Appendix: Answers to Exercises

pffiffiffiffiffi 1 1 1. Chapter 1 Exercise 2.4. FðÞ¼x x kf þ ; 1 0 x x x pffiffiffiffiffi 1 1 1 1 0 SI x kf ¼ V. 0 x x x Exercise 1.1. Physical law. 1 1 0 x2v x2v Exercise 1.2. (a), (b), (f), (g), (i), (j) are quantities. (c), Exercise 2.5. d ¼ a ab ;SI a ab ¼ m. (d), (e), (n) are units. (k) is neither. (l), (m) could be kq xavab kq xavab Exercise 2.6. i2 ¼10 mA. quantities if we could devise suitable definitions and = Exercise 2.7. w1 ¼ w0r1 r0. measurements. (h) could be a quantity if we define Exercise 2.8. 14.4 nm. color in terms of frequency. Exercise 2.9. 1 kO. Exercise 1.3. (a) WR (b) J (c) A (d) W (e) s (f) Nm ¼ J. Exercise 2.10. (a) 9:5 O (b) 127 O. 1 t0 0 0 Exercise 1.4. SI L 1 vðt Þdt ¼ SI½¼i A. Exercise 2.11. 49:9kO from the E96 series. Exercise 1.5. (a) No, (b) Yes. Exercise 2.12. (a) 4:3kO Æ 5% (b) 459 kO Æ 0:1%. Exercise 1.6. (a) 104 As1, (b) 104 Vs1, (c) 105 VA, Exercise 2.13. Various answers (d) 1010ms2. m Exercise 2.14. Because 298 K ¼ 25 C, 298 K is the 1 5 V 5nV m Exercise 1.7. (a) 5mVs ¼ ms ¼ ms , (b) 25kA s¼ reference temperature. 1 1 1 25Ams¼25MAns, (c) 100 mJ ms ¼ 100mJms ¼ Exercise 2.15. a10 ¼ 0:0042 K . 100 Js1. Exercise 2.16. 4. Exercise 1.8. (a) No space, milli (b), (e) first m is Exercise 2.17. 10.2. meter, second is milli (c), (d) meter Exercise 2.18. (a) (i) 0:42 O, (ii) 4:08 O; (b) (i) 0:33 O, : O Exercise 1.9. Av ffi 1=b; b determines the gain. (ii) 3 23 Exercise 1.10. x ffi 5:064. Exercise 2.19. Exercise 1.11. (a) 0:495, (b) 5, (c) 1, (d) 3:02, (e) 50. 10 Exercise 1.12. (a) i0, (b) 0. Exercise 1.13. (a) 2:015 Â 103, (b) 16:38 Â 106, 1 (c) 759 Â 103, (d) 462 Â 106, (e) 4:792 Â 103.

0.1 2. Chapter 2

0.01 = Exercise 2.1.I ¼ 2Nqp T where qp is the charge of a 1 proton. SI 2Nqp=T ¼ Cs ¼ A. 0.001 3 4 5 6 7 8 9 Exercise 2.2. h ¼ u2=ð2gÞ;SI½¼u2=ð2gÞ m. 10 10 10 10 10 10 10 f pffiffiffiffiffiffiffiffiffiffiffi hipffiffiffiffiffiffiffiffiffiffiffi (Hz) d d Exercise 2.3. q ¼ d f =k;SI d f =k ¼ C. 20 (mm) 100 (mm)

T.H. Glisson, Introduction to Circuit Analysis and Design, 745 DOI 10.1007/978-90-481-9443-8, # Springer ScienceþBusiness Media B.V. 2011 746 Appendix: Answers to Exercises

3. Chapter 3 Exercise 4.2. 2 2 R1ð2R0 þ 4R0R2 þ R2 Þ Req ¼ 2 2 . 2R0 þ 4R0R2 þ R2 þ 2R1ðÞR0 þ R2 Exercise 3.1. vbd ¼ 15 V; vcd ¼ 5V; vda ¼20 V; v1 ¼5V; vb ¼ 15V. Exercise 4.3. a ¼2; b ¼ 19=5; g ¼ 6=5; d ¼1. Exercise 3.2. (a) The series connection of elements Exercise 4.4. 8:77 O. 1 and 2 is in parallel with the series connection of R0R1R2 elements 3 and 4. (b) The parallel connection of Exercise 4.5. (a) vx ¼ ðÞi0 i1 R ðÞþR þ R R R elements 1, 2, and 3 is in series with element 4. 1 0 2 0 2 R1R2 : % Exercise 3.3. vab ¼ v1 v2 v3, vac ¼ v1 v3, vbd ¼ (b) vx ffi i1 (c) 1 06 . R1 þ R2 v þ v . 2 3 : : : Exercise 3.4. No. i ¼ i þi so the sources aren’t Exercise 4.6. (a) RT ¼ 0 91R, vT ¼ 0 91Ri0 0 48v0 2 1 3 : : = : independent. (b) RN ¼ 0 91R, iN ¼ i0 0 53ðÞv0 R RnRm ; R23I0 ; Exercise 3.5. Let Rnm ¼ I1 ¼ 4v1 þ 12Ri0 12 Rn þ Rm R1 þ R23 Exercise 4.7. vT ¼ v0 þ , RT ¼ R: R13I0 R12I0 7 7 I2 ¼ ; I3 ¼ . R2 þ R13 R12 þ R3 7v0 þ 4v1 12 Exercise 4.8. iN ¼ i0 þ ; RN ¼ R. R1 12R 7 Exercise 3.6. v1 ¼ v0. R1 þ R2 ðÞR2 R1 v2i1 R2v2 R1R2i1 Exercise 3.7. Exercise 4.9. vT ¼ , RT ¼ . R2i1 v2 R2i1 v2 R3v2 ðÞR2 þR3 v1 R2 R3v1 ðÞR1 þR3 v2 i1 ¼ ; i3 ¼ . R R þR R þR R R R R þR R þR R R2ðÞ3R1 þ 2R2 1 2 1 3 2 3 3 1 2 1 3 2 3 Exercise 4.10. RN ¼ . R1 þ R2 Exercise 3.8. vac ffi0:015v0; vbc ffi 0:463v0. va va ðÞvd v2 v1 Exercise 3.9.ðÞ a i1 þ þ ¼ 0; R1 R2 v v v v v v v 5. Chapter 5 ðÞd d þ d 2 þ d 2 1 a ¼ 0. R4 R3 R2 v R i v v þ v Exercise 3.10. ðbÞ b 1 1 þ c þ c 2 ¼ 0; Exercise 5.1. Answer given in problem statement. R1 þ R2 R3 R4 vc vc þ v2 vb R1i1 Exercise 5.2. ðcÞ þ þ ¼ 0. R3 R4 R1 þ R2 2 2 Exercise 3.11. Choose node c as reference node. pR ¼ð1=9RÞðÞV0 2RI0 cos ðÞo0 t 0 R R R i i v 1 2 3ðÞþ1 þ 2 1 2 2 vb ¼ . p2R ¼ð2=9RÞðV0 þ RI0Þ cos ðÞo0 t 0 R1R2 þ R1R3 þ R2R3

Exercise 3.12. No. v3 ¼ v1þv2 so the sources aren’t 2 2 pi ¼ð2=3ÞðV0I0 þ RI0 Þ cos ðÞo0 t 0 independent. R R v R v ðÞ2 þ 3 a 3 b 2 2 Exercise 3.13. i1 ¼ . pv ¼ð1=3RÞðV0 2RI0V0Þ cos ðÞo0 t R1R2 þ R1R3 þ R2R3 Exercise 3.14. p > 0 and p > 0 so the resistors consume energy. R v R R R R i R 2R B 0 þ ðÞ2 3 2 4 0 ; < i1 ¼ 2 pi 0, so the current source produces energy. If RARB R2 2 V0 2RI0V0 > 0, then Pv < 0 and the voltage source R2v0 þ RAðÞR3 5R4 i0 2 ; produces energy. If V0 2RI0V0 < 0, then pv > 0 and i2 ¼ 2 RARB R2 the voltage source consumes energy. RA ¼ R1 þ R2; RB ¼ R2 þ R3 þ R4: Exercise 5.3. P ¼ 25 mW; V ¼ 5V; w ¼ 50 mJ. Exercise 5.4. P ﬃ 547 mW: Exercise 3.15. Answer given in problem statement. Exercise 5.5. Answer given in problem statement. Exercise 5.6. 4. Chapter 4 P y p^ ¼ 0 ½¼1 þ cosðÞy P cos2 2 0 2 3 1 3v0 Exercise 4.1. ia ¼ þ va þ i0. Exercise 5.7. Answer given in problem statement. 2R0 R1 2R0 Appendix: Answers to Exercises 747

Exercise 5.8. 6. Chapter 6

RS V + I R S – S S Exercise 6.1. Rin ¼ R1kðÞR2 þ R3kRL ¼ R1ðÞR2R3 þ R2RL þ R3RL VS VS = 12.6 V, RS = 0.1 Ω IS == 126 A, RS = 0.1 Ω R1R3 þ R1RL þ R2R3 þ R2RL þ R3RL RS 2 2 Rout ¼ R3kðÞR2 þ R1kRS ¼ VS IS RS P = = ≅ 397 W; I = I = 126 A L max R L max S R ðÞR R þ R R þ R R 4 S 4 3 2 1 2 S 1 S , R1R3 þ R1RS þ R2R1 þ R2RS þ R3RS Exercise 5.9. There are two solutions: (b) and (c) same as (a). 179O 189V 1:06A Exercise 6.2. g ¼ m=R. R ffi ; V ffi ) I ffi . T 0:557O T 10:6V N 19:0A RoRL RiRS Exercise 5.10. 1=2. Exercise 6.3. vab ¼ gis. Ro þ RL Ri þ RS Exercise 5.11. 1 þ cosðÞy1 y2 . R1 Exercise 5.12. Let x ðÞ¼t x ðÞ¼t cosðÞ)o t Exercise 6.4. Rin ¼ . 1 2 1 þ b 2 1 ; x1x2 ¼ cos ðÞ¼o t 2 x1 x2 ¼ cosðÞo t cosðÞ¼o t 0. RoRL RiRS ; Exercise 5.13. 1. Exercise 6.5. vT ¼ gis 2 2 Ro þ RL Ri þ RS I0R I0R Exercise 5.14. ptðÞ¼ þ cosð4pftÞ; p^¼ 1:5kW; RoRL 2 2 RT ¼ . P ¼ 750W. Ro þ RL Exercise 5.15. Exercise 6.6. R2 R2 2; 2; 2 ðÞ7m 1 v ðÞ1 þ 4m v þ ðÞ11b 4m 1 Ri PR1 ¼ ðÞi0 i1 PR2 ¼ ðÞi0 i1 ðÞi0 i1 1 2 1 R1 R2 vout ¼ . m 19 1 2 2 ¼ I0 þI1 2I0I1 cosðÞy 2 Exercise 6.7. m1 ¼ 0. I2R I I R I2R I I R Exercise 6.8. m ¼ gR0. P ¼ 0 þ 0 1 cosðÞy ; P ¼ 1 þ 0 1 cosðÞy : i0 2 2 i1 2 2 Exercise 6.9. Exercise 5.16. ðaÞ 25 mA ðbÞ 5VðcÞ p10ffiffi V ðdÞ p10ffiffi mA. 2 2 Ri Rf ðÞþRL þ Ro RLRo Rin ¼ ; Exercise 5.17. Rf ðÞþRL þ Ro RLRo þ Ri½Ro þ RLðÞm þ 1 2 2 ÀÁ V0 þ ðÞI0R ; 1 2 ; PR ¼ Pi ¼ V0I0 þ I0R Ro Rf ðÞþRi þ RS RiRS 4R 4 R ¼ ÀÁ : 2 out V V I R Rf þ Ro ðÞþRi þ RS ðÞm þ 1 RiRS P ¼ 0 0 0 ; P þ P þ P ¼ 0: v 4R R i v 7% Exercise 5.18. (a) 4 s, (b) 10 . pffiffiffi Exercise 6.10. Exercise 5.19. Yes. Measure peak and divide by 2. Ro Exercise 5.20. 1:037 kO. + Exercise 5.21. The load voltages are equal and given v R + mv 1 i – 1 V – by V ¼ 0 : L 2 2 2 Ro1Ro2 V0 ðÞR þR0 V0 Exercise 5.22. P ; P (a) Ri ¼ Ri1, Ro ¼ Ro2, m ¼ b2g1, max ¼ V0 ¼ R R 4RR0 2R0 o1 þ i2 R R 2R V2 R 2R 2 V2R o1 þ 0 ; 0 þ 0 ; 0 (b) VCCS: Ri ¼ Ri1, Ro ¼ Ro2, g ¼ b2g1, PR0 ¼ PR ¼ . R þ R 4 2 o1 i2 R þR0 R0 R þR0 4ðÞR þ R0 R R R CCVS: R ¼ R , R ¼ R , r ¼ i1 o1 o2 b g Exercise 5.23. 0.5. ÀÁ i i1 o o2 R R 2 1 2 2 2 o1 þ i2 2RL R0I0 þ V0 Exercise 5.24. P ¼ ¼ 25 mW: Ri1Ro1 L 2 CCCS: R ¼ R , R ¼ R , b ¼ b g . ðÞR0 þ R1 þ 2RL i i1 o o2 2 1 Ro1 þ Ri2 748 Appendix: Answers to Exercises Exercise 6.11. 4RS RiRoRLg APdB ¼ 10 log þ 20 log : RL D RT = 7.5 Ω Exercise 6.31. 114 dB. + V = 15 V I = 2A R = 7.5 Ω – T N N 7. Chapter 7

Exercise 6.12. No. Magnitudes only. Exercise 6.13. 7.5 W. Exercise 7.1. (a) (ii), (b) (iv), (c) (i), (d) (iii). Exercise 6.14. Two. Exercise 7.2. (a) (iii), (b) (iv), (c) (i), (d) (ii). Exercise 6.15. RL=RS 19: R4 R2 Exercise 7.3. vL ¼ 1 þ vS. Exercise 6.16. RL=RS 0:053: R3 R1 Exercise 6.17. 8=9 ffi 0:889: Exercise 7.4. vL ¼ v1 þ v2. Exercise 6.18. 0:63 RL=RS 1:58: Exercise 7.5. Rin¼ Exercise 6.19. D ¼ ðÞRi þ RS ðÞRo þ RL ðÞm0þ1 RLRiR1þðÞR1þRi ½þR2ðÞþRLþRo RoRL RiR1Ro: mRiRL mRiRSRL R ðÞþR þR R ðÞþR þR R R Hv ¼ , Hr ¼ ; 2 o L L o 1 1 o D D Av Exercise 7.6. Rout ffi Ro: mRi mRiRS m0 þ Av Hg ¼ ; Hi ¼ : D D Exercise 7.7. Exact: Rin ¼ 974 GO; Rout ¼ 808 mO O; mO Exercise 6.20. D ¼ ðÞRS þ Ri ðÞRo þ RL , Approx: Rin ¼ 990 G Rout ¼ 808 . rR rR Exercise 7.8. Answer given in problem statement. A ¼ L , A ¼ S , ; v D i D Exercise 7.9. Inverting amp: Exact Av ¼ 99 Rin ¼ 10 kO; Rout ¼ 10:01 mO. r rRSRL Ag ¼ , Ar ¼ . Approx: A ffi 99:01; R ffi 10 kO; R ffi 9:9mO. D D v in out Non-inverting amp: Av ¼ 101; Rin ¼ 97:1GO; Rout ¼ Exercise 6.21. D ¼ ðÞRS þ Ri1 ðÞRo1 þ Ri2 ðÞRo2 þ RL 10:1mO rbR R rbR R Approx: Av ffi 101; Rin ffi 99 GO; Rout ffi 10:1mO: A ¼ o2 L ; A ¼ S o2 ; v D i D Exercise 7.10. Answer given in problem statement. rbR rbR R R Exercise 7.11. PA max ¼ 78:1 mW (virtually any op A ¼ o2 ; A ¼ S o2 L : g D r D amp will satisfy this requirement). Exercise 7.12. R0 ¼ R kðÞffiR þ R 998 O; = = L L 1 2 Exercise 6.22. Av ¼ AiRL RS, Ag ¼ Ai RS, Ar ¼ RLAi. V2 = ; = ; CC Exercise 6.23. (a) a ¼ 1 RL (b) a ¼ 1 RS PA max ¼ 0 ffi 157 mW. 4RL (c) a ¼ 1=RS; (d) a ¼ 1: Exercise 6.24. Results follow immediately from relations among the gains (see Example 7.12). Exercise 6.25. Usually no, because increasing the out- 8. Chapter 8 put resistance of the source increases the power wasted (decreases the power transferred to the load). Exercise 8.1. Exercise 6.26. No, they are consistent because A ¼ i v RSAv=RL. 10 Exercise 6.27. AP ¼ 6:31 Â 10 : Exercise 6.28. PA ¼ 0:5mW: Exercise 6.29. Follows immediately from Exercise 6.24. Exercise 6.30. D ¼ ðÞRo þ RL ðÞRS þ Ri RiRoRLg RSRiRog AvdB ¼ 20 log ; AidB ¼ 20 log ; 0 t D D 0 Appendix: Answers to Exercises 749

Exercise 8.2.

v (t) dv 5V 0 ≤ t < 4ms: C = (20 pF) = 25 nV dt 4ms 50 nV dv 4ms ≤ t < 5ms: C = 0 25 nV dt dv –10 V t (ms) 5ms ≤ t < 7ms: C = (20 pF) = –100 nV dt 2ms dv 5V 10ms ≤ t < 12ms: C = (20 pF) = 50 nV dt 2ms

–100 nV

Exercise 8.3. itðÞ¼2pfVCcosðÞ 2pft , maxjj¼itðÞ 2pfVC¼ 31:42A: Exercise 8.4.

v(t)(V)

6 V 1 ∫ t i t ′ dt ′ = 10 mA × t = × 6 × t ≤ t < μ 0 ( ) 2 10 ; 0 3 s 4 C 5 nF s etc...

12 t (µs) 3 4 6 8

–2

Exercise 8.5. lim vCðÞ¼t RI0: Exercise 8.10. vC; i2. t!1 Exercise 8.6. The model is unrealistic if i0 has a Exercise 8.11. 4C. dc component, because the charge on the capacitor Exercise 8.12. No, because the models would still allow would grow without bound (mathematically). Also, if capacitor voltages to change instantaneously. the voltage vC is the quantity of interest, the resistor is Exercise 8.13. irrelevant, and one must wonder why it is in the X5R )55Cto85C; Æ 15%; model. We cannot find the voltage vCðÞt1 at any Y5V )30Cto85C; þ 22%; 82%; time t1 unless we know (or are given) vtðÞ0 for some Z5U ) 10Cto85C; þ 22%; 56%: time t t andanexpressionforthecurrenti ðÞt for 0 1 0 Exercise 8.14. Any combination ______, where the t t t . 0 1 first character is X, or Y, the second is 5 or 7, and the O: Exercise 8.7. R ffi 101 third is A,B,C,D,E,F,P,R,S,T, or U. : : Exercise 8.8. C ffi 39 1pF Exercise 8.15. m ; m : Exercise 8.9. ffi 50 sat1 ffi 250 sat2 0.3 5

4.5

3.5 0.2

3 w (fJ ) v (V) 2.5

2 0.1 1.5

0.5

0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 0 0123456789101112 t (µs) t (ms) 750 Appendix: Answers to Exercises pffiffiffi Exercise 8.16. t¼RC; expðÞ¼t=t 1= 2 )tffi866ns. Exercise 9.4. vtðÞ¼2 p fI1 cosð2 p ftþ p=4Þ: Exercise 8.17. v ðÞ¼t 14:14 þ31:6cosðÞ 2pf t V; f ¼ C 0 0 Exercise 9.5. À ÀÁ 20kHz: t=t itðÞ¼0; t < 0; itðÞ¼ V0t=LÞ 1 e ; t 0: : : Exercise 8.18. (a) ffi 10 9 MHz, (b) ffi 11 V Exercise 9.6. Exercise 8.19. i (t) (µA) 0; t 0 1.5 vout ¼ V0 sinðÞ 2 p ft =ð2 p fRCÞ; t > 0 1.0

Exercise 8.20. (a) VCC 25 V, (b) ffi 5ns: Exercise 8.21. C ffi 563 mF. 12 0 t (µs) V 3684 Exercise 8.22. V ¼ IR; 99I ¼ 99 ¼ CVo0 ) C ¼ 99 R ffi 158 mF: –0.5 o0R Exercise 8.23. f <18:4Hz: Exercise 9.7. Answer implied by problem statement. R2V0 t Exercise 8.24. Exercise 9.8. vL ¼ exp . Other ans- ðÞR1 þ R2 t wers implied by problem statement. Td : Exercise 9.9. t ¼ D = lnðÞ 1 2 V Vmax t L iLðÞ¼t I0; t 0; iLðÞ¼t I0 exp ; t > 0; t ¼ t R LI t v ðÞ¼t 0; t 0; v ðÞ¼t 0 exp ; t > 0: L L t t 9. Chapter 9 Exercise 9.10. L. 1 Exercise 9.11. wtðÞ¼ LI2½1 þ cosðÞ 4pft : 4 Exercise 9.1. H. Exercise 9.12. 0 V; V =R: Exercise 9.2. 97:18 mm; 0:724 O: 0 Exercise 9.3. Exercise 9.13. 0 V; I0: Exercise 9.14. f ¼ 0:01R=ðÞ2pL : 6 Exercise 9.15. Vac rms ffi 1:2mV; g ffi 2:65 Â 10 : di 5mA Exercise 9.16. L ffi 31:8mH: L = (10mH)=12.5mV; 0 £ t <4ms, etc... dt 4ms Exercise 9.17. Answer implied by problem v(t) statement. Exercise 9.18. (a) Answer implied by problem statement. 25 mV (b) No. i1 could have dc component. 12.5 mV Exercise 9.19. (a) Dots at tops of coils. (b) Reverse 57 t (ms) one winding. pffiffiffiffiffiffiffiffiffiffiffiffiffi = : 0 4 10 12 Exercise 9.20. n ¼ RpL ffiffiffiffiffiffiffiffiffiffiffiffiffiRS Exercise 9.21. n1n2 ¼ RL=RS: nvðÞ t : Exercise 9.22. itðÞ¼ 2 2R þ n RL –50 mV Appendix: Answers to Exercises 751

10. Chapter 10 Exercise 11.7. Answer implied by problem statement. Exercise 11.8. d !1: Exercise 11.9. Answer implied by problem statement. Exercise 10.1. Answers given in problem statement. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi Exercise 11.10. Exercise 10.2. z real ) Re2ðÞþz Im2ðÞz ¼ z2 ¼jjz . Remaining exercises: answers given in problem a 6 statement.

vC (V)

11. Chapter 11 2

Exercise 11.1. itðÞ¼I0 þ ðÞI1 I0 utðÞþ t0 0 0 20 40 60 80 100 120 140 160 I I ut t : ðÞ2 1 ðÞ 1 b t (µs) V0 ; ; : 5 Exercise 11.2. IL ¼ VC1 ¼ R2I VC2 ¼ 0 R1 þ R2 Exercise 11.3. vLðÞ¼t vLðÞ1 ½1 expðÞt=t utðÞ; t ﬃ 148:52 ms; vLðÞﬃ1 334:75 V: vC (V) 0

400 –5 0 20 40 60 80 100 120 140 160 c t (µs) 300 10

vL (V) 200 0

vC (V)

–10 100

–20 0 0 20 40 60 80 100 120 140 160 –200 0 200 400 600 800 t (µs) t (µs) Exercise 11.11.

2 d vC L dvC 3 2 3 2 Exercise 11.4. LC þ þ vC ¼ 0; t > 0; d vo d vo dvo d vS d vS dvS dt2 R dt 4 þ14 þ8 þvo ¼4 þ4 þ4 þvS SI½¼LC s2; SI½¼L=R s: dt3 dt2 dt dt3 dt2 dt 0 1 Exercise 11.5. (a) R ffi 79:1kO; R < R ) 1 0 1 0 0 B C 2:823 = B 8 C @ A overdamped (b) R ¼ 10R0 ) vCðÞ¼t 2jjY exp ðÞt t v:¼ st :¼ PolyrootsðÞ v st ¼ 0:5 : @14A cosðÞo tþy utðÞ; 2jjY ffi503mV;tffi1:58ms; 0:177 0 4 6 1 o0 ffi 6:29 Â 10 s ; y ffi1:57; R ¼ R0 ) vCðÞ¼t Yt expðÞt=t utðÞ; Y ffi 6:33 Â 107 Vs1; t ffi 158ns Characteristic roots real and negative, circuit is overdamped. The circuit cannot be underdamped because R ¼ R0=10 ) vCðÞ¼t Y½expðÞt=t1 expðÞt=t2 utðÞ; Y ffi 5:03V; t ffi 3:16ms; t ffi 7:93ns: it consists of only resistors and capacitors. 1 2 t ; : ; ; Largest time constant is t ffi ffi5:65t¼ 56:5s: Exercise 11.6. n cycles ffi 1 1 6 4 8; Yes. 3 0:177 752 Appendix: Answers to Exercises

12. Chapter 12 Exercise 12.12. I~ffi 6:37ffp=2mA ) itðÞffi6:37 cosðÞo t p=2 mA. Exercise 12.1. 2p=3. Im Exercise 12.2. Lags. ~ ; m ; V Exercise 12.3. v2 ¼ V2 cosðÞ 2pftþ y V2 ¼ 500 V Re f ¼ 1 kHz; y ffi 1:26: Exercise 12.4. 50ffðÞp=4 mA: ~ Exercise 12.5. V ¼ V0ff ðÞf p=2 : ~ Exercise 12.6. I¼50ffp=3mAffiðÞ25þj43:3 mA; leads: I~ Exercise 12.7. itðÞ¼I0 cosðÞ 2pftþ y ; I0 ¼ 50 mA; f ¼ 4 MHz; y ¼ p=3: O: Exercise 12.8. Exercise 12.13. 178pffiffiffiffiffi k : : I~ Exercise 12.14. 10 ffi 3 16 Im Exercise 12.15. vðtÞffi146 cosðÞo t 0:350 mV: ~ ~ ~ Exercise 12.16. I ¼ I1 þ I2 ffi 14:1ff0:065 mA ) itðÞffi14:1 cosðÞo t 0:065 mA; f ¼ 100 kHz: 2 ðÞoL R1 R3 Exercise 12.17. Z¼R2 þ 2 þ 2 þ p /3 V~ R2 oL 1 oCR 2 1 þðÞ þðÞ3 Re "# –p /4 oLR2 oCR2 j 1 3 : 2 2 2 R1 þðÞoL 1þðÞoCR3 ~ 2 ~ V 1o LCþjoCRðÞ1 þR2 Exercise 12.18. I¼ ¼ V0; V~ Z ðÞR1 þjoL ðÞ1þjoCR2 1 itðÞ¼I0 cosðÞotþy ; I0 ¼1:14mA; y¼0:44: Exercise 12.19. f ¼ 5 kHz; R ffi 469 O; L ffi 3:85 mH; Exercise 12.9. Answer implied by problem statement. f ¼ 1 kHz; R ffi 540 O; C ffi 52:7 mF: Exercise 12.10. Answer implied by problem statement. Exercise 12.20. f ¼ 5 kHz; R ffi 500 O; L ffi 61:7 mH; ~ : = Exercise 12.11. I ffi 20 7ffp 2mA f ¼ 1 kHz; R ffi 540 O; C ffi 1:65 nF: : : ) itðÞffi20 7 sinðÞ 2pf0 t mA Exercise 12.21.

Im (Z) (Ω) 540 Re (Z )(Ω) 121

–3.02 Re (Z )(Ω) Im (Z)(Ω) 469 f = 5 kHz f = 1 kHz

1.20 Re (Y ) (mS) Im (Y) (mS) 0.010

–0.516 Re (Y ) (mS) Im (Y ) (mS) 1.85 f = 5 kHz f = 1 kHz Appendix: Answers to Exercises 753

Exercise 12.22. and RL ¼ RT if RT 6¼ 0. If RT ¼ 0 (not realistic), then ~ VT ffi ðÞ2:47ff0:251 V; ZT ffi ðÞ107ff1:34 O: RL should be the value that draws the maximum cur- Exercise 12.23. Resonant frequency is rent from the (current-limited) source. R þ R Exercise 13.9. (a) Answer implied by problem statement. o ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 if R2C > L. Otherwise circuit is 2 2 1 (b) No, (c) Yes. LR1C L not resonant. Exercise 13.10. (a) f ¼ 0ðdcÞ (b). Exercise 12.24. Two resonant frequencies: f1 ffi 1.25 67:1kHz; f2 ffi 13:3kHz:

1 13. Chapter 13

0.75 Exercise 13.1. Load: S ffi ðÞ112þj298 VA; P ffi 112W Source impedance: S ffi ðÞ90:1þj68:0 VA; P ffi 90:1W 0.5 Voltage source: S ffi ðÞ202 þ j365 VA; P ffi 202 W:

Exercise 13.2. Source: ffi 790 VAR, Capacitor: 0.25 ffi802 VAR, Inductor: ffi 12:2 VAR: Exercise 13.3. pf ¼ 1. 0 Exercise 13.4. See figure below. Each side of the 105 106 107 108 109 2 f power triangle equals Vrms times the corresponding (Hz) h f side of the admittance triangle. Pn ( f ) ( ) Exercise 13.5. (a) ffi 17:5 kVAR (b) pf ffi 0:87. Exercise 13.11. 18 dB.

2 Q = V 2 Y 14. Chapter 14 |S | =Vrms |Y | rms Im ( ) 2 = BYVrms

q Exercise 14.1. Answer implied by problem statement. Exercise 14.2. No. P = V 2 Re (Y ) = G V 2 rms Y rms Exercise 14.3. Both are zero. (a) S = P + jQ

|Y| 15. Chapter 15

Im (Y ) = BY : : q Exercise 15.1. vL ﬃ 8 40 cosðÞo1 t þ 0 82 þ12:23 cosðÞþo2 t 0:06 9:00 cosðÞo3 t 0:75 : Re (Y ) = GY Exercise 15.2. (b) Y = GY + jBY

Hv Hi Hz Hy 1 1 Exercise 13.6. Answer implied by problem statement Hv 1 ZS ZL ZS ZL 1 1 Exercise 13.7. Answer implied by problem statement Hi ZL ZS 1 ZL ZS Hz ZS ZL 1 ZLZS Exercise 13.8. The equivalent series reactances XL and 1 1 1 Hy ZL ZS ðÞZLZS 1 XT must be non-negative (capacitor doesn’t pass dc) 754 Appendix: Answers to Exercises

Exercise 15.3. Current transfer: Zin ! 0; Zout !1; Exercise 16.7. Transimpedance: Z ! 0; Z ! 0; Transadmittance: in out f ðÞkHz 4 8 12 16 20 24 28 32 Zin !1; Zout !1. Ak ðÞmA 0.00 45.02 31.83 15.01 0.00 9.00 10.61 6.43 ZS K Exercise 15.4. HiðÞ¼jo1 . yk 0.00 0.00 0.00 0.00 0.00 3.14 3.14 3.14 ZLðÞjo1 1 þ jo1=o0 Exercise 15.5. 10 W m1. Exercise 16.8. Answer implied in problem statement. Exercise 15.6. A ffi 33 dB; A ffi 50 dB. v i Exercise 16.9. a ¼ x ; b ¼ 0: Exercise 15.7. jjffiY 17:8mS: 0 dc 0 Exercise 16.10. Exercise 15.8. Because 1 Æ f =f0 is not a linear factor (no j). K0123

ak ðÞV 2.0 1.0 0.4 0.2 bk ðÞV 0.0 1.0 0.8 0.6 16. Chapter 16 V0 ; ; ; ; ; Exercise 16.11. Ak ¼ 2 yk ¼ kp k ¼ 1 2 ÁÁÁ V k X ¼ x ; X 0 ðÞ1 k; k ¼Æ1; Æ2; ÁÁÁ; a ¼ x ; Exercise 16.1. Average value: xdc ¼ A0ffy0 ¼ 500 mV 0 dc k 2k2 0 dc 1 V0 k Third harmonic: f ¼ 3f ¼ 6kHz; T ¼ ffi 167ms; b0 ¼ 0; ak ¼ ðÞ1 ; bk ¼ 0; k ¼ 1; 2; ÁÁÁ: 0 3 k2 500 3p p f0 A3 ¼ pffiffiffiffiffi mV; y3 ¼ ¼ : Exercise 16.12. 10 6 2 A5 500 Fifth harmonic: A5 rms ¼ pffiffiffi ¼ pffiffiffipffiffiffiffiffi mV: 2 2 26 k 0123 p 500 Ak ðÞmV 1.00 1.41 0.89 0.63 Fundamental: f0 ¼ 2 kHz; y1 ¼ ; A1 rms ¼ pffiffiffipffiffiffi 6 yk 0.00 0.52 1.05 1.57 mV 250 mV: 2 2 ¼ ak ðÞmV 1.00 1.23 0.45 0.00 : Exercise 16.2. 5f0 ¼ 1 5 kHz ) f0 ¼ 300 Hz ) T ¼ bk ðÞmV 0.00 0.71 0.78 0.63 1 ffi 3:33 ms: f kp 0 Exercise 16.13. X ¼ V sa : k ¼ 0; Æ1; Æ2; ÁÁÁ: Exercise 16.3. Only the third, whose frequencies are k 0 2 harmonics of a fundamental. T t < T t : ; m ; Exercise 16.14. t þ . Exercise 16.4. P1 ¼ 3 125 mW P2 ¼ 500 W P3 ¼ 2 2 2 2 m ; : m ; : m : 125 W P4 ¼ 43 25 W P5 ¼ 18 49 W x0t 2 kpt x0t 2 kt Exercise 16.15. Xk ¼ sa ¼ sinc ; P ¼ 3:812 mW;¼ 0:994: 2T 2T 2T 2T 2 kp 2 k Xk ¼ x0sa ¼ x0sinc : Exercise 16.5. X0 ¼10 mV, X1 ¼ 25 mV, X3 ¼ 2 2 : : : : ðÞ12 5ff0 785 mV, X5 ¼ ðÞ6 25ff1 57 mV, all Exercise 16.16. others ¼ 0. Exercise 16.6. Average value ¼ dc component ¼ k 123

X0 ¼ 1V Ak ðÞV 6.37 3.18 2.12 Period T ¼ 2 ms Fundamental frequency yk 1.57 1.57 1.57 1 f ¼ ¼ 500 Hz 0 T Exercise 16.17. Third harmonic: A3 ¼ 2jjX3 ffi 632 mV; R2ðÞ1 þ joR1C 2 kp 1 1 (a) Yk ¼ V0 sa ,(b). y ¼ ffi1:25; ffi 667 ms R1ðÞ1 þ joR2C 2 3 1 þ j3 3f 0ffiffiffi A5 p Fifth harmonic: pffiffiffi ¼ 2jjX5 ¼ 277 mV k 01 23 45 67 2 Ak ðÞmV 0.00 405.29 0.00 45.03 0.00 16.21 0.00 8.27 Fundamental: f ¼ 500 Hz; y ¼ X ffi0:785; pffiffiffi pffiffiffi 0 1 1 Bk ðÞV 0.00 1.46 0.00 0.83 0.00 0.02 0.00 0.01 A1= 2 ¼ 2jjX1 ¼ 1V. Appendix: Answers to Exercises 755

Exercise 16.18. 17. Chapter 17 5

4.167 Exercise 17.1. 120 3.333

Ak ()mV 2.5 100 1.667

0.833 Av (dB) 80

0 0 246 8 10 12 14 16 f 60 10Hz 0 40 1 10 100 1.103 – 0.1 f (Hz)

– 0.2 θk π Exercise 17.2. Answer implied by problem statement. – 0.3 pffiffiffi Exercise 17.3. (a) AvðÞ¼0 100; AvðÞ¼W 100 2; O; : O – 0.4 (b) ZinðÞffi0 10 k jjffiZinðÞj2pW 14 14 k ; ZoutðÞ0 ffi 0:038 O; jjZoutðÞj2pW ffi 39:78 O: – 0.5 0 246 8 10 12 14 16 f 10Hz Exercise 16.19.

Xk ()mV

0 –100 0 100 f ()Hz

0.6

0.4

0.2 θ k 0 π –0.2

–0.4

–0.6 –100 0 100 f ()Hz 756 Appendix: Answers to Exercises

Exercise 17.4.

Inverting Amplifier 50 20

40 0

30 –20 Zin Z ()dB o ()dB R R i 20 o –40

10 –60

0 –80 0.1 10 1´103 1´105 1´107 0.1 10 1´103 1´105 1´107 f ()Hz f ()Hz Non-Inverting Amplifier 80 20

60 0

–20 Z 40 Z in ()dB o ()dB R R i 20 o –40

0 –60

–20 –80 0.1 10 1´103 1´105 1´107 0.1 10 1´103 1´105 1´107 f ()Hz f ()Hz Voltage Follower 150 50

100 0 Z in ()dB Z R o ()dB i R 50 o –50

0 –100

–50 –150 0.1 10 1´103 1´105 1´107 0.1 10 1´103 1´105 1´107 f ()Hz f ()Hz

: : Exercise 17.5. C ﬃ 3 98 pF v(t – t ) (V) : 0 Exercise 17.6. Both ratios ﬃ 1 5

18. Chapter 18 0 75 125 t (μs) (a) 2 2 Exercise 18.1. FsðÞ¼2a=ðÞs a ; jsj

Exercise 18.7. Answer implied by problem statement. 25 75 t (μs) Exercise 18.8. (b) Appendix: Answers to Exercises 757 Exercise 18.21. I ¼ b I ¼ b b I ¼ b b I No. 1 expðÞst ; L 2 2 1 2 1 1 2 S vtðÞ¼V0½)utðÞutðÞ t VsðÞ¼V0 I s L 1 expðÞst : I R I I R LfgvtðÞ t0 ¼ V0 expðÞst0 S S 1 2 L s b I b I 1 1 2 2

1 1 Exercise 18.9. SI½¼b3 s ; SI½¼a2 s : 1 Exercise 18.10. SI½¼K As . Exercise 18.22. Answer implied by problem 3 2 c3s þc2s þc1sþc0 ; statement; Yes. Exercise 18.11. IsðÞ¼K 1þ 4 3 2 s þa3s þa2s þa1sþa0 Exercise 18.23. 3 2 c3 ¼ b3a3 ¼5s ; c2 ¼ 1s ; c1 ¼2s; c0 ¼ 0: Im(s) Exercise 18.12. Answer implied by problem statement. 103s-1 Exercise 18.13. Answer implied by problem statement. 1.83 Exercise 18.14. 1 ; RL ; Hi ¼ 2 Hv ¼ Hi s LC þ ðÞL=RS þ RLC s þ 1 þ RL=RS RS Hz ¼ RLHi. Exercise 18.15. First answer implied by problem − statement. Yes. 0.8 Re(s) VL 1 RS 3 -1 Exercise 18.16. Hv ¼ ¼ ; Hi ¼ Hv; −2 −1 1 10 s VS ðÞ1þst RL þjXL Hv Hy ¼ ; Hz ¼ RSHv: ðÞRL þjXL ðÞ1þst Exercise 18.17. t ¼ RC; p ¼1=t

gvðÞ¼t expðÞpt utðÞ¼expðÞt=t utðÞ; −1.83 SI½¼gvðÞt 1ðdimensionlessÞ = = ; hvðÞ¼t dðÞt ðÞ1 t expðÞt t utðÞ Exercise 18.24. Same as in the referenced example. 1 SI½¼hvðÞt SI½dðÞt SI½¼ 1=t s : 19. Chapter 19 Exercise 18.18. Exercise 19.1. Three, one of which must be minimum R2 ðÞ1 þ jf=f0 1 1 Hv ¼ ; f0 ¼ ; f1 ¼ passband input resistance. The other two must specify R0 ðÞ1 þ jf=f 2pR0C 2pR C 1 2 the poles, either directly or indirectly. RSR2 ðÞ1 þ jf=f0 RSR2 ðÞ1 þ jf=f0 H ¼ ; H ¼ ; Exercise 19.2. R ¼ 24 kO, C1 ffi 663 pF; C2 ffi 166 pF: i 0 = z 0 = RLR ðÞ1 þ jf f1 R ðÞ1 þ jf f1 Exercise 19.3. Answer implied by problem statement. R ðÞ1 þ jf=f H ¼ 2 0 : Exercise 19.4. Answer implied by problem statement. y 0 = RLR ðÞ1 þ jf f1 Exercise 19.5. Lowpass: Hv LPðÞ¼0 k þ 1; Highpass: Hv HPðÞ¼1 k þ 1: Exercise 18.19. Answer implied by problem statement. Exercise 19.6. Answer implied by problem statement. Exercise 18.20. Answer implied by problem statement. Exercise 19.7. VCC ffiÆ15 V: Index

A Asymptotic gain plots, 555–565 AC and DC, 36 error in, 558 AC resistance, 37–39 linear factors, 555 Active circuit, 141 low-and high-frequency asymptotes, 557 Active device, 141 quadratic factors, 562 Adder, inverting, 205 Asymptotic phase plots, 565–569 Admittance, 395 linear factors, 566 of circuit elements (table), 393 procedure, 567 comparing magnitudes of (convention), 396 quadratic factors, 567 expressed in dB, 396–397 Available power, 175, 177. See also Power, available as a function of frequency, 396 Available voltage and current, 175 generalized, or s-domain, 669 Average power. See also Power, average normalized, in dB, 396 conservation of, 126 SI unit, 395 defined, 126 triangle, defined, 414 dissipated by a resistor, 126, 132 Alternating current (AC), 36 American wire gauge (AWG), 35–36 Ammeter, symbol for, 72 B Amplifiers, 197 Balanced load, 333 capacitance coupled, 626 branch currents and line voltages in, 529 in cascade, 624 Balanced power, 332–333 bandwidth, 625 Balanced wye-delta transformations, 528 difference, 206 Balun, 333 effective load on, 220 Bandwidth input bias-current compensation, 218, 627, 633 amplifiers in cascade, 625 inverting, 204 filter, 569 input and output resistance, 213 half-power (3 dB), defined, 571 voltage transfer characteristic, 213 other definitions, 572 non-inverting, 204 reactive-feedback circuit, 617 input and output resistance, 212 reference gain for, 571 voltage transfer characteristic, 214 resistive-feedback amplifiers, 612 transconductance, 207, 210 signal, 569–570 transresistance, 206 slew rate limitations, 622 Amplitude distortion, 570 Basic approximation, 11 Angle, units of, 5 Biquad. See Filter, biquadratic Angular frequency of a sinusoidal signal, 383 Bode plot. See also Asymptotic gain plot Apparent power asymptotic approximation to, 555–556 defined, 484 defined, 555 delivered to a balanced load, 530 and pole-zero plot, 700 and line current, 484, 532 relation to pole-zero plot, 692 SI unit (VA), 484 Branch currents and line voltages, relations among, in balanced Approximations three-phase loads, 529 asymptotic, 13 Breakdown of a dielectric, 240 basic, 11 Buffer. See Voltage follower and checking results, 93 Bypass capacitor, 266–267

759 760 Index

C Charge Capacitance, 239 conservation of, 19 junction, 255 electron and proton, 19 parasitic, 255 properties, 19 physical basis, 237 SI unit, 19 required for power-factor correction, 532 Checking your work, 432–435, 675 residual, 255 Circuit analysis, s-domain, 669–670 sheet, 240 Circuit diagram, 49–81 SI unit, 239 annotating, 51–52 stray, 255 and schematics, 49 temperature coefficient, 258, 259 series and parallel connections, 53 variation with temperature, 258, 259 Coils wire-to-wire and wire-to-ground, 256 air-core, inductance of, 303 Capacitance coupling air-core, Wheeler’s formula for, 335 amplifiers, 626 magnetically coupled, 319 cascaded stages, 273 quality factor of, 439, 442 input-bias-current compensation, 276, 627 radio-frequency (rf), 442 input impedance, 626 short air core, inductance of, 303 non-inverting amplifier, 627 coupled, 322 vs. direct coupling, 626 Commutativity Capacitive load, 408 elements in series and parallel, 88 and leading power factor, 490 transfer functions, 691 Capacitors Compensated voltage divider, 255 admittance, 395, 680 Compensation, by pole cancellation, 698 applications, 262 Complex frequency, 653–654 bypass, 266–267 Complex-frequency domain, 653 digital systems, 270 Complex numbers, 345–353 in rectifier circuits, 268 angle of, 349 specifying, 266 arithmetic using, 345 charge on, 239 conjugate of, 346 circuit model for, 454 dimensioned, 345 construction, 239 magnitudes of, 347, 349 continuity of voltage across, 243, 245 magnitudes, ordering, 345–347 coupling, 271 polar form, 348–349 dissipation factor for, 455 radial and angular coordinates of, 349 electrolytic, 240 real and imaginary parts of, 345 energy storage and power dissipation in, 260 rectangular form, 348 equivalent series inductance, 454 relation of rectangular to polar form, 348, 349 equivalent series resistance, 262, 454 Complex plane, 348 impedance, 392–393, 681 Complex power, 479–520 loss angle, 455 angle of, 483 marking, 240 calculating, 482 in parallel, 253 conservation of, 486–487 parallel-plate, 239 dissipated (passive sign convention), 486 power dissipation in, 454 dissipated in a balanced three-phase load, 530 for power-factor correction, 491–496, 531–534 expressions for, 480 quality factor for, 454 and resonance, 487 self-resonance in, 454 SI unit, 480 in series, 252–253 superposition of, 496 standard (E-series) values, 240 Complex representation of a sinusoidal signal, 388 switched, 279 Conductance, 26–27 symbols for, 242 of a resistor, 55 terminal characteristics, 242 Conductivity, 27 thin-film, 240 Conductor, 50 Cascaded circuits, 174, 691 Conservation of power Characteristic equation apparent (not conserved), 483 first-order differential equation, 357 average power, 126 second-order differential equation, 360 complex, 486–487 Characteristic roots instantaneous power, 128 first-order differential equation, 357 peak power (not conserved), 120 passive RC and RL circuits, 368 reactive, 483 Index 761

Controlled source. See Dependent source Distortionless transmission, 570 Corner frequency, transfer function, 552 Dominant pole. See Pole, dominant Cosine, why used as standard form for a sinusoidal signal, 383 Dot convention Coulomb’s law, 19 coupled coils, 321, 326 Coupled coils, ideal transformer, 327 dot convention, 321, 326 transformer, 325 Coupling capacitor, 271, 626 Double-subscript notation, 52 Coupling coefficient, 320 defined, 320 E Crest factor, 157 Effective conductance, 412 defined, 152 Effective load, 220, 628 Critically damped. See Second-order circuit inverting and non-inverting amplifiers, 629 Current, 20 Effective resistance, defined, 408 conduction, 5 Effective rms value, 132 defined, 20 Electric field, 22 displacement, 5 energy exchange with, 23 how annotated on circuit diagrams, 51–52 SI unit, 24 Current divider, 58, 66–67 Electric potential, 23 Current gain, 180 Engineering notation, 14 in dB, 185 Equivalent circuits, 85 Current loading factor, 176 elements in series and parallel, 85 Current-to-voltage converter, 206 significance of, 101 input-bias-current compensation, 218 source transformations, 91 Current transfer, 176, 186 Thevenin and Norton source models, 93–100 Equivalent resistance D approximating, 90–91 dB (decibel) using known source, 89–90 acoustic power expressed in, 547 Equivalent series resistance of a capacitor, 262 current and voltage ratios, 547 E series, 29 current gain and voltage gain in, 547 standard tolerances, 30 defined, 184 table, 31 normalized transadmittance and transimpedance in, 549 ESL. See Capacitor, equivalent series inductance power ratios expressed in, 546 ESR. See Capacitor, equivalent series resistance DC and AC, 36 Euler’s identity, 349–351, 388, 389 DC gain Exponential order, 657 from differential equation, 358, 362 from transfer function, 552 F DC resistance, 37 Faraday, Michael, 302 (fn) DC steady state, defined, 354, 404 Faraday’s law of induction, 302, 319 Decibel. See dB Feedback, 207 Delta function, 658 advantages of, 207 Laplace transform of, 659 negative, linear stable operation, 208 Dependent source positive and negative, 207 defined, 166 and stability, 207, 208 intrinsic parameters of, 167 Feedback amplifiers types and terminal characteristics, 167 input bias-current compensation, 627 Derating. See Power dissipation inverting vs. non-inverting, 634 Design of linear op-amp circuits, guidelines for, 222, 637 precision of external resistors, 635 Dielectric, 239, 253 rules and guidelines for design, 633, 634, 636 breakdown, 240 small or fractional gain, 634 materials, 240 specifying the feedback resistor, 629 Differentiating circuits, 262 Filter Digital logic circuit, power dissipation in, 282 active, digital, and passive, 723 Dimensionless quantities, 5 analog, defined, 723 Dimensions, 5 bandwidth of, 569, 724 Diode, in half-wave rectifier, 269 Bessel, 736 Dirac delta. See Delta function biquadratic, 732–736 Dirac, Paul, 658 (fn) Butterworth, 736 Direct current (DC), 36 Chebyshev I, 736 Dirichlet, Peter Gustave, 589 (fn), 597 classification by gain, 569, 723–724 Displacement current, 238 desirable properties of, 725 762 Index

Filter (cont.) specifying, 636 distortionless, 725 Gibbs, Josiah Willard, 599–600 (fn) narrowband, 724 Gibbs’ phenomenon, 599–600 passbands, transition bands and stopbands, 559, 569, 724 Greek alphabet, 10 passive, defined, 723 Ground-loop current, 331 sharp-cutoff, 724 Ground symbol, as potential reference in types, described, 569 circuit diagrams, 52 VCVS Group delay, 724 design procedure, 737 Gyrator, 334, 444 gain and group delay, 729 input impedance, 730 H output impedance, 731 Half-power bandwidth. See Bandwidth transfer functions, 728 Half-wave rectifier, 268 Filter capacitor, 268–270 Heaviside, Oliver, 664 (fn) Finite-time integrator, 265 First-order circuit, 355 I RC and RL, 358 Ideal op amp. See Op amp Follower. See Voltage follower Ideal transformer, 326 Forced response, 356, 357, 404, 683, 694 as model for a real transformer, 327 Fourier coefficients Impedance, 391 amplitude-scaled signal, 600 angle of, 393 composite waveforms, 595 of circuit elements (table), 393, 669 DC shift, 599 comparing magnitudes (convention), 394 defined, 585 expressed in dB, 396–397 integral formula for, 588 as a function of frequency, 394 negated signal, 596 generalized, or s-domain, 669 operational properties, 597 normalized, in dB, 396–397 SI unit of, 585 SI unit, 391 superposition of, 596 Impedance matching table of, 592–595 for maximum power transfer, 504 Fourier series using L sections, 506–510 AC and DC components of, 583 using transformers, 504 amplitude-phase form, defined, 583 Impedance triangle, 414 circuit analysis using, 600–601 and power triangle, 491 convergence of, 589, 597–599 Impulse, as a model for a current or voltage pulse, 684. See also exponential form, 585 Delta function forms of, summary, 586 Impulse response, 684–688 fundamental component of, 583 Independent source, 56 fundamental frequency and period of, 583 Inductance, 302 harmonics, 583 of an air-core coil, 303 interval of expansion, 589 of a coil, 302 mean-squared amplitude of, 598 limits of lumped-constant model, 315 quadrature form of, 586 mutual (see Mutual inductance) relations among forms of, 588 parasitic, 314–316 Four-quadrant inverse tangent. See Inverse tangent self, 302 Frequency-domain, 402–403 of a wire, 314 analysis, 653 Inductive kick, 318–319 Frequency of a sinusoidal signal, 383 Inductive load, 408 Frequency response and lagging power factor, 490–491 defined, 572 Inductors relation to voltage transfer function, 573 air-core and iron-core, 304 measuring, 572–573 circuit-diagram symbols, 304 circuit model for, 451 G continuity of current in, 305 Gain, 545 energy storage and power dissipation in, 313 in dB, 546 in parallel, 312–313 intrinsic, 167 parasitic capacitance in, 451 overall, 182 quality factor of, 451 Gain-bandwidth product reducing ripple using, 316–318 of a feedback amplifier, 612, 616 self-resonance in, 451 of an op amp, 610 in series, 312 Index 763

terminal characteristics, 305–307 convergence of, 655–656 variable, 304 of current or voltage, 654 Initial phase of a sinusoidal signal, 383 dimension of, 654 Initial tolerance of a resistor, 137 inversion using partial fractions, 663 Input bias current compensation one-sided, 657 in capacitively coupled amplifiers, 276–279 operational properties, 659 direct-coupled and capacitance-coupled amplifiers, 627 region of convergence, 655 Input bias currents, 217 shorthand notation for, 657 Input impedance, inverting and non-inverting amplifiers, 627 SI unit, 686 Input offset voltage, 216–217 table of pairs, 659 Input resistance, 164 two-sided, 653 for maximum transfer, 174–178, 186 Leakage current, 254 op amp, 201 Leakage resistance, 253–255 op-amp circuits, 210, 213–215 parallel-plate capacitor, 254 two-port circuits, 172–174, 210 Left-sided function, 656 Insertion gain, 504 Lenz’s law, 302 Insertion loss, 503–504 Linear factor Instantaneous amplitude of a sinusoidal signal, 383 angle of, 566 Instantaneous phase of a sinusoidal signal, 383 angle of, piecewise-linear approximation, 566 Instantaneous power standard-form transfer function, 552 defined, 114 Line voltage and branch currents, relations among in balanced dissipated, 114 three-phase loads, 529 dissipated by a resistor, 116 Loading factor Institute of Electrical and Electronic Engineering, 1 current and voltage, 176 Integrating circuits, 263–265 input, interstage, output, 178 Internal impedance, 421 Look-back method, for finding Thevenin or Norton equivalent Internal resistance, 96, 122 resistance, 99 independent sources, 55, 98 Loop, 63 Inverse tangent Lossless circuit model, in sinusoidal steady state, 405 four-quadrant, 348 L sections, for impedance matching, 506 in software and pocket calculators, 348 Lumped-parameter model, 447 two-quadrant, 348 limits on applicability, 315, 447 Inverting amplifier bandwidth, 612 M capacitively coupled, 276 in cascade, 624 Magnetic field, 301 circuit diagram and parameters, 611 Magnetic flux, 301–303, 323 dc voltage gain, 612 Matching transformer. See Transformer, matching effective load, 629 Mathematical notation, 9–10 gain <10, 634 Maximum power transfer, 496 gain-bandwidth product, 612–615, 633 impedance matching for, 504 output impedance, 615 broad maximum, 145 power-conversion efficiency, 632 condition for, 496 reactive-feedback, 617 and design (discussion), 143 rules and guidelines for design, 636 matching networks, 144 s-domain transfer function, 690 and Thevenin equivalent source, 144 specifying op-amp power dissipation, 636 and Thevenin source models, 147 voltage-transfer function, 617 Maximum power transfer efficiency, 147, 496, 504 Isolation transformers. See Transformer, isolation Mean squared amplitude, 126 when additive, 147 J Mesh, 63 Joule’s law, 116, 480 Mesh analysis, described, 63 Mesh current, defined, 63 K Metric wire gauge (MWG), 36 Kirchhoff, Gustav, 56 (fn) Modern filter design, 736 Kirchhoff’s current law, 56 Multiple switching times, 250 Kirchhoff’s voltage law, 62 Mutual inductance, 319–320 dot convention, 321 L parasitic, 323–324 Laplace transform sign of, 321 condition for existence, 657 sign of, for transformers, 325 764 Index

N s-domain intrinsic voltage transfer function, 689 Negative resistance, using a switched capacitor, 280 s-domain model, 689 Node, 57 saturated, 200, 202 Node analysis, 58 slew rate, 621 Non-inverting amplifier specifying, 630, 631 bandwidth, 612 symmetric power supply for, 199 capacitance coupled and input-bias compensation, 627 terminals and voltage reference, 199 capacitively coupled, 277 typical parameter values, 201, 616–617 in cascade, 624 unity-gain frequency, 610 circuit diagram and parameters, 611 Open circuit, 53 dc voltage gain, 612 Open-loop dc voltage gain, 207 effective load, 629 Operational amplifier. See Op amp gain-bandwidth product, 612 Operator notation, 375 input and output impedance, 613, 615, 627 Order power conversion efficiency, 632 of a circuit, 353 reactive-feddback, 617 of a differential equation, 353 rules and guidelines for design, 636 Out of phase, 386 s-domain transfer function, 690 Output current limit, 216 specifying op-amp power dissipation, 636 specifying, 637 voltage-transfer function, 617 Output impedance Non-physical element, power dissipated by, 139 op-amp circuits (table), 613 Norton equivalent, 96, 421 The´venin and Norton source models, 422 ac circuits, 421–432 Output resistance, 96 circuit, 98, 100 for maximum transfer, 186 current and resistance, 96 op-amp circuits, 210 as current divider (ac circuits), 429 Thevenin and Norton source models, 165 impedance, 421 two-port circuits, 163 resistance, 96 Output swing s-domain, 670 design guideline, 623 op amp, 619 O and rms amplitude, 620 Ohm’s law, 25 and supply voltage, 636 generalized, 392, 669 Output transformer. See Transformer, output in terms of conductance, 27 Overdamped. See Second-order circuit One-sided Laplace transform. See Laplace transform Op amp, 198ff P ac model, 609 Parallel connection, 55 bandwidth and slew rate, compared, 622 equivalent admittance of, 399 circuit-diagram symbol, 199 sinusoidal current sources, 402 compensation capacitor, 609 Parasitic mutual inductance, 323–324 constraints on supply voltages, 199 Parasitic or residual properties, 447 dc model, 201 Partial-fraction expansion gain-bandwidth product, 610, 616 complex-conjugate poles, 665 ideal, defining properties, 203 distinct poles, 664 ideal, use in design, 207 repeated poles, 667 idle power dissipation, 632 Passband. See Filters input impedance, 610 Passive circuit, 141 integrated, 617 stability of, 695 internal construction (BJT, JFET, MOSFET), 614–617, 639 Passive device, 141 intrinsic dc voltage gain, 198, 610–611 Passive RC and RL circuits, characteristic roots of, 368 intrinsic parameters, 609 Passive sign convention, 114, 116 intrinsic voltage gain in dB, 611 Peak amplitude of a sinusoidal signal, 383 intrinsic voltage transfer function, 609 Peaking factor, 552 kinds of, 638 Peak power, 120–121 output current limit, 636 Period of a sinusoidal signal, 384 output impedance, 610 Permittivity, 239 output swing, 619 Phase delay, 384 power dissipation, 628–632 how measured, 573 power supply constraints, 199 Phase distortion, 570 rail-to-rail, 200, 620, 636 Phase lag and lead, 386 range of linear operation, 200 Phase opposition, 386 Index 765

Phase quadrature, 386 specifying, 629–631 Phase reference, 386 Power factor Phase shift, 545 for capacitive load, 491 Phasor, 388 defined, 489 current and voltage, 388 for inductive load, 491 diagram, 390 lagging and leading, 490 and Kirchhoff’s laws, 389–390, 397 of a motor, 495 limitations of, 394 Power-factor correction magnitude of, 481 bulk and local, 494 polar form, 388 capacitors for, how rated, 494, 532 rectangular form, 389 example, 492, 532 rms amplitude, 481 fundamental principle, 492 Piecewise-constant source, 250 practical considerations, 495 Polar arithmetic, 349–351 residential, 534 Polar form of a complex number, 348 three-phase load, 533 Pole Power formulas table of, 133 defined, 663, 692 Power gain dominant, 697 in dB, 185 finite and infinite, 692 defined, 183 on imaginary axis, 693 Power transfer, 500–503 LHP, 693 dependence on frequency, 503 RHP, 693 efficiency of, 144, 496–497, 503 Pole-zero cancellation, 697 and internal losses, 147 Pole-zero plot maximum, 186, 500–501 and Bode plot, 700 Proximity effect, 41, 448 defined, 692 relation to Bode plot, 694 Q Port, defined, 163 Quadratic factor Power, 113 angle of, 566, 567 apparent, 484 asymptotic approximation, 562–563, 568 available, 122, 142 maximum magnitude of, 564 average, 126 in standard-form transfer function, 552 complex, 479 when factorable, 552–553, 568 consumption, average residential, 524 Quality factor, 438 in dB, 184 capacitor and inductor, 438–439 delivered by a balanced three-phase source, 530 rf coil, 442 delivered vs. dissipated, 114, 116 factor (see Power factor) R peak, 120 RC circuit, voltages and currents in, 249 pulsating, in a single-phase system, 534 Reactance, defined, 408–411 reactive (see Reactive power) Reactive power, 483 SI unit, 114 dissipated by a capacitor, 495 superposition of, 147 dissipated by an inductor, 490 transfer, 121, 142 vs. real power, 484 transfer efficiency, 142 sign of, 483 triangle, 491 SI unit, 484 why important, 113 Rectifier, half-wave, 268 Power conversion efficiency, 634 Reference potential, defined, 25 Power dissipation, 219 Region of convergence. See Laplace transform average, residential, 524 Relative phase of a sinusoidal signal, 383 balanced three-phase load, 530, 534–535 Residential wiring, 524 derating, 135, 640 Residual properties, 447 in digital logic circuits, 283 Resistance and effective load, 628 ac and dc, 37 in feedback amplifiers, 219–222, 635 defined, 25 as fraction of maximum, 631 ratio of ac to dc, 38 in non-physical elements, 139, 497 sheet, defined, 27 op amps and op-amp circuits, 219–222, 628–632 SI unit, 25 in physical components, 116 variation with temperature, 32 in a resistor, 116, 126, 131 Resistivity, 26 slowly varying currents or voltages, 630 metals, variation with temperature, 33 766 Index

Resistivity (cont.) critically damped response, 367 SI unit, 26 damping factor, 370–371 superconductors, 34 differential equation, standard-form, 370 temperature coefficient, 32 oscillations of underdamped response, 369 variation with temperature, 32 overdamped, dominant time constant of, 369 Resistor, 27 overdamped response, 367 chip, 448 summary, 365–368 circuit model, 449 underdamped response, 366 color codes, 31 Self-contradictory circuit, 56 composition, 448 Self heating, 135–137 construction of, 27 Self-inductance. See Inductance electrical noise in, 450 Self-resonance impedance of, 450 inductor, 444, 451 labeling in circuit diagrams, 55 capacitor, 454 planar, 449 Series connection properties, 31 defined, 53 surface-mount, 448 equivalent impedance of, 399 temperature coefficients, 450 Sheet capacitance, 240 thin-film, 27 Sheet resistance, 27 Resonance, 435 Short circuit, 53 of common configurations, 439 SI and energy exchange, 436 function, 7 parallel, 435 prefixes, 8 and reactive power, 487 symbols and units, 6 self, of a coil, 442 system of units, 5 series, 435 Signal, 383 useful (loop) approximation, 439, 440 Sine-cardinal function, 591 working definitions, 437 Sinusoidal signal Resonant circuits, power relations for, 487 complex representation, defined, 388 Resonant frequency, formulas for, 438 frequency, 383 rf coil, 441 initial phase, 383 Right-hand rule, 301, 303 instantaneous amplitude, 383 Right-sided function, 656 instantaneous phase, 383 Ring frequency, 364 peak amplitude, 383 Ripple period, 384 and bypass capacitors, 266 phase delay, 384 in half-wave rectifier, 269 phase reference for, 385–388 in rectifier circuits, 270 phasor representation, 388 reducing with inductance, 316 relative phase, 385 Ripple factor rms amplitude of, 131 defined, 266, 316 standard form, 383 half-wave rectifier, 270 Sinusoidal steady state, 404 RMS amplitude analysis, procedure for, 405 defined, 130 and time origin, 385–388 of a Fourier series, 598 Skin depth, 37 measurement of, 133 Skin effect, 37, 448 notation, in electric power industries, 132 ac and dc resistance, 38 as a pseudo unit, 133 Slew rate, 621 of a sinusoid, 131 and bandwidth limitations, 622 time required for measurement, 133, 134 in design example, 636 design guideline, non-sinusoidal input, 623 sinusoidal input, 621–622 S and spike suppression, 624 Sallen-Key filter. See Filter, VCVS Snubber, 319 Sampling function, 591 Sound intensity, 184 Schematic, 49 Source transformations s domain, 653 dc sources, 91 s-domain circuit analysis, 670ff dependent sources, 173 common errors in, 677 isolated sources, 91 relation to frequency-domain analysis, 679, 689 s-domain, 670 Second-order circuit, 360 sinusoidal sources, 421 Index 767

Spectrum sources and loads, 83 dimension of, 603 THD. See Total harmonic distortion line, 604 The´venin equivalent one-sided, 603 experimental determination, 98, 427–431 two-sided, 607 frequency-domain, 421 s plane, 692–693 non-physical nature of, 94, 96, 422, 430 Stability resistive circuit, 94–100 BIBO, 694 s-domain, 679 op-amp circuits, 207–209 as voltage divider, 430 passive circuit, 695 The´venin’s theorem, 93–95, 421–431 s-plane condition, 694 Three-phase circuits, 521–535 Standard form line and branch currents, line and phase voltages, 525 first-order differential equation, 356 neutral line, 522 second-order differential equation, 370 why important, 521 sinusoidal signal, 383 Three-phase load transfer function, 552 balanced, 525 Steady-state Y and D connected, 525 extraneous dc in inductor, 404–405 Three-phase power generation and distribution (diagram), as forced response, 357, 362 523–524 sinusoidal (ac), 385, 404 Three-phase source step response, 355, 362 abc and acb sequence, 521–525 underdamped step response, 370 balanced, 522 Stefan–Boltzmann law, 118 wye-and delta-connected, 523 Step function, 684 Tight coupling, 322 Step response Time average, 123 first-order circuit, 356–357 alternate definitions, 123 forced and unforced components, 356, 362 of a constant, 124 formal definition (voltage, current, etc,), 684 existence, 124 Laplace transform of, 685–686 linearity, 124 second-order circuits (summary), 365–366 powers and products of sinusoids, 130 SI unit, 686 properties, 124 steady-state and transient components, 355, 357, 365 of a sinusoid, 124 Step-Up and Step-Down Transformers. See Transformers Time constant Stopband. See filters first-order differential equation, 357 Strain gage, 195 RC circuit, 246 Strength of a delta function. See Delta function RL circuit, 308 Superposition, 374 second-order response, 364, 365 of power, 147, 496 Time domain, 402–403, 653 principle of, 66, 415 Time invariance, 374 proof, 416 Time origin, 385–388 when useful, 420 Time translation Susceptance, 412 Fourier series, 595 Switched capacitor, 279–284 s-domain operational property, 661 power dissipation in, 281–284 sinusoid, 384 resistor, 279–281 Total harmonic distortion(problem), 608 Switched source, modeling using steps, 682 Transadmittance Switching frequency, in a digital logic circuit, 283 normalized, in dB, 549 transfer function, 540, 679 T Transconductance Tantalum nitride, 28 amplifier (op-amp based), 197 Temperature coefficient of resistance (TCR), 34, 137 amplifier (voltage-to-current converter), 227 Temperature coefficient of resistivity, 32 dependent source (intrinsic), 167 error in linear approximation for selected metals, 35 s-domain operational property, 661 in parts per million, 32 two-port, 180 selected metals, 32 Transfer characteristics, 179, 539 Terminal characteristics Transfer function, 539–540, 679 defined, 54, 83 angle of, 565 reference direction and polarity for current and voltage, 83 and available current and voltage, 543 resistor and independent sources, 54 commutativity, 691 s-domain, 669 dependence on source and load, 540, 544–545, 552, 679 significance of, 101 dimension and unit, 680 768 Index

Transfer function (cont.) U frequency-domain, 539ff Unbalanced three-phase load, 332 gain and phase shift, 545 Underdamped. See Second-order circuit linear and quadratic factors of, 552 Unforced response, 356, 405, 683, 694 polar form, 545 Unit impulse, 657–659, 684 quadratic factor, 552 Unit step function, 353 s-domain, 678–679 s-domain, relation to frequency domain, 689 V standard form, 550, 552 VA. Volt-ampere Transfer ratio, 179 See VAC, 133 Transformer, 324–326 VAR. Reactive power, SI unit adjustable, 324 See air-core, 324 Variable-frequency drive, 496 VCVS filter. Filter, VCVS center-tapped, 324 See VDC, 133 choosing reference directions for currents and reference polarities for voltages, 327 Voltage defined, 24 circuit-diagram symbols, 324 drop and rise, 25 coupling coefficient, 320, 322 how annotated on a circuit diagram, 51 dot convention, 325 ideal, 326–327 Voltage divider, 59 buffered, 635 iron-core, 324 compensated, 255 matching, 328 resistive, 66 output, 328 source and load transformation using, 329 Voltage follower, 205 bandwidth, 612 step-up and step-down, 330–331 circuit diagram and parameters, 611 turns ratio, 327 dc voltage gain, 612 Transient response gain-bandwidth product, 612 first-order circuits, 355ff as impedance buffer, 616 and pole-zero plot, 697 input and output impedance, 613–615 second-order circuits, 360ff and unforced response, 356, 697 s-domain transfer function, 690 Voltage gain, 180 Transimpedance in dB, 185 normalized, in dB, 549 Voltage loading factor, 176 transfer function, 540, 679 Transresistance Voltage source, constant (dc), symbol, 56 Voltage-to-current converter, 207, 210 amplifier (op-amp based), 197 Voltage transfer, 176 amplifier (current-to-voltage converter), 206 maximizing, 186 dependent source (intrinsic), 167 two-port, 180 Voltage transfer function, reactive-feedback circuit, 617 Volt-ampere. See Apparent power, SI unit s-domain operational property, 661 Volt-ampere-reactive. Reactive power, SI unit Triangle inequality, 347 See Tuned circuit, gyrator-based, 445 Voltmeter, symbol for, 72 Turns ratio, 327 Two-port W bilateral and unilateral, 172 Weight, and mass, 8 in cascade, 174 Wheeler’s formula, 335 controlling and controlled current or voltage, 172 Wiring diagram, 49 four types, 173 Work, 21, 113 input and output resistance, 164, 172 intrinsic parameters, 167 models and circuits, 163 Z Two-quadrant inverse tangent. See Inverse tangent Zeros, 663, 692 Two-sided function, 656 finite and infinite, 692