http://www.alumni.caltech.edu/~croft/notes.txt David Wallace Croft Compilation copyright released to the public domain by the author. 1998-01-18 ---------------------------------------------------------------------- Miscellaneous ---------------------------------------------------------------------- The SCA also attempts to create an atmosphere embodying those lost ideals that are found in medieval romances: chivalry, courtesy, and honor. We recreate the Middle Ages as they ought to have been: doing away with the strife and pestilence and emulating the beauty, grace, chivalry and brotherhood. -- The Society for Creative Anachronism (\SCA\PAMPHLET.SCA rev. 6/94) Reminder of important documents to have ready in a secure place (01) A current will (02) Permanent file of military/business documents (03) Birth certificates -- yours, your spouse's, and dependents' (04) Marriage certificates (05) Health records -- yours, your spouse's, and dependents' (06) Income Tax returns -- last four years (07) Social Security Numbers (SSN) -- yours, your spouses, and dependents' (08) Real estate deeds, mortgages, and insurance (09) Insurance papers -- life and health (10) Bank Accounts -- type numbers and locations (11) Stocks and bonds ---------------------------------------------------------------------- American Heritage Dictionary, 2nd College Edition, 1982 ---------------------------------------------------------------------- elitism: n. 1.a. Belief in rule by an elite. paroxysm: n. 1. A sudden outburst of emotion or action: a paroxysm of laughter. 2. Pathol. a. A crisis in or recurrent intensification of a disease. b. A spasm or fit; convulsion. paternalism: n. A policy or practice of treating or governing people in a fatherly manner, esp. by providing for their needs without giving them responsibility. pithy: adj. 2. Precisely meaningful; cogent and terse. surmise: v. --tr. To infer (something) without sufficiently conclusive evidence; guess. --intr. To make a guess or conjecture. ---------------------------------------------------------------------- Algebra Serway, Raymond A. Physics for Scientists & Engineers, 3/e, Volume 2, Updated Printing. 1992. ---------------------------------------------------------------------- The general form of a quadratic equation is a * ( x ** 2 ) + b * x + c = 0 where x is the unknown quantity and a, b, and c are numerical factors referred to as coefficients of the equation. The quadratic equation has two roots, given by ___________________________ -b ñ û ( b ** 2 ) - ( 4 * a * c ) x = ---------------------------------. 2 * a The roots will be real in the quadratic equation, if ( b ** 2 ) >= ( 4 * a * c ). ---------------------------------------------------------------------- Linear Algebra (from the class notes of Caltech's CNS 185 "Collective Computation") ---------------------------------------------------------------------- Multiplying m x n matrix A by n x p matrix B produces an m x p matrix C = A * B whose elements are defined to be n C_ik = ä A_ij * B_jk. j = 1 Matrix multiplication can only be performed between two matrices A and B if (columns of A) = (rows of B). Like ordinary multiplication, matrix multiplication is associative and distributive, but unlike ordinary multiplication, it is not commutative. A * B /= B * A Transpose ( A * B ) = Transpose ( B ) * Transpose ( A ) The inner product (also known as the dot product) of n-dimensional vectors x and y is defined as Transpose ( x ) * y which is scalar. The inner product is the sum of the products of corresponding elements from the two vectors: n Transpose ( x ) * y = ä x_i * y_i. i = 1 If the inner product of two vectors is zero, they are said to be orthogonal, which has the usual geometric connotation of perpendicularity. The diagonal of an n x n square matrix A are the elements A_ii running diagonally from the top left corner to the bottom right. A diagonal matrix is a matrix which has zeroes everywhere off the diagonal. The symbol I is reserved for a particular diagonal matrix known as the identity matrix, which has ones along its diagonal and zeroes elsewhere. It is the multiplicative identity for matrix multiplication of square matrices. A * I = I * A = A. The n x n square matrix A is called invertible if there exists a matrix denoted A ** -1 which satisfies A * ( A ** -1 ) = ( A ** -1 ) * A = I. If A ** -1 exits, it is called the inverse matrix. If A ** -1 does not exist, A is called a singular matrix. ( Transpose ( A ) ) ** -1 = Transpose ( A ** -1 ) ( A * B ) ** -1 = ( B ** -1 ) * ( A ** -1 ) All square matrices have a particular scalar value associated with them, known as the determinant, which is written as: det A = | A |. For two dimensions, the formula for calculating the determinant is | a_11 a_12 | det A = | A | = | | = a_11 * a_12 - a_12 * a_21. | a_21 a_22 | If the determinant of a matrix is zero, the matrix is singular. iff == "if and only if" A simple way to find if a matrix A is invertible or not is to find its determinant, since ( det A = 0 ) iff ( A is not invertible ). If A is invertible, the only vector x satisfying A * x = 0 is x = 0. If A is not invertible, there can be interesting non-zero solutions for x. If vector x satisfies A * x = lambda * x, so does à * x. So you won't be able to solve for a unique x, just for the direction that x should lie in. That direction is the eigenvector direction, and all vectors parallel to it are eigenvectors with the same eigenvalue. ---------------------------------------------------------------------- Feller, William. An Introduction to Probability Theory and Its Applications. Vol I, 3rd Ed. 1960. Chapter 1. ---------------------------------------------------------------------- A compound (or decomposable) event is an aggregate of certain simple events. Example: the compound event of "sum of six" in rolling 2d6 (two 6-sided dice) is composed of the 5 simple events "(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)". In set theory, "A * B = 0" means that A and B are mutually exclusive. U == or, union, logical sum, + ï == and, intersection, simultaneous realization, * _ |_ (sideways U) == "is contained in" _ _| (sideways U) == "contains" î == "is a member of" _ A implies B == A |_ B _ B is implied by A == B _| A Truth Table for "Implies": Z = A Imp B = A' + B A B Z ------------- 0 0 1 0 1 1 1 0 0 1 1 1 A - A * B = A * B' = A occurs but not B A sample space is called discrete if it contains only finitely many points or infinitely many points which can be arranged into a simple sequence E1, E2, .... The probability of any event is between 0 and 1 inclusive: 0 <= P { A } <= 1 P { A1 U A2 } <= P { A1 } + P { A2 } Boole's Inequality P { A1 U A2 U A3 U ... } <= P { A1 } + P { A2 } + P { A3 } + ... P { A1 U A2 } = P { A1 } + P { A2 } - P { A1 ï A2 } ---------------------------------------------------------------------- Feller, William. An Introduction to Probability Theory and Its Applications. Vol I, 3rd Ed. 1960. Chapter 2. ---------------------------------------------------------------------- Pairs. With m elements a_1, ..., a_m and n elements b_1, ..., b_n, it is possible to form m * n pairs ( a_j, b_k ) containing one element from each group. Multiplets. Given n_1 elements a_1, ..., a_n_1 and n_2 elements b_1, ..., b_n_2, etc., up to n_r elements x_1, ..., x_n_r; it is possible to from n_1 * n_2 * ... * n_r ordered r-tuplets ( a_j1, b_j2, ..., x_jr ) containing one element of each kind. Sampling without replacement notation (n)_r = n * ( n - 1 ) * ( n - 2 ) * ... * ( n - r + 1 ) (n)_r = 0 for integers r, n such that r > n For a population of n elements and a prescribed sample size r, there exist n ** r different samples with replacement and (n)_r samples without replacement. In sampling without replacement a sample of size n includes the whole population and represents a reordering (or permutation) of its elements. The number of different orderings of n elements is (n)_n = n * ( n - 1 ) * ... * 2 * 1 = n! Whenever we speak of random samples of fixed size r, the adjective random is to imply that all possible samples have the same probability, namely, n ** -r in sampling with replacement and 1 / (n)_r in sampling without replacement, n denoting the size of the population from which the sample is drawn. If n is large and r relatively small, the ratio (n)_r / ( n ** r ) is near unity. This leads us to expect that, for large populations and relatively small samples, the two ways of sampling are practically equivalent. In sampling without replacement the probability for any fixed element of the population to be included in a random sample of size r is 1 - ( n - 1 )_r / ( n )_r = 1 - ( n - r ) / n = r / n. In sampling with replacement the probability that an element be included at least once is 1 - ( 1 - ( 1 / n ) ) ** r. A random sample of size r with replacement is taken from a population of n elements. Assuming that all arrangements have equal probability, we conclude that the probability of no repetition in our sample is p = (n)_r / ( n ** r ) = n * ( n - 1 ) * ... * ( n - r + 1 ) / n ** r. If n balls are randomly placed into n cells, the probability that each cell will be occupied equals n! / n ** n. For n = 7, is is only 0.00612. The probability that in a group of r people no one has the same birthday as another is ( 365 )_r / 365 ** r. For 23 people the probability that at least 2 people have a common birthday exceeds 0.5. log ( 1 - x ) ~= -x 1 + 2 + 3 + ... + ( r - 1 ) = r * ( r - 1 ) / 2 We use the term "population of size n" to denote an aggregate of n elements without regard to their order. Two populations are considered different only if one contains an element not contained in the other. The standard notation for the binomial coefficients is / n \ n * ( n - 1 ) * ... * ( n - r + 1 ) | | = ( n )_r / r! = ----------------------------------- \ r / 1 * 2 * ... ( r - 1 ) * r ("n choose r"), the number of subpopulations of size r in a population of size n. Theorem 1. A population of n elements possesses "n choose r" different subpopulations of size r <= n. In other words, a subset of r elements can be chosen in "n choose r" different ways. "n choose r" = "n choose n - r" since there are as many subpopulations of size r as there are subpopulations with ( n - r ) elements not belonging to them. / n \ n! ( n )_r ( n - r )! | | = ------------- = ------- * ---------- \ r / r! ( n - r )! r! ( n - r )! n * ( n - 1 ) * ... * ( n - r + 1 ) * ( n - r )! = ------------------------------------------------ r! * ( n - r )! "n choose 0" = 1 0! = 1 ( n )_0 = 1 a "choose" b = 0 whenever b > a x "choose" r = 0 whenever ( r < 0 ) or ( r > n ) Binomial Distribution: the probability that a specified cell contains exactly k balls (k = 0, 1, 2, ..., r ) is / r \ 1 p_k = | | * ---------- * ( n - 1 ) ** ( r - k ) \ k / ( n ** r ) / r \ 1 = | | * ---------- * ( 1 - 1 / n ) ** ( r - k ). \ k / ( n ** k ) Theorem 2. Let r_1, r_2, ..., r_k be integers such that r_1 + r_2 + ... + r_k = n, r_i >= 0. The number of ways in which a population of n elements can be divided into k ordered parts (partitioned into k subpopulations) of which the first contains r_1 elements, the seconds r_2 elements, etc., is n! / ( r_1! * r_2! * ... * r_k! ). (multinomial coefficients) The number of distinguishable distributions, that is, the number of different solutions of r_1 + r_2 + ... + r_n = r, is / n + r - 1 \ / n + r - 1 \ A_r,n = | | = | |. \ r / \ n - 1 / The number of distinguishable distributions in which no cell remains empty is ( r - 1 ) "choose" ( n - 1 ). Example: there are ( r + 5 ) "choose" 5 distinguishable results of a throw with r indistinguishable dice. 2d6 ==> 7 choose 5 = 7! / ( 5! * ( 7 - 5 )! ) = 7 * 6 / 2 = 21. 11 12 22 13 23 33 14 24 34 44 15 25 35 45 55 16 26 36 46 56 66 Maxwell-Boltzmann statistics: ? Bose-Einstein statistics: 1 / A_r,n ? Fermi-Dirac statistics are based on these hypotheses: (1) it is impossible for two or more particles to be in the same cell, and (2) all distinguishable arrangements satisfying the first condition have equal probabilities. Misprints. A book contains n symbols (letters), of which r are misprinted. The distribution of misprints corresponds to a distribution of r balls in n cells with no cell containing more than one ball. It is therefore reasonable to suppose that, approximately, the misprints obey the Fermi-Dirac statistics. In any ordered sequence of elements of two kinds, each maximal subsequence of elements of like kind is called a run. For example, the sequence aaabaabbba opens with an "a" run of 3; it is followed by runs of length 1, 2, 3, 1, respectively. The "a" and "b" runs alternate so that the total number of runs is always one plus the number of conjunctions of unlike neighbors in the given sequence. Hypergeometric Distribution In a population of n elements n_1 are red and n_2 = n - n_1 are black. A group of r elements is chosen at random. The probability q_k that the group so chosen will contain exactly k red elements is q_k = = ( n_1 choose k ) * ( ( n - n_1 ) choose ( r - k ) ) / ( n choose r ) = ( r choose k ) * ( ( n - r ) choose ( n_1 - k ) ) / ( n choose r ) For any particular set of observations n_1, r, k, the value of n for which q_k ( n ) is largest is denoted by n^ and is called the maximum likelihood estimate of n. This notion was introduced by R. A. Fisher. q_0 + q_1 + q_2 + ... = 1. / n \ / r \ / n - r \ / r \ / n - r \ | | = | | * | | + | | * | | + ... \ n_1 / \ 0 / \ n_1 / \ 1 / \ n_1 - 1 / / r \ / n - r \ + | | * | | for any positive integers n, n_1, and r. \ n_1 / \ 0 / Bridge. The population of 52 cards consists of four classes (suits), each of 13 elements. The probability that a hand of 13 cards consists of 5 spades, 4 hearts, 3 diamonds, and 1 club is ( 13 choose 5 ) * ( 13 choose 4 ) * ( 13 choose 3 ) * ( 13 choose 1 ) / ( 52 choose 13 ). -1 "choose" r = ( -1 ) ** r -2 "choose" r = ( ( -1 ) ** r ) * ( r + 1 ) / x + 1 \ / x \ / x \ | | = | | + | | \ r / \ r - 1 / \ r / Newton's Binomial Formula For any number a (integer) and all values -1 < t < 1, ( 1 + t ) ** a / a \ / a \ / a \ = 1 + | | * t + | | * t ** 2 + | | * t ** 3 + .... \ 1 / \ 2 / \ 3 / If a is a positive integer, all terms to the right containing powers higher than t ** a vanish automatically and the formulat is correct for all t. If a is not a positive integer, the right side represents an infinite series. Geometric Series ( -1 < t < 1 ) ( 1 + t ) ** -1 = 1 - t + t ** 2 - t ** 3 + t ** 4 - t ** 5 +- .... Taylor expansion of the natural logarithm ( -1 < t < 1 ) (integral of the geometric series) log ( 1 + t ) = t - ( t ** 2 ) / 2 + ( t ** 3 ) / 3 -+ .... If -1 < t < 1, then log ( 1 / ( 1 - t ) ) = t + ( t ** 2 ) / 2 + ( t ** 3 ) / 3 + .... Stirling's formula n! ~= ( ( 2 * Pi ) ** 0.5 ) * ( n ** ( n + 0.5 ) ) * exp ( -n ) where the sign ~= is used to indicate that the ratio of the two sides tends to unity as n approaches infinity. log ( n! ) = log ( 1 ) + log ( 2 ) + ... + log ( n ) ---------------------------------------------------------------------- Hoppensteadt, F. C. An Introduction to the Mathematics of Neurons. 1986. Chapter 1. ---------------------------------------------------------------------- VCON == Voltage-Controlled Oscillator Neuron PLL == Phase-Locked Loop Ohm's Law: V = I * R Inductors. These are coils of wire wrapped around a metal core. Current through the coil induces a magnetic field in the core that creates a voltage. V = L * I' The constant L is called the inductance, and it is measured in units of henrys. Capacitors. A capacitor is a device that accumulates charge on plates separated by a non-conductor. The constant C is called the capacitance, and it is measured in units of farads. In most of the circuits used here the appropriate units are micro-farads (1.0e-6 farads). Electro-motive force. A power supply, like a battery or an alternating voltage, applies an electro-motive force (voltage) to a circuit. We denote an electro-motive force by E. RLC-circuits are linear. Kirchhoff's Laws 1. The total voltage measured around any closed loop that can be drawn in the circuit is zero. 2. The total current into any circuit node is zero. The "isoclines" on a plot of I vs. V are where both V' = 0 and I' = 0. Solutions in closed form can only be found for linear circuits. Harmonic Oscillator of an LC circuit: L * C * V'' + V = E with natural frequency ( 1 / ( L * C ) ) ** 0.5 LC-circuits, RLC-circuits with no resistance, behave like timers. Filters are important since they sort out, and allow to pass only certain frequencies. Their purpose is to eliminate noise from a signal or to restrict a signal to a size that meets tolerances of circuit elements farther down stream. V_In *------/\/\/\/-------*---------* V_Out R | | Low-Pass ------- Filter ------- C | | ----- --- Ground - | | V_In *------| |-------*---------* V_Out | | | | High-Pass \ Filter / \ R / \ | | ----- --- Ground - ---------------------------------------------------------------------- Hoppensteadt, F. C. An Intoduction to the Mathematics of Neurons. 1986. Chapter 8. ---------------------------------------------------------------------- Radioactive Decay Given x' = -alpha * x -- decay rate is proportional to its mass) x ( Time => 0 ) = A -- original mass at time 0 dx / x = -alpha * dt ln ( x / A ) = -alpha * t x ( t ) = A * exp ( -alpha * t ) Harmonic Oscillator x'' + ( omega ** 2 ) * x = 0 x ( 0 ) = A x' ( 0 ) = B Omega is a known constant called the free frequency. Given any two linearly independent solutions, say x1 ( t ) and x2 ( t ), _any_ solution of the harmonic oscillator can be written as a linear combination of them. That is, the form x ( t ) = a1 * x1 ( t ) + a2 * x2 ( t ), where a1 and a2 are free constants, covers all possible solutions. Pi / 2 radians = 90 degrees Externally Forced Harmonic Oscillator x'' + ( omega ** 2 ) * x = f ( t ) where f ( t ) == external forcing function ---------------------------------------------------------------------- Mead, Carver. Analog VLSI and Neural Systems. 1989. ---------------------------------------------------------------------- epsilon_0 = 8.85e-12 farads/meter. When epsilon is stated in terms of epsilon_0, it is called the dielectric constant. Force of gravity f_g = G * m1 * m2 / ( r ** 2 ) The force of gravity on Earth (weight ) is f_g = m * ( M * G / ( r ** 2 ) ) = m * g. The gravitational field due to the mass M (force per unit mass) is g = M * G / ( r ** 2 ). For values of height (h) much smaller than r, on Earth, the potential energy PE == V = m * g * h. The gravitational potential, g * h, is the energy per unit mass of matter at height h above the Earth's surface. The raise the potential of 1 coulomb of charge by 1 volt requires 1 joule of energy. The electron volt (e_V) is the energy required to raise the potential of one electron of charge by 1 volt: 1 e_V = 1.6e-19 joules. The unit of capacitance, coulombs per volt (C/V), is called the farad. The unit of resistance R, volts per ampere, is called the ohm (Omega). Q = C * V I = dQ / dt I = C * dV / dt V = I * R ==> R = V / I mho = siemens = G = 1 / R = I / V Devices with gain are called active devices. Newton's Law: Force = mass * acceleration == F = m * a The distance s traveled in time t by a particle starting from rest with acceleration a is s = 0.5 * a * ( t ** 2 ). delta h = 0.5 * a * ( t_f ** 2 ) = ( f / ( 2 ** m ) ) * ( t_f ** 2 ). The average drift velocity (v_drift) of a large collection of particles subject to the force f per particle is just the net change in position delta h per average time t_f between collisions: v_drift = delta h / t_f = f * t_f / ( 2 * m ). The force on each particle with charge q in the presence of an electric field E is f = q * E. The drift velocity in the electric field E is v_drift = f * t_f / ( 2 * m ) = q * t_f * E / ( 2 * m ) = mu * E where the constant mu = q * t_f / ( 2 * m ) is called the mobility of the particle. There is a diffusion of particles from regions of higher density to lower density. The flow rate (J), given in particles per unit area per second, can be viewed as a movement of all particles to the right with some effective diffusion velocity (v_diff): J = N * v_diff. For electrically charged particles, J usually is given in terms of charge per second per unit area (current per unit area), and is called the current density. J = ( N * q ) * v_diff. In a one-dimensional model, a particle in thermal equilibrium has a mean kinetic energy that defines its temperature (T): m * v_Tý / 2 = k * T / 2. Here k is the Boltzmann's constant, m is the mass of the particle, and v_T is called the thermal velocity. At room temperature, k * T = 0.025 electron volt. v_diff = ( -1 / ( 2 * N ) ) * ( dN / dh ) * k * T * t_f / m = -D * ( 1 / N ) * ( dN / dh ) The quantity D = k * T * t_f / ( 2 * m ) is called the diffusion constant of the particle. The mobility and diffusion constants are related: D = ( k * T / q ) * mu. This result is called the Einstein relation; it was discovered by Einstein during his study of Brownian motion. It reminds us that drift and diffusion are not separate processes, but rather are two aspects of the behavior of an ensemble of particles dominated by random thermal motion. If the temperature were reduced to absolute zero, the entire atmosphere would condense into a solid sheet about 5 meters thick. The density of molecules per unit volume in the atmosphere decreases exponentially with altitude above the earths surface (w is weight): N = N_0 * exp ( - w * h / ( k * T ) ). For charged particles, the potential energy is q * V. The voltage V developed in response to a gradient in the concentration of a charged species, and exhibiting the logarithmic dependence on concentration shown below, is called the Nernst potential. V = -( k * T / q ) * ln ( N / N_0 ) In the electrochemistry and biology literature, k * T / q is written R * T / F. The Boltzmann distribution describes the exponential decrease in density of particles in thermal equilibrium with a potential gradient. N = N_0 * exp ( -q * V / ( k * T ) ) ---------------------------------------------------------------------- Mead, Carver. Analog VLSI and Neural Systems. 1989. Chapter 3: Transistor Physics ---------------------------------------------------------------------- The active devices in electronic systems are called transistors. Their function is to control the flow of current from one node based on the potential at another node. For the planet Earth, the gravitational attraction, temperature, and molecular weight are such that the atmospheric density decreases by a factor of e for each approximately 20 km increase in elevation. The total number of electrons in orbit around an atom of a particular element is called the atomic number of that element. Simplified Periodic Table of the Elements I II III IV V VI VII Zero ------------------------------------------------------------------- H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Zn Ga Ge As Se Br Kr Rb Cd In Sn Sb Te I Xe The Group IV elements form a diamond lattice. Silicon is by far the most commonly used semiconductor. Boron, aluminum, and gallium are acceptor impurities in silicon; phosphorous are arsenic are donors (of electrons). Group III elements can combine with Group V elements to form diamondlike crystals in which alternate lattice sites are occupied by atoms of each element. The best known of thes Group III-V semiconductors is gallium arsenide, which is used for microwave transistors and light-emitting diodes. Group II-VI crystals also are semiconductors. Zinc sulfide is a common phosphor in television display tubes, and cadmium sulfide was the earliest widely used photosensitive material. Group 0 elements such helium, neon, and argon are inert gases. Covalent bonds are based on sharing electrons such that all elements can have a full shell, even though some of the electrons taht fill the outer shell are shared with neighbors. Small communal aggregates of this sort are called molecules. Three elements in Group IV of the periodic table crystallize naturally into a remarkable structure called the diamond lattice: carbon, silicon, and germanium. Each atom is covalently bonded to four neighbors arranged at the corners of a regular tetrahedron. A pure crystal formed from Group IV atoms is called an intrinsic semiconductor; it is an electrical insulator, because there are no charged particles free to move around and to carry current. If we alter the crystal by replacing a small fraction of its atoms with impurity atoms of Group V, the crystal becomes conductive. The addition of impurities is called doping. Group V doping atoms are called donors, because they donate a free electron to the crystal. The free electrons are negative, and a semiconductor crystal doped with donors is said to be n-type. When an electron leaves its donor, the donor is said to be ionized. An ionized donor has a positive charge because it has lost one electron. Group III dopants are called acceptors. The absence of one electron in a bond is called a hole. Doping a semiconductor with acceptors renders it conductive, the current being carried by positive holes. Such a crystal is called p-type. Heavily doped n-type material is called n+, and heavily doped p-type material is called p+. The density of impurity atoms is always small compared to the approximately 5e+22 atoms per cubic centimeter in the crystal itself. Because electrons and holes are both charged, and are both used to carry current, we refer to them generically as charge carriers. SiO2 == Silicon Oxide == Quartz is an excellent electrical insulator. Current flows from the source to drain in the region just under the gate oxide called the channel. This structure was first described by a freelance inventor named Lilienfeld in a patent issued in 1933. MOS transistor: Metallic gate, Oxide insulator, Semiconductor channel polycrystalline silicon == polysilicon == poly The Fermi level measures the energy of the charge carrier. The general form of the MOS transistor current is I = I_0 * exp ( -q * Vg / ( k * T ) ) * ( exp ( q * Vs / ( k * T ) - exp ( q * Vd / ( k * T ) ) ). For a transistor with its source connected to the power supply rail, Vs is equal to zero and I = I_0 * exp ( -q * Vgs / ( k * T ) ) * ( 1 - exp ( q * Vds / ( k * T ) ) ) where Vgs and Vds are the gate-to-source and drain-source-voltages. Because there are charge carriers with positive as well as negative charge, there are two kinds of MOS transistor: Those using electrons as their charge carriers are called n-channel, whereas those using holes are called p-channel; the technology is thus called complementary MOS, or CMOS. For positive q (p-channel device), the current increases as the gate voltage is made negative with respect to the source; for negative q (n-channel device), the opposite occurs. k * T is the thermal energy per charge carrier, so the quantity k * T / q has the units of potential; it is called the thermal voltage, and its magnitude is 25 mV at room temperature. A carrier must slide down a potential barrier of k * T / q to raise its energy by k * T. Drain Source | | | | |__| |__| | | | | Gate ----| | n-Channel Gate ---o| | p-Channel | |__ transistor | |__ transistor | | | | | | | | Source Drain We put a bubble on the gate of the p-channel symbol to remind us that the transistor turns on as we make the gate more negative relative to the source. We normally will draw the positive supply at the top of the diagram, and the most negative supply at the bottom. For this reason, the sources of p-channel devices usually are located at the top, whereas those of n-channel devices normally are at the bottom. For a given gate voltage, the drain current increases with Vds and then saturates after a few k * T / q. The current in the flat part of the curves is nearly independent of Vds and is called the saturation current, I_sat. ---------------------------------------------------------------------- Serway, Raymond A. Physics for Scientists & Engineers, 3/e, Volume 2, Updated Printing. 1992. ---------------------------------------------------------------------- The naturally occurring stone magnetite (Fe3O4) is attracted to iron. "Elektron" is the Greek word for amber. The coulomb is the unit of electric charge. Electric charge exists in discrete packets: q = N * e. The force between charges varies as the inverse square of their separation: F ~= 1 / ( r ** 2 ). Charging a body by induction requires no contact with the body inducing the charge, unlike conduction. Coulomb's Law The magnitude of electric force between two charges is F = k * abs q1 * abs q2 / ( r ** 2 ) where k is the Coulomb constant. Current: 1 ampere (A) = 1 coulomb (C) / second. The Coulomb constant k ~= 9.0e+9 N * ( m ** 2 ) / ( C ** 2 ). The Coulomb constant k = 1 / ( 4 * Pi * epsilon_0 ) where epsilon_0 is the permittivity of free space. The permittivity of free space epsilon_0 = 8.8542e-12 ( C ** 2 ) / ( N * ( m ** 2 ) ). The smallest unit of charge known in nature is the charge on an electron or proton with a magnitude abs e = 1.60219e-19 C. 1 C = 1 / e = 6.3e+18 electrons (1e-6 C ~= charged glass rod). A metal atom, such as copper, contains one or more outer electrons, which are weakly bound to the nucleus. When many atoms combine to form a metal, the so-called free electrons are these outer electrons, which are not bound to any one atom. These electrons move about the metal in a manner similar to gas molecules moving in a container. Table 23.1 Charge and Mass of the Electron, Proton, and Neutron Particle Charge (C) Mass (kg) ------------ -------------- ----------- Electron (e) -1.6021917e-19 9.1095e-31 Proton (p) +1.6021917e-19 1.67261e-27 Neutron (n) 0.0 1.67492e-27 Force is a vector quantity with magnitude and direction (not a scalar). Coulomb's law applies exactly only to point charges or particles.