Transcription

1

2

3 e jωt e jωt e jωt

4 OFF ( 1) ( 2) ( 1) RC ( 2) RC ( 3) RL ( 4) RL R, L, C

5 ( Q ) Q (Quality Factor) RLC Q R RLC Q R Q LC RLC () () () (Dot convention) k () ()

6 () Y, Z, K, H, G Y Z K ( 1) H ( 2) G

7 RL RC RLC A 227 A A A A.4 j A A A.7 e jθ A.8 e x A.9 e jθ

8

9 9 1 () R V I V = RI. (1.1) 1.3 R 1, R 2, R R S R S = R 1 + R 2 + R 3. (1.3) 1 I = V 1 R 1, I = V 2 R 2, I = V 3 R 3. (1.4) 1.2 I R 1 V 1 = R 1 I Ω, R V R 2 V 2 = R 2 I S, G R 3 V 3 = R 3 I G = 1 R. (1.2) I I V = RI V I = V/R V R S V = R S I

10 10 1 V I I I I 1 V = R 1 I 1 E V R L J V R L I 2 R 1 V = R 2 I 2 DC voltage source Load Resistance DC current source Load Resistance I 3 R 2 V = R 3 I V R 3 R P V = R P I (1.6) R P = I R P = R 2R 3 + R 1 R 3 + R 1 R 2 R 1 + R 2 + R 3. (1.9) V = R P I R P 1.3 V = V 1 + V 2 + V 3. (1.5) R S V = R S I (1.3) ( ) 1.4 R 1, R 2, R R P 1 R P = 1 R R R 3. (1.6) V = R 1 I 1, V = R 2 I 2, V = R 3 I 3. (1.7) I = I 1 + I 2 + I 3. (1.8) ( ) 1.5 G 1, G 2, G 3, G P (1) (2) I 1 = G 1 V, I 2 = G 2 V, I 3 = G 3 V. (1.10) I = I 1 + I 2 + I 3 I = (G 1 +G 2 +G 3 )V. (1.11) G P = G 1 +G 2 +G 3. (1.12)

11 () I V R L E R i V i I V R L R L V ( ) R L I = V /R L (R L = 0 () ) R L I ( ) R L V = R L J (R L = () ) R i I V i = R i I E V R L R i DC voltage source Load Resistance 1.5 R L R i E R i R L E = V i + V (1.13) R i R L V i = R i I, V = R L I. (1.14) E = (R i + R L )I, I = I V = R L I V = R L R i + R L E = E R i + R L. (1.15) R i R L E (1.16) V R L R L R i R i /R L 1 V E (1.17) V R L

12 V E 0.1 Ω R i I V R L (a) Ω 1.5 V Voltage (V) Current (A) E = 1.5 V R i = 0.1 Ω V I R i = 0.1 Ω E = 1.5 V 1.6 V V = I R L R i + R L E. (1.18) I = V R L (1.19) Voltage (V) R L (Ω) (b) R L (Ω) Current (A) R L V I V 2.8 Ω V 1.49 V R i V I i = V /R i J I Ω 1.5 V V I R (a) (b) R L R i R L R i

13 I J I i R i V R L DC current source Load Resistance 1.8 J R i I ( R L ) J = I i + I (1.20) R i R L I i = V R i, I = V R L. (1.21) ( 1 J = + 1 ) V, V = R i R L J R i R L (1.22) V I = V /R L I = 1 R L J R i R L = R L R i J (1.23) I R L R L R i R L /R i 1 I J (1.24) I R L

14 (a) c d i i > 0 c d d c 1.9 (b) i i > (a) c d v v > 0 c d v d c v 1.10 (b) v + c + d v v > 0 c d v v < 0 d c v c d v i i c d c d (a) (b) 1.9 v v c d c d (a) (b) 1.10 (=) (=) ( ) v +

15 15 i source i i i (a) Normal combination of the directions for current and voltage in the case of voltage source i e drop v i i i (b) Normal combination of the directions for current and voltage in the case of passive circuit elements e example v example 1.11 (a) (b) + v e

16 16 1 Imag. z [1] R V (t) 1 I(t) 45 o V (t) = RI(t). O Real 1.12 z = cos(45 ) + jsin(45 ) [2] R 1 R 2 R S R P R S 1/R P R 1 R 2 R S = R 1 + R 2, 1 R P = 1 R R 2. 1/3 cm *2 [3] j *1 z 1 = j2.0, z 2 = j1.0 z 1 z 2 z 1 /z 2 z 1 z 1 x+jy 2 z1 z 1 z 1 () z 1 z 2 = j3.0, z 1 z 2 = j0.50, z 1 = 2.2, ( 2 ) 0.33 cm cm cm [4] z = cos(45 ) + jsin(45 ) 1.12 z 1 = 1.0 j2.0. [5] f (t) = cos(ωt) + jsin(ωt) f (t) ω( 0) t *1 j + i i +i + j *2 1/4

17 17 d f (t) = ωsin(ωt) + jωcos(ωt) dt { } = jω cos(ωt) + jsin(ωt) = jωf (t), f (t) dt = 1 ω sin(ωt) j ω cos(ωt) = 1 { } cos(ωt) + jsin(ωt) jω = 1 jω f (t). [6] f (t) = e at g(t) = e bt a b f (t)g(t), f (t) g(t), d dt f (t), f (t)g(t) = e at e bt = e (a+b)t, f (t) g(t) = eat e bt = e(a b)t, d dt f (t) = d dt eat = ae at, f (t) dt = e at dt = 1 a eat. f (t) dt. Amplitude (arb. units) 1 f(t) 0 g(t) ωt (degree) 1.13 [7] f (t) = sin(ωt) g(t) = sin(ωt + 90 ) f (t) g(t) f (t) g(t) g(t) f (t)

18

19 19 2 ω *1 V m I m R V m = R I m ( ) L V m = ωl I m 90 C V m = I m ωc 90 *1 ω i(t) v(t) = Ri(t) 2.1 [1] (resistor) (resistor) 2.1 [1] v(t) i(t) v(t) = Ri(t). (2.1) R (resistance) Ω, Ohm (conductance) ( S (Siemens)) (inductor) (inductor) 2.1 [2]

20 第 2 章 交流回路素子とその性質 抵抗 コイル コンデンサ 20 i(t) i(t) di(t) v(t) = L dt v(t) = 1 i(t) dt C 図 2.2 コイル (インダクタ) [2] 図 2.3 コンデンサ (キャパシタ) [3] はインダクタである コイルの両端に印加された電圧 費電力が負である とは 電力がその回路素子から供給 v(t) と抵抗に流れる電流 i(t) の間には 以下の関係があ されることを意味する 無から電力が供給されることは り ファラデーの電磁誘導の法則から導き出されるもの 無いので この状況は 回路素子に投入した電力がその である 回路素子で反射されてしまうことを意味する 本節で di(t) v(t) = L. dt (2.2) は 抵抗 コイル コンデンサの各素子に対してこのこ とを検証する ここで L をインダクタンス (inductance) という 単位 は H (ヘンリー, Henry) である なお 交流回路では この反射を抑制し 効率良く電 力を負荷に供給するための方策をとることになる この 電磁誘導による電圧は 電磁気学的には 誘導起電 力 即ち 起電力 である 従って 電磁気学的に見れ 方策を理解するためには 本講義で学ぶ交流回路理論の 学習が必要なのである ば コイルは電源のような能動素子として扱うべき素子 である しかし 電気回路では コイルを抵抗と同じ範 疇の受動素子として扱い そこに発生する誘導起電力を 受動素子の両端の電圧 即ち 電圧降下 として扱う 抵抗 抵抗 R に流れる電流を i(t) とするとき 抵抗での消 費電力 p R (t) は次式で与えられる このように扱う理由については 第 8 章の相互インダク p R (t) = R i(t)2. タンスの豆知識を参照されたし 従って 抵抗での消費電力は常に正であることがわかる コンデンサ (capacitor) コンデンサ (capacitor) は 図 2.3 の写真に示すよう な回路素子であり [3] 電流の積分に比例した電圧が端 子間に現れる素子である 日本語ではコンデンサである が 英語ではキャパシタである コンデンサの両端の電 圧 v(t) とそこに流れる電流 i(t) の間には 以下の関係が コイルとコンデンサ コイル L に流れる電流を i(t) とするとき コイルで の消費電力 p L (t) は 天下り的であるが 次式で与えら れる p L (t) = ある いわゆるコンデンサの充電の式である v(t) = 1 C i(t) dt. (2.4) (2.3) ここで C をキャパシタンス (capacitance) という 単 位は F (ファラッド, Farad) である 2.2 回路素子における電力とエネルギー ) ( d 1 Li(t)2. dt 2 (2.5) また コンデンサ C の電圧を v(t) とするとき コンデ ンサでの消費電力 p C (t) は 天下り的であるが 次式で 与えられる ) ( d 1 2 Cv(t). p C (t) = dt 2 (2.6) これらの式より 具体的な i(t) や v(t) の波形がわって いなくても コイルとコンデンサについては 抵抗と異 抵抗の場合には 電力は消費されるだけ 即ち電力は なり i(t) や v(t) の時間的変化の仕方によっては消費電 常に正であるが コイルとコンデンサの場合には 電力 力が負になり得る ということが読み取れると思う 即 が消費されるだけとは限らず 負になることもある 消 ち 抵抗では交流の場合も電力は消費だけであるが コ

21 v(t) = V m sin ωt i(t) = v(t) R v(t) = V m sin ωt i(t) = 1 L v(t) dt Current (A) Voltage (V) Voltage Freq. = 60 Hz R = 1 kohms Phase (degree) Power Current Power (W) Current (A) Voltage (V) Voltage Freq. = 60 Hz L = 100 mh Current Phase (degree) 270 Power Power (W) 2.5 V m = 100 V f = ω/(2π) = 60 Hz R = 1 kω 2.7 V m = 100 V f = ω/(2π) = 60 Hz L = 100 mh v(t) v(t) = V m sinωt (2.7) i(t) i(t) = v(t) R (2.8) = V m sinωt R (2.9) = I m sinωt (2.10) ω I m = V m R θ = v(t) v(t) = V m sinωt (2.11) i(t) i(t) = 1 v(t) dt (2.12) L = V m cosωt (2.13) ωl ( = I m sin ωt π ) (2.14) 2 cos sin ω

22 22 2 v(t) = V m sin ωt i(t) = C dv(t) dt v(t) 2.8 v(t) = V m sinωt (2.15) i(t) Voltage Current 40 i(t) = C d v(t) (2.16) dt Current (A) Voltage (V) 50 0 Power 20 0 Power (W) = ωcv m cosωt (2.17) ( = I m sin ωt + π ) (2.18) 2 sin cos Freq. = 60 Hz C = 1000 uf Phase (degree) ω I m = ωc V m 2.9 V m = 100 V f = ω/(2π) = 60 Hz C = 1000 µf I m = V m ωl θ = π 2 = θ = + π 2 = * 2 *2 () () = 90 (t = 0 )

23 i(t) = I m sin ωt v(t) = R i(t) i(t) = I m sin ωt v(t) = L di(t) dt Current (A) Voltage (V) Freq. = 60 Hz R = 10 Ohms 90 Current 180 Phase (degree) Power Voltage Power (W) Current (A) Voltage (V) Freq. = 60 Hz L = 10 mh 90 Current 180 Power Phase (degree) Voltage Power (W) 2.11 I m = 1 A f = ω/(2π) = 60 Hz R = 10 Ω 2.13 I m = 1 A f = ω/(2π) = 60 Hz L = 10 mh 2.4 * R 2.10 i(t) = I m sinωt (2.19) v(t) v(t) = R i(t) (2.20) = R I m sinωt (2.21) = V m sinωt (2.22) *3 ω V m = R I m θ = 0 = L 2.12 i(t) = I m sinωt (2.23) v(t) v(t) = L d i(t) (2.24) dt = ωl I m cosωt (2.25) ( = V m sin ωt + π ) (2.26) 2 cos sin

24 i(t) = I m sin ωt v(t) = i(t) dt C ω V m = I m ωc θ = π 2 = Current (A) Voltage (V) Current Freq. = 60 Hz C = 1000 uf Voltage Phase (degree) 270 Power I m = 1 A f = ω/(2π) = 60 Hz C = µf ω V m = ωl I m θ = + π 2 = C 2.14 Power (W) i(t) = I m sinωt (2.27) v(t) v(t) = 1 i(t) dt (2.28) C = 1 ωc I m cosωt (2.29) ( = V m sin ωt π ) (2.30) i(t) v(t) v(t) = Ri(t) + L d dt i(t) + 1 i(t) dt. (2.31) C i(t) = I m sinωt v(t) v(t) = v f + v s. (2.32) cos sin v f = A 1 e s 1t + A 2 e s 2t (2.33) v s = V m sin(ωt + θ) (2.34)

25 v(t) v R (t) v L (t) v C (t) R L C i(t) 2.16 v f t 0 v s t v(t) = V m sin(ωt + θ) V m θ i(t) (2.31) ( v(t) = I m [R sinωt + ωl 1 ) ] cosωt. (2.35) ωc V m θ ( V m = I m R 2 + ωl 1 ) 2, (2.36) ωc ( ωl 1 ) θ = tan 1 ωc (2.37) R

26 26 2 sin θ cos θ θ definition of sin and cos L [4] The term inductance was coined by Oliver Heaviside in February It is customary to use the symbol L for inductance, in honor of the physicist Heinrich Lenz. In the SI system the measurement unit for inductance is the henry, H, named in honor of the scientist who discovered inductance, Joseph Henry. (current) I i [5] The conventional symbol for current is I, which originates from the French phrase intensite de courant, or in English current intensity. This phrase is frequently used when discussing the value of an electric current, but modern practice often shortens this to simply current. The I symbol was used by Andre-Marie Ampere, after whom the unit of electric current is named, in formulating the eponymous Ampere s force law which he discovered in The notation travelled from France to Britain, where it became standard, although at least one journal did not change from using C to I until sin cos sin, cos x y sin cos d dθ sinθ = cosθ = sin (θ + π 2 ). (2.38) -1 θ -cos θ π sin(- + θ ) 2 π 2 θ θ π sin( + θ ) 2-1 cos θ π 2 θ 1 π -cos θ = sin(θ - ) 2 π cos θ = sin(θ + ) cos sin 90 sin cos ( sinθ dθ = cosθ = sin θ π ). (2.39) 2 90 inductor capacitor

27 inductor 2 capacitor condensare ( ) condenser potassium kalium ( ) sodium natrium ( ) titanium 3 aluminum 3 magnesium 3 germanium neon 3 xenon 3 uranium 3 ion anion cation (kation)

28 28 2 [1] L i(t) v(t) [3] sinωt t cos sin cos cos sin ( ) v(t) = L d i(t). (2.40) dt 8 d (ωt dt sinωt = ωcosωt = ωsin + π ). 2 (2.44) sinωt dt = 1 ω cosωt = 1 ( ω sin ωt π ). 2 (2.45) ±90 [2] C i(t) v(t) v(t) q(t) q(t) = Cv(t). (2.41) q(t) = i(t) dt. (2.42) v(t) = 1 i(t) dt. (2.43) C

29 29 1. () R L C v(t) i(t) R v(t) = Ri(t) 2. () R L C i(t) = I m sinωt L R v(t) = RI sinωt v(t) = L d dt i(t) C v(t) = 1 C i(t) dt L ( v(t) = ωl I m sin ωt + π ) 2 90 C v(t) = 1 ( ωc I m sin ωt π ) 2 90

30

31 31 [1] [2] [3] [4] [5]

32

33 33 3 a(t) = A m sin(ωt + θ) a(t) = A m e j(ωt+θ) a(t) = A m e j(ωt+θ) jωt A m A e = A m / 2 a(t) A *1 A = A e e jθ V I v(t) = R i(t) V = R I v(t) = L di dt V = jωl I v(t) = 1 C i dt V = 1 jωc I e jωt i(t) = I m sinωt i(t) = I m cosωt (3.1) i(t) = I m e jωt = I m exp(jωt) (3.2) *2 e jωt 3.2 e jωt e jωt = exp(jωt) = cosωt + jsinωt. (3.3) sin cos exp sin exp exp sin cos exp exp cos *1 A = A e θ θ *2 exp() e ()

34 34 3 : R : sin cos exp sin cos cos sin 90 cos sin i(t) = I m e jωt v(t) = L d i(t) (3.8) dt exp ( cos(ωt) = sin ωt + π ) 2 e j(ωt+ π 2 ) = e jωt e j π 2 = je jωt. (3.4) v(t) = jωl I m e jωt (3.9) v(t) = jωl i(t) (3.10) cos sin sin cos 90 sin cos exp ( sin(ωt) = cos ωt π ) 2 e j(ωt π 2 ) = e jωt e j π 2 = je jωt. (3.5) 3.3 e jωt e jωt sin cos (3.10) : ωl : 90 * v(t) = 1 C i(t) = I m e jωt i(t) dt (3.11) v(t) = 1 jωc I me jωt (3.12) exp v(t) = [ ]i(t) (3.6) v(t) = 1 jωc i(t) (3.13) (3.13) : 1 ωc : 90 * 4 v(t) = Ri(t) (3.7) R exp *3 j *4 j

35 exp e jωt e jωt e jωt i(t) = I m e j(ωt+θ) (3.14) I = I m e jθ (3.15) ω 3.5 V = R I, (3.16) V = jωl I, (3.17) V = 1 I. jωc (3.18) i(t) = I m e jωt I = I m (3.19) v(t) = V m e j(ωt+θ) V = V m e jθ (3.20) v(t) = R i(t) (3.21) exp V m e jθ e jωt = R I m e jωt (3.22) e jωt V m e jθ = R I m (3.23) V = R I. (3.24) v(t) = jωl i(t) (3.25) exp e jωt V m e jθ e jωt = jωl I m e jωt (3.26) V m e jθ = jωl I m (3.27) V = jωl I. (3.28) v(t) = 1 i(t) (3.29) jωc exp V m e jθ e jωt = 1 jωc I me jωt (3.30)

36 36 3 e jωt V m e jθ = 1 jωc I m (3.31) V = 1 jωc I. (3.32) 3.6 () A e A m 2 A e = A m 2. (3.33) i(t) = I m e j(ωt+θ) i(t) = I m e j(ωt+θ) I = I e e jθ, I e = I m 2 (3.34) V I v(t) = R i(t) V = R I, (3.37) v(t) = L di V = jωl I, (3.38) dt v(t) = 1 i dt V = 1 I. (3.39) C jωc 3 3 v(t) = V m sin(ωt + θ) v(t) = V m e j(ωt+θ). V = V e e jθ, V e = V m 2. (3.35) i(t) = I m sin(ωt + ϕ) i(t) = I m e j(ωt+ϕ). I = I e e jϕ, I e = I m 2. (3.36)

37 ωl ωl /ωC /ωC i(t) v(t) I V R R i(t) = I m sin(ωt + θ) v(t) = RI m sin(ωt + θ) Im V R I I = I e e jθ 3.1 i(t) v(t) I V L L i(t) = I m sin(ωt + θ) V jωl I Re v(t) = ωl I m sin(ωt + θ + 90 o ) Im I I e e jθ 3.2 i(t) i(t) = I m sin(ωt + θ) v(t) I V C Im Re C v(t) I = m sin(ωt + θ 90 o ) ωc I I e e jθ V I jωc Re 3.3

38 R V I P P = V I = RI 2 (3.40) 2 1/2 (= ) *5 (=) R v(t) = V m sinωt i(t) = I m sinωt p(t) p(t) = v(t)i(t) (3.41) ( T = 2π/ω) P ( ) P = 1 T T 0 1 = V m I m T 1 = V m I m 2 v(t)i(t) dt T 0 sin 2 ωt dt (3.42) 1/2 *5 1/2

39 39 cos v(t) v(t) = ωla sin(ωt + θ). (3.49) sin exp 1: sin exp i(t) = A sin(ωt + θ) v(t) sin exp v(t) v(t) = L di dt (3.43) sin v(t) exp v(t) = ωla cos(ωt + θ). (3.44) i(t) = Ae j(ωt+θ) (3.45) v(t) = jωlae j(ωt+θ) (3.46) sin exp exp v(t) = jωla cos(ωt + θ) ωla sin(ωt + θ) (3.47) exp sin sin exp 2: cos exp i(t) = A cos(ωt + θ) v(t) cos exp v(t) exp i(t) = Ae j(ωt+θ) (3.50) v(t) = jωlae j(ωt+θ) (3.51) cos exp exp v(t) = jωla cos(ωt + θ) ωla sin(ωt + θ) (3.52) exp cos sin exp 3: i(t) = A sin(ωt + θ) i(t) ( w(t) = K L di ) 2 (3.53) dt sin w(t) = K(ωLA) 2 cos 2 (ωt + θ) (3.54) 2 1 cos[2(ωt + θ)] = K(ωLA) 2 (3.55) exp w(t) = K(jωLA) 2 exp[2j(ωt + θ)] (3.56) = K(ωLA) 2 cos[2(ωt + θ)] (3.57) jk(ωla) 2 sin[2(ωt + θ)] (3.58) sin sin v(t) = L di dt (3.48)

40 40 3 j j j j j π/2 (90 ) j π/2 (90 ) jωl j j exp ( j = e j π 2 = exp j π ) 2 j (3.59) v(t) = jωl i(t) (3.60) = e j π 2 ωl I m e jωt (3.61) = ωl I m e j(ωt+ π 2 ) (3.62) π/2 1/(jωC) j 1 ( j = π e j 2 = exp j π ) 2 (3.63) π/2 sin(ωt+π/2) sin(ωt π/2) j π/2 (90 ) 1/ 2 2 P = (1/2)V m I m 1/2 1/2 P = V I () P = V I V m I m V m = [ ]I m V = [ ]I V = V m /2 I = I m P = V I V = [ ]I V = [ ]I (root-mean-square: rms) 2 OK v(t) = V m f (t) i(t) = I m g(t) f (t) g(t) T P = 1 T T 0 = V m I m 1 T v(t)i(t) dt T 0 V e = V m / 2 I e = I m / 2 f (t)g(t) dt. (3.64)

41 41 1 V e = V m T 1 I e = I m T T 0 T 0 f (t)g(t) dt, (3.65) f (t)g(t) dt (3.66) R f (t) = g(t) 1 V e = V m T 1 I e = I m T T 0 T 0 f (t) 2 dt, (3.67) g(t) 2 dt (3.68) RMS 1 V e = T 1 I e = T T 0 T 0 v(t) 2 dt, (3.69) i(t) 2 dt (3.70) () (root-mean-square: rms) RMS RMS RMS RMS

42 42 3 [1] ( 1) i(t) = I m sinωt L d dt i(t) i(t) 90 1 i(t) dt i(t) 90 C 1 C L d dt i(t) = ωl I m cosωt ( = ωl I m sin ωt + π ), 2 i(t) dt = 1 ωc I m cosωt = 1 ωc I m sin [2] ( 2) ( ωt π 2 i(t) = I m cosωt L d dt i(t) i(t) 90 1 i(t) dt i(t) 90 C 1 C ). L d dt i(t) = ωl I m sinωt ( = ωl I m cos ωt + π ), 2 i(t) dt = 1 ωc I m sinωt [3] j = 1 ωc I m cos ( ωt π 2 () j () j ). j = e j π 2 π 2 90 j = e j π 2 π 2 90 [4] e jωt i(t) = I m e jωt 1 C L d i(t) = jωl i(t), dt 1 i(t) dt = 1 C jωc i(t) L d ( I m e jωt) = jωl I m e jωt = jωl i(t). dt (3.71) ( I m e jωt) dt = 1 jωc I me jωt = 1 i(t). jωc (3.72) [5] [1] [2] [4] i(t) i(t) = I m e jωt = I m cosωt + ji m sinωt (3.73) L d i(t) = jωl i(t) dt = e +j π 2 ωl I m e jωt = ωl I m e j(ωt+ π 2 ) ( = ωl I m cos ωt + π ) 2 ( +jωl I m sin ωt + π ), (3.74) 2 1 C i(t) dt = 1 jωc i(t) = e j π 1 2 ωc I me jωt = 1 ωc I me j(ωt π 2 ) = 1 ( ωc I m cos ωt π ) 2 1 ( +j ωc I m sin ωt π ) (3.75) 2

43 43 V = V e θ, I = I e ϕ. 1. v(t) = V m sin(ωt + θ), i(t) = I m sin(ωt + ϕ) v(t) = V m e j(ωt+θ), i(t) = I m e j(ωt+ϕ) 2. R L v(t) = Ri(t) = V = R I v(t) = d i(t) = V = jωl I dt 2. e jωt C v(t) = i(t) dt = V = 1 jωc I V = V e e jθ, I = I e e jϕ V e = V m 2, I e = I m 2.

44

45 45 4 V I V = ZI Z Z Y I = Y V : V = RI R V I () Z sin cos 4.1 V = R I (4.1) V = Z I (4.2) Z Z Z (impedance) Z Ω (, Ohm) 1 R, ωl, Ω ωc ω [rad s 1 ] [s 1 ] L [H]

46 46 4 I V 1 = Z 1 I V 2 = Z 2 I V 3 = Z 3 I V V = Z I Z 1 Z 2 Z I V = V 1 + V 2 + V 3 ( ) [H] v = Ldi/dt [V] = [H][A][s] 1 = [H] = [V][A] 1 [s]. (4.3) V = Z S I Z S ωl [s 1 ][L] = [V][A] 1 = [Ω] (4.4) I V C [F] () [F] v = 1 i dt C [V] = [F] 1 [A][s] = [F] = [V] 1 [A][s]. (4.5) 1 ωc = (ωc) 1 ([s] 1 [F]) 1 = [V][A] 1 = [Ω] (4.6) 4.2 R jωl 1 jωc R Z (1) 1 I = V 1 Z 1, I = V 2 Z 2, I = V 3 Z 3. (4.7) (2) V = V 1 + V 2 + V 3. (4.8) V = Z S I Z S Z S = Z 1 + Z 2 + Z 3. (4.9) R Z (1) V = Z 1 I 1, V = Z 2 I 2, V = Z 3 I 3. (4.10)

47 Z = R + jωl I Z 1 = R 1 Z = R + jωc V Z 2 = jωl 1 Z = R + j ( ωl ) ωc Z 3 = 1 jωc V 4.5 I I I 1 I 2 I 3 V = Z 1 I 1 Z 1 V = Z 2 I 2 Z 2 V = Z 3 I 3 Z 3 V V = Z P I Z P Z = R + jx. (4.13) R ( (resistance)) X ( (reactance)) X > 0: X < 0: () () (2) I = I 1 + I 2 + I 3. (4.11) V = Z P I Z P 1 Z P = 1 Z Z Z 3. (4.12) 4.5 V I V = RI R

48 48 4 V = ZI Z( ) V 1 V 2 V 3 I = Y 1 V 1 I = Y 2 V 2 I = Y 3 V 3 Y 1 Y 2 Y 3 I V = V 1 + V 2 + V (admittance) I = Y V. (4.14) I = Y S V I V Y S V Y Z Y = 1 Z. (4.15) S (, Siemens) Y S = 1 Y Y Y 3 (4.17) Y P = Y 1 + Y 2 + Y 3 (4.18) 4.8 Y = G + jb (4.16) G (conductance) B (susceptance) B B > 0: B < 0: z = x + jy

49 4.11. 極座標形式の計算例 49 V ಐᑙṾ ٳ ǽ ಏǽɪʀɑǛ ǃಏ FGITGG DŽǺǹ DzǵǓȚǢǷȡ ǽ I I 1 = Y1 V ǢǷ V Y1 I 2 = Y2 V V I 3 = Y3 V ಐᑙṾǷ ᏩಐᑙṾǽ ၁ ɤɇɻ ᏩಐᑙṾ Y2 V Y3 కકȡકȅȑǽǾ ǃሱ ᄋઆDŽȡ ᕧǺǨȚǢǷ V I 図 4.8 関数電卓による直角座標系と極座標系の変換例 I = YPV 4.11 V YP 極座標形式の計算例 極座標形式の計算例を以下に示す 例えば かけ算の 場合には 図 4.7 アドミタンスの並列合成 (4 15 ) (2 30 ) = 8 45 (4.19) となる 割り算の場合には 指数関数形式 4 15 = z = rejθ = r exp[jθ ] これは極座標形式の一種であるが 指数関数を使う ので 本講義では 指数関数形式 と呼ぶことにす る この場合の θ の単位はラジアンが用いられる 数値を扱わない理論計算の時には この形式が良く 用いられるが 実際の工学的問題の場合には 偏角 が π 何倍などという形では出てこないので 次に示 すような角度を度で表した方式が良く用いられる 例えば 偏角が 1 ラジアン と言われても どれく (4.20) となる 関数電卓によっては こうした極座標表記と直角座標 表記の変換をしてくれるものがある 電気回路を扱う場 合には そのような電卓を良く使う 最近では スマホ アプリで無料のものがあるので利用するとよい なお 関数電卓で角度を扱うときには 電卓の角度の単位の設 定に気をつけること らいの角度なのかがすぐにわからないハズである 4.12 極座標形式 z = r θ この場合の θ の単位は度 が用いられる 度を用 いるのは数値を扱う工学的問題を対象とするからで ある 先ほどの 1 ラジアンは 度で表すと 約 57 である これならば だいたい 60 なので 三角定 規を思い浮かべれば どれくらいの角度なのかがす ぐにピンとくるハズである 交流電源の内部インピーダンスと内部 アドミタンス 現実の直流電源には内部抵抗なるものが潜んでいるこ とを以前に述べた 交流の電源の場合にも 現実の交流 電源には 内部インピーダンスが潜んでいる 図 4.9 に示すように 交流電源を交流電圧源としてみ た場合 電源端子から電源側を見たときに V = Z I を満 たす Z を電源の内部インピーダンスという なお 内部 インピーダンスがいくらか ということを計算したりす

50 50 4 Power Source V I E Z V I E Z V I OFF Z V I (a) OFF in the case of a voltage source 4.9 ( ) J Y V I OFF Y V I I I Power Source V J Y V (a) OFF in the case of a current source 4.10 ( ) "OFF" 4.10 I = Y V Y "OFF" 4.13 OFF OFF E = (a) J = (b) 4.14 V = ZI OFF (a) ( ) (b) ( ) I I V? V Z = 1 Y 4.12 E = ZJ, Y = 1 (4.21) Z 4.15

51 V = ZI, I = Y V (4.22) Z Y

52 Z 2 Z 2 Z 2 = 0 Z 2 Z 2 Z 2 = 0 Z 2 = Z 1 Z 2 Z 3 Remove Z 2 Z 1 Z 3 Z 2 = 0 Z 1 Z 2 Z 3 Remove Z 2 Z 1 Z 2 = Z

53 53 [3] R 1 R 2 R 3 R S [1] i(t) = I m sin(ωt + θ) R S = R 1 + R 2 + R 3 i(t) [4] R 1 R 2 R 3 R P I = I e e jθ 1 (/ 2) I e = I m [2] v(t) = V m sin(ωt + θ) v(t) V = V e e jθ = R S R 1 R 2 R 3 [5] z = re jθ z = r θ r = 2.0 θ = π 4 = 45 2 (/ 2) V e = V m [6] z = Imag. Imag. I m 2 θ I V m 2 θ V O Real O Real 4.14 i(t) = I m sin(ωt + θ) 4.15 v(t) = V m sin(ωt + θ)

54 π

55 55 j2 A. j 1. R = 1 Ω L = 1 mh C = 500 µf ω = 1000 rad/s 2 R R = 1.0 Ω L jωl = j (1000) ( ) = j 1.0 Ω C 1 jωc = j 1 (1000) ( ) = j 2.0 Ω 2. R, L, C Z 2 Z Z = (1.0 j1.0) Ω Z = 2 45 = ( ) Ω j -j R = 1 Ω L = 1 mh C = 500 µf ω = 1000 rad/s Z Z Z V m = 2.0 V ω = 1000 rad/s Z () () = = 2, = = V V e = V m 2 = 2 2 = 2 = 1.4 V V = 2 0 = ( ) V Z = ( 2 45 ) Ω I = V Z = = ( ) A Z

56 56 4 j2 j I V j -j R = 1 Ω L = 1 mh C = 500 µf ω = 1000 rad/s V m = 2 V V = ( ) V, I = ( ) A, V e = V = 1.4 V, V m = 2V e = 2.0 V, θ = 0, I e = I = 1.0 A, I m = 2I e = 1.4 A, ϕ = Voltage (V) and Current (A) 3 2 v(t) 1 i(t) Phase (degree) 4.18 R = 1 Ω L = 1 mh C = 500 µf ω = 1000 rad/s V m = 2 V V = V e θ v(t) = V m sin(ωt + θ), I = I e ϕ i(t) = I m sin(ωt + ϕ) 4.18

57 57 B. 1. R = 1 Ω L = 0.5 mh C = 1000 µf ω = 1000 rad/s 2 R 1 R = 1.0 S L 1 jωl = j 1 (1000) ( ) = j 2.0 S C jωc = j (1000) ( ) = j 1.0 S 2. R L C Y Y Y = (1.0 j1.0) S Y = 2 45 = ( ) S Y 4.19 j2 j j -j R = 1 Ω L = 0.5 mh C = 1000 µf ω = 1000 rad/s Y 3. Y V m = 2.0 V ω = 1000 rad/s Y () () = = 2, = = V V e = V m 2 = 2 2 = 2 = 1.4 V V = 2 0 = ( ) V Y = ( 2 45 ) S I = V Z = V Y = ( 2 0 ) ( 2 45 ) = ( ) A Y

58 58 4 j2 j V j I -j R = 1 Ω L = 0.5 mh C = 1000 µf ω = 1000 rad/s V m = 2.0 V V = ( ) V, I = ( ) A, V e = V = 1.4 V, V m = 2V e = 2.0 V, θ = 0, I e = I = 2.0 A, I m = 2I e = 2.8 A, ϕ = Voltage (V) and Current (A) 3 2 i(t) 1 v(t) Phase (degree) 4.21 R = 1 Ω L = 0.5 mh C = 1000 µf ω = 1000 rad/s V m = 2 V V = V e θ v(t) = V m sin(ωt + θ), I = I e ϕ i(t) = I m sin(ωt + ϕ) 4.21

59 Z Y Z Y 4 E V p I C R V R θ E a b p I V C = jωc V R =RI V b V a V C I Z = R + 1 jωc 2 1 Z = R + ( ωc ) 2 arg Z = tan 1 1 ( ωcr) 1 jωc θ R Z RC 5.1 R C Z Z = R + 1 jωc = R j 1 ωc. (5.1) Z Z arg Z θ ( ) 1 2 Z = R 2 +, (5.2) ωc ( arg Z = tan 1 1 ). (5.3) ωcr 5.1 RC RC 5.2 R C Y Y = 1 + jωc. (5.4) R Y Y argy θ ( ) 1 2 Y = + (ωc) 2, (5.5) R argy = tan 1 (ωcr). (5.6)

60 60 5 E I I R R C I R = E/R I C = jωce I Y = 1 C R + jωc Y = ( 2 1 R ) + (ωc) 2 arg Y = tan 1 (ωcr) E I I R R L I R = E/R I L = E/( jωl ) 1 1 Y = I L R + jωl Y = ( + R ) arg Y = tan 1 R ( ωl ) ( ωl ) 2 I θ I R I C E jωc θ Y 1/R θ I I R I L E 1 jωl θ Y 1/R 5.2 RC 5.4 RL E θ I E R L V R V R = RI Z = R 2 + (ωl) 2 V L = jωli arg Z = tan 1 ωl ( R ) V L I Z = R + jωl jωl 5.3 RL RL 5.3 R L Z θ Z R Z = R + jωl. (5.7) Z Z arg Z θ Z = arg Z = tan 1 ( ωl R R 2 + (ωl) 2, (5.8) ). (5.9) RL 5.4 R L Y Y = 1 R + 1 jωl = 1 R j 1 ωl. (5.10) Y Y argy θ ( ) 1 2 ( ) 1 2 Y = +, (5.11) R ωl ( argy = tan 1 R ). (5.12) ωl R C RC I E R R V R R I C C V C C I

61 E I C R a b p I V C = jωc V R =RI E I a R V R =RI b I C V C = jωc 5.5 CR RC p (= 0 V) b b p C R R C a b p a b p I V C = jωc V R =RI V R =RI I V C = jωc V p (a) V p V b (b) V R θ E V R θ E V b I V C V a I V C Va p a b b *1 p b V bp p b V b p V bp V b p V bp V b p V bp p b b p *2 5.6(a) 5.6(b) R C I (a) (b) R C V R = RI, (5.13) V C = 1 I. jωc (5.14) *1 b b b b ) *2 5.6 R C R C b b C R (a) 5.6(b) p a a V ap ( ) 5.6(a) p R 5.6(b) C R (5.13) C (5.14) b b

62 (a) p b V bp = V R I V bp 5.6(b) p b V b p = V C 90 I V b p 90 a C R E b V b R C p 5.8 (Phase-Shifter) V b V R V C V p E V a V C V V R V b Vb Vb Vp Va Vp Va Vb Vb Vb V b Vp Va Vp Va 5.3 ( 1) Vb Vb (phase-shifter) 5.8 R ωc E V 5.9 V R V C p b V bp p b V b V = V bp V b p = E V E V E V 5.9 R C 1 V E o b 2 R R ( 2) 5.4 ( 2) 5.10 V bo () E E /2 V bo E R 1 V bo = R E E 2 jωc 1 R 1 1 jωc 1 E = R jωc 1 = jωc 1R 1 1 E jωc 1 R (5.15)

63 Z 1 Z 3 I 5 Z 5 V 5 Z 2 Z 4 E A R 1 R 3 v 2 v 1 B v BC v 3 v 4 R 2 R 4 C D 5.11 E 5.5 V bo = E 2 (5.16) 5.11 Z 5 Z 1 Z 2 Z 3 Z 4 Z 1 Z 2 = Z 3 Z 4. (5.17) (Wheatstone bridge) (Maxwell bridge) (Wien bridge) R 1 R 2 = R 3 R 4 (5.18) 5.12 R 1 R 2 R 3 R 4 R 1 = R 3 R 4 R 2 (5.19) R v 3 v 4 R 3 v 3 = E, R 1 + R 3 (5.20) R 4 v 4 = E. R 2 + R 4 (5.21) v BC v BC = v 3 v 4. (5.22) v BC = 0 R 3 R 1 + R 3 R 4 R 2 + R 4 = 0. (5.23) R 1 R 2 = R 3 R 4. (5.24) R 1 R 4 = R 2 R 3

64 64 5 R 1 A R 3 C 3 R 2 C 4 R 4 E ( & ) [1] 1 Ω 10 MΩ ( 4 ) R i R i 5.13 [1] R 1 = A.BCD 10 E Ω R 1 R 2 = A.BCD 1 A 2 B 3 C 4 D R 3 /R 4 R 1 E R 1 4 R 1 = A.BCD 10 E Ω B C ( ) ω = 1 R 3 C 3 (5.25) R 1 = 2R 2 R 3 = R 4 C 3 = C ( R )( ) 1 + jωc 4 = R 1. (5.26) jωc 3 R 4 R 2 R 3 + C ( 4 + j ωr 3 C 4 R 4 C 3 1 ωr 4 C 3 ) = R 1 R 2. (5.27) R 3 R 4 + C 4 C 3 = R 1 R 2, (5.28) ωr 3 C 4 = 1 ωr 4 C 3 (5.29) R 1 = 2R 2 R 3 = R 4 C 3 = C 4 ω = 1 R 3 C 3 (5.30) R 3 C 3 ω [2] R 2 R 3 R 4 C 4 Z = R 1 + jωl 1 R 1 L 1 ω

65 L 1 R 1 A R 3 Z s L 1 C 1 1 Z s = jωl 1 + jωc1 1 = j ( ωl 1 ωc1 ) R 2 C 4 E R LC 5.15 Z p C 2 L 2 1 Z p = 1 jωc 2 + jωl2 ωl = j ( 2 1 ω ) 2 L 2 C LC [2] R 1 + jωl 1 = R 1 2 R 3, (5.31) + jωc 4 R 4 R 1 R 4 + jωl 1 R 4 = R 2 R 3. (5.32) 1 + jωc 4 R 4 R 1 R 4 + jωl 1 R 4 = R 2 R 3 (1 + jωc 4 R 4 ). (5.33) R 1 = R 2R 3, R 4 (5.34) L 1 = C 4 R 2 R 3 (5.35) R 1 L 1 R 2 R 3 R 4 C LC LC L C 5.17 LC Z s Z s = jωl jωc 1 ( = j ωl 1 1 ) ωc 1 (5.36) () 5.18 LC Z p 1 Z p = jωc jωl 2 ( ) ωl 2 = j 1 ω 2 L 2 C 2 (5.37) () 5.17 LC 5.18 LC

66 L 2 = 100 mh C 2 = 10 uf Series Zs L1 C 1 Reactance X ( ) Frequency (Hz) 5.19 LC L 1 = 100 mh C 1 = 10 uf Parallel Reactance X ( ) Zp C 2 L Frequency (Hz) LC L 1 = 100 mh C 1 = 10 µf L 2 = 100 mh C 2 = 10 µ F Z S Z P f = 1 khz

67 5.7. ( 1) RC ( 1) RC 5.21 E e(t) = E m sinωt E m = 10 2 V ω = 5000 rad/s C = 10 µf R = 10 Ω 3 *3 E I C R V C V R 5.21 RC 1. e(t) E E r θ 2. C R Z r θ 3. I r θ 4. C ()V C r θ 5. R ()V R r θ 6. E V C V R 7. I i(t) 8. e(t) i(t) 9. Z E I 1 e(t) = E m sin(ωt + θ) E = Em 2 θ E * E = = 10 0 = ( ) V. Z Z = R + 1 jωc Z 3 1 Z = 10 + j(5000) ( ) 1 = 10 j = 10 j = 10 j = 10 j20 = = ( ) Ω. I = E/Z E = (10 0 ) V Z = ( ) Ω I 4 I = E Z = = = = ( ) A. (5.38) C Z C C V C = Z C I Z C Z C = 1 jωc 1 = = j20 j = ( ) Ω V C V C = Z C I = (20 90 ) ( ) = = ( ) V.

68 V V R E V V C V (a) V R V C E (b) Voltage (V) Voltage (V) Current Current (A) 5.22 RC ( ) (a) o (b) Phase (degree) 5 R V R = RI V R 6 V R = Z R I = = = ( ) V. E V C V R E = ( ) V, V C = ( ) V, V R = ( ) V. E = V C + V R ; I = I m 2 ϕ i(t) i(t) = I m sin(ωt + ϕ). I = ( ) A I m ϕ I m = = A, ϕ = RC 10.0 Ω A I j20.0 Ω V Z Ω = (10.0 j20.0) Ω 5.24 RC Z E I 8 e(t) i(t) e(t) = 10 2sinωt = 14.1sinωt V, i(t) = 0.632sin(ωt ) A Z E I Z = 10.0 j20.0 = ( ) Ω, E = ( ) V, I = ( ) A E i(t) i(t) = 0.632sin(ωt ) A.

69 5.8. ( 2) RC ( 2) RC 5.25 E e(t) = E m sinωt E m = 10 2 V ω = 5000 rad/s C = 100 µf R = 10 Ω 3 I Y Y = j(5000) ( ) = j = j0.5 = = ( ) S. E I R R C I C 3 I = Y E E = (10 0 ) V Y = ( ) S I 5.25 RC 1. e(t) E E r θ 2. C R Y r θ 3. I r θ 4. C ()I C r θ 5. R ()I R r θ 6. I I C I R 7. I i(t) 8. e(t) i(t) 9. Y E I 1 e(t) = E m sin(ωt + θ) E = Em 2 θ E 2 E = = ( ) V. Y Y = 1 R + jωc 4 I = Y E = ( ) (10 0 ) = = ( ) A. C Y C C I C = Y C E Y C Y C = jωc = j = j0.5 = ( ) S. I C 5 I C = Y C E = ( ) (10 0 ) = ( ) A. (5.39) R I R = E/R I R 6 I R = E R = = ( ) A. (5.40) I I C I R I = ( j5.00) = ( ) A, I C = j5.00 = ( ) A, I R = 1.00 = ( ) A.

70 70 5 I A I C A I R A Voltage (V) 5.26 RC ( ) I = I R + I C Voltage (V) 10 0 Current 5 0 Current (A) o 7 ; Phase (degree) I = I m 2 ϕ i(t) 5.27 RC i(t) = I m sin(ωt + ϕ). I = ( ) A I m ϕ I m = = 7.21 A, ϕ = i(t) i(t) = 7.21sin(ωt ) A. 8 e(t) i(t) e(t) = 10 2sinωt = 14.1sinωt V, i(t) = 7.21sin(ωt ) A Y E I j0.500 S S = ( j0.500) S Y S A I 78.7 E V 5.28 RC Y E I Y = ( j0.500) = ( ) S, E = ( ) V, I = ( ) A. 5.28

71 5.9. ( 3) RL ( 3) RL 5.29 E e(t) = E m sinωt E m = 10 2 V ω = 5000 rad/s L = 10 mh R = 10 Ω 3 I Z Z = R + jωl. Z = 10 + j(5000) ( ) = 10 + j50 = = ( ) Ω. E L R V L V R 3 I = E/Z E = (10 0 ) V Z = ( ) Ω I 5.29 RL 1. e(t) E E r θ 2. L R Z r θ 3. I r θ 4. L ()V L r θ 5. R ()V R r θ 6. E V L V R 7. I i(t) 8. e(t) i(t) 9. Z E I 4 I = E Z = = = = ( ) A. (5.41) L Z L L V L = Z L I Z L Z L = jωl = j = j50.0 = ( ) Ω. V L V L = Z L I = (50 90 ) ( ) = = ( ) V. (5.42) 1 e(t) = E m sin(ωt + θ) E = Em 2 θ E E = = ( ) V 5 R V R = RI V R V R = Z R I = 10 ( ) = = ( ) V. 2 6

72 V V L 20 Voltage (V) 1.0 E V R V V 5.30 RL ( ) Voltage (V) 10 0 Current Current (A) E V L V R o Phase (degree) E = ( ) V, V L = ( ) V, V R = ( ) V. (5.43) E = V L + V R ; 5.31 RL j50.0 Ω 10.0 Ω Ω = ( j50.0) Ω Z V E 78.7 I A I = I m 2 ϕ 5.32 RL Z E I i(t) i(t) = I m sin(ωt + ϕ). I = ( ) A I m ϕ I m = = A, ϕ = Z E I Z = ( j50.0) = ( ) Ω, E = ( ) V, I = ( ) A i(t) i(t) = 0.277sin(ωt 78.7 ) A. 8 e(t) i(t) e(t) = 10 2sinωt = 14.1sinωt V, i(t) = 0.277sin(ωt 78.7 ) A. 5.31

73 5.10. ( 4) RL ( 4) RL 5.33 E e(t) = E m sinωt E m = 10 2 V ω = 5000 rad/s L = 1 mh R = 10 Ω 3 Y Y = j(5000) ( ) = 0.1 j0.2 = = ( ) S. E I I R R L I L 3 I = Y E E = 10 0 V Y = S I 5.33 RL I = Y E = ( ) (10 0 ) = = ( ) A. (5.44) 1. e(t) E E r θ 2. L R Y r θ 3. I r θ 4. L ()I L r θ 5. R ()I R r θ 6. I I L I R 7. I i(t) 8. e(t) i(t) 9. Y E I 1 e(t) = E m sin(ωt + θ) E = Em 2 θ E 2 E = = ( ) V Y Y = 1 R + 1 jωl. 4 L Y L L I L = Y L E Y L Y L = 1 jωl = = j0.2 j = ( ) S. I L 5 I L = Y L E = ( ) (10 0 ) = 2 90 = ( ) A. R I R = E/R I R 6 I R = E R = = ( ) A. I I L I R I = ( ) A, I L = ( ) A, I R = ( ) A.

74 74 5 I R A 20 Voltage (V) 10 I L 10 Current A I A Voltage (V) 0 0 Current (A) 5.34 RL ( ) o I = I L + I R ; I = I m 2 ϕ i(t) i(t) = I m sin(ωt + ϕ). I = ( ) A I m ϕ I m = = 3.16 A, ϕ = i(t) Phase (degree) 5.35 RL S V E j0.200 S I A Y S = (0.100 j0.200) S 5.36 RL Y E I 5.36 i(t) = 3.16sin(ωt 63.4 ) A. 8 e(t) i(t) e(t) = 10 2sinωt = 14.1sinωt V, i(t) = 3.16sin(ωt 63.4 ) A Y E I Y = ( j0.200) = ( ) S, E = ( ) V, I = ( ) A.

75 75 Z () ( 5.23 ) () I V P k B f (x)... I m = 1 A V e = 1 V sinθ dx... I m m I m V e e V e j( i) j i e jθ e e jθ k B B Botlzmann k B di dt di dt R... R... R... R... R (...) R R R R L L L L C C C C () 5.37

76 (BOSS PH-3) [3].

77 77 [1] i(t) = I m sin(ωt + θ) I = I m 2 e jθ I = I m 2 θ [2] v(t) = V m sin(ωt + θ) [3] R, L, C Z Z V I V I Z [4] Z = R + jωl + 1 jωc V = ZI R, L, C Y Y V I V I Y V = V m 2 e jθ V = V m 2 θ Y = 1 R + 1 jωl + jωc I = Y V Imag. I m 2 θ I O Real 5.38 i(t) = I m sin(ωt + θ) Imag. V m 2 θ V O Real 5.39 v(t) = V m sin(ωt + θ)

78 78 5 A. v(t) = V m sinωt V m = 10 2 V ω = 5000 rad/s R = 1 Ω C = 400 µf RC 3 1. v(t) V V (r θ ) Z = R + 1 jωc = (1.00 j0.500) Ω Z = = ( ) Ω Z I I V I V m V e V e = V m 2 = 10.0V V = ( ) V I = V Z = = ( ) = = ( ) A RC Z Z 4. I i(t) j10 j2 j5 I -2-1 j j0.5 Z V j5 -j -j10 -j ω = 5000 rad/s R = 1 Ω C = 400 µf Z 5.41 ω = 5000 rad/s R = 1 Ω C = 400 µf Z V I

79 79 Voltage v(t) and current i(t) v(t) i(t) 26.6 o -2-1 j2 j j0.5 Y -j Phase (degree) 360 -j ω = 5000 rad/s R = 1 Ω C = 400 µf Z v(t) i(t) 5.43 ω = 5000 rad/s R = 1 Ω L = 0.4 mh Y 2. RL Y Y I m = I e 2 = = = 12.7 A i(t) = 12.7sin(ωt ) A Y = 1 R + 1 jωl = (1.00 j0.500) S 5.41 B. i(t) = I m sinωt I m = 10 2 A ω = 5000 rad/s R = 1 Ω L = 0.4 mh RL 3 Y = = ( ) S Y V V I V 1. i(t) I I (r θ ) I m I e I e = I m 2 = 10.0A V = I Y = = ( ) = = ( ) V I = ( ) A

80 80 5 j10 j5 V I j5 -j ω = 5000 rad/s R = 1 Ω L = 0.4 mh Y I V 20 Current i(t) and voltage v(t) i(t) v(t) 26.6 o Phase (degree) ω = 5000 rad/s R = 1 Ω L = 0.4 mh Y i(t) v(t) V v(t) V m = V e 2 = = = 12.7 V v(t) = 12.7sin(ωt ) V 5.45

81 81 [1] [2] [3]

82

83 Power S E I I : V(V), I(A), P(W) θ = 0 o Current S = EI Voltage 1.0 θ = + 60 o E V S = V I S V(V), I(A), P(W) Current Power P Voltage Voltage Current Q * 1 cosθ V(V), I(A), P(W) Power cos 100 % θ = + 90 o Time (ms) , 60, 90, *1 Reactive Power 6.1

84 84 6 I i(t) E Z jx Z θ R p(t) = e(t)i(t) (6.4) = 2 E I sinωt sin(ωt θ) (6.5) = EI cosθ EI cos(2ωt θ) (6.6) ω E Z = R + jx = Z exp(jθ) e(t) i(t) p(t) = e(t)i(t) I = E Z = E E = Z ejθ Z e jθ = I e jθ (6.1) e(t) i(t) e(t) = E m sinωt, (6.2) i(t) = I m sin(ωt θ) (6.3) E m = 2 E, I m = 2 I e(t) EI cosθ EI cos(2ωt θ) Z R, L, C 6.2 R, L, C e(t)i(t) p(t) = EI cosθ EI cos(2ωt θ) (6.7) = EI cosθ R EI cosθ cos2ωt EI sinθ sin2ωt (6.8) θ = 0, cosθ = 1, sinθ = 0 p(t) = EI EI cos2ωt (6.9)

85 (a) R E(V), I(A), P(W) θ = 0 o Power Current p(t) = 0 (6.12) 6.3(b) -1.0 Voltage (b) L E(V), I(A), P(W) Voltage Current Power C θ = π, cosθ = 0, sinθ = 1 2 θ = - 90 o Voltage Current p(t) = + EI sin2ωt (6.13) (c) C E(V), I(A), P(W) Power p(t) = 0 (6.14) θ = + 90 o Time (ms) (c) L 6.3 R, L, C p(t) = EI (6.10) L C 6.3(a) L θ = + π, cosθ = 0, sinθ = +1 2 p(t) = EI sin2ωt (6.11) 6.3 p(t) = EI cosθ EI cosθ cos2ωt EI sinθ sin2ωt (6.15)

86 86 6 EI cosθ EI cosθ cos2ωt = EI cosθ (1 cos2ωt) = EI cosθ (2sin 2 ωt) (6.16) cosθ % θ EI sinθ sin2ωt EI EI cosθ EI sinθ 6.4 S S = EI = ZI 2 = E 2 /Z (6.17) [W] [VA] Q = S sinθ (6.20) [var] *2 sinθ (6.21) 6.7 S P Q θ 6.4(a) 6.4(b) P = EI cosθ = S cosθ (6.18) [W] cosθ (6.19) 6.8 *2 v Var

87 (a) (b) Q S θ P S θ P Q S S P Q cos θ 6.4 (a) S P Q θ (b) S = EĪ S = EI (6.22) Ī I I E = E e j0 = E Z = Z e jθ I = I e jθ E I EI = E I e jθ (6.23) EI EI = E I e jθ (6.24)

88 (a) ( ) 6.5(b) (I m sinωt) 2 = 1 2 I2 m (1 cos2ωt) cos2ωt I 2 m 6.5(c) v v/r () i v v i v i R ± v + N i ± S (a) moving coil meter i (c) wattmeter (b) moving iron meter ± Current coil ± Voltage coil i + v 6.5 [1,2] (a) (b) (c) Z L

89 89 [A] [B] [A] ( ) R E I R I R = E R 6.6(a) () ; 2 E 2 IR = R ( ) (b) R jx R jx I L I L = I R + ji X I L 2 2 IL = 2( I R 2 + I X 2 ) EI = E I L E I L E I L ( ) ( 1 )

90 90 6 I R = E/R 2 E 2 IR e(t) E I R = E/R R R jx i R (t) (a) I = E/R je/x 2 E R jx jx 2 I 2 I R e(t) i E X (t) I R = E/R R I X = je/x jx i R (t) i(t) (b) (a) R (b) R jx [3] 6 () 85 ( (4) ) ( ) ( 1) ( 2) 1 6.9(a)(b) 1 R ωl R 2ωL 1

91 91 L L I S = I L ΔV I S = I L (a) R (b) R C E E V I R I L R L jx L ji X Source Transmission line Z S = R S + jx S Load Z L = R L // jx L Z 1 Z = jωl + 1/R + jωc = jωl + R 1 + jωcr ( R = 1 + ω 2 C 2 R 2 + jω CR 2 ) L 1 + ω 2 C 2 R 2. (6.25) 1 Z 1 C CR 2 L 1 + ω 2 C 2 = 0. (6.26) R2 C (ω 2 R 2 L) C 2 R 2 C + L = 0. (6.27) C C = R2 ± (R 2 ) 2 4(ω 2 R 2 L)L 2(ω 2 R 2 L) = R2 ± R R 2 (2ωL) 2 2(ω 2 R 2 L) = R2 ± R (R + 2ωL)(R 2ωL) 2(ω 2 R 2. (6.28) L) (R 2ωL) C R ωl R 2ωL. (6.29) [B] [4] 6.10 *3 ( ) E Z S R S jx S 6.10 R L jx I S E V *3

92 92 6 V = E V V = (R S + jx S ) I S E ΔV ΔV X I L 6.10 I S = I L O θ I L = I S V R S I S φ θ jx S I S ΔV R E V V I 6.11 V = (R S + jx S ) I L R L I R jx ji X I I L = I R ji X V = (R S + jx S ) (I R ji X ) = (R S I R + X S I X ) + j (X S I R R S I X ) = V R + j V X V R = R S I R + X S I X, V X = X S I R R S I X P jq P = V I R, Q = V I X V R V X P Q V R = R SP + X S Q, V V X = X SP R S Q. V V P Q 6.11 V V E V E V E V E V V = E V = V R + j V X V V = V E = V + V R + j V X E 2 = ( V + V R ) 2 + V 2 X V E A = R S P + X S Q, (6.30) B = X S P R S Q (6.31) { E 2 = V + A } 2 { } B 2 + (6.32) V V W = V 2 W 2 + (2A E 2 )W + (A 2 + B 2 ) = 0.

93 93 I S = I L + I C ΔV I S = I L + I C I L I C E jx S I S E Source E Transmission line Z S = R S + jx S V I R R L jx L Load Z L = R L // jx L ji X Compensator O I C I S θ V ΔV φ R S I S I L 6.12 E = V W V 6.13 a = 1, b = 2A E 2, c = A 2 + B 2 V 2 = W = b ± b 2 4ac 2a E 2 = V 2 Q (6.33) E 2 = V 2 Q aq 2 + bq + c = 0. V = W = b ± b 2 4ac ± 0 E 2a a b c a = R 2 S + X 2 S, (6.34) b = 2 V 2 X S, (6.35) c = ( V 2 + R S P) 2 + X 2 S P2 V 4. (6.36) Q E = V Q = b ± b 2 4ac. (6.37) 2a ± E = V Q C *4 Q = Q +Q C E 2 = { V + R SP + X S Q } 2 { XS P R S Q } 2 + (6.33) V V *4 Q Q C 6.13 V Q C ( I C ) I S E = V

94 94 6 *5 1 E = V V 2 = kv 2 V = kv (+ ) V 2 = kv 2 V = kv ( ) 0 50 Mvar 0 E 10 kv ( 1) [4] E = 10 kv P = 25 MW Q = +50 MVar 25/ = Q > 0 R S = Ω X S = Ω ( ) *6 V V A B (6.30) (6.31) A = R S P + X S Q = kv 2, B = X S P R S Q = kv 2 E = 10 kv (6.32) V + 10 kv 3.2 kv V = 6.78 kv A B V V = A V + j B = ( j0.8678) kv V E E = 10 kv V V V E E E = V + V = ( j0.8678) kv = ( ) kv V 5 E V 5 I L I L = P jq V = (3.686 j7.373) ka = ( ) ka PF ( ) PF = cos( ) = () 6.14(a) *5 *6 () 4 3 V E 10 kv Q C

95 95 (6.37) Q (6.34) (6.36) a = R 2 S + X 2 S = Ω 2, b = 2 V 2 X S = 2 ( ) = V 2 Ω c = ( V 2 + R S P) 2 + X 2 S P2 V 4 = (( ) ( )) 2 +( ) 2 ( ) 4 = V 4. (6.37) + Q = Mvar (+ ) = Mvar ( ). Q C = Q Q = = Mvar (+ ) = = Mvar ( ) Q ( ) Q (+ ) Q C = 56.4 Mvar I L I C I C I C = jq C V Mvar = j = j5.635 ka 10 kv I L V = 10 kv I L = P jq V = (2.500 j5.000) ka = ( ) ka I L I C ( I S ) I S = I L + I C = ( j0.635) ka = ( ) ka PF = cos14.25 = () P 25 PF = = = P 2 +Q ( 6.354) V A = R S P + X S Q 0 = kv 2 B = X S P R S Q 0 = kv 2 V = A V + j B V = = ( j1.030) kv = ( ) kv + j E E = V + V = ( j1.030) kv = ( ) kv

96 96 6 V E 6 E (b) Q Q Q = Mvar Q C = Q Q = Mvar I C I C = j Q C Mvar = j = j53.40 ka V 10 kv I C 10 I C I L V Q C I L = (2.500 j5.000) ka = ( ) ka I S I S = I L + I C = ( j48.40) ka = ( ) ka PF = cos87.04 = E V V A = R S P + X S Q 0 = kv 2 B = X S P R S Q 0 = kv 2 V = A V + j B V = = ( j4.775) kv = ( ) kv + j E E = V + V = ( j4.775) kv = ( ) kv E V ( 2) [4] 1 Q C = Q Q 0 = 0 1 V (6.30) (6.31) A = R S P = kv 2 B = X S P = kv 2 E = 10 kv (6.32) V 2 = kv 2 V = kv E = 10 kv kv 10 kv 2.5%

97 97 2.5% 1 V V R = kv, V X = kv. V = ( j1.006) kv = ( ) kv E E = V + V = ( j1.006) kv = ( ) kv I L V = kv I L = P jq V = (2.565 j5.129) ka = ( ) ka Q C = Q I C = +j5.129 ka 6.14(c) (1 m) [5] L S(1 m) = µ { 0 log 2π ( 2h a ) + µ } S. (6.38) 4 h a µ S (µ S = 1) µ 0 (µ 0 = 4π 10 7 H/m) h = 10 m a = 5.64 mm L S(1m) = H/m 1 km R S(1 km) = Ω/km, L S(1 km) = H/km 60 Hz ω = 2πf = 377 rad/s R S(1 km) = Ω/km, X S(1 km) = Ω/km (6.39) ( 2) ( j0.3922) Ω [4] 100 mm m ( ρ = Ωm) S = 100 mm 2 ( a = 5.64 mm) (1 m) R S R S(1 m) = ρ 1 S = Ω/m

98 98 6 (a) 63.4 o E = 10.0 kv 4.98 o V = 6.78 kv I R = 3.69 ka V V R = 3.18 kv V X = kv I X = 7.37 ka I L = I S = 8.24 ka I C = 5.64 ka (b) 14.3 o I S 5.91 o 63.4 o I R = 2.50 ka E = 10.0 kv V = 10.0 kv V I X = 5.00 ka I = 5.59 ka I C = 5.13 ka (c) 63.4 o 5.77 o E = 10.0 kv I S = I R = 2.57 ka V = 9.75 kv V I X = 5.13 ka I = 5.74 ka 6.14 (a) (b) (b) 1

99 99 [1]

100 V = 10 0 V I = A S () S S = V I = (10 0 ) (50 60 ) = ( ) VA S = 500 VA VA 2. % cos( 60 ) = 0.5 = 50.0% 3. P P = S cos( 60 ) = = 250 W W

101 101 [1] C. K. Alexander and M. N. O. Sadiku: Fundamentals of Electric Circuits 5th Ed. (McGraw-Hill, New York, NY, 2013) pp [2] S. Tumanski: Principles of Electrical Measurement (Taylor & Francis, New York, NY, 2006) Chapter 3 Classic electrical measurement, pp [3] : ( ) (,, 2015) p [4] T. J. E. Miller: in Reactive Power Control In Electric Systems, Ed. Timothy J. E. Miller (John Wiley & Sons, New York, NY, 1982) Chapter 1 The theory of load compensation, pp [5], : (,, 1970) p. 276.

102

103 103 7 ω ω ω 7.1 0, () Q, 7.2 ( Q ) 7.1 ω Z = R +jx Z 1/ Z 7.2 ω 7.2 Abs. admittance (x10-3 S) RLC Series Circuit Y R = 20 Ω L = 100 mh C = 10 uf Q = Angular frequency (rad/s) 2000 V(ω) Z = R + jx 7.1 ω Z 7.2 () R = 20 Ω L = 100 mh C = 10 µf ()

104 104 7 V(ω) R L C Z s = R + j ( ωl 1 ) ωc 7.3 RLC L C ω 0 = 1 LC. (7.4) ( ) 1 f 0 = 2π LC. (7.5) L C 7.3 V () Z s I = V Z s (7.1) Y s = 1/Z s *1 Z s Z s Y s Y s Z s ( Z s = R + j ωl 1 ) ωc (7.2) Z s j( ) Z s Z s (=R) Z s ω 0 ω 0 L 1 = 0. (7.3) ω 0 C * ω 0 = 1/ LC Z s Z s R Y s 1/R I V /R R, L, C L = 100 mh C = 10 µf R L C ω 0 = 1 1 = (7.6) LC = 1000 rad/s (7.7) R R 0 Ω 10 Ω 20 Ω 50 Ω R ω 0 = 1000 rad/s Y s R

105 Abs. admittance (x10-3 S) RLC Series Circuit Y L = 100 mh C = 10 uf R = 0 Ω R = 10 Ω R = 20 Ω R = 50 Ω Angular frequency (rad/s) 2000 I(ω) R L C Y p = 1 R + j ( ωc 1 ωl ) 7.6 RLC Z p 7.4 RLC ( ) Abs. impedance (Ω) RLC Series Circuit Z L = 100 mh C = 10 uf R = 50 Ω R = 30 Ω R = 10 Ω R = 0 Ω Angular frequency (rad/s) RLC ( ) Q L C 7.6 I () V = Z p I (7.8) Y p = 1/Z p Y p Y p Y p Y p = 1 ( R + j ωc 1 ) ωl (7.9) Y p j( ) Y p Y p (=1/R) Y p ω 0 ω 0 C 1 ω 0 L = 0 (7.10) L C ω 0 = 1 LC. (7.11) ( ) 1 f 0 = 2π LC. (7.12)

106 106 7 Abs. admittance (x10-3 S) RLC Parallel Circuit Y L = 100 mh C = 10 uf R = 100 Ω R = 500 Ω R = 1000 Ω R = Ω Angular frequency (rad/s) RLC ( ) Abs. impedance ( ) RLC Parallel Circuit Z L = 100 mh C = 10 uf R = Ω R = 1000 Ω R = 500 Ω R = 100 Ω Angular frequency (rad/s) 2000 ω 0 = 1 1 = (7.13) LC = 1000 rad/s (7.14) R R 100 Ω 500 Ω 1000 Ω Ω R ω 0 = 1000 rad/s Z p R Q 7.8 RLC ( ) ω 0 = 1/ LC Y p 1/R Z p R V R I R, L, C L = 100 mh C = 10 µf R L C I = Y s V V = Z p I 7.9 *2 *2

107 7.7. RLC Q R 107 Input ω ω 0 Good filter ω Output ω 1 1/ 2 Q = ω 0 ω 2 ω 1 ω 1 ω 0 ω 2 ω Input ω ω 0 Poor filter ω Output ω 7.10 Q 7.9 ω 1 ω 2 ( ), Q (Quality Factor) 7.6 Q (Quality Factor) 7.10 Q Q = ω 0 ω 2 ω 1. (7.15) ω 0 ω 1 ω 2 1/ RLC Q R 7.11 L = 100 mh C = 10 µf R = 10 Ω 20 Ω 50 Ω RLC RLC R R Q R = 10 Ω 20 Ω 50 Ω Q = 10, Q = 5, Q = RLC Q R 7.12 L = 100 mh C = 10 µf R = 1000 Ω 500 Ω 100 Ω RLC RLC R R Q R = 1000 Ω 500 Ω 100 Ω Q = 10 Q = 5 Q = Q R Q R

108 108 7 (a) Abs. admittance (x10-3 S) RLC Series Circuit Y R = 10 Ω L = 100 mh C = 10 uf Q = 10 (a) Abs. impedance ( ) RLC Parallel Circuit Z R = 1 k L = 100 mh C = 10 uf Q = Angular frequency (rad/s) Angular frequency (rad/s) 2000 (b) Abs. admittance (x10-3 S) RLC Series Circuit Y R = 20 Ω L = 100 mh C = 10 uf Q = 5 (b) Abs. impedance ( ) RLC Parallel Circuit Z R = 500 L = 100 mh C = 10 uf Q = Angular frequency (rad/s) Angular frequency (rad/s) 2000 (c) Abs. admittance (x10-3 S) RLC Series Circuit Y R = 50 Ω L = 100 mh C = 10 uf Q = 2 (c) Abs. impedance ( ) RLC Parallel Circuit Z R = 100 L = 100 mh C = 10 uf Q = Angular frequency (rad/s) Angular frequency (rad/s) (L = 100 mh) (C = 10 µf) (R = 10 Ω 20 Ω 50 Ω) 7.12 (L = 100 mh) (C = 10 µf) (R = 1000 Ω 500 Ω 100 Ω) RLC Q Q = ω 0L R = 1 ω 0 CR = 1 L R C. (7.16) RLC Q Q = ω 0 CR = R ω 0 L = R C L. (7.17) RLC 7.14 RLC Q Q

109 R L V(ω) C L R L L ω 0 = 1 LC Q = 1 R L C (a) ideal (b) real 7.13 RLC ω 0 Q 7.15 I(ω) R L C C R C C ω 0 = 1 LC Q = R C L (a) ideal (b) real RLC ω 0 Q R R = 0 Ω R R = Ω LC LC Q *3 * TDK [1] 1 mh 1 Ω

110 110 7 Leakage current = V R C = 2 MΩ 7.17 (TDK) [1] 7.18 () [2] [2] 1000 V 0.5 ma 2 MΩ 7.12 LC RLC LC 7.19 RLC LC 7.20 RLC Q X Q Q X 7.15 Q X

111 7.12. LC RLC 111 L C V(ω) (a) Ideal LC series circuit C R L L R C V(ω) (b) Real LC series circuit R s L C V(ω) (c) Equivalent circuit for real LC series circuit 7.19 (a) LC (b) LC (c) LC Q ω L C R Q X ω Q X ω () ω ω 0 () *4 Q X Q L = ω 0L R L, (7.20) Q C = ω 0C G C. (7.21) I(ω) I(ω) I(ω) (a) Ideal LC series circuit L (b) Real LC series circuit R p L R L L R C C C (c) Equivalent circuit for real LC series circuit 7.20 (a) LC (b) LC (c) LC Q L = C = ωl R L. (7.18) 7.16 Q X Q C = = ωc G C. (7.19) G C = 1/R C (a) 7.20(a) 7.19(b) 7.20(b) RLC RLC 7.19(b) 7.20(b) 7.19(c) 7.20(c) LC RLC 7.21 LC RLC 7.22 *4 x.xx 10 y 4 ( 5 )

112 112 7 C R C R C(s) C 7.21 LC RLC R L L L R L(p) 7.22 LC RLC 7.23 (b) 7.23 (a) 7.23 (b) 1 jx Z = 1 jx + 1 = 1 + j 1 X Q X X Q X X (7.22) Q X 1 *5 1/Q 1 X/ X 1 w 1 w ( Z jx 1 j (1 + w) 1 1 w (7.23) X Q X X ) = X Q X + jx (7.24) 7.23 (a) X /Q X jx Q X X jx (a) (b) 7.23 Q X 7.23 ( ) Q X 7.23 (a), (b) LC RLC R C(s) R C(s) = R C Q 2. (7.25) C 7.22 R L(p) R L(p) = Q 2 L R L. (7.26) LC RLC ( ) R s R p *5 ωl 1/ωC

113 7.12. LC RLC 113 (a) Abs. admittance (S) Series circuit L = 1 mh R L = 1 Frequency (khz) C = 1000 pf R C = 2 M (a) Abs. impedance (10 6 ) Parallel circuit L = 1 mh R L = 1 Frequency (khz) C = 1000 pf R C = 2 M (b) Abs. admittance (S) Series circuit (equivalent) L = 1 mh R L = 1 Frequency (khz) C = 1000 pf R C(s) = R C /Q C (b) Abs. impedance (10 6 ) Parallel circuit (equivalent) Frequency (khz) L = 1 mh R L(p) = R L Q L 2 C = 1000 pf R C = 2 M (a) LC (b) RLC 7.25 (a) LC (b) RLC R s = R L + R C(s) = R L + R C Q 2, C (7.27) 1 = = R p R C R L(p) R C Q 2 L R. L (7.28) RLC LC L = 1 mh R L = 1 Ω C = 1000 pf R C = 2 MΩ L C ω 0 = 1/ LC = 10 6 rad/s f 0 = ω 0 /(2π) = 159 khz 160 khz 7.24 (a) (b) LC 7.19 (c) 7.25 (a) (b) LC 7.20 (c)

114 114 7 RLC Q R L C ω 0 ( ) 1/ 2 ( ) ω 1 ω 2 Q (7.15) RLC Z s = R 2 + ( ωl 1 ) 2 (7.29) ωc ω 0 Y s = 1 Z s 1 = ( R 2 + ωl 1 ). (7.30) 2 ωc ω 0 L 1 ω 0 C = 0 (7.31) Y s Y s0 = 1 R Q ω = ω 1 ω 2 (7.32) Y s Y s0 = 1 2 (7.33) ω 1 ω 2 (ω 1 < ω 2 ) Q Y s / Y s0 Y s Y s0 = R ( R 2 + ωl 1 ) 2 ωc 1 = ( ωl 1 + R 1 ) (7.34) 2 ωcr ω ωl R 1 = ±1. (7.35) ωcr ωl R 1 = +1 (7.36) ωcr ω ω 2 R L ω 1 LC = 0 (7.37) ω = 1 ( ) R R 2 2 L ± + 4 L LC (7.38) ω ω ω ± + ω = 1 ( ) R R 2 2 L L LC. (7.39) ω Q ω 1 ω 2 ω (7.4) ω 0 = 1/ LC 4/LC 4 1/ LC ω = 1 2 ( R 1 L + 2 ) R L LC (7.40) ω ω 0 = 1/ LC ω Q ω 2 ω 2 = 1 ( ) R R 2 2 L L LC (7.41) ωl R 1 = 1 (7.42) ωcr ω ( ω 1 ) ω 2 + R L ω 1 = 0. (7.43) LC ω = 1 ( 2 R ) R 2 L ± + 4 L LC. (7.44)

115 115 ω ± + ω ω = 1 ( 2 R ) R 2 L L LC. (7.45) ω 2 (> ω 0 ) ω ω 1 (< ω 0 ) ω = 1 2 ( R 1 L + 2 ) R L LC (7.46) ω 1/ LC ω Q ω 1 ω 1 = 1 ( 2 R ) R 2 L L LC (7.47) ω 1 ω 2 ω 2 ω 1 ω 2 ω 1 = R L (7.48) Q (7.15) Q = ω 0 ω 2 ω 1 = ω 0L R ω ω 0 ω 0 L = 1 ω 0 C (7.49) (7.50) Q = ω 0L R = 1 ω 0 CR. (7.51) (7.4) ω 0 = 1/ LC Q = ω 0L R = 1 R L C. (7.52) RLC Q R L C RLC Y p = ( 1 R 2 + ωc 1 ) 2 (7.53) ωl Z p = 1 Y p = 1 R 2 + ω 0 1 ( ωc 1 ωl ) 2. (7.54) ω 0 C 1 ω 0 L = 0 (7.55) Z p Z p0 = R (7.56) Q ω = ω 1 ω 2 Z p Z p0 = 1 2 (7.57) ω 1 ω 2 (ω 1 < ω 2 ) Q Z p / Z p0 Z p Z p0 = 1 ( 1 R R 2 + ωc 1 ) 2 ωl 1 = ( 1 + ωc 1 ) (7.58) 2 R 2 ωl ω ( R ωc 1 ) = ±1. (7.59) ωl ( R ωc 1 ) = +1 (7.60) ωl ω ω 2 LCR ωl R = 0 (7.61)

116 ω = 2LCR { } L ± L 2 + 4LCR 2 (7.62) ω 2 (> ω 0 ) ω ω 1 (< ω 0 ) ω ω ω ± + 1 ω = 2LCR { } L + L 2 + 4LCR 2. (7.63) ω Q ω 1 ω 2 ω (7.11) ω 0 = 1/ LC 4/LC 4 1/ LC ( ω = 1 ) 2CR CR LC (7.64) ω ω 0 = 1/ LC ω Q ω 2 1 ω 2 = 2LCR { } L + L 2 + 4LCR 2 (7.65) ( R ωc 1 ) = 1 (7.66) ωl ω ( ω 1 ) ( ω = 1 ) 2CR CR LC (7.70) ω ω 0 = 1/ LC ω Q ω 1 1 ω 1 = 2LCR { } L + L 2 + 4LCR 2 (7.71) ω 1 ω 2 ω 2 ω 1 ω 2 ω 1 = 1 CR (7.72) Q (7.15) Q = ω 0 ω 2 ω 1 = ω 0 CR (7.73) ω ω 0 ω 0 L = 1 ω 0 C (7.74) Q = ω 0 CR = R ω 0 L. (7.75) (7.11) ω 0 = 1/ LC C Q = ω 0 CR = R L. (7.76) ω 2 LCR + ωl R = 0. (7.67) 1 ω = 2LCR { } L ± L 2 + 4LCR 2. (7.68) ω ± + ω 1 ω = 2LCR { } L + L 2 + 4LCR 2. (7.69)

117 117 LC ( ) d f 0 = L[mH] C[pF] [MHz] (7.77) 7.23 () d d = 1 Q X (7.78) d 7.23 (a) cosθ = R R 2 + X 2 (7.79) R = X /Q X cosθ = = X 2 R 2 1 +Q 2 X 1 = 1 +Q 2 X 1 ( X X ) 2 (7.80) Q X 1 cosθ 1 = 1 (7.81) Q 2 Q X X 1/ 2 1/ 2 1/2 (full width at half maximum: FWHM) 1/2 1/ 2 FWHM 1/2 1/2 1/2 *6 FWHM 1/2 1/ 2 f (t) = a 0 + { a n cos(nω 0 t) + b n sin(nω 0 t) }. (7.82) n=1 f (t) = a 0 + A n cos(nω 0 t + ϕ n ). (7.83) n=1 A n = a 2 n + b 2 n, (7.84) ). (7.85) ϕ n = tan 1 ( bn a n *6

118 118 7 f(t) n = 0 n = 1 n = 1~2 t t t t n = 1~3 n = 1~4 n = 1~ t t t 7.28 n = 0 n = 1 n = 5 sin n = 0~10 t 7.27 sin f (t) = n= c n e jnω 0t. (7.86) c n = a n jb n = c n ϕ n, (7.87) 2 c n = A n 2 = a 2 n + b 2 n, (7.88) ( 2 ) ϕ n = tan 1 bn. (7.89) a n 7.26 f (t) = π 1 sin[(2n 1)πt] 2n 1 n=1 = π sin(πt) + 2 3π sin(3πt) + 2 sin(5πt) + (7.90) 5π 7.29 n = 0 n = 10 sin 7.29 f (t) = sin[(2n 1)πt] π n=1 2n 1 = a 0 + A n cos(nω 0 t + ϕ n ) (7.91) n= n = 0,1,2,3,4, n = 0 n = 10 A n = a 2 n + b 2 n = b n { 2/(nπ) (n = odd) = 0 (n = even) (7.92)

119 119 A n π f(t) 1 2 3π 2 5π t A n π 2 3π 2 5π 0 π 2π 3π 4π 5π 6π ω φ n 0 π 2π 3π 4π 5π 6π ω n 0 π 2π 3π 4π 5π 6π ω 0 90 π 2π 3π 4π 5π 6π ω [3] 7.30 ( ) ϕ n = tan 1 bn = a n { 90 (n = odd) 0 (n = even) (7.93) ω(= nω 0 ) A n ϕ n A n ϕ n 7.30 *7 (spectrum) 7.31 [3] (spectrum analyzer) Band-pass filter = Select desired-frequency component(s) Noise Reduc on or Signal Detec on Original Signal (ω 0 ) Noise ω 1 <ω 0 Noise ω 3 >ω 0 Another Signal ω 2 <ω 0 Signal +Noises Noise ω 4 >ω 0 Transmi ance High-Q Band-Pass Filter ω1 ω2 ω0 ω3 ω4 Noise Reduc on 7.32 * rad/s, 1000 rad/s, 1500 rad/s 1000 rad/s ( *7 *8

120 I1 (ω 1 = 500 rad/s) V1 (ω 1 = 500 rad/s) I2 (ω2 = 1000 rad/s) V2 (ω 2 = 1000 rad/s) I(ω) R L C V(ω) I3 (ω 3 = 1500 rad/s) Abs. impedance ( ) RLC Parallel Circuit Z R = 1 k L = 100 mh C = 10 uf Q = V3 (ω 3 = 1500 rad/s) Angular frequency (rad/s) I = I1 + I2 + I V = V1 + V2 + V Time (x10-3 s) Time (x10-3 s) 7.33 ) (1000 rad/s) 1000 rad/s Q 1000 rad/s AM *9 * rad/s 1000 rad/s ()

121 121 [1] R, L, C Z ω () ω 0 L, C Z = R + jωl + 1 jωc ( = R + j ωl 1 ). ωc ω 0 = 1 LC. ω = ω 0 () R [2] R, L, C Y ω ( ) ω 0 L, C Y = 1 R + 1 jωl + jωc = 1 ( R + j 1 ) ωl + ωc. ω 0 ω 0 = 1 LC. ω = ω 0 () 1/R

122 122 7 A. RLC 1. RLC RLC Z s ω ( Z s = R + j ωl 1 ). ωc B. RLC 1. RLC RLC Y p ω Y p = 1 ( R + j ωc 1 ). ωl 2. RLC RLC 2. RLC RLC ω 0 Z s ω ω 0 = 1 LC. 3. RLC Q RLC Q Q R, L C 1/ Z s ω 0 Q 1/ Z s 1/ 2 ω 1 ω 2 (ω 1 < ω 2 ) Q Q = ω 0 ω 2 ω 1. R L C Q = 1 R L C. ω 0 Y p ω ω 0 = 1 LC. 3. RLC Q RLC Q Q R, L C 1/ Y p ω 0 Q 1/ Y p 1/ 2 ω 1 ω 2 (ω 1 < ω 2 ) Q Q = ω 0 ω 2 ω 1. R L C C Q = R L. R R

123 123 [1] [2] [3]

124

125 125 8 () ( ) 1 2 V, I, N (primary) (secondary) 1 2 V 1 I 1 = V 2 I 2, V 2 V 1 = N 2 N 1. V 1 = ±jωl 1 I 1 ± jωmi 2, V 2 = ±jωmi 1 ± jωl 2 I 2, ± 8.1 () () 8.1 [1 3] 8.2 ( ) 8.1 (, transformer, ) [1 3] i 1 i flux Φ 2 v 1 primary secondary v 2 ferromagnetic core (a) i 1 M i 2 v 1 L 1 L 2 v 2 (b) 8.2 Load

126 126 8 () (p s ) ( ) V 2 V 1 = I 1 I 2 = N 2 N 1. (8.1) V 1 V 2 I 1 I 2 N 1 N kv 100 V *1 1 V 1 I 1 = V 2 I 2 (8.2) 8.3 () ( ) v 1 = L 1 di 1 dt + M di 2 dt, (8.3) v 2 = M di 1 dt + L 2 di 2 dt. (8.4) i M 1 i 2 v 1 L 1 L 2 v L 1 L 2 M di v 1 1 L 1 di 1 dt v 2 2 L 2 2 dt * 2 v 1 2 M di 2 dt v 2 1 M di 1 dt * (a) (b) : L k di k dt (k = 1,2) (8.5) *1 *2

127 8.4. (DOT CONVENTION) 127 i i i 1 i 1 v 1 t v 1 t i 1 i 1 Φ (a) Φ (b) i 2 i 2 v 2 v 2 M di 1 L di 2 = + dt 2 dt v 2 t v 2 M di 1 L di 2 = + dt 2 dt (a) ( ) () di 2 v 2 = L 2 dt + M di 1 dt t t t (8.6) 8.4 (b) () () di 2 v 2 = L 2 dt M di 1 dt M (8.7) ( ) 8.4 (Dot convention) J. W. Nilsson and S. A. Riedel [6] When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal. When the reference direction for a current leaves the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is negative at its dotted terminal. = = 8.5 = =

128 128 8 () 8.5 How to ( ) L 1 L 2 V 1 = ±jωl 1 I 1 ± jωmi 2, (8.8) V 2 = ±jωmi 1 ±jωl 2 I 2. (8.9) M 8.6 ( ) V 1 = ±jωl 1 I1 ±jωmi 2, (8.10) V 2 = ±jωmi1 ± jωl 2 I 2. (8.11) *3 () + *3 v v > 0 () () ( (> 0) )

129 (a) i 1 M i 2 v 1 L 1 L 2 v 2 di v 1 = 1 L 1 + dt di 1 v 2 = M + dt di 2 M dt di 2 L 2 dt Loss I M 1 I 2 (b) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I k High loss in magnetic flux Low loss in magnetic flux 8.7 jωmi jωmi k < 1 k 1 Loosely coupled Tightly coupled 8.9 () 8.6 di/dt ω i(t) I di = jωi (8.12) dt 8.7 (a) (b) 8.7 k V 1 = jωl 1 I 1 + jωmi 2, (8.13) V 2 = jωmi 1 + jωl 2 I 2. (8.14) k M k L 1 L 2 M = k L 1 L 2 ( k 1). (8.15) k < 1 M k k < 1 k < 1 k (a) () 8.10 (b) 8.10 (b) 8.10 (a)

130 130 8 () i 1 i 2 M I 1 I 2 (a) v 1 v 2 L1 L 2 (a) V 1 L 1 L 2 V 2 Z 2 M (b) I 1 I 2 L 1 M L 2 M V 1 M I 1 +I 2 V 2 I 1 (b) V 1 Z 1 = jωl 1 + ω 2 M 2 jωl 2 + Z (a) (b) 8.11 (a) (b) I M 1 I 2 V 1 = jωl 1 I 1 + jωmi 2, (8.16) V 2 = jωmi 1 + jωl 2 I 2. (8.17) (a) R 1 R 2 V 1 L 1 L 2 V 2 Z 2 V 1 = jω(l 1 M)I 1 + jωm(i 1 + I 2 ), (8.18) V 2 = jω(l 2 M)I 2 + jωm(i 1 + I 2 ). (8.19) (b) I 1 V 1 Z 1 = R 1 + jωl 1 + ω 2 M 2 jωl 2 + R 2 + Z (b) ( ) Z 1 = V 1 /I 1 V 1 = jωl 1 I 1 + jωmi 2, (8.20) V 2 = jωmi 1 + jωl 2 I 2, (8.21) V 2 = Z 2 I 2. (8.22) Z 1 = V 1 I 1 = jωl 1 + ω2 M 2 jωl 2 + Z 2. (8.23) 8.12 (a) (b) Z 2 = Ω Z 1 = jωl 1. (8.24) Z 2 = 0 Ω ) Z 1 = jω (L 1 M2. (8.25) M 2 = k 2 L 1 L 2 k = 1 Z 1 = L 2 ω 2 M 2 Z 1 = R 1 + jωl 1 +. (8.26) jωl 2 + R 2 + Z 2

131 8.11. () 131 (a) L M 1 L 2 L = L + L 2M (b) L M 1 L 2 L = L + L 2M [5] 8.14 (k ) M [4] 8.16 [7] 8.10 ( ) 8.13 M L + L M [4] 8.15 [5] 8.11 () 8.16 [7] 8.17 (a) (b)

132 132 8 () V 1 I 1 I 2 V 2 I 1 V 1 I 1 I 2 I 2 L 2 V 2 V 1 M V 2 L 1 I 2 I 1 L 1 +M V 1 I 2 V 2 L 1 L 1 +L 2 +2M (a) (b) I 2 I 2 J 2 L 2 W 2 M W 1 J 1 L 1 L 1 +M I 1 L 2 +M M V 2 V 2 L 1 +L 2 +2M V 1 L 1 +M I 1 V 1 L W 1 = V 1, (8.27) W 2 = V 2 V 1 (8.28) J 1 = I 1 I 2 (8.29) J 2 = I 2. (8.30) 8.18 W 1 = jωl 1 J 1 + jωmj 2, (8.31) W 2 = jωmj 1 + jωl 2 J 2. (8.32) V 1 = jωl 1 (I 1 + I 2 ) + jωmi 2, (8.33) V 2 V 1 = jωm(i 1 + I 2 ) + jωl 2 I 2. (8.34) V 1 = jωl 1 I 1 + jω(l 1 + M)I 2, (8.35) V 2 = jω(l 1 + M)I 1 + jω(l 1 + L 2 + 2M)I 2 (8.36) T 8.10 T 8.20 (b) 8.20 (a) V 2 = nv 1, (8.37) I 2 = I 1 n. (8.38) n n = N 2 /N 1 n L 1 L 2 M () 1 () R L C

133 M (a) L 1 L 2 =M 2 /L 1 L 1 :M 8.21 (b) L (c) M:L 2 L V 1 I 1 = V 2 n ni 2 = V 2I 2 (8.39) 8.22 (a) (b) L 1 L 1 : M (c) L 2 M : L 2 M (a) L 1 L 2 Z 1 = V 1 I 1 = 1 n 2 V 2 I 2 = 1 n 2 Z 2 (8.40) (b) L 1 M 2 /L 2 = (1 k 2 )L 1 M L 1 =M 2 /L 2 =k 2 L 1 L (c) L 2 M 2 /L 1 = (1 k 2 M )L 2 L 1 L 2 =M 2 /L 1 =k 2 L (a) (b) (c)

134 134 8 () L 1 M 2 /L 2 = (1 k 2 )L 1 M:L 2 (a) L 1 =M 2 /L 2 =k 2 L 1 (b) L 1 M 2 /L 2 M:L 2 L 2 M:L 2 (L 1 L 2 M 2 )L 2 /M 2 (c) L ( 8.23) (a) (b) (c)

135 (1a) (1b) 8.25 (a) 8.25 (b) +jωl 1 I 1 +jωl 2 I 2 +jωl 1 I 1 *4 jωl 2 I 2 +jωmi 2 +jωmi 2 V 1 = +jωl 1 I 1 +jωmi 2 (8.41) V 1 = +jωl 1 I 1 +jωmi 2 (8.43) +jωmi 1 jωmi 1 V 2 = +jωl 2 I 2 +jωmi 1 (8.42) V 2 = jωl 2 I 2 jωmi 1 (8.44) I 1 M I 2 (a) V 1 L 1 L 2 V 2 I 1 M I 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 (b) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 jωl 2 I (1) *4 ()

136 136 8 () (2a) (2b) 8.26 (a) 8.26 (b) +jωl 1 I 1 +jωl 2 I 2 +jωl 1 I 1 *5 jωl 2 I 2 jωmi 2 jωmi 2 V 1 = +jωl 1 I 1 jωmi 2 (8.45) V 1 = +jωl 1 I 1 jωmi 2 (8.47) jωmi 1 +jωmi 1 V 2 = +jωl 2 I 2 jωmi 1 (8.46) V 2 = jωl 2 I 2 +jωmi 1 (8.48) I 1 M I 2 (a) V 1 L 1 L 2 V 2 I 1 M I 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 (b) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 jωl 2 I (2) *5 ()

137 (3a) (3b) 8.27 (a) 8.27 (b) +jωl 1 I 1 *6 jωl 2 I 2 +jωl 1 I 1 +jωl 2 I 2 jωmi 2 jωmi 2 V 1 = +jωl 1 I 1 jωmi 2 (8.49) V 1 = +jωl 1 I 1 jωmi 2 (8.51) +jωmi 1 V 2 = jωl 2 I 2 +jωmi 1 (8.50) jωmi 1 V 2 = +jωl 2 I 2 jωmi 1 (8.52) I 1 M I 2 (a) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 jωl 2 I 2 I 1 M I 2 (b) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 + jωl 2 I (3) *6 ()

138 138 8 () (4a) 8.28 (a) 8.28 (b) ( 8.28 (a)) +jωl 1 I 1 +jωl 2 I 2 +jωmi 2 V 1 = +jωl 1 I 1 +jωmi 2 (8.53) +jωmi 1 V 2 = +jωl 2 I 2 +jωmi 1 (8.54) L 1, L 2, M V 2 = nv 1 (8.55) V 1 = jωl 1 I 1 + jωmi 2, (8.55) V 2 = jωmi 1 + jωl 2 I 2 (8.56) I 1 = V 1 jωmi 2 jωl 1 (8.57) (8.56) V 2 = jωl 2 I 2 + M L 1 V 1 jωm2 L 1 I 2 (8.58) (M = L 1 L 2 ) L1 L 2 V 2 = jωl 2 I 2 + V 1 jωl 1L 2 I 2 L 1 L 1 L 2 = V 1 = nv 1 (8.59) L 1 n = L 2 /L 1 *7 I 2 = I 1 /n I 1 = V 1 jωmi 2 jωl 1 (8.60) I 1 = V 1 jωmi 2 jωl 1 = V 1 jωl 1 M L 1 I 2 (8.61) (a) I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 I 1 M I 2 (M = L 1 L 2 ) n = L 2 /L 1 I 1 = V 1 L 2 I 2 jωl 1 L 1 = V 1 jωl 1 ni 2 (8.62) (b) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 L 1 I 1 = ni 2 (8.63) 8.28 (4) *7

139 n 0 n = L2 /L 1 L 1 L 2 L 2 M = L 1 L 2 M

140 140 8 () e = L di dt. (8.64) i L e ( ) (8.64) 8.29 v = L di dt (8.65) 8.30 (a) i dt di + (b) e (c) di e i e (d) i e (a) i dt (b) i+di Positive direction of the voltage should be like this, if the voltage of this element is regarded as electromotive force. (a) (b) (c) B i i B dt B+dB B+dB e i+di i+di di dt > 0 db dt > 0 di e = L dt Electromotive force to supress increase of i 8.29 Which is correct? (c) (d) i+di di e = L dt i+di di e = L dt If we write like this, e does not suppress increase of i. This does not correspond to the self-induction theory, in which e should suppress the increase of i. Since e should suppress the increase of i, e should be expressed like this. 8.30

141 141 (a) i di e = L dt The voltage is treated as electromotive force. (b) i di v = L dt The voltage is treated as voltage drop (a) e = L di dt (8.66) 8.31 (b) v = L di dt (8.67) ( ) : ± M di k dt (k = 1,2) (8.68) + [1] di 1 dt > 0 [2] +

142 142 8 () [1] di 2 dt > 0 [2] + M M 8.32 M M (a) (b) I 1 MI 2 V 1 L 1 L 2 V 2 I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jω( M)I 2 V 2 = jω( M)I 1 + jωl 2 I 2 V 1 = jωl 1 I 1 + jωm I 2 V 2 = jωm I 1 + jωl 2 I 2 M = M < (5) [6] 1. Arbitrarily select one terminal say, the D terminal of one coil and mark it with a dot. 2. Assign a current into the dotted terminal and label it i D. 3. Use the right-hand rule to determine the direction of the magnetic field established by i D inside the coupled coils and label this field ϕ D. 4. Arbitrarily pick one terminal of the second coil say, terminal A and assign a current into this terminal, showing the current as i A. 5. Use the right-hand rule to determine the direction of the flux established by i A inside the coupled coils and label this flux ϕ A. 6. Compare the directions of the two fluxes ϕ D and ϕ A. If the fluxes have the same reference direction, place a dot on the terminal of the second coil where the test current (i A ) enters. (In the Figure, the fluxes ϕ D and ϕ A have the same reference direction, and therefore a dot goes on terminal A.) If the fluxes have different reference directions, place a dot on the terminal of the second coil where the test current leaves.

143 143 B 8.33 [6] I

144 144 8 () [1] ( 1) 8.35 v 2 v 2 L 2 di 2 /dt L k di k /dt (k = 1,2) 1. (di 1 /dt > 0) ±Mdi 1 /dt 6. ( ) v 2 > v 2 v 2 > 0 v 2 +Mdi 1 /dt [2] ( 2) 8.35 v 2 v 2 v 1 i 1 Φ i 2 v 2 v 1 i 1 Φ i 2 v 2 i i 1 t v 2 M di 1 L di 2 = + dt 2 dt t t i i 1 t v 2 M di 1 L di 2 = + dt 2 dt t t 8.35 ( 1) 8.36 ( 2)

145 (di 1 /dt > 0) ±Mdi 1 /dt 6. ( ) v 2 > v 2 v 2 > 0 v 2 Mdi 1 /dt

146 146 8 () A V 1 V 2 I 1 I 2 jωm j1 Ω 4 Ω I 1 I V V 1 j8 Ω j5 Ω V 2 10 Ω (8.74) (10 + j5) I 1 = I 2 = (5 j10)i 2. (8.75) j (8.75) (8.73) j6 = (4 + j8)(5 j10)i 2 ji 2 = (100 j)i 2 100I j *8 I 2 = j6 100 = j0.06 = A ma. (8.75) V 1 = j8i 1 j1i 2, (8.69) V 2 = j1i 1 + j5i 2. (8.70) B V 1 V 2 I 1 I 2 3 (8.69), (8.70) 4 4 I 1 = (5 j10) j0.06 = j0.3 = = A ma. V 2 V 2 = 10I 2 = j0.6 = V. V 1 (8.71) V 1 = j6 4I 1 = j6 4 (0.6 + j0.3) = j4.8 = = V. j6 V 1 = 4I 1, (8.71) V 2 = 10( I 2 ). (8.72) (8.71) (8.69) V 1 (8.72) (8.70) V 2 j6 = (4 + j8)i 1 ji 2, (8.73) 0 = ji 1 + (10 + j5)i 2. (8.74) *8 A + B B A 1/100 A + B = A

147 147 [1] [2] [3] [4] ishida96/ih-aichi/isan_wo_aruku/1996/ _yosamisoshinsho.html [5] [6] J. W. Nilsson and S. A. Riedel: Electric circuits 9th Edition (Prrentice Hall, 2011) p.190. [7]

148

149 149 9 ( 1 ) ( 2 ) 9.2 ( 1 ) ( 2 ) (node) (branch) (loop) (graph) i 2 i 1 i 3 i 1 + i 2 + i 3 = i 4 i ( 1 ) Kirchhoff s current law (KCL) v 3 node branch e 4 v 2 v 1 + v 2 + v 3 = e 4 loop v (graph) (node) (branch) (loop) 9.3 ( 2 ) Kirchhoff s voltage law (KVL)

150 150 9 Z 1 Z 2 V 1 I 1 Z 3 I 2 V KVL KVL ( ) 1 I 1 2 I 2 () V (V > 0) ( ) + *1 *1 > 0 > KVL KVL 1 2 ( ) V 1 Z 1 I 1 + Z 3 (I 1 I 2 ) Z 3 1 I ( ) V 2 ( ) Z 2 I 2 + Z 3 (I 2 I 1 ) 2 1 KVL V 1 = Z 1 I 1 + Z 3 (I 1 I 2 ), V 2 = Z 3 (I 2 I 1 ) + Z 2 I 2. V 1 = (Z 1 + Z 3 ) I 1 + ( Z 3 ) I 2, V 2 = ( Z 3 ) I 1 + (Z 2 + Z 3 ) I 2. [ V1 V 2 ] [ Z1 + Z 3 Z 3 = Z 3 Z 2 + Z 3 ][ I1 I 2 (9.1) (9.2) ]. (9.3) [I 1, I 2 ] [ I1 I 2 ] [ = ]. (9.4)

151 I 2 V 1 Y 2 V 2 Y 1 Y 3 V a V ab = V a V b V b I ab = YV ab or I ab I ab = V ab Z I 1 V 0 = 0 V 9.6 a Y b KCL KCL () () 9.6 a ( V a ) b ( V b ) a b I ab a b V ab V a V b ab Y ( Z = 1/Y ) a (0 V) *2 I ab = Y V ab = Y (V a V b ), (9.5) I ab = V ab Z = V a V b Z (9.6) KCL KCL () * KCL (a) 9.8 (a) 9.7 (b) 9.8 (b) 1 ( 9.7 ) = 1 + 2

152 152 9 V 2 I 2 V 1 I 1 V 1 Y 2 V 2 Y 2 I 2 V 0 V 2 Y 1 Y 3 Y 1 I 1 V 0 = 0 V V 0 (a) (b) 9.7 ( 9.5) 1 V 1 I 2 KCL I 1 I 2 = Y 1 V 1 + Y 2 (V 1 V 2 ), I 2 = Y 2 (V 2 V 1 ) + Y 3 V 2. I 1 I 2 = (Y 1 + Y 2 ) V 1 + ( Y 2 ) V 2, I 2 = ( Y 2 ) V 1 + (Y 1 + Y 2 ) V 2. [ I1 I 2 I 2 ] [ Y1 + Y 2 Y 2 = Y 2 Y 1 + Y 3 ][ V1 V 2 (9.7) (9.8) ]. (9.9) [V 1,V 2 ] [ V1 V 2 ] [ = ]. (9.10) V 1 Y 2 V 2 V 1 Y 2 I 2 V 2 Y 1 Y 3 Y 3 I 1 V 0 (a) (b) V ( 9.5) 2 I 1 + ( I 2 ) I 2 = Y 1 (V 1 V 0 ) + Y 2 (V 1 V 2 ) 3 Y 1 (V 1 V 0 ) V 0 = 0 Y 1 V 1 2 ( 9.8 ) = 1 I 2 = Y 2 (V 2 V 1 ) + Y 3 (V 2 V 0 )

153 I 1 I 2 I 1 I 2 3 j10 Ω 40 Ω j20 Ω 40 0 V I 1 I V 9.9 = j j2 1 = j2 4 j2 = 2 j4 = , (9.19) 4 j2 5 4 j2 = 16 + j2 = , (9.20) 2 = j 4 j2 5 = j3 = (9.21) I 1 = 1 = = , (9.22) 40 0 = j10i 1 + ( j20)(i 1 I 2 ), (9.11) 50 0 = 40I 2 + ( j20)(i 2 I 1 ). (9.12) I 1 I 2 I 2 = 2 = = (9.23) I 1 I 2 I 1 = ( ) A, (9.24) I 2 = ( ) A (9.25) 40 = j10i 1 + j20i 2, (9.13) 50 = j20i 1 + (40 j20)i 2. (9.14) 4 = ji 1 + j2i 2, (9.15) 5 = j2i 1 + (4 j2)i 2. (9.16) [ 4 5 ] [ j j2 = j2 4 j2 ][ I1 I 2 ]. (9.17) [I 1, I 2 ] [ I1 I 2 ] [ j j2 = j2 4 j2 ] 1 [ 4 5 ]. (9.18)

154 V 1 V 2 V 1 V Ω 10 Ω V 1 V A 30 Ω j20 Ω j10 Ω = V V 1 V 2, 10 (9.26) 0 = V 2 V 1 + V j20 + V j10. (9.27) 150 = 4V 1 3V 2, (9.28) 0 = 2V 1 + 3V 2. (9.29) V 1 = 75.0 V, (9.30) V 2 = 50.0 V (9.31)

155 [ y1 y 2 ] [ a b = c d ][ x1 [x 1, x 2 ] x 2 ]. (9.32) x 1 = 1, (9.33) x 2 = 2. (9.34) = a b c d = ad bc, (9.35) 1 = y 1 b y 2 d, (9.36) 2 = a y 1 c y 2 (9.37) 3 3 y 1 y 2 y 3 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 x 1 x 2 x 3 [x 1, x 2, x 3 ] = 1 = 2 = 3 =. (9.38) x 1 = 1, (9.39) x 2 = 2, (9.40) x 3 = 3. (9.41) a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 y 1 a 12 a 13 y 2 a 22 a 23 y 3 a 32 a 33 a 11 y 1 a 13 a 21 y 2 a 23 a 31 y 3 a 33 a 11 a 12 y 1 a 21 a 22 y 2 a 31 a 32 y 3, (9.42), (9.43), (9.44) (9.45)

156 () MATLAB a i j ( 1) i+ j (i, j)- M i j M M M = a 1 j M 1 j + a 2 j M 2 j + + a n j M n j (9.46) M = a i1 M i1 + a i2 M i2 + + a in M in (9.47) j i n n

157

158 158 9 [1] M = { } 1 0 = { } = 2 { 1 + ( 1) } +3 { ( 1) + 1 } = 0 [2] [3] [4] 3 3 M M M =

159 159 A I 1 I 2 I 3 I 1 I 2 I Ω j1 Ω I 3 j1 Ω 1 Ω 1 0 V I 1 I V = 1 = 2 = 3 = 1 j 1 j j j j j j j 1 + j j 0 j 1 1 j 1 j 1 j j j j j j j j 0 = 1, = 2 + j3, = 3 + j2, = 1 + j. I 1 = 1 I 2 = 2 I 3 = 3 = ( j3.00) A, = ( j2.00) A, = ( j1.00) A = (1 j)i 1 + ( I 2 ) + ( j)( I 3 ), 1 90 = ( I 1 ) + (1 + j)i 2 + j( I 3 ), 0 = ( j)( I 1 ) + j( I 2 ) + (1 + j j)i 3. B I 13 1 Ω j1 Ω j1 Ω V 2 V 1 V 3 1 = (1 j)i 1 I 2 + ji 3, j = I 1 + (1 + j)i 2 ji 3, 0 = ji 1 ji 2 + I V I 12 1 Ω I V 1 j 0 = 1 j 1 j j j j j 1 I 1 I 2 I 3 I 1, I 2, I 3, 1, 2, I 12, 2 3 I 23, 1

160 I 13 I 12, I 23, I 13 I 1, I 2, I I 12, I 23, I 13 V 1, V 2, V 3 3 V 1, V 2, V 3 4 1, 2, 3 I 1, I 2, I 3 1 I 12 = I 1 I 3, (9.48) I 23 = I 2 I 3, (9.49) I 13 = I 3. (9.50) 0 = V 2 V 1 j + V V 2 V 3. j * 3 (9.56) 0 = jv 1 + V 2 + jv 3. (9.57) (9.58) (9.54) (9.55) (9.57) V 1, V 3 0 = j1 + V 2 + j( j), = j + V V 2 V 2 = j 1. V 1 = 1.00 V, V 2 = ( j1.00) V, V 3 = j1.00 V. 2 ( )=() ( ) 3 I 12 = V 1 V 2, (9.51) j I 23 = V 2 V 3, (9.52) j I 13 = V 1 V 3. (9.53) V 1 = 1, (9.54) V 3 = j. (9.55) 2 4 (9.51), (9.52), (9.53) I 12 = V 1 V 2 j 1 (j 1) = j I 23 = V 2 V 3 j (j 1) ( j) = j = 2 j j I 13 = V 1 V 3 1 = 1 ( j) = 1 + j. 1 = j2 1 j = 1 + j2, = 2 + j, (9.48), (9.49), (9.50) I 1 I 3 = 1 + j2, (9.59) I 2 I 3 = 2 + j, (9.60) I 3 = 1 + j. (9.61) *3 0 = V 1 + jv 2 V 3

161 161 (9.59) (9.61) I 1 = 1 + j2 + I 3 = 1 + j j = 2 + j3. (9.60) (9.61) I 2 = 2 + j + I 3 = 2 + j j = 3 + j2. I 1 = ( j3.00) A, I 2 = ( j2.00) A, I 3 = ( j1.00) A I 1, I 2, I 3

162

163 a. b. Linear 2-terminal circuit (a) Internal source ON (c) V o V o Internal source OFF (d) (b) V i I i = Z i 10.1 (a) (b) (c) V o (d) Z i Z i I i V i ( )1 1 1 ( ) (Thevenin) V o Z i V o Z i (Thevenin) 10.2 (a) 10.2 (b) OFF cd OFF 10.3 ( ) R i = 4 Ω (10.1)

164 Ω 1 Ω c 4 Ω c 32 V 12 Ω 2 A R L 30 V R L (a) d d R i c 10.5 V o R L (b) d Ω 1 Ω c 12 Ω R i d 10.3 OFF 4 Ω x 1 Ω c 32 V 12 Ω 2 A V o d 10.4 ON cd c 1 Ω (= ) c c () 1 Ω 2 A 1 Ω *1 cd V cd V o 1 Ω V cd V xd V o V xd d d V d 0 V xd V xd = V x V d = V x V x x = ( ) 2 = V x 32 4 V d = 0 + V x V d. (10.2) 12 V x = 30 V (10.3) V o 10.5 V o ON ( ) cd Ω c (Norton) I s Z i I s Z i *1

165 Linear 2-terminal circuit I s Z i 8 Ω c 4 Ω 2 A 5 Ω R L 12 V (a) Internal source ON I s (b) Internal source OFF I i 8 Ω (a) c d (c) V i I i = Z i (d) V i I s R i d R L (b) 10.6 (a) (b) (c) I s (d) Z i Y i = 1/Z i I s Y i I s Y i (Norton) 10.7 (a) 10.7 (b) OFF cd OFF 10.8 ( ) R i = 4 Ω (10.4) I s ON c d Ω c 4 Ω 5 Ω R i 8 Ω d 10.8 OFF 8 Ω 2 A I 1 4 Ω 12 V 5 Ω 8 Ω c I2 I s d 10.9 ON Ω cd c d I cd I s 5 Ω 10.9

166 c R i I 1 A 4 Ω R L (a) V o V R L d P = VI = R L I V o = 10 V R i = 50 2 I 2 I s 1 2 A 2 A I 1 = 2 A 2 12 = 4(I 2 I 1 ) + 8I 2 + 8I 2 (10.5) I 1 = 2 A I 2 I 2 = 1 A (10.6) () () () (a) (R i ) (R L ) R i = R L (10.7) (b) R L I 2 (W) R i = R L R L ( ) (a) R i R L (b) R i = 50 Ω R i = R L ( ) R i = 50 Ω R L R L 0 Ω 300 Ω R L (b) R L = R i (C L) Z i = R i + jx i, (10.8) Z L = R L + jx L (10.9) ( ) ( )

167 10.2. () 167 Z i = R i + jx i I I Z i = R i + jx i V o V Z L = R L + jx L V o Z L = R i jx i Z i = Z L R i 100% Variable capacitor Inductor Matching box (a) Z i SWR meter (b) Z i = Z L. (10.10) R i = R L X i = X L. (10.11) SWR Meter Matching box (c) Z L () Z i = R i + jx i (10.12) Z L = R i jx i (10.13) V o Z i + Z L = 2R i (10.14) 100% [1, 2] 100% () ( )

168 (a) [1] (c) π SWR (b) [2] SWR Standing Wave Ratio ( ) SWR / ( ) V S ON V R ON V S I S ON V R(1) OFF V R = V R(1) + V R(2) V R(2) ON OFF ON () ON(ON OFF) ( ) OFF OFF OFF I S

169 I (a) I V 1 I 2 1 V 2 (a) Z (b) E 1 I 1 I 2 I 2 I+ΔI (b) Z ΔZ (c) I 1 I 1 I 2 E 2 ΔI ( ) (c) Z IΔZ V o Z i V o Z i I s Y i I s Y i ( ) (a) 4 E 1 I 2 E 2 I 1 ( ) * I Z I OFF Z I I Z E 1 I 2 = E 2 I 1. (10.15) *2

170 KCL KVL () I L I L < I max RI R R L V L = E RI L E I L IL < I max J max R L R i I max E R L 1 V L E V L = E R i I L (10.16) I L R i R i I L I max ( ) ( ) E 1 E 2 R 1 R 2 I 1max I 2max V L = E 1 + E 2 (R 1 + R 2 )I L. (10.17) I L 2

171 I L I L (< 2I max ) R 1 I L E 1 R 1 I L I L (< I 1max ) I L < min(i 1max, I 2max ) RI L /2 E R I L /2 RI L /2 E R I L /2 R L V L R L V L = E 1 + E 2 R 2 I L R 1 I L (< I 2max ) R 1 I L R 2 I L I L /2(< I max ) I L /2(< I max ) E 2 I L I max I L = V L E 1 + E 2 = (10.18) R L R 1 + R 2 + R L R L I L I L I L 1 2 * ( ) *3 OFF * E R I max I L I L /2 I L /2 I max 2I max V L V L = E R I L 2 (10.19) * () *4 *5

172 I L = I 3 V R I R1 2 0 V R2 R 1 I L E 1 E 2 J 1 I 1 J 1 I L R 1 I L I 3 R L V L () J 2 J 2 I L I 2 R 2 I L I 0 E 1 E 2 = R 1 I 0 + R 2 I 0 (10.20) I 0 = E 1 E 2 R 1 + R 2 (10.21) E 1 = E 2 I 0 = 0 E 1 E 2 I 0 0 (R 1 + R 2 )I 2 0 ( ) () () I 1 = J 1, (10.22) I 2 = J 2. (10.23) 3 I 3 (I L ) 0 = R L I 3 + R 1 (I 3 I 2 ) + R 2 (I 3 I 2 ). (10.24)

173 J 1 J 2 I L 0 = R L I L + R 1 (I L J 1 ) + R 2 (I L J 2 ). (10.25) I L I L = R 1J 1 + R 2 J 2 R 1 + R 2 + R L (10.26) (R L = 0) I L = R 1J 1 + R 2 J 2 R 1 + R 2 (10.27) (R = R 1 = R 2 ) I L = J 1 + J 2. (10.28) pn pn ( ) I I (a) (a') V J V Dark Photo I I (b) V V I I (c) V V D V V D I I (d) V V D J V V J D pn pn (a) (a ) J *6 (b) (b) (a) *6

174 I I J J J J J V D a V D =0 =0 V Solar cell 1 J Solar cell 2 J I I V D V V D V Solar cell 1+2 I 2V D J V J J+ ΔJ J J J V D a V D =0 V =ΔJ Solar cell 1 J Solar cell 2 J+ ΔJ I I V D V V D V Solar cell 1+2 I 2V D J V (c) ( ) (c) (d) () J J [ ] [ ] J a *7 * (=V D ) V D J

175 a J V D J+ ΔJ J+ ΔJ J+ ΔJ J J V D V D I =0 =0 V Solar cell 1 J+ ΔJ Solar cell 2 J+ ΔJ I I V D V V D V J+ ΔJ Solar cell 1+2 I 2V D V

176 Z i = R i + jx i I V o V Z L = R i jx i ( ) Z i = R i + jx i Z L = R i jx i P max P max = R i I 2 = R i V o 2 Z i + Z L 2 (10.29) = V o 2 4R i (10.30) P o P o = 2P max (10.31) P max 1 R T T + R = P P = ( T) P max (10.33) = (1 R) P max (10.34) R = ρ 2 (10.35) T = 1 R P max 1 ρ 2 = 1 ρ ρ (10.36) = 1 Z L Z i Z L Z i Z L + Z i Z L + Z i (10.37) = (Z L + Z L )(Z i + Z i ) (Z L + Z i )(Z L + Z i ) (10.38) = P P max (10.39) P max P ρ = Z L Z i Z L + Z i. (10.32) ρ = 0 Z L = Z i P max

177 177 a I V Linear 2-terminal circuit V S1, V S2, I S1, I S2. (10.40) ab ab V = A 0 I + A 1 V S1 + A 2 V S2 + A 3 I S1 + A 4 I S2. (10.41) A i V = A 0 I + B 0, (10.42) B 0 = A 1 V S1 + A 2 V S2 + A 3 I S1 + A 4 I S2. (10.43) ab I = 0 V = B 0 V o = B 0. (10.44) I 0 OFF V S1 = V S2 = I S1 = I S2 = 0 (10.45) B 0 = 0 V = A 0 I. (10.46) V = Z i I. (10.47) V = A 0 I + B 0 V o Z i V = Z i I + V o. (10.48) b a Z i I V V o b V S1, V S2, I S1, I S2. (10.49) a I = C 0 V + C 1 V S1 + C 2 V S2 + C 3 I S1 + C 4 I S2. (10.50) C i I = C 0 V + D 0, (10.51) D 0 = C 1 V S1 + C 2 V S2 + C 3 I S1 + C 4 I S2. (10.52) ab V = 0 I = D 0 I s = D 0. (10.53)

178 V I a Linear 2-terminal circuit I s = D 0 I s = D b I a V Y i I s b () R L R i V o P = R L I 2 = R L ( V o R i + R L ) 2. (10.58) P R L (b) V I a b Y i I s R L P R L = R i P dp/dr L = 0 R L = R i P R L dp = V 2 R i R L o dr L (R i + R L ) 3. (10.59) dp/dr L = 0 R L = R i V 0 OFF V S1 = V S2 = I S1 = I S2 = 0 (10.54) D 0 = 0 I = C 0 V. (10.55) I = Y i V. (10.56) I = C 0 V + D 0 I s Y i I = Y i V + I s. (10.57) ( ) P = R L I 2 (10.60) V o 2 = R L Z i + Z L 2 (10.61) = R L V o 2 (R i + R L ) 2 + (X i + X L ) 2 (10.62) R L

179 179 X L (a) I V 1 I 2 1 V 2 P R L = 0, P X L = 0. (10.63) R L X L R L = R i, X L = X i (10.64) (b) E 1 I 1 I 2 I 2 (10.62) X L { } P = 2 V o 2 R L (X i + X L ) [ X L (Ri + R L ) 2 + (X i + X L ) 2] 2 (10.65) (c) I 1 I 1 I 2 E 2 P/ X L = 0 X L (10.65) X L + X i = 0 (10.66) R L X i X L *8 P ( ) X L = X i (10.67) P R L (10.62) X L = X i (10.62) P = R L V o 2 (R i + R L ) 2. (10.68) R L P/ R L = 0 R L = R i (10.69) R L = R i, X L = X i. (10.70) Z L = Z i (10.71) *8 R L = 0 P/ X L = 0 P = (a) I 1 = y 11 V 1 + y 12 V 2, (10.72) I 2 = y 21 V 1 + y 22 V 2 (10.73) y 12 = y 21 (10.74) (b) V 1 = E 1 V 2 = 0 I 2 = y 21 E 1 (10.75) (c) ] V 1 = 0 V 2 = E 2 I 1 = y 12 E 2 (10.76) y 12 = y 21 E 1 I 2 = E 2 I 1 (10.77)

180 180 10

181 181 [1] ( 1) P R L = R i V o R i P(R L ) = R L ( V o R i + R L ) 2. () [2] ( 2) P R L = R i X L = X i V o R i X i P(R L, X L ) = R L ( V o 2 (R i + R L ) 2 + (X i + X L ) 2 ). ( )

182 Ω V 1 6 Ω V 2 a A V 2 A 4 Ω V o b V o. 2. R L 12 V 6 Ω 6 Ω 2 A 4 Ω R L R i () R i R i ab () OFF ab () OFF ( ) OFF ( ) R i R i 4 Ω = 12 a b Ω 1.2 V o /( 1 R i = ) = 3 Ω. 12 V o ab V o V 1 V 2 2 = V = V 2 V V 1 V 2, 6 + V = 2V 1 V 2, (10.78) 0 = 2V 1 + 5V 2. (10.79) V 1 V 2 V o (10.78) (10.79) 24 = 4V 2 V 2 = V o = 6 V Ω a 6 Ω 6 Ω a 6 V R L 4 Ω R i b b R i.

183 183 3 Ω a 3 Ω 3 Ω a 6 V V L I L 3 Ω 18 V 4 A 6 Ω R L b b Ω 3 Ω a Ω R i R L = 3 Ω. b V L I L P Max = V L I L = V 2 L R L V L 3 Ω V L = = 3 V. P Max = V 2 L R L = 32 3 = 3 W B R L R i R i R i. 18 V 3 Ω V 1 3 Ω 4 A 6 Ω I s I s. R i ab OFF ab OFF OFF R i R i 6 Ω 3+3 = 6 Ω 1.1 I s /( 1 R i = ) = 3 Ω. 6 I s ab I s ab 6 Ω 6 Ω a b

184 a 3 Ω V 1 3 Ω a 5 A 3 Ω R L 18 V I 1 I 4 A s I s b b A a 3 Ω V L b I L 3 Ω V 1 4 = V V 1 = 15 V. + V ab I s I s = V 1 3 = 15 3 = 5 A R L = 3 Ω I L 5 A 3 Ω I L = 3 5 = 2.5 A I s I 1 18 = 3I 1 +? () V 1 18 = 3I 1 + V 1, V 1 = 3I s 4 A I s I 1 I s I 1 = 4. P Max = R L I 2 L = = = = 18.8 W

185 185 [1] [2]

186

187 Y, Z, K, H, G (a) ( ) 11.1 (b) I (a) V I I 1 I 2 (b) input output V 1 port port V Y 11.1 (b) I 1 = y 11 V 1 + y 12 V 2, (11.1) I 2 = y 21 V 1 + y 22 V 2. (11.2) [ ] [ ][ ] [ ] I1 y11 y 12 V1 V1 = = Y. (11.3) I 2 y 21 y 22 V 2 V Y Y 11.2 y 11 = I 1, (11.4) V 1 V2 =0 y 21 = I 2 V 1 V2 =0 y 12 = I 1 V 2 V1 =0 y 22 = I 2 V 2 V1 =0, (11.5), (11.6). (11.7) I 1 I (a)1 (b) Y Y 11.3 (a)

188 Y, Z, K, H, G I 1 I 2 I 1 I 2 = 0 I 1 V 1 y 11 = I 1 / V 1 y 21 = I 2 / V 1 V 2 = 0 V 1 z 11 = V 1 / I 1 z 21 = V 2 / I 1 V 2 I 1 I 2 I 1 = 0 I 2 V 1 = 0 y 12 = I 1 / V 2 y 22 = I 2 / V 2 V 2 I 2 V 1 z 12 = V 1 / I 2 z 22 = V 2 / I 2 V Y 11.5 Z (a) (b) I 1 I 2 V 1 y 11 y 12 V 2 y 21 V 1 y 22 V 2 I 1 I 2 y 12 V 1 y 11 +y 12 y 22 +y 12 V Y I 1 I 1a V 1a N a I 2a V 2a I Z 11.1 (b) V 2 V 1 = z 11 I 1 + z 12 I 2, (11.9) V 2 = z 21 I 1 + z 22 I 2. (11.10) [ ] [ ][ ] [ ] V1 z11 z 12 I1 I1 = = Z. (11.11) z 21 z 22 I 2 I 2 V 1 I 1b I 2b V 2 V 1b N b V 2b 11.4 Y 11.3 (b) Y Y 11.4 N a N b Y Y = N a + N b. (11.8) Z Z 11.5 z 11 = V 1, (11.12) I 1 I2 =0 z 21 = V 2 I 1 I2 =0 z 12 = V 1 I 2 I1 =0 z 22 = V 2 I 2 I1 =0, (11.13), (11.14). (11.15) Z z 11.6 (a) 11.6 (b)

189 11.4. K 189 I 1 I 2 z 11 z 22 I 1 I 2 (a) V 1 z 12 I 2 z 21 I 1 V 2 V 1 V 2 (b) I 1 I 2 z 11 z 12 z 22 z 12 V 1 z 12 V K 11.6 Z Z I 1 I 1a V 1a N a I 2a V 2a I K 11.9 V 1 I 1b I 2b V 2 V 1 = AV 2 + BI 2, (11.19) V 1b N b V 2b I 1 = CV 2 + DI 2. (11.20) 11.7 Z I 1 1:n I 2 V 1 V ( ) Z Z 11.7 N a N b Z Z = N a + N b. (11.16) Z 11.8 V 1 = 1 n V 2, (11.17) I 1 = ni 2. (11.18) (b) [ ] [ ][ ] [ ] V1 A B V2 V2 = = K. (11.21) C D I 1 I K K A = V 1, (11.22) V 2 I2 =0 B = V 1 I 2 V2 =0 C = I 1 V 2 I2 =0 D = I 1 I 2 V2 =0 I 2, (11.23), (11.24). (11.25)

190 Y, Z, K, H, G I 1 I 2 = 0 I 1 I 2 A = V 1 / V 2 V 1 C = I 1 / V 2 V 2 V 1 V 2 I 1 I 2 I 2 I 1 V 1 B = V 1 / I 2 D = I 1 / I 2 V 2 = 0 V 2 V K K I 1 V 1 I 1a V 1a A a B a C a D a I 2a I 1b V 2a V 1b A b B b C b D b I 2b I 2 V 2b V K K [ Aa B a N a = C a D a [ Ab B N b = b C b D b ], (11.26) ] (11.27) K [ Aa B a K = N a N b = C a D a K ][ Ab B b C b D b ]. (11.28) K K K [ V1 I 1 ] [ A B = C D ][ V2 I 2 ] [ V2 = K I 2 ] (11.29) [ V2 I 2 ] = 1 K [ D B C A ][ V1 I 1 ]. (11.30) K = 1 D A [ ] [ ][ ] V2 D B V1 =. (11.31) C A I ( 1) H 11.1 (b) I 1 V 1 = h 11 I 1 + h 12 V 2, (11.32) I 2 = h 21 I 1 + h 22 V 2. (11.33) [ ] [ ][ ] [ ] V1 h11 h 12 I1 I1 = = H. (11.34) I 2 h 21 h 22 V 2 V 2

191 11.6. ( 2) G 191 I 1 I 2 I 1 I 2 = 0 V 1 h 11 = V 1 / I 1 h 21 = I 2 / I 1 V 2 = 0 V 1 g 11 = I 1 / V 1 g 21 = V 2 / V 1 V 2 I 1 = 0 I 2 I 1 I 2 V 1 h 12 = V 1 / V 2 h 22 = I 2 / V 2 V 2 V 1 = 0 g 12 = I 1 / I 2 g 22 = V 2 / I 2 V H G I 1 I 2 h 11 I 1 I 2 g 22 V 1 h 12 V 2 h 21 I 1 h 22 V 2 V 1 g 11 g 12 I 2 g 21 V 1 V H G H H h 11 = V 1 h 21 = I 2 I 1 V2 =0 I 1 V2 =0 h 12 = V 1 V 2 I1 =0 h 22 = I 2 V 2 I1 =0, (11.35), (11.36), (11.37). (11.38) H H ( 2) G 11.1 (b) G I 1 = g 11 V 1 + g 12 I 2, (11.39) V 2 = g 21 V 1 + g 22 I 2 (11.40) [ I1 V 2 ] [ g11 g 12 = g 21 g 22 ][ V G I 2 ] [ V1 = G I 2 ]. (11.41) G g 11 = I 1 V 1 I2 =0 g 21 = V 2 V 1 I2 =0 g 12 = I 1 I 2 V1 =0 g 22 = V G I 2 V1 =0, (11.42), (11.43), (11.44). (11.45) G E, B, C

192 Y, Z, K, H, G I c I d V b I b B C E V c V g I g G D S V d (FET) i b h ie i c G i g =0 i d D v b h re v c h fe i b h oe v c v g g m v g v d S S FET I b i b I c h fe i c = h fe i b H v b = h ie i b + h re v c, (11.46) i c = h fe i b + h oe v c. (11.47) h h h ie : ( 6 kω) h re : ( ) h fe : ( 200) h oe : ( 8 µs) h fe i d = g m v g FET Y i g = 0, (11.48) i d = g m v g. (11.49) g m (Y y 21 ) FET FET FET y 11 ( ) y 12 y 22 y 21 V 1 = g m vg y S, D, G V g v g I d g m

193 193 [1] (Y ) I 1 = y 11 V 1 + y 12 V 2, I 2 = y 21 V 1 + y 22 V 2. (V 2 = 0) I 1 V 1 I 2 y 11 y 21 (V 1 = 0) I 2 V 2 I 1 y 12 y 22 I 1 I 2 V 1 y 11 y 12 V 2 y 21 V 1 y 22 V Y V 2 = 0 I 1 = y 11 V 1, I 2 = y 21 V 1. I 1 V 1 I 2 y 11 y 21 y 11 = I 1 V 1, y 21 = I 2 V 1. V 1 = 0 I 1 = y 12 V 2, I 2 = y 22 V 2. I 2 V 2 I 1 y 12 y 22 y 12 = I 1 V 2, y 22 = I 2 V 2. [2] (H ) V 1 = h 11 I 1 + h 12 V 2, (11.50) I 2 = h 21 I 1 + h 22 V 2. (11.51) V 1 h 11 I 1 h 12 V 2 h 11 I 1 I 1 h 11 h 12 V 2 V 2 h I 2 h 21 I 1 h 22 V 2 h 21 I 1 I 1 h 21 h 22 V 2 V 2 h I 1 I 2 h 11 V 1 h 12 V 2 h 21 I 1 h 22 V H

194 Y, Z, K, H, G A I 1, I 2 z [ ] 40 Ω j20 Ω I 1 = (2 0 ) A, I 2 = (1 90 ) A. B y j30 Ω 50 Ω 2I 1 I 1 I 2 I 1 8 Ω 4 Ω I V V 1 V 2 10 Ω V 1 2 Ω V z 12 z 21 V 1 = 40I 1 + j20i 2, V 2 = j30i I 2. (I 2 ) V 1 = 100, V 2 = 10I = 40I 1 + j20i 2, (11.52) 10I 2 = j30i I 2. (11.53) y 11 = I 1 V 1 y 21 = I 2 V V 0 I 1, I 2, V 1, V 2 V 0 y i j V 0 1 I 1 8 Ω 2I 1 I 1 2I 1 = I 1 = V V 0 2 = 3 4 V 0. (11.55) 8 Ω I 1 (11.53) I 1 = j2 I 2. (11.54) 2I 1 (11.52) I 1 8 Ω 1 V 0 4 Ω 2 I = j80i 2 + j20i 2 I 2 = 100 j100 = j. I 1 V 1 2 Ω V 2 =0 (11.54) I 1 = j2 ( j) = y 11 y 21.

195 195 I 1 2I 1 8 Ω 1 V 0 4 Ω 2 I 2 I 1 = 0 V 0 8 = V 0 8 (11.59) V 1 =0 2 Ω V 2 I y 12 y 22. () (11.55) I 1 = V 1 V 0. (11.56) 8 V 1 V 0 = V 0, V 1 V 0 = 6V 0, (11.56) V 1 = 5V 0. (11.57) I 1 = 5V 0 V 0 8 (11.57) = 0.75V 0. (11.58) y 11 = I 1 V 1 = 0.75V 0 5V 0 = 0.15 S. I 2 + 2I 1 = 0 V 0 4 = 0.25V 0. (11.58) I 1 V 0 I V 0 = 0.25V 0, I 2 = 1.25V 0. V 0 8 = V 0 V 2 + V 0 4 2, V 0 = 2V 0 2V 2 + 4V 0, 2V 2 = V 0 + 2V 0 + 4V 0 = 5V 0, V 2 = 2.5V 0. (11.60) y 12 = I 1 V 2 = V 0 / 8 2.5V 0 = 0.05 S. I 2 + 2I 1 = V 2 V 0. 4 (11.59) (11.60) I 1 V 2 V 0 I 2 V 0 4 = 2.5V 0 V 0. 4 I 2 = V 0 = 0.625V 0. y 22 = I 2 V 2 = 0.625V 0 2.5V 0 = 0.25 S. y 11 = 0.15 S, y 12 = 0.05 S, y 21 = 0.25 S, y 22 = 0.25 S. y 12 y 21 y 21 = I 2 V 1 = 1.25V 0 5V 0 = 0.25 S. y 12 y I 1 2I 1 = V 0 V 2 4 I 1 = V 0 V V 0 2, + V 0 2.

196

197 V 2 V 1 I 2 I 1 (12.1) V 2 /V 1 I 2 /I 1 ( ) θ v = ln θ i = ln input ( V1 V ( 2 I1 I 2 ) = ln ) = ln V 1 V 2 I 1 I 2 I 1 I 2 V 1 V 2 ( ) + jarg V1, (12.2) V ( 2 ) + jarg I1. (12.3) I 2 output (12.1) / 1 () () ( ) θ = α + jβ V 1 = V 1 e jϕ 1 V 2 = V 2 e jϕ 2 V 2 = e θ V 1 (12.4) V 2 = e α V 1, (12.5) arg(v 2 ) = e j(ϕ1 β). (12.6) 12.1 e α ( ) β

198 input (a) Z Z* Voltage (V) output (b) Z Z* Z Z* Phase (degree) 12.2 α + jβ (c) Z* Z* Z Z* Z Z 12.4 (a) (b) (c) 12.3 α+jβ (a) 12.4 (a) Z Z 12.4 (b) 12.4 (c)

199 I I I I (a) Z K1 V 1 ZK1 V 2 Z K1 Z K1 V 1 ZK1 V 2 Z K1 I I (b) Z K2 V 1 Z K2 V 2 Z K (a) Z K1 θ 1K Z K2 (1) Z K1 θ 2K Z K2 (2) Z K1 θ 3K Z K2 (3) Z K1 (b) Z K1 θ 1K + θ 2K + θ 3K Z K2 Z K () () (a) Z K1 = 1 [ ] (A D) ± (A D) 2 + 4BC 2C (12.7) 12.5 (b) Z K2 = 1 [ ] (D A) ± (D A) 2 + 4BC 2C (12.8) 12.7 D A e θ K = A + D 2 + (A + D) 2 4 1, (12.9) coshθ K = A + D. (12.10) Z K1 Z K2 θ K i Z K1 Z K2 θ ik 12.7 Z K1 (12.11)

200 Z K2 (12.12) θ ik (12.13) i Z K1 = Z, (12.14) Z K2 = Z (12.15) 12.4

201 201 K 12.8 V 1 = AV 2 + BI 2, (12.16) V 2 = CV 2 + DI 2, (12.17) Z K1 = V 2 I 2. (12.18) A B C D V 1 I 1 = Z K1 (12.19) (12.16) (12.17) V 1 = A V 2 + B, I 2 I 2 (12.20) I 1 = C V 2 + D. I 2 I 2 (12.21) (12.18) Z K1 = V 2 /I 2 V 1 = AZ K1 + B, I 2 (12.22) I 1 = CZ K1 + D. I 2 (12.23) I 2 A B C D V 1 = AZ K1 + B I 1 CZ K1 + D = Z K1 (12.24) Z K1 Z K1 = 1 [ ] (A D) ± (A D) 2 + 4BC 2C I 1 I 2 (12.25) AD BC = 1 (12.26) Z K1 = 1 [ ] (A D) ± (A + D) 2 4 2C (±) (12.27) [ ] [ ][ ] V1 A B V2 = C D I 1 I 2 (12.28) K [ ] [ V2 D B = C A I 2 ][ V1 I 1 ]. (12.29) A D Z K1 = 1 [ ] (D A) ± (D A) 2 + 4BC. (12.30) 2C AD BC = 1 (12.31) Z K1 = 1 [ ] (D A) ± (D + A) 2 4 2C (K ) (12.32) V 1 = AV 2 + BI 2, (12.33) I 1 = CV 2 + DI 2. (12.34) Z K1 V 1 A B C D V 2 Z K1 θ K I 2 e θ K I 1, (12.35) V 2 e θ K V 1 (12.36) 12.8 (K ) AD BC = 1 (12.37)

202 (12.34) (12.35) I 1 I 2 = C e θ K D V 2 (12.38) (12.33) V 1 = AV 2 + BC e θ K D V 2 (12.39) (12.37) V 1 = AV 2 + AD 1 e θ K D V 2 (12.40) V 1 (12.36) e θ K = V 1 V 2 = A + AD 1 e θ K D (12.41) (12.42) e θ K + e θ K = A + D, 2 2 (12.42) coshθ K = A + D 2 (12.43) x = e θ K (12.44) x 2 (A + D)x + 1 = 0 (12.45) x = e θ K = A + D 2 ( ) A + D (12.46) 2

203 203 (db) e log 10 B A (12.47) (B) 10 10log 10 B A (12.48) (db) 10log 10log 10log 10 P 2 P 1 (12.49) ( V2 V 1 ( I2 I 1 ) 2 = 20log 10 ) 2 = 20log 10 V 2 V 1, (12.50) I 2 I 1. (12.51) 10log 10 20log 10 P = RI 2 = V 2 R (12.52) 10 (db) 20log 10 V 2 V 1 20log 10 I 2 I 1 10log 10 P 2 P 1 (12.53) e (Np) Weber-Fechner ln V 2 V 1 (12.54) ln I 2 I 1 (12.55) 1 2 ln P 2 P 1 (12.56) Weber- Fechner 1,2,3, 1,2,4,8, 1,10,100,1000, Ernst Heinrich Weber ( ) *1 Gustav Theodor Fechner ( ) *2 Weber-Fechner db db db (db) (Np) 10 e ln *1 A German physician who is considered as a founder of experimental psychology. *2 A German experimental psychologyst.

204 ln natural logarithm nl logarithmus naturalis ln sinhθ = eθ e θ, 2 (12.57) coshθ = eθ + e θ. 2 (12.58) sinθ = ejθ e jθ, (12.59) j2 cosθ = ejθ + e jθ. (12.60) 2 10log 10 ( ) ( ) Pmax 10log 10 P (db) (12.61) P max P () Z G = R G + jx G E V 1 I 1 Z i I 2 Z L = R L + jx L 12.9 Z o V 2 K 12.9 Z i = V 1 I 1 (12.62) () Z o = V 1 I 1 (12.63) Z T = V 2 I 1 (12.64)

205 205 I 1 I 2 Z I1 (1) (1) (2) Z I1 V 1 Z I1 Z I2 V 2 Z I2 θ I (1) Z I1 (1) Z I2 (1) θ I (2) Z I1 (2) Z I2 (2) Z I2 (2) (a) Z I1 (1) (1+2) I 1 I 2 θ I (1) + θ I (2) Z I1 (1) Z I2 (2) Z I2 (2) Z I1 V 1 Z I1 Z I2 V 2 Z I2 (b) () () Z I1 Z I2 θ I Z (1) I2 = Z(2) I Z (1) I1 Z(2) I2 θ(1) I + θ (2) I (12.65) (12.66) AB Z I1 = CD, (12.65) BD Z I2 = AC. (12.66) (K ) [ V1 I 1 ] [ A B = C D ][ V2 I 2 ] (12.70) θ I V2 I 2 = e θ I V1 I 1. (12.67) e θ I = AD + BC, (12.68) AD cothθ I = BC. (12.69) Z I1 = V 1 I 1 [ V2 I 2 = AV 2 + BI 2 CV 2 + DI 2 = A(V 2/I 2 ) + B C(V 2 /I 2 ) + D = AZ I2 + B CZ I2 + D. (12.71) ] [ D B = C A ][ V1 I 1 ] (12.72)

206 Z I2 = DZ I1 + B CZ I1 + A. (12.73) CZ I1 Z I2 + DZ I1 AZ I2 B = 0, (12.74) CZ I1 Z I2 DZ I1 + AZ I2 B = 0. (12.75) (12.76) V 1 /V 2 I 1 /I 2 e θ I = AD + BD (12.85) Z I1 Z I2 = B C, (12.77) Z I1 Z I2 = A D. (12.78) A B C D AB Z I1 = CD, (12.79) DB Z I2 = CA. (12.80) V 1 V 2 = A + BI 2 V 2 = A + B Z I2 = A + B DB = A D CA ( AD + BD ) (12.81) I 1 I 2 = CV 2 I 2 + D = CZ I2 + D = C = D A DB CA + D ( AD + BD ) (12.82) V2 I 2 = e θ I V1 I 1 (12.83) e θ I = V 2 I 2 V 1 I 1 (12.84)

207 ON ON RL RC * R v(t) = Ri(t) (13.1) 13.2 RL 13.1 RL t = 0 S i(t) = i R (t) = i L (t) R v R (t) L v L (t) t = 0 R L v R (t) = Ri(t), (13.4) v L (t) = L d i(t). (13.5) dt v R (t) + v L (t) = E, (13.6) i R (t) = i L (t) = i(t). (13.7) L d i(t) + Ri(t) = E. (13.8) dt L C v(t) = L d i(t) (13.2) dt v(t) = 1 C i(t) dt (13.3) i(t) i(t) = E R τ = L R ( 1 e t/τ) (13.9) R L v R (t) = Ri(t) ( = E 1 e t/τ), (13.10) * II II II v L (t) = L d dt i(t) = Ee t/τ. (13.11) 13.2 τ (1 e 1 ) (= 0.63 = 63%)

208 v R v R S R i R i L L S R i R i C C E i v L E i v C 13.1 RL 13.3 RC Current (A) approx. 60% E = 1 V R = 10 Ω L = 100 mh Current (A) approx. 60% down E = 1 V R = 10 Ω C = 1000 µf 0.02 τ = L / R = 10 ms 0.02 τ = RC = 10 ms Time (s) Time (s) v C (t) Voltage (V) v R (t) E = 1 V R = 10 Ω L = 100 mh Voltage (V) approx. 60% of max. E = 1 V R = 10 Ω C = 1000 µf 0.2 v L (t) 0.2 v R (t) Time (s) Time (s) RL 13.4 RC RL τ = L R 13.3 RC 13.3 RC t = 0 S i(t) R v R (t) C v C (t) t = 0 R C v R (t) = Ri(t), (13.12) v C (t) = 1 i(t) dt. (13.13) C v R (t) + v C (t) = E, (13.14) i R (t) = i C (t) = i(t). (13.15) 1 C i(t) dt + Ri(t) = E. (13.16)

209 13.4. RLC 209 τ = RC i(t) = E R e t/τ (13.17) R C v R (t) = Ri(t) = Ee t/τ, (13.18) v C (t) = 1 i(t) dt C ( = E 1 e t/τ). (13.19) 13.4 τ v C (t) (1 e 1 ) (= 0.63 = 63%) RC τ = RC 13.4 RLC R L C R L C R R R L L L (open) (short) C C C ( ) (short) (open) 13.4

210 High freq. DC or Low freq. Large dv/dt or Large di/dt Small dv/dt or Small di/dt R 13.5 L High freq. C DC or Low freq. ω L Z L Z L = jωl (13.20) ω Z L = jωl ω Z L (13.21) ω (open) ω Z L = jωl ω 0 Z L 0 (13.22) ω (short) ω C Z C 13.6 ω Z C = 1 jωc ω Z C 0 (13.24) ω (short) ω Z C = 1 jωc ω 0 Z C (13.25) ω (open) * Z C = 1 jωc (13.23) *2

211 211 V D = 0.7 V D 1 V T1 V T2 V RL R L 13.4 D V 12 V (a) without a reserver capacitor V D = 0.7 V V T1 V T2 C 1 V RL 100 V 12 V (b) with a reserver capacitor R L * 3 * (a) (b) (a) V T2 (b) V RL w/o C 1 (c) V RL w C V 0 V 17 V V 0 V ripple 0 V 13.8 *3 choke a choke coil, a choking coil, a choke * C C 1 R L V RL V ( 12 V ) 13.8(b) 0.7 V 13.8(a) 0.7 V

212 V D = 0.7 V R 1 V D = 0.7 V L 1 D 1 V T1 V T2 C 1 C 2 V RL R L D 1 V T1 V T2 C 1 C 2 V RL R L 13.9 RC LC * (b) 13.7(b) 13.8(c) *6 (ripple) V * 7 V = V m f C 1 R L (13.26) f V m 13.8(b) V m = 16.3 V f = 60 Hz R L = 10 kω C 1 = 4700 µf V = 5.8 mv *5 0.7 V *6 * RC R 1 C 2 R 1 C 2 V V V = V X C2 R X 2 C2 (13.27) X C2 = 1 ωc 2 (13.28) V = 5.8 mv R 1 = 100 Ω C 2 = 1000 µf V = 1.5 mv V RL R 1 V RL R 1 L 1 = 10 H R 1 ωl 1 LC V = 4 µv

213 213 (a1) AC t (a2) AC t (a3) pulsated (a4) quasi-dc v(t) = L d i(t) (13.29) dt AC 100 V AC (N 2 /N 1 ) x 100 V t t R L V 1 : V 2 = N 1 : N 2 diode bridge di/dt (b1) AC 100 V AC t (b2) (a) AC-DC converter with a smoothing capacitor AC AC (N 2 /N 1 ) x 100 V t (b3) pulsated t (b4) DC t choking coil 1 V 1 : V 2 = N 1 : N 2 diode bridge *8 LC v in v out *9 *8 4 *9 () (b) AC-DC converter with smoothing capacitors and a choking coil (a) (b) LC B B B V CC = +10 V R 1 R V v in v in B v in 0 V v in B B 0 V h FE

214 V CC +10 V v in 100 uv 0 R 1 10 kω C 1 R kω +1.8V R C 3.6 kω R E 1 kω +6.04V +1.1V 0 V R L 100 kω B E C C 3 C 2 v out B v in C 1 C 1 v in B v in B C 1 B v in B (+1.8 V) v in C 2 C R L 0 V C R L C 2 C C R L R L 0 V C 3 E R E R E * 10 R E E R E B R E B E * 11 C 3 R E E R E R E R E *10 R E *11 R E

215 215 High freq. noise Low freq. signal (differential-mode noise) Low freq. signal High freq. noise (a) Normal-mode noise High freq. noise Low freq. signal R L (common-mode noise) 13.15(a) Low freq. signal High freq. noise (b) Reduction of normal-mode noise R L (a) (b) (Schaffner FN9222) [1] RL (normal-mode noise) 13.13(a) () 13.13(b) RL 13.13(b) R L 13.15(b) 13.16

216 High freq. noise High freq. noise Low freq. signal Low freq. signal floating Low freq. signal High freq. noise (a) Common-mode noise Low freq. signal floating High freq. noise Common-mode choke floating floating (b) Reduction of common-mode noise (a) (b) A B For common mode current (noise) A Flux is added. B Work as an inductor for both A and B, i.e. For differential mode current (signal) A Flux is canceled. B Does not work as an inductor R L R L Work for noise reduction. for both A and B, i.e. Work as simple lines CPU (central processing unit) (ultra large scale integrated circuits; ULSI) CPU CPU GPU (graphics processing unit) "0" "1" CPU CPU "0" "1" (metal oxide semiconductor field effect transistor (MOS FET)) ON OFF MOS FET () (Schaffner RB series) [1] Intel Core i7 [2]

217 217 Gate width, Z Polysilicon or metal Oxide n-type semiconductor 60 nm gate n-source Gate n-drain Metal source contact S L p-substrate (a) Gate, G n-type polysilicon Deposited insulator Metal source contact D 1.5 nm gate oxide nm MOS FET [3] SiO 2 n + d ox n + SiO 2 Source Drain Field oxide Channel region L p Silicon dioxide Channel length p-type body, B (b) D G B S (c) MOS FET (a) (b) (c) } MOS FET ( ) MOS FET ( ) 60 nm MOS FET [3] Feature size (nm) (1971) 8080 (1974) 8086 (1978) (1982) (1985) Pentium (1993) PowerPC 603 (1994) UltraSparc II (1997) Pentium 4 (2000) Xeon 5400 (2007) Core i7 (2009) Core M (2014) Year Clock frequency (MHz) CPU [4] * CPU [2] *12

218 第 13 章 過渡現象の基礎 218 v(t) input 1 level (a) 0 level T time v(t) output ( RC << T ) 1 level (b) 0 level 23.1 µm time v(t) output ( RC ~ T ) 1 level 図 IBM の多層配線の電子顕微鏡写真 [4] エッチ ングによって配線間の絶縁膜を除去した後の写真 (c) 0 level time 図 配線間容量による信号遅延が"0"/"1"情報伝達に 及ぼす影響 dielectric 処理を行うが 集積回路では トランジスタの ON/OFF metal によって変化する電圧信号を他のトランジスタ等に伝達 (a) (b) することによってこの情報処理を行う このとき 信号 伝達用の配線は 既に示したように 必ず図 に示 図 簡単化した多層配線断面の模式図 した構造になる t = 0 でこの回路の入力端子の電圧が 0 から E に変化した場合 入力端子側では t = 0 で論理 metal input 値が"0"から"1"に変化したことになる しかし 本章で dielectric output 学習したように この回路において 入力側の電圧が 0 から E に変化したとしても 出力側の端子間の電圧は すぐには E に到達せず 次式のように変化する R input v1 output C v2 ( ) v2 (t) = E 1 e t/τ. (13.30) ここで τ = RC である 即ち 入力端子側の信号の変 化が出力端子側に反映されるのに遅延時間が伴う この ような遅延のことを RC 遅延と呼んでいる delay E 1 level E v1 0 1 level v2 0 level t 0 0 level t 上記のような RC 遅延時間を伴う信号伝達回路の場 合 クロック周波数の周期 T が τ = RC よりも十分に大 きい RC T の場合には 図 13.25(b) に示すように 多少の遅れ時間を伴うが 出力側でも正常に"0"と"1"の 切り替えがなされる しかし 高周波数化によって T が 図 多層配線の基本構造の等価回路 RC に近づくと 図 13.25(c) に示すように 入力側の変 化が出力に反映されなくなる 即ち 情報処理デバイス とになる 但し 図 に示した回路のままでは解析 として機能しなくなる これが CPU のクロック周波 が困難である ここでは 上下 または左右で隣り合っ 数の頭打ちの原因である た二つの配線だけに注目する すると 図 のよう な回路 即ち 本章で学習した RC 直列回路となる 論理回路は "0"と"1"の情報をやりとりすることで情報 こうした頭打ちを打開するために各種の施策が実施さ れた その中で 現在の CPU に採用されている施策内 容を以下に紹介する RC 遅延については 電気回路に

219 219 passivation global Cu wire via low-k dielectric etch stop layer dielectric diffusion barrier Cu RC intermediate barrier/seed layer (Ta/TaN) R C local CMOS W plug isolation (STI, USG) p-si wafer buried oxide R R ρ S L R = ρs (13.31) L R L ULSI S ULSI [3] ρ * 13 * ρ = 2.8 µω cm Al Al Au (2.4 µω cm) Cu (1.7 µω cm) Ag (1.6 µω cm) Cu [5] RC [6] C C C C = ε rε 0 S d (13.32) ε r ε 0 d S ()

220 C S d ε r 4 SiO 2 (low-k ) d * 14 R C * 15 "1" () R τ = RC *14 C d *15 R C τ = RC C Delay time (ps) Cu/Low-k + Gate NMOS gate delay Al/SiO 2 + Gate Cu/Low-k RC delay Al/SiO 2 RC delay Feature size (µm) RC [7] 2 3 SiO 2 SiO Al (ρ = 2 µω cm) SiO 2 (ε r = 4) Cu (ρ = 3 µω cm) Low-k (ε r = 2) [7] RC Al SiO µm Cu ε r = 2 Low-k 0.2 µm 0.2 µm CPU CPU CPU CPU CPU

221 (a) 13.28(b) v(t) V m t 1 Δt ~ T t 2 ΔV T (a) t (a) V = V m f RC (13.33) v(t) V m Δt ~ T/2 t 1 t 2 ΔV (b) V = V m 2f RC (13.34) T/2 T (b) t f 13.29(a) [8] t 1 t 1 () t 2 t 2 () V C v(t) R L (a) C v(t) R L V (b) (a) (b) (a) (b) RC RC t = t 1 v = V m t = t 2 t = t t 1 RC ( v(t) = V m exp t ). (13.35) RC RC 2 v(t) = V m ( 1 t RC ( ) t 2 ) RC ( = V m 1 t ). (13.36) RC V t = t 1 t = t 2 T V = V m RC (13.37) f = 1/T V = V m f RC. (13.38)

222 (b) T T/2 V = V m 2f RC. (13.39) 10:1 C RC R RC Z Z = R 2 + X 2 (13.42) C 10: (a) R C 10:1 [9] X C = R = 0.1R (13.43) 10 Z = R 2 + (0.1R) 2 = 1.005R (13.44) R X C < R 10 (13.40) R Z = 1 = = 99.5%. (13.45) X C = 1 ωc (13.41) ω C Z R 10: (b) f = 20 Hz ω = 126 rad/s R = 2 kω (a) v AC C R C C > 40 µf (13.46) v DC short (b) v AC R 13.31(a) R C open (c) v DC R 10:1 * (a) (b) (c) X C < R 10 X C = 1 ωc *16 (13.47) (13.48)

223 223 (a) v AC v DC r E R C 1/X C Y = 10/R = = 99.5%. (13.52) 10.05/R (b) v AC (c) v DC r r E R E R open short Y C 10: (b) R f = 20 Hz ω = 126 rad/s R = 1 kω C (a) (b) (c) ω C r E r 10:1 C RC C RC Y Y = 1 R X 2 C (13.49) C > 80 µf (13.53) R r R R r * 17 v AC R v AC * 18 R v AC 10:1 X C = R 10 (13.50) Y = 1 R R 2 = 101 R R (13.51) *17 *18 R

224 q(t) [1] i(t) L d i(t) + Ri(t) = E. (13.54) dt t = 0 i(t) = 0 R L E t ( 0) 1 Ri E di = 1 dt. (13.55) L t 1 R R Ri E di = 1 L dt. (13.56) 1 R ln(ri E) = 1 t + ln K. (13.57) L K Ri E = Ke R L t (13.58) t = 0 i(t) = 0 E = K (13.59) i(t) i(t) = E R ( 1 e R L t). (13.60) R d dt q(t) + 1 q(t) = E. (13.62) C 1 dq = 1 dt. (13.63) 1 C q E R t C 1 C dq = 1 1 C q E R dt. (13.64) ( ) 1 C ln C q E = 1 t + ln K. (13.65) R K 1 C q E = t Ke RC (13.66) t = 0 q(t) = 0 E = K (13.67) q(t) ) q(t) = CE (1 e t RC. (13.68) i(t) = d dt q(t) i(t) i(t) = E R e t RC. (13.69) [2] i(t) = d dt q(t) i(t) Ri(t) + 1 C i(t) dt = E. (13.61) t = 0 q(t) = 0 R C E t ( 0)

225 225 [1] [2] CPU [3] S. Thompson et al.: 130 nm logic technology featuring 60 nm transistors, low-k dielectrics, and Cu interconnects, Intel Technol. J. 6 (May 2002) pp [4] [5] D. Edelstein et al.: Full copper wiring in a sub-0.25 µm CMOS ULSI technology, IEDM Tech. Digest (1997) pp [6] K. Ohashi et al.: On-chip optical interconnect, Proc. IEEE 97, (2009). [7] : ULSI, 68, (1999). [8] J. Millman and C. C. Halkias: Integrated Electronics: Analog and Digital Circuits and Systems (McGraw- Hill Kogakusha, Tokyo, 1972) pp [9] Albert Malvino and David Bates: Electronic Principles 8th Ed. (McGraw-Hill Edutation, New York, NY, 2016) pp

226

227 227 A i j j π/2 (90 ) e jθ = cosθ + jsinθ A.1 i j j 1 ( ) j j ( ) () π/2 j A sin(ωt +θ) A () θ () e jθ = cosθ + jsinθ (A.1) j e A.2 j 2 = 1 A = 5 (A.2) + = (A.3) ( ) = 5 A

228 228 A A.1 A.3 ( 2) ( 3) = A.2 A = 6 (A.4) = (A.5) ( 2) ( 3) = 6 (A.6) 2 3 = 6 A ( ) 20 2 ( ) 3 ( 2) ( 3) = 6 A ( ) 20 b a 0 A.4 () 2 ( ) 3 ( ) ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) A.3 A.4 a b A.3.1 a b a + b a + b 0a a b

229 A ab a y + b y b y a y a b a + b b θ a x b x a x + b x a φ θ A ab A.7 a 2 b a a A.6 A.8 A.5 a b (, ) (a x, a y ) (b x, b y ) a + b (a x + b x, a y + b y ) A.3.2 a b ab ab 0 1 ( ) ab 0 a ( ) b A.6 a b A.7 () a, b ( ) θ,ϕ ( ) ab a b θ + ϕ A.3.3 aa = a 2 a ( ) a0 a ( ) a A.8 0,1, a 0, a, a 2 0, a a 2

230 230 A j j A.9 j 1 j a + j a j A.10 j a + j A.12 j j = 1 a j π/2 j a A.11 j π/2 (90 ) ( π/2 (90 ) ) 0 1 A.11 j aj n n n n j j (3) j 2 aj a = j j 2 1 π (180 ) A.12 A.4 j j 2 = 1 (A.7) ( ) 1 A.9 j 1 π/2 (90 ) j A.4.1 j (1) a j a + j A.10 a j A.4.2 j (2) a j aj aj = a j = a θ + π/2 j = 1 + j

231 A jy j j sin θ e jθ cos θ θ 1 x + jy r ( cos θ + j sin θ ) re jθ x r e jθ exp(jθ) (A.10) e jθ = cosθ + jsinθ (A.11) j A.13 e jθ A.5 x + jy (A.8) j j = 1 j x ( )jy ( y ) ( ) ( ) () r θ r(cosθ + jsinθ) (A.9) sin cos ( 1) n sinθ = n=0 (2n + 1)! θ2n+1, (A.12) ( 1) n cosθ = (2n)! θ2n (A.13) n=0 e x e x ( 1) n = n! n=0 x n (A.14) e x x jθ (A.11) [3] (A.11) A.6 A.6.1 e x sin x cos x u = e x v = cos x w = sin x

232 232 A 0 u v w 1 u w v 2 u v w 3 u w v 4 u v w 0 0 *1 y = f (x) f f (x), y, d dx f (x), = dy dx cos +sin ( sin ) v + w *2 0 u v w v + w 1 u w v v + w 2 u v w v + w 3 u w v v + w 4 u v w v + w cos +sin *1 4 cos sin 4 j j 2 = 1 j 3 = j j 4 = j *2 cos sin 0 u v w v + w 1 u w v v w 2 u v w v w 3 u w v v + w 4 u v w v + w cos +sin sin cos j 4 j y = v +jw 0 u v w v + jw y 1 u w v w + jv jy 2 u v w v jw j 2 y 3 u w v w jv j 3 y 4 u v w v + jw j 4 y z = e kx z y 0 z y 1 kz jy 2 k 2 z j 2 y 3 k 3 z j 3 y 4 k 4 z j 4 y k = j e jx = cos x + jsin x v+jw

233 A.7. E Jθ 233 (A.11) A.7 e jθ e jθ ( ) (A.11) A.7.1 f (x) = a x x a x a x a x x 0.5 x 0 a x a 0 = a x a 0 = 1 x a x a x = a 0 = 1 a x = 1/a x x m/n () a m/n = ( n a ) m x x m/n x a m/n a x x () A.7.2 f (x) x f (x + y) = f (x)f (y). (A.15) x a (a 0) f (x) = a x (A.16) f (x) (f (x)) f (x) (A.15) (A.15) (x + y) f (x)f (y) f (x) = a x a () a

234 234 A A.7.3 f (x + y) = f (x)f (y) (A.15) (A.15) f (x) f (x) = a x a (A.15) x f (x) 1 y = 1 f (x + 1) = f (x)f (1) (A.17) 1 1/n n f (1) = f ( 1 n + 1 ) n + = f f (1) = a ( ) 1 n (A.22) n ( ) 1 f = n a = a 1 n (A.23) n n x = m/n x 1/n m ( m f n ) ( 1 = f n + 1 ) n + = f ( ) 1 m = ( n a ) m n (A.24) x f (1) f (x) = a x f (1) = a (A.18) a x x x 1 x f (x) = f ( ) = f (1)f (1) = f (1) x = a x (A.19) x m/n x f (m/n) f (x) x (A.15) f (x) (A.16) f (x) x 0 0 a a 0 = 1 (A.15) f (0) = 1 y = 0 f (x) = f (x + 0) = f (x)f (0) (A.20) f (x) f (0) = 1 (A.21) A.7.4 f (x + y) = f (x)f (y) y = f (x) x x + x x x y y y/ x x x x 0 () (A.15) f (x) f (x) f (x) x x 1 1

235 A.8. E X 235 y y y f(x + 1) = f(x)f(1) f(x) slope f(x) { f(1) 1 } f(x + Δx) = f(x)f(δx) f(x) x x x Δx Δy Δx x x + Δx slope Δy Δx x Δx 0 f(x) x slope dy dx x A.14 f (x) y y x = f (x + 1) f (x) 1 = f (x) {f (1) 1} (A.25) f (1) 1 f (x) f (1) 1 1 a = f (1) = 1 + r r r x x 0 x f (x) f (x) (= dy/dx) f f (x + x) f (x) (x) = lim x 0 x f (x)f ( x) f (x) = lim x 0 x = f (x) lim x 0 f ( x) 1 x f f ( x) 1 (0) = lim x 0 x k f (x) = k f (x) f (x) f (x) (A.26) (A.27) (A.28) (A.29) (A.30) f (x) (A.26) (A.28) (A.15) (A.30) f (0) = 1 (A.21) e kx (A.30) k = j e jθ k = 1 (e x ) e A.8 e x (A.30) k = f (0) = 1 f (x) a x a a *3 f (0) = 1 f ( x) 1 a x 1 lim = lim = 1 (A.31) x 0 x x 0 x a a( x) x 1 = 1 (A.32) x a( x) a( x) x 0 a a *3 e e

236 236 A n = n = 10 n = 5 n = 3 n = 2 n = ( A ) m (m = 0,1, n) n = 2,3,4,5,10,100 n ( n ) m n e n e a( x) = (1 + x) 1/ x a = lim a( x) = lim (1 + x)1/ x (A.34) x 0 x (A.33) x 1/n x 0 n ( a = lim ) n (A.35) n n n a e ( e = lim ) n = (A.36) n n n A.15 n e f (x + y) = f (x)f (y) f (0) = 1 e f (x) = e x (A.37) (A.36) 1/n x/n ( 1 + x ) n (A.38) n lim n n = kx k ( lim 1 + x ) n = lim n n k = lim = k { lim k ( ) kx k { ( ( k ) k } x k ) } k x = e x (A.39) (A.38) e x e x = lim (1 + x ) n (A.40) n n OK e x 1 + x/n n n e A.9 e jθ e (A.15) (A.30) (A.40) x (A.30) f (x) = k f (x), f (0) = 1 (A.41) f (x) = e kx (A.42)

237 A.9. E Jθ 237 x (A.41) k j g (θ) = jg(θ), g(0) = 1 (A.43) g(θ) e jθ A.9.1 (A.43) g(θ) = cosθ + jsinθ (A.44) (A.43) g (θ) = sinθ + jcosθ = jg(θ) (A.45) (A.43) k = j g(0) g(0) = cos0 + jsin0 = 1 (A.46) (A.43) g(θ) e jθ e jθ = cosθ + jsinθ (A.47) A.16 e jθ 1 θ e jθ j e j sin θ jθ θ 1 cos θ 1 j A.16 e jθ cos sin A.9.2 cos sin (1) (A.41) (A.43) 1 k 1 j x θ f (x) g(θ) t f (t) = d f (t) = 1f (t), f (0) = 1 (A.48) dt g (t) = d g(t) = jg(t), g(0) = 1 (A.49) dt f (t) g (t) f (t) g(t) k = 1 : f (t) A.17 f (0) = 1 f (t) = f (t) k = j :

238 238 A 0 f(0)=1 f'(0)=1 f(t) f'(t)=f(t) A.17 f (t) = 1f (t) 0 g'(t) = j g(t) g(t) g'(0) = j g(0) = 1 A.18 g (t) = jg(t) j g(t) 0 g(t) g (t) j 1 A t ( ) t g(t) cos t sin t A.9.3 cos sin (2) k 1 j k k = 1 f (t) = e t = ( e 1) t e 1 k = j g(t) = e jt = ( e j) t e j k = f (0) f (t) k = 1 f ( t) 1 lim = 1 (A.50) t 0 t f (t) = a t a ( a = lim ) n (A.51) n n a e = e 1 e t = ( e 1 ) t 1/n t/n ( e t = lim 1 + t ) n (A.52) n 0 n e t e t 1 + t/n n n k = j g( t) 1 lim = j (A.53) t 0 t g(t) = b t b ( b = lim 1 + j ) n (A.54) n n b e j e jt = ( e j ) t A.19 ( 1 + j n ) m, m = 0,1, n (A.55) n = 2,3,5,10,100 n = j 1 (= ) 1 n = j 1/ ,1,1 + j/2 n = j 1/3 3 3

239 A.9. E Jθ n = 1 n = 2 n = e jt ( e jt = lim 1 + jt ) n (A.56) n n n = 5 n = 10 n = t cos t sin t ( A j ) m (m = 0,1, n) n = 2,3,5,10,100 n ( n 1 + j ) m n 1 n 1 1 n = 4 n = 5 n 1 1 1/n 1 1/n n n A.9.4 e x (A.40) x n f (x) = n! n=0 (A.57) e x e x e x (A.57) e x x x e jθ cos sin e jθ cos sin e jθ cos sin e jθ

240 240 A cos sin e jθ reexp θ imexp θ

241 241 [1] ( ) (1959, ). [2] ( ) (1960, ). [3] (2000, ). [4] (2012, ).