Time Constant Methods: Open-Circuit Time Constants

Time Constant Methods: Open-Circuit Time Constants

The frequency domain/transform methods for finding delay and risetime are particularly valuable for design optimization because they identify dominant time constants. Once the time constants are found, the designer can make efforts to change biases, component values, or optimize the design of the transistors themselves to improve the performance through addressing the relevant bottleneck in performance. The drawback in the above technique is that a network function must be derived. This becomes tedious and time consuming if the network is of even modest complexity. An alternate technique was developed [5,6] that also can provide reasonable estimates for delay, but with much less computational difficulty. The open-circuit time constant (OCTC) method is widely used for the analysis of the bandwidth of analog electronic circuits only for this reason. It is just as applicable for estimating the delay of very high-speed digital circuits.

The basis for this technique again comes from the transfer or network function F(s) = Vo(s)/Vi(s). Considering low-pass transfer functions containing only poles, the function can be written as

of the time constants and b2 the product of all the time constants. Often, the first-order term dominates the frequency response. In this case, the 3 dB bandwidth is then estimated by w3 dB = 1/b1. The higher order terms are neglected. The accuracy of this approach is good when the circuit has a dominant pole. If all poles have the same frequency, the error in delay predicted by this method is about 25%. Much worse errors can occur however if the poles are complex or if there are zeros in the transfer function as well. This case will be discussed later.

Elmore [2] has once again provided the connection needed to obtain delay and risetime estimates from the network function. In the more general case where there could be a first-order zero in the numerator, a1, the Elmore delay is given by

In this equation, a2 and b2 correspond to the coefficients of the second-order zero and pole, respectively. At this point, it would appear that nothing has been gained since finding the time constants associated with the poles and zeros is well known to be difficult. Fortunately, it is possible to obtain the b1 and b2 coefficients directly by a much simpler method, OCTC or method of time constants (MOTC). It has been shown that [5,6]

the sum of the time constants tjo, defined as the product of the effective open-circuit resistance Rjo across each capacitor Cj when all other capacitors are open-circuited, equals b1. These time constants are very easy to calculate since open-circuiting all other capacitors greatly simplifies the network by decoupling many other components. Consider Figure 74.2(a). A network is represented by the box and all internal capacitors are brought to the outside of the box. Figure 74.2(b) then illustrates how the driving point resistance, R1o = V1/I1, is determined for each capacitor with all others removed from the network.

Dependent sources must be considered in the calculation of the Rjo open-circuit resistances. Note that these open-circuit time constants are not equal to the pole time constants, but their sum gives b1. It should also be noted that the individual OCTCs give the time constant of the network if the jth capacitor were the only capacitor. Thus, each time constant provides information about the relative contribution of that part of the circuit to the bandwidth or the delay [5]. If one of these is much larger than the rest, this is the place to begin working on the circuit to improve its speed.

The b2 coefficient can also be found by a similar process [7], taking the sum of the product of time constants of all possible pairs of capacitors. A short-circuit time constant is also required. Figure 74.3 illustrates how the resistance is calculated while short-circuiting one of the other ports. For example, in a three-capacitor circuit, b2 is given by

where the Ri resistance is the resistance across capacitor C calculated when capacitor C is short-circuited and all other capacitors open-circuited. The superscript indicates which capacitor is to be shorted. So,

R2 is the resistance across C when C is short-circuited and C is open-circuited. Note that the first time constant in each product is an open-circuit time constant that has already been calculated. In addition, for any pair of capacitors in the network, we can find an OCTC for one and an SCTC for the other. The order of choice does not matter because

so one can choose whichever combination minimizes the computational effort [7].