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2.A: Appendix- Physical Constants and Material Properties

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    2.A.1 SI Units

    The main SI units used in RF and microwave engineering are given in Table \(\PageIndex{1}\). Symbols for SI units\(^{1}\) (from the French name Systeme International d’Unites) are written in upright roman font. (Source: U.S. National Institute of Standards and Technology (2006) [26], and 2002 CODATA recommended values of constants [27].)

    The International Organization for Standardization (ISO) maintains the International System of Quantities (ISQ) defines the quantities that are measured in SI units [28]. These are set out in the ISO 8000 standard (which is jointly published with the International Electrotechnical Commission (IEC) as the IEC 8000).

    • The fundamental units of the SI system are meter, kilogram, second, candela, mole, and Kelvin.
    • The unit of length is spelled meter in the United States and metre in other countries.
    • The unit designations, such as m for meter, is called a symbol and not an abbreviation.
    • Symbols for units are written in lowercase unless the symbol is derived from the name of a person. For example, the symbol for the unit of force is \(\text{N}\) as it is named after Isaac Newton. An exception is the use of \(\text{L}\) for liter to avoid possible confusion with \(\text{l}\), which looks like the numeral one and the letter \(\text{i}\).
    • A space separates a value from the symbol for the unit (e.g., \(5.6\text{ kg}\)). There is an exception for degrees, with the symbol \(^{\circ}\). For example, \(45\) degrees is written \(45^{\circ}\).

    SI Unit Combinations

    When SI units are multiplied a center dot is used. For example, newton meters is written \(\text{N}\cdot\text{m}\). When a unit is derived from the ratio of symbols then either a solidus \((/)\) or a negative exponent is used; the symbol for velocity (meters per second) is either \(\text{m/s}\) or \(\text{m}\cdot\text{s}^{−1}\). The use of multiple solidi for a combination symbol is confusing and must be avoided. So the symbol for acceleration is \(\text{m}\cdot\text{s}^{−2}\) or \(\text{m/s}^{2}\) and not \(\text{m/s/s}\). Another example is the thermal conductivity of aluminum at room temperature which is \(k = 237\text{ kW}\cdot\text{m}^{−1}\cdot\text{K}^{−1}\) and not \(k = 237\text{ kW/m/K}\) or \(237\text{ kW/m}\cdot\text{K}\). However, \(237\text{ kW}/\text{(m}\cdot\text{K)}\) is sometimes used.

    Consider calculation of the thermal resistance of a rod of cross-sectional area \(A\) and length \(\ell\):

    \[\label{eq:1}R_{\text{TH}}=\frac{\ell}{kA} \]

    If \(A = 0.3\text{ cm}^{2}\) and \(\ell = 2\text{ mm}\), the thermal resistance is

    \[\begin{align}\label{eq:2}R_{\text{TH}}&=\frac{(2\text{ mm})}{(237\text{ kW}\cdot\text{m}^{-1}\cdot\text{K}^{-1})\cdot(0.3\text{ cm}^{2})}=\frac{(2\cdot 10^{-3}\text{ m})}{237\cdot (10^{3}\cdot\text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1})\cdot 0.3\cdot (10^{-2}\cdot\text{m})^{2}} \\ \label{eq:3} &=\frac{2\cdot 10^{-3}}{237\cdot 10^{3}\cdot 0.3\cdot 10^{-4}}\cdot\frac{\text{m}}{\text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}\cdot\text{m}^{2}} \\ \label{eq:4} &=2.813\cdot 10^{-4}\text{ K}\cdot\text{W}^{-1}=281.3\:\mu\text{K/W}\end{align} \]

    This would be an error-prone calculation if the thermal conductivity was taken as \(237\text{ kW/m/K}\).

    The use of SI units initially means that calculations can be undertaken without the tedium of tracing units through calculations. This requires that the SI unit of the final result be known and assigned. Repeating the above calculation for the thermal resistance of a rod using Equation \(\eqref{eq:1}\), first express the

    SI unit Name Usage In terms of fundamental units
    \(\text{A}\) ampere current (abbreviated as amp) Fundamental unit
    \(\text{cd}\) candela luminous intensity Fundamental unit
    \(\text{C}\) coulomb charge \(\text{A}\cdot\text{s}\)
    \(\text{F}\) farad capacitance \(\text{kg}^{-1}\cdot\text{m}^{-2}\cdot\text{A}^{-2}\cdot\text{s}^{4}\)
    \(\text{g}\) gram weight \(=\text{kg}/1000\)
    \(\text{H}\) henry inductance \(\text{kg}\cdot\text{m}^{2}\cdot\text{A}^{-2}\cdot\text{s}^{-2}\)
    \(\text{J}\) joule unit of energy \(\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{-2}\)
    \(\text{K}\) kelvin thermodynamic temperature Fundamental unit
    \(\text{kg}\) kilogram SI fundamental unit Fundamental unit
    \(\text{m}\) meter length Fundamental unit
    \(\text{mol}\) mole amount of substance Fundamental unit
    \(\text{N}\) newton unit of force \(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\)
    \(\Sigma\) ohm resistance \(\text{kg}\cdot\text{m}^{2}\cdot\text{A}^{-2}\cdot\text{s}^{-3}\)
    \(\text{Pa}\) pascal pressure \(\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}\)
    \(\text{s}\) second time Fundamental unit
    \(\text{S}\) siemen admittance \(\text{kg}^{-1}\cdot\text{m}^{-2}\cdot\text{A}^{2}\cdot\text{s}^{3}\)
    \(\text{V}\) volt voltage \(\text{kg}\cdot\text{m}^{2}\cdot\text{A}^{-1}\cdot\text{s}^{-3}\)
    \(\text{W}\) watt power \(J\cdot\text{s}^{-1}\)

    Table \(\PageIndex{1}\): Main SI units used in RF and microwave engineering. The SI units used in electromagnetics are given in Table 1.5.1.

    quantities in SI units: \(\ell = 2\cdot 10^{−2}\text{ m};\: k = 2.37\cdot 10^{5}\text{ W/(m}\cdot\text{K)};\) and \(A = 0.3\cdot 10^{−4}\text{ m}^{2}\), then

    \[\label{eq:5}R_{\text{TH}}=\frac{2}{2.37\cdot 10^{5}\cdot 0.3\cdot 10^{-4}}=2.813\cdot 10^{-4} \]

    The SI unit of \(R_{\text{TH}}\) is \(\text{K/W}\), so that \(R_{\text{TH}} = 2.813\cdot 10^{−4}\text{ K}\cdot\text{W}^{−1} = 281.3\:\mu\text{K/W}\).

    SI Prefixes

    A prefix before a unit indicates a multiple of a unit (e.g., \(1\text{ pA}\) is \(10^{−12}\text{ amps}\)). (Source: 2015 ISO/IEC 8000 [28].) In 2009 new definitions of the prefixes for bits and bytes were adopted [28] removing the confusion over the earlier use of quantities such as kilobit to represent either \(1,000\text{ bits}\) or \(1,024\text{ bits}\). Now kilobit (\(\text{kbit}\)) always means \(1,000\text{ bits}\) and a new term kibibit (\(\text{Kibit}\)) means \(1,024\text{ bit}\). Also the now obsolete usage of kbps is replaced by \(\text{kbit/s}\) (kilobit per second). The prefix \(\text{k}\) stands for kilo (i.e. \(1,000\)) and \(\text{Ki}\) is the symbol for the binary prefix \(\text{kibi}\)- (i.e. \(1,024\)). (Note that “\(\text{K}\)” is sometimes used as an abbreviation for \(1,024\) but this is non-standard.) The symbol for byte (\(= 8\text{ bits}\)) is “\(\text{B}\)”.

    SI Prefixes SI Prefixes Prefixes for bits and bytes
    Symbol Factor Name Symbol Factor Name Name
    \(10^{-24}\) \(\text{y}\) yocto \(10^{1}\) \(\text{da}\) deca kilobit \(\text{kbit}\) \(1000\text{ bit}\)
    \(10^{-21}\) \(\text{z}\) zepto \(10^{2}\) \(\text{h}\) hecto megabit \(\text{Mbit}\) \(1000\text{ kbit}\)
    \(10^{-18}\) \(\text{a}\) atto \(10^{3}\) \(\text{k}\) kilo gigabit \(\text{Gbit}\) \(1000\text{ Mbit}\)
    \(10^{-15}\) \(\text{f}\) femto \(10^{6}\) \(\text{M}\) mega terabit \(\text{Tbit}\) \(1000\text{ Gbit}\)
    \(10^{-12}\) \(\text{p}\) pico \(10^{9}\) \(\text{G}\) giga kibibit \(\text{Kibit}\) \(1024\text{ bit}\)
    \(10^{-9}\) \(\text{n}\) nano \(10^{12}\) \(\text{T}\) tera mebibit \(\text{Mibit}\) \(1024\text{ Kibit}\)
    \(10^{-6}\) \(\mu\) micro \(10^{15}\) \(\text{P}\) peta gibibit \(\text{Gibit}\) \(1024\text{ Mibit}\)
    \(10^{-3}\) \(\text{m}\) milli \(10^{18}\) \(\text{E}\) exa tebibit \(\text{Tibit}\) \(1024\text{ Gibit}\)
    \(10^{-2}\) \(\text{c}\) centi \(10^{21}\) \(\text{Z}\) zetta kilobyte \(\text{kB}\) \(1000\text{ B}\)
    \(10^{-1}\) \(\text{d}\) deci \(10^{24}\) \(\text{Y}\) yotta kibibyte \(\text{KiB}\) \(1024\text{ B}\)

    Table \(\PageIndex{2}\): SI prefixes.

    Physical and Mathematical Constants

    Physical and mathematical constants in SI units. Source: U.S. National Institute of Standards and Technology (2006) [26], and 2002 CODATA recommended values of constants [27].

    Parameter Value Description
    \(c\) \(299792458\text{ m}\cdot\text{s}^{−1}\) Speed of light in a vacuum (free space)
    \(e\) \(1.6021765310^{−19}\text{ C}\) Elementary charge (negative of an electron’s charge)
    \(\text{e}\) \(2.718281828459045\) Natural log base
    \(\gamma\) \(0.577215664901532\) Euler's ratio
    \(\phi\) \(1.618033988749894\) Golden ratio
    \(\varepsilon_{0}\) \(8.854187817\times 10^{−12}\text{ F}\cdot\text{m}^{−1}\) Permittivity of a vacuum (free space)
    \(h\) \(6.6260693\times 10^{−34}\text{ J}\cdot\text{s}\) Planck constant (alt. \(\overline{h} = h/(2π)\))
    \(k\) \(1.3806505\times 10^{−23}\text{ J}\cdot\text{K}^{−1}\) the Boltzmann constant
    \(m_{e}\) \(9.1093826\times 10^{−31}\text{ kg}\) Electron mass
    \(\mu_{0}\) \(12.566370614\times 10^{−7}\text{ N}\cdot\text{A}^{-2}\) Permeability of free space \(= 4π\times 10^{−7}\text{ N}\cdot\text{A}^{−2}\)
    \(\pi\) \(3.14159265358979323846264\) Pi, ratio of circumference to diameter of a circle
    \(\text{P}\) \(101325\text{ Pa}\) Standard atmosphere (pressure)
    \(\eta\) \(376.730313461\:\Omega\) Characteristic impedance of vacuum (free space)
    \(\text{J}\) \(6.241509\times 10^{18}\text{ eV}\) \(1\) joule of energy in terms of electron-volts
    \(\text{eV}\) \(1.602176\times 10^{−19}\text{ J}\) \(1\text{ eV}\) of energy in joules (the energy required to move a charge \(e\) through a potential of \(1\text{ V}\))

    Table \(\PageIndex{3}\): Physical and mathematical constants.

    Accuracy and Precision

    Precision is a description of statistical variability or random error while accuracy includes systematic errors in a measurement or calculation (and these are not random) combined with statistical variability. For example, consider the calculation of the resistance of a uniform length of metal using dimensions (length \(\ell\), width \(w\), thickness \(t\), which are known accurately) and resistivity \(\rho\) where the \(\rho\) may not be known accurately. The resistance \(R = \rho\ell /(wt)\).

    The resistance calculation would not be accurate if the resistivity is not known accurately. However the calculation would be precise if the resistivity of the metal is known to be fixed (and not statistically variable). If the resistivity of the metal varies from place to place, i.e. it is not homogeneous, then there is statistical variability of the resistivity depending on the section of metal chosen and so both the precision and the accuracy of the resistance calculation would be poor. You can have a precise answer that is not accurate, but you cannot have an accurate result that is not precise.

    The number of significant digits in a calculation or measurement implies the accuracy and/or precision of a number. Sometimes the accuracy is explicitly stated, for example \(4.01\pm 0.02\text{ m}\), but more commonly in engineering the number of significant digits implies the accuracy and/or precision. Error is taken to be one half of the last significant digit. So \(1200\text{ m}\) implies an accuracy of \(0.5\text{ m}\). While \(1.2\text{ km}\) would imply an accuracy of \(0.05\text{ km}\) or \(20\text{ m}\). Usually \(4\) is enough but there are departures from this.

    One common situation is when we are talk about the center frequency of a carrier in a communication system. That is because we can precisely set the frequency of a carrier and can put communication bands very close together. For example, if we have frequency bands that are \(25\text{ kHz}\) wide on a carriers near \(900\text{ MHz}\). We would need to indicate the frequency of the carrier to a fraction of a kilohertz. So there is a significant difference between specifying a carrier as being at \(1.000000\text{ GHz}\) or at \(1.000001\text{ GHz}\). Sometimes however it is not important to specify frequency so accurately. In the great majority of situations four digits of accuracy is enough, in any case it is very difficult to fabricate something with better than \(0.1\%\) accuracy.

    When using decibels, a logarithmic scale, two digits after the decimal point is the usually sufficient and required. One digit after the decimal point is not enough accuracy. For example, \(0\text{ dB} = 100 = 1.000\) and \(0.1\text{ dB} = 100.01 = 1.023\). So if there is only one digit after the decimal point is used then the accuracy implied is \(1\%\). Now \(0.01\text{ dB} = 100.001 = 1.0023\) and so with two digits after the decimal point the implied accuracy is \(0.1\%\) which is about four digits of precision. With four digits after the decimal digit, \(0.0001\text{ dB} = 100.00001 = 1.000023\). which is the same as claiming around \(6\) digits of accuracy. Achieving this is highly unlikely except in circumstances involving frequency or when circuits are tuned after fabrication.

    There are a few exceptions to the accuracy implied by the number of digits used with decibels. When a number is written as \(0\text{ dB}\) or \(10\text{ dB}\), for example, with no digits after the decimal point, then the implication is that it is exactly \(0\text{ dB}\: (= 1.00000000)\) or \(10\text{ dB}\: (= 10.0000000)\). Another exception is a factor of \(2\) which in decibels is \(3.01\text{ dB}\). It is common to write this as \(3\text{ dB}\) and when one sees \(3\text{ dB}\) then most people automatically recognize this as a factor of exactly \(2\). Similarly for \(6\text{ dB}\) the implication is that it refers to a factor of exactly \(4\).

    Using more significant digits in a result than is justified can leave an impression of a lack of understanding, that the result was simply a matter of plugging numbers into a calculation rather than understanding what was being done. When an engineer presents results, the observer, usually another engineer, wants to develop trust for the engineering process. Engineering is pragmatic and abstractions are made, you always want to create an impression that you are in control and know what you are doing. Sure interim results can have many significant digits of accuracy but final results should have reasonable accuracy.

    Standard Temperatures

    Description Value In Terms of Fundamental Units
    Absolute zero temperature \(0\text{ K}\) Fundamental unit \(= −273.15^{\circ}\text{C}\)
    Room temperature \(290-298\text{ K}\) \(19–25^{\circ}\text{C}\), generally used as an imprecise measurement implying that properties are unchanged over a few degrees variation.
    Standard temperature \(290\text{ K}\) In microwave engineering [29], different in other disciplines.
    Available noise of a resistor at room temperature   \(−174\text{ dBm/Hz}\). (e.g., in \(2\text{ Hz}\) bandwidth the available noise of a resistor at room temperature is \(−171\text{ dBm}\)). (At \(290\text{ K}\) the available noise power is \(−173.97\text{ dBm/Hz}\), at \(293\text{ K}\) it is \(−173.93\text{ dBm/Hz}\), at \(298\text{ K}\) it is \(−173.86\text{ dBm/Hz}\).)

    Table \(\PageIndex{4}\): Temperature constants.

    2.A.2 Greek Alphabet and Additional Characters

    Greek Alphabet
    Name Uppercase Lowercase
    alpha \(\text{A}\) \(\alpha\)
    beta \(\text{B}\) \(\beta\)
    gamma \(\Gamma\) \(\gamma\)
    delta \(\Delta\) \(\delta\)
    epsilon \(\text{E}\) \(\epsilon\)
    zeta \(\text{Z}\) \(\zeta\)
    eta \(\text{H}\) \(\eta\)
    theta \(\Theta\) \(\theta\)
    iota \(\text{I}\) \(\iota\)
    kappa \(\text{K}\) \(\kappa\)
    lambda \(\Lambda\) \(\lambda\)
    mu \(\text{M}\) \(\mu\)
    nu \(\text{N}\) \(\nu\)
    xi \(\Xi\) \(\xi\)
    omicron \(\text{O}\) \(\omicron\)
    pi \(\Pi\) \(\pi\)
    rho \(\text{P}\) \(\rho\)
    sigma \(\Sigma\) \(\sigma\)
    tau \(\text{T}\) \(\tau\)
    upsilon \(\Upsilon\) \(\upsilon\)
    phi \(\Phi\) \(\phi\)
    chi \(\text{X}\) \(\chi\)
    psi \(\Psi\) \(\psi\)
    omega \(\Omega\) \(\omega\)

    Table \(\PageIndex{5}\)

    Additional Characters
    nabla \(\nabla\)
    cross \(×\)
    times \(\times\)
    varepsilon \(\varepsilon\)
    varphi \(\varphi\)
    varpi \(\varpi\)
    varrho \(\varrho\)
    varsigma \(\varsigma\)
    vartheta \(\vartheta\)
    aleph \(\aleph\)

    Table \(\PageIndex{6}\)

    2.A.3 Conductors, Dielectrics, and Magnetic Materials

    Electrical and thermal properties of RF and microwave materials are given in the tables below. A parameter listed as a range indicates that the parameter depend on the formulation of the alloy. \(\perp\) indicates the property in the direction perpendicular to the crystal axis. \(//\) indicates the property in the direction parallel to the crystal axis.

    Material data from several sources including the Standard Reference Data database of the U.S. National Institute of Standards and Technology [26], the CODATA databases of the International Council for Science, Committee on Data for Science and Technology [27], and references [30, 31, 32, 33]. Electrical and especially thermal properties are functions of temperature; properties at temperatures other than \(300\text{ K}\) should be researched.

    Material Relative Permeability \(\mu_{r}\)
    Aluminum \(1.00000065\)
    Cobalt \(60\)
    Copper \(0.999994\)
    Ferrite (NiZn) \(16-640\)
    Gold \(0.999998\)
    Iron \(5,000-6,000\)
    Lead \(0.999983\)
    Magnesium \(1.00000693\)
    Manganese \(1.000125\)
    Mumetal \(20,000-1000,000\)
    Nickel \(50-600\)
    Palladium \(1.0008\)
    Permalloy 45 \(2,500\)
    Platinum \(1.000265\)
    Silver \(0.99999981\)
    Steel \(100-40,000\)
    Superconductors \(0\)
    Supermalloy \(100,000\)
    Tungsten \(1.000068\)
    Wood (dry) \(0.99999942\)

    Table \(\PageIndex{7}\): Relative permeability of metals.

    Some calculations require the use of volumetric heat capacity, \(c_{v}\) obtained from

    \[\label{eq:6} c_{v}=c_{p}\rho \]

    but ensure the use of SI units, i.e. convert the density \(\rho\) to \(\text{kg}\cdot\text{m}^{-3}\).

    The electrical resistivities listed in Table \(\PageIndex{10}\) for single-element metals are those of single-crystal metals. The resistivity of the best fabricated metal with multiple crystal grains tends to be up to \(5\%\) above that of a single-crystal. Poorly fabricated metals can have a resistivity twice as high.

    Material Resistivity \((\text{M}\Omega\cdot\text{m}\:\rho,\) at \(300\text{ K})\) Relative permittivity \((\varepsilon_{r}\) at \(1\text{ GHz})\) Loss tangent \((\tan\:\delta\) at \(1\text{ GHz})\)
    Air (dry, sea level) \(4\times 10^{7}\) \(1.0005\) \(0.000\)
    \(99.5\%\) \(> 10^{6}\) \(9.8\) \(0.0001-0.0002\)
    \(96\%\) \(> 10^{6}\) \(9.0\) \(0.0006\)
    \(85\%\) \(> 10^{6}\) \(8.5\) \(0.0015\)
    Aluminum nitride \(10^{6}\) \(8.9\) \(0.001\)
    Bakelite \(1–100\) \(4.74\) \(0.022\)
    Beryllium oxide (toxic) \(> 10^{8}\) \(6.7\) \(0.004\)
    Diamond \(10^{5}–10^{10}\) \(5.68\) \(<0.0001\)
    Ferrite (MnZn) \(0.1–10\:\Omega\cdot\text{m}\) \(13-16\) \(0.0004\)
    Ferrite (NiZn) \(0.1-12.4\) \(13-16\) \(0.0004\)
    FR-4 circuit board \(8\times 10^{5}\) \(4.3-4.5\) \(0.01\)
    GaAs \(1.0\) \(12.85\) \(0.0006\)
    InP Up to \(0.001\) \(12.4\) \(0.001\)
    Glass \(2\times 10^{8}\) \(4-7\) \(0.002\)
    Mica \(2\times 10^{5}\) \(5.4\) \(0.0006\)
    Mylar \(10^{10}\) \(3.2\) \(0.005\)
    Paper, white \(3.5\times 10^{6}\) \(3\) \(0.008\)
    Polyethylene \(> 10^{7}\) \(2.26\) \(0.0002\)
    Polyimide \(10^{10}\) \(3.2\) \(0.005\)
    Polypropylene \(> 10^{7}\) \(2.25\) \(0.0003\)
    Quartz (fused) \(7.5\times 10^{11}\) \(3.8\) \(0.00075\)
    \(//\) \(> 10^{6}\) \(11.6\) \(0.00004-0.00007\)
    \(\perp\) \(> 10^{6}\) \(9.4\) \(0.00004-0.00007\)
    Polycrystalline \(> 10^{6}\) \(10.13\) \(0.00004-0.00007\)
    Silicon (undoped)      
    Low resistivity (used in CMOS) \(50\:\mu\Omega\cdot\text{m}\) \(11.68\) \(0.005\)
    High resistivity \(300\:\text{m}\Omega\cdot\text{m}\) \(11.68\) \(0.005\)
    Carbide (SiC) \(100\) \(10.8\) \(0.002\)
    Dioxide (SiO\(_{2}\)) \(5.8\times 10^{7}\) \(3.7-4.1\) \(0.001\)
    Nitride (Si\(_{3}\)N\(_{4}\)) \(10^{7}\) \(7.5\) \(0.001\)
    Poly \(0.1-10\text{ k}\Omega\cdot\text{m}\) \(11.7\) \(0.005\)
    Teflon (PTFE) \(10^{10}\) \(2.1\) \(0.0003\)
    Vacuum \(\infty\) \(1\) \(0\)
    Distilled \(182\) \(80\) \(0.1\)
    Ice (\(273\text{ K}\)) \(1\) \(4.2\) \(0.05\)
    Wood (dry oak) \(3\times 10^{11}\) \(1.5-4\) \(0.01\)
    Zirconia (variable) \(10^{4}\) \(28\) \(0.0009\)

    Table \(\PageIndex{8}\): Electrical properties of dielectrics and nonconductors.

    Material Thermal conductivity, \(k\:(\text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1})\) Specific heat capacity, \(c_{p}\) \((\text{kJ}\cdot\text{kg}^{-1}\cdot\text{K}^{-1})\) Density, \(\rho\) \((\text{g}\cdot\text{cm}^{-3})\) Speed of sound, \(c_{s}\) \((\text{m}\cdot\text{s}^{-1})\)
    (at \(300\text{ K}\)) (at \(25^{\circ}\text{C}\)) (at \(25^{\circ}\text{C}\)) (at \(25^{\circ}\text{C}\))
    Air (dry, sea level) \(0.026\) \(1.005\) \(0.0018\) \(343\)
    Alumina       \(9,900-10,520\)
    \(100\%\) \(30\) \(0.78\) \(3.8\)  
    \(99.5\%\) \(26.9-30\) \(0.78\) \(3.8\)  
    \(96\%\) \(24.7\) \(0.78\) \(3.8\)  
    \(85\%\) \(16\) \(0.92\) \(3.5\)  
    Aluminum nitride \(285\) \(0.74\) \(3.28\) \(11,000\)
    Bakelite (wood filler) \(0.2-1.4\) \(1.38\) \(1.25-1.36\)  
    Beryllium oxide (toxic) \(64-210\) \(1.75\:(@0^{\circ}\text{C})\) \(1.85-2.85\)  
    Diamond \(1,000-2,000\) \(0.52-0.63\) \(3.50-3.53\) \(12,000\)
    Ferrite (MnZn) \(3.5-5\) \(0.7-0.8\) \(4.9\)  
    Ferrite (NiZn) \(3.5-5\) \(0.75\) \(4.5\)  
    FR-4 circuit board \(0.16-0.3\) \(0.6\) \(1.3-1.8\)  
    Graphite \(25-470\) \(0.71-0.83\) \(1.3-2.27\) \(1,200\)
    GaAs \(50-59\) \(0.37\) \(5.32\) \(4,730\)
    InP \(68\) \(0.31\) \(4.81\)  
    Glass \(0.8-1.2\) \(0.5-0.84\) \(2.0-8.0\) \(3,950-5,640\)
    Mica \(260-750\) \(0.5\) \(0.72\)  
    Mylar (polyethylene-terephtalate) \(0.08\) \(1.19\) \(1.4\) \(1,900-2,430\)
    Paper, (white bond) \(40-90\) \(1.4\) \(0.72\)  
    Polyethylene \(0.42-0.51\) \(2.3-2.9\) \(2.30\) \(1,900-2,430\)
    Polyimide \(0.12\) \(1.09-1.15\) \(1.43\)  
    Polypropylene \(0.35-0.40\) \(1.7-2.0\) \(0.855\) \(2,740\)
    Quartz (fused) \(1.30-1.44\) \(0.67-0.74\) \(2.2\) \(5,800\)
    Sapphire       \(11,100\)
    \(//\) \(35\) \(0.74-0.78\) \(4.05\)  
    \(\perp\) \(32\) \(0.74-0.78\) \(4.05\)  
    Polycrystalline \(31-33\) \(0.74-0.78\) \(3.97-4.05\)  
    Silicon (undoped)        
    low resistivity (used in CMOS) \(149\) \(0.705\) \(2.34\) \(8,433\)
    high resistivity \(149\) \(0.705\) \(2.34\) \(8,433\)
    carbide (SiC) \(350-490\) \(0.75\) \(2.55\) \(13,060\)
    dioxide (SiO\(_{2}\)) \(1.4\) \(1.0\) \(2.27-2.63\)  
    nitride (Si\(_{3}\)N\(_{4}\)) \(28\) \(0.711\) \(3.44\) \(11,000\)
    polysilicon \(12.5-157\) \(0.71-0.75\) \(2.2-2.3\)  
    Teflon (PTFE) \(0.20-0.25\) \(0.97\) \(2.1-2.2\) \(1,400\)
    Vacuum \(0\) \(0\) \(0\)  
    Distilled \(580\) \(4.18\) \(0.997\) \(1,480\)
    Ice (at \(273\text{ K}\)) \(2.22\) \(2.05\) \(0.917\) \(4,000\)
    Wood (dry oak) \(170\) \(2\) \(0.6-0.9\) \(3,960\)
    Zirconia (variable) \(1.7-2.2\) \(0.40-0.50\) \(5.6-6.1\)  

    Table \(\PageIndex{9}\): Thermal properties of dielectrics and nonconductors.

    Material Electrical resistitivity, \(\rho\) \((\text{n}\Omega\cdot\text{m})\) Thermal conductivity, \(k\:(\text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1})\) Specific heat capacity, \(c_{p}\) \((\text{kJ}\cdot\text{kg}^{-1}\cdot\text{K}^{-1})\) Density, \(\rho\) \((\text{g}\cdot\text{cm}^{-3})\) Thermal coefficient of resistance \((\text{K}^{-1})\) Speed of sound, \(c_{s}\) \((\text{m}\cdot\text{s}^{-1})\)
    (at \(20^{\circ}\text{C}\)) (at \(300\text{ K}\)) (at \(25^{\circ}\text{C}\)) (at \(25^{\circ}\text{C}\)) (at \(20^{\circ}\text{C}\)) (at \(20^{\circ}\text{C}\))
    Aluminum \(26.50\) \(237\) \(0.897\) \(2.70\) \(0.004308\) \(6420\)
    Brass Variable \(120\) \(0.38\) \(8.4-8.7\) \(0.0015\) \(3,500-4,700\)
    Bronze Variable \(110\) \(0.38\) \(7.4-8.9\)    
    Chromium \(125\) \(93.9\) \(0.450\) \(7.15\)   \(5,490\)
    Constantan \(500\) \(19.5\) \(0.39\) \(8.9\) \(\pm 0.00003\)  
    Copper \(16.78\) \(401\) \(0.39\) \(8.94\) \(0.004041\) \(3,560-4,700\)
    Gold \(22.14\) \(318\) \(0.129\) \(19.30\) \(0.003715\) \(3,240\)
    \(//\) c-axis \(1,200\) \(1,950\) \(0.71\) \(2.09-2.23\) \(-0.0002\)  
    \(\perp\) c-axis \(41,000\) \(5.7\) \(0.71\) \(2.09-2.23\) \(-0.0002\)  
    Iridium \(47.1\) \(147\) \(0.131\) \(22.6\)    
    Iron (cast, hard) \(96.1\) \(80.2\) \(0.449\) \(7.87\) \(0.005671\) \(5,600-5,900\)
    Lead \(208\) \(35.3\) \(0.127\) \(11.3\)   \(1,160-2,200\)
    Manganin \(430-480\) \(22\) \(0.406\) \(8.4\) \(\pm 0.000015\)  
    Mercury \(961\) \(8.34\) \(0.139\) \(13.53\) \(0.0089\)  
    Nickel \(69.3\) \(90.9\) \(0.445\) \(8.90\) \(0.0058-0.0064\) \(5,600\)
    NiChrome \(1,100\) \(11.3\) \(0.432\) \(8.40\) \(0.00017\)  
    Palladium \(105.4\) \(71.8\) \(0.244\) \(12.0\)   \(3,070\)
    Platinum \(105\) \(71.6\) \(0.133\) \(21.5\) \(0.0037-0.0038\) \(3,300\)
    Silver \(15.87\) \(429\) \(0.235\) \(10.49\) \(0.003819\) \(3,600-3,650\)
    tin-lead Pb, Sn \(17.2\) \(34\) \(0.167\) \(8.89\)    
    \(50\%\) Pb            
    lead-free \(170\) \(53.5\) \(0.23\) \(7.25\)    
    \(77.2\%\) Sn            
    \(2.8\%\) Ag            
    \(20\%\) In            
    Steel, stainless \(720\) \(16\) \(0.483\) \(7.48-8.00\)   \(5,740-5,790\)
    Steel, carbon            
    (standard) \(208\) \(46\) \(0.49\) \(7.85\) \(0.003-0.006\) \(4,880-5,050\)
    Tantalum \(133\) \(57.5\) \(0.14\) \(16.69\) \(0.0038\)  
    Tin \(115\) \(66.8\) \(0.227\) \(7.27\)   \(3,300\)
    Titanium \(4,200\) \(21.9\) \(0.522\) \(4.51\)   \(6,070-6,100\)
    Tungsten \(52.8\) \(173\) \(0.132\) \(19.3\) \(0.004403\) \(5,200\)
    Zinc \(59.0\) \(116\) \(0.388\) \(7.14\) \(0.0037-0.0038\) \(4,200\)

    Table \(\PageIndex{10}\): Thermal and electrical properties of conductors.


    [1] The older metric systems used different fundamental units; for example, the mks metric system used meter, kilogram, and second as fundamental units; the cgs metric system used centimeter, gram, and second as fundamental units.

    This page titled 2.A: Appendix- Physical Constants and Material Properties is shared under a not declared license and was authored, remixed, and/or curated by Michael Steer.

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