10. Magnetically coupled networks

Transformer, Inductor Transformer Used for changing AC voltage levels. Transmission line : high voltage levels are used to decrease power loss due to resistance of copper wires. The smaller magnitude of current, the less power loss, when transmitting the same power. Used for Impedance matching. Transformers are used to change magnitudes of impedances to achieve maximum power transfer condition. Inductors or transformers are difficult to integrate in an IC. (occupies large areas)

Two important laws on magnetic field Current B-field Current generates magnetic field (Biot-Savart Law) Current Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law) B-field Top view Electric field

Magnetic flux Current Magnetic flux :

Self inductance Current

Inductor circuit (S : cross-section area of a coil, μ : permeability) Magnetic field Total magnetic flux linked by N-turn coil Ampere’s Law (linear model) Faraday’s Induction Law Assumes constant L and linear models! Ideal Inductor The current flowing through a circuit induces magnetic field (Ampere’s law). A sudden change of a magnetic field induces electric field that opposes the change of a magnetic field (Faraday’s law), which appear as voltage drops across an inductor terminals.

Mutual Inductance (1) When the secondary circuit is open The current flowing through the primary circuit generates magnetic flux, which influences the secondary circuit. Due to the magnetic flux, a repulsive voltage is induced on the secondary circuit.

Nomenclatures Primary circuit Secondary circuit Primary coil Secondary coil

Secondary voltage and current with different coil winding directions

Two-coil system (both currents contribute to flux) (2) Current flowing in secondary circuit Self-inductance Self-inductance Mutual-inductance Mutual-inductance (From reciprocity)

The ‘DOT’ Convention Dots mark reference polarity for voltages induced by each flux

Example 10.2 Mesh 1 Voltage terms

Mesh 2 Voltage Terms

Example 10.4 1. Coupled inductors. Define their voltages and currents 2. Write loop equations in terms of coupled inductor voltages 3. Write equations for coupled inductors 4. Replace into loop equations and do the algebra

Example E10.3 WRITE THE KVL EQUATIONS 1. Define variables for coupled inductors 2. Loop equations in terms of inductor voltages 3. Equations for coupled inductors 4. Replace into loop equations and rearrange

Example 10.6 DETERMINE IMPEDANCE SEEN BY THE SOURCE 1. Variables for coupled inductors 2. Loop equations in terms of coupled inductors voltages 3. Equations for coupled inductors 4. Replace and do the algebra

10.2 Energy analysis

Coupling coefficient ; Coefficient of coupling

Example 10.7 Compute the energy stored in the mutually coupled inductors Assume steady state operation We can use frequency domain techniques Merge the writing of the loop and coupled inductor equations in one step Circuit in frequency domain

10.3 The ideal transformer Insures that ‘no magnetic flux goes astray’ First ideal transformer equation Since the equations are algebraic, they are unchanged for Phasors. Just be careful with signs Ideal transformer is lossless Second ideal transformer equations Circuit Representations

Reflecting Impedances For future reference Phasor equations for ideal transformer

Non-ideal transformer To build ideal transformers, following two conditions are needed. (1) k=1; (2) ZL<<jωL;

Example 10.8 Determine all indicated voltages and currents SAME COMPLEXITY Strategy: reflect impedance into the primary side and make transformer “transparent to user.” CAREFUL WITH POLARITIES AND CURRENT DIRECTIONS!

Thevenin’s equivalents with ideal transformers Replace this circuit with its Thevenin equivalent Reflect impedance into secondary Equivalent circuit with transformer “made transparent.” One can also determine the Thevenin equivalent at 1 - 1’ To determine the Thevenin impedance...

Thevenin’s equivalents from primary Equivalent circuit reflecting into primary Equivalent circuit reflecting into secondary Thevenin impedance will be the the secondary mpedance reflected into the primary circuit

Example 10.9 Draw the two equivalent circuits Equivalent circuit reflecting into secondary Equivalent circuit reflecting into primary

Example E10.8 Equivalent circuit reflecting into primary Notice the position of the dot marks

Example E10.9 Transfer to secondary

Safety considerations Houses fed from different distribution transformers Braker X-Y opens, house B is powered down When technician resets the braker he finds 7200V between points X-Z Good neighbor runs an extension and powers house B when he did not expect to find any