Bài giảng Electric circuit theory - Chapter XII: Magnetically Coupled Circuits - Nguyễn Công Phương

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 2 Magnetically Coupled Circuits 1. Mutual Inductance 2. Dot Convention 3. Analysis of Magnetically Coupled Circuits 4. Energy in a Coupled Circuit 5. Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 3 Mutual Inductance (1) it() + + – v  – N turns d ddi di Faraday’s law: vN  N  L dt di dt dt d LN di Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 4 Mutual Inductance (2) it1() L1 L2 12 + + 11 + di – 1 v1 v2  M 21 – – dt N1 turns N2 turns 11112  d d vN 1 vN 12 11dt 22dt ddi11 di 1 ddi12 1 di 1 NL11 NM221 di1 dt dt di1 dt dt Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 5 Mutual Inductance (3) it1() L1 L2 12 + + di1 11 vL1  + dt – v1 v2 di – – vM 1 221dt N1 turns N2 turns L1 L2 it2 () 21 + di2 + vM112  + dt 22 v1 v2 – – di – vL 2 2 dt N1 turns N2 turns Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 6 Mutual Inductance (4) it1() L1 L2 12 + + 11 + di1 – vM221 v1 v2 – – dt N1 turns N2 turns it1() 12   v v 1  2  11  Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 7 Mutual Inductance (5) it1() 12 it1() 12     di1 di1 v1 vM2  v1 vM2  11 dt 11     dt it()  it1() 12 1 12    di  di 1 v vM 1 v1 vM2  1  2 11 dt 11 dt     Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 8 Magnetically Coupled Circuits 1. Mutual Inductance 2. Dot Convention 3. Analysis of Magnetically Coupled Circuits 4. Energy in a Coupled Circuit 5. Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 9 Dot Convention (1) it() M • If a current enters a dotted 1 terminal of one coil, it induces + + a positive voltage at the dotted – L v 1 L2 2 terminal of the second coil – di1 • If a current leaves a dotted vM2  terminal of one coil, it induces dt a negative voltage at the dotted terminal of the second coil Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 10 Dot Convention (2) it() M • If a current enters a dotted 1 terminal of one coil, it induces + + a positive voltage at the dotted – L v 1 L2 2 terminal of the second coil – di1 • If a current leaves a dotted vM2  terminal of one coil, it induces dt a negative voltage at the dotted terminal of the second coil Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 11 Dot Convention (3) it() M • If a current enters a dotted 1 terminal of one coil, it induces + + a positive voltage at the dotted – L v 1 L2 2 terminal of the second coil – di1 • If a current leaves a dotted vM2  terminal of one coil, it induces dt a negative voltage at the dotted terminal of the second coil Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 12 Dot Convention (4) it() M • If a current enters a dotted 1 terminal of one coil, it induces + + a positive voltage at the dotted – L v 1 L2 2 terminal of the second coil – di1 • If a current leaves a dotted vM2  terminal of one coil, it induces dt a negative voltage at the dotted terminal of the second coil Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 13 Dot Convention (5) M M it1() it2 () it1() it2 () + + – + + + R R R1 2 R1 2 – – L L 1 L2 1 L2 – – – + e1 e1 di di di di vL21 M vL21 M 22dt dt 22dt dt M M it1() it2 () it1() it2 () + + – + + R + R R1 2 R1 2 – L – L 1 L2 1 L2 – – + – e1 e1 di di di di vL21 M vL21 M 22dt dt 22dt dt Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 14 Magnetically Coupled Circuits 1. Mutual Inductance 2. Dot Convention 3. Analysis of Magnetically Coupled Circuits 4. Energy in a Coupled Circuit 5. Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 15 Analysis of Magnetically Coupled Circuits (1) M jM it1() it2 () I1 I2 + + + + R R R1 2 R1 2 – – L 1 L2 jL 1 jL 2 – e1 – E1 di di vL12 M VIIjL  jM 11dt dt 1112 di di vL21 M VIIjL  jM 22dt dt 2221 ZM  jM Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 16 Analysis of Magnetically Coupled Circuits (2) Ex. 1 o jM E1  100 0 V;  100 rad/s; I1 I2 – LLM120.2 H; 0.3H; 0.1H; + + R R RR30 ;  40 ; Find currents? 1 2 12 – jL 1 jL 2 + E1 – VIM 12 jM VI111L  jL RjLjM11IIIE 11  2  1 VVVRLM11 1  E 1 VIM 21 jM VI222L  jL RjLjM22III 22  1 0 VVVRLM22 2 0 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 17 Analysis of Magnetically Coupled Circuits (3) Ex. 1 o jM E1  100 0 V;  100 rad/s; I1 I2 – LLM120.2 H; 0.3H; 0.1H; + + R R RR30 ;  40 ; Find currents? 1 2 12 – jL 1 jL 2 + E1 – RjLjM11IIIE 11 2  1 RjLjM22III 22 1 0 o 30III112jj 100 0.2 100 0.1 100 0 I1  10A     I  j20A 40III221jj 100 0.3 100 0.1 0  2 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 18 Analysis of Magnetically Coupled Circuits (4) Ex. 1 o jM E1  100 0 V;  100 rad/s; I1 I2 – LLM120.2 H; 0.3H; 0.1H; + + R R RR30 ;  40 ; Find currents? 1 2 12 – jL 1 jL 2 + E – VIVIMM1221jM ; jM 1 VVVRLM11 1  E 1  VVVRLM22 2 0 1. Write voltages of mutual inductance RjLjM11IIIE 11 2  1   2. Assign signs at dotted terminals RjLjM22III 22 1 0 (using dot convention) 3. Write KVL equations I1  10A   4. Write the set of equations & solve for it I2  j20A Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 19 Analysis of Magnetically Coupled Circuits (5) Ex. 2 + + – Find current? – 0.2 H 0.4 H j62.8 j125.6 + 0.1H + j31.4 – 311cos314t V – 60 311 60 I jjj62.8II 31.4 125.6 III  j 31.4 60 311 1. Write voltages of mutual inductance o I 2.23  64.5 A 2. Assign signs at dotted terminals (using dot convention) 3. Write KVL equations it2.23cos(314  64.5o ) A 4. Write the set of equations & solve for it Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 20 Analysis of Magnetically Coupled Circuits (6) Ex. 3 Find current? 0.2 H 0.4 H + 0.1H – 60 311cos314t V 60 I 1A 60 DC 60 (jjj 62.8 31.4 125.6  j 31.4 60)I AC  311 + – 60 I o 60 DC I AC 2.23  64.5 A + + – – o itAC 2.23cos(314  64.5 ) A j62.8 j125.6 + j31.4 o – iIDC  i AC 1 2.23cos(314 t  64.5 ) A 311 60 I AC Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 21 Analysis of Magnetically Coupled Circuits (7) Ex. 4 V1M Z I1 – + abI3 3 VZI12MM Z I E 1 + 4 1 Z2 VZI + 21MM Z V2M – M – Z4 J III1230 + I2 IIJ0 – 34 E2 c VVZMZM11 VV 2 2 EE 12 ZI ZI  ZI  ZI 1. Write voltages of mutual inductance 11MM 2 2 2 1 2. Assign signs at dotted terminals  EE12 (using dot convention) 3. Write KVL equations VVZM22 VVE 34 2 4. Write the set of equations & solve ZI22 ZM I 1  ZI 33 ZI 44 E 2 for it Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 22 Analysis of Magnetically Coupled Circuits (8) Ex. 5 V1M Z I1 + – abI3 3 Z1 – I4 VZI12MM E 1 Z2 + Z V2M – M + Z4 J VZI21MM + I2 E2 – c III1230 IIJ340 VVZMZM11 VV 2 2 EE 12ZI11 ZIMM 2  ZI 2 2  ZI 1  E 1 E 2 ZI Z I  ZI ZI E VVZM22 VVE 34 2 22M 1 33 44 2 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 23 Analysis of Magnetically Coupled Circuits (9) Ex. 6 V3M I1 abI3– + Z Z1 + 3 I4 VZI23MM E 1 Z2 + V2M Z – M – Z4 J VZI32MM + I2 E2 – c III123 0 IIJ34 0 ZI11 ZI 2 2 ZM I 3 E 1 E 2 ZI22 ZMM I 3 ZI 33 Z I 2  ZI 44 E 2 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 24 Analysis of Magnetically Coupled Circuits (10) Ex. 4 V1M + – abZ3 Z E 1 + VZII()1 Z2 1M MA B + Z V2M – M – Z J VZI 4 2M MA + IA IB E2 – c VVZMZM11 VV 2 2 EE 12 ZI1212AMAB  Z()() I  I Z I AB  I Z MA I  E E VVZM22 VVE 34 2 ZII2342()BA ZIZIZIJE MAB () B Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 25 Analysis of Magnetically Coupled Circuits (11) Ex. 5 V1M – + abZ3 Z E 1 – VZII()1 Z2 1M MA B + Z V2M – M + Z J VZI 4 2M MA + IA IB E2 – c VVZMZM11 VV 2 2 EE 12 ZI1212A  Z MAB()() I  I Z I AB  I Z MA I  E E VVZM22 VVE 34 2 ZII2342()BA ZIZIZIJE MAB () B Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 26 Analysis of Magnetically Coupled Circuits (12) Ex. 6 V + 3M – Z Z3 E 1 + VZI 1 Z2 2M MB + V2M Z – – M Z J VZII() 4 3M MA B + IA IB E2 – c VVV122M  EE 12 ZI12AABMB Z() I  I Z I E 12 E VVZMZM22 VV 33 VE 4 2 ZI233442()BA I ZI MBZBMAB ZIZ ()() I  I ZI J E Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 27 Analysis of Magnetically Coupled Circuits (13) 1. Write voltages of mutual inductance 2. Assign signs at dotted terminals (using dot convention) 3. Write KVL equations (branch current method or mesh current method) 4. Write the set of equations & solve for it Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 28 Analysis of Magnetically Coupled Circuits (14) Ex. 7 I j10 Find the Thevenin equivalent subcircuit? 1 + + + 30 VI j10 – M 21 j20 j30 E – eq 100 – EVeq M 21j10 I (30j 20)I 100 1 Zeq + I1 2.31 j 1.54 A – Eeq Eeq 15.38 j 23.08V Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 29 Analysis of Magnetically Coupled Circuits (15) Ex. 8 I j10 Find the Thevenin equivalent subcircuit? 1 + E – Z  eq + 30 eq – J j20 + j30 eq Jeq – 100 (30jj 20)IJ1 10eq 100  jj10IJ1 30eq  0 Zeq Jeq 0.85 j 0.47 A + 15.38 j 23.08 – Zeq 2.31 j 28.46  E 0.85 j 0.47 eq EZeq 15.38jj 23.08V; eq 2.31 28.46 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 30 Magnetically Coupled Circuits 1. Mutual Inductance 2. Dot Convention 3. Analysis of Magnetically Coupled Circuits 4. Energy in a Coupled Circuit 5. Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 31 Energy in a Coupled Circuit M it1() it2 () + + R R1 2 – L 1 L2 e1 – 11 wLiLiMii22 2211 2 2 12 M  kLL12 01 k  Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 32 Magnetically Coupled Circuits 1. Mutual Inductance 2. Dot Convention 3. Analysis of Magnetically Coupled Circuits 4. Energy in a Coupled Circuit 5. Transformers Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 33 Transformers (1) VII1112jL jM  L VIIjL  j M 2 N N  2121VVV211 n 1 2 L1 IfkMLL 1  12 V L 22 n V11L I1 I2 + d + vN11 N N dt vN22 V22N 1 2 V  n n V1 2 – d vN V N – vN 11 11 22dt iv21 I21N 1 p12pvivi 1122     iv12 I12Nn Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 34 Ex. 1 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) I1 I2 I1 I2 I1 I2 + + + + + + N1 N2 N1 N2 N1 N2 V1 V2 V1 V2 V1 V2 – – – – – – VINN 2221;  VII1112jL jM VII1112jL jM   VI1112NN VII2121jL  jM VII2121jL jM I1 I2 M  LL12 + +  V22N p12 p n  V N N1 N2 V N  11 V V  22   1 2 n –  I21N 1 –  V11N       I Nn I N 1  12 VINN  21 2221;   I12Nn VI1112NN Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 36 Transformers (4) I1 I2 I1 I2 • If v & v are both + + + 1 2 + N N positive or both N N 1 2 V 1 2 V V1 2 negative at the V1 2 – – – – dotted terminals, VI1 VI1 22n; use +n. 22 n; VIn VI11n Otherwise, use –n 11 I I I I 1 2 • If i1 & i2 both 1 2 + + + + enter into or both N1 N2 leave the dotted N1 N2 V1 V2 V1 V2 – – – – terminals, use –n. VI1 Otherwise, use +n VI1 22n; 22 n; VI11n VI11n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 37 Transformers (5) I1 I2 + a + + R1 Vab V1 – – Zin  – Z E1 b 2 II11 I1 I2 + VV / n + 12 a Zin II12 n – b – Z2 VZI222 I1 Z2 a + R Zin 1 2 – Z n in E1 b Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 38 Ex. 2 Transformers (7) I1 I2 + Given an ideal transformer, find currents if + o + R E1  100 0 V;nR 5;12 6 ; R 100 ? R1 2 – V V 1– 2 Method 1 – E1 R11IVE 1 1 VIR 0 222 I1 I2 + 6100IV 0o +  11 N N V 1 2 V   I 1 2 – 1 – 5V1 100 0  5 VI221 I1  n;  I 10A I 2A VI11n 1 2 5 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 39 Ex. 2 Transformers (8) I1 I2 Given an ideal transformer, find currents if o + R E1  100 0 V;nR 5;12 6 ; R 100 ? R1 2 Method 2 – E1 R 100 Z 2 4 in n2 25 I1 E1 100 I1 10A RZ64 + 1 in R1 – Zin E II11 1 n 52AI2   I2 5 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 40 I I Ex. 3 Transformers (9) 1 2 + Given an ideal transformer, find currents if + + R R2 o 1 E1  100 0 V;nR 5;12 6 ; R 100 ; – V V 1– 2 E – Z3 j20 ? 1 R11IVZII 1 3() 1  2  E 1 II Z3 VIZII222312R ()0   12  I1 o 6IV11j 20 I 1   100 0 I I  5 1 2 +   + II  11 N N 5VI11 100j 20   0 V 1 2 V  55 1 2  – – I1 3.79 j 4.85A VI221 I1  n;  I2 1.90 j 2.43A VI11n 5 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 41 Ex. 4 Transformers (10) R4 I1 I2 + R R1 2 – E1 Z3 Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 42 Transformers (11) I d 1 vNN() 112dt V NN + 112 d N I2 vN V21N 1 22 + dt V1 I N N2 V 12 – 2 pp12 – I212NN I2 I + VI11112NNN 1 ; N1 VINN N V 2122 1 + N 2 V 2 – 1 – Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 43 Ex. 5 Transformers (12) I Given an ideal autotransformer, find currents 1 o + if E1  100 0V;Zj2  5 10? I2 + 10 turns V112NN10 90 + 10 – V V N 10 1 21 E 90 turns V Z – 2 2 1 – V 100 0o I V 1  10 0Vo 3 2 10 10 o V2 10 0 I2  0.40j 0.80A Z2 510 j I12N 90 0.9 II120.9  0.36 j 0.72 A I212NN10 90 III3120.040 j 0.080A Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 44 Transformers (13) 1:n + + I primary Isecondary Vprimary Vsecondary – – VVsecondary n primary I I  primary secondary n Magnetically Coupled Circuits - sites.google.com/site/ncpdhbkhn 45 Transformers (14) I primary Isecondary + + Vprimary Vsecondary – – VVsecondary n primary I I  primary secondary 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