EE 221 Circuits II. Chapter 13 Magnetically Coupled Circuits

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1 EE Circuits II Chapter 3 Magnetically Coupled Circuits

2 Magnetically Coupled Circuits 3. What is a transformer? 3. Mutual Inductance 3.3 Energy in a Coupled Circuit 3.4 inear Transformers 3.5 Ideal Transformers 3.6 Applications

3 3. What is a transformer? It is an electrical device designed on the basis of the concept of magnetic coupling. It uses magnetically coupled coils to transfer energy from one circuit to another. It is the key circuit element for stepping up or stepping down ac voltages and currents, impedance matching, isolation, etc. 3

4 3. Mutual Inductance It is the ability of one inductor to induce a voltage across a neighboring inductor, Mutual inductance is measured in henrys (H). di dt v di dt v M di v dt di v M dt 4

5 3. Mutual Inductance If a current enters the dotted terminal of the first coil, the reference polarity of the voltage in the second coil is positive at the dotted terminal. Illustration of the dot convention. 5

6 3. Mutual Inductance Series connection of two mutually coupled inductors; (a) series-aiding connection, (b) series-opposing connection. v di di ( M ) ( dt dt di dt M di dt ) v di di ( M ) ( dt dt di dt M di ) dt M (series - aiding connection) M (series - opposingconnection) 6

7 Problem : 7

8 3. Mutual Inductance Time-domain analysis of a circuit containing coupled coils. Frequency-domain analysis of a circuit containing coupled coils 8

9 3. Mutual Inductance Example Calculate the phasor currents I and I in the circuit shown below. Ans: I A; I.94A 9

10 3.3 Energy in a Coupled Circuit () The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0 k. k M The instantaneous energy stored in the circuit is given by w i i (+) if both currents enter or leave the dotted sides (-) if one current enters the dotted side and the other current leaves dotted side. 0 Mi i

11 3.3 Energy in a Coupled Circuit Example Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = sec if v = 60cos(4t +30 ) V. Ans: k=0.56; w()=0.73j

12 Problem 0:

13 3.4 inear Transformer Reflected Impedance 3 inear Transformer refers to a case where the inductances and and mutual inductance M are constant. V V ref in Z R Z R M R Z I V MI I I Z R MI I R I V ) ( ) ( ) ( ) ( ) (

14 3.4 inear Transformer Example 3 In the circuit below, calculate the input impedance and current I. Take Z =60-00Ω, Z =30+40Ω, and Z =80+60Ω. Ans: Zin ; I 0.53.A 4

15 3.4 inear and Ideal Transformer 5 Voltage Ratio (ignore resistances) Current Ratio In case of unity coupling (i.e., k =) and very large inductances, ) ( ) ( M Z MZ V V Z M I I Z Z V V I I

16 3.4 Ideal Transformer 6 The inductance is proportional to the number of turns squared. In case of an ideal Transformer, n N N an an Z Z V V n N N I I =0-3 N r /(8r+54l)

17 3.5 Ideal Transformer An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite selfinductances. V V N N n I N I N n (a) (b) Ideal Transformer Circuit symbol V>V step-up transformer V<V step-down transformer 7

18 3.5 Ideal Transformer Example 4 An ideal transformer is rated at 400/0V, 9.6 kva, and has 50 turns on the secondary side. Calculate: (a) the turns ratio, (b) the number of turns on the primary side, and (c) the current ratings for the primary and secondary windings. Ans: (a) (b) (c) This is a step-down transformer, n=0.05 N = 000 turns I = 4A and I = 80A 8

19 Problem 36 9

20 3.6 Applications Transformer as an isolation device to isolate ac supply from a rectifier 0

21 3.6 Applications Transformer as an isolation device to isolate dc between two amplifier stages.

22 3.6 Applications Transformer as a matching device Using an ideal transformer to match the speaker to the amplifier Equivalent circuit

23 3.6 Applications Example 5 Calculate the turns ratio of an ideal transformer required to match a 00Ω load to a source with internal impedance of.5kω. Find the load voltage when the source voltage is 30V. Ans: n = 0.; V = 3V 3

24 3.6 Practical Electric Utility transformers Used to step-up or step-down the voltage 4

25 Ideal Auto-Transformer Auto-transformers are used in cases where the voltage ratio is less than. Note that there is only one winding, the primary and secondary side share part of this winding. There is no electrical isolation between the primary and secondary sides. The apparent power rating of an auto-transformer is often much higher than a two-winding transformer of the same size (see example 3.0)

26 3-Phase Transformer

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