Magnetically Coupled Circuits

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1 DR. GYURCSEK ISTVÁN Magnetically Coupled Circuits Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN: Ch. Alexander, M. Sadiku: Fundamentals of Electric Circuits, 6th Ed., McGraw Hill NY 2016, ISBN: W. M. Flanagan, Handbook of Transformer Design and Applications, 2nd ed. (New York: McGraw-Hill, 1993) Mayergoyz - Lawson: Basic Electric Circuit Theory (ISBN13: ) Simonyi K.: Villamosságtan. AK Budapest 1983, ISBN: Dr. Selmeczi K. Schnöller A.: Villamosságtan 1. MK Budapest 2002, TK szám: 49203/I Dr. Selmeczi K. Schnöller A.: Villamosságtan 2. TK Budapest 2002, ISBN: Zombory L.: Elektromágneses terek. MK Budapest 2006, ( 1 gyurcsek.istvan@mik.pte.hu

2 Coupled Inductances Linear Transformers Ideal Transformers Three-Phase Transformers Transformer Applications 2 gyurcsek.istvan@mik.pte.hu

3 Self-Inductance (Recall) v = N dφ = N dφ di di = L di Self-inductance L = N dφ di 3 gyurcsek.istvan@mik.pte.hu

4 Mutual Inductance (Recall) Φ 1 = Φ 11 + Φ 12 Φ 11 : links only coil 1, Φ 12 : links both coils dφ 1 v 1 = N 1 = N dφ 1 di 1 1 di 1 = L di 1 1 v 2 = N 2 dφ 12 Mutual inductance dφ 12 di 1 = N 2 di 1 = M di 1 21 M 21 = N 2 dφ 12 di 1 OC mutual (induced) voltage v 2 = M 21 di 1 Φ 2 = Φ 21 + Φ 22 dφ 2 v 2 = N 2 = N dφ 2 di 2 2 di 2 = L di 2 2 v 1 = N 1 dφ 21 dφ 21 di 2 = N 1 di 2 = M di 2 12 OC mutual (induced) voltage v 1 = M 12 di 2 Mutual inductance M 12 = N 1 dφ 21 di 2 M 12 = M 21 = M 4 gyurcsek.istvan@mik.pte.hu

5 Dot Convention [current enters at dot] [mutual voltage positive at dot] 5 gyurcsek.istvan@mik.pte.hu

6 Dot Convention - Examples Series-aiding connection L = L 1 + L 2 + 2M Series-oppositing connection L = L 1 + L 2 2M 6 gyurcsek.istvan@mik.pte.hu

7 Time and Frequency Domain Analysis di 1 v 1 = i 1 R 1 + L 1 + M di 2 di 2 v 2 = i 2 R 2 + L 2 + M di 1 V 1 = R 1 + jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + R 2 + jωl 2 I 2 V = Z 1 + jωl 1 I 1 jωmi 2 0 = jωmi 1 + Z L + jωl 2 I 2 7 gyurcsek.istvan@mik.pte.hu

8 Energy in Coupled Circuit Approaching in two steps di 1 1 : i 1 = 0 I 1, i 2 = 0 p 1 t = v 1 i 1 = i 1 L 1 I 1 w 1 = න p 1 = L 1 න i 1 di 1 = 1 2 L 2 1I : i 1 = I 1, i 2 = (0 I 2 ) p 2 t = i 1 M di 2 + i di 2 2L 2 I 2 I 2 w 2 = න p 2 = MI 1 න di 2 + L 2 න i 2 di 2 = MI 1 I L 2 2I Total w (both of currents max value) w = w 1 + w 2 = 1 2 L 1I L 2I 2 2 ± MI 1 I 2 (± dot convention!) 8 gyurcsek.istvan@mik.pte.hu

Coupling Coefficient w = 1 2 L 1i 1 2 + 1 2 L 2i 2 2 ± Mi 1 i 2 Passive circuit 1 2 L 1i 1 2 + 1 2 L 2i 2 2 Mi 1 i 2 0 a b 2 = a 2 2ab + b 2 Coupling coefficient Perfekt coupling.

9 Coupling Coefficient w = 1 2 L 1i L 2i 2 2 ± Mi 1 i 2 Passive circuit 1 2 L 1i L 2i 2 2 Mi 1 i 2 0 a b 2 = a 2 2ab + b 2 Coupling coefficient Perfekt coupling... (k = 1) Tight coupling (k > 0.5) Loose coupling (k < 0.5) 1 2 i 1 L 1 i 2 L i1 i 2 L 1 L 2 M 0 L 1 L 2 M 0 M L 1 L 2 M = k L 1 L 2 k measure of the magn. coupling bw. coils (0-1) k = Φ 12 Φ 1 = Φ 12 Φ 12 + Φ 11 = Φ 21 Φ 2 = Φ 21 Φ 21 + Φ 22 9 gyurcsek.istvan@mik.pte.hu

10 Coupled Inductances Linear Transformers Ideal Transformers Three-Phase Transformers Transformer Applications

Linear Transformers Linear transformer Φ~I V = R

impedance Z R 2 + jωl 2 + Z R = L ω 2 M 2 R 2 +

11 Linear Transformers Linear transformer Φ~I V = R 1 + jωl 1 I 1 jωmi 2 0 = jωmi 1 + R 2 + jωl 2 + Z L I 2 Primary impedance Z PR = R 1 + jωl 1 Z in = V I 1 = R 1 + jωl 1 + ω 2 M 2 Reflected (coupled) impedance Z R 2 + jωl 2 + Z R = L ω 2 M 2 R 2 + jωl 2 + Z L 11 gyurcsek.istvan@mik.pte.hu

12 Equivalent T - Circuit Equivalent circuits convenient analysis V 1 V 2 = jωl 1 jωm jωm jωl 2 I 1 I 2 mesh analysis V 1 V 2 = jω L a + L c jωl c jωl c jω L b + L c I 1 I 2 L a = L 1 M, L b = L 2 M, L C = M 12 gyurcsek.istvan@mik.pte.hu

13 Equivalent П - Circuit Recall maths M = a 11 a 12 a 21 a 22 = a 11 a 22 a 12 a 21 M 1 = 1 adjm = V 1 V 2 = jωl 1 jωm jωm jωl 2 a 22 a 12 a 21 I 1 I 2 I 1 mesh analysis a 11 I 2 = I 1 I 2 = L 2 adjm = a 22 a 12 a 21 a 11 M jω L 1 L 2 M 2 jω L 1 L 2 M 2 M L 1 jω L 1 L 2 M 2 jω L 1 L 2 M jωl A jωl C jωl C jωl C jωl B jωl C V 1 V 2 V 1 V 2 L A = L 1L 2 M 2 L 2 M, L B = L 1L 2 M 2 L 1 M, L C = L 1L 2 M 2 M 13 gyurcsek.istvan@mik.pte.hu

14 Coupled Inductances Linear Transformers Ideal Transformers Three-Phase Transformers Transformer Applications

15 Ideal Transformers Let s see k = 1 μ 1 : V 1 = jωl 1 I 1 + jωmi 2 2 : V 2 = jωmi 1 + jωl 2 I : V 2 = jωl 2 I 2 + MV 1 L 1 jωm2 I 2 L 1 k = 1 M = L 1 L 2 I 1 = V 1 jωmi 2 jωl 1 V 2 = jωl 2 I 2 + L 1L 2 V 1 L 1 jωl 1L 2 I 2 L 1 = L 2 L 1 V 1 = n V 1 n turns ratio = constant, even ifl 1, L 2, M Ideal transformer Coils have very large reactances L 1, L 2, M Coupling coefficient is equal to unity k = 1 Primary and secondary coils are lossless R 1 = R 2 = 0 v 1 = N 1 dφ v 2 v 2 = N 2 dφ Isolation transformer n = 1, (V 2 = V 1 ) Step-up transformer n > 1, (V 2 > V 1 ) Step-down transformer n < 1, (V 2 < V 1 ) = V 2 = N 2 = n v 1 V 1 N 1 15 gyurcsek.istvan@mik.pte.hu

16 Ideal Transformers Power conservation v 1 i 1 = v 2 i 2 V 1 I 1 = V 2 I 2 I 1 I 2 = V 2 V 1 = n V 2 V 1 = N 2 N 1, I 2 I 1 = N 1 N 2 Sign convention (four figures ) Complex power S 1 = V 1 I 1 = V 2 n ni 2 = V 2 I 2 = S 2 Impedance transform (imp. matching i.e. for max PWR transfer) Z in = V 1 = 1 V 2 I 1 n 2 = Z L I 2 n 2 16 gyurcsek.istvan@mik.pte.hu

17 Ideal Transformers Equivalent Circuit 1 (1) Reflecting secondary to the primer circuit (Thevenin equivalence) I 1 = 0 I 2 = 0 V Th = V 1 = V 2 n = V S2 n Z Th = V 1 I 1 = V 2 n ni 2 = Z 2 n 2, V 2 = Z 2 I 2 17 gyurcsek.istvan@mik.pte.hu

18 Ideal Transformers Equivalent Circuit 2 (1) Reflecting secondary to the primer circuit (result) (2) Reflecting primer to the secondary circuit 18 gyurcsek.istvan@mik.pte.hu

19 Ideal Autotransformers Autotransformer single coil with (variable) tap Electric isolation is lost Step-down transformer Step-up transformer V 1 V 2 = N 1 + N 2 N 2 = 1 + N 1 N 2, S 1 = S 2 I 1 I 2 = N 2 N 1 + N 2 V 1 V 2 = N 1 N 1 + N 2, S 1 = S 2 I 1 I 2 = N 1 + N 2 N 1 = 1 + N 2 N 1 19 gyurcsek.istvan@mik.pte.hu

20 Coupled Inductances Linear Transformer Ideal Transformer Three-Phase Transformers Transformer Applications

21 Three-Phase Transformers 1 Three-phase transformer Smaller and cheaper than transformer bank Four connections (YY, DD, DY, YD) For any connection P total = S total cos Θ = 3V L I L cos Θ Q total = S total sin Θ = 3V L I L sin Θ S total = 3V L I L 21 gyurcsek.istvan@mik.pte.hu

22 Three-Phase Transformers 2 22 gyurcsek.istvan@mik.pte.hu

23 Coupled Inductances Linear Transformer Ideal Transformer Three-Phase Transformers Transformer Applications

24 App. Isolation Device

25 App. Matching Device Matching transformer for maximum power transfer Z Th = Z L n 2 n = Z L Z Th I p = V Th Z Z Th + L n 2 I s = I p n = V Th n Z Z Th + L n 2 With matching transformer Equivalent circuit P L = I s 2 Z L = V Th n Z Z Th + L n 2 2 Z L = nv Th n 2 Z Th + Z L 2 Z L Without matching transformer P L = I 2 Z L = V Th Z Th + Z L 2 Z L 25 gyurcsek.istvan@mik.pte.hu

26 App. - Power Distribution P loss = I 2 R w = V 2 R w V = V send V rec Step-up transforming V send V rec P loss 0 26 gyurcsek.istvan@mik.pte.hu

27 Questions

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