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(BQ) Part 2 book Introductory circuit analysis has contents: The basic elements and phasors, series and parallel AC circuits, series parallel AC networks, methods of analysis and related topics, pulse waveforms and the RC response, nonsinusodial circuits, polyphase systems,...and other contents.
14 The Basic Elements and Phasors 14.1 INTRODUCTION The response of the basic R, L, and C elements to a sinusoidal voltage and current will be examined in this chapter, with special note of how frequency will affect the “opposing” characteristic of each element Phasor notation will then be introduced to establish a method of analysis that permits a direct correspondence with a number of the methods, theorems, and concepts introduced in the dc chapters 14.2 THE DERIVATIVE In order to understand the response of the basic R, L, and C elements to a sinusoidal signal, you need to examine the concept of the derivative in some detail It will not be necessary that you become proficient in the mathematical technique, but simply that you understand the impact of a relationship defined by a derivative Recall from Section 10.11 that the derivative dx/dt is defined as the rate of change of x with respect to time If x fails to change at a particular instant, dx ϭ 0, and the derivative is zero For the sinusoidal waveform, dx/dt is zero only at the positive and negative peaks (qt ϭ p/2 and p in Fig 14.1), since x fails to change at these instants of time The derivative dx/dt is actually the slope of the graph at any instant of time A close examination of the sinusoidal waveform will also indicate that the greatest change in x will occur at the instants qt ϭ 0, p, and 2p The derivative is therefore a maximum at these points At and 2p, x increases at its greatest rate, and the derivative is given a positive sign since x increases with time At p, dx/dt decreases at the same rate as it increases at and 2p, but the derivative is given a negative sign since x decreases with time Since the rate of change at 0, p, and 2p is the same, the magnitude of the derivative at these points is the same also For various values of qt between these maxima and minima, the derivative will exist and will have values from the minimum to the maximum inclusive A plot of the derivative in Fig 14.2 shows that the derivative of a sine wave is a cosine wave 576 THE BASIC ELEMENTS AND PHASORS dx = dt x dx = max dt p Sine wave p 2p 3 p qt dx = dt FIG 14.1 Defining those points in a sinusoidal waveform that have maximum and minimum derivatives dx = dt dx dt dx = dt max p p max p 2p Cosine wave qt max FIG 14.2 Derivative of the sine wave of Fig 14.1 The peak value of the cosine wave is directly related to the frequency of the original waveform The higher the frequency, the steeper the slope at the horizontal axis and the greater the value of dx/dt, as shown in Fig 14.3 for two different frequencies x1 f1 > f2 x2 Less slope Steeper slope dx1 dt Higher peak dx2 dt Lower peak Smaller negative peak Negative peak FIG 14.3 Effect of frequency on the peak value of the derivative RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT 577 Note in Fig 14.3 that even though both waveforms (x1 and x2) have the same peak value, the sinusoidal function with the higher frequency produces the larger peak value for the derivative In addition, note that the derivative of a sine wave has the same period and frequency as the original sinusoidal waveform For the sinusoidal voltage e(t) ϭ Em sin(qt Ϯ v) the derivative can be found directly by differentiation (calculus) to produce the following: d ᎏ e(t) ϭ qEm cos(qt Ϯ v) dt ϭ 2pfEm cos(qt Ϯ v) (14.1) The mechanics of the differentiation process will not be discussed or investigated here; nor will they be required to continue with the text Note, however, that the peak value of the derivative, 2pfEm, is a function of the frequency of e(t), and the derivative of a sine wave is a cosine wave 14.3 RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT Now that we are familiar with the characteristics of the derivative of a sinusoidal function, we can investigate the response of the basic elements R, L, and C to a sinusoidal voltage or current Resistor i For power-line frequencies and frequencies up to a few hundred kilohertz, resistance is, for all practical purposes, unaffected by the frequency of the applied sinusoidal voltage or current For this frequency region, the resistor R of Fig 14.4 can be treated as a constant, and Ohm’s law can be applied as follows For v ϭ Vm sin qt, Vm sin qt v Vm iϭᎏϭᎏϭᎏ sin qt ϭ Im sin qt R R R Vm Im ϭ ᎏᎏ R where + R v – FIG 14.4 Determining the sinusoidal response for a resistive element (14.2) vR In addition, for a given i, Vm Im v ϭ iR ϭ (Im sin qt)R ϭ Im R sin qt ϭ Vm sin qt where Vm ϭ Im R (14.3) iR p 2p qt A plot of v and i in Fig 14.5 reveals that for a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law FIG 14.5 The voltage and current of a resistive element are in phase 578 THE BASIC ELEMENTS AND PHASORS Inductor + velement – e + i – Opposition FIG 14.6 Defining the opposition of an element to the flow of charge through the element + – e + iL vL – L Opposition a function of f and L FIG 14.7 Defining the parameters that determine the opposition of an inductive element to the flow of charge For the series configuration of Fig 14.6, the voltage velement of the boxed-in element opposes the source e and thereby reduces the magnitude of the current i The magnitude of the voltage across the element is determined by the opposition of the element to the flow of charge, or current i For a resistive element, we have found that the opposition is its resistance and that velement and i are determined by velement ϭ iR We found in Chapter 12 that the voltage across an inductor is directly related to the rate of change of current through the coil Consequently, the higher the frequency, the greater will be the rate of change of current through the coil, and the greater the magnitude of the voltage In addition, we found in the same chapter that the inductance of a coil will determine the rate of change of the flux linking a coil for a particular change in current through the coil The higher the inductance, the greater the rate of change of the flux linkages, and the greater the resulting voltage across the coil The inductive voltage, therefore, is directly related to the frequency (or, more specifically, the angular velocity of the sinusoidal ac current through the coil) and the inductance of the coil For increasing values of f and L in Fig 14.7, the magnitude of vL will increase as described above Utilizing the similarities between Figs 14.6 and 14.7, we find that increasing levels of vL are directly related to increasing levels of opposition in Fig 14.6 Since vL will increase with both q (ϭ 2pf ) and L, the opposition of an inductive element is as defined in Fig 14.7 We will now verify some of the preceding conclusions using a more mathematical approach and then define a few important quantities to be employed in the sections and chapters to follow For the inductor of Fig 14.8, we recall from Chapter 12 that iL = Im sin qt + L vL diL vL ϭ L ᎏ dt and, applying differentiation, – FIG 14.8 Investigating the sinusoidal response of an inductive element diL d ᎏ ϭ ᎏ(Im sin qt) ϭ qIm cos qt dt dt Therefore, diL vL ϭ L ᎏ ϭ L(qIm cos qt) ϭ qLIm cos qt dt vL ϭ Vm sin(qt ϩ 90°) or where Vm ϭ qLIm Note that the peak value of vL is directly related to q (ϭ 2pf ) and L as predicted in the discussion above A plot of vL and iL in Fig 14.9 reveals that for an inductor, vL leads iL by 90°, or iL lags vL by 90° If a phase angle is included in the sinusoidal expression for iL, such as iL ϭ Im sin(qt Ϯ v) then vL ϭ qLIm sin(qt Ϯ v ϩ 90°) RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT L: vL leads iL by 90° vL Vm iL Im –p 90° 3p p p 2p qt FIG 14.9 For a pure inductor, the voltage across the coil leads the current through the coil by 90° The opposition established by an inductor in a sinusoidal ac network can now be found by applying Eq (4.1): cause Effect ϭ ᎏᎏ opposition which, for our purposes, can be written cause Opposition ϭ ᎏ effect Substituting values, we have Vm qLIm Opposition ϭ ᎏ ϭ ᎏ ϭ qL Im Im revealing that the opposition established by an inductor in an ac sinusoidal network is directly related to the product of the angular velocity (q ϭ 2pf ) and the inductance, verifying our earlier conclusions The quantity qL, called the reactance (from the word reaction) of an inductor, is symbolically represented by XL and is measured in ohms; that is, XL ϭ qL (ohms, ⍀) (14.4) In an Ohm’s law format, its magnitude can be determined from Vm XL ϭ ᎏ Im (ohms, ⍀) (14.5) Inductive reactance is the opposition to the flow of current, which results in the continual interchange of energy between the source and the magnetic field of the inductor In other words, inductive reactance, unlike resistance (which dissipates energy in the form of heat), does not dissipate electrical energy (ignoring the effects of the internal resistance of the inductor) Capacitor Let us now return to the series configuration of Fig 14.6 and insert the capacitor as the element of interest For the capacitor, however, we will determine i for a particular voltage across the element When this approach reaches its conclusion, the relationship between the voltage 579 580 THE BASIC ELEMENTS AND PHASORS and current will be known, and the opposing voltage (velement) can be determined for any sinusoidal current i Our investigation of the inductor revealed that the inductive voltage across a coil opposes the instantaneous change in current through the coil For capacitive networks, the voltage across the capacitor is limited by the rate at which charge can be deposited on, or released by, the plates of the capacitor during the charging and discharging phases, respectively In other words, an instantaneous change in voltage across a capacitor is opposed by the fact that there is an element of time required to deposit charge on (or release charge from) the plates of a capacitor, and V ϭ Q/C Since capacitance is a measure of the rate at which a capacitor will store charge on its plates, for a particular change in voltage across the capacitor, the greater the value of capacitance, the greater will be the resulting capacitive current In addition, the fundamental equation relating the voltage across a capacitor to the current of a capacitor [i ϭ C(dv/dt)] indicates that for a particular capacitance, the greater the rate of change of voltage across the capacitor, the greater the capacitive current Certainly, an increase in frequency corresponds to an increase in the rate of change of voltage across the capacitor and to an increase in the current of the capacitor The current of a capacitor is therefore directly related to the frequency (or, again more specifically, the angular velocity) and the capacitance of the capacitor An increase in either quantity will result in an increase in the current of the capacitor For the basic configuration of Fig 14.10, however, we are interested in determining the opposition of the capacitor as related to the resistance of a resistor and qL for the inductor Since an increase in current corresponds to a decrease in opposition, and iC is proportional to q and C, the opposition of a capacitor is inversely related to q (ϭ 2pf ) and C + – e + iC – vC C Opposition inversely related to f and C FIG 14.10 Defining the parameters that determine the opposition of a capacitive element to the flow of the charge iC = ? + C vC = Vm sin qt We will now verify, as we did for the inductor, some of the above conclusions using a more mathematical approach For the capacitor of Fig 14.11, we recall from Chapter 10 that – dvC iC ϭ C ᎏ dt and, applying differentiation, FIG 14.11 Investigating the sinusoidal response of a capacitive element d dvC ᎏ ϭ ᎏ(Vm sin qt) ϭ qVm cos qt dt dt RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT 581 2p qt Therefore, dvC iC ϭ C ᎏ ϭ C(qVm cos qt) ϭ qCVm cos qt dt iC ϭ Im sin(qt ϩ 90°) or where Im ϭ qCVm Note that the peak value of iC is directly related to q (ϭ 2pf ) and C, as predicted in the discussion above A plot of vC and iC in Fig 14.12 reveals that C: iC leads vC by 90° * for a capacitor, iC leads vC by 90°, or vC lags iC by 90° Vm If a phase angle is included in the sinusoidal expression for vC, such as vC ϭ Vm sin(qt Ϯ v) then –p iC ϭ qCVm sin(qt Ϯ v ϩ 90°) vC iC Im 90° p p 3p Applying FIG 14.12 The current of a purely capacitive element leads the voltage across the element by 90° cause Opposition ϭ ᎏ effect and substituting values, we obtain Vm Vm Opposition ϭ ᎏ ϭ ᎏ ϭ ᎏ Im qCVm qC which agrees with the results obtained above The quantity 1/qC, called the reactance of a capacitor, is symbolically represented by XC and is measured in ohms; that is, XC ϭ ᎏ qC (ohms, ⍀) (14.6) In an Ohm’s law format, its magnitude can be determined from Vm XC ϭ ᎏ Im (ohms, ⍀) (14.7) Capacitive reactance is the opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of the capacitor Like the inductor, the capacitor does not dissipate energy in any form (ignoring the effects of the leakage resistance) In the circuits just considered, the current was given in the inductive circuit, and the voltage in the capacitive circuit This was done to avoid the use of integration in finding the unknown quantities In the inductive circuit, diL vL ϭ L ᎏ dt *A mnemonic phrase sometimes used to remember the phase relationship between the voltage and current of a coil and capacitor is “ELI the ICE man.” Note that the L (inductor) has the E before the I (e leads i by 90°), and the C (capacitor) has the I before the E (i leads e by 90°) 582 THE BASIC ELEMENTS AND PHASORS Ύ iL ϭ ᎏ vL dt L but (14.8) In the capacitive circuit, dvC iC ϭ C ᎏ dt Ύ vC ϭ ᎏ iC dt C but (14.9) Shortly, we shall consider a method of analyzing ac circuits that will permit us to solve for an unknown quantity with sinusoidal input without having to use direct integration or differentiation It is possible to determine whether a network with one or more elements is predominantly capacitive or inductive by noting the phase relationship between the input voltage and current If the source current leads the applied voltage, the network is predominantly capacitive, and if the applied voltage leads the source current, it is predominantly inductive Since we now have an equation for the reactance of an inductor or capacitor, we not need to use derivatives or integration in the examples to be considered Simply applying Ohm’s law, Im ϭ Em /XL (or XC), and keeping in mind the phase relationship between the voltage and current for each element, will be sufficient to complete the examples EXAMPLE 14.1 The voltage across a resistor is indicated Find the sinusoidal expression for the current if the resistor is 10 ⍀ Sketch the curves for v and i a v ϭ 100 sin 377t b v ϭ 25 sin(377t + 60°) Solutions: a Eq (14.2): Vm 100 V Im ϭ ᎏ ϭ ᎏ ϭ 10 A R 10 ⍀ (v and i are in phase), resulting in i ϭ 10 sin 377t The curves are sketched in Fig 14.13 Vm = 100 V vR In phase Im = 10 A iR p FIG 14.13 Example 14.1(a) 2p ␣ RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT b Eq (14.2): Vm 25 V Im ϭ ᎏ ϭ ᎏ ϭ 2.5 A R 10 ⍀ (v and i are in phase), resulting in i ϭ 2.5 sin(377t ؉ 60°) The curves are sketched in Fig 14.14 vR Vm = 25 V iR In phase p Im = 2.5 A – p2 60° 3p 2p ␣ p FIG 14.14 Example 14.1(b) EXAMPLE 14.2 The current through a 5-⍀ resistor is given Find the sinusoidal expression for the voltage across the resistor for i ϭ 40 sin(377t ϩ 30°) Solution: Eq (14.3): Vm ϭ ImR ϭ (40 A)(5 ⍀) ϭ 200 V (v and i are in phase), resulting in v ϭ 200 sin(377t ؉ 30°) EXAMPLE 14.3 The current through a 0.1-H coil is provided Find the sinusoidal expression for the voltage across the coil Sketch the v and i curves a i ϭ 10 sin 377t b i ϭ sin(377t Ϫ 70°) Solutions: a Eq (14.4): Eq (14.5): XL ϭ qL ϭ (377 rad/s)(0.1 H) ϭ 37.7 ⍀ Vm ϭ ImXL ϭ (10 A)(37.7 ⍀) ϭ 377 V and we know that for a coil v leads i by 90° Therefore, v ϭ 377 sin(377t ؉ 90°) The curves are sketched in Fig 14.15 Vm = 377 V vL v leads i by 90° Im = 10 A 90° – p2 p p FIG 14.15 Example 14.3(a) iL 3p 2p ␣ 583 584 THE BASIC ELEMENTS AND PHASORS b XL remains at 37.7 ⍀ Vm ϭ ImXL ϭ (7 A)(37.7 ⍀) ϭ 263.9 V and we know that for a coil v leads i by 90° Therefore, v ϭ 263.9 sin(377t Ϫ 70° ϩ 90°) and v ϭ 263.9 sin(377t ؉ 20°) The curves are sketched in Fig 14.16 Vm = 263.9 V vL iL Im = A 20° 70° p p 3p 2p ␣ 90° v leads i by 90° FIG 14.16 Example 14.3(b) EXAMPLE 14.4 The voltage across a 0.5-H coil is provided below What is the sinusoidal expression for the current? v ϭ 100 sin 20t Solution: XL ϭ qL ϭ (20 rad/s)(0.5 H) ϭ 10 ⍀ Vm 100 V Im ϭ ᎏ ϭ ᎏ ϭ 10 A 10 ⍀ XL and we know that i lags v by 90° Therefore, i ϭ 10 sin(20t ؊ 90°) EXAMPLE 14.5 The voltage across a 1-mF capacitor is provided below What is the sinusoidal expression for the current? Sketch the v and i curves v ϭ 30 sin 400t Solution: Eq (14.6): Eq (14.7): 1 106 ⍀ XC ϭ ᎏ ϭ ᎏᎏᎏ ϭ ᎏ ϭ 2500 ⍀ Ϫ6 qC (400 rad/s)(1 ϫ 10 F) 400 Vm 30 V Im ϭ ᎏ ϭ ᎏ ϭ 0.0120 A ϭ 12 mA XC 2500 ⍀ and we know that for a capacitor i leads v by 90° Therefore, i ϭ 12 ؋ 10؊3 sin(400t ؉ 90°) APPENDIX C ϭ 1[6 Ϫ 1] ϩ 3[Ϫ(6 Ϫ 3)] ϩ 2[2 Ϫ 6] ϭ ϩ 3(Ϫ3) ϩ 2(Ϫ4) ϭ5Ϫ9Ϫ8 ϭ ؊12 0 6 4 b D ϭ 2 5 ϭ ϩ Ϫ 4 8 0 6 4 ϩ8 ϩ 0 0 6 5 ϭ ϩ 2[Ϫ(0 Ϫ 24)] ϩ 8[(20 Ϫ 0)] ϭ ϩ 2(24) ϩ 8(20) ϭ 48 ϩ 160 ϭ 208 1205 Appendix D COLOR CODING OF MOLDED TUBULAR CAPACITORS (PICOFARADS) Color Significant Figure Decimal Multiplier Tolerance ؎% Black Brown Red Orange Yellow Green Blue Violet Gray White 10 100 1000 10,000 105 106 — — — 20 — — 30 40 — — — 10 Note: Voltage rating is identified by a single-digit number for ratings up to 900 V and a two-digit number above 900 V Two zeros follow the voltage figure 1st 2nd Capacitance significant figure Multiplier Tolerance Voltage significant figure 1st 2nd (If required) FIG D.1 1206 in pF Appendix E THE GREEK ALPHABET Letter Capital Lowercase Used to Designate Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu A B G D E Z H V I K L M a b g d e z h v i k l Area, angles, coefficients Angles, coefficients, flux density Specific gravity, conductivity Density, variation Base of natural logarithms Coefficients, coordinates, impedance Efficiency, hysteresis coefficient Phase angle, temperature Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega N Y O P R ⌺ T ⌼ ⌽ X W ⍀ m n y o p r j t v f x w q Dielectric constant, susceptibility Wavelength Amplification factor, micro, permeability Reluctivity 3.1416 Resistivity Summation Time constant Angles, magnetic flux Dielectric flux, phase difference Ohms, angular velocity 1207 Appendix F MAGNETIC PARAMETER CONVERSIONS SI (MKS) ⌽ B webers (Wb) Wb Wb/m2 Wb/m2 A m2 CGS maxwells ϭ 108 maxwells 1208 Hg 7.97 ϫ 105Bg (At/m) lines/in.2 ϭ 6.452 ϫ 104 lines/in.2 ϭ 104 cm2 ϭ 1550 in.2 Ᏺ NI (ampere-turns, At) 0.4pNI (gilberts) At ϭ 1.257 gilberts NI/l (At/m) At/m lines ϭ 108 lines gauss (maxwells/cm2) ϭ 104 gauss mo 4p ϫ 10Ϫ7 Wb/Am ϭ gauss/oersted H English ϭ 3.20 lines/Am NI (At) gilbert ϭ 0.7958 At 0.4pNI/l (oersteds) NI/l (At/in.) ϭ 1.26 ϫ 10Ϫ2 oersted ϭ 2.54 ϫ 10Ϫ2 At/in Bg (oersteds) 0.313Bg (At/in.) Appendix G MAXIMUM POWER TRANSFER CONDITIONS Derivation of maximum power transfer conditions for the situation where the resistive component of the load is adjustable but the load reactance is set in magnitude.* For the circuit of Fig G.1, the power delivered to the load is determined by V R2 P ϭ ᎏL RL ZTh I XTh I RTh RL ZL + ZT ETh XL – FIG G.1 Applying the voltage divider rule: RLETh VRL ϭ ᎏᎏᎏᎏ RL ϩ RTh ϩ XTh Є90° ϩ XL Є90° The magnitude of VRL is determined by RLETh VRL ϭ ᎏᎏᎏᎏ ͙ෆ (Rෆ ϩෆ Rෆ )2ෆ ϩෆ(ෆ Xෆ ϩෆ Xෆ L ෆ Thෆ Thෆ L )ෆ and R 2L E Th V R2L ϭ ᎏᎏᎏ (RL ϩ RTh )2 ϩ (XTh ϩ XL )2 with RL ETh VR2L P ϭ ᎏ ϭ ᎏᎏᎏ RL (RL ϩ RTh)2 ϩ (XTh ϩ XL)2 Using differentiation (calculus), maximum power will be transferred when dP/dRL ϭ The result of the preceding operation is that RL ϭ ͙R ෆTh ෆෆ ϩෆ(X ෆෆ ϩෆ XLෆ)2ෆ Thෆ [Eq (18.21)] The magnitude of the total impedance of the circuit is 2 ZT ϭ ͙(R ෆෆ ϩෆ Rෆ ෆෆ(X ෆෆ ϩෆ Xෆ Thෆ L )ෆϩ Thෆ L )ෆ Substituting this equation for RL and applying a few algebraic maneuvers will result in ZT ϭ 2RL(RL ϩ RTh) *With sincerest thanks for the input of Professor Harry J Franz of the Beaver Campus of Pennsylvania State University 1209 1210 APPENDIXES and the power to the load RL will be 2 E Th E Th RL ᎏᎏ P ϭ I 2RL ϭ ᎏ R ϭ L ZT 2RL(RL ϩ RTh) E 2Th ϭ ᎏᎏ RL ϩ RTh ᎏᎏ E Th ϭᎏ 4Rav with RL ϩ RTh Rav ϭ ᎏ Appendix H ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS Chapter h CGS MKS ϭ CGS ϭ 20°C K ϭ SI ϭ 293.15 11 45.72 cm 13 (a) 15 ϫ 103 (b) 30 ϫ 10Ϫ3 (c) 7.4 ϫ 106 (d) 6.8 ϫ 10Ϫ6 (e) 402 ϫ 10Ϫ6 (f) 200 ϫ 10Ϫ12 15 (a) 104 (b) 10 (c) 109 (d) 10Ϫ2 (e) 10 (f) 1031 17 (a) 10Ϫ1 (b) 10Ϫ4 (c) 109 (d) 10Ϫ9 (e) 1042 (f) 103 19 (a) 106 (b) 10Ϫ2 (c) 1032 (d) 10Ϫ63 21 (a) 10Ϫ6 (b) 10Ϫ3 (c) 10Ϫ8 (d) 109 (e) 10Ϫ16 (f) 10Ϫ1 23 (a) 0.006 (b) 400 (c) 5000, 5, 0.005 (d) 0.0003, 0.3, 300 25 (a) 90 s (b) 144 s (c) 50 ϫ 103 ms (d) 160 mm (e) 120 ns (f) 41.898 days (g) 1.02 m 27 (a) 2.54 m (b) 1.219 m (c) 26.7 N (d) 0.1348 lb (e) 4921.26 ft (f) 3.2187 m (g) 8530.17 yd 29 670.62 ϫ 106 mph 31 2.045 s 33 67.06 days 35 $900 37 345.6 m 39 47.29 min/mile 41 (a) 4.74 ϫ 10Ϫ3 Btu (b) 7.098 ϫ 10Ϫ4 m3 (c) 1.2096 ϫ 105 s (d) 2113.38 pints 43 5.000 45 2.949 Chapter (a) 18 mN (b) mN (c) 180 mN (a) 72 mN (b) Q1 ϭ 20 mC, Q2 ϭ 40 mC 3.1 A 11 90 C 13 0.5 A 15 1.194 A > A (yes) 17 (a) 1.248 million (b) 0.936 million, sol ϭ (a) 19 252 J 21 C 23 3.533 V 25 A 27 25 h 29 0.773 h 31 60 Ah:40 Ah ϭ 1.5:1, 50% more with 60 Ah 33 545.45 mA, 129.6 kJ 43 600 C Chapter (a) 500 mils (b) 10 mils (c) mils (d) 1000 mils (e) 240 mils (f) 3.937 mils (a) 0.04 in (b) 0.03 in (c) 0.2 in (d) 0.025 in (e) 0.00278 in (f) 0.009 in 73.33 ⍀ 3.581 ft (a) Rsilver > Rcopper > Raluminum (b) silver 9.9 ⍀, copper 1.037 ⍀, aluminum 0.34 ⍀ 11 (a) 21.71 m⍀ (b) 35.59 m⍀ (c) increases (d) decreases 13 942.28 m⍀ 15 (a) #8: 1.1308 ⍀ #18: 11.493 ⍀ (b) #18: #8 ϭ 10.1641:1 ഡ 10:1 #18: #8 ϭ 1:10.164 ഡ 1:10 17 (a) 1.087 mA/CM (b) 1.384 kA/in.2 (c) 3.6127 in.2 19 (a) 21.71 m⍀ (b) 35.59 m⍀ 21 0.15 in 23 2.409 ⍀ 25 3.67 ⍀ 27 0.046 ⍀ 29 (a) 40.29°C (b) Ϫ195.61°C 31 (a) a20 ഡ 0.00393 (b) 83.61°C 33 1.751 ⍀ 35 142.86 41 Ϫ30°C: 10.2 k⍀ 100°C: 10.15 k⍀ 43 6.5 k⍀ 47 (a) Red Red Brown Silver (b) Yellow Violet Red Silver (c) Blue Gray Orange Silver (d) White Brown Green Silver 49 yes 51 (a) 0.1566 S (b) 0.0955 S (c) 0.0219 S 57 (a) 10 fc: k⍀ 100 fc: 0.4 k⍀ (b) neg (c) no—log scales (d) Ϫ321.43 ⍀/fc Chapter 11 13 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 15 V k⍀ 72 mV 54.55 ⍀ 28.571 ⍀ 1.2 k⍀ (a) 12.632 ⍀ (b) 4.1 MJ 800 V 1W (a) 57,600 J (b) 16 ϫ 10Ϫ3 kWh 2s 196 mW 4A 9.61 V 0.833 A, 144.06 ⍀ (a) 0.133 mA (b) 66.5 mAh (c) ഡ 70.7 mA (a) 12 kW (b) 10,130 W < 12,000 W (yes) 16.34 A (a) 238 W (b) 17.36% (a) 1657.78 W (b) 15.07 A (c) 19.38 A 65.25% 80% (a) 17.9% (b) 76.73%, 328.66% increase (a) 1350 J (b) W doubles, P the same 6.67 h (a) 50 kW (b) 240.38 A (c) 90 kWh $2.19 Chapter (a) (b) (c) (d) 20 ⍀, A 1.63 M⍀ , 6.135 mA 110 ⍀, 318.2 mA 10 k⍀, 12 mA 1211 1212 APPENDIXES (a) 16 V (b) 4.2 V (a) 0.388 A (CW) (b) 2.087 A (CCW) (a) V (b) 70 V 3.28 mA, 7.22 V 11 (a) 70.6 ⍀, 85 mA (CCW), V1 ϭ 2.8045 V, V2 ϭ 0.4760 V, V3 ϭ 0.850 V, V4 ϭ 1.870 V (b)–(c) P1 ϭ 0.2384 W, P2 ϭ 0.0405 W, P3 ϭ 0.0723 W, P4 ϭ 0.1590 W (d) all W 13 (a) 225 ⍀, 0.533 A (b) W (c) 15 V 15 All Vab ϩϪ 17 19 21 23 25 27 29 31 33 35 (a) 66.67 V (b) Ϫ8 V (c) 20 V (d) 0.18 V (a) 12 V (b) 24 V (c) 60 ⍀ (d) 0.4 A (e) 60 ⍀ (a) Rs ϭ 80 ⍀ (b) 0.2 W < W R1 ϭ k⍀, R2 ϭ 15 k⍀ (a) R1 ϭ 0.4 k⍀, R2 ϭ 1.2 k⍀, R3 ϭ 4.8 k⍀ (b) R1 ϭ 0.4 M⍀, R2 ϭ 1.2 M⍀, R3 ϭ 4.8 M⍀ (a) I (CW) ϭ 6.667 A, V ϭ 20 V (b) I (CW) ϭ A, V ϭ 10 V (a) 20 V, 26 V, 35 V, Ϫ12 V, V (b) Ϫ6 V, Ϫ47 V, V (c) Ϫ15 V, Ϫ38 V V0 ϭ V, V4 ϭ 10 V, V7 ϭ V, V10 ϭ 20 V, V23 ϭ V, V30 ϭ Ϫ8 V, V67 ϭ V, V56 ϭ Ϫ6 V, I(up) ϭ 1.5 A 2⍀ 100 ⍀ 1.52% Chapter (a) 2, 3, (b) 2, (c) 1, (a) ⍀, 0.1667 S (b) k⍀, mS (c) 2.076 k⍀, 0.4817 mS (d) 1.333 ⍀, 0.75 S (e) 9.948 ⍀, 100.525 mS (f) 0.6889 ⍀, 1.4516 S (a) 18 ⍀ (b) R1 ϭ R2 ϭ 24 ⍀ 120 ⍀ (a) 0.8571 ⍀, 1.1667 S (b) Is ϭ 1.05 A, I1 ϭ 0.3 A, I2 ϭ 0.15 A, I3 ϭ 0.6 A (d) P1 ϭ 0.27 W, P2 ϭ 0.135 W, P3 ϭ 0.54 W, Pdel ϭ 0.945 W (e) R1, R2 ϭ W, R3 ϭ W 11 (a) 66.67 mA (b) 225 ⍀ (c) W 13 (a) Is ϭ 7.5 A, I1 ϭ 1.5 A (b) Is ϭ 9.6 mA, I1 ϭ 0.8 mA 15 1260 W 17 (a) mA (b) 24 V (c) 18.4 mA 19 (a) I1 ϭ mA, I2 ϭ mA, I3 ϭ 1.5 mA (b) I2 ϭ mA, I3 ϭ 1.5 mA, I4 ϭ 5.5 mA, I1 ϭ mA 21 (a) R1 ϭ ⍀, R2 ϭ 10 ⍀ (b) E ϭ 12 V, I2 ϭ 1.333 A, I3 ϭ A, R3 ϭ 12 ⍀, I ϭ 4.333 A (c) I1 ϭ 64 mA, I3 ϭ 16 mA, I2 ϭ 20 mA, R ϭ 3.2 k⍀, I ϭ 36 mA (d) E ϭ 30 V, I1 ϭ A, I2 ϭ I3 ϭ 0.5 A, R2 ϭ R3 ϭ 60 ⍀, PR2 ϭ 15 W 23 (a) I1 ϭ A, I2 ϭ A (b) I1 ϭ A, I2 ϭ A, I3 ϭ A, I4 ϭ 1.333 A (c) I1 ϭ 272.73 mA, I2 ϭ 227.27 mA, I3 ϭ 90.91 mA, I4 ϭ 500 mA (d) I2 ϭ 4.5 A, I3 ϭ 8.5 A, I4 ϭ 8.5 A 25 (a) I ϭ A, I2 ϭ A, I1 ϭ A 27 R1 ϭ k⍀, R2 ϭ 1.5 k⍀, R3 ϭ 0.5 k⍀ 29 I ϭ A, R ϭ ⍀ 31 (a) 6.13 V (b) V (c) V 33 (a) V (b) 3.997 V (c) 3.871 V (d) V (e) Rm large as possible 35 No! 4-V supply reversed (b) 11 13 15 17 19 21 23 25 27 29 31 Chapter (a) series: E, R1, and R4, parallel: R2 and R3 33 35 37 series: E and R1, parallel: R2 and R3 (c) series: E, R1, and R5; R3 and R4 parallel: none (d) series: R6 and R7, parallel: E, R1, and R4; R2 and R5 (a) yes (KCL) (b) A (c) yes (KCL) (d) V (e) ⍀ (f) A (g) P1 ϭ 12 W, P2 ϭ 18 W, Pdel ϭ 50 W (a) ⍀ (b) Is ϭ A, I1 ϭ A, I2 ϭ A (c) V I1 ϭ A, I2 ϭ 16 A, I3 ϭ 0.8 A, I ϭ 22 A (a) A (b) I2 ϭ 1.333A, I3 ϭ 0.6665A (c) Va ϭ V, Vb ϭ V (a) ⍀, 16 A (b) IR2 ϭ A, I3 ϭ I9 ϭ A (c) I8 ϭ A (d) 14 V (a) VG ϭ 1.9 V, Vs ϭ 3.65 V (b) I1 ϭ I2 ϭ 7.05 mA, ID ϭ 2.433 mA (c) 6.268 V (d) 8.02 V (a) 0.6 A (b) 28 V (a) I2 ϭ 1.667 A, I6 = 1.111 A, I8 ϭ A (a) 1.882 ⍀ (b) V1 ϭ V4 ϭ 32 V (c) A ← (d) 1.882 ⍀ (a) 6.75 A (b) 32 V 8.333 ⍀ (a) 24 A (b) A (c) V3 ϭ 48 V, V5 ϭ 24 V, V7 ϭ 16 V (d) P(R7) ϭ 128 W, P(E) ϭ 5760 W 4.44 W (a) 64 V (b) RL2 ϭ k⍀, RL3 ϭ k⍀ (c) R1 ϭ 0.5 k⍀, R2 ϭ 1.2 k⍀, R3 ϭ k⍀ (a) yes (b) R1 ϭ 750 ⍀, R2 ϭ 250 ⍀ (c) R1 ϭ 745 ⍀, R2 ϭ 255 ⍀ (a) mA (b) Rshunt ϭ m⍀ (a) Rs ϭ 300 k⍀ (b) 20,000 0.05 mA APPENDIX H Chapter 28 V (a) I1 ϭ 12 A, Is ϭ 11 A (b) Vs ϭ 24 V, V3 ϭ V (a) A, ⍀ (b) 4.091 mA, 2.2 k⍀ (a) A (b) A 9.6 V, 2.4 A 11 (a) 5.4545 mA, 2.2 k⍀ (b) 17.375 V (c) 5.375 V (d) 2.443 mA 13 (I) CW: IR1 ϭ 1.445 mA; down: IR3 ϭ 9.958 mA; CCW: IR2 ϭ 8.513 mA (II) CW: IR1 ϭ 2.0316 mA; left: IR2 ϭ 0.8 mA; CW: IR3 ϭ IR4 ϭ 1.2316 mA 15 (d) left: 63.694 mA 17 (a) CW: IR1 ϭ Ϫ A; CW: IR2 ϭ Ϫ A IR3 ϭ A (down) (b) CW: IR1 ϭ Ϫ3.0625 A; CW: IR3 ϭ 0.1875 A IR2 ϭ 3.25 A (up) 19 (I) CW: I1 ϭ 1.8701 A; CW: I2 ϭ Ϫ8.5484 A; Vab ϭ Ϫ22.74 V (II) CW: I2 ϭ 1.274 A; CW: I3 ϭ 0.26 A; Vab ϭ Ϫ0.904 V 21 (a) 72.16 mA, Ϫ4.433 V (b) 1.953 A, Ϫ7.257 V 23 (a) All CW I1 ϭ 0.0321 mA I2 ϭ Ϫ0.8838 mA I3 ϭ Ϫ0.968 mA I4 ϭ Ϫ0.639 mA (b) All CW I1 ϭ Ϫ3.8 A I2 ϭ Ϫ4.2 A I3 ϭ 0.2 A 25 (a) CW, I1 ϭ Ϫ A, I2 ϭ Ϫ A (b) CW, I1 ϭ Ϫ3.0625 A, I2 ϭ 0.1875 A 27 (I) (a) CW (b) I1 ϭ 1.871 A, I2 ϭ Ϫ8.548 A (c) IR1 ϭ 1.871 A, IR2 ϭ Ϫ8.548 A, IR3 ϭ 10.419 A 29 I5⍀ (CW) ϭ 1.9535 A, Va ϭ Ϫ7.26 V 31 (a) All CW, I1 ϭ 0.0321 mA, I2 ϭ Ϫ0.8838 mA, I3 ϭ Ϫ0.968 mA, I4 ϭ Ϫ0.639 mA 33 35 37 39 41 43 45 47 49 51 53 (b) All CW, I1 ϭ 3.8 A, I2 ϭ Ϫ4.2 A, I3 ϭ 0.2 A (I) (b) V1 ϭ Ϫ14.86 V, V2 ϭ Ϫ12.57 V (c) VR1 ϭ VR4 ϭ V1 ϭ Ϫ14.86 V, VR2 ϭ V2 ϭ Ϫ12.57 V, VR3 ϭ 9.71 V (ϩ Ϫ) (II) (b) V1 ϭ Ϫ2.556 V, V2 ϭ 4.03 V (c) VR1 ϭ V1 ϭ Ϫ2.556 V, VR2 ϭ VR5 ϭ V2 ϭ4.03 V, VR4 ϭ VR3 ϭ V2 Ϫ V1 ϭ 6.586 V (I) V1 ϭ 7.238 V, V2 ϭ Ϫ2.453 V, V3 ϭ 1.405 V (II) V1 ϭ Ϫ6.64 V, V2 ϭ 1.288 V, V3 ϭ 10.676 V (a) V1 ϭ 10.083 V, V2 ϭ 6.944 V, V3 ϭ Ϫ17.056 V (b) V1 ϭ 48 V, V2 ϭ 64 V (b) (I) V1 ϭ Ϫ14.86 V, V2 ϭ Ϫ12.57 V (II) V1 ϭ Ϫ2.556 V, V2 ϭ 4.03 V (c) (I) VR1 ϭ VR4 ϭ Ϫ14.86 V, VR2 ϭ Ϫ12.57 V VR3 ϭ V1 ϩ 12 Ϫ V2 ϭ 9.71 V (II) VR1 ϭ Ϫ2.556 V, VR2 ϭ VR5 ϭ 4.03 V VR3 ϭ VR4 ϭ V2 Ϫ V1 ϭ 6.586 V (I) V1 ϭ Ϫ5.311 V, V2 ϭ Ϫ0.6219 V, V3 ϭ 3.751 V VϪ5A ϭ Ϫ5.311 V (II) V1 ϭ Ϫ6.917 V, V2 ϭ 12 V, V3 ϭ 2.3 V V5A ϭ V2 Ϫ V1 ϭ 18.917 V, V2A ϭ V3 Ϫ V2 ϭ Ϫ9.7 V (b) VR5 ϭ 0.1967 V (c) no (d) no (b) IRs ϭ A (c) no (d) no (a) 3.33 mA (b) 1.177 A (a) 133.33 mA (b) A (b) 0.833 mA 4.2 ⍀ 1213 Chapter (a) CW: IR1 ϭ A, IR2 ϭ A, CW: IR3 ϭ A (b) E1: 5.33 W, E2: 0.333 W (c) 8.333 W (d) no (a) down: 4.4545 mA (b) down: 3.11 A (a) ⍀, V (b) ⍀: 0.75 A, 30 ⍀: 0.1667 A, 100 ⍀: 0.0566 A (I) ⍀, 84 V (II) 1.579 k⍀, Ϫ1.149 V (I) 45 ⍀, Ϫ5 V (II) 2.055 k⍀, 16.772 V 11 4.041 k⍀, 9.733 V 13 (I): 14 ⍀, 2.571 A, (II): 7.5 ⍀, 1.333 A 15 (a) 9.756 ⍀, 0.95 A (b) ⍀, 30 A 17 (a) 10 ⍀, 0.2 A (b) 4.033 k⍀, 2.9758 mA 19 (I) (a) 14 ⍀ (b) 23.14 W (II) (a) 7.5 ⍀ (b) 3.33 W 21 (a) 9.756 ⍀, 2.2 W (b) ⍀, 450 W 23 ⍀ 25 500 ⍀ 27 39.3 mA, 220 mV 29 2.25 A, 6.075 V 35 (a) 0.357 mA (b) 0.357 mA (c) yes Chapter 10 11 13 15 17 19 21 ϫ 103 N/C 70 mF 50 V/m ϫ 103 V/m 937.5 pF mica (a) 106 V/m (b) 4.96 mC (c) 0.0248 mF 29,035 V (a) 0.5 s (b) 20(1 Ϫ eϪt/0.5) (c) 1t: 12.64 V, 3t: 19 V, 5t: 19.87 V (d) iC ϭ 0.2 ϫ 10Ϫ3eϪt/0.5 v R ϭ 20eϪt/0.5 (a) 5.5 ms Ϫ3 (b) 100(1 Ϫ eϪt/(5.5ϫ10 )) (c) 1t: 63.21 V, 3t: 95.02 V, 5t: 99.33 V (d) Ϫ3 iC ϭ 18.18 ϫ10Ϫ3eϪt/(5.5ϫ10 ) Ϫ3 vR ϭ 60eϪt/(5.5ϫ10 ) (a) 10 ms 1214 25 27 29 31 33 35 37 39 41 43 45 47 49 51 Ϫ3 50(1 Ϫ eϪt/(10ϫ10 )) Ϫ3 10 ϫ 10Ϫ3eϪt/(10ϫ10 ) vC Х 50 V, iC ϭ A Ϫ3 vC ϭ 50eϪt/(4ϫ10 ) Ϫ3 Ϫt/(4ϫ10Ϫ3) iC ϭ Ϫ25 ϫ 10 e Ϫ6 (a) 80(1 Ϫ eϪt/(1ϫ10 )) Ϫ3 Ϫt/(1ϫ10Ϫ6) (b) 0.8 ϫ 10 e Ϫ6 (c) vC ϭ 80eϪt/(4.9ϫ10 ) Ϫ3 Ϫt/(4.9ϫ10Ϫ6) iC ϭ 0.163 ϫ 10 e (a) 10 ms (b) kA (c) yes (a) vC ϭ 52 V Ϫ 40 V eϪt/123.8ms iC ϭ 2.198 mA eϪt/123.8ms 1.386 ms R ϭ 54.567 k⍀ (a) vC ϭ 60(1 Ϫ eϪt/0.2s), 0.5 s: 55.07 V, s: 59.596 V iC ϭ 60 ϫ 10Ϫ3 eϪt/0.2s 0.5 s: 4.93 mA, s: 0.404 mA vR1 ϭ 60 eϪt/0.2s 0.5 s: 4.93 V, s: 0.404 V (b) t ϭ 0.405 s, 1.387 s longer (a) 19.634 V (b) 2.31 s (c) 1.155 s (a) vC ϭ 3.275(1 Ϫ eϪt/52.68ms) iC ϭ 1.216 ϫ 10Ϫ3 eϪt/52.68ms (a) vC ϭ 27.2 Ϫ 25.2 eϪt/18.26ms iC ϭ 3.04 mA eϪt/18.26ms 0–4 ms: 0.3 mA, 4–6 ms: 0.9 mA, 6–7 ms: mA, 7–10 ms: mA, 10–13 ms: Ϫ3.2 mA, 13–15 ms: 1.8 mA 0–4 ms: V, 4–6 ms: Ϫ8 V, 6–16 ms: 20 V, 16–18 ms: V, 18–20 ms: Ϫ12 V, 20–25 ms: V V1 ϭ 10 V, Q1 ϭ 60 mC, V2 ϭ 6.67 V, Q2 ϭ 40 mC, V3 ϭ 3.33 V, Q3 ϭ 40 mC (a) 56.54 V (b) 42.405 V (c) 14.135 V (d) 43.46 V (e) 433.44 ms 8640 pJ (a) J (b) 0.1 C (c) 200 A (d) 10 kW (e) 10 s (b) (c) (d) (e) 23 APPENDIXES Chapter 11 ⌽: ϫ 104 maxwells, ϫ 104 lines B: gauss, 51.616 lines (a) 0.04 T 952.4 ϫ 103 At/Wb 2624.67 At/m 2.133 A 11 (a) N1 ϭ 60 t (b) 13.34 ϫ 10Ϫ4 Wb/Am 13 2.687 A 15 1.35 N 17 (a) 2.028 A (b) Х2 N 19 6.1 ϫ 10Ϫ3 Wb 21 (a) B ϭ 1.5(1 Ϫ eϪH/700At/m) (c) H ϭ Ϫ700 loge(1 Ϫ B/1.5 T) (e) Eq: 40.1 mA 33 Chapter 12 35 11 13 15 17 19 21 23 4.25 V 14 turns 15.65 mH (a) 2.5 V (b) 0.3 V (c) 200 V 0Ϫ3 ms: V 3Ϫ8 ms: 1.6 V 8Ϫ13 ms: Ϫ1.6 V 13Ϫ14 ms: V 14Ϫ15 ms: V 15Ϫ16 ms: –8 V 16Ϫ17 ms: V 0Ϫ5 ms: mA 10 ms: Ϫ8 mA 12 ms: mA 12Ϫ16 ms: mA 24 ms: mA (a) 2.27 ms (b) 5.45 ϫ 10Ϫ3(1 Ϫ eϪt/2.27ms) (c) vL ϭ 12eϪt/2.27ms vR ϭ 12(1 Ϫ eϪt/2.27ms) (d) iL: 1t ϭ 3.45 mA, 3t ϭ 5.179 mA, 5t ϭ 5.413 mA vL: 1t ϭ 4.415 V, 3t ϭ 0.598 V, 5t ϭ 0.081 V (a) iL ϭ 4.186 mA Ϫ 3.814 mA eϪt/13.95ms) vL ϭ Ϫ32.8 V eϪt/13.95ms (a) vL ϭ 20 V eϪt/1ms iL ϭ mA(1 Ϫ eϪt/1ms ) (b) iL ϭ mA eϪt/0.5ms vL ϭ Ϫ40 V eϪt/0.5ms (a) iL ϭ mA(1 Ϫ eϪt/0.5ms ) vL ϭ 12 V eϪt/0.5ms (b) iL ϭ 5.188 mA eϪt/83.3ns vL ϭ Ϫ62.256 V eϪt/83.3ns 25.68 ms (a) iL ϭ 3.638 ϫ 10Ϫ3 (1 Ϫ eϪt/6.676ms) vL ϭ 5.45 eϪt/6.676ms 25 27 29 31 37 39 (b) 2.825 mA, 1.2186 V (c) iL ϭ 2.825 ϫ 10Ϫ3eϪt/2.128ms vL ϭ Ϫ13.27 eϪt/2.128ms (a) 0.243 V (b) 29.47 V (c) 18.96 V (d) 2.025 ms (a) 20 V (b) 12 mA (c) 5.376 ms (d) 0.366 V iL ϭ Ϫ3.478 mA Ϫ 7.432 mA eϪt/173.9ms vL ϭ 51.28 V eϪt/173.9ms (a) H (b) H L: H, H R: 5.7 k⍀, 9.1 k⍀ V1 ϭ 16 V, V2 ϭ V, I1 ϭ mA V1 ϭ 10 V I1 ϭ A I2 ϭ 1.33 A WC ϭ 360 mJ WL ϭ 12 J Chapter 13 (a) 10 ms (b) (c) 100 Hz (d) amplitude ϭ V, Vp-p ϭ 6.67 V 10 ms, 100 Hz (a) 60 Hz (b) 100 Hz (c) 29.41 Hz (d) 40 kHz 0.25 s T ϭ 50 ms 11 (a) p/4 (b) p/3 (c) p (d) p (e) 0.989p (f) 1.228p 13 (a) 3.14 rad/s (b) 20.94 ϫ 103 rad/s (c) 1.57 ϫ 106 rad/s (d) 157.1 rad/s 15 (a) 120 Hz, 8.33 ms (b) 1.34 Hz, 746.27 ms (c) 954.93 Hz, 1.05 ms (d) 9.95 ϫ 10Ϫ3 Hz, 100.5 s 17 104.7 rad/s 23 0.4755 A 25 11.537°, 168.463° 29 (a) v leads i by 10° (b) i leads v by 70° (c) i leads v by 80° (d) i leads v by 150° 31 (a) v ϭ 25 sin(qt ϩ 30°) (b) i ϭ ϫ 10Ϫ3 sin(6.28 ϫ 103t Ϫ 60°) 33 ms 35 0.388 ms 37 (a) 0.4 ms APPENDIX H 39 41 43 45 47 49 (b) 2.5 kHz (c) Ϫ25 mV (a) 1.875 V (b) Ϫ4.778 mA (a) 40 ms (b) 25 kHz (c) 17.13 mV (a) sin 377t (b) 100 sin 377t (c) 84.87 ϫ 10Ϫ3 sin 377t (d) 33.95 ϫ 10Ϫ6 sin 377t 2.16 V 0V (a) T ϭ 40 ms, f ϭ 25 kHz, Vav ϭ 20 mV, Vrms ϭ 34.6 mV (b) T ϭ 100 ms, f ϭ 10 kHz, Vav ϭ Ϫ0.3 V, Vrms ϭ 367 mV Chapter 14 (a) 3770 cos 377t (b) 452.4 cos(754t ϩ 20°) (c) 4440.63 cos(157t Ϫ 20°) (d) 200 cos t (a) 210 sin 754t (b) 14.8 sin(400t Ϫ 120°) (c) 42 ϫ 10Ϫ3 sin(qt ϩ 88°) (d) 28 sin(qt ϩ 180°) (a) 1.592 H (b) 2.654 H (c) 0.8414 H (a) 100 sin(qt ϩ 90°) (b) sin(qt ϩ 150°) (c) 120 sin(qt Ϫ 120°) (d) 60 sin(qt ϩ 190°) 11 (a) sin(qt Ϫ 90°) (b) 0.6 sin(qt Ϫ 70°) (c) 0.8 sin(qt ϩ 10°) (d) 1.6 sin(377t ϩ 130°) 13 (a) ϱ ⍀ (b) 530.79 ⍀ (c) 265.39 ⍀ (d) 17.693 ⍀ (e) 1.327 ⍀ 15 (a) 9.31 Hz (b) 4.66 Hz (c) 18.62 Hz (d) 1.59 Hz 17 (a) ϫ 10Ϫ3 sin(200t ϩ 90°) (b) 33.96 ϫ 10Ϫ3 sin(377t ϩ 90°) (c) 44.94 ϫ 10Ϫ3 sin(374t ϩ 300°) (d) 56 ϫ 10Ϫ3 sin(qt ϩ 160°) 19 (a) 1334 sin(300t Ϫ 90°) (b) 37.17 sin(377t Ϫ 90°) (c) 127.2 sin 754t (d) 100 sin(1600t Ϫ 170°) 21 (a) C (b) L ϭ 254.78 mH (c) R ϭ ⍀ 25 318.47 mH 27 5.067 nF 29 (a) W (b) W (c) 122.5 W 31 192 W 33 40 sin(qt Ϫ 50°) 35 (a) sin(157t Ϫ 60°) (b) 318.47 mH (c) W 37 (a) i1 ϭ 2.828 sin(104t ϩ 150°), i2 ϭ 11.312 sin(104t ϩ 150°) (b) is ϭ 14.14 sin(104t ϩ 150°) 39 (a) Є36.87° (b) 2.83 Є45° (c) 16.38 Є77.66° (d) 806.23 Є82.87° (e) 1077.03 Є21.80° (f) 0.00658 Є81.25° (g) 11.78 ЄϪ49.82° (h) 8.94 Є153.43° (i) 61.85 ЄϪ104.04° (j) 101.53 ЄϪ39.81° (k) 4326.66 Є123.69° (l) 25.495 ϫ 10Ϫ3 ЄϪ78.69° 41 (a) 15.033 Є86.19° (b) 60.208 Є4.76° (c) 0.30 Є88.09° (d) 2002.5 ЄϪ87.14° (e) 86.182 Є93.73° (f) 38.694 ЄϪ94° 43 (a) 11.8 ϩ j (b) 151.9 ϩ j 49.9 (c) 4.72 ϫ 10Ϫ6 ϩ j 71 (d) 5.2 ϩ j 1.6 (e) 209.3 ϩ j 311 (f) Ϫ21.2 ϩ j 12 (g) 7.03 ϩ j 9.93 (h) 95.698 ϩ j 22.768 45 (a) ЄϪ50° (b) 0.2 ϫ 10Ϫ3 Є140° (c) 109 ЄϪ230° (d) 76.471 ЄϪ80° (e) Є0° (f) 0.71 ЄϪ16.49° (g) 4.21 ϫ 10Ϫ3 Є161.1° (h) 18.191 ЄϪ50.91° 47 (a) x ϭ 4, y ϭ (b) x ϭ (c) x ϭ 3, y ϭ or x ϭ 6, y ϭ (d) 30° 49 (a) 56.569 sin(377t ϩ 20°) (b) 169.68 sin 377t (c) 11.314 ϫ 10Ϫ3 sin(377t ϩ 120°) (d) 7.07 sin(377t ϩ 90°) (e) 1696.8 sin(377t Ϫ 120°) (f) 6000 sin(377t Ϫ 180°) 51 i1 ϭ 2.537 ϫ 10Ϫ5 sin(qt ϩ 96.79°) 53 iT ϭ 18 ϫ 10Ϫ3 sin 377t Chapter 15 (a) 6.8 ⍀ Є0° (b) 754 ⍀ Є90° 11 13 15 17 19 25 27 29 1215 (c) 15.7 ⍀ Є90° (d) 265.25 ⍀ ЄϪ90° (e) 318.47 ⍀ ЄϪ90° (f) 200 ⍀ Є0° (a) 88 ϫ 10Ϫ3 sin qt (b) 9.045 sin(377t ϩ 150°) (c) 2547.02 sin(157t Ϫ 50°) (a) 4.24 ⍀ ЄϪ45° (b) 3.04 k⍀ Є80.54° (c) 1617.56 ⍀ Є88.33° (a) 10 ⍀ Є36.87° (c) I ϭ10 A ЄϪ36.87°‚ VR ϭ 80 V ЄϪ36.87°, VL ϭ 60 V Є53.13° (f) 800 W (g) 0.8 lagging (a) 1660.27 ⍀ ЄϪ73.56° (b) 8.517 mA Є73.56° (c) VR ϭ 4.003 V Є73.56°, VL ϭ 13.562 V ЄϪ16.44° (d) 34.09 mW, 0.283 leading (a) 3.16 k⍀ Є18.43° (c) 3.18 mF, 6.37 H (d) I ϭ 1.3424 mA Є41.57°, VR ϭ 4.027 V Є41.57°, VL ϭ 2.6848 V Є131.57°, VC ϭ 1.3424 V ЄϪ48.43° (g) 5.406 mW (h) 0.9487 lagging (a) 40 mH (b) 220 ⍀ (a) V1 ϭ 37.97 V ЄϪ51.57°, V2 ϭ 113.92 V Є38.43° (b) V1 ϭ 55.80 V Є26.55°, V2 ϭ 12.56 V ЄϪ63.45° (a) I ϭ 39 mA Є126.65°, VR ϭ 1.17 V Є126.65°, VC ϭ 25.86 V Є36.65° (b) 0.058 leading (c) 45.63 mW (g) ZT ϭ 30 ⍀ Ϫ j 512.2 ⍀ ZT ϭ 3.2 ⍀ ϩ j 2.4 ⍀ (a) ZT ϭ ⍀ ϩ j ⍀, YT ϭ 41.1 mS Ϫ j 109.5 mS (b) ZT ϭ 60 ⍀ Ϫ j 70 ⍀, YT ϭ 7.1 mS ϩ j 8.3 mS (c) ZT ϭ 200 ⍀ Ϫ j 100 ⍀, YT ϭ mS ϩ j mS (a) YT ϭ 538.52 mS ЄϪ21.8° (c) E ϭ 3.71 V Є21.8°, IR ϭ 1.855 A Є21.8°, IL ϭ 0.742 A ЄϪ68.2° (f) 6.88 W (g) 0.928 lagging (h) e ϭ 5.25 sin(377t ϩ 21.8°), iR ϭ 2.62 sin(377t ϩ 21.8°), iL ϭ 1.049 sin(377t Ϫ 68.2°), is ϭ 2.828 sin 377t (a) YT ϭ 129.96 mS ЄϪ50.31° (c) Is ϭ 7.8 A ЄϪ50.31°, IR ϭ A Є0° IL ϭ A ЄϪ90° 1216 31 33 39 41 43 APPENDIXES (f) 300 W (g) 0.638 lagging (h) e ϭ 84.84 sin 377t, iR ϭ 7.07 sin 377t, iL ϭ 8.484 sin(377t Ϫ 90°), is ϭ 11.03 sin(377t Ϫ 50.31°) (a) YT ϭ 0.416 mS Є36.897° (c) L ϭ 10.61 H, C ϭ 1.326 mF (d) E ϭ 8.498 V ЄϪ56.897°, IR ϭ 2.833 mA ЄϪ56.897°, IL ϭ 2.125 mA ЄϪ146.897°, IC ϭ 4.249 mA Є33.103° (g) 24.078 mW (h) 0.8 leading (i) e ϭ 12.016 sin(377t Ϫ 56.897°), iR ϭ sin(377t Ϫ 56.897°), iL ϭ sin(377t Ϫ 146.897°), iC ϭ sin(377t ϩ 33.103°) (a) I1 ϭ 18.09 A Є65.241°, I2 ϭ 8.528 A ЄϪ24.759° (b) I1 ϭ 11.161 A Є0.255°, I2 ϭ 6.656 A Є153.690° (a) Rp ϭ 94.73 ⍀, Xp ϭ 52.1 ⍀ (C) (b) Rp ϭ k⍀, Xp ϭ k⍀ (C) (a) E ϭ 176.68 V Є36.44°‚ IR ϭ 0.803 A Є36.44°, IL ϭ 2.813 A ЄϪ53.56° (b) 0.804 lagging (c) 141.86 W (f) IC ϭ 1.11 A Є126.43° (g) ZT ϭ 142.15 ⍀ ϩ j 104.96 ⍀ R ϭ ⍀, XL ϭ 3.774 ⍀ Chapter 16 (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (a) (b) (c) (a) (b) (c) 1.2 ⍀ Є90° 10 A ЄϪ90° 10 A ЄϪ90° I2 ϭ A ЄϪ90°, I3 ϭ A ЄϪ90° 60 V Є0° ZT ϭ 3.87 ⍀ ЄϪ11.817°, YT ϭ 0.258 S Є11.817° 15.504 A Є41.817° 3.985 A Є82.826° 47.809 V ЄϪ7.174° 910.71 W 0.375 A Є25.346° 70.711 V ЄϪ45° 33.9 W 1.423 A Є18.259° 26.574 V Є4.763° 54.074 W YT ϭ 0.099 S ЄϪ9.709° V1 ϭ 20.4 V Є30°, V2 ϭ 10.887 V Є58.124° 1.933 A Є11.109° 11 33.201 A Є38.89° 13 139.71 mW Chapter 17 (a) Z ϭ 21.93 ⍀ ЄϪ46.85°, E ϭ 10.97 V Є13.15° (b) Z ϭ 5.15 ⍀ Є59.04°, E ϭ 10.3 V Є179.04° (a) 5.15 A ЄϪ24.5° (b) 0.442 A Є143.48° (a) 13.07 A ЄϪ33.71° (b) 48.33 A ЄϪ77.57° Ϫ3.165 ϫ 10Ϫ3 V Є137.29° 11 I1k⍀ ϭ 10 mA Є0° I2k⍀ ϭ 1.667 mA Є0° 13 IL ϭ 1.378 mA ЄϪ56.31° 15 (a) V1 ϭ 19.86 V Є43.8°, V2 ϭ 8.94 V Є106.9° (b) V1 ϭ 19.78 V Є132.48°, V2 ϭ 13.37 V Є98.78° 17 V1 ϭ 220 V Є0° V2 ϭ 96.664 V ЄϪ12.426° V3 ϭ 100 V Є90° 19 (left) V1 ϭ 14.62 V ЄϪ5.86° (top) V2 ϭ 35.03 V ЄϪ37.69° (right) V3 ϭ 32.4 V ЄϪ73.34° (middle) V4 ϭ 5.677 V Є23.53° 21 V1 ϭ 4.372 V ЄϪ128.66° V2 ϭ 2.253 V Є17.628° 23 V1 ϭ Ϫ10.667 V Є0° V2 ϭ Ϫ6 V Є0° 25 Ϫ2451.92Ei 27 (a) No (b) 1.76 mA ЄϪ71.54° (c) 7.03 V ЄϪ18.46° 29 Balanced 31 Rx ϭ R2R3/R1 Lx ϭ R2L3/R1 33 (a) 11.57 A ЄϪ67.13° (b) 36.9 A Є23.87° Chapter 18 (a) 6.095 A ЄϪ32.115° (b) 3.77 A ЄϪ93.8° i ϭ 0.5A ϩ 1.581 sin(qt Ϫ 26.565°) 6.261 mA ЄϪ63.43° Ϫ22.09 V Є6.34° 19.62 V Є53° 11 Vs ϭ 10 V Є0° 13 (a) ZTh ϭ 21.312 ⍀ Є32.196° ETh ϭ 2.131 V Є32.196° (b) ZTh ϭ 6.813 ⍀ ЄϪ54.228° ETh ϭ 57.954 V Є11.099° 15 (a) ZTh ϭ ⍀ Є90° ETh ϭ V ϩ 10 V Є0° (b) Iϭ0.5 Aϩ 1.11AЄϪ26.565° 17 (a) ZTh ϭ 4.472 k⍀ ЄϪ26.565° ETh ϭ 31.31 V ЄϪ26.565° (b) I ϭ 6.26 mA Є63.435° 19 ZTh ϭ 4.44 k⍀ ЄϪ0.031° ETh ϭ Ϫ444.45 ϫ 103 I Є0.255° 21 ZTh ϭ 5.099 k⍀ ЄϪ11.31° ETh ϭ Ϫ50 V Є0° 23 ZTh ϭ Ϫ39.215 ⍀ Є0° ETh ϭ 20 V Є53° 25 ZTh ϭ 607.42 ⍀ Є0° ETh ϭ 1.62 V Є0° 27 (a) ZN ϭ 21.312 ⍀ Є32.196°, IN ϭ 0.1 A Є0° (b) ZN ϭ 6.813 ⍀ ЄϪ54.228°, IN ϭ 8.506 A Є65.324° 29 (a) ZN ϭ 9.66 ⍀ Є14.93°, IN ϭ 2.15 A ЄϪ42.87° (b) ZN ϭ 4.37 ⍀ Є55.67°, IN ϭ 22.83 A ЄϪ34.65° 31 (a) ZN ϭ ⍀ Є0°, IN ϭ 1.333 A ϩ 2.667 A Є0° (b) 12 V ϩ 2.65 V ЄϪ83.66° 33 ZN ϭ 5.1 k⍀ ЄϪ11.31°, IN ϭ Ϫ1.961 ϫ 10Ϫ3 V Є11.31° 35 ZN ϭ 5.1 k⍀ ЄϪ11.31°, IN ϭ 9.81 mA Є11.31° 37 ZN ϭ 6.63 k⍀ Є0° IN ϭ 0.792 mA Є0° 39 (a) ZL ϭ 8.32 ⍀ Є3.18°, 1198.2 W (b) ZL ϭ 1.562 ⍀ ЄϪ14.47°, 1.614 W 41 40 k⍀, 25 W 43 (a) ⍀ (b) 20 W 45 (a) 1.414 k⍀ (b) 0.518 W 49 25.77 mA Є104.4° Chapter 19 (a) (b) (c) (d) (a) (c) (a) (b) (d) (a) (b) (c) (d) (f) 120 W QT ϭ VAR, ST ϭ 120 VA 0.5 A I1 ϭ 1⁄6 A, I2 ϭ 1⁄3 A 400 W, Ϫ400 VAR (C), 565.69 VA, 0.7071 leading 5.66 A Є135° 500 W, Ϫ200 VAR (C), 538.52 VA 0.928 leading 10.776 A Є21.875° R: 200 W, L,C: W R: VAR, C: 80 VAR, L: 100 VAR R: 200 VA, C: 80 VA, L: 100 VA 200 W, 20 VAR (L), 200.998 VA, 0.995 (lagging) 10.05 A ЄϪ5.73° APPENDIX H (a) R: 38.99 W, L: W, C: W (b) R: VAR, L: 126.74 VAR, C: 46.92 VAR (c) R: 38.99 VA, L: 126.74 VA, C: 46.92 VA (d) 38.99 W, 79.82 VAR (L), 88.83 VA, 0.439 (lagging) (f) 0.31 J (g) WL ϭ 0.32 J, WC ϭ 0.12 J 11 (a) Z ϭ 2.30 ⍀ ϩ j 1.73 ⍀ (b) 4000 W 13 (a) 900 W, VAR, 900 VA, (b) A Є0° (d) Z1: R ϭ ⍀, XC ϭ 20 ⍀ Z2: R ϭ 2.83 ⍀, X ϭ ⍀ Z3: R ϭ 5.66 ⍀, XL ϭ 4.717 ⍀ 15 (a) 1100 W, 2366.26 VAR, 2609.44 VA, 0.4215 (leading) (b) 521.89 V ЄϪ65.07° (c) Z1: R ϭ 1743.38 ⍀, XC ϭ 1307.53 ⍀ Z2: R ϭ 43.59 ⍀, XC ϭ 99.88 ⍀ 17 (a) 7.81 kVA (b) 0.640 (lagging) (c) 65.08 A (d) 1105 mF (e) 41.67 A 19 (a) 128.14 W (b) a–b: 42.69 W, b–c: 64.03 W, a–c: 106.72 W, a–d: 106.72 W, c–d: W, d–e: W, f–e: 21.34 W 21 (a) ⍀, 132.03 mH (b) 10 ⍀ (c) 15 ⍀, 262.39 mH Chapter 20 (a) qs ϭ 250 rad/s, fs ϭ 39.79 Hz (b) qs ϭ 3535.53 rad/s, fs ϭ 562.7 Hz (c) qs ϭ 21,880 rad/s, fs ϭ 3482.31 Hz (a) XL ϭ 40 ⍀ (b) I ϭ 10 mA (c) VR ϭ 20 mV, VL ϭ 400 mV, VC ϭ 400 mV (d) Qs ϭ 20 (high) (e) L ϭ 1.27 mH, C ϭ 0.796 mF (f) BW ϭ 250 Hz (g) f2 ϭ 5.125 kHz, f1 ϭ 4.875 kHz (a) BW ϭ 400 Hz (b) f2 ϭ 6200 Hz, f1 ϭ 5800 Hz (c) XL ϭ XC ϭ 45 ⍀ (d) PHPF ϭ 375 mW (a) Qs ϭ 10 (b) XL ϭ 20 ⍀ (c) L ϭ 1.59 mH, C ϭ 3.98 mF (d) f2 ϭ 2100 Hz, f1 ϭ 1900 Hz L ϭ 13.26 mH, C ϭ 27.07 nF f2 ϭ 8460 Hz, f1 ϭ 8340 Hz 11 (a) fs ϭ MHz (b) BW ϭ 160 kHz (c) R ϭ 720 ⍀, L ϭ 0.7162 mH, C ϭ 35.37 pF (d) Rl ϭ 56.25 ⍀ 13 (a) fp ϭ 159.155 kHz (b) VC ϭ V (c) IL ϭ IC ϭ 40 mA (d) Qp ϭ 20 15 (a) fs ϭ 11,253.95 Hz (b) Ql ϭ 1.77 (no) (c) fp ϭ 9,280.24 Hz, fm ϭ 10,794.41 Hz (d) XL ϭ 5.83 ⍀, XC ϭ 8.57 ⍀ (e) ZTp ϭ 12.5 ⍀ (f) VC ϭ 25 mV (g) Qp ϭ 1.46, BW ϭ 6.356 kHz (h) IC ϭ 2.92 mA, IL ϭ 3.54 mA 17 (a) XC ϭ 30 ⍀ (b) ZTP ϭ 225 ⍀ (c) IC ϭ 0.6 A Є90°, IL Х 0.6 A ЄϪ86.19° (d) L ϭ 0.239 mH, C ϭ 265.26 nF (e) Qp ϭ 7.5, BW ϭ 2.67 kHz 19 (a) fs ϭ 7.118 kHz, fp ϭ 6.647 kHz, fm ϭ kHz (b) XL ϭ 20.88 ⍀, XC ϭ 23.94 ⍀ (c) ZTP ϭ 55.56 ⍀ (d) Qp ϭ 2.32, BW ϭ 2.865 kHz (e) IL ϭ 99.28 mA, IC ϭ 92.73 mA (f) VC ϭ 2.22 V 21 (a) fp ϭ 3558.81 Hz (b) VC ϭ 138.2 V (c) P ϭ 691 mW (d) BW ϭ 575.86 Hz 23 (a) XL ϭ 98.54 ⍀ (b) Ql ϭ 8.21 (c) fp ϭ 8.05 kHz (d) VC ϭ 4.83 V (e) f2 ϭ 8.55 kHz, f1 ϭ 7.55 kHz 25 Rs ϭ 3.244 k⍀, C ϭ 31.66 nF 27 (a) fp ϭ 251.65 kHz (b) ZTp ϭ 4.444 k⍀ (c) Qp ϭ 14.05 (d) BW ϭ 17.91 kHz (e) 20 nF: fp ϭ 194.93 kHz, ZTp ϭ 49.94 ⍀, Qp ϭ 2.04, BW ϭ 95.55 kHz (f) nF: fp ϭ 251.65 kHz, ZTp ϭ 13.33 k⍀, Qp ϭ 21.08, BW ϭ 11.94 kHz 1217 (g) Network: L/C ϭ 100 ϫ 103 part (e): L/C ϭ ϫ 103 part (f): L/C ϭ 400 ϫ 103 (h) yes, L/C , BW Chapter 21 (a) 0.2 H (b) ep ϭ 1.6 V, es ϭ 5.12 V (c) ep ϭ 15 V, es ϭ 24 V (a) 158.02 mH (b) ep ϭ 24 V, es ϭ 1.8 V (c) ep ϭ 15 V, es ϭ 24 V (a) 3.125 V (b) 391.02 mWb 56.31 Hz 400 ⍀ 11 12,000t 13 (a) (b) 2.78 W 15 (a) 360.56 ⍀ Є86.82° (b) 332.82 mA ЄϪ86.82° (c) VRe ϭ 6.656 V ЄϪ86.82°, VXe ϭ 13.313 V Є3.18°, VXL ϭ 106.50 V Є3.18° 19 1.354 H 21 I1(R1 ϩ j XL1) ϩ I2( j Xm) ϭ E1 I1( j Xm) ϩ I2( j XL2 ϩ RL) ϭ 23 (a) 20 (b) 83.33 A (c) 4.167 A (d) a ϭ , Is ϭ 4.167 A, Ip ϭ 83.33 A 25 (a) 25 V Є0°, A Є0° (b) 80 ⍀ Є0° (c) 20 ⍀ Є0° 27 (a) E2 ϭ 40 V Є60°, I2 ϭ 3.33 A Є60°, E3 ϭ 30 V Є60°, I3 ϭ A Є60° (b) R1 ϭ 64.52 ⍀ 29 [Z1 ϩXL1]I1 ϪZM12I2 ϩZM13I3 ϭ E1, ZM12I1 Ϫ [Z2 ϩ Z3 ϩ XL2]I2 ϩ Z2I3 ϭ 0, ZM13I1 Ϫ Z2I2 ϩ [Z2 ϩ Z4 ϩ XL3]I3 ϭ Chapter 22 (a) (c) (a) (c) (a) (b) 120.1 V (b) 120.1 V 12.01 A (d) 12.01 A 120.1 V (b) 120.1 V 16.98 A (d) 16.98 A v2 ϭ Ϫ120°, v3 ϭ 120° Van ϭ 120 V Є0°, Vbn ϭ 120 V ЄϪ120°, Vcn ϭ 120 V Є120° (c) Ian ϭ A ЄϪ53.13°, Ibn ϭ A ЄϪ173.13°, Icn ϭ A Є66.87° (e) A (f) 207.85 V Vf ϭ 127 V, If ϭ 8.98 A, IL ϭ 8.98 A 1218 APPENDIXES (a) EAN ϭ 12.7 kV ЄϪ30°, EBN ϭ 12.7 kV ЄϪ150°, ECN ϭ 12.7 kV Є90° (b) Ian ϭ 11.285 A ЄϪ97.54°, Ibn ϭ 11.285 A ЄϪ217.54°, Icn ϭ 11.285 A Є22.46° (c) IL ϭ If (d) Van ϭ 12,154.28 V ЄϪ29.34°, Vbn ϭ12,154.28VЄϪ149.34°, Vcn ϭ 12,154.28 V Є90.66° 11 (a) 120.1 V (b) 208 V (c) 13.364 A (d) 23.15 A 13 (a) v2 ϭ Ϫ120°, v3 ϭ ϩ120° (b) Vab ϭ 208 V Є0°, Vbc ϭ 208 V ЄϪ120°, Vca ϭ 208 V Є120° (d) Iab ϭ 9.455 A Є0°, Ibc ϭ 9.455 A ЄϪ120°, Ica ϭ 9.455 A Є120° (e) 16.376 A (f) 120.1 V 15 (a) v2 ϭ Ϫ120°, v3 ϭ 120° (b) Vab ϭ 208 V Є0°, Vbc ϭ 208 V ЄϪ120°, Vca ϭ 208 V Є120° (d) Iab ϭ 86.67 A ЄϪ36.87°, Ibc ϭ 86.67 A ЄϪ156.87°, Ica ϭ 86.67 A Є83.13° (e) 150.11 A (f) 120.1 V 17 (a) Iab ϭ 15.325 A ЄϪ73.30°, Ibc ϭ 15.325 A ЄϪ193.30°, Ica ϭ 15.325 A Є46.7° (b) IAa ϭ 26.54 A ЄϪ103.31°, IBb ϭ 26.54 A Є136.68°, ICc ϭ 26.54 A Є16.69° (c) EAB ϭ 17,013.6 V ЄϪ0.59°, EBC ϭ 17,013.77 V ЄϪ120.59°, ECA ϭ 17,013.87 V Є119.41° 19 (a) 208 V (b) 120.09 V (c) 7.076 A (d) 7.076 A 21 Vf ϭ 69.28 V, If ϭ 2.89 A, IL ϭ 2.89 A 23 Vf ϭ 69.28 V, If ϭ 5.77 A, IL ϭ 5.77 A 25 (a) 440 V (b) 440 V (c) 29.33 A (d) 50.8 A 27 (a) v2 ϭ Ϫ120°, v3 ϭ ϩ120° (b) Vab ϭ 100 V Є0°, Vbc ϭ 100 V ЄϪ120°, Vca ϭ 100 V Є120° (d) Iab ϭ A Є0°, Ibc ϭ A ЄϪ120°, Ica ϭ A Є120° (e) 8.66 A 29 (a) v2 ϭ Ϫ120°, v3 ϭ 120° (b) Vab ϭ 100 V Є0°, Vbc ϭ 100 V ЄϪ120°, Vca ϭ 100 V Є120° 31 33 35 37 39 41 43 45 49 (d) Iab ϭ 7.072 A Є45°, Ibc ϭ 7.072 A ЄϪ75°, Ica ϭ 7.072 A Є165° (e) 12.25 A 2160 W, VAR, 2160 VA, Fp ϭ 7210.67 W, 7210 67 VAR (C), 10,197.42 VA, 0.707 leading 7.263 kW, 7.263 kVAR, 10.272 kVA, 0.707 lagging 287.93 W, 575.86 VAR (L), 643.83 VA, 0.4472 lagging 900 W, 1200 VAR (L), 1500 VA, 0.6 lagging Zf ϭ 12.98 ⍀ Ϫ j 17.31 ⍀ (a) 9237.6 V (b) 80 A (c) 1276.8 kW (d) 0.576 lagging (e) IAa ϭ 80 A ЄϪ54.83° (f) Van ϭ 7773.45 VЄϪ4.87° (g) Zf ϭ 62.52 ⍀ ϩ j 74.38 ⍀ (h) Fp (entire system) ϭ 0.576, Fp (load) ϭ 0.643 (both lagging) (i) 93.98% (b) PT ϭ 5899.64 W, Pmeter ϭ 1966.55 W (a) 120.09 V (b) Ian ϭ 8.492 A, Ibn ϭ 7.076 A, Icn ϭ 42.465 A (c) 4928.5 W, 4928.53 VAR (L), 6969.99 VA, 0.7071 lagging (d) Ian ϭ 8.492 A ЄϪ75° Ibn ϭ 7.076 A ЄϪ195° Icn ϭ 42.465 A Є45° (e) IN ϭ 34.712 A ЄϪ42.972° Chapter 23 (a) left: 1.54 kHz, right: 5.623 kHz (b) bottom: 0.2153 V, top: 0.5248 V (a) 1000 (b) 1012 (c) 1.585 (d) 1.096 (e) 1010 (f) 1513.56 (g) 10.023 (h) 1,258,925.41 1.681 Ϫ0.301 (a) 1.845 (b) 18.45 11 13.01 13 38.49 15 24.08 dBs 19 (a) 0.1fc: 0.995, 0.5fc: 0.894, fc: 0.707, 2fc: 0.447, 10fc: 0.0995 21 23 25 27 29 31 33 35 37 (b) 0.1fc: Ϫ5.71°, 0.5fc: Ϫ26.57°, fc: Ϫ45°, 2fc: Ϫ63.43°, 10fc: Ϫ84.29° C ϭ 0.265 mF, 250 Hz: Av ϭ 0.895, v ϭ Ϫ26.54°, 1000 Hz: Av ϭ 0.4475, v ϭ Ϫ63.41° (a) fc ϭ 3.617 kHz, fc: Av ϭ 0.707, v ϭ 45°, 2fc: Av ϭ 0.894, v ϭ 26.57° 0.5fc: Av ϭ 0.447, v ϭ 63.43° 10fc: Av ϭ 0.995, v ϭ 5.71° ⁄10 fc: Av ϭ 0.0995, v ϭ 84.29° R ϭ 795.77 ⍀ → 797 ⍀, fc: Av ϭ 0.707, v ϭ 45° kHz: Av ϭ 0.458, v ϭ 63.4° kHz: Av Х 0.9, v ϭ 26.53° (a) fc1 ϭ 795.77 Hz, fc2 ϭ 1989.44 Hz fc1: Vo ϭ 0.656Vi, fc2: Vo ϭ 0.656Vi fcenter ϭ 1392.60 Hz: Vo ϭ 0.711Vi 500 Hz: Vo ϭ 0.516Vi, kHz: Vo ϭ 0.437Vi (b) BW Х 2.9 kHz, fcenter ϭ 1.94 kHz (a) fs ϭ 100.658 kHz (b) Qsϭ 18.39, BW ϭ 5473.52 Hz (c) fs: Av ϭ 0.93 f1 ϭ 97,921.24 Hz, f2 ϭ 103,394.76 Hz, f ϭ 95 kHz: Av ϭ 0.392, f ϭ 105 kHz: Av ϭ 0.5 (d) f ϭ fs, Vo ϭ 0.93 V, f ϭ f1 ϭ f2, Vo ϭ 0.658 V (a) Qs ϭ 12.195 (b) BW ϭ 410 Hz, f2 ϭ5205 Hz, f1 ϭ 4795 Hz (c) fs: Vo ϭ 0.024Vi (d) fs: Vo still 0.024Vi (a) fp ϭ 726.44 kHz (stop-band) f ϭ 2.013 MHz (pass-band) (a–b) fc ϭ 6772.55 Hz (c) fc: Ϫ3 dB, fc: Ϫ6.7 dB, 2fc: Ϫ0.969 dB, ᎏᎏ f : Ϫ20.04 dB, 10 c 10fc: Ϫ0.043 dB (d) fc: 0.707, fc: 0.4472, 2fc: 0.894 (e) fc: 45°, fc: 63.43°, 2fc: 26.57° (a–b) fc ϭ 13.26 kHz (c) fc: Ϫ3 dB, fc: Ϫ0.97 dB, 2fc: Ϫ6.99 dB APPENDIX H fc: Ϫ0.043 dB, 10fc: Ϫ20.04 dB (d) fc: 0.707, fc: 0.894, 2fc: 0.447 (e) fc: Ϫ45°, fc: Ϫ26.57°, 2fc: Ϫ63.43° (a) f1 ϭ 663.15 Hz, fc ϭ 468.1 Hz < f < fc: ϩ6 dB/octave, f > fc: Ϫ3.03 dB (b) f1: 45°, fc: 54.78°, f1: 63.43°, 2f1: 84.29° (a) f1 ϭ 19,894.37 Hz fc ϭ 1,989.44 Hz < f < fc: dB, fc < f < f1: Ϫ6 dB/octave, f > f1: Ϫ20 dB (b) fc: Ϫ39.29°, 10 kHz: Ϫ52.06°, f1: Ϫ39.29° (a) f1 ϭ 964.58 Hz, fc ϭ 7,334.33 Hz < f < f1: Ϫ17.62 dB, f1 < f < fc: ϩ6 dB/octave, f > fc: dB (b) f1: 39.35°, 1.3 kHz: 43.38°, fc: 39.35° (a) f ϭ 180 Hz Х Ϫ3 dB, f ϭ 18 kHz: Ϫ3.105 dB (b) 100 Hz: 97°, 1.8 kHz: 0.12° Х 0°, 18 kHz: Ϫ61.8° Av ϭ Ϫ120/[(1 Ϫ j 50/f )(1 Ϫ j 200/f )(1 Ϫ jf/36 kHz)] fc ϭ kHz, < f < fc: dB, f > fc: Ϫ6 dB/octave f1 ϭ kHz, f2 ϭ kHz, f ϭ kHz < f < f1: dB, f < f < f2: ϩ6 dB/octave f < f < f3: ϩ12 dB/octave, f > f3 : 13.06 dB (a) woofer: 0.673, tweeter: 0.678 (b) woofer: 0.015, tweeter: 0.337 (c) mid-range: 0.998 Х 1 ᎏᎏ 10 39 41 43 45 47 49 51 53 Chapter 24 (a) positive-going (b) V (c) 0.2 ms (d) V (e) 6.5% (a) positive-going (b) 10 mV (c) 3.2 ms (d) 20 mV (e) 3.4% V2 of (V1 Ϫ V2)/V ϭ 0.1 is 13.571 mV (a) 120 ms (b) 8.333 kHz (c) maximum ϭ 440 mV, minimum ϭ 80 mV prf ϭ 125 kHz, duty cycle ϭ 62.5% 11 (a) ms (b) ms (c) 125 kHz (d) V (e) 3.464 mV 13 18.88 mV 15 117 mV 17 vo ϭ 4(1 ϩ eϪt/20ms) 19 iC ϭ Ϫ8 ϫ 10Ϫ3eϪt 21 iC ϭ ϫ 10Ϫ3eϪt/0.2ms (a) 5t ϭ T/2 (b) 5t ϭ (T/2) (c) 5t ϭ 10(T/2) 23 Ϫ T/2: vC ϭ 20 V, T/2 Ϫ T: vC ϭ 20eϪt/t, T Ϫ T: vC ϭ 20(1 Ϫ eϪt/t) T Ϫ T: vC ϭ 20eϪt/t 25 Zp ϭ 4.573 M⍀ ЄϪ59.5°, Zs ϭ 0.507 M⍀ ЄϪ59.5° Chapter 25 (I) a no b no c yes d no e yes (II) a yes b yes c yes d yes e no (III) a yes b yes c no d yes e yes (IV) a no b no c yes d yes e yes (a) 19.04 V (b) 4.53 A 71.872 W 11 (a) i ϭ ϩ 2.08 sin(400t Ϫ 33.69) ϩ 0.5 sin(800t Ϫ 53.13°) (b) 2.508 A (c) vR ϭ 24 ϩ 24.96 sin(400t ϩ 33.69°) ϩ sin(800t Ϫ 53.13°) (d) 30.092 A (e) vL ϭ 16.64 sin(400t ϩ 56.31°) ϩ sin(800t ϩ 36.87°) (f) 13.055 V (g) 75.481 W 13 (a) i ϭ 1.2 sin(400t ϩ 53.13°) (b) 0.848 A (c) vR ϭ 18 sin(400t ϩ 53.13°) (d) 12.73 V (e) vC ϭ 18 ϩ 23.98 sin(400t Ϫ 36.87°) (f) 24.73 V (g) 10.79 W 15 vo ϭ 2.257 ϫ 10Ϫ3 sin(377t ϩ 93.66°) ϩ 1.923 ϫ 10Ϫ3 sin(754t ϩ 1.64°) 1219 17 iT ϭ 30 ϩ 30.27 sin(20t ϩ 7.59°) ϩ 0.5 sin(40t Ϫ 30°) Chapter 26 Zi ϭ 986.84 ⍀ (a) Ii1 ϭ 10 mA (b) Zi2 ϭ 4.5 k⍀ (c) Ei3 ϭ 6.9 V Zo ϭ 44.59 k⍀ Zo ϭ 10 k⍀ (a) Av ϭ Ϫ392.98 (b) AvT ϭ Ϫ320.21 11 (a) AvNL ϭ Ϫ2398.8 (b) Ei ϭ 50 mV (c) Zi ϭ k⍀ 13 (a) AG ϭ 6.067 ϫ 104 (b) AGT ϭ 4.94 ϫ 104 15 (a) AvT ϭ 1500 (b) AiT ϭ 187.5 (c) Ai1 ϭ 15, Ai2 ϭ 12.5 (d) AiT ϭ 187.5 17 (a) z11 ϭ (Z1Z2 ϩ Z1Z3)/ (Z1 ϩ Z2 ϩ Z3), z12 ϭ Z1Z3/(Z1 ϩ Z2 ϩ Z3), z21 ϭ z12, z22 ϭ (Z1Z3 ϩ Z2Z3)/ (Z1 ϩ Z2 ϩ Z3) 19 (a) y11 ϭ (Y1Y2 ϩ Y1Y3)/ (Y1 ϩ Y2 ϩ Y3), y12 ϭ ϪY1Y2/(Y1 ϩ Y2 ϩ Y3), y21 ϭ y12, y22 ϭ (Y1Y2 ϩ Y2Y3)/ (Y1 ϩ Y2 ϩ Y3) 21 h11 ϭ Z1Z2/(Z1 ϩ Z2), h21 ϭ ϪZ1/(Z1 ϩ Z2), h12 ϭ Z1/(Z1 ϩ Z2), h22 ϭ (Z1 ϩ Z2 ϩ Z3)/ (Z1Z3 ϩ Z2Z3) 23 h11 ϭ (Y1 ϩ Y2 ϩ Y3)/ (Y1Y2 ϩ Y1Y3), h21 ϭ ϪY2/(Y2 ϩ Y3), h12 ϭ Y2/(Y2 ϩ Y3), h22 ϭ Y2Y3/(Y2 ϩ Y3) 25 (a) 47.62 (b) Ϫ99 27 Zi ϭ 9,219.5 ⍀ ЄϪ139.4°, Zo ϭ 29.07 k⍀ ЄϪ86.05° 29 h11 ϭ 2.5 k⍀, h12 ϭ 0.5, h21 ϭ Ϫ0.75, h22 ϭ 0.25 mS ... C2: and C2 ϭ X2 ϩ j Y2 X1 ϩ j Y1 X2 ϩ j Y2 ᎏᎏ X1X2 ϩ j Y1X2 ϩ j X1Y2 ϩ j 2Y1Y2 ᎏᎏᎏᎏ X1X2 ϩ j (X1Y1X2 ϩ X1Y2) ϩ Y1Y2(Ϫ1) and C1 ⋅ C2 ϭ (X1X2 Ϫ Y1Y2) ϩ j (Y1X2 ϩ X1Y2) (14. 32) In Example 14 .22 (b),... imaginary parts collected That is, if C1 ϭ X1 ϩ jY1 then and C2 ϭ X2 ϩ jY2 C1 (X1 ϩ jY1)(X2 Ϫ jY2) ᎏ ϭ ᎏᎏᎏ C2 (X2 ϩ jY2)(X2 Ϫ jY2) (X1X2 ϩ Y1Y2) ϩ j(X2Y1 Ϫ X1Y2) ϭ ᎏᎏᎏᎏ X 22 ϩ Y 22 and C1 X2Y1 Ϫ X1Y2... and C2 ϭ Z2 Єv2 we write C1 ⋅ C2 ϭ Z1Z2 / v1 ϩ v2 (14.33) EXAMPLE 14 .23 a Find C1 ⋅ C2 if C1 ϭ 20 ° and b Find C1 ⋅ C2 if C1 ϭ ЄϪ40° and C2 ϭ 10 Є30° C2 ϭ Єϩ 120 ° Solutions: a C1 ⋅ C2 ϭ (5 20 °)(10