MASTER THESIS. Martin Horváth Dimensional Analysis for Hardware Description Languages

Transcription

1 Charles University in Prague Faculty of Mathematics and Physics MASTER THESIS Martin Horváth Dimensional Analysis for Hardware Description Languages Department of Software Engineering Supervisor: RNDr. Dávid Bednárek Computer Science 2007

2 I would like to express my gratitude to all those who helped me to complete this thesis. Especially to my superviser RNDr. David Bednárek for his academic guidance and to Martin Vlach, Ph.D., CEO of Lynguent, Inc. for his inspiration in the development of the thesis. I declare that I have worked out this thesis on my own, using only the resources stated. I agree that the thesis may be made publicly available. Prague, August 8, 2007 Martin Horváth 1

3 Contents 1 Introduction 5 2 Theory Hardware Description Languages and ModLyng TM Principles of Dimensional Analysis Example Dimensionless Quantities DA for Dimensionless Quantities Thermodynamic Temperature Functions Formal Model of DI Dimensional Information (DI) DI Algebra Model of HDL Program Quantity Expression Function DA Algorithm Main Algorithm Expanding of Program s Expressions Transforming Expression over DI Deriving of Dimensional Information Making Two DI Equivalent Computing DI Result of DA Algorithm

4 6 Design and Implementation Use Cases Requirements Software Design Implementation Notes Conclusion 65 Bibliography 68 A Contents of the Compact Disc 69 A.1 Delivered Contents A.2 Limitations of the Delivered Contents A.3 System Requirements A.4 Installation Instructions A.5 Running Test

5 Názov práce: Dimenzionálna Analýza pre HDL jazyky Autor: Martin Horváth Katedra: Katedra softwarového inženýrství Vedúci diplomovej práce: RNDr. David Bednárek vedúceho: Abstrakt: Na vývoj elektrických obvodov v súčasnosti používajú inžinieri špeciálne popisné jazyky, ktoré im umožňujú vytvoriť modely týchto obvodov a otestovať ich pred samotnou konštrukciou. Tieto jazyky však neposkytujú dostatočne komplexné prostriedky na definovanie fyzikálnych dimenzií a jednotiek pre veličiny používané v modeloch a kontrolu ich konzistencie vo výpočtoch a rovniciach. Nekonzistencia v použití fyzikálnych jednotiek môže byť spôsobená drobnou typografickou chybou, ale taktiež môže predstavovať hlbšie konceptuálne chyby. Hľadanie takýchto chýb môže byť veľmi náročné, keďže modely bežne obsahujú tisíce riadkov kódu. Táto diplomová práca si kladie dva ciele. Prvým je analyzovať problematiku fyzikálnych dimenzií, formalizovať ich vlastnosti a navrhnúť algoritmus ich automatickej kontroly v modeloch napísaných v jazykoch na popis hardwaru. Druhým cieľom je navrhnúť a implementovať systém pre anotáciu modelov dimenzionálnou informáciou a implementovať algoritmus dimenzionálnej analýzy do komerčného produktu ModLyng. Tento produkt je integrovaným prostredím pre vývoj AMS integrovaných obvodov. Použitie automatickej analýzy by malo inžinierom pomôcť navrhovať korektné modely a urýchliť ich vývoj. Kľúčové slová: fyzikálne dimenzie, jednotky, HDL, ModLyng TM Title: Dimensional Analysis for Hardware Description Languages Author: Martin Horváth Department: Department of Software Engineering Supervisor: RNDr. David Bednárek Supervisor s address: David.Bednarek@mff.cuni.cz Abstract: The engineers today use special description languages for developing electronic circuits and these enable them to test their models prior to their industrial construction. However, none of these languages provides sufficiently complex means to define physical dimensions and units for quantities used in the models and to check their consistency in all equations and computations. The inconsistency in the usage of physical units can be caused by plain typing errors but sometimes represents deep conceptual flaws. The tracking of these errors can be very difficult as the models commonly consist of thousands of lines of code. This master s thesis sets itself two goals. The first is to analyze the problem of physical dimensions, formalize their properties and devise an algorithm for automatic checking of their usage in the models written in hardware description languages. The second is to design and implement the system for annotating models with dimensional information and to implement the algorithm of dimensional analysis as a module into commercial tool ModLyng which is the integrated environment for developing analog/mixed-signal integrated circuits. The usage of this automatic analysis will help engineers in writing correct models and will speed up their development. Keywords: physical dimensions, units, HDL, ModLyng TM 4

6 Chapter 1 Introduction The physical sciences are quantitative disciplines. The measurement and observation of quantitative properties of material entities is an essential part of their research. Quantity refers to any type of measurable properties or attributes of objects. Quantities which pertain to physical science are called physical quantities (such as distance, speed, electric charge, etc.). A value which represents a physical quantity has both pure magnitude and physical unit (hereafter simply called unit). A dimension of a physical quantity is the expression of the class of units used to measure a quantity. Within a given dimension, quantity is measured by comparing it to a quantity which magnitude is defined to be a unit. For example, a quantity with twice the mass of a unit sample is said to have the mass of 2. Several unity quantities may be defined in one dimension. They are distinguished by unique names and represent different magnitudes, e.g. there are several units in the dimension of physical quantity length: meter, millimeter, foot, inch, angstrom, etc. Measured quantities do not usually stand alone. They appear in the equations which characterize other quantities or form various physical statements and hypothesis. The existence of physical dimensions imposes strong and useful constraints on the types of these equations. A violation of these constraints leads to conceptual or computational errors. An example of the constraint could be that two quantities can be added only if they have the same physical dimension. There is no physical or engineering problem for which the violation of this constraint would be ever physically meaningful, like adding 5 meters to 2 seconds. More common problems result from usage of the different units within the same dimension, like computing with meters and feet in the same model. A shining example of the most expensive bugs resulting from the 5

7 confusion of units is the case of NASA s Mars orbiter 1. The spacecraft completed 10-month journey to Mars before it was lost on September 23, 1999, resulting in the loss of US$ 125 million of American tax payers money. The problem was that one engineering team used metric units while another used English units for a key spacecraft operation. The spacecraft thus moved too close to the planet s atmosphere where it presumably burned and broke into pieces. The mission failed on a day when engineers had expected to celebrate the craft s entry into Mars orbit. Dimensional analysis, written as DA throughout the rest of this thesis, is a conceptual tool which deals with checking the constraints imposed by the physical dimensions in physical equations. It is worth to note that this meaning is different than the meaning of dimensional analysis based on the Buckingham theorem [2] which yields statements about irrelevance of some quantities in a problem, or the need for additional parameters. This thesis presents a solution of DA for the field of electronics which concerns with formal description of electronic circuits in hardware description languages (HDL). These languages can describe circuit s operation and test its behaviour by means of simulation. The most used representatives of HDL languages are the languages Verilog and VHDL. In particular, the system for annotating physical quantities by the dimensional information as well as the algorithm of DA will be implemented into the commercial tool ModLyng TM. This tool is developed by the company Lynguent, Inc. 2 which was still in the beta version as this thesis was being written. ModLyng is an integrated development environment for analog/mixed-signal (AMS) integrated circuit and systems design. It can read AMS models written in Verilog, VHDL and MAST languages. A model is internally stored and manipulated in the AIREbased intermediate representation [11]. The implementation of the DA presented by the thesis works upon this representation. The thesis is organized in chapters as follows. The principles of DA are discussed in the second chapter. They are demonstrated on a short example - a model of a diode written in Verilog language. Later in the chapter, an extension to the usual view of physical dimensions, i.e. a classification of dimensionless units, is presented and two approaches for its handling in the dimensional analysis are proposed. The importance of this extension and benefits of incorporating it into the dimensional analysis is also well documented. 1 see more details in the article at 2 see their home page at 6

8 In Chapter 3, the physical dimensions and the operations with them are formalized by using a mathematical notation. Chapter 4 contains an abstraction of HDL languages which is to our belief sufficient for the needs of dimensional analysis. Chapter 5 presents an algorithm of dimensional analysis which operates on this abstraction (Chapter 4) together with the formal model of physical dimensions (Chapter 3). Design and implementation of the system for dimensional analysis in ModLyng tool is described in Chapter 6. The results and the conclusion are stated in the last chapter. The instructions for installation and usage of the enclosed binaries and the testing programs are described in detail in the Appendix A. 7

9 Chapter 2 Theory 2.1 Hardware Description Languages and ModLyng TM As summarized in [10]: a hardware description language or HDL is any language from a class of computer languages for formal description of electronic circuits. It can describe the circuit s operation, its design and organization, and tests to verify its operation by means of simulation. HDL is a standard text-based expression of the temporal behaviour and/or (spatial) circuit structure of an electronic system. In contrast to a software programming language, the syntax and semantics of a HDL language include explicit notations for expressing time and concurrency which are the primary attributes of hardware. HDLs are used to write executable specifications of some part of hardware. A simulation program, designed to implement the underlying semantics of the language statements, coupled with simulating the progress of time, provides the hardware designer with the ability to model a piece of hardware before it is created physically. It is this executability that gives the illusion of HDLs being a programming language. Simulators capable of supporting discrete event (digital), and continuous time (analog) modeling exist and HDLs targeted for each are available. 8

10 Figure 2.1: ModLyng features diagram It is certainly possible to represent hardware semantics using traditional programming languages such as C++ (and augmented with extensive and unwieldy class libraries.) However, the C++ language does not include any capability for expressing time explicitly and consequently is not a proper hardware description language. The most used modern HDLs are Verilog, VHDL and MAST languages. They provide very limited support for ascribing units to physical variables that represent values of physical quantities. Therefore they put a burden on the engineer to keep track of the units used in any equation or routine and to control that they do not violate constraints resulting from the existence of physical dimensions. A comprehensive example of HDL model and the benefits resulting from the dimensional analysis are shown in Section 2.3. ModLyng is an integrated environment for analog modeling. Its main features are illustrated by the Diagram 2.1. ModLyng is conceptually placed above the simulating environment. It reads models written in HDL languages, translates them into internal AIRE-based intermediate representation and allows the users to run various analysis tools. The models can be also created directly in ModLyng with its graphical tools 9

11 and then exported into supported HDLs. The implementation of the dimensional analysis presented in this thesis works upon its intermediate representation, which can store dimensional information. This representation can be created from / exported to a proprietary XML-based language called CMX. This language is also a place where physical quantities can be annotated by dimensional information. Use case scenarios for the usage of dimensional analysis in ModLyng are listed in Section 6.1 in the chapter about implementation. 2.2 Principles of Dimensional Analysis As already outlined in the introduction, a complete and unambiguous representation of the value of a physical quantity must contain not only a pure magnitude, but also either an explicit or implicit statement of the units employed, which in turn must be consistent with the underlying dimensions. The numerical representation of the quantity s pure magnitude depends on the selected unit; however, the product of the quantity s magnitude and its dimensions represents an invariant physical quantity. The conversion between different units is nothing more than multiplication of the magnitude by a constant. Consider the speed of light in vacuum, the distance that the light travels per unit of time. Its value can be expressed as m/s or km/h. Both expressions represent the same physical speed and the only difference is the choice of units for expressing the physical dimensions of length divided by time. The choice of units and even dimensions used in physical and engineering problems and computations is a matter of convention. Physical dimensions can be divided into two groups - base and derived. Base dimensions are assumed to be mutually independent and such that every other physical dimension can be derived from them in the process of multiplication and division with other base or derived dimension. Every base dimension is bound with a base unit, which is usually the most used unit in the dimension. Other units in the dimension can be converted to its base unit or among themselves by multiplying with a conversion factor. In addition, units can be usually prefixed with a symbol which specifies a multiplication factor of the unit. Units in the space of derived dimensions are derived units. A derived dimension can be bound with a reference unit. The choice of the base dimensions is also a matter of convention. For example, traditional physical base dimensions for physical quantities are 10

12 Table 2.1: Base dimension defined by SI symbol symbol physical for unit for dimension base unit quantity quantity symbol length l L meter m mass m M kilogram kg time t T second s electric current i I ampere A thermodynamic temperature T O kelvin K amount of substance n N mole mol luminous intensity I J candela cd mass, length, time and temperature. The dimension for the physical quantity of speed is a derived dimension and may be expressed as a quotient of length divided by time. Some physicists, nevertheless, do not recognize temperature as a base dimension of physical quantities since temperature is simply the energy per particle per degree of freedom and can thus be expressed in terms of mass, length and time. Seven base dimensions are defined in the International System of Units (SI) [3]. They are shown (with base units and units symbols) in the table 2.1. This system of base dimensions and units is well standardized and established by now and it will be used in this thesis. The standard also specifies reference units for some derived dimensions such as for the physical quantities: frequency - Hz, pressure - Pa, power - W, etc. All units in the SI system of units can be prefixed with any of the several decimal prefix defined by the standard. The system of base dimensions should not be confused with the system of units. The system of units can be simply viewed as a set of units from an arbitrary number of dimensions. The International System of Units mentioned above provides units sufficient for all natural sciences. Another well-known example is the English system of units, however it provides units only for the base dimensions of length and mass and for the derived dimensions of area and volume. There are one special types of quantities which do not have physical dimensions, they are dimensionless. These quantities are defined in such 11

13 a way that all units cancel themselves out or they can not be described in the terms of the base quantities. Dimensionless quantities are analyzed in depth in the Section 2.4 and the proposed approaches for their consistency checking are in the Section 2.5. The constraints on the equations using physical quantities are these: 1. Two physical quantities can be added, subtracted or compared only if they have the same dimension and unit within this dimension. For example, 6 kg + 3 s is not valid, but 6 kg + 3 kg is valid. The equation 6 kg + 3 g is not valid too according to our definition. This does not hold if a necessary conversion is performed. Then it is valid and the result equals to kg. 2. The result of the consistent addition and subtraction of two physical quantities have the same dimension and unit as its operands. 3. Two physical quantities can be multiplied and divided only if the base units in their common base dimensions are the same. For example, 6 kg 3 s is valid as well as 6 kg m/3 kg is valid, but 6 kg m/3 g is not. This does not hold if a necessary conversion is performed. Then it is valid and the result equals to 2000 m or 2 km. 4. The result of the consistent multiplication has dimension in which each base dimension appears in dimension of one of its operands and if both operands have that dimension, their powers are added, otherwise the power is the same. The unit within each base dimension is the same as the unit of its operands. For example, the result of the equation 6 kg 3 m 3 kg 2 has unit [kg m]. 5. The result of the consistent division has the dimension which is counted similarly as in case of multiplication with the difference that the powers are subtracted. For example, the result of the equation 6 kg 3 m/3 kg 2 has dimension [kg 5 m]. 6. Transcendental functions, such as trigonometric functions, exponent, logarithm, etc. can take only a dimensionless quantity as an argument 1. 1 A transcendental function is a function which is not algebraic, i.e. can t be constructed using only a finite number of elementary operations together with the inverses of functions capable of being so constructed. The requirement that their argument must be a dimensionless quantity is clear when one takes the Taylor expansion of 12

14 7. The result of the raising of the quantity to the exponent (which is dimensionless according to the previous rule) has a dimension in which each base dimension is a base dimension of the argument quantity with the power multiplied by the exponent. The unit within each base dimension is the same as the unit of its argument quantity. For example, the result of the equation (6 kg 2 m 3 ) 2 has unit [6 kg 4 m 6 ]. All these rules are formally written in Chapter 3. In this formalization we do not take into account the possibilities of a unit conversion which allows more equations to be valid. There is one good reason for this: Units are not a part of the models written in HDL languages and simulators are unaware of them. They operate only with magnitudes of the quantities. This is a principal problem. For example, in the equation a = 6 kg m/3 g the simulator computes that quantity a has the value of 2. The analysis can derive that the unit of a is [km] to match the resulting value of 2. Consider now an equation a = 6 kg m/3 g 1/2. What is the unit of a? The first option is to let a have the unit [kg g 1/2 m], but this can lead to very long and confusing units in long equations which are common in the models. Another option is to perform a unit conversion: [g] is converted to [kg] what results in an additional factor - a would equal 2 kg 1/2 m with the factor 31, 62. This factor could be included into the equation, but this would break one of the main requirements on the implementation of the analysis in the target environment - that it can not modify the model. The analysis is meant to be read-only and at most it can add other units to the quantities which were not annotated in the model. Therefore we impose stronger rules on the units consistency. 2.3 Example Consider the simple model written in the HDL language Verilog-A which describes the basic behavior of a diode: include " d i s c i p l i n e. h" module diode ( anode, cathode ) ; inout anode, ca tho de ; e l e c t r i c a l anode, cathode ; these functions. For example, the logarithm of 3 kg is undefined even though the logarithm of 3 is nearly An attempt to compute ln 3 kg would produce 3 kg 9 kg2 2 + which is dimensionally incompatible. 13

15 //Area s c a l i n g f a c t o r parameter real area = 1 from ( 0 : i n f ) ; // Saturation current [A] parameter real i s =1e 14 from ( 0 : i n f ) ; // Emission c o e f f i c i e n t parameter real n=1 from ( 0 : i n f ) ; //Zero b ias junction capacitance [F] parameter real cjo=0 from [ 0 : i n f ) ; // Grading c o e f f i c i e n t parameter real m=0.5 from [ 0 : i n f ) ; //Body p o t e n t i a l [V] parameter real phi =0.7 exclude 0 ; //Forward b ias capacitance f a c t o r parameter real f c =0.5 from ( 0 : 1 ] ; // Transit time [ s ] parameter real t t=1p from [ 0 : i n f ) ; // Reverse breakdown v o l t a g e [V] parameter real bv=1.0e+100 from [ 0 : i n f ) ; // S e r ies r e s i s t a n c e [Ohm] parameter real rs=0 from [ 0 : i n f ) ; real Vd, Id, Qd; real f1, f2, f3, fcp ; real ibv ; analog i n i t i a l _ s t e p ) begin f1 = ( phi /(1 m) ) (1 pow((1 f c ), m) ) ; f2 = pow ((1 f c ), (1 + m) ) ; f3 = 1 f c (1 + m) ; fcp = f c phi ; ibv = i s bv/$vt ; end Vd = V( anode, cathode ) ; Id = I ( anode, cathode ) ; // i n t r i n s i c diode. i f (Vd < 5 n $vt ) begin i f (Vd == bv ) I ( anode, cathode ) <+ area ibv ; else i f (Vd > bv ) I ( anode, cathode ) <+ area i s ; else 14

16 I ( anode, cathode ) <+ area i s ( exp( (bv + Vd) / $vt ) 1 + bv/$vt ) ; end else I ( anode, cathode ) <+ area i s ( exp ( (Vd rs Id ) /(n $vt ) ) 1) ; // capacitance ( junction and d i f f u s i o n ). i f (Vd <= fcp ) Qd = t t Id + area cjo phi (1 pow ((1 Vd/ phi ), (1 m) ) ) /(1 m) ; else Qd = t t Id + area cjo ( f1 + (1/ f2 ) ( f3 (Vd fcp ) + ( 0. 5 m/ phi ) (Vd Vd fcp fcp ) ) ) ; I ( anode, cathode ) <+ ddt (Qd) ; end endmodule A number of parameters are defined at the beginning of the model. Some of them represent physical quantities and their physical units are expressed in the comments. Others are factors or quotients which are dimensionless quantities. Verilog-A does not provide any significant support for dimensions and units which could be used for the automatic dimensional analysis. The only information about physical units is stored in natures which are used to define disciplines 2. The electrical discipline is used in our example as a type for module ports (anode and cathode) and its definition is written in the file discipline.h as follows: // Cu rrent i n amperes nature Current units = "A" ; a c c e s s = I ; idt_nature = Charge ; i f d e f CURRENT_ABSTOL a b s t o l = CURRENT_ABSTOL ; e l s e a b s t o l = 1.0 e 12; endif endnature // P otential in v o l t s nature Voltage units = "V" ; a c c e s s = V; idt_nature = Flux ; i f d e f VOLTAGE_ABSTOL 2 For more about these language constructs in Verilog-A see [1] 15

17 a b s t o l = VOLTAGE_ABSTOL ; e l s e a b s t o l = 1e 6; endif endnature // Conservative d i s c i p l i n e d i s c i p l i n e e l e c t r i c a l p o t e n t i a l Voltage ; flow Current ; e n d d i s c i p l i n e The only constraint imposed by natures is that the connection between two ports of some disciplines can be created only if they have compatible natures, hence the same physical quantities are used. By way of several examples of a model of a diode, it will be attempted to illustrate the problems which are meant to be solved by the dimensional analysis and how it can automatically check the plausibility of derived equations and computations: 1. When declaring local variables Vd, Id, f1, fcp, etc. it is not clear what are their dimensions since it is not clear how exactly they will be used. However, the intended purpose of these variables is apparent from their first assignment and their dimensions can be inferred from the dimension of the expressions on the right side of the assignment. This information is used in dimensional analysis in later equations. f3 = 1 - fc*(1 + m) Only dimensionless quantities (fc is a factor and m is a coefficient) are used in this assignment. The result of algebraic operations with the dimensionless quantities has no dimension, therefore f3 will be dimensionless. fcp = fc phi phi represents a body potential (which has dimension electric potential with unit volt - V). Multiplication of physical quantity by dimensionless physical quantity retains the dimension therefore fcp will have the dimension electric potential. These examples demonstrate that it is not necessary to annotate explicitly all variables holding physical quantities. Dimensional analysis is capable of inferring accurate dimensions. 16

18 2. if (Vd < 5 n $vt) A comparison of two physical quantities is admissible only if they have the same dimension and the same physical unit within this dimension. Two quantities with dimension electric potential are compared in this relation: Vd gets its dimension in the assignment Vd = V(anode, cathode) where electric potential between two ports is measured in volts 5 n $vtis a multiplication of dimensionless quantities and a thermal voltage which has dimension electric potential Consider a situation in which the thermal voltage (acquired by the built-in function $vt) is to be measured in millivolts instead of volts. In this case the comparison would be not plausible and dimensional analysis would report an error: comparing quantities with different units. 3. Qd = tt Id + area cjo phi (1 pow((1 Vd/phi), (1 m)))/(1 m) This is a rather complicated expression but very common in models. Writing it correctly requires serious attention and automatic dimensional analysis could make it significantly easier. Let s look at the possible risks. At the top level of the expression we have addition. The leftside expression tt Id has the dimension time electric current with unit [s A] (tt is transit time and Id gets its dimension in the assignment Id = I(anode) where electric current in the shortcircuit branch from the anode to ground is measured). To make the addition correct, the right side has to have the same dimension as the left side. Any other dimension would cause dimensional analysis to report an error. The first three operands on the right side of the addition (area cjo phi) form dimension capacitance electric potential with the unit [F V] (area is a dimensionless scaling factor, cjo is a zero-biased junction capacitance and phi is a body potential). This is a derived dimension and it could be expressed in a reduced form by base dimensions: capacitance is L -2 M -1 T 4 I 2 (expressed in symbols defined in Table 2.1) and potential is L 2 MT 3 I 1. After the reduction we get TI which is time electric current - the same dimension as the 17

19 left side. This implies that the rest of the expression must be dimensionless; otherwise the multiplication would alter the resulting dimension of the expression. And truly, (1 m) is dimensionless (subtraction of a dimensionless number and a dimensionless grading coefficient); Vd/phi is the division of quantities with the same dimension, which results in a dimensionless quotient; and so we get division of two dimensionless quantities (1 pow((1 Vd/phi), (1 m))) and (1 m) which all results in a dimensionless quantity. It is important to mention the function pow(x,y). It represents exponentiation x y and, as mentioned before, it can only use a pure (dimensionless) number as the exponent y. Dimensional analysis is capable of controlling the usage of all functions and reports any violation of the limitations imposed by the dimensions of arguments and result. 4. I(anode, cathode) <+ ddt(qd) An electrical current is contributed to the branch between anode and cathode in this equation. The value of the contribution comes from the time derivative of the variable Qd and it must have as its physical dimension electrical current. The derivative of time alters the dimension of input by reducing its dimension time by one power. This implies that the input has to have dimension TI which is true in this case as shown for Qd in the previous example. The derivative of time transforms the dimension of the input and therefore behaves differently than a normal function. The dimension of the result in a normal function is known in advance and does not depend on the dimensions of the input arguments as it is in the case of derivative. Other transforms which behave similarly as the derivative figure also in engineering problems, e.g. the integral of time or Laplace transform. The usage of transforms represents the same risk of dimensional inconsistency as the usage of functions and can also be automatically checked by dimensional analysis. For more about functions see Section 2.7. As shown by these examples, the dimensional analysis can detect any operations on physical quantities with unintentionally different dimensions or units, and it strictly controls passing of dimensionally correct arguments to the functions and the transforms as well as the usage of 18

20 their results. Thus it can reveal the conceptual errors in modeling or plain typographical errors, which are even harder to track. 2.4 Dimensionless Quantities A dimensionless quantity is a quantity of dimension one. That means it is defined either in such a way that all units cancel themselves out or it can not be described in terms of the SI base quantities 3. We identify the following classes of dimensionless quantities: 1. dimensionless ratios 2. counts 3. logarithmic quantities (a) logarithmic ratio quantities (b) logarithmic counts In the following sections are all the classes described Dimensionless Ratios A dimensionless ratio is a quantity defined by a ratio of quantities which units cancel themselves out. They are either simple ratios, for example, refractive index which is the ratio of a velocity of light in a vacuum and its velocity in a specified medium, or more complex where the equation consists of several quantities, for example the Fourier number (Fo) is a dimensionless quantity which characterizes heat conduction and is defined as Fo = αt (where α is the thermal diffusivity [m 2 /s], t is the d 2 characteristic time [s] and d is the length through which the conduction occurs [m]). In some cases it is useful to distinguish between dimensionless ratios of different natures and to give special names to the dimensionless unit number one: 3 A quantity, which is dimensionless in one system of units, doesn t need to be dimensionless in other system of units. Consider a physical quantity permittivity of free space ε 0 (also called vacuum permittivity). In cgs system of units, the unit for dimension electric charge is statcoloumb (statc) and is defined in such a way that permittivity of free space (vacuum) is ε 0 = 1 4Π what is dimensionless. In SI system of units, it is ε 0 = F/m what is dimensioned quantity. 19

21 1. radian [rad] which is a unit for the dimensionless quantity of plane angel and is equal to an angle at the center of a circle whose arc is equal in length to the radius (i.e. ratio of two lengths [m/m]). Plane angle can be measured also in other units which are submultiples of radian. These are: a degree [ ], a minute [ ] and a second [ ]. An example of a quantity which uses plain angle in its definition is angular velocity with the unit radian per second [rad/s]. 2. steradian [sr] which is a unit for dimensionless quantity of solid angel and is equal to an angle at the center of a sphere subtended by a part of the surface equal in area to the square of the radius (i.e. ration of two areas [m 2 /m 2 ]). An example of a quantity which uses solid angle in its definition is radiant intensity with the unit watt per steradian [W/sr] Counts A count is a quantity which can not be described in terms of the SI base quantities. They are numbers, such as a number of molecules or degeneracy in quantum mechanics (the number of independent states of the same energy) Logarithmic Quantities A logarithmic quantity is a quantity whose magnitude is measured as a logarithm of another dimensionless (but non-logarithmic) quantity 4. A unit of a logarithmic quantity, called logarithmic unit, is defined by the base of the logarithm and a multiplicative constant 5. Therefore a bel is defined as B = log 10 10, a decibel as db = 0.1 log 10 10, a neper as Np = log e e, etc. Logarithmic ratio quantities. Quantities defined as logarithm of ratio of two quantities of the same kind (i.e. all units cancel themselves out) fall into this class. Values of logarithmic ratio physical quantities 4 Recall that it is not physically meaningful to pass a dimensioned quantity into a logarithm as it is a transcendental function. 5 All disparate types of logarithmic quantities, such as power levels in physics, tone intervals on a musical scale or information entropy, can be understood as dealing with fundamentally the same underlying system of logarithmic units. These various quantities with their units are all exactly comparable with each other, e.g. 7 bits can be compared with 2 decibels. They are distinguished only by the application of these quantities and units in describing different domains of study. More on this see [5]. 20

22 are expressed in logarithmic units of the neper or the bel and decimal multiples and submultiples of the neper and the bel formed by attaching SI prefixes to them, such as the millineper [mnp] and the decibel [db]. So the statement L X = m db = m B (where m is a number) is interpreted 10 as log 10 ( X X 0 ) = m, where X 10 0 is a reference level. Thus when L X = 1 B then X X 0 = 10, and when L X = 1 db then X X 0 = It is worthwhile to remind that the values of logarithmic ratio quantities as well as the values of all logarithmic quantities are independent on the logarithmic unit which expresses that value. They can be expressed in either the neper or the bel or other (sub)multiples of them and they can be converted from one to another simply by multiplying by the appropriate conversion factor 6. This is the same situation as converting length from meters to millimeters. Logarithmic ratio quantities typically try to capture human perception of some quantitative property. Sound pressure level (SPL or L p ) is a well-known example. It is defined as L p = log( p2 where p p0), 2 0 is the reference sound pressure (reference level is X 0 = p 2 0 ) and p is sound pressure being measured. The reference sound pressure in air is p 0 = 20 µpa, in water p 0 = 1 µpa. When reporting a type or a value of logarithmic ratio quantity, one must always specify the reference level. Suggested notation[9] is L x (rex ref ) where x ref is value of reference quantity (reference level) 7. For instance, L p (re 20 µpa) = 25 db means that the level of a certain sound pressure is 25 db above the reference pressure p 0 = 20 µpa. Or the following condensed notation can be used instead: 25 db(20 µpa). Logarithmic counts. Logarithmic counts are quantities defined as logarithm of dimensionless count quantities. This is for example entropy which is defined as a logarithm of a number of possible distinct states that a system could (statistically) be in. 6 For example, 1 B = ln10 2 Np. For the computation of this conversion factor see [9]. 7 This rule preclude, for example, the use of the symbol dbm to indicate a reference level of power of 1 mw. SI[3] does not permit adding of attachments to unit symbols. 21

23 2.5 DA for Dimensionless Quantities There are two basic approaches when dealing with dimensionless quantities in a dimensional analysis: 1. Treating all dimensionless quantities as having the same type, so that they all behave as quantities of dimension one 8. The operations then have the following semantics: (a) The result of multiplication and division of two quantities of dimension one has dimension one. (b) The quantity of dimension one can be added, subtracted or compared only with the quantity of dimension one. (c) The result of multiplication and division of the quantity of dimension one by dimensioned quantity has a dimension of the dimensioned quantity (or its reciprocal). So dimension one behaves as an algebraic identity element, i.e. it is neutral to these operations. (d) The result of raising a quantity with dimension one to a power has dimension one. 2. Differentiating dimensionless quantities according to their class and the underlying quantity, if they have any (which is not true for counts). So two dimensionless quantities with different meaning would behave as having different dimensions. This will affect the semantics of operations. In next sections we analyze the second approach for each class of dimensionless quantities. They do not contain a comprehensive list of all additions to dimensional constraints listed in Section 2.2, but rather contain benefits of this approach for the engineer developing in HDLs DA for Dimensionless Ratios If the second approach for dimensionless ratio quantities is used, there are the following consequences: 8 The dimension one needs to be represented in a way that it would not be confused with a quantity with no dimension. A dimension for a quantity with no dimension is unknown and is waiting to be deduced from expressions where quantity emerges. 22

24 It can be checked if dimensionless ratio quantities with different underlying quantities are not added, subtracted or compared. For example, if we have two dimensionless ratio quantities U 1 and U 2 where U 1 is a refractive index of some medium, what is a ratio of two speeds, and U 2 is a mass ratio, what is a ratio of two masses, then the expression U 1 < U 2 would cause an error and an engineer will realize that he wanted to compare two refractive indices instead. It can not be differentiated between two dimensionless ratio quantities with the same underlying quantity, but different meaning. For instance, a mass fraction and a mass ratio 9 are both defined as ratios of two masses, but they have different physical meaning. It is unclear what would be the resulting dimension of a multiplication or division of two dimensionless ratio quantities. A simple solution is that the result has dimension one. However, it is thinkable that the dimension is derived from the operation, i.e. if there are two dimensionless ratios R 1 = U 1 U 1 and R 2 = U 2 U 2, where U 1 and U 2 are dimensioned units and U 1 U 2, then R 1 R 2 = U 1 U 2 U 1 U 2. Deriving a dimension for division is also theoretically thinkable, i.e. U 1 R 1 U R 2 = 1 U 2, but the level of ratio can grow into indefinite depth. It is U 2 questionable if such a strong DA is needed. This point can be the subject of the future elaboration. Angular quantities. A special approach can be undertaken for dimensionless ratio quantities of plane and solid angles. As already mentioned, unit 1 has special names for these quantities and they are often used. It would be convenient to add specialized base dimensions for these quantities - a dimension for a plane angel with the radian as a reference unit and a dimension for a solid angle with the steradian as a reference unit. Calculus with these dimensions would be the same as with other base or derived dimensions. The following consequences would be brought about by this approach: The angles can have several units. This is particularly the case of a plane angle which value can be expressed not only in radians, 9 In aerospace engineering, the mass fraction is a measure of a vehicle s performance and mass ration is a measure of the vehicle s proportion. 23

25 but also in degrees, minutes, seconds and their multiples and submultiples. This way the value of a plane angle could be converted between different units. The dimensional consistency can be checked when operating with angular quantities, e.g. radians can be added only to radians. Quantities which are described also by angular quantities can be well defined and thus their true physical nature can be expressed, e.g. angular velocity [rad/s], angular acceleration [rad/s 2 ], radiant intensity [W/sr], radiance [W/m 2 sr], etc. It can be checked that only a plane angle quantity with a value in radians is passed as an argument to trigonometric functions (sinus, cosines, etc.) DA for Counts Because the counts can not be described as derived quantities of SI base quantities, the second approach of DA is not applicable to them. Hence, counts will be always of dimension one DA for Logarithmic Quantities It is important to distinguish the class of a dimensionless quantity used as an argument of a logarithm: dimensionless ratio or count. The fact that the logarithmic quantity is a logarithm of some dimensionless quantity adds one distinctive property - it is measured on a logarithmic scale and it can be perceived as having a special kind of dimension - logarithmic dimension with logarithmic units. DA for logarithmic ratio quantities. As mentioned before, when reporting a value of a logarithmic ratio quantity, one must always specify a reference level, which figures as a denominator in the dimensionless ratio. Thus the type of a dimensionless logarithmic quantity is determined by: 1. the kind of the dimensionless ratio quantity which is passed to a logarithm as an argument, that means the dimension of a underlying quantity in the dimensionless ratio and 2. the reference level. 24

26 There are the following consequences brought about by this approach: Errors can be detected when adding or subtracting logarithmic quantities with a different type of dimensionless ratio or a different reference value, as they represent different physical quantities. For example, it is an error to add the electric field strength level to the sound pressure level or to add the sound pressure level in water to the sound pressure level in air. It can checked that logarithmic quantities of the same type are added only if they have the same logarithmic unit, e.g adding bels only to bels and not to decibels. A value of a logarithmic quantity can be converted between different logarithmic units (e.g. bels to nepers). Physical quantity which is a ratio or a product of a logarithmic quantity and a dimensioned quantity can be expressed. For example, this is the case of a physical quantity attenuation of a cable. It is the reduction in amplitude and intensity of a signal per unit of length and is measured in decibels per unit of length. DA for logarithmic counts. DA for logarithmic counts can only check if logarithmic count quantity is not added, subtracted or compared to logarithmic count quantity with a different logarithmic unit or other dimensionless or dimensioned quantity. In other cases, the logarithmic count quantity behaves in the same way as the dimensionless quantity of dimension one. However, these constraints will be probably checked very rarely in DA for HDLs as there are only a few logarithmic counts (like entropy) in physics DA Between Different Classes of Quantities To prevent dealing with very complicated dimensions, it is convenient to declare that a dimensionless quantity of any class which occurs in an operation of multiplication or division with other dimensioned or dimensionless quantity behaves as quantity of dimension one. The direct consequence of this is that the dimensionless quantity of class other than count (dimensionless ratio, logarithmic quantity) can originate only by explicit declaration and not by deriving from an expression with other quantities. This point can be the subject of the future elaboration. 25

27 2.6 Thermodynamic Temperature The base dimension of thermodynamic temperature with its units deserves a more extensive comment. The base unit of temperature is Kelvin and it is defined[3] as the fraction 1/ of the thermodynamic temperature of the triple point of water. It follows that the thermodynamic temperature of the triple point of water is exactly kelvins. Because of the manner in which temperature scales used to be defined, it remains common practice to express a thermodynamic temperature, symbol T, in terms of its difference from the reference temperature T 0 = K, the ice point. This difference is called the Celsius temperature, symbol t, which is defined by the quantity equation: t = T T 0 The unit of Celsius temperature is the degree Celsius, symbol C, which is by definition equal in magnitude to the kelvin. A difference or interval of temperature may be expressed in kelvins or in degrees Celsius, the numerical value of the temperature difference being the same. However, the numerical value of a Celsius temperature expressed in degrees Celsius is related to the numerical value of the thermodynamic temperature expressed in kelvins by the relation t/ C = T/K The important fact about thermodynamic temperature is the difference between the absolute temperature and the relative temperature when expressed in Kelvins or degrees Celsius. For instance, a decrease of temperature of 1 K is equivalent to a decrease of temperature of 1 C. In this case temperature is relative and the conversion factor is 1. But when expressing temperature of triple point of water, which is 0 C or K, the temperature is absolute, conversion factor is also 1 but the offset is K. Hence, the intended usage of temperature (either absolute or relative) in expressions is not known without additional information. However, this problem affects only the conversion functionality of the system and the dimensional calculus remains unaffected whether Kelvins or degrees Celsius are used or not. 26

28 Fahrenheit scale is another widely used temperature scale 10. Fahrenheit is defined similarly as degree Celsius, but a different conversion factor and an offset figure in equation: t/ F = T/K Therefore the same holds for the Fahrenheit scale as for the Celsius scale. The problem of the absolute vs. relative quantities is irrelevant for other quantities since no other units are defined by a dimension s reference unit with use of an offset in addition to a conversion factor. 2.7 Functions In this section, functions are categorized according to characteristics important for dimensional analysis Normal Functions These are the functions where the input and output parameters have clear dimensional information: both the dimension and the unit. Consider a simple Verilog function: analog function real ohm_law ; input voltage, r e s i s t a n c e ; real voltage, r e s i s t a n c e ; begin ohm_law = voltage / r e s i s t a n c e ; end endfuntion It is clear what are the dimensions and the units of the input and output parameters: variable voltage represents the derived quantity of electric potential with the unit volt [V], variable resistance represents the derived quantity of electric resistance with unit ohm [Ω] and the result has a dimension of the base quantity of electric current with the unit ampere [A]. Hence, the function can be declared with the complete dimensional information for the parameters and the result. 10 There is a number of other temperature scales which are rather historical and are not used in today s physics, e.g. Rankine scale, Newton scale, Delisle scale, Reaumur scale, etc. 27

29 2.7.2 Generic Functions A generic function is a function which does not specify the exact dimensions and units of the input parameters. The dimension and unit of the function s result thus depends on the dimensions and units of the input parameters and an expression which assigns a value to the result. Consider a simple Verilog function. analog function real multiply ; input n1, n2 ; real n1, n2 ; begin multiply = n1 n2 ; end endfunction This function takes two numbers and returns its product. It behaves like a C++ template - quantities with any dimensions can be the inputs, the only constraint is that they must form a consistent multiplication. This kind of functions can not be declared with any dimensional information. The unit of the result can be computed only in the place of function s call where the units of parameters are known. It is also meaningful to declare a function for which only dimensions of parameters are known, but not the units within those dimensions. Consider the function ohm_law with specified dimensions for voltage and for resistance but with no particular units specified for them. DA for function calls of such function can control dimensions of parameters and computes the unit of the result from the units of the parameters Functions with Definitions If a function has definition (that means its body is defined), DA can treat every function call as inlined function body. This means that a function call will expand into several expressions. Thus DA can check dimensional information for parameters and the consistency of the body s expressions in case of normal functions or can compute dimensional information for the result in case of generic functions Functions without Definitions Function definitions for some functions do not have to be reachable in HDLs. For example, this is the case of built-in functions or transform 28