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, w_{H} (as well as the equivalent low frequency limit, w_{L}) 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 w_{H} is:
or even w_{H} @ w_{p1}, if there is a dominant pole (i.e. w_{p1} << w_{p2}, ..., w_{z1}, w_{z2}, ...). In the same way, a good approximation to w_{L} is:
or even w_{L} @ w_{p1}, if it is dominant for low frequency (i.e. w_{p1} >> w_{p2}, ..., w_{z1}, w_{z2}, ...). 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 A_{M} is the middle frequency gain, F_{L}(s) is the low frequency response and F_{H}(s) is the high frequency response. We can re-write F_{H}(s) as follows:
For F_{H}(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 C_{i} with all the other capacitors open circuited. If there is a dominant pole P_{1} then and, therefore i.e. we can approximately determine w_{H} through the expression: We call this method the open circuit time constant method to determine the high frequency band limit. F_{L}(s) can be re-written as follows
For F_{L}(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 C_{i} with all the other capacitors short-circuited. If there is a dominant pole P_{1} then and, therefore i.e. we can approximately determine w_{L} 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 b_{2} of the denominator of F_{H}(s). But b_{2} is the summation of q terms of the form where q is equal to the number of combinations of n_{H} (number of high frequency poles, equal to the number of independent capacitors) in groups of two. is the resistance seen by capacitor C_{i} with all the others open circuited, and it is what we have referred to as used to compute b_{1}, and is the resistance seen by C_{j} with C_{i} 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 |