Electric Circuits, 9th Edition P23. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 196 Inductance Capacitance and Mutual Inductance Figure The magnetically coupled coils of Fig. with coil 2 excited and coil 1 open. shown in Fig. . Again the polarity reference assigned to t is based on the dot convention. The total flux linking coil 2 is 02 022 012- The flux 02 and its components 022 and 0 2 are related to the coil current z2 as follows 02 9 2N2Z2 022 St 22 2z2 012 12 2 2- The voltages v2 and Vt are v2 Nl dt dt - 7 dt dXi d di - 7 1 12 7. dt dt dt The coefficient of mutual inductance that relates the voltage induced in coil 1 to the time-varying current in coil 2 is the coefficient of di2 dt in Eq. Mn NXN29 2. For nonmagnetic materials the permeances 9 12 and 9 2 are equal and therefore M12 M2 M. Hence for linear circuits with just two magnetically coupled coils attaching subscripts to the coefficient of mutual inductance is not necessary. Mutual Inductance in Terms of Self-Inductance The value of mutual inductance is a function of the self-inductances. We derive this relationship as follows. From Eqs. and L N u L N f2 respectively. From Eqs. and LiL2 N2N29i92. A Closer Look at Mutual Inductance 197 We now use Eq. and the corresponding expression for SA2 to write LxL2 0 21 l 22 SP12 . But for a linear system SA21 SA12 so Eq. becomes J nV L L2 NjTV n 2 1 1 12 12 Replacing the two terms involving permeances by a single constant expresses Eq. in a more meaningful form Substituting Eq. into Eq. yields M2 k2LxL2 or M kvTm Relating self-inductances and mutual inductance using coupling coefficient where the constant k is called the coefficient of coupling. According to Eq. 1 k2 must be greater than 1 which means that k must be less than 1. In fact the coefficient of coupling must lie between 0 and 1 or 0 k 1. The coefficient of coupling is 0 when the two coils have no common flux that is when f 12 21 0. This
