters

2.2 Noise

Signal detection is more difficult in the presence of noise In addition to thedesired signal, the receiver is also picking up noise from the rest of the universe

9

Any matter above 0K contains thermal energy This thermal energy movesatoms and electrons around in a random way, leading to random currents incircuits, which are also noise Noise can also come from man-made sourcessuch as microwave ovens, cell phones, pagers, and radio antennas Circuitdesigners are mostly concerned with how much noise is being added by thecircuits in the transceiver At the input to the receiver, there will be some noisepower present that defines the noise floor The minimum detectable signal must

be higher than the noise floor by some signal-to-noise ratio (SNR) to detect

signals reliably and to compensate for additional noise added by circuitry Theseconcepts will be described in the following sections

We note that to find the total noise due to a number of sources, therelationship of the sources with each other has to be considered The mostcommon assumption is that all noise sources are random and have no relationshipwith each other, so they are said to be uncorrelated In such a case, noise power

is added instead of noise voltage Similarly, if noise at different frequencies isuncorrelated, noise power is added We note that signals, like noise, can also

be uncorrelated, such as signals at different unrelated frequencies In such acase, one finds the total output signal by adding the powers On the otherhand, if two sources are correlated, the voltages can be added As an example,correlated noise is seen at the outputs of two separate paths that have the sameorigin

2.2.1 Thermal Noise

One of the most common noise sources in a circuit is a resistor Noise inresistors is generated by thermal energy causing random electron motion [1–3].The thermal noise spectral density in a resistor is given by

where T is the Kelvin temperature of the resistor, k is Boltzmann’s constant

density is expressed using volts squared per hertz (power spectral density) Inorder to find out how much power a resistor produces in a finite bandwidth,

also be written equivalently as a noise current rather than a noise voltage:

Thermal noise is white noise, meaning it has a constant power spectraldensity with respect to frequency (valid up to approximately 6,000 GHz) [4].The model for noise in a resistor is shown in Figure 2.1.

2.2.2 Available Noise Power

Thus, available power is kT, independent of resistor size Note that kT is

watts, multiply by the bandwidth, with the result that

2.2.3 Available Power from Antenna

The noise from an antenna can be modeled as a resistor [5] Thus, as in theprevious section, the available power from an antenna is given by

Pavailable=kT =4 × 10−21 W/Hz (2.6)

Figure 2.1 Resistor noise model: (a) with a voltage source, and (b) with a current source.

More commonly, the noise floor would be expressed in dBm, as in thefollowing for the example shown above:

Thus, we can now also formally define signal-to-noise ratio If the signal

has a power of S, then the SNR is

Thus, if the electronics added no noise and if the detector required a

The minimum detectable signal in a receiver is also referred to as the receiversensitivity However, the SNR required to detect bits reliably (e.g., bit error

on a variety of factors, such as bit rate, energy per bit, IF filter bandwidth,detection method (e.g., synchronous or not), and interference levels Suchcalculations are the topics for a digital communications course [6, 7] and will

about 7 dB for quadrature phase shift keying (QPSK), about 12 dB for 16

quadrature amplitude modulation (QAM), and about 17 dB for 64 QAM, though

often higher numbers are quoted to leave a safety margin It should be noted

in an SNR requirement of 11 dB or more for QPSK Thus, the input signal

level must be above the noise floor level by at least this amount Consequently,

dBm (assuming no noise is added by the electronics)

2.2.4 The Concept of Noise Figure

Noise added by electronics will be directly added to the noise from the input.Thus, for reliable detection, the previously calculated minimum detectable signallevel must be modified to include the noise from the active circuitry Noise

from the electronics is described by noise factor F, which is a measure of how

much the signal-to-noise ratio is degraded through the system We note that

by the electronic circuitry, then we can write:

N o (total)= N o (source) +N o (added) (2.13)Noise factor can be written in several useful alternative forms:

a filter with 3 dB of loss has a noise figure of 3 dB This is explained by notingthat output noise is approximately equal to input noise, but signal is attenuated

by 3 dB Thus, there has been a degradation of SNR by 3 dB

2.2.5 The Noise Figure of an Amplifier Circuit

We can now make use of the definition of noise figure just developed and apply

it to an amplifier circuit [8] For the purposes of developing (2.14) into a moreuseful form, it is assumed that all practical amplifiers can be characterized by

an input-referred noise model, such as the one shown in Figure 2.2, where the

how to take a practical amplifier and make it fit this model.) In this model, all

If the amplifier has finite input impedance, then the input current will

We can expand both current and voltage into these two explicit parts:

In addition, the correlated components will be related by the ratio

The noise figure can now be written as

Thus, the noise figure is now written in terms of these parameters:

after several pages of math:

2.2.6 The Noise Figure of Components in Series

For components in series, as shown in Figure 2.3, one can calculate the total

determine the noise figure

Figure 2.3 Noise figure in cascaded circuits with gain and noise added shown in each.

The input noise is

The total output noise is

N o (total)= N i (source) G1G2G3 + N o1(added) G2G3

+N o2(added) G3 +N o3(added) (2.33)The output noise due to the source is

N o (source) =N i (source) G1G2G3 (2.34)Finally, the noise factor can be determined as

of each block is typically determined for the case in which a standard inputsource (e.g., 50⍀) is connected The above formula can also be used to derive

an equivalent model of each block as shown in Figure 2.4 If the input noisewhen measuring noise figure is

and noting from manipulation of (2.14) that

Figure 2.4 Equivalent noise model of a circuit.

Example 2.1 Noise Calculations

Figure 2.5 shows a 50-⍀ source resistance loaded with 50⍀ Determine howmuch noise voltage per unit bandwidth is present at the output Then, for any

Solution

290K, which, after the voltage divider, becomes one half of this value, or

Now, for maximum power transfer, the load must remain matched, so

to the load In this case,

4R L =Pin(available)

Figure 2.5 Simple circuit used for noise calculations.

Pin(available) = v o2

uncorre-lated)

Example 2.2 Noise Calculation with Gain Stages

In this example, Figure 2.6, a voltage gain of 20 has been added to the originalcircuit of Figure 2.5 All resistor values are still 50⍀ Determine the noise atthe output of the circuit due to all resistors and then determine the circuit noisefigure and signal-to-noise ratio assuming a 1-MHz bandwidth and the input is

a 1-V sine wave

Solution

the input sources, as well as the noise from the two output resistors, all see avoltage divider Thus, one can calculate the individual components For the

The noise from the source can be determined from this equation:

Figure 2.6 Noise calculation with a gain stage.