5.9 The Bandwidth of an Impedance Transformation Network

Chapter 5.9 The Bandwidth of an Impedance Transformation Network

Radio Frequency Integrated Circuit Design Second Edition Book
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Radio Frequency Integrated Circuit Design Second Edition Book

  • 118 Impedance MatchingTaking the imaginary part of this expression, the inductance seen looking into the primary Leff can be found, making use of (5.21) to express the results in terms of the coupling coefficient k: ωω--+=-+2 22 22eff2 222(1)(1)s ppLpsLL LkR LLLkR (5.27)When k = 1, or when k = 0, then the inductance is simply Lp. When k has a value between these two limits, then the inductance will be reduced slightly from this value, depending on circuit values. Thus, a transformer can be made to resonate and have a zero reactive component at a particular frequency using a capacitor on either the primary Cp or secondary Cs: effeff11op ps sLCLCω--== (5.28)where Leff-s is the inductance seen looking into the secondary.The exact resistance transformation can also be extracted and is given by 2222 2eff2(1)pspLL sR LL LkRR L kω--= (5.29)Note again that if k = 1, then Reff = RLN and goes to infinity as k goes to zero. 5.9  The Bandwidth of an Impedance Transformation Network Using the theory already developed, it is possible to make most matching networks into equivalent parallel or series inductance, resistance, and capacitance (LRC) cir-cuits, such as the one shown in Figure 5.22. The transfer function for this circuit is determined by its impedance, which is given by out2in( )11( )VsssI sC sRCLCæöç÷= ç÷++ç÷èø (5.30)Figure 5.22  An LC resonator with resistive loss.