7.4 Linearity in Amplifiers 2057.4 Linearity in AmplifiersNonlinearity analysis will follow the same basic principles as discussed in Chapter 2, with power series expansions and nonlinear terms present in the amplifier. These will now be discussed in detail.7.4.1 Exponential Nonlinearity in the Bipolar TransistorIn bipolar transistors, one of the most important nonlinearities present is the basic exponential characteristic of the transistor itself, illustrated in Figure 7.28.Source resistance improves linearity. As an extreme example, if the input is a current source, RS = ¥, then ic = βib. This is as linear as β is. It can be shown that a resistor in the emitter of value RE has the same effect as a source or base resistor of value RE β. The transistor base has a bias applied to it and an ac signal super-imposed. Summing the voltages from ground to the base and assuming that ie = ic: ()sSbeBEE CcvVvVR Ii+=+++ (7.82)where VBE and vbe are the dc and ac voltages across the base emitter junction of the transistor.Extracting only the small signal components from this equation gives: =+sbeE cvvR i (7.83)Also from the basic properties of the junction: ++ ===BEbeBEbebeTTTTVvVvvvvvvccSSCIiI eI eeI e (7.84)where from Chapter 4, vT = kT/q. Solving for vbe gives: lnæö=+ç÷èø1cbeTCivvI (7.85)Figure 7.28 Bipolar common-emitter amplifier for linearity analysis.