Chapter 13 Magnetically Coupled Circuits. Chapter Objectives:

Transcription

1 Chapter 13 Magnetically Coupled Circuits Chapter Objectives: Understand magnetically coupled circuits. Learn the concept of mutual inductance. Be able to determine energy in a coupled circuit. Learn how to analyze circuits involving linear and ideal transformers. Be familiar with ideal autotransformers. Learn how to analyze circuits involving three-phase transformers. Be able to use PSpice to analyze magnetically coupled circuits. Apply what is learnt to transformer as an isolation device and power distribution Payam Huseyin zarbakhsh Bilgekul EElE301 Eeng224 Circuit Theory II Department of of Electrical and Electronic Engineering Cyprus Eastern International Mediterranean university University

2 Mutual Inductance Transformers are constructed of two coils placed so that the charging flux developed by one will link the other. The coil to which the source is applied is called the primary coil. The coil to which the load is applied is called the secondary coil. Three basic operations of a transformer are: Step up/down Impedance matching Isolation

3 Mutual Inductance Devices Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

4 Mutual Inductance When two coils are placed close to each other, a changing flux in one coil will cause an induced voltage in the second coil. The coils are said to have mutual inductance M, which can either add or subtract from the total inductance depending on if the fields are aiding or opposing. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor. d 1 d( 11 21) v1 N 1 N 1 dt dt d 2 d( 12 22) v2 N 2 N 2 dt dt

5 Mutual Inductance di 1 v2 M21 dt a) Magnetic flux produced by a single coil. b) Mutual inductance M 21 of coil 2 with respect to coil 1. di 2 v1 M12 dt c) Mutual inductance of M 12 of coil 1 with respect to coil 2.

6 Mutual Inductance Mutual inductances M 12 and M 21 are equal. They are referred as M. We refer to M as the mutual inductance between two coils. M is measured in Henry s. Mutual inductance exists when two coils are close to each other. Mutual inductance effect exist when circuits are driven by time varying sources. Recall that inductors act like short circuits to DC. M M M Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

7 Dot Convention If the current ENTERS the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is POSITIVE at the dotted terminal of the second coil. If the current LEAVES the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is NEGATIVE at the dotted terminal of the second coil. di 1 v2 M dt v 1 di M dt 2 v 2 di1 M dt v 1 di M dt 2

8 Dot Convention

9 Coils in Series The total inductance of two coupled coils in series depend on the placement of the dotted ends of the coils. The mutual inductances may add or subtract. a) Series-aiding connection. L=L 1 +L 2 +2M b) Series-opposing connection. L=L 1 +L 2-2M

10 Time-domain and Frequency-domain Analysis j M V 1 I 1 j L 1 j L 2 I 2 V 2 a) Time-domain circuit b) Frequency-domain circuit Time Domain di1 di2 v1 i1r 1 L1 M dt dt di di v2 i2r2 L2 M dt dt Frequency Domain 2 1 V ( R j L ) I j MI V j MI ( R j L ) I

11 Induced mutual voltages Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

12 Induced mutual voltages

13 P.P.13.2 Determine the phasor currents + - Mesh =(5+j2+j6-j3 2)I j6i j3i Mesh 2 0=(j6-j4)I j6i j3i j3i 2 j3i j3i 1 Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

14 Mutually Induced Voltages To find I 0 in the following circuit, we need to write the mesh equations. Let us represent the mutually induced voltages by inserting voltage sources in order to avoid mistakes and confusion. -j50 I o I 3 + j20i c V I a j40 j10i b + j30i c + j80 I 1 I 2 I b + + j30i b j10i a j60 I c + j20i a 100 I a = I 1 I 3 I b = I 2 I 1 I c = I 3 I 2 I o = I 3 Blue Voltage due to I a Red Voltage due to I c Green Voltage due to I b

15 Mutually Induced Voltages To find I 0 in the following circuit, we need to write the mesh equations. Let us represent the mutually induced voltages by inserting voltage sources in order to avoid mistakes and confusion.

16 Energy in a Coupled Circuit The total energy w stored in a mutually coupled inductor is: Positive sign is selected if both currents ENTER or LEAVE the dotted terminals. Otherwise we use Negative sign. 1 1 w L i L i Mi i Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

17 Coupling Coefficient The Coupling Coefficient k is a measure of the magnetic coupling between two coils 0 k 1 k 1 Perfect Coupling k 0.5 Loosly Coupling k 0.5 Tightly Coupling a) Loosely coupled coil b) Tightly coupled coil 0 k 1 k M LL 1 2 Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

18 Linear Transformers A transformer is generally a four-terminal device comprising two or more magnetically coupled coils. The transformer is called LINEAR if the coils are wound on magnetically linear material. For a LINEAR TRANSFORMER flux is proportional to current in the windings. Resistances R 1 and R 2 account for losses in the coils. The coils are named as PRIMARY and SECONDARY.

19 Reflected Impedance for Linear Transformers Let us obtain the input impedance as seen from the source, Z R Z R V ( R j L ) I j MI j MI ( R j L Z ) I L V M Z in R j L R j L Z I R j L Z L 2 2 M R j L Z 2 2 L REFLECTED IMPEDANCE Secondary impedance seen from the primary side is the Reflected Impedance. R

20 Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

21 Equivalent T Circuit for Linear Transformers The coupled transformer can equivalently be represented by an EQUIVALENT T circuit using UNCOUPED INDUCTORS. a) Transformer circuit b) Equivalent T circuit of the transformer L L M, L L M, L M a 1 b 2 c Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

22 Equivalent П Circuit for Linear Transformers The coupled transformer can equivalently be represented by an EQUIVALENT П circuit using uncoupled inductors. a) Transformer circuit b) Equivalent Π circuit of the transformer L L M L L L L M L L L M A, B, C L2 M L1 M M Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university

23 L L M a b c 1 L L M L 2 M