agnetically Coupled Circuits utual Inductance: This is the magnetic flux coupling of coils where the current in one coil causes a voltage to be induced in the other coil. st I d like to emphasize that mutual inductance is very important in electric circuits of all ratings (from low power level mother board electronics to high power transmission on the utility grid). utual inductance sets up the operation of a transformer for: ) Voltage step up /step down, for efficient transfer/use of electric power ( the primacy reason AC was chosen over DC (for the generation, transmission, distribution and utilization of electric power) in 890. (AC was favored by Westinghouse and Tesla, DC was favored by Edison.) ) Impedance matching 3) Isolation (galvanic) 4) Filtering 96
* In addition, we can see how oise can couple from one circuit branch to another. So how do we create mutual inductance. Consider a circuit with a resistor and a single inductor, a coil with turns. st a review of self inductance R φ + The changing current i(t) causes a changing i magnetic field, or a magnetic flux φ,which i ( t) v means (by Faraday s law) that a voltage will be induced across the coil. Turns Faraday s law: Voltage induced in coil φ is induced by i (t), so v or dφ ( so a function of the number of turns and d φ d di φ di dφ di v ( ) di the change in flux) di L ( v Ri) remember the inductance L relates the induced voltage to the changing current. dφ inductance L self inductance di 97
ow let s move coils physically close together so that some flux issuing from one coil links with the other. R + + i ( t ) v v φ R ow a voltage will be induced in the nd coil due to the nd coil experiencing the changing magnetic field φ due to a current in the st coil. If the circuit is closed, a current will flow. 98
To look at this in more practical detail: Inductors are often wound on a magnetic core to provide a better magnetic path than air for flux to flow to achieve better coupling between coils. i i + + v φ φ v φ flux linking due to the current in coil primary secondary turns turns φ linking coil due to the current in coil i i Transformer: -we can transform (step up/down) v depending on the number of turns on the winding ) note: p v*i (conservation of power, so if v goes up, i goes down) 99
φ φ linking coil due to current in coil As resistance R impedes current flow in an electric circuit, reluctance R impedes magnetic flux flow in a magnetic circuit. Just as a copper wire is a far better conductor of electric current than air, Iron is a far better magnetic path for flux than air ( low reluctance R ). Practical note on eddy current losses and need for laminations. As the flux φ varies with time, there will certainly be an induced electric field in the core, and if it is conducting, currents will flow. These are called eddy currents, and will cause heating, and represent an energy loss. They are minimized by decreasing the conductivity of the iron by making the core from a stack of thin laminations insulated from each other. 00
ow we can use Faraday s law to derive expressions for v and v due to the current in coil. Again v v dφ dφ di di L di dφ dφ di di Polarities discussed later. di Self inductance of coil Will determine v in a transformer m is the mutual inductance relating the voltage induced in coil, due to current in coil. Likewise if we energize coil with a current source and leave coil open. di v L, di v (relating the voltage in coil due to the current in coil ) (reciprocal of Reluctance R) 0
For nonmagnetic materials, permeances P and Pare equal. ( can use as approximation) Where we can find the mutual inductance in terms of the self inductances. Where k k L L coefficient of coupling 0 k coils have no common flux φ φ 0 all the flux that links coil also links coil φ φ 0 0
In reality, winding coils such that they share precisely the same flux is physically impossible. Though advanced designs approach unity. if both sides energized by current and the total voltages are: (transformer equations) v v L L di di ± ± di di m relating the voltage induced in coil due to the current in coil m m 03
Thus we can use a Dot Convention to establish corresponding terminals. (to take care of the + and - signs) v v i + v( t) L L v v v i v( t) L L v i 0 i 0 + k L L, v + v v L di v + v L di v k coefficient of 0 < k < coupling 04
This can be summarized in equivalent ways: a) A current entering a dotted (undotted) terminal in one winding induces a voltage ( di the other winding. ) with positive (negative) polarity at the dotted terminal of b) If both currents enter (or leave) the dotted terminals of the coils, the mutual and self inductances for each terminal pair have the same sign, else have the opposite signs. 05
The linear Transformer a c v transformer v source b d load Can be wound on either a nonmagnetic core ( air core) or on a magnetic core (iron). Typically, with a magnetic core, since permeance P varies with the flux ( φ P i ).There is a nonlinear relationship between φ and i.but in the case of a linear transformer on a magnetic core, we only operate in the linear part of the B-H curve, i.e. where the flux and current are proportional, with < Flux density B,φ,λ H, i, ideal linear inductor non-ideal magnetic material i intensity k. 06
ow our model looks as follows: source impedance Z S a R R jω I jωl jωl I V S c Z L not electrically connected R resistance of primary winding R L L b d source primary secondary load linear transformer resistance of secondary winding self inductance of the primary winding self inductance of the secondary winding the mutual inductance 07
The Ideal Transformer Consists of magnetically coupled coils having and turns respectively with the following characteristics: - no winding resistance - no core loss - no leakage, k, φ φ 0 (all the flux that links coil also links coils ) - no energy storage - no magnetization requirement, i.e. negligible current is needed to develop flux. Z S a : c can represent therelationsh ip Z L of v tov with theturn ratios b primary winding ideal d secondary load winding 08
Keep sight of how this is related to Turns ratio a ( V ( dueto I) jω I) P To summarize ( from textbook derivations) ) v v ± v a v Where the polarity depends on the polarity of the windings, It is positive, if v and v are both positive, w.r.t dots and negative Otherwise. i i ± i i For the impedance, it follows: v Where the polarity is negative if i and i are both into or out of the dotted terminals, else positive. Z v Z Z Z i Z i v i a 09
Equivalent Circuits For magnetically Coupled Coils (transformer) In power engineering we often simplify magnetically coupled circuits by replacing them with equivalent circuits that are not magnetically coupled. I R R a c + + v L v L I b d 0
The T equivalent circuit is commonly used. (note: no dots) R a L L c R + + i v v i b d Which satisfies di di v L + di di v L + can verify by writing the mesh equations. ote we must be able to short terminals b and d without disturbing the voltages and currents in the original circuit.
v g 40cos 4000t V Example 8Ω a c 50Ω + + + v L 0mH L 0mH b d lossless VL 6.5μF Transformer coefficient of coupling k 0.5 a) specify the frequency domain T equivalent circuit 3 3 k L L 0.5 (0 0 )(0 0 ) 0mH 0mH j(4000)(0 0 a 3 ) j40ω 0mH 0mH ( j 40Ω) ( j0ω) L L c b 0mH ( j40ω) d
b) find v L 8 a j40 c 50 + Find I V g 40 0 (not rms) j40 I b d I j40 V L 40 I (8 + j40 + j40) - I (j40) 0 Det. I (j40) + I (50 - j40 + 8 + j80 - j40 Δ j40 50 j40) use I (col. sub. with LHS vector matrix) Δ (8 + j80)50 - (-j40)(-j40) 3000 + j4000 3
8 + j80 40 ( j40)40 j40 0 I.9 36.87 Δ A j9600 V L j40i 76.8-53.3 V v L 76.8cos(4000t 53.3 ) V 4