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Dot convention for coils in series; the sign indicates the polarity of the mutual voltage; (a) series-aiding connection, (b) series-opposing connection.
Calculate the phasor currents I1 and I2 in the circuit shown below.
A04.1491.2I A;39.4901.13I 21 Ans:
*Refer to textbook pp. 561-562
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13.3 Energy in a Coupled Circuit (1)13.3 Energy in a Coupled Circuit (1)
• The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0≤k≤1.
• The instantaneous energy stored in the circuit is given by
21LLkM
21222
211 2
1
2
1IMIiLiLw
Positive sign for mutual term if both currents enter or leave the dotted terminals. Negative sign if otherwise.
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13.3 Energy in a Coupled Circuit (2)13.3 Energy in a Coupled Circuit (2)
Example 13.3 Consider the circuit below. Determine the coupling
coefficient. Calculate the energy stored in the coupled inductors at time t = 1s if v=60cos(4t +30°) V.
Ans: k=0.56; w(1)=20.73J
Refer to textbook pp. 566-567
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13.5 Ideal Transformer (1)13.5 Ideal Transformer (1)• A transformer is said to be ideal if:
1. Coils have very large reactances (L1, L2, M )2. Coupling coefficient is equal to unity (k = 1)3. Primary and secondary coil are loss less (R1 = 0 = R2)
• An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite self-inductances.
•Input impedance is also known as reflected impedance, since it appears as if load impedance is reflected to primary side.
•Common practice in analyzing circuit with ideal transformer is to eliminate transformer by reflecting impedances and sources from one side of the transformer to the other.
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13.5 Ideal Transformer (8)13.5 Ideal Transformer (8)•Example: Want to reflect secondary side of circuit below to primary side:
1. Find Thevenin equivalent of circuit to the right of terminals a-b. (i.e. Obtain VTh as open-circuit voltage at terminals a-b).
2. Get ZTh by removing voltage source at secondary winding and insert a unit source at terminals a-b.
13.5 Ideal Transformer (11)13.5 Ideal Transformer (11) The general rule for eliminating the transformer and reflecting the secondary circuit to the primary side is: divide the secondary impedance by n2, divide the secondary voltage by n, and multiply the secondary current by n.
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13.5 Ideal Transformer (12)13.5 Ideal Transformer (12) The general rule for eliminating the transformer and reflecting the primary circuit to the secondary side is: multiply the primary impedance by n2, multiply the primary voltage by n, and divide the primary current by n.