7.6 Differential Amplifiers 217intersects the axis at vL given by the ratio of the nominal bias current IC (equal to IEE/2) to the effective transconductance gmeff (equal to IEE/4vT), or ====»22250 mVCEETLEETmeffmEEIIvvIvggI (7.115)7.6.2 Linearity in Bipolar Differential PairsAs shown in the previous section, the equation for differential output current in a bipolar differential pair with a total current bias of Io is given by 1tanh2ooTviIv= (7.116)The hyperbolic tangent can be expanded with 3512tanh...315x xxx= -+- (7.117) 3512tanh...223 215 2TTTTvvvvvvvvæöæöæö=-+-ç÷ç÷ç÷èøèøèø (7.118) oo33...224oTTIIivvvv=-+ (7.119) ====oo1IP33324423324TTLTIkvvvvIkv (7.120)Thus it can be seen that vIP3 is equal to 4vT or 2vL. Thus, no matter what the transistor size or bias current, vIP3 is equal to about 100 mV. To get increased linear-ity, add resistive degeneration in series with the emitters and then use the new value for vL to estimate the improved linearity. æö===+=+=+ç÷èøIP3422242(2)22CEETEELETEE EeEmeffEEIIvIvvRvI RrRgI (7.121)where RE is the total resistance between the two emitters.The input one-tone 1-dB compression voltage is ideally lower than the input vIP3 by 9.66 dB or a linear factor of 3.02. Thus, v1dB is equal to 0.662vL. Because the resistor provides linear feedback, for large values of RE linearity is expected to be better than predicted above, and a common estimate is that vL is equal to the 1-dB compression point, in which case vIP3 would be equal to 3.02 vL.