13. Magnetically Coupled Circuits

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1 13. Magnetically Coupled Circuits The change in the current flowing through an inductor induces (creates) a voltage in the conductor itself (self-inductance) and in any nearby conductors (mutual inductance) The mutual inductance forms the basis for an extremely important device called a transformer The transformer consists of two coils of wire separated by a small distance, and is used to convert ac voltages to higher or lower values depending on the application 1

2 13.1 Mutual Inductance The inductor's voltage and current relationship is v t =L di t dt where L is the self-inductance of the inductor The current flowing through the inductor creates magnetic flux The time rate of change the current induces a voltage in the inductor Coefficient of Mutual Inductance A current flowing in one coil establishes a magnetic flux about that coil and also about a second coil nearby The time-varying flux surrounding the second coil produces a voltage across the terminals of the second coil 2

3 3 v2 t =M 21 di1 t dt v 1 t =M 12 di 2 t dt M21 and M12 are the coefficients of mutual inductance The subscripts on M21 indicates that the voltage response at L2 is produced by a current source at L1 The subscripts on M12 indicates that the voltage response at L1 is produced by a current source at L2 The double headed arrow indicates that these inductors are coupled

4 The Dot Convention M21=M21=M and is always a positive quantity The induced voltage M di/dt may be positive or negative The dot convention determines the sign of the mutual voltage as follows: If the current enters the dotted terminal of one coil, the voltage will be positive at the dot on the second coil 4

5 A current entering the undotted terminal of one coil provides a voltage that is positively sensed at the undotted terminal of the second coil 5

6 Example: (a) determine v1 if i2 = 5 sin 45t A and i1 = 0; (b) determine v2 if i1= 8e t A and i2 = 0 The current i2 (entering undotted terminal) results in a positive reference for the voltage induced across the left coil is the undotted terminal v1 t = M di2 t = 450 cos 45 t V dt The current i1 (entering dotted terminal) results in a positive reference for the voltage induced across the right coil is the dotted terminald di1 t v2 t = M = 16 e t V dt 6

7 Combined Mutual and Self-Induction Voltage This mutual voltage is present independently of and in addition to any voltage of self-induction Since the pairs v1, i1 and v2, i2 each satisfy the passive sign convention, the voltages of selfinduction are both positive Since i1 and i2 each enter dotted terminals, and since v1 and v2 are both positively sensed at the dotted terminals, the voltages of mutual induction are also both positive di1 t di2 t v1 t =L1 M dt dt v2 t =L2 di2 t di t M 1 dt dt 7

8 Since the pairs v1, i1 and v2, i2 are not sensed according to the passive sign convention, the voltages of self-induction are both negative Since i1 enters the dotted terminal and v2 is positively sensed at the dotted terminal v2 t = L 2 di2 t di t M 1 dt dt Since i2 enters the undotted terminal and v1 is positively sensed at the undotted terminal di1 t di2 t v1 t = L1 M dt dt The same considerations lead to identical choices of signs for excitation by a sinusoidal source operating at frequency ω 8

Physical Basis of the Dot Convention The dot convention is made by examining the way in which both coils are physically wound and applying Lenz s law in conjunction with the

upper terminals of coils will be of the same dot sign Dots may be placed either on the upper terminal of each coil or on the lower terminal of each coil Using the right hand rule, the

9 Physical Basis of the Dot Convention The dot convention is made by examining the way in which both coils are physically wound and applying Lenz s law in conjunction with the right-hand-rule Using the right hand rule, the flux direction at the left-hand side is upward and the right-hand side is downward Since the flux at both coils has the same direction, the upper terminals of coils will be of the same dot sign Dots may be placed either on the upper terminal of each coil or on the lower terminal of each coil Using the right hand rule, the flux direction at the left-hand side is upward and the right-hand side is upward Since the flux at both coils has the apposite direction, the upper terminals of coils will be of the opposite dot notation 8

10 9 Example: find the ratio of the output voltage across the 400 Ohm resistor to the source voltage, expressed using phasor notation Transform the circuit into phasor ZL=j L ZM =j M

11 11 Mesh 1 Since I2 enters the undotted terminal of L2, the mutual voltage across L1 must have the positive reference at the undotted terminal. Thus, 1 I1 j10 I1 j 90 I 2 =10 0 o Mesh 2 Since I1 enters the dot-marked terminal of L1, the mutual voltage across L2 must have the positive reference at the dotted terminal. Thus, 400 I 2 j 1000 I 2 j 90 I 1=0 By solving the equations, we find Thus, I2 = o A V o = V1 10 0o

12 P1: Calculate the mesh currents in the shown circuits 12

13 P2: Calculate the mesh currents in the shown circuits 13

14 13.2 Energy in a Coupled Circuit The energy stored in an inductor is given by 1 wl= L i2l 2 We now consider the energy stored in magnetically coupled coils, which leads to 1 1 W = L1 I 12 L 2 I 22 M I 1 I If one current enters a dot-marked terminal while the other leaves a dotmarked terminal, the sign of the mutual energy term is reversed 1 1 W = L1 I 12 L 2 I 22 M I 1 I

15 13 Since I1 and I2 are arbitrary values, they may be replaced by i1(t) andi2(t), which gives the instantaneous energy stored in the circuit 1 1 w t = L1 i21 t L2 i22 t ±M i12 t i22 t 2 2 Since w(t) represents the energy stored within a passive network, it cannot be negative for any values of i1, i2, L1, L2, or M this implies that M L1 L 2 The ratio M / L1 L 2 represents the coupling coefficient, which is a measure of the magnetic coupling between two coils k= or M L1 L 2 M =k L1 L 2 where 0 k 1 or equivalently 0 M L1 L2 For k < 0.5, coils are said to be loosely coupled; and for k > 0.5, they are said to be tightly coupled

16 Example: Determine the coupling coefficient. Calculate the stored energy in the coupled inductors at t=1 s if v=60 cos(4t+30) V The coupling coefficient is Thus, the inductors are tightly coupled To find the energy stored, we need to obtain the frequency-domain equivalent of the circuit. 14

17 We now apply mesh analysis. For mesh 1, For mesh 2, or Solving the two equations 17

18 In the time-domain, At time t = 1 s, 4t = 4 rad = 229.2, and The total energy stored in the coupled inductors is, 18

19 P3: Determine the coupling coefficient. Calculate the stored energy in the coupled inductors at t=1.5 s 17

13.3 The Linear Transformer A transformer is generally a four-terminal device comprising two (or more) magnetically coupled coils The coil that is directly connected to the voltage source is called

20 13.3 The Linear Transformer A transformer is generally a four-terminal device comprising two (or more) magnetically coupled coils The coil that is directly connected to the voltage source is called the primary winding The coil connected to the load is called the secondary winding The resistances R1 and R2 are included to account for the losses (power dissipation) in the coils The transformer is said to be linear if the coils are wound on a magnetically linear material 18

21 19 Transformer: Reflected Impedance We shall compute the input impedance Zin as seen from the source Applying KVL to the two meshes We get the input impedance as The input impedance comprises two terms: The first term, Z11=R1 + jωl1, is the primary impedance The second term is due to the coupling between the primary and secondary windings The second term is known as the reflected impedance ZR Z22=R2 + jωl2+zl

22 Transformer: The T Equivalent Network The voltage-current relationships for the primary and secondary coils give the matrix equation For the T (or Y) network, mesh analysis provides the terminal equations as Then, 20

23 Transformer: The Delta Equivalent Network The current-voltage relationships for the primary and secondary coils give the matrix equation For the Delta network, nodal analysis provides the terminal equations as Then, 21

24 P4: Find the input impedance and the current from the voltage source 22

25 13.4 Ideal Transformers An ideal transformer is one with perfect coupling (k = 1) It consists of two (or more) coils with a large number of turns wrapped on a common core of high permeability The ideal transformer is the limiting case of two coupled inductors where the inductances approach infinity and the coupling is perfect In the frequency domain, thus, Since M = L1 L 2 where n = L2/L1 and is called the turns ratio 23

core as distinct from the air core used in linear transformer According to Faraday s law, the voltage across

26 1. Coils have very large reactances (L1, L2, M ) 2. Coupling coefficient is equal to unity (k = 1) 3. Primary and secondary coils are lossless (R1 = 0 = R2) The vertical lines between the coils indicate an iron core as distinct from the air core used in linear transformer According to Faraday s law, the voltage across the primary winding and the secondary winding are Then, we have where n is, again, the turns ratio or transformation ratio 24

27 27 Using phasor voltages V1 and V2, we may write The energy supplied to the primary equals to the energy absorbed by the secondary Thus, we have The primary and secondary currents are related to the turns ratio as n = 1, we have an isolation transformer n > 1, we have a step-up transformer n < 1, we have a step-down transformer

28 Voltage Polarities and Current Directions 1. If V1 and V2 are both positive or both negative at the dotted terminals, use +n. Otherwise, use n 2. If I1 and I2 both enter into or both leave the dotted terminals, use n. Otherwise, use +n 25

29 28 Ideal Transformer: The Complex Power Express V1 in terms of V2 and I1 in terms of I2 The complex power in the primary winding is Ideal Transformer: Reflected Impedance The input impedance as seen by the source Since ZL=V2/I2, we have (Impedance Matching)

30 Ideal Transformer: Thevenin Equivalent 29

31 P5: Find the Thevenin equivalent of the circuit to the left of the terminals c-d 31

Electric Circuits II Magnetically Coupled Circuits. Dr. Firas Obeidat

Electric Circuits II Magnetically Coupled Circuits Dr. Firas Obeidat 1 Table of contents 1 Mutual Inductance 2 Dot Convention 3 Analyze Circuits Involving Mutual Inductance 4 Energy in a Coupled Circuit