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 tem￾perature coefficient of electrical resistance is the change in electrical resistance of a resistor per unit change in temperature. The tempera￾ture 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 tem￾perature; 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