Draft issue: 1-5-1 LCD: 9-14-11 Chapter 1 EVIEW OF NETWOK ELEMENTS S.P. Chan and H.T. ussell, Jr. OPL tx ll rights reserved 198, 1 1.1 Introduction The purpose of an electrical network is to transfer and/or transform energy. The study (analysis) of how networks achieve this involves the transformation of the physical behavior of electrical network elements into accurate mathematical descriptions. Such a transformation is known as modeling and the description that results is the mathematical model. The more accurate the model is to the element s actual behavior, the more accurate the results of the analysis. review of the mathematical models of various two-terminal elements commonly found in physical (i.e., practical) networks will be performed in this chapter. This review will be instrumental in laying a firm foundation for the subsequent development of powerful network analysis methods. It will be shown the power of these methods lies in certain algorithmic (step-by-step) procedures that excludes all human judgment and intuition, and therefore lends these methods readily applicable to the computer. The computer is a very powerful tool in the hands of a competent engineer or scientist. s such, any network analysis method that can be computer implemented is in itself an even more powerful tool. 1. Network Elements Before the discussion of graph theory and its application to electrical networks can begin, a brief review of basic ideal two-terminal network elements is necessary. In this review, each element will be characterized by a functional equation relating terminal parameters such as voltage and current, charge and voltage, and flux linkage and current. These equations serve as the foundation for network branch voltage-current-relationship equations. For more details into these elements, the interested reader is referred to the textbooks listed at the end of this chapter. [,3,4] The types of ideal network elements used in this text will be divided into two classes passive and active. Those elements termed passive cannot generate energy; rather, they either store or dissipate energy. Furthermore, the passive elements in this text will be restricted to be linear (the relationship between terminal parameters is described by a function that obeys the laws of mathematical linearity) rather than non-linear, lumped (as opposed to distributive), finite (singularities describing the functional relationship between terminal variables are finite in number) as opposed to infinite, and time-invariant (the element values are fixed with respect to time). s such, these elements are known as LLFTI (linear, lumped, finite, time-invariant) passive elements and consist of resistors, conductors, capacitors, inductors, and sets of mutually coupled inductors. Conversely, the class of basic elements termed active contains those which can generate energy and are likewise restricted to be linear, lumped, and finite but not necessarily time-invariant. These elements consist of voltage sources and current sources both of which - 1 -
may be independent or dependent in value. 1..1 esistance and esistors Time-domain current and voltage polarities for the general two-terminal element E are shown in Figure 1.1. These polarities are consistent with the voltage drop convention used throughout this textbook. The mathematical expression relating the current through the element to the voltage across the element is given by ( ) ( ) v t = f i t (1.1) e r e v e (t) i e (t) E Figure 1.1 Two-terminal element E where v e (t) is voltage in volts (V), i e (t) is current in amperes (), and f r [ ] is the function that transforms i e (t) into v e (t). This equation is a mathematical model that defines the voltage-current relationship (VC) for E. The incremental resistance parameter r of E is defined as the change in voltage produced by a change in current. Incremental resistance is also known as small-scale resistance. In point form where these changes are reduced to extremely small values, r is found from the derivative of v e (t) with respect to i e (t) evaluated at a given current I e ; that is, () t () e() () d v d f e r i t r = = (1.) di t di t e Ie esistance r has units of volt per ampere identified as ohms and represented by the Greek letter omega or Ω. By definition, an ohm is one volt per one ampere. If E is an LLFTI element such that the function f r [ ] in (1.1) can be replaced by the linear operation e ( ) ( ) Ie fr ie t = ie t (1.3) then E is known as a linear resistor or simply a resistor. The character symbol for a resistor is the letter while its electronic symbol is shown in Figure 1.. The VC for the resistor is written as ( ) ( ) v t = i t (1.4) v (t) i (t) Figure 1. Electronic symbol for a resistor - -
which is known as Ohm s law. pplication of the derivative operation in equation (1.) to equation (1.4) reveals that the resistor has a resistance of ohms which is positive for passive, physically realizable resistors and negative for active resistors. For either type, is calculated from () () ( t) () () () d v d t i v t = = = (1.5) di t di t i t The graphical representation of the VC in equation (1.4) is known as a voltage-current plot or VI plot and the slope of the plot is the resistance. For example, the VI plot of a typical passive 1,5 ohm (1.5KΩ) resistor is shown in Figure 1.3 where the slope is a positive constant value of 3V per m or 1,5 ohms. 1 8 4 v (V) 4 8 1 4 3 1 1 3 4 i (m) Figure 1.3 VI plot of a 1.5KΩ resistor t any time t, the power p (t) in watts (W) delivered to a resistor is the product of its voltage and current; that is, v () () () () which is positive for all values and polarities of resistor current and voltage. Therefore, it is certain that energy can never be generated by or stored in a resistor. Energy delivered to a resistor is only absorbed and consequently converted into other forms such as heat or light. esistors are divided into three types based upon the manner of resistance variation. These types consist of resistors whose values are constant or fixed, resistors with manually adjustable values, and resistor sensors whose values dependent on or affected by environmental or external conditions. - 3 - ( t) p t = v t i t = i t = (1.) Since p (t) is positive for all terminal voltages and currents, it is commonly referred to as the power dissipated by the resistor which is converted into heat. The energy w (t) in Joules (J) absorbed by a resistor during the time interval t o to t 1 is determined from the integral of p (t) where t t t 1 t w( t) = p( τ ) dτ = v( τ) i( τ) dτ = i( τ) dτ = v( τ) dτ t t t t (1.7) o o o o
1..1.1 Fixed esistors Physically realizable fixed resistors are the most common elements found in electronic networks and systems. pplications of these devices include current limiting and steering, transforming voltage into current and current into voltage, voltage and current dividers, and signal attenuators, to name a few. esistors are fabricated from chemical composites (such as carbon and metal composition), chemical compounds (such as ceramic and metal paste known as cermet material), films (such as metal film, metal oxide film, and carbon film), metal alloy foils, resistive wire (for wirewound resistors), and semiconductor materials such as silicon and tantalum. Typical resistors used in modern electronic industry are built from solid bars of resistive materials with metal caps attached to its ends for contacts. This arrangement produces a structure with leads such as the axial-lead and the radial-lead resistor, and a lead-less structure known as a surface-mount device (SMD) which includes the rectangular chip resistor and the cylindrical metal electrode leadless face (MELF) resistor. xial-lead resistors like those shown in Figure 1.4 have wire leads attached to the contacts that leave the body axially or parallel to the body. Wire leads of the radial-lead resistors exit the body in a radial manner rather than parallel as shown in Figure 1.5. xial and radial-lead resistors are often referred to as through-hole resistors since their leads are easily soldered through the holes on a printed circuit board. SMD resistors are being applied in Figure 1.4 xial-lead resistors Figure 1.5 adial-lead resistors - 4 -
increasingly more applications due to their small size and low tolerance. s shown in Figures 1. and 1.7, these resistors have no wire leads and are constructed from thick film or thin film chips with metal end caps exposed for flow soldering onto printed circuit boards. [, 3] Figure 1. SMD rectangular chip resistors Figure 1.7 SMD cylindrical MELF resistors 1..1.1.1 Characteristic Parameters esistors are generally characterized with respect to value, tolerance, temperature coefficient, and power rating. (a.) Value. Commercially available resistors have values that range from milli-ohms (1-3 Ω) to giga-ohms (1 9 Ω). The value of a resistor made from a bar of resistive material is a function of the bar s dimensions and a physical property known as resistivity. esistivity is a parameter that indicates how well the material resists current flow and is strongly dependent on the content of impurities in the material. Consider, for example, the structural diagrams shown in Figure 1.8 for the resistors discussed above where all dimensions are in centimeters (cm). The resistance of the rectangular structures is calculated from while the resistance of the cylindrical structures is calculated from rect ρl ρl = = (1.8) HW - 5 -
cyl ρl ρl = = (1.9) π r In both expressions, is the bar s cross-sectional area perpendicular to the direction of current, L is the length of the bar, and ρ is the resistivity of the material in ohm-cm. Table 1.1 provides a compilation of resistivity along with temperature coefficient for several common materials for comparison. Table 1.1 Material properties at C Temperature coefficient α (1/ C) Temperature coefficient α (1/ C) esistivity ρ esistivity ρ Material (Ω-cm) Material (Ω-cm) Silver 1.59 1 -.1 Mercury 9.84 1-5.89 Copper 1.7 1 -.8 Nichrome 1.1 1-4.4 Gold.4 1 -.34 Carbon 3.5 1-3 -.5 luminum. 1 -.43 Germanium 4. -.5 Iron 9.71 1 -.5 Silicon.4 1 4 -.7 metal caps resistive material ρ (Ω-cm) wire connector metal caps wire connector H wire connector W L wire connector r L resistive material ρ (Ω-cm) metal caps (a) resistive material ρ (Ω-cm) (b) metal caps H wire connector W L wire connector wire connector r L resistive material ρ (Ω-cm) wire connector (c) (d) metal caps thick or thin film chip ρ (Ω-cm) metal caps resistive material ρ (Ω-cm) H W L r L (e) (f) Figure 1.8 Dimensions for (a, b) axial-lead, (c, d) radial-lead, and (e, f) SMD resistor structures - -
Manufactured resistors are assigned standard values based on the international standard IEC 3 preferred number series commonly referred to as the E series [xx]. This series is applied to produce a scale factor that subdivides the logarithmic interval from 1 to 1 into evenly spaced segments based on integer multiples of with the formula SF ( j E) j 1 E, = 1 (1.1) where E is E for segments, E1 for 1 segments, E4 for 4 segments, and so on, and j is series index from 1 to E. The scale factor is then combined with a base value to produce the preferred value denoted here as k ( ) base ( ) ( ) ( ) k, je, = k SF je, = 1 Ω 1 (1.11) base (k) is the resistor base value in ohms calculated from the exponent k that ranges typically from 3 to +9 for a base value from 1mΩ to 1GΩ, respectively. Standard resistor values are determined from preferred values rounded up or down within the subdivision to reduce the number of digits. These are the values published in resistor catalogs and product lists. For example, with k = 3, j = 5, and E = 1 for E1, the preferred value is calculated from ( ) ( ) 5 1 3 1 3, 5,1 = 1 Ω 1 =,154.435Ω (1.1) which is rounded up to,ω for the standard value. Table 1. provides a comparison of the calculated preferred and standard values for E4 with base = 1Ω (k = ) and the index j spanning 1 to 4. The difference (δ j ) in per-cent between these values is also included. Standard values for other E series resistors with an base of 1Ω are shown at the end of this chapter. j 1 E Table 1. Preferred and standard values for E4 with base = 1Ω j j (,j,4) (preferred value in Ω) j (standard value in Ω) δ j (%) j j (,j,4) (preferred value in Ω) j (standard value in Ω) δ j (%) 1 1. 1.. 13 3.178 3.3 4.35513 1.194 1.1 -.37 14 3.4871 3. 3.47454 3 1.1158 1. -.951498 15 3.831187 3.9 1.79131 4 1.33351 1.3 -.513753 1 4.195 4.3 1.999 5 1.47799 1.5.19381 17 4.41589 4.7 1.5843 1.15598 1. -.9549 18 5.1897 5.1 -.17559 7 1.77879 1.8 1.1439 19 5.3413 5. -.41353 8 1.95734..179395.18958..178 9.154435..114954 1.8191.8 -.1895 1.371374.4 1.711 7.49894 7.5.1417 11.1157.7 3.4445 3 8.544 8. -.5473 1.87985 3. 4.4118 4 9.8517 9.1.1317-7 -
(b.) Tolerance. Before a resistor exits the factory as a finished product, its resistance must be trimmed or adjusted to fit within a certain tolerance of its standard value. esistor tolerance is determined from the difference between adjacent preferred segment values in a given E range. This allows a resistor s actual value to overlap those of most of its neighbors to produce an almost continuous series of resistance within a given decade. For example, the per-cent difference (δ) between preferred resistor values and 7 in Table 1. is calculated from δ ( ) ( ) ( ) ( ), 7, 4,, 4 1.77879Ω 1.15598Ω,, 4 1.15598Ω 7 % = 1% = 1% = 1.7% Dividing δ by gives the tolerance (Tol) with a ± value such that for E4, the tolerance is (%) (1.13) δ Tol (%) = = ± 5% (1.14) With this tolerance, it is possible the standard values for resistors in the E4 series will have variations that mostly overlap into adjacent segment values. This is demonstrated in Table 1.3 for the standard values generated in Table 1. where the tolerance value is calculated from (%) Tol i ( Tol (%)) = i ( nom) 1+ 1 For example, a resistor with a nominal value ( i (nom)) of 3.9Ω and a ±5% tolerance (Tol(%)) will exhibit a range of resistance between a minimum of 3.75Ω (for 5%)and a maximum of 4.95Ω (for +5%). These values are seen to overlap into the adjacent values of 3.Ω and 4.3Ω since the minimum value is less than the maximum value of 3.Ω while the maximum value is greater than the minimum value of 4.3Ω. However, a resistor with a nominal value of 1.5Ω with the same tolerance will exhibit a range of resistance that does not completely overlap into the adjacent value of 1.3Ω. Table 1.4 lists the manufacturing tolerances specified for resistors in the other E series. (c.) Temperature coefficient. The variation of a resistor s value over temperature is specified with its temperature coefficient of resistance TC. TC (also known as α) is defined as the fractional change in resistance with respect to a change in temperature; that is, Δ 1 TC = SF ΔT nom nom is nominal resistance and SF is a scale factor applied to eliminate leading zeros. Values for the scale factor are given as 1 no units or one part per one part 1 percent (%) (one part per one hundred parts) SF = (1.17) 1 ppm (one part per one million parts) 9 1 ppb (one part per one billion parts) Depending on the type of resistive material used, typical TCs range from ±1ppm/ C (±.1%/ C) to ±1,ppm/ C (±1%/ C). Material temperature coefficients are given in Table 1.1. The TC is applied in a simple first-order equation that models resistor temperature variation with (1.15) (1.1) - 8 -
Table 1.3 Standard resistance values for E4 with base = 1Ω and ±5% tolerance j j ( 5%) Ω j (nom) Ω j (+5%) Ω j j ( 5%) Ω j (nom) Ω j (+5%) Ω 1.95 1. 1.5 13 3.135 3.3 3.45 1.45 1.1 1.155 14 3.4 3. 3.78 3 1.14 1. 1. 15 3.75 3.9 4.95 4 1.35 1.3 1.35 1 4.85 4.3 4.515 5 1.45 1.5 1.575 17 4.45 4.7 4.935 1.5 1. 1.8 18 4.845 5.1 5.355 7 1.71 1.8 1.89 19 5.3 5. 5.88 8 1.9..1 5.89..51 9.9..31 1.4.8 7.14 1.8.4.5 7.15 7.5 7.875 11.55.7.835 3 7.79 8. 8.1 1.85 3. 3.15 4 8.45 9.1 9.555 Table 1.4 esistor tolerances E Tolerance (%) ± 1 ±1 4 ±5 48 ± 9 ±1 19 ±.5 TC T ( ) = nom ( To ) 1 + ( T To ) SF (1.18) where T o is the reference temperature usually given as room temperature of 7 C and (T) is resistance at temperature T. lthough actual resistors have temperature variations that do not exactly follow a straight line, the first-order model provides a bracketed domain that simplifies the resistor s temperature behavior. This is illustrated in the resistance versus temperature (T) plot in Figure 1.9 where the actual resistance (T) is constrained by the pair of lines with slopes of ±m. The value of m is determined from equation (1.1) for Δ nomtc m = = (1.19) ΔT SF - 9 -
esistance (Ω) nom (T o ) slope = +m slope = -m (T) T o Temperature ( o C) Figure 1.9 T plot for a fixed resistor (d.) Power rating. resistor s power rating specifies the maximum amount of power the resistor can dissipate before it is destroyed by excessive heat. The overall size of the resistor generally determines its power rating. Physically larger resistors have larger volumes and are capable of dissipating more heat than smaller resistors. Furthermore, the maximum current density of resistive material is another factor governing power rating. If the current through the resistor yields a current density greater than the material maximum, it is possible to produce areas of excessive heat (hot spots) that cause catastrophic failure. Consider, for example, the resistor structure shown in Figure 1.1 which has the cross-sectional area and a maximum current density of J max. The maximum current the resistor can safely carry (I max ) without failure is determined from I max = J (1.) max resistive material ρ and J max L I max Figure 1.1 esistor structure This current is used to calculate the maximum power P max from ρl Pmax = Imax = ( Jmax ) = ( ρ Jmax ) L (1.1) which is proportional to the resistor volume L. For example, the relative sizes of a 1W and a 5W resistor are shown in Figure 1.11 where it is seen that the 5W resistor is roughly five times larger. Power ratings for resistor types and various materials are given in Table 1.5 [3]. - 1 -
Figure 1.11 elative resistor sizes Table 1.5 esistor Power ating esistor Type Material Power ating (W) Carbon composition.15 to 1. Carbon film.15 to. xial and radial lead SMD Metal film.5 to 1. Metal foil.5 to 1. Power metal strip.1 to 5. Wire-wound.5 to 1. ectangular (chip).5 to 1. Cylindrical (MELF).3 to 1. 1..1.1.3 Markings esistor characteristics are marked on the resistor body with either of two methods. xial-lead and MELF resistors are marked with a pattern of colored bands to indicate standard value, tolerance, and, in some cases, temperature coefficient. The color bands and corresponding values are listed in Table 1.. The letters next to the tolerances in column are used for designating tolerances of military grade resistors. For tolerances of ±% or greater, four color bands (excluding a band for the TC) are usually painted on the resistor. n axial-lead metal film resistor with a four-band color marking or code is shown in Figure 1.1 where the bands 1 and (brown and black) denote the two significant digits (d 1 and d ) of the resistance, the third band (yellow) is the multiplier band that denotes the power of ten multiplier (m), and the fourth band (gold) is the tolerance band that indicates tolerance (tol). ccording to Table 1., the value and tolerance of this resistor is calculated from an equation that utilizes the brown, black, yellow, gold color pattern for ( ) ( ) = d 1 + d 1 m± tol = 1 1 + 1 1KΩ± 5% = 1KΩ± 5% (1.) 1 Five color bands are used on resistors with tolerance less than %. The extra color is used for low tolerance resistors to add a significant digit to the value. resistor marked with five bands is shown in Figure 1.13 where the first three colors - 11 -
denote the three significant digits of the value, the fourth color denotes the multiplier, and the fifth color denotes the tolerance. For the color marking of orange, orange, white, black, brown, the resistor value and tolerance are computed from ( ) ( ) = d 1 + d 1 + d 1 m± tol = 3 1 + 3 1 + 9 1 1Ω± 1% = 339Ω± 1% (1.3) 1 3 Table 1. esistor color bands Band 1 Band Band 3 Multiplier Color (d 1 ) (d ) (d 3 ) (m) Black 1Ω Tolerance (tol) Brown 1 1 1 1Ω ±1% (F) ±1ppm ed 1Ω ±% (G) ±5ppm Orange 3 3 3 1KΩ ±15ppm Yellow 4 4 4 1KΩ ±5ppm Green 5 5 5 1KΩ ±.5% (D) Blue 1MΩ ±.5% (C) Violet 7 7 7 1MΩ ±.1% (B) Grey 8 8 8 ±.5% () White 9 9 9 Gold.1Ω ±5% (J) Silver.1Ω ±1% (K) None ±% (M) TC band 1 brown, d 1 = 1 band black, d = tolerance band gold, tol = 5% multiplier band yellow, m = 1KΩ Figure 1.1 Four-band resistor marking - 1 -
band 1 orange, d 1 = 3 band orange, d = 3 band 3 white, d 3 = 9 band 5 brown, tol = 1% band 4 black, m = 1Ω Figure 1.13 Five-band resistor marking Color bands cannot be used on surface mount chip resistors due to their small size. ather, the values of these resistors are printed or laser scribed with a code similar to that of the color code. This is the second most common method of marking resistors where a three or four digit number is inscribed to denote resistor value. For tolerances greater than ±1%, three digits are applied where the first two are the two significant digits of the value while the third digit is the power-of-1 ohms multiplier. For ±1% tolerances or less, four digits are used where three significant digits of the resistor s value are determined from the first three digits while the fourth digit denotes the multiplier. For example, values for the four SMD resistors shown in Figure 1.14 are determined from 8 = 8 1 Ω = 8Ω 3 = 1 Ω = KΩ 3 15 = 15 1 Ω= 15Ω 51 = 5 1 Ω = 5Ω 1 (1.4) Figure 1.14 SMD chip resistors Finally, SMD resistors with values less than 1Ω use the letter or the ohm symbol Ω to indicate the position of the decimal point. Several examples are given below. - 13 -
Ω 1=.1Ω 5= 5.Ω Ω 8=.8Ω =.Ω (1.5) 1..1.1.3 esistance and esistor Examples Example 1.1. The VC for the non-linear element E 1 is described by a parabolic function of current ( ) ( ) ( ) = r = 1. V/ (1.) V I f I I The VI plot for this function is shown in Figure 1.15 where current is swept from zero to 5m. Derive an expression for the incremental resistance r as a function of current and apply this function to generate the graph of r versus I. 1 9 8 7 V(I) (V) 5 4 3 1.5 1 1.5.5 3 3.5 4 4.5 5 I (m) Figure 1.15 VI plot of E 1 Solution. The incremental resistance as a function of I is derived by applying equation (1.) to the VC to yield ( ) r I d fr = = = 1 ( I.) Ω (1.7) di di ( ) d1 I ( I.) which is the slope of the VI plot evaluated at the current I. The plot of r(i) as a function of current is shown in Figure 1.1 which indicates the resistance exhibits a straight-line relationship to I. - 14 -
5 4 3 r(i) (ohms) 1 1 3 4.5 1 1.5.5 3 3.5 4 4.5 5 I (m) Figure 1.1 Small-scale resistance r of E 1 Example 1.. The physical dimensions of a cylindrical axial-lead resistor made from conductive metal paste are shown in Figure 1.17. The length L is millimeters (mm) while the resistivity ρ of the paste is 95Ω-cm. Calculate values for the cylinder resistance (a) for d of 1.5mm, and (b) for d varying from.5mm to 3mm. L ρ Figure 1.17 xial-lead resistor Solution. (a) For a diameter of 1.5mm, the resistor value is computed from equation (1.9) for ( cm) ( mm) π ( 1.5mm) ρl ρl 4ρL 4 95Ω 1mm = = = = d π d 1cm π 4 = 3.55KΩ (b) With L and ρ fixed at the values above, the cylinder resistance as a function of diameter d in mm is (1.8) 4ρL 1mm 757.45 d ( ) = π d = Ω 1cm (1.9) d With d swept from.5mm to 3mm, the cylinder resistance ranges from 8Ω to 3KΩ as shown in the graph of Figure 1.18. - 15 -
1 1 (Kohms) 1.1.5.75 1 1.5 1.5 1.75.5.5.75 3 d (mm) Figure 1.18 esistance versus diameter Example 1.3. 1V, 1KHz sinusoidal voltage is forced across a 1Ω resistor. Generate time-domain plots for the resistor voltage v (t), current i (t), power p (t), and energy w (t) over four cycles of the sinusoidal frequency; that is, from zero sec. to 4.msec. ssume the energy absorbed by is zero at t = sec. Solution. Time-domain expressions for resistor voltage, current, and power are written as () = msin ( π ) = 1sin ( π1 ) = 1sin ( 83.185 ) v() t Vmsin ( π ft) 1sin ( π1t) V () = = = = 1sin ( 83.185 ) v t V ft t V t V i t t 1Ω v t V ft t V () sin ( π ) 1sin ( π1 ) m p() t = v() t i() t = = = = 1sin ( 83.185t) W Plots of these variables versus time are shown in Figures 1.19 and 1.. 1Ω (1.3) 1 8 4 v(t) (V) & i(t) () 4 8 1.5 1 1.5.5 3 3.5 4 time (msec.) voltage current Figure 1.19 esistor voltage and current - 1 -
1 9 8 7 p(t) (W) 5 4 3 1.5 1 1.5.5 3 3.5 4 time (msec.) Figure 1. Power dissipated by The energy w (t) absorbed by is derived from t V t 1 t m V w() t = p ( ) d sin ( ) d sin ( 1 ) d τ τ = f π τ τ = π τ τ 1Ω V sin ( 4 ft) sin ( 15.37t m t π t ) w () t = = 1 J 8π f 513.74 and plotted versus time in Figure 1.1. (1.31) 18 1 14 w(t) (mj) 1 1 8 4.5 1 1.5.5 3 3.5 4 time (msec.) Figure 1.1 Energy absorbed by - 17 -