EE310 Lecture 11.pptx - EE310 Lecture 11 13.3 Energy in a...

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EE310 Lecture 11
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13.3 Energy in a Coupled Circuit As we know the energy stored in an inductor is given by We now want to determine the energy stored in magnetically coupled coils. Consider the circuit in Fig. 13.14. We assume that currents and are zero initially, so that the energy stored in the coils is zero. If we let increase from zero to while maintaining , the power in coil 1 is Fig. 13.14 ) 23 . 13 ( 2 1 2 Li w 2 1 i i 1 i 1 I 0 2 i ) 24 . 13 ( 1 1 1 1 1 1 dt di L i i v t p
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If we now maintain and increase from zero to the mutual voltage induced in coil 1 is , while the mutual voltage induced in coil 2 is zero, since does not change. The power in the coils is now and the energy stored in the circuit is ) 25 . 13 ( 2 1 1 0 2 1 1 1 1 1 1 1 I I L di i L dt p w 1 1 I i 2 i 2 I dt di M / 2 12 1 i ) 26 . 13 ( 2 2 2 2 12 1 2 2 2 12 1 2 dt di L i dt di M I v i dt di M i t p ) 27 . 13 ( 2 1 2 2 2 2 1 12 0 2 2 2 0 2 1 12 2 2 2 2 I L I I M di i L di I M dt p w I I
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The total energy stored in the coils when both and have reached constant values is If we reverse the order by which the currents reach their final values, that is, if we first increase from zero to and later increase from zero to the total energy stored in the coils is Since the total energy stored should be the same regardless of how we reach the final conditions, comparing Eqs. (13.28) and (13.29) leads us to conclude that And 2 1 i i ) 28 . 13 ( 2 1 2 1 2 1 12 2 2 2 2 1 1 2 1 I I M I L I L w w w 2 i 2 I 1 i 1 I ) 29 . 13 ( 2 1 2 1 2 1 21 2 2 2 2 1 1 I I M I L I L w ) 30 . 13 ( 21 12 a M M M ) 30 . 13 ( 2 1 2 1 2 1 2 2 2 2 1 1 b I MI I L I L w
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This equation was derived based on the assumption that the coil currents both entered the dotted terminals. If one current enters one dotted terminal while the other current leaves the other dotted terminal, the mutual voltage is negative, so that the mutual energy is also negative. In that case, Also, since and are arbitrary values, they may be replaced by and which gives the instantaneous energy stored in the circuit the general expression The positive sign is selected for the mutual term if both currents enter or leave the dotted terminals of the coils; the negative sign is selected otherwise. 2 1 I MI ) 31 . 13 ( 2 1 2 1 2 1 2 2 2 2 1 1 I MI I L I L w I 1 I 2 I 1 i 2 i ) 32 . 13 ( 2 1 2 1 2 1 2 2 2 2 1 1 i Mi i L i L w
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We will now establish an upper limit for the mutual inductance M . The energy stored in the circuit cannot be negative because the circuit is passive. This means that the quantity must be greater than or equal to zero: To complete the square, we both add and subtract the term on the right-hand side of Eq. (13.33) and obtain The squared term is never negative; at its least it is zero. Therefore, the second term on the right- hand side of Eq. (13.34) must be greater than zero; that is, 2 1 2 2 2 2 1 1 2 / 1 2 / 1 i Mi i L i L ) 33 . 13 ( 0 2 1 2 1 2 1 2 2 2 2 1 1 i Mi i L i L 2 1 2 1 L L i i ) 34 . 13 ( 0 2 1 2 1 2 1 2 2 2 1
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