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
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean 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

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.

5. b) Mutual inductance M21 of coil 2 with respect to coil 1.
a) Magnetic flux produced by a single coil.
b) Mutual inductance M21 of coil 2 with respect to coil 1.
c) Mutual inductance of M12 of coil 1 with respect to coil 2.

6. Mutual Inductance Mutual inductances M12 and M21 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.

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.

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.
Series-aiding connection.
L=L1+L2+2M
b) Series-opposing connection.
L=L1+L2-2M

10. Time-domain and Frequency-domain Analysis
jM V1 I1
jL1
jL2 I2 V2
b) Frequency-domain circuit
a) Time-domain circuit

11. Induced mutual voltages

12. Induced mutual voltages

13. P.P.13.2 Determine the phasor currents+ -
j3I1
j3I2

14. Mutually Induced Voltages
To find I0 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. + 
 + I1 I2 Io
j20Ic
100 
500 V I3
+ 
j10Ib
j40
j30Ic
j80
j10Ia
j20Ia
j60
j30Ib
-j50 Ia Ib Ic
Ia = I1 – I3
Ib = I2 – I1
Ic = I3 – I2
Io = I3
Blue Voltage due to Ia
Red Voltage due to Ic
Green Voltage due to Ib

15. Mutually Induced Voltages
To find I0 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.

17. Coupling Coefficient
The Coupling Coefficient k is a measure of the magnetic coupling between two coils
Loosely coupled coil b) Tightly coupled coil

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 R1 and R2 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, ZR
Secondary impedance seen from the primary side is the Reflected Impedance.

21. Equivalent T Circuit for Linear Transformers
The coupled transformer can equivalently be represented by an EQUIVALENT T circuit using UNCOUPED INDUCTORS.
Transformer circuit b) Equivalent T circuit of the transformer

22. Equivalent П Circuit for Linear Transformers
The coupled transformer can equivalently be represented by an EQUIVALENT П circuit using uncoupled inductors.
Transformer circuit b) Equivalent Π circuit of the transformer

24. Homework 2
Problem 13.79
X23
Submit your results by giving in the following results similar to the form given below by May 2, 2007.
Your origiinal schematic diagram
The print page of your results
Repeat the calculation for 2 other values of X23=j0, j10, j15 Ω

25. Homework 2
Schematic Diagram