5.8 Tuning a Transformer 117Note that here we have defined N as the inductance ratio, but traditionally it is defined as a turns ratio. Since in an integrated circuit turns and inductance are not so easily related, this alternative definition is used. Now if the secondary is loaded with impedance Rs, then the impedance seen in parallel on the primary side Rp will be: psspsspsVV NVRN R NIIIN==== (5.25)Thus, the impedance on the primary and secondary are related by the induc-tance ratio. Therefore, placing a transformer in a circuit provides the opportunity to transform one impedance into another. However, the above expressions are only valid for an ideal transformer where k = 1. Also, if the resistor is placed in series with the transformer rather than in parallel with it, then the resistor and inductor will form a voltage divider, modifying the impedance transformation. In order to prevent the voltage divider from being a problem, the transformer must be tuned or resonated with a capacitor so that it provides an open circuit at a particular fre-quency at which the match is being performed. Thus, there is a trade-off in a real transformer between near-ideal behavior and bandwidth. Of course, the losses in the winding and substrate cannot be avoided.5.8 Tuning a TransformerUnlike the previous case where the transformer was assumed to be ideal, in a real transformer there are losses. Since there is inductance in the primary and secondary, this must be resonated out if the circuit is to be matched to a real impedance. To do a more accurate analysis, we start with the equivalent model for the transformer loaded on the secondary with resistance RL, as shown in Figure 5.21. Next, we find the equivalent admittance looking into the primary. Through circuit analysis, it can be shown that 223222in42 22 22()()()Ls ps pLps p Ls ps pRL LMjL LMj R LL L RYL LMR Lωωωωωω-----+=-+ (5.26)Figure 5.21 Real transformer used to transform one resistance into another.