LECTURE 6 MUTUAL INDUCTANCE

Transcription

1 ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 6 MUTUAL INDUCTANCE Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter Sauer s ECE 330 lecture notes. Some slides are taken from Ali Bazi s presentations Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook. 1/31/018

2 SOURCE ON PRIMARY SIDE When two coils are magnetically coupled there is mutual inductance. Assuming an ac voltage source on one side and an open circuit on the other (i = 0): v N 1 d d 1 N i 1 (t) v 1 (t) 1 l1 N 1 N v (t) 1/31/018

3 SOURCE ON PRIMARY SIDE Let the flux and current be linearly related (no saturation): M i, L i Then, di1 di1 v M 1, and v 1 L1. M 1 is the mutual inductance. L 1 is the self-inductance of coil 1. 1/31/018 3

4 SOURCE ON SECONDARY SIDE Assuming an ac voltage source on one side and an open circuit on the other (i 1 = 0):. l 1 d d N 11, v 1 N 1 Let the flux and current be linearly related (no saturation): M 1 is the mutual inductance and L is the self-inductance of coil. M i, L i. 1 1 di di v L, and v1 M1. N 1 N 1/31/018 4 v 1 (t) 1 l i (t) v (t)

5 SOURCES ON BOTH SIDES Using superposition: Let M 1 = M 1 = M, then: 1 l l N N L i M i N N M i L i L i M i M i L i 1 v 1 (t) i 1 (t) l1 1 N 1 N 1 d1 di1 di v1 L1 M d di di v L M l 1 i (t) v (t) 1/31/018 5

6 COUPLING COEFFICIENT Define the coupling coefficient as: If k = 0, no coupling and M = 0. k If k =1, ideal coupling with zero leakage: M LL 1 k M LL 1 11 l1 1 l 1 Therefore, M is always 0. and 0k 1 0 M L1L. 1/31/018 6

7 What if i is reversed? What if v is reversed? COUPLING COEFFICIENT d1 di1 di ' v1 L1 M d di ' di v L M 1 d1 di1 di ' v1 L1 M d di ' di v ' L M 1 1/31/018 7 i v v i

8 EQUIVALENT CIRCUIT Ni 11 1 l1 1 l l Ni 1/31/018 8

9 DOT MARKINGS Dots relate the flux direction between coils. If two fluxes are in the same direction, they add, otherwise, they subtract. Depending which ends you connect the load to the secondary coil you either get an Output voltage in sync. With the input voltage or in reverse phase. v 1 (t) N 1 N i (t) v (t) 1/31/018 9 i 1 (t) l1 1 l

10 DOT MARKINGS The polarity markings are assigned such that a positively increasing current in the dotted terminal in one winding induces a positive voltage at the dotted terminal of the other winding i 1 (t) i (t) v 1 (t) l1 1 l N 1 N v (t) 1/31/018 10

11 DOT MARKINGS Assume the following configuration. 1) Select one coil and one terminal and place a dot on that terminal. ) Assume a current is flowing and determine the flux direction. 1 i 1 (t) 1 1' 3) Place a test current in the second coil and determine its flux direction. 1/31/ '

12 DOT MARKINGS i 1 (t) 1 1' 1 i (t) ' 1 4) If 1 and add, then place dot on (where test current enters). 5) If they subtract, then place dot on (where test current leaves). 1 1' ' 1/31/018 1

13 DOT MARKINGS Practical determination of dot locations: 1) Build the following circuit. -Turn the switch on => pulse is generated where di/ is not zero on the secondary side. -If the pulse causes the meter (V) to read positive, then the dot on the secondary is on the top terminal. -If (V) reads a negative pulse, then the dot is on the lower side. 1/31/ R + V dc V -

14 EXAMPLES The dots in the circuit below are as shown. i 1 (t) 1 1' i (t) ' 1/31/018 14

15 WRITING EQUATIONS WITH MUTUALLY COUPLED COILS Suppose we have two mutually coupled coils and the dot markings are as shown The self induced voltage due to the self inductance is in the direction of the current and is a voltage drop. The polarity of the mutually induced voltage depends on the dot marking 1/31/018 15

16 WRITING EQUATIONS WITH MUTUALLY COUPLED COILS di v 1 = i1r1 L1 M di 1 Source: jacobs-university.de di v = i R L M di 1 1/31/018 16

17 WRITING EQUATIONS WITH MUTUALLY COUPLED COILS di v 1 = i1r1 L1 M di 1 Source: jacobs-university.de di v = ir L M di 1 1/31/018 17

18 WRITING EQUATIONS WITH MUTUALLY COUPLED COILS Source: jacobs-university.de 1/31/018 18

19 WRITING EQUATIONS WITH MUTUALLY COUPLED COILS If the reference current in a coil leaves the dotted (undotted) terminal, then the voltage induced at the dotted (undotted) terminal of the other coil has a negative sign. 1/31/018 19