1 10 Magnetically coupled networks EMLAB Transformer Inductor
1 10. Magnetically coupled networks EMLAB
Transformer, Inductor 2 1. Transformer ① Used for changing AC voltage levels. ② 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. ③ Used for Impedance matching. Transformers are used to change magnitudes of impedances to achieve maximum power transfer condition. 2. Inductors or transformers are difficult to integrate in an IC. (occupies large areas) EMLAB
3 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 EMLAB
4 Magnetic flux Current Magnetic flux : EMLAB
5 Self inductance Current EMLAB
6 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. EMLAB
7 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. EMLAB
8 Nomenclatures Primary circuit Secondary circuit Primary coil Secondary coil EMLAB
Secondary voltage and current with different coil winding directions 9 EMLAB
Two-coil system 10 (both currents contribute to flux) (2) Current flowing in secondary circuit Self-inductance Mutual-inductance (From reciprocity) EMLAB
11 The ‘DOT’ Convention Dots mark reference polarity for voltages induced by each flux EMLAB
12 Example 10. 2 Write mesh equations for the circuit using the assigned mesh currents. Mesh 1 : Mesh 2 : EMLAB
Example 10. 4 We wish to determine the output voltage Vo 13 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 EMLAB
Example E 10. 3 14 Write the KVL equations in standard form for the network 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 EMLAB
Example 10. 6 15 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 EMLAB
16 10. 2 Energy analysis EMLAB
17 Coupling coefficient ; Coefficient of coupling EMLAB
Example 10. 7 Compute the energy stored in the mutually coupled inductors at 5 ms. 18 Circuit in frequency domain EMLAB
19 10. 3 The ideal transformer Insures that ‘no magnetic flux goes astray’ First ideal transformer equation (Ideal transformer is lossless) Second ideal transformer equations Since the equations are algebraic, they are unchanged for Phasors. Just be careful with signs EMLAB
Reflecting Impedances 20 Complex power (for future reference) (Both + signs at dots) Phasor equations for ideal transformer EMLAB
21 Non-ideal transformer EMLAB
Example 10. 8 22 Determine all indicated voltages and currents Strategy: reflect impedance into the primary side and make transformer “transparent to user. ” Careful with polarities and current directions! EMLAB
Thevenin’s equivalents with ideal transformers 23 Replace this circuit with its Thevenin equivalent Equivalent circuit with transformer “made transparent. ” EMLAB
Thevenin’s equivalents from primary 24 Thevenin impedance will be the secondary impedance reflected into the primary circuit Equivalent circuit reflecting into primary EMLAB
Example 10. 9 25 Draw the two equivalent circuits Equivalent circuit reflecting into secondary Equivalent circuit reflecting into primary EMLAB
26 Example E 10. 15 Equivalent circuit reflecting into primary Notice the position of the dot marks EMLAB
27 Example 10. 16 Form an equivalent circuit for the transformer and primary, and use the resultant network to find Vo. Transfer to secondary EMLAB
28 Safety considerations Houses fed from different distribution transformers Breaker X-Y opens, house B is powered down When technician resets the breaker he finds 7200 V between points X-Z, which endangers him. Good neighbor runs an extension and powers house B EMLAB
- Slides: 28