Lecture Electric circuit theory - Magnetically coupled circuits presents the following content: Mutual inductance, Dot convention, analysis of magnetically coupled circuits, energy in a coupled circuit, transformers.
Nguyễn Công Phương Electric Circuit Theory Magnetically Coupled Circuits 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 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Magnetically Coupled Circuits Mutual Inductance Dot Convention Analysis of Magnetically Coupled Circuits Energy in a Coupled Circuit Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Mutual Inductance (1) i (t ) + + v – – N turns di d di d Faraday’s law: v N L N dt di dt dt d LN di Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Mutual Inductance (2) i1 (t ) L1 12 + + 11 L2 – + v1 – – N1 turns di1 v2 M 21 dt N turns 1 11 12 d1 v1 N1 dt d1 di1 di1 N1 L1 di1 dt dt d12 v2 N dt d12 di1 di1 N2 M 21 di1 dt dt Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Mutual Inductance (3) i1 (t ) L1 12 v2 – + L2 21 i2 (t ) 22 v2 + v1 N turns + – – di2 v2 L dt di1 dt – – L1 di2 dt v2 M 21 – + v1 N1 turns v1 M 12 di1 v1 L dt + + 11 L2 N1 turns N turns Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Mutual Inductance (4) i1 (t ) L1 12 + + 11 L2 + v1 v2 – – – N turns N1 turns i1 (t ) v1 di1 v2 M 21 dt 12 11 v2 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Mutual Inductance (5) i1 (t ) v1 i1 (t ) v1 12 i1 (t ) 11 di1 v2 M dt 12 v1 i1 (t ) 11 di1 v v2 M dt 12 11 di1 dt v2 M 12 11 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn di1 v2 M dt Magnetically Coupled Circuits Mutual Inductance Dot Convention Analysis of Magnetically Coupled Circuits Energy in a Coupled Circuit Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Dot Convention (1) + L1 L2 v2 – – • If a current leaves a dotted terminal of one coil, it induces a negative voltage at the dotted terminal of the second coil i1 (t ) M + • If a current enters a dotted terminal of one coil, it induces a positive voltage at the dotted terminal of the second coil di1 v2 M dt Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 10 Magnetically Coupled Circuits Mutual Inductance Dot Convention Analysis of Magnetically Coupled Circuits Energy in a Coupled Circuit Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 33 Transformers (1) If k M L1 L2 N1 L2 n L1 N2 I1 I2 + + v N V N n n V1 v1 N1 V1 N1 N1 – d v1 N1 dt d v2 N dt V2 V1 L2 V1 nV1 L1 p1 p2 v1i1 v2i2 N2 V2 – V1 j L1I1 j MI V2 j L1I j MI1 V2 i2 v1 I N i1 v2 I1 N n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 34 Ex Transformers (2) Given an ideal step-down transformer rated at 22/0.4 kV, 1000 turns on the primary side Find: a) The turn ratio? b) The number of turns on the secondary side? Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 35 Transformers (3) V2 V1 N2 V2 N I N1 V2 ; N1 I1 N V1 I1 I2 V1 N1 N2 V2 – – – V1 j L1I1 j MI V2 j L1I j MI1 – V2 N V N n 1 I N1 I1 N2 n N1 + – – V2 N V N n 1 I N1 I1 N n N2 + V1 j L1I1 j MI V2 j L1I j MI1 M L1L2 p1 p2 N1 + V2 V1 I2 + N2 I1 + N1 I2 + + + V1 I1 – I2 – I1 N I N V2 2; N1 I1 N2 V1 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 36 Transformers (4) I2 V2 V2 I2 n; V1 I1 n I1 I2 V1 N1 N2 V2 – – – – – – n N2 – I1 N1 + + V1 n; V1 + + + I2 + + + • If v1 & v2 are both positive or both N1 N2 V2 V1 negative at the dotted terminals, V2 I use +n n; V1 I1 n Otherwise, use –n I1 I2 • If i1 & i2 both enter into or both N1 N2 leave the dotted V2 V1 terminals, use –n Otherwise, use +n V I I1 – I1 V2 I n; n V1 I1 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 37 Transformers (5) – I1 + Z2 – – I1 – b I2 + Zin a Z2 – b + – Z2 Zin n E1 + V2 Z 2I R1 a I2 + I1 nI + Vab V1 Zin I1 I1 V1 V2 / n I1 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn R1 a E1 Zin b 38 Transformers (7) I1 V2 R2I I1 I2 + + V1 N1 N2 V2 – – 6I1 V1 100 0o I1 5V1 100 I1 I1 10 A I A V2 – R1I1 V1 E1 E1 V1 R2 – – Method R1 + E1 100 0o V; n 5; R1 6; R2 100 ? + Given an ideal transformer, find currents if I2 + Ex V2 I n; V1 I1 n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 39 Transformers (8) Ex I1 I2 Given an ideal transformer, find currents if + E1 100 0o V; n 5; R1 6; R2 100 ? – Method R2 R1 E1 R2 100 Zin 4 n 25 I1 + 100 E1 I1 10 A R1 Z in – I1 I1 n I2 A I2 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn R1 Zin E1 40 Transformers (9) V1 V2 R2 – – E1 I2 + – Z j 20 ? R1 + Given an ideal transformer, find currents if E1 100 0o V; n 5; R1 6; R2 100; I1 + Ex R1I1 V1 Z3 ( I1 I ) E1 + + Z3 I1 I V2 R2I Z3 ( I1 I ) I1 o I V j 20 I 100 I1 I2 5 5V 100 I1 j 20 I I1 N1 N2 1 V2 V 5 I1 3.79 j 4.85A V2 I I1 n; I 1.90 j 2.43A V1 I1 n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 41 – – Transformers (10) Ex R4 I1 I2 + R2 R1 E1 – Z3 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 42 Transformers (11) I1 V1 + N2 V2 – I1 N2 p1 p2 I N1 N I2 N1 – V1 N1 N N1 V2 + d v1 ( N1 N ) dt d v2 N dt + I1 V1 – Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn N1 N2 V2 – V1 I1 N1 N N1 ; V2 N1 N I N1 + I2 43 Transformers (12) I1 I2 V1 90 turns – – V1 100 V2 10 0o V 10 10 o V2 10 I2 0.40 j 0.80 A Z j10 E1 10 turns + o + V1 N1 N 10 90 10 V2 10 N1 + Given an ideal autotransformer, find currents if E1 100 0o V; Z j10 ? I3 V2 Z2 – Ex I1 N2 90 0.9 I1 0.9I 0.36 j 0.72 A I N1 N 10 90 I I1 I 0.040 j 0.080 A Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 44 Transformers (13) 1: n + + I primary I secondary Vprimary Vsecondary – – Vsecondary nVprimary I secondary I primary n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 45 Transformers (14) I secondary I primary + + Vprimary Vsecondary – – Vsecondary nVprimary I secondary I primary n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 46 Transformers (15) • Applications: – – – – Power supply transformers Transformers in power systems Isolation applications Impedance matching Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 47 ... Three-phase Circuits XII Magnetically Coupled Circuits XIII.Frequency Response XIV.The Laplace Transform XV Two-port Networks Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn Magnetically. .. sites.google.com/site/ncpdhbkhn Magnetically Coupled Circuits Mutual Inductance Dot Convention Analysis of Magnetically Coupled Circuits Energy in a Coupled Circuit Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn... 28.46 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 30 Magnetically Coupled Circuits Mutual Inductance Dot Convention Analysis of Magnetically Coupled Circuits Energy in a Coupled