Pecht, M, Lall, P,, Ballou, G, Sankaran, C, Angelopoulos, N."Passive Components The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Pecht, M., Lall, P., Ballou, G., Sankaran, C., Angelopoulos, N. “Passive Components” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
Passive components Michael Pecht 1.1 Resistors University of maryland Resistor Characteristics.Resistor Types 1.2 Capacitors and Inductors Capacitors. Types of Capacitors. Inductors motorola 1.3 Transformers Glen ballou Types of Transformers. Principle of sformation. Electromagnetic Equation. Transformer Core· Transformer losses· Transformer C. Sankaran Connections Transformer Impedance 1.4 Electrical fuses atings. Fuse Performance. Selective Nick Angelopoulos Coordination· Standards· Products· Standard- Class h Gould Shawmut Company 1 Resistors Michael Pecht and Pradeep lall The resistor is an electrical device whose primary function is to introduce resistance to the flow of electric current. The magnitude of opposition to the flow of current is called the resistance of the resistor. A larger resistance value indicates a greater opposition to current flow The resistance is measured in ohms. An ohm is the resistance that arises when a current of one ampere is passed through a resistor subjected to one volt across its terminals. The various uses of resistors include setting biases, controlling gain, fixing time constants, matching and loading circuits, voltage division, and heat generation. The following sections discuss resistor characteristi and various resistor types. Resistor characteristics Voltage and Current Characteristics of Resistors The resistance of a resistor is directly proportional to the resistivity of the material and the length of the resistor and inversely proportional to the cross-sectional area perpendicular to the direction of current flow. The resistance R of a resistor is given by R A where p is the resistivity of the resistor material( Q2.cm), l is the length of the resistor along direction of current flow(cm), inherent property of materials. Good resistor materials typically have resistivities between 2 x 10- and 200x c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 1 Passive Components 1.1 Resistors Resistor Characteristics • Resistor Types 1.2 Capacitors and Inductors Capacitors • Types of Capacitors • Inductors 1.3 Transformers Types of Transformers • Principle of Transformation • Electromagnetic Equation • Transformer Core • Transformer Losses • Transformer Connections • Transformer Impedance 1.4 Electrical Fuses Ratings • Fuse Performance • Selective Coordination • Standards • Products • Standard— Class H • HRC • Trends 1.1 Resistors Michael Pecht and Pradeep Lall The resistor is an electrical device whose primary function is to introduce resistance to the flow of electric current. The magnitude of opposition to the flow of current is called the resistance of the resistor. A larger resistance value indicates a greater opposition to current flow. The resistance is measured in ohms. An ohm is the resistance that arises when a current of one ampere is passed through a resistor subjected to one volt across its terminals. The various uses of resistors include setting biases, controlling gain, fixing time constants, matching and loading circuits, voltage division, and heat generation. The following sections discuss resistor characteristics and various resistor types. Resistor Characteristics Voltage and Current Characteristics of Resistors The resistance of a resistor is directly proportional to the resistivity of the material and the length of the resistor and inversely proportional to the cross-sectional area perpendicular to the direction of current flow. The resistance R of a resistor is given by (1.1) where r is the resistivity of the resistor material (W · cm), l is the length of the resistor along direction of current flow (cm), and A is the cross-sectional area perpendicular to current flow (cm2 ) (Fig. 1.1). Resistivity is an inherent property of materials. Good resistor materials typically have resistivities between 2 ¥ 10–6 and 200 ¥ 10–6 W · cm. R l A = r Michael Pecht University of Maryland Pradeep Lall Motorola Glen Ballou Ballou Associates C. Sankaran Electro-Test Nick Angelopoulos Gould Shawmut Company
The resistance can also be defined in terms of sheet resistivity. If the sheet resistivity is used, a standard sheet thickness is assumed and factored into resistivity. Typically, resistors are rectangular in shape; therefore the length I divided by the width w gives the number of squares within the resistor(Fig. 1. 2). The number of squares multiplied by the resistivity is the resistance Cross-sectional area(A) FIGURE 1.1 Resistance of a rectangular l heet p sheet (1.2) cross-section resistor with cross-sectional where Psheet is the sheet resistivity(Q2/square), I is the length of resistor(cm), w is the width of the resistor(cm), resistance(Q2) The resistance of a resistor can be defined in terms of the voltage drop across the resistor and current through V (1.3) where R is the resistance(Q2), Vis the voltage across the resistor(V), and I is the current through the resistor (A). Whenever a current is passed through a resistor, a voltage is dropped across the ends of the resistor Figure 1.3 depicts the symbol of the resistor with the Ohms law relation. All resistors dissipate power when a voltage is applied. The power dissipated by the resistor is represented by (1.4) R where P is the power dissipated (W), V is the voltage across the resistor(V), and R is the resistance(Q2).An ideal resistor dissipates electric energy without storing electric or magnetic energy Resistor Networks Resistors may be joined to form networks. If resistors are joined in series, the effective resistance(Rr) is the m of the individual resistances(Fig 1.4) R WWWW 1-R THE ABOVE RESISTOR IS 3. 25 SQUARES FIGURE 1.3 A resistor with IFp=100, THEN R=325ax1000=3259 flowing through it will have a FIGURE 1.2 Number of squares in a rectangular resistor. oltage drop of IR across it
© 2000 by CRC Press LLC The resistance can also be defined in terms of sheet resistivity. If the sheet resistivity is used, a standard sheet thickness is assumed and factored into resistivity. Typically, resistors are rectangular in shape; therefore the length l divided by the width w gives the number of squares within the resistor (Fig. 1.2). The number of squares multiplied by the resistivity is the resistance. (1.2) where rsheet is the sheet resistivity (W/square), l is the length of resistor (cm), w is the width of the resistor (cm), and Rsheet is the sheet resistance (W). The resistance of a resistor can be defined in terms of the voltage drop across the resistor and current through the resistor related by Ohm’s law, (1.3) where R is the resistance (W), V is the voltage across the resistor (V), and I is the current through the resistor (A). Whenever a current is passed through a resistor, a voltage is dropped across the ends of the resistor. Figure 1.3 depicts the symbol of the resistor with the Ohm’s law relation. All resistors dissipate power when a voltage is applied. The power dissipated by the resistor is represented by (1.4) where P is the power dissipated (W), V is the voltage across the resistor (V), and R is the resistance (W). An ideal resistor dissipates electric energy without storing electric or magnetic energy. Resistor Networks Resistors may be joined to form networks. If resistors are joined in series, the effective resistance (RT) is the sum of the individual resistances (Fig. 1.4). (1.5) FIGURE 1.1 Resistance of a rectangular cross-section resistor with cross-sectional area A and length L. R l w sheet = rsheet R V I = P V R = 2 R R T i i n = = Â 1 FIGURE 1.2 Number of squares in a rectangular resistor. FIGURE 1.3 A resistor with resistance R having a current I flowing through it will have a voltage drop of IR across it
R1 R WWwr-o0···WWW0···0WW FIGURE 1. 4 Resistors connected in series If resistors are joined in parallel, the effective resistance(Rr)is the reciprocal of the sum of the reciprocals of individual resistances(Fig. 1.5 (16) Ry Temperature Coefficient of Electrical Resistance The resistance for most resistors changes with temperature. The tem perature coefficient of electrical resistance is the change in electrical WW resistance of a resistor per unit change in temperature. The tempera- ture coefficient of resistance is measured in S/C. The temperature coefficient of resistors may be either positive or negative. A positive temperature coefficient denotes a rise in resistance with a rise in tem erature; a negative temperature coefficient of resistance denotes a decrease in resistance with a rise in temperature. Pure metals typically have a positive temperature coefficient of resistance, while some metal alloys such as constantin and manganin have a zero temperature coef- ficient of resistance. Carbon and graphite mixed with binders usually fIGuRE 15 Resistors connected exhibit negative temperature coefficients, although certain choices of binders and process variations may yield positive temperature coeff.parallel. cients. The temperature coefficient of resistance is given by R(T2)=R(T1)[1+an(T2-T1) (1.7) where an is the temperature coefficient of electrical resistance at reference temperature T, R(T))is the resistance (32), and R(T,) is the temperature T,(S2). Th aken to be 20oC. Because the variation in resistance between any two temperatures is usually not linear as predicted by Eq(1.7), common practice is to apply the equation between temperature increments and the to plot the resistance change versus temperature for a number of incremental temperatures High-Frequency Effects Resistors show a change in their resistance value when subjected to ac voltages. The change in resistance with voltage frequency is W known as the Boella effect. The effect occurs because all resistors have some inductance and capacitance along with the resistive component and thus can be approximated by an equivalent circuit shown in Fig. 1.6. Even though the definition of useful frequency FIGURE 1.6 Equivalent circuit for a resistor range is application dependent, typically, the useful range of the resistor is the highest frequency at which the impedance differs from the resistance by more than the tolerance of the resistor The frequency effect on resistance varies with the resistor construction. wire-wound resistors typically exhibit an increase in their impedance with frequency. In composition resistors the capacitances are formed by the many conducting particles which are held in contact by a dielectric binder. The ac impedance for film resistors remains constant until 100 MHz(1 MHz =10 Hz) and then decreases at higher frequencies( Fig. 1.7).For film resistors, the decrease in dc resistance at higher frequencies decreases with increase in resistance. Film resistors have the most stable high-frequency performance. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC If resistors are joined in parallel, the effective resistance (RT) is the reciprocal of the sum of the reciprocals of individual resistances (Fig. 1.5). (1.6) Temperature Coefficient of Electrical Resistance The resistance for most resistors changes with temperature. The temperature coefficient of electrical resistance is the change in electrical resistance of a resistor per unit change in temperature. The temperature coefficient of resistance is measured in W/°C. The temperature coefficient of resistors may be either positive or negative. A positive temperature coefficient denotes a rise in resistance with a rise in temperature; a negative temperature coefficient of resistance denotes a decrease in resistance with a rise in temperature. Pure metals typically have a positive temperature coefficient of resistance, while some metal alloys such as constantin and manganin have a zero temperature coef- ficient of resistance. Carbon and graphite mixed with binders usually exhibit negative temperature coefficients, although certain choices of binders and process variations may yield positive temperature coeffi- cients. The temperature coefficient of resistance is given by R(T2) = R(T1)[1 + aT1(T2 – T1)] (1.7) where aT1 is the temperature coefficient of electrical resistance at reference temperature T1, R(T2) is the resistance at temperature T2 (W), and R(T1) is the resistance at temperature T1 (W). The reference temperature is usually taken to be 20°C. Because the variation in resistance between any two temperatures is usually not linear as predicted by Eq. (1.7), common practice is to apply the equation between temperature increments and then to plot the resistance change versus temperature for a number of incremental temperatures. High-Frequency Effects Resistors show a change in their resistance value when subjected to ac voltages. The change in resistance with voltage frequency is known as the Boella effect. The effect occurs because all resistors have some inductance and capacitance along with the resistive component and thus can be approximated by an equivalent circuit shown in Fig. 1.6. Even though the definition of useful frequency range is application dependent, typically, the useful range of the resistor is the highest frequency at which the impedance differs from the resistance by more than the tolerance of the resistor. The frequency effect on resistance varies with the resistor construction.Wire-wound resistors typically exhibit an increase in their impedance with frequency. In composition resistors the capacitances are formed by the many conducting particles which are held in contact by a dielectric binder. The ac impedance for film resistors remains constant until 100 MHz (1 MHz = 106 Hz) and then decreases at higher frequencies (Fig. 1.7). For film resistors, the decrease in dc resistance at higher frequencies decreases with increase in resistance. Film resistors have the most stable high-frequency performance. FIGURE 1.4 Resistors connected in series. 1 1 1 R R T i i n = = Â FIGURE 1.5 Resistors connected in parallel. FIGURE 1.6 Equivalent circuit for a resistor
Frequency (MHz) FIGURE 1.7 Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors. The smaller the diameter of the resistor the better is its frequency response. Most high-frequency resistors have a length to diameter ratio between 4: 1 to 10: 1. Dielectric losses are kept to a minimum by proper choice of base material Voltage Coefficient of Resistance Resistance is not always independent of the applied voltage. The voltage coefficient of resistance is the change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. The voltage coefficient is given by the relationship Voltage coefficient 100(R1-R2) (18) R2(V-V2) where R, is the resistance at the rated voltage Vi and R2 is the resistance at 10% of rated voltage V2. Resistors exhibit electrical noise in the form of small ac voltage fluctuations when dc voltage is applied. Noise in a resistor is a function of the applied voltage, physical dimensions, and materials. The total noise is a sum of Johnson noise, current flow noise, noise due to cracked bodies, resistors the noise can also be caused by the jumping of a moving contact over turns and by an imperfect electrical path between the contact and resistance element. The Johnson noise is temperature-dependent thermal noise(Fig. 1. 8). Thermal noise is also called"white noise" because the noise level is the same at all frequencies. The magnitude of thermal ERMs (V), is dependent on the resistance value and the temperature of the resistance due to thermal agitation kRT△f where ERMs is the root-mean-square value of the noise voltage(V), R is the resistance(]2), K is the Boltzmann constant(1.38x 102J/K), Tis the temperature( K), and Af is the bandwidth(Hz)over which the noise energy Figure 1. 8 shows the variation in current noise versus voltage frequency. Current noise varies inversely with frequency and is a function of the current flowing through the resistor and the value of the resistor. The magnitude of current noise is directly proportional to the square root of current. The current noise magnitude is usually expressed by a noise index given as the ratio of the root-mean-square current noise voltage(ERMs
© 2000 by CRC Press LLC The smaller the diameter of the resistor the better is its frequency response. Most high-frequency resistors have a length to diameter ratio between 4:1 to 10:1. Dielectric losses are kept to a minimum by proper choice of base material. Voltage Coefficient of Resistance Resistance is not always independent of the applied voltage. The voltage coefficient of resistance is the change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. The voltage coefficient is given by the relationship (1.8) where R1 is the resistance at the rated voltage V1 and R2 is the resistance at 10% of rated voltage V2. Noise Resistors exhibit electrical noise in the form of small ac voltage fluctuations when dc voltage is applied. Noise in a resistor is a function of the applied voltage, physical dimensions, and materials. The total noise is a sum of Johnson noise, current flow noise, noise due to cracked bodies, and loose end caps and leads. For variable resistors the noise can also be caused by the jumping of a moving contact over turns and by an imperfect electrical path between the contact and resistance element. The Johnson noise is temperature-dependent thermal noise (Fig. 1.8). Thermal noise is also called “white noise” because the noise level is the same at all frequencies. The magnitude of thermal noise, ERMS (V), is dependent on the resistance value and the temperature of the resistance due to thermal agitation. (1.9) where ERMS is the root-mean-square value of the noise voltage (V), R is the resistance (W), K is the Boltzmann constant (1.38 ¥ 10–23 J/K), T is the temperature (K), and Df is the bandwidth (Hz) over which the noise energy is measured. Figure 1.8 shows the variation in current noise versus voltage frequency. Current noise varies inversely with frequency and is a function of the current flowing through the resistor and the value of the resistor. The magnitude of current noise is directly proportional to the square root of current. The current noise magnitude is usually expressed by a noise index given as the ratio of the root-mean-square current noise voltage (ERMS) FIGURE 1.7 Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors. Voltage coefficient = 100( ( 1 2 1 R R R V V – ) – ) 2 2 E kRT f RMS = 4 D