Bài giảng Electric circuit theory - Chapter X: Sinusoidal Steady State Analysis - Nguyễn Công Phương

Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits

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Nguyễn Công Phương Electric Circuit Theory Sinusoidal Steady-State Analysis Contents I. Basic Elements Of Electrical Circuits II. Basic Laws III. Electrical Circuit Analysis IV. Circuit Theorems V. Active Circuits VI. Capacitor And Inductor VII. First Order Circuits VIII.Second Order Circuits IX. Sinusoid and Phasors X. Sinusoidal Steady State Analysis XI. AC Power Analysis XII. Three-phase Circuits XIII.Magnetically Coupled Circuits XIV.Frequency Response XV. The Laplace Transform XVI.Two-port Networks Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 2 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 3 Sinusoidal Steady-State Analysis (1) 10sin5t V20Ω 6H 0.02F – + di 1 20iidtt 6  10sin 5 i dt 0.02  o 1 10 0 j30 di d2 i i 20 j0.1 20 6 50cos5t dt dt 2 0.02 – + iIm sin(5 t ) 100Itmm cos(5 ) 150 It sin(5 ) I 50Im sin(5tt ) 50cos5 o 10 0 o oo I   0.35  45 A 2 2It sin(5 135 ) sin(5 t 90 ) 1 m 20j 30  j0.1 22Im  1 Im  0.35     oo 45o  135 90  it0.35sin(5  45o ) A it0.35sin(5  45o ) A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 4 Sinusoidal Steady-State Analysis (2) 10sin5t V20Ω 6H 0.02F – + i o 1 1. Transform to phasor 10 0 j30 20 j0.1 domain – + 2. Solve the problem using I dc circuit analysis 10 0o I   0.35  45o A 3. Transform the resulting 1 20j 30 phasor to the time- j0.1 domain. it0.35sin(5  45o ) A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 5 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 6 Ohm’s Law (1) V VI R R R R I V V VI jL L jL  Z VZI L I I I V 1 V  C Z: impedance (Ω) C jC I jC 1 Admittance (S): Y  Z Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 7 Ohm’s Law (2) V  Z I V 1 R  R Z R Y  I R R R V 1  j L  jL Z jL Y  I L L jLL V 1 1  j C  Z  Y  jC I jC C jC  C C Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 8 Ohm’s Law (3)  j Z  jL Z  L C C Z  0 Z    0 L C Short circuit Open circuit Z  Z  0    L C Open circuit Short circuit Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 9 Ohm’s Law (4) I Z +–V Z  R  jX R: resistance X: reactance Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 10 Ohm’s Law (5) L C 1 Z jL jC Z 0 1112LC If jL00  jC jC LC 1 L jL jC LC/ Z  11 jL jL C jC jC Z 1112LC If jL00  jC jC LC Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 11 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 12 Kirchhoff’s Law (1) vv12 ... vn  0 VtVtmm1122sin(  ) sin( ) ... Vt mnn sin( ) 0 VV12... Vn 0 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 13 Kirchhoff’s Law (2) ii12 ... in 0 ItItmm1122sin(  ) sin( ) ... It mnn sin( ) 0 II12... In 0 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 14 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 15 Impedance Combinations (1) a a Z1 Z2 ZZZeq  12 b b Z1 VV1  ab ZZ12 ZZZeq 12... Z n Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 16 Impedance Combinations (2) a a ZZ12 Z1 Z2 Zeq  b b ZZ12 Z2 II1  ab ZZ12 111 1 ... ZZZeq12 Z n Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 17 Impedance Combinations (3) Zc a b Z 1 Z2 Zb Za Z3 ZZbc ZZ ZZ ZZ Z  12 23 31 1 Za  ZZZabc Z1 c ZZca ZZ12 ZZ 23 ZZ 31 Z2  Zb  ZZZabc Z2 ZZ ZZ ZZ ZZ Z  ab 12 23 31 3 Zc  ZZZabc Z3 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 18 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 19 Branch Current Method (1) Ex. 1 I1 Z1 abI3 Z3 + – + – I V1 – I2 V3 4 + V + 1 Z – 2 E1 A + B V4 Z4 J + – E2 – c 1. Find: I1 + I2 – I3 = 0 nKCL = n – 1, and nKVL = b – n + 1 I3 – I4 = –J 2. Apply KCL at n nodes KCL Z1I1 – Z2I2 = E1 – E2 3. Apply KVL at n loops KVL Z I + Z I + Z I = E 4. Solve simultaneous equations 2 2 3 3 4 4 2 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 20 Branch Current Method (2) Ex. 2 Z2 E + – b :0IIIc 23 I2 βI1 c :0IIJ13 ab A:0ZI ZI ZI E 11 33 2 2 Ic Z1 I1 I Z3 II  3 c 1 J c III1230  IIJ13 0  ZI11 ZI 33 ZI 2 2 E 0 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 21 Ex. 3 Branch Current Method (3) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – Z2 J Z3 IIIJ1230 ZI11 ZI 2 2 E 1 ZI22 ZI 33 E 3 o  II12 I 32  30   10II12j 20 30  o  jj20II23 (5 10)  45 15  II123;; I 12 3 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 22 Ex. 3 Branch Current Method (4) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – o Z2 J Z3  II12 I 32  30  10II12j 20 30  o  jj20II23 (5 10)  45 15  II123;; I 12 3 11 1  j20 0 1 1 1 1 10 j 20 0 1100 jjjjj20 5 10 20 5 10 20 0 020510j  j  250j 200 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 23 Ex. 3 Branch Current Method (5) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – o Z2 J Z3  II12 I 32  30  10II12j 20 30 123  o II12;; I 3  jj20II23 (5 10)  45 15   2 30o 1 1 30 j 20 0 45 15o j 20 5 j 10 I   1.04 j 3.95  4.09 75.2o A 1 250j 200 o it1 4.09sin(  75.2 ) A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 24 Ex. 3 Branch Current Method (6) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – o Z2 J Z3  II12 I 32  30  10II12j 20 30 123  o II12;; I 3  jj20II23 (5 10)  45 15   12 30o 1 10 30 0 04515o 5 j 10 I   1.98 j 0.98  2.20 26.4o A 2 250j 200 o it2 2.20sin(  26.4 ) A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 25 Ex. 3 Branch Current Method (7) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – o Z2 J Z3  II12 I 32  30  10II12j 20 30 123  o II12;; I 3  jj20II23 (5 10)  45 15   11 2 30o 10 j 20 30 02045j 15o I   4.75 j 3.93  6.16 39.6o A 3 250j 200 o it3 6.16sin(  39.6 ) A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 26 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 27 Node Voltage Method (1) Ex. 1 I1 Z1 abI3 Z3 1. the reference node – – + V1 I2+ V3 I4 2. the sum of the reciprocals – + V + of all impedances 1 Z – E 2 V Z J connected to each node 1 + 4 4 + – 3. the negative sum of the – reciprocals of the E2 impedances of all branches c joining each pair of node 111 1 EE 4. current source(s) for each VV  12 node ab ZZZ123   Z 3  ZZ 12 5. node voltage equations  111   6. node voltages  VVJ  ab   7. branch currents  ZZZ334   Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 28 Node Voltage Method (2) 11 1 E Ex. 2 Z2 E VVJIabc + – ZZ12 Z 2 Z 2  I 111 E 2 βI  VVI 1  abc ab  ZZZZ2232 I V Z I c Z a 1 1 I3 3 IIc 1 Z1 J c 11 1 1 E  VVJab   ZZ12 Z 1 Z 2 Z 2   11 11 E   VV    ab  ZZ21 ZZ 23 Z 2 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 29 Ex. 3 Node Voltage Method (3) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – J o Z2 Z3 11 1 30 o 45 15 Va 2 30 10jj 20 5 10 10 510 j Va 19.57 j 39.50 V  30 (19.57j 39.50) o I1 1.04j 3.95 4.09 75.2 A  10 it4.09sin( 75.2o ) A  1  (19.57j 39.50) o  o  I2 1.98j 0.98 2.20 26.4 A it2 2.20sin(  26.4 ) A j20  it6.16sin( 39.6o ) A  o  3 45 15 (19.57j 39.50) o I3  4.75j 3.93  6.16 39.6 A  510 j Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 30 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 31 Mesh Current Method (1) Ex. 1 I1 Z1 abI3 Z3 + V – I + V – I 1 – 2 3 4 J + V + 1 Z – 2 V Z I E1 IA + IB 4 4 C + – E2 – c Z1IA + Z2(IA – IB) = E1 – E2 Z2(IB – IA) + Z3IB + Z4(IB + J) = E2 (Z1+ Z2) IA – Z2IB = E1 – E2 – Z2IA + (Z2 + Z3 + Z4)IB = E2 – Z4J Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 32 Mesh Current Method (2) Ex. 2 Z2 E + – ZI12ADB ZI ZI 3 E 0 I2 ID βI1 IIBA  J ab II  I I BD c Z I c Z 1 1 IA I3 3 IB IIc   1 J c ()()ZZZI123A  EZZJZI 23  21 IIA   1 ()()ZZZ123 ZIEZZJ 21  23 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 33 Ex. 3 Mesh Current Method (3) ZZ10 ; jj 20  ; Z  5 10  ; Z1 I1 I3 12 3 – + o o E E1330V;E 45 15 V;J  2  30 A; 1 I2 E3 + Find currents? – I A I IB C J  o Z2 Z3 10IIIAABj 20( 2 30 ) 30  o o  j20(IIBA 2 30 ) (5j 10)IB 45 15  o  (10jj 20)IIAB 20 30 j 20 2  30 IA 1.04j 3.95 A     o o I 4.75j 3.93 A jjj20IIAB (5 20) 20 2 30 45 15  B II1.04 j 3.95  4.09 75.2o A o 1 A it1 4.09sin( 75.2 ) A    o it2.20sin(  26.4o ) A  IIIJ2 AB   2.20 26.4 A  2  o  o it6.16sin( 39.6 ) A II4.75 j 3.93  6.16 39.6 A  3  3 B Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 34 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 35 Superposition Theorem (1) Ex. 1 e = 10sin10t V; j = 4sin(50t + 30o) V; e = 6 V L + 1 2 + R1 R2 – (DC); L = 1 H; R1 = 1 Ω; R2 = 5 Ω; C = 0.01 F; – vR1 = ? C e1 j e2 – – + – + Step 1 v Step 2 v Step 3 + v R1 e2 R1 e1 R1 j L L L R1 R2 + + R1 R2 R1 R2 – – C C C e2 e1 j v  ? v  ? v  ? R1 e2 R1 e1 R1 j Step 4: vv  vv  RR11eej21 R 1 R 1 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 36 Superposition Theorem (2) Ex. 1 e = 10sin10t V; j = 4sin(50t + 30o) V; e = 6 V L + 1 2 + R1 R2 – (DC); L = 1 H; R1 = 1 Ω; R2 = 5 Ω; C = 0.01 F; – vR1 = ? C e1 j e2 Step 1 – v + R1 e2 L 6 R1 R2 + – i 1A e2 15 C e2 vR1  ? v 11 1V e2 R1 e2 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 37 Superposition Theorem (3) Ex. 1 e = 10sin10t V; j = 4sin(50t + 30o) V; e = 6 V L + 1 2 + R1 R2 – (DC); L = 1 H; R1 = 1 Ω; R2 = 5 Ω; C = 0.01 F; – vR1 = ? C e1 j e2 + V – R1 e1 Step 2 + v – R1 e1 j10 5( j 10) + 1 5 Z j10 1 L – 510 j + R1 R2 o – –j10 o 10 0V 5j 8  9.43 58  C e1 E 10 0 I 1 1.06  580 A v  ? R1 e1 o R1 e1 Z 9.43 58 VIR 11.06 58o 1.06  58o V RR111ee11 vt1.06sin(10  58o ) V R2 e1 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 38 Superposition Theorem (4) Ex. 1 e = 10sin10t V; j = 4sin(50t + 30o) V; e = 6 V L + 1 2 + R1 R2 – (DC); L = 1 H; R1 = 1 Ω; R2 = 5 Ω; C = 0.01 F; – vR1 = ? C e1 j e2 – – + + – V + V Step 3 v R1 j R1 j R1 j j50 j50 1 5 + 1 L R R – 1 2 Z 4 30o V –j2 E j C 200 120o v  ? I   4.14 32o A R1 j j jj50 1 0.69  1.72 E  (j 50)(4 30o ) 200 120o V V 14.1432o  4.14 32o V j R1 j 5( j 2) Z 0.69j 1.72 vt4.14sin(50  32o ) V 52 j R1 j Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 39 Superposition Theorem (5) Ex. 1 e = 10sin10t V; j = 4sin(50t + 30o) V; e = 6 V L + 1 2 + R1 R2 – (DC); L = 1 H; R1 = 1 Ω; R2 = 5 Ω; C = 0.01 F; – vR1 = ? C e1 j e2 – – + – + Step 1 v Step 2 v Step 3 + v R1 e2 R1 e1 R1 j L L L R1 R2 + + R1 R2 R1 R2 – – C C C e2 e1 j v 1V vt1.06sin(10 58o ) V vt4.14sin(50 32o ) V R1 e2 R1 e1 R1 j vv  vv  1  1.06sin(10 t  58oo )  4.14sin(50 t  32 ) V RR11eej21 R 1 R 1 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 40 Superposition Theorem (6) vv  vv  1  1.06sin(10 t  58oo )  4.14sin(50 t  32 ) V RR11eej21 R 1 R 1 v 1V R1 e2 V 1.06  58o V R1 e1 V  4.14 32o V R1 j o o VR1 11.06  58 4.14 32 1  (0.56 jj 0.90)  (3.51  2.19) 3.07j 1.29  3.33 22.8o V o vR1 3.33sin(?  22.8 ) V Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 41 Superposition Theorem (7) Ex. 2 e1 = 45V (DC); e4 = 60V (DC); j = 10sin(100t) A; C L + R + R = 5Ω; R = 10Ω; C = 2mF; L = 0.1H; v = ? 1 – 1 3 C j – e1 R3 e4 Step 1  C  L + R1 + – v – e1 C ee1, 2 R3 Step 3 e4 vv v CCee1, 2 C j Step 2 R  C  L 1 v C j R j 3 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 42 Superposition Theorem (8) Ex. 2 e1 = 45V (DC); e4 = 60V (DC); j = 10sin(100t) A; C L + R + R = 5Ω; R = 10Ω; C = 2mF; L = 0.1H; v = ? 1 – 1 3 C j – e1 R3 e4 Step 1  C  L + R1 + – v – e1 C ee1, 2 R3 e4 vee45  60  15V C ee1, 2 14 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 43 Superposition Theorem (9) Ex. 2 e1 = 45V (DC); e4 = 60V (DC); j = 10sin(100t) A; C L + R + R = 5Ω; R = 10Ω; C = 2mF; L = 0.1H; v = ? 1 – 1 3 C j – e1 R3 e4 R J 15101 V 1 C j 1RjL3 ( )jC 1 10( j 100 0.1) j100 0.002 R1 5   jC R3  jL j100 0.002 10 j 100 0.1 j25.00  25.00  90o V Step 2  C  L R1 o v vtC 25sin(100  90 ) V C j R j j 3 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 44 Superposition Theorem (10) Ex. 2 e1 = 45V (DC); e4 = 60V (DC); j = 10sin(100t) A; C L + R + R = 5Ω; R = 10Ω; C = 2mF; L = 0.1H; v = ? 1 – 1 3 C j – e1 R3 v 15V e4 C ee1, 2 Step 1  C  L + R1 + – v – e1 C ee1, 2 R3 Step 3 e4 vv v CCee1, 2 C j Step 2 R  C  L 1 v 15  25sin(100t  90o ) V C j R j 3 vt25sin(100 90o ) V C j Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 45 Ex. 3 Superposition Theorem (11) j1 o e = 45V (DC); j1 = 6sin(100t + 15 ) A; j2 = 10sin(100t) A; + e C R1 = 5Ω; R2 = 10Ω; C = 2mF; L = 0.1H; iR1 = ? – L R2 j1 j2 R1 + e C – e 45 i 9A R1 e i L R2 R1 5 R1 e j2 1 R1 JJ R jC 122 jL I  R1 jj1, 2 1()RjL 1 R1  jL i R2  R1 j1,j 2 j1 jC R1  jL C 6.39j 2.79  6.97 23.6o A L R2 j o 2 itR1 9 6.97sin(100  23.6 ) A R1 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 46 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 47 Source Transformation (1) Z E J  + Z E – J Z EZJ EV VEZI I   ZZ Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 48 Source Transformation (2) Ex. 1 E  20 45o V;J 5 60o A; + Z1 ZZ1212 ; jj 10  ; Z 3  16 ; – Z2 Z3 Find the current of Z2? E J o E 20  45 o J1   1.67  45 A Z1 12 Z2 Z3 J J1 Z1 Z E J  + Z E – J Z EZJ Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 49 Source Transformation (3) Ex. 1 E  20 45o V;J 5 60o A; + Z1 ZZ1212 ; jj 10  ; Z 3  16 ; – Z2 Z3 Find the current of Z2? E J o E 20  45 o J1   1.67  45 A Z1 12 ZZ 12( j 16) Z Z Z 13 7.68j 5.76 2 3 13 ZZ12j 16 J 13 J1 Z1 o o JJJeq 1 5 60 1.67 45 1.32 j 5.51A Z 7.68 j 5.76 IJ13 (1.32j 5.51) 2 eq Z2 Z13 ZZ213jj10 7.68 5.76 Jeq 6.09j 1.16 A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 50 Source Transformation (3) Ex. 2 o o E1  100 30 V;E4  80 45 V;J 5A; Z Z + Z 1 2 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find the current of Z ? J 2 E1 E4 o E1 100 30 o J1   10 30 A Z1 10 Z2 J4 o Z E 80  45 1 Z J 4 2.26j 2.26A 3 4 Z  j25 J J 4 1 Z4 o JJJeq 1 10 30 5 13.66 j 5.00A Z2 J4 Z1 ZZ jj20( 25) Z34 Z 34 j100 34 Jeq ZZ34jj20 25 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 51 Source Transformation (4) Ex. 2 o o E1  100 30 V;E4  80 45 V;J 5A; Z Z + Z 1 2 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find the current of Z ? J 2 E1 E4 JZJeq 13.66jj 5.00A;34  100 ;4 2.26 j 2.26 A EZJeq1 eq 10(13.66 jj 5.00)  136.6 50 V Z2 J4 Z EZJ34 34 4 226j 226V 1 Z J 3 J1 EEeq  34 Z4 I2 1.19j 3.81A ZZZ1234 Z Z Z Z J4 + 1 2 34 + 2 – – Z1 Z E 34 Eeq 34 Jeq Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 52 J Source Transformation (5) 3 Ex. 3 o o  E  100 30 V;JJ 5A; 8  45 A; E1 Z3 1 34  ZZZ10 5 ;jj 20  Z 25 ; 123 4 Z2 Z4 Find the current of Z ? Z1 3 J4 100 30o E1 o ZZ12 J1   10 30 A Z12  3.33 Z1 10 ZZ12 ZIJ12()()()0 1 ZIJ 3  3 ZIJ 4  4 I 7.68 j 28.94 A IIJ3312.68 j 28.94A J3 J3 J1 Z3 J1 Z3 Z12 Z4 Z2 Z4 J4 J4 Z1 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 53 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 54 Thévenin & Norton Equivalent Circuits (1) I Zeq I + + + Z – V V L Eeq ZL – – I I + + V V ZL Jeq Zeq ZL – – Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 55 Thévenin & Norton Equivalent Circuits (2) I + V ZL – Zeq I I + + + Eeq = Zeq Jeq – V V Eeq ZL Jeq Zeq ZL – E – Z  eq eq J V eq Z open-circuit Eeq = Vopen-circuit eq Ishort-circuit Jeq = Ishort-circuit Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 56 Ex. 1 Thévenin & Norton Equivalent Circuits (3) E  20 45o V;J 5 60o A; + Z1 ZZ1212 ; jj 10  ; Z 3  16 ; – Z2 Z3 Find the current of Z2? E J Z 1 Zeq Z3 Zeq I2 + – Eeq Z2 ZZ13 12( j 16) Zeq  7.68j 5.76 ZZ1312j 16 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 57 Ex. 1 Thévenin & Norton Equivalent Circuits (4) E  20 45o V;J 5 60o A; + Z1 ZZ1212 ; jj 10  ; Z 3  16 ; – Z2 Z3 Find the current of Z2? E J a + + Z1 – Eeq Z3 Zeq I2 E – J + – Eeq Z2 E 20  45o  J  5 60o Z EV1  12  54.38 140.4o V eq a 11 11   ZZ13 12 j 16 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 58 Ex. 1 Thévenin & Norton Equivalent Circuits (5) E  20 45o V;J 5 60o A; + Z1 ZZ1212 ; jj 10  ; Z 3  16 ; – Z2 Z3 Find the current of Z2? E J Zeq  7.68j 5.76 Zeq I2 o Eeq  54.38 140.4 V + – Eeq Z2 o Eeq 54.38 140.4 o I2   6.20 169.3 A ZZeq  2 7.68jj 5.76 10 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 59 Ex. 2 Thévenin & Norton Equivalent Circuits (6) o o E1  100 30 V;E4  80 45 V;J 5A; Z Z + Z 1 2 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find the current of Z ? J 2 E1 E4 Zeq ZZ jj20( 25) ZZ 34 10 10j 100 eq 1 ZZjj20 25 Z1 Z4 34 Z3 Evveq a b ZJvEvEZJ1111aa 137 j 50V a b E4  vZIZb 33 3 226j 226V  Eeq ZZ Z 34 + Z 4 + 1 Evv363 j 176V – – eq a b Z3 J E 363 j 176 E1 E4 eq I2  1.19j 3.81A ZZeq 2 10j 100 5 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 60 J Ex. 3 Thévenin & Norton Equivalent Circuits (7) 3 o o  E  100 30 V;JJ 5A; 8  45 A; E1 Z3 1 34  ZZZ12310 5 ;jj 20  Z 4 25 ; Z2 Z4 Z Find the current of Z3? 1 J4 Zeq ZZ12 10 5 ZZeq  4 jj25 3.33 25 ZZ1210 5 Z2 Z4 Z1 Eeq I3  J3 ZZeq  3 a b    Evveq a b 12.68j 28.94 A E1 Eeq   Z Z 11 E1 2 4  vJa 3 Z  v 45,53j 16,67 V 1 J ZZ12 Z 1  a  4    1 vb 141,4  j 16,4V  vJJ  Z b 34  4 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 61 Ex. 4 Thévenin & Norton Equivalent Circuits (8) a b o o E1  100 30 V;E4  80 45 V;J 5A; Z + Z 1 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find Z ? J ab E1 E4 Z ab Method 1 Z1 Z4 Z3 ZZ34 jj20( 25) ZZab 1 10  10j 100  ZZ34jj20 25 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 62 Ex. 4 Thévenin & Norton Equivalent Circuits (9) a b o o E1  100 30 V;E4  80 45 V;J 5A; Z + Z 1 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find Z ? J ab E1 E4 a b   Voc Method 2 Vopen-circuit Z Zab  + Z 1 4 + Ishort-circuit – Z3 – V  363 j 176V J oc E1 E4 IIJsc  1   1, 39j 3, 77 A EZ// JEZ a b v 11 4 4150j 88V a 1/ZZZ 1/ 1/ Z 134 + I Z 1 1 I 4 + sc IEvZ11 (a ) / 1 6.39j 3.77A – Z3 – J E E1 4 Zab 10j 100 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 63 Ex. 4 Thévenin & Norton Equivalent Circuits (10) a b o o E1  100 30 V;E4  80 45 V;J 5A; Z + Z 1 4 + – ZZZ10 5 ;jj 20  Z 25 ; – 123 4 Z3 Find Z ? J ab E1 E4 E + – Method 3 Z  in Z ab Z1 Iin 4 Iin Ein Z3 Ein  100V 100 100 Iin  0.099j 0.099A ZZ34 jj20( 25) Z1 10 ZZ34jj20 25 100 Z 10j 100 ab 0.099 j 0.099 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 64 Thévenin & Norton Equivalent Circuits (11) Ex. 5 o V j6 16 0V open-circuit + – Find Zeq? Zeq  Ishort-circuit I2 j8 2I a 1 b + (VVcb ) 16j 6 I21 4 I 0 I I c 4 1 Voc VVV  o oc b c 2 30 A – c VIIoc 16j 621 4 o I1  2 30 o II21c 222 I 30 o o Voc 16j 6 2 2 30 4 2 30 21.07  j 24.78 V Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 65 Thévenin & Norton Equivalent Circuits (12) Ex. 5 j6 16 0Vo Vopen-circuit Find Z ? Z  + – eq eq I Ishort-circuit j8 2 2I1 o a b I1  2 30Isc 0 o Ic Isc 2 30  I1 4 I1 Isc o o 2 30 A j6416II21 0 c o II212 30Ic 0 o I1 0.67 j 0.41 A II212 30 2I1 0 o Isc 2 30 (0.67j 0.41) 32II 30o 12 1.06j 1.41 A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 66 Thévenin & Norton Equivalent Circuits (13) Ex. 5 j6 16 0Vo + – Find Zeq? Method 1 j8 2I a 1 b Ic 4 I1 Vopen-circuit 2 30o A Zeq  c Ishort-circuit 21.07j 24.78 Zeq Voc 21.07  j 24.78 V 1.06 j 1.41  4.00j 18.00 Isc 1.06j 1.41 A Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 67 Thévenin & Norton Equivalent Circuits (14) Ex. 5 j6 16 0Vo 100 + – Find Zeq? Method 2 Zeq  Iin j8 2I j6 a 1 b j8 I 2I1 in I a c b 4 I1 o Ic + 2 30 A 4 I1 – c 100 V c j6(IIrg ) 4 I r 100 IIrin1.18 j 5.29 A  IIIg 221 r 100 Z 4.00 j 18.00  eq 1.18 j 5.29 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 68 Sinusoidal Steady-State Analysis 1. Sinusoidal Steady-State Analysis 2. Ohm’s Law 3. Kirchhoff’s Laws 4. Impedance Combinations 5. Branch Current Method 6. Node Voltage Method 7. Mesh Current Method 8. Superposition Theorem 9. Source Transformation 10. Thévenin & Norton Equivalent Circuits 11. Op Amp AC Circuits Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 69 Op Amp AC Circuits (1) Ex. 1  j1000 Find Vo? 2k V a – + V + + b o – 10 V 5 0 Vo o o 0 5 0V – 2000 j 1000 5 0o V j1000  j2.5 02.5o  90o V o 2000 Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 70 Op Amp AC Circuits (2) Ex. 2 R b 2 + Find vo? a – + EVaaoab V V V V a :  C1 C2 + R ZZR 4 CC12 2 – c e R1 vo R3 VV V – b : ab b ZC21R R3 c : VVVcob RR34 VEoCCfRRRR(,,,,,1234 ZZ 1 , 2 ) Sinusoidal Steady-State Analysis - sites.google.com/site/ncpdhbkhn 71

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