Bài giảng Electric circuit theory - Chapter IV: Circuit Theorems - Nguyễn Công Phương

Nguy ễn Công Ph ươ ng Electric Circuit Theory Circuit Theorems 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. Sinusoidal Steady State Analysis X. AC Power Analysis XI. Three-phase Circuits XII. Magnetically Coupled Circuits XIII.Frequency Response XIV.The Laplace Transform XV. Two-port Networks Circuit Theorems - sites.google.com/site/ncpdhbkhn 2 Circuit Theorems 1. Source Transformation 2. Linearity and Superposition 3. Thévenin Equivalent Subcircuits 4. Norton Equivalent Subcircuits 5. Maximum Power Transfer Circuit Theorems - sites.google.com/site/ncpdhbkhn 3 Source Transformation (1) R E J = + R E – J R E= RJ E v E= Ri + v i = − RR Circuit Theorems - sites.google.com/site/ncpdhbkhn 4 Source Transformation (2) Ex. 1 a J = 2 A; R = 4 Ω; R = 6 Ω; find i ? 1 2 2 i2 J R1 R2 = =×= E R1 J 4 2 8V b E 8 i = = = 0.8A 2 + + RR1 2 4 6 a E R J = + 1 R – E R2 + R i2 – J R E b E= RJ Circuit Theorems - sites.google.com/site/ncpdhbkhn 5 Source Transformation (3) Ex. 2 R1 aR3 b Find the current i 3. i3 + R – 2 E1 R4 J + E2 – c Circuit Theorems - sites.google.com/site/ncpdhbkhn 6 Circuit Theorems 1. Source Transformation 2. Linearity and Superposition 3. Thévenin Equivalent Subcircuits 4. Norton Equivalent Subcircuits 5. Maximum Power Transfer Circuit Theorems - sites.google.com/site/ncpdhbkhn 7 Linearity and Superposition (1) E R E – + 1 – + i i + J R1 R2 – R2 R1J RJ+ E RJ E i =1 = 1 + + + + RR12 RR 12 RR 12 =i + i E=0 J = 0 E – + i i E=0 J =0 J R1 R2 R1 R2 Circuit Theorems - sites.google.com/site/ncpdhbkhn 8 Linearity and Superposition (2) = + ++ yax11 ax 22 ... axn n = + + + y1 y 2 ... y n = ≠ yi y= , ki xk 0 Circuit Theorems - sites.google.com/site/ncpdhbkhn 9 Linear and Superposition (3) + + deactivated – v = 0 – i = 0 deactivated Circuit Theorems - sites.google.com/site/ncpdhbkhn 10 Linear and Superposition (4) R1 a R3 b + R2 J – i2 + R4 E1 E2 – c To deactivate E1? R1 a R3 b R2 J i2 + R4 E2 – c Circuit Theorems - sites.google.com/site/ncpdhbkhn 11 Linear and Superposition (5) R1 a R3 b + R2 J – i2 + R4 E1 E2 – c To deactivate E2? R1 a R3 b + R2 J – i2 R4 E1 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 12 Linear and Superposition (6) R1 a R3 b + R2 J – i2 + R4 E1 E2 – c To deactivate J? R1 a R3 b + R2 J – i2 + R4 E1 E2 – c Circuit Theorems - sites.google.com/site/ncpdhbkhn 13 Ex. 1 Linear and Superposition (7) R = 10 Ω, R = 20 Ω, E = 30V, J = 2A, find the current of R ? R1 1 2 2 + – 1. Deactivate J, find i2|E 2. Deactivate E, find i2|J E J R2 R1 R1 + i i 3. Find i | + i | 2 E 2 J 2 E2 2 J – E R2 J R2 = + i2 i 2 i 2 E R J 3 j i = i = 1 2 E 2 E i = + R+ R R+ R 2 J 1 0.67 1 2 1 2 R1 × + 30 10 2 – = = = 1A = R 1.67 A 10+ 20 10+ 20 2 = 0.67A R1 J Circuit Theorems - sites.google.com/site/ncpdhbkhn 14 Ex. 2 Linear and Superposition (8) R1 aR3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; i2 R3 = 2 Ω; R4 = 10 Ω; find i2? + R – 2 E1 R4 J + E2 – c 1. Deactivate E2 & J, find i2|E1 2. Deactivate E1 & J, find i2|E2 3. Deactivate E1 & E2, find i2|J 4. Find i2 = i2|E1 + i2|E2 + i2|J Circuit Theorems - sites.google.com/site/ncpdhbkhn 15 Ex. 2 Linear and Superposition (9) R1 aR3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; i2 R3 = 2 Ω; R4 = 10 Ω; find i2? + R – 2 E1 R4 J 1. Deactivate E & J, find i | + 2 2 E1 – E2 R (R + R ) 2(6 +10 ) R = 2 3 4 = = 4Ω 234 + + + + c R2 R3 R4 6 2 10 R1 R3 = + = + = Ω R1234 R1 R234 4 4 8 i2 E1 16 i = = = 2 A + 1 E1 R R 8 – 2 1234 E1 R4 R i 4× 2 i =−234 1 E1 =− =− 1.33A 2 E1 i2 R2 6 E1 Circuit Theorems - sites.google.com/site/ncpdhbkhn 16 Ex. 2 Linear and Superposition (10) R1 aR3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; i2 R3 = 2 Ω; R4 = 10 Ω; find i2? + R – 2 E1 R4 J 2. Deactivate E & J, find i | + 1 2 E2 – E2 R (R + R ) (4 2 +10 ) R = 1 3 4 = = 3Ω R c R 134 + + + + 1 3 R1 R3 R4 4 2 10 i = + = + = Ω 2 R2134 R2 R134 6 3 9 R2 E 9 i2|E2 R4 i =2 = = 1A 2 E2 + R2134 9 E2 – Circuit Theorems - sites.google.com/site/ncpdhbkhn 17 Ex. 2 Linear and Superposition (11) R1 aR3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; i2 R3 = 2 Ω; R4 = 10 Ω; find i2? + R – 2 E1 R4 J 3. Deactivate E & E , find i | + 1 2 2 J – E2 E= R J =10 ×= 2 20V 4 4 c R1 R3 R R 4× 6 R =1 2 = =Ω2,4 12 R+ R 4 + 6 1 2 i2 E 20 i =4 = = 1.39 A R 3 J ++ ++ 2 R J R12 R 3 R 4 2.4 2 10 4 R i 2.4× 1.39 i2|J i =−12 3 J =− =− 0.56A 2 J R 6 2 Circuit Theorems - sites.google.com/site/ncpdhbkhn 18 Ex. 2 Linear and Superposition (12) R1 aR3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; i2 R3 = 2 Ω; R4 = 10 Ω; find i2? + R – 2 E1 R4 J 4. Find i = i | + i | + i | + 2 2 E1 2 E2 2 J – E2 c i = − 1.33A 2 E1 i = 1A → i = –1.33 + 1 – 0.56 = – 0.89A 2 E 2 2 i = − 0.56A 2 J Circuit Theorems - sites.google.com/site/ncpdhbkhn 19 Circuit Theorems 1. Source Transformation 2. Linearity and Superposition 3. Thévenin Equivalent Subcircuits 4. Norton Equivalent Subcircuits 5. Maximum Power Transfer Circuit Theorems - sites.google.com/site/ncpdhbkhn 20 Thévenin Equivalent Subcircuits (1) Req + Eeq – Circuit Theorems - sites.google.com/site/ncpdhbkhn 21 Thévenin Equivalent Subcircuits (2) i Req i + + + v – v RL Eeq RL – – Eeq : the open-circuit voltage at the terminals Req : the input or equivalent resistance at the terminals when the independent sources are deactivated Circuit Theorems - sites.google.com/site/ncpdhbkhn 22 Ex. 1 Thévenin Equivalent Subcircuits (3) R = 10 Ω, R = 20 Ω, R = 30 Ω, E = 30V. 1 2 3 R + 1 Find the current of R3? – Eeq : the open-circuit voltage at the terminals = = E R2 R3 + E v R i R eq R 2 2 2 + 1 = E – R Eeq 2 + R1 R 2 R – E 2 = 20V R + eq R : the input or equivalent resistance at the terminals – eq Eeq R when the independent sources are deactivated 3 R × 1 = R1 R 2 Req E R+ R = eq = 20 1 2 i3 R2 R+ R 6.67+ 30 =6.67 Ω eq 3 = 0.55A Circuit Theorems - sites.google.com/site/ncpdhbkhn 23 Ex. 2 Thévenin Equivalent Subcircuits (4) R1 = 10 Ω, R2 = 20 Ω, E = 30V, J = 2A. Find the current of R2? R1 + Eeq : the open-circuit voltage at the terminals – + + = Ri1 1 Etd E E J R2 R1  + i1 = − i1 J – E eq →E = E + RJ J – td 1 E = 50V R + eq R : the input or equivalent resistance at the terminals – eq Eeq R when the independent sources are deactivated 2 R 1 = R =10 Ω Req 1 E 50 i = eq = 2 + + Req R 2 10 20 = 1.67 A Circuit Theorems - sites.google.com/site/ncpdhbkhn 24 Thévenin Equivalent Subcircuits (5) a b Ex. 3 R R + 1 2 E = 16 V; J = 2 A; R1 = 4 Ω; – R R EJ 3 iL L R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find iL? c Eeq : the open-circuit voltage at the terminals b R + eq a b – R Eeq iL L + R1 R2 + c – EJ R3 Eeq c – Circuit Theorems - sites.google.com/site/ncpdhbkhn 25 Thévenin Equivalent Subcircuits (6) Ex. 3 = = Eeq v3 Ri 3 3 E = 16 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; 1 1  E +  v = + J find iL? + a RRR123  R 1 Eeq : the open-circuit voltage 1 1  16 at the terminals →+  v =+ 2 468+  a 4 →v = 18.67 V a b a i + va 18.67 R1 R2 3 →= = = + i 1.33 A 3 + + – R R 6 8 EJ R3 Eeq 2 3 →E =×8 1.33 = 10.67 V c – eq Circuit Theorems - sites.google.com/site/ncpdhbkhn 26 Thévenin Equivalent Subcircuits (7) a b Ex. 3 R R + 1 2 E = 16 V; J = 2 A; R1 = 4 Ω; – R R EJ 3 iL L R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find iL? c Req : the input or equivalent resistance at the terminals b when the independent sources are deactivated R + eq – E R a b eq iL L c R1 R2 Req R3 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 27 Thévenin Equivalent Subcircuits (8) Ex. 3 = + E = 16 V; J = 2 A; R1 = 4 Ω; Req ( RR1 2 )// R 3 R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find i ? L ( R+ R) R = 1 2 3 Req : the input or equivalent R+ R + R resistance at the terminals 1 2 3 when the independent + sources are deactivated = (4 6)8 a b 4+ 6 + 8 R1 R2 Req =4.44 Ω R3 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 28 Thévenin Equivalent Subcircuits (9) a b Ex. 3 R R + 1 2 E = 16 V; J = 2 A; R1 = 4 Ω; – R R EJ 3 iL L R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find iL? c = Eeq 10.67 V b R = Ω 10.67 + eq Req 4.44 → = = i 1.13A – L E i RL E 4.44+ 5 eq L i = eq c L + RReq L Circuit Theorems - sites.google.com/site/ncpdhbkhn 29 Thévenin Equivalent Subcircuits (10) a b Ex. 4 R R 2 i2 + 1 E = 16 V; J = 2 A; R1 = 4 Ω; – EJ R3 RL R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find i2? c Req a b i R1 2 a R3 RL R + eq c – R Eeq i2 2 = + Req( RR3 // L ) R 1 b R R 8× 5 =3 L + R = + = Ω + 1 4 7.08 R3 R L 8+ 5 Circuit Theorems - sites.google.com/site/ncpdhbkhn 30 Thévenin Equivalent Subcircuits (11) a b Ex. 4 R R 2 i2 + 1 E = 16 V; J = 2 A; R1 = 4 Ω; – EJ R3 RL R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find i2? c + – a Eeq b R + 1 a – EJ R3 RL R c + eq – R Eeq i2 2 = − Eeq v a v b b →E = v = = eq a → = vb v c 0 Eeq 24V − + =→ = +×= RJ1 va E v a 16 4 2 24V Circuit Theorems - sites.google.com/site/ncpdhbkhn 31 Thévenin Equivalent Subcircuits (12) a b Ex. 4 R R 2 i2 + 1 E = 16 V; J = 2 A; R1 = 4 Ω; – EJ R3 RL R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; find i2? c = Eeq 24 V a = Ω Req 7.08 24 R → = = + eq i2 1.84 A + – E R E 7.08 6 eq i2 2 i = eq 2 + b RReq 2 Circuit Theorems - sites.google.com/site/ncpdhbkhn 32 Ex. 5 Thévenin Equivalent Subcircuits (13) R1 a R3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; R3 = 2 Ω; R4 = 10 Ω; find i2? + R2 J – i2 + R4 E1 E2 – c R2 a R2 a R3 b + + i2 Req i2 R1 J – – + + R – 4 E2 Eeq E2 E1 – c c Circuit Theorems - sites.google.com/site/ncpdhbkhn 33 Ex. 5 Thévenin Equivalent Subcircuits (14) R2 a R3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; R3 = 2 Ω; R4 = 10 Ω; find i2? + R1 J – i2 a R3 b + R4 E2 E1 – Req R1 c R4 R2 a c + i2 Req = + – Req ( RR3 4 ) // R 1 – (R+ R ) R (2+ 10)4 E + = 3 4 1 = =3 Ω 2 Eeq + + + + (R3 R 4 ) R 1 (2 10) 4 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 34 Ex. 5 Thévenin Equivalent Subcircuits (15) R2 a R3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; R3 = 2 Ω; R4 = 10 Ω; find i2? + R1 J – i2 a R3 b + R – 4 E2 E1 – R1 J c Eeq i1 + R4 – + − = E1 Ri1 1 Eeq E 1 R2 a c → = − Eeq Ri1 1 E 1 1 1  E + + =1 + i2 Req v J –   b = − – RRR+ RR + 17 V 134  13 + E E− v 2 Eeq →v = 17.5V →=i 1 b =− 0.25A b 1 + c R1 R 3 Circuit Theorems - sites.google.com/site/ncpdhbkhn 35 Ex. 5 Thévenin Equivalent Subcircuits (16) R2 a R3 b E1 = 16 V; E2 = 9 V; J = 2 A; R1 = 4 Ω; R2 = 6 Ω; R3 = 2 Ω; R4 = 10 Ω; find i2? + R1 J – i2 + R4 E2 E1 – c = − Eeq 17 V R =3 Ω 9− 17 R2 a eq →=i =− 0.89 A + 2 3+ 6 EE2 eq + = i2 Req – i – 2 + RReq 2 + E2 Eeq c Circuit Theorems - sites.google.com/site/ncpdhbkhn 36 Circuit Theorems 1. Source Transformation 2. Linearity and Superposition 3. Thévenin Equivalent Subcircuits 4. Norton Equivalent Subcircuits 5. Maximum Power Transfer Circuit Theorems - sites.google.com/site/ncpdhbkhn 37 Norton Equivalent Subcircuits (1) i i + + v v RL Jeq Req RL – – Jeq : the short-circuit through the terminals Req : the input or equivalent resistance at the terminals when the independent sources are deactivated Circuit Theorems - sites.google.com/site/ncpdhbkhn 38 Ex. 1 Norton Equivalent Subcircuits (2) R = 10 Ω, R = 20 Ω, R = 30 Ω, E = 30V. 1 2 3 R + 1 Find the current of R3? – Jeq : the short-circuit through the terminals E R2 R3 R + 1 = = E = 30 – J iR2 eq R 10 R 1 E 2 = 3A R R Req : the input or equivalent resistance at the terminals eq 3 when the independent sources are deactivated Jeq R R× R 1 1  1 R = 1 2 +  v = J eq +   eq R1 R 2 Req R 3  R → = 2 =6.67 Ω v 13.36V → =v = i3 0.55A Circuit Theorems - sites.google.com/site/ncpdhbkhn R3 39 Norton Equivalent Subcircuits (3) Ex. 2 a b E = 16 V; J = 2 A; R1 = 4 Ω; R1 R2 R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; + find the Norton equivalent – E J R3 subcircuit? c Jeq : the short-circuit through the terminals b a b Jeq Req R1 R2 + – EJ R3 Jeq c c Circuit Theorems - sites.google.com/site/ncpdhbkhn 40 Norton Equivalent Subcircuits (4) Ex. 2 = E = 16 V; J = 2 A; R1 = 4 Ω; Jeq i 2 R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; find the Norton equivalent 1 1  E +v = + J subcircuit?   a R1 R 2  R 1 J : the short-circuit through eq →+1 1  =+ 16 the terminals   va 2 4 6  4 i2 → = va 14.40 V a b R1 R2 va 14.40 + →= = = i2 2.40 A – EJ R3 Jeq R2 6 → = c Jeq 2.40 A Circuit Theorems - sites.google.com/site/ncpdhbkhn 41 Norton Equivalent Subcircuits (5) Ex. 2 a b E = 16 V; J = 2 A; R1 = 4 Ω; R1 R2 R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; + find the Norton equivalent – E J R3 subcircuit? c Req : the input or equivalent resistance at the terminals when the independent b sources are deactivated J R a b eq eq R1 R2 Req c R3 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 42 Norton Equivalent Subcircuits (6) Ex. 2 E = 16 V; J = 2 A; R = 4 Ω; 1 = + R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; Req ( RR1 2 )// R 3 find the Norton equivalent subcircuit? ( R+ R) R = 1 2 3 Req : the input or equivalent R+ R + R resistance at the terminals 1 2 3 when the independent + sources are deactivated = (4 6)8 + + a b 4 6 8 R1 R2 Req =4.44 Ω R3 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 43 Norton Equivalent Subcircuits (7) Ex. 2 a b E = 16 V; J = 2 A; R1 = 4 Ω; R1 R2 R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; + find the Norton equivalent – E J R3 subcircuit? c b = Jeq 2.40 A J R = Ω eq eq Req 4.44 c Circuit Theorems - sites.google.com/site/ncpdhbkhn 44 Norton Equivalent Subcircuits (8) Ex. 3 a b E = 16 V; J = 2 A; R1 = 4 Ω; R R 2 i2 + 1 R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; – EJ R3 RL find i2? c Req a b i R1 2 R3 RL a i2 c Jeq Req R2 R=( RR // ) + R eq3 L 1 b R R 8× 5 =3 L + R = + = Ω + 1 4 7.08 R3 R L 8+ 5 Circuit Theorems - sites.google.com/site/ncpdhbkhn 45 Norton Equivalent Subcircuits (9) Ex. 3 a b E = 16 V; J = 2 A; R1 = 4 Ω; R R 2 i2 + 1 R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; – EJ R3 RL find i2? c a b R1 + i1 Jeq – EJ R RL 3 a c i2 = + Jeq Req R2 Jeq i1 J   1++ 1 1 =+E → =   va J va 10.43V b RRR  R 1 3L 1 E− v →i =a = 1.39 A → = += 1 J eq 1.39 2 3.39A R1 Circuit Theorems - sites.google.com/site/ncpdhbkhn 46 Norton Equivalent Subcircuits (10) Ex. 3 a b E = 16 V; J = 2 A; R1 = 4 Ω; R R 2 i2 + 1 R2 = 6 Ω; R3 = 8 Ω; RL = 5 Ω; – EJ R3 RL find i2? c = J eq 3.39 A R =7.08 Ω a eq a E= R J R i eq eq eq + eq 2 – E R J R R = 24V eq i2 2 eq eq 2 b b E 24 i =eq = = 1.84A 2 + + Req R 2 7.08 6 Circuit Theorems - sites.google.com/site/ncpdhbkhn 47 Thévenin & Norton Equivalent Subcircuits (1) i + v RL – Req i i + E = R J + eq eq eq + – v R v Eeq L Jeq Req RL – E – R = eq eq J eq v →R = open-circuit Eeq = vopen-circuit eq ishort-circuit Jeq = ishort-circuit Circuit Theorems - sites.google.com/site/ncpdhbkhn 48 Thévenin & Norton Equivalent Subcircuits (2) a Ex. 1 b R1 R2 E = 16 V; J = 2 A; R1 = 4 Ω; + – E J R3 R2 = 6 Ω; R3 = 8 Ω; Rt = 5 Ω; find R ? eq c v E =open-circuit = eq Req ishort-circuit J eq 10.67 →R = =4.44 Ω = eq 2.40 Eeq 10.67 V = Jeq 2.40 A Circuit Theorems - sites.google.com/site/ncpdhbkhn 49 Circuit Theorems 1. Source Transformation 2. Linearity and Superposition 3. Thévenin Equivalent Subcircuits 4. Norton Equivalent Subcircuits 5. Maximum Power Transfer Circuit Theorems - sites.google.com/site/ncpdhbkhn 50 Maximum Power Transfer (1) Req 2 i p= i R 2 L L L E  + →p = eq  R + Eeq L L – v = +  Eeq RL iL Req R L  R+ R – eq L dp L = 0 dR L dp (R+ R )2(2 − RR + R ) →L = E 2 eq L L eq L eq + 4 dRL( RR eq L ) RRR+ −2 RR − =E2eq L L = E 2 eq L = 0 eq+3 eq + 3 (RReq L ) ( RR eq L ) = RL R eq Circuit Theorems - sites.google.com/site/ncpdhbkhn 51 Maximum Power Transfer (2) Ex. 1 E = 16 V; R1 = 4 Ω; R2 = 6 Ω; R3 = 2 Ω; R4 = 10 Ω; R1 R2 Find the value of RL for maximum power transfer? E + – R = L RL R eq R3 R4 R R R R R =1 2 + 3 4 eq R+ R R + R R1 R2 1 2 3 4 E Req i Req =4.6 + 2.10 4+ 6 2 + 10 + + – v =4,07 Ω Eeq RL R3 R4 – → = Ω RL 4.07 Circuit Theorems - sites.google.com/site/ncpdhbkhn 52