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

Magnetically coupled circuits Lecture 5 Magnetically coupled circuits

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

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

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.

Positive and negative coupling Positive coupling Negative coupling

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

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

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

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

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

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.

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)

The circuit after elimination of coupling

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

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

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.

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

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

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

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

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

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.

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

Solution Symbolic values of parameters Circuit description

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