MULTISTAGE DIFFERENTIAL AMPLIFIERS Franclim F. Ferreira, Pedro Guedes de Oliveira, Vítor G. Tavares
 Time constants method
 The time constant method enables an easy approximate computation of the -3 dB high frequency limit of an amplifier frequency response, wH (as well as the equivalent low frequency limit, wL) when it is not possible to determine, by direct inspection, the values of the zeros and poles of the frequency response. In fact, if we can determine the zeros and poles, a good approximation to wH is: or even wH @ wp1, if there is a dominant pole (i.e. wp1 << wp2, ..., wz1, wz2, ...). In the same way, a good approximation to wL is: or even wL  @ wp1, if it is dominant for low frequency (i.e. wp1 >> wp2, ..., wz1, wz2, ...). There are many situations, however, where an easy evaluation of poles and zeros is not possible, e.g. if there are interacting capacitors. The transfer function of an amplifier may be written as    ,    where AM is the middle frequency gain, FL(s) is the low frequency response and FH(s) is the high frequency response. We can re-write FH(s) as follows: For FH(s), the zeros normally occur at much higher frequencies than the first pole or set of poles. But       and it can be shown [Gray and Searle, 1969] that       where is the resistance seen by capacitor Ci with all the other capacitors open circuited. If there is a dominant pole P1 then        and, therefore    i.e. we can approximately determine wH through the expression:    We call this method the open circuit time constant method to determine the high frequency band limit. FL(s) can be re-written as follows For FL(s), the zeros normally occur at much lower frequencies than the higher frequency pole or set of poles. But       and it can be shown that        where is the resistance seen by capacitor Ci with all the other capacitors short-circuited. If there is a dominant pole P1 then         and, therefore    i.e. we can approximately determine wL through the expression:    We call this method the short-circuit time constant method to determine the low frequency band limit. While the low frequency response is determined by the coupling capacitors, chosen by the circuit designer, the high frequency response is determined by transistor intrinsic capacitances, the values of which are not controlled by the designer. That is why the high frequency response analysis is of the utmost importance. In general it is important to determine at least the first two high frequency poles, being the second one determined by the coefficient b2 of the denominator of FH(s). But b2 is the summation of q terms of the form        where q is equal to the number of combinations of nH (number of high frequency poles, equal to the number of independent capacitors) in groups of two. is the resistance seen by capacitor Ci with all the others open circuited, and it is what we have referred to as used to compute b1, and is the resistance seen by Cj with Ci short circuited, and all the others open. We should note that    . This fact allows us to choose between and the easier to evaluate. Let us take as an example a circuit with three independent capacitors and, therefore, three poles: With two poles, we will have just    When we have only two poles it is easy to verify if a dominant pole exists (considering that the first pole is dominant if it is, at least, a decade below the second pole), using the following practical rule: If       then     and so       and

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