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2 Electric source Voltage source Current source Kirchhoff s law Kirchhoff s voltage law Kirchhoff s current law Resistor Ohm s law Capacitor 6 23 Inductor 6 24 Switch Power 8 32 Maximum power Phasor Inpedance, Admittance Resonant circuit Series resonant circuit Parallel resonant circuit Power Maximum power transfer theorem ZImpedance parameters,z-parameters YAdmittance parameters,y-parameters HHybrid parameters,h-parameters FABCD-parameters 16 1

3 Bridge circuit Y- Y- transform Thevenin s theorem Norton s theorem Compensation theorem Millman s Theorem Self inductance Mutual inductance Transformer Coupling coefficient Equivalent circuit 22 2

4 1 11 Electric source 111 Voltage source 0, E + + et + et 11: 112 Current source 0, + jt + jt 12: 113 E = N k=1 E k E 1 E 2 E N 13: 3

5 E 1 E 2 E N E = E 1 = E 2 = = E N 14: J 1 J 2 J N J = J1 = J 2 = = J N 15: N J = k=1 J k J 1 J 2 J N 16: 12 Kirchhoff s law 121 Kirchhoff s voltage law node 0 N i k = 0 11 k=1 122 Kirchhoff s current law loop 0 N v k = 0 12 k=1 4

6 2 21 Resistor 211 Ohm s law v = Ri, R : resistance[ω, ohm] 21 i = Gv, G : conductance[s, siemens] 22 R = 1 G, G = 1 R 23 i i v R v R 21: 212 R 1 R 2 R N R = N k=1 R k 22: series connection G 1 G 2 G N G = 23: paralell connection N k=1 G k 5

7 22 Capacitor capacitancec[f, farad],,v q = Cv 24 i = C dv dt 25 vt = v0 + 1 C t 0 iξdξ 26 v i C 24: 23 Inductor inductancel[h, henry],,i ϕ = Li 27 v = L di dt 28 it = i0 + 1 L t 0 vξdξ 29 v i L 25: 6

8 24 Switch v i v open close i 26: 7

9 3 31 Power pt = vtit : instantaneous electric power[w, watt] 31 W = t2 t 1 ptdt : W [Wh] 32 pt = dw dt 33 P = 1 T T 0 ptdt : average power[w, watt] Maximum power P = R R + r 2 E2 35 R = r P max = E2 4r 36 r i E v R 31: matching P = G G + g 2 J 2 37 G = g P max = J 2 4g 38 8

10 i J g v G 32: matching 9

11 4 41 Phasor { } sin vt = V m ωt + ϕ = { } sin 2V e ωt + ϕ 41 cos cos V m : peak amplitude[v] V e : RMS value[v] ω = 2πf : angular frequency[rad/s] ϕ : initial phase angle[rad] 42 ωt + ϕ : phase[rad] T = 2π/ω = 1/f : period[s] f = 1/T : frequency[hz] Im V = V m e jϕ vt = V e jωt Re Im V = V e e jϕ vt = 43 2V e jωt Re 42 Inpedance, Admittance ω ω Z = R + jxz : inpedance, R : resistance, X : reactance 44 Y = G + jby : admittance, G : conductance, B : susceptance 45 Y = 1 Z, Z = 1 Y 46 Z = V I = V I ejϕ V ϕ I 47 10

12 43 Resonant circuit V I Series resonant circuit : v R + v L + v C = et V R + V L + V C = E : v R = Ri, v L = L di dt, i = C dv C V R = RI, V L = jωli, V C = I dt jωc 49 R C dv C dt + L d dt C dv C dt + vc = et : v C Ri + L di t iξdξ = et : i dt + 1 C Z = I = Z = Z e jθ R 2 + ωl 1 ωc θ = arctan ωl 1 ωc R E R + jωl + 1 jωc 2 = E Z ejϕ θ ω 0 = 1 LC : f 0 = 2π LC : 414 Q = V L E = C V ω=ω0 E ω=ω0 = 1 R L C : Q quality factor[ ]

13 i I R v R V R et E jωl v L V L 1 jωc v C V C 41: 432 Parallel resonant circuit : i G + i L + i C = jt J G + J L + J C = J : i G = Gv, v = L di L dt, i C = C dv dt J G = GV, J L = V jωl, J C = jωcv 416 G L di L dt + il + C d dt Gv + 1 L t 0 L di L dt = jt : il vξdξ + C dv dt = jt : v 417 Y = V = Y = Y e jθ G 2 + ωc 1 ωl θ = arctan ωc 1 ωl G J G + jωc + 1 jωl 2 = J Y ejϕ θ ω 0 = 1 LC : f 0 = 2π LC : 421 Q = C I J = L I ω=ω0 J ω=ω0 = 1 G C : Q quality factor[ ] 422 L 12

14 jt J G i G J G i L J L i C J C jωc 1 jωl v V 42: 44 Power pt = vtit : instantaneous electric power[w, watt] 423 P = 1 T T 0 ptdt = EI cos θ = Re[S] = S cos θ 424 E I S = EI = P + jq 425 Q = Im[S] = S sin θ 426 θ :, cos θ : power factor P :, average power,effective power,real power[w, watt] Q : reactive power[var, volt ampere reactive] S : complex power[va, volt ampere] S : apparent power[va, volt ampere] Maximum power transfer theorem P L = R L Z L + Z i 2 E Z L = Z i P max = E 2 4R i

15 Z i I E V Z L 43: P L = G L Y L + Y i 2 J Y L = Y i P max = J 2 4G i 431 I J Y i V Y L 44: 14

16 5 I 1 I 2 V 1 N I 1 I 2 51: two-port network ZImpedance parameters,z-parameters V = ZI 51 V 1 Z 11 Z 12 = Z 21 Z 22 I 1 I 2 52 Z 11 = V 1 I 1 I2 =0 Z 21 = I 1 I2 =0 Z 12 = V 1 I 2 I1 =0 Z 22 = I 2 I1 = Z 12 = Z 21 : reciprocity YAdmittance parameters,y-parameters I = YV 56 I 1 I 2 Y 11 Y 12 = Y 21 Y 22 V

17 Y 11 = I 1 V 1 V2=0 Y 21 = I 2 V 1 V2=0 Y 12 = I 1 V1=0 Y 22 = I 2 V1= Y 12 = Y 21 : reciprocity 510 Z = Y 1, Y = Z HHybrid parameters,h-parameters V 1 I 2 h 11 h 12 = h 21 h 22 I h 11 = V 1 h 21 = I 2 I 1 V2=0 I 1 V2=0 h 12 = V 1 I1=0 h 22 = I 2 I1= h 12 + h 21 = 1 : reciprocity FABCD-parameters V 1 I 1 = A C B D Î 2 Î2 = I A = V 1 Î2 =0 C = I 1 Î2 =0 B = V 1 D = I 1 Î 2 V2 =0 Î 2 V2 = det F = 1 : reciprocity 519 = F 1 V 1 Î 2 I 1 D = C B A V 1 I

18 52 Z Y H F = = = = Z 11 [Ω] 1 det Y Z 21 [Ω] 1 h 22 1 C Z 12 [Ω] Z 22 [Ω] Y 22 Y 12 Y 21 Y 11 det H h 12 h 21 1 A det F 1 D 1 det Z 1 h 11 Z 22 Z 12 Y 11 [S] 1 B Y 21 [S] D Z 21 Z 11 Y 12 [S] Y 22 [S] 1 h 12 det H h 21 1 det F A 1 Z 22 1 Y 11 det Z Z 12 Z Y 12 det Y Y 21 h 11 [Ω] h 21 [ ] 1 D B 1 h 12 [ ] h 22 [S] det F C 1 Z 21 Z 11 det Z 1 Z 22 1 Y 22 1 Y 21 det Y Y 11 1 h 21 A[ ] C[S] det H h 11 h 22 1 B[Ω] D[ ] Z 12 = Z 21 Y 12 = Y 21 h 12 + h 21 = 0 det F = 1 17

19 6 61 Bridge circuit Z 5 : Z 1 Z 2 = Z 3 Z 4 61 Z 1 Z 4 Z 5 Z 2 Z 3 E 61: 62 Y- Y- transform Z 1 = Y 1 /S Z 2 = Y 2 /S S = Y 1 Y 2 + Y 2 Y 3 + Y 3 Y 1 62 Z 3 = Y 3 /S Y 1 = Z 1 /S Y Y 2 = Z 2 /S Y S Y = Z 1 Z 2 + Z 2 Z 3 + Z 3 Z 1 63 Y 3 = Z 3 /S Y Z 3 Z 2 Z 1 Y 1 Y 2 Y 3 62: Y 18

20 63 63: 631 Thevenin s theorem Z s : 64 V s : 65 Z s I = V Z s +Z V s Z 64: 632 Norton s theorem Y s : 66 I s : 67 19

21 I s Y s V = I Y Y s +Y 65: 64 Compensation theorem I Z ZI 65 Millman s Theorem Y k E k N V N Y k E k V = k=1 68 N Y k k=1 Y 1 E 1 Y 2 E 2 Y N E N V 66: 20

22 7 71 Self inductance v = dϕ dt ϕ = Li v = L di V = jωli 71 dt v : Electromotive force,emf[v] ϕ : magnetic flux[wb, weber] 72 L : [H, henry] L n turn = n 2 L 1 turn Mutual inductance ψ 1 ψ 2 L 1 M 12 = M 21 L 2 i 1 i Transformer i 1 M i 2 v 1 L 1 L 2 v 2 71: 731 v 1 v 2 L 1 M di1 12 dt = di M 21 L 2 2 dt V 1 L 1 M 12 = jω M 21 L 2 I 1 I

23 732 Coupling coefficient k = M L1 L 2 : [ ] 76 K 1 = K 2, M 12 = M 21 n = n 2 L2 = :, turns ratio[ ] 77 n 1 L 1 close-coupled transformer :, v 2 = nv 1 k = 1 i 1 = ni 2 + i L1 78 i 2 = 1 n i 1 + i L2 i 1 1 : n i 2 i 1 1 : n i 2 i L1 v 1 L 1 v 2 v 1 L 2 i L2 v 2 72: v 2 = nv 1 ideal transformer :, i 2 = 1 n i 1 79 i 1 1 : n i 2 v 1 v 2 73: 733 Equivalent circuit i 1 M i 2 v 1 L 1 L 2 v 2 L 1 M L 2 M M 74: 22

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

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