7.2 Amplifiers with Feedback 183where ωo is the frequency of interest. Note that nodal analysis will not result in the same expression unless the input impedance is set equal to RS and in practice this requires the addition of an input inductor. The input impedance has the same form as the common collector amplifier and is also given by: in(1)EmZZZg Zππ=++ (7.39)Of particular interest is the product of ZE and Zp. If the emitter impedance is inductive, then when this is reflected into the base, this will become a real resistance. Thus, placing an inductor in the emitter or source tends to raise the real part of the input impedance of the circuit, so it is very useful for matching purposes. (Con-versely, placing a capacitor in the emitter or source will tend to reduce the input impedance of the circuit and can even make it negative.)7.2.2 The Common-Emitter/Source with Shunt FeedbackApplying shunt feedback to a common-emitter or common-source amplifier is a good basic building block for broadband amplifiers. This technique allows the am-plifier to be matched over a broad bandwidth while having minimal impact on the noise figure of the stage. A basic common-emitter amplifier with shunt feedback is shown in Figure 7.10. The analysis is the same for a CMOS amplifier noting that Zp for a bipolar transistor is rp in parallel with Cp while for CMOS the equivalent is simply Cgs. Resistor Rf forms the feedback and capacitor Cf is added to allow for independent biasing of the base and collector. Cf can normally be chosen so that it is large enough to be a short circuit over the frequency of interest. Note that this circuit can be modified to become a cascode amplifier if desired.Ignoring the Miller effect and assuming Cf is a short circuit (1/ωCf << Rf), the gain is given by: 11Lm Lfom LvLLiffRg RRvg RARRvRR--==»++ (7.40)Thus, feedback has reduced the gain compared to the original gain (-gmRL) without feedback. Figure 7.10 A common-emitter amplifier with shunt feedback.