1. CIRCUITS and SYSTEMS – part I
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

2. Magnetically coupled circuits
Lecture 5
Magnetically coupled circuits

3. Phenomena of magnetically coupled coils
A magnetically (inductively) coupled inductor circuit consists of more than one coil of inductive wire wound on the same magnetic core. The flux linkage of the z-turn coil is defined as flux Φ times the number of turns z. Each coils has its own flux linkage. The total flux linking each coil is the sum or difference of two compounds: the leakage flux ψii produced by current through the coil itself and the mutual flux ψij produced by the current through the other coil j. For two magnetically coupled coils we have

4. Self- and mutual inductance
For magnetically coupled coils we have .
Self-inductances L1, L2
Mutual inductance (M12=M21=M)
Coefficient of coupling k

5. Voltage on magnetically coupled coils
The voltage of magnetically coupled coils
Dot (marking convention)
The self-induced voltage and the mutually induced are additive (positive coupling) , i.e. have the same polarity if both coil current enter/leave the dotted or undotted ends of the coils. They are subtractive (negative coupling), i.e. opposite to each other, if one coil current enters/leaves the dotted end of the coil while the other coil current enters/leaves the undotted end of the other coil.

6. Positive and negative coupling
Positive coupling
Negative coupling

7. Symbolic representation of magnetically coupled coils
At sinusoidal excitation the symbolic complex form equations are of the form
Sign plus for positive coupling and minus for negative one.
Self inductive reactance of the coil
Mutual inductive reactance

8. Voltage over magnetically coupled coils
RMS complex values of voltages across the magnetically coupled coils
ZL1=jω L1, ZL2=jω L2 – the complex self-impedances of the coils
ZM= jω M – the complex mutual inductance of the coils

9. Elimination of magnetic coupling
Elimination of magnetic couplings is done directly by inspection of circuits and takes into account only the dot points of coupled coils (the direction of coil currents has no influence on the elimination).
We recognize two types of couplings:
coils of identical positions of dots with respect to common node
coils of reverse positions of dots with respect to common node

10. Elimination of magnetic coupling (cont.)
Dots identically situated to the common node
Dots differently situated to the common node

11. Rules of elimination of coupling:
Dots identically situated with respect to the common node
Dots situated in a reverse way with respect to the common node

12. Example 1
The original circuit with magnetic coupling (a) and after elimination of couplings (b)
The circuit after elimination of magnetic coupling is equivalent to the original circuit only with respect to the currents! The voltages inside the circuits may change.

13. Example 2
Calculate the currents and voltages of the circuit with magnetic coupling. Assume: R=5Ω, L1=2H, L2=2H, M=1H, i(t)=5 sin(t+45o)

14. The circuit after elimination of coupling

15. The succeding steps of calculations
Symbolic values of the circuit elements
Input impedance of the circuit

16. The succeding steps of calculations (cont.)
Voltage UAB
Currents
Voltages of the coupled coils

17. Transformer – principle of operation
A transformer is a device that transfers electrical enegy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magentic flux in the transformer's core, and thus a varying magnetic field through the secondary winding. This varying magnetic field induces a varying electromotive force or voltage in the secondary winding.
The main role of transformer is change the value of voltage from one level to another one. Change of voltages change also the levels of currents in both coils.

18. Transformer – principle of operation (cont.)
Transmission of energy from primary coil to the secondary one is done through the magnetic field.
Illustration of transformer performance
Primary windings
Secondary windings

19. Ideal transformer Assumptions: No losses of power
Perfect coupling of coils (k=1)
Turn ratio=voltage ratio
Graphical symbol of ideal transformer

20. Mathematical relations for ideal transformer
Voltage ratio (U1/U2) is equal to the turn ratio (z1/z2)
Because of lossless operation we have
Matrix description of ideal transformer

21. Practical realization of transformer
Practical implementation of transformer is made by using ferromagnetic core, for which we have (k≈1).
Electrical model of transformer applying magnetically coupled coils

22. Mathematical description
Equations for 2 magnetically coupled coils
Output voltage
At ideal coupling (k≈1)
Output voltage of transformer

23. Voltage and current relations at ideality
Reactances of coils are approximately depending on the number of turns according to relations
K- construction coefficient
At ideal coupling the secondary (output) voltage of transformer is dependent only on turn ratio.
The minus sign is a result of the assumed directions of voltages and dot convention.

24. Example
Calculate the currents in the circuit with ideal transformer at n=2. Assume: e(t)=14.1sin(t), R=5Ω, C=0.2F.

25. Solution
Symbolic values of parameters
Circuit description

26. The numerical results After subsituting the numerical values we get
It is easy to show that