114 Impedance Matchingwhich is just the series combination of the two inductors if the resistor is absent. The same type of analysis can be performed on network Figure 4.16(a). In this case 212eq1CCRRC+æö» ç÷èø (5.19) 1eq1211CCC-æö»+ç÷èø (5.20)5.6 The Concept of Mutual InductanceAny two coupled inductors that affect each other’s magnetic fields and transfer energy back and forth form a transformer. How tightly they are coupled together affects how efficiently they transfer energy back and forth. The amount of coupling between two inductors can be quantified by defining a coupling factor k, which can take on any value between one and zero. Another way to describe the coupling be-tween two inductors is with mutual inductance. For two coupled inductors of value Lp and Ls, coupling factor k, and the mutual inductance M as shown in Figure 5.17 are related by p sMkL L= (5.21)The relationship between voltage and current for two coupled inductors can be written out as follows : pp psss spVj L Ij MIVj L Ij MIωωωω=+=+ (5.22)Note that dots in Figure 5.17 are placed such that if current flows in the indi-cated direction, then fluxes will be added . Equivalently, if Ip is applied and Vs is 0V, current will be induced opposite to Is, to minimize the flux. For transformers, it is necessary to determine where to place the dots. We il-lustrate this point with Figure 5.18 where voltages V1, V2, and V3 generate flux through a transformer core. The currents are drawn so that the flux is reinforced. The dots are placed appropriately to agree with Figure 5.17. Figure 5.17 A basic transformer structure.