Time Constant Delay Methods: Elmore Delay and Risetime

Time Constant Delay Methods: Elmore Delay and Risetime

Time constant delay estimation methods are very useful when the wiring capacitance is quite small or the charging current is quite high. In this situation, typical of very high-speed SSI and MSI circuits that push the limits of the device and process technology, the circuit delays are dominated by the devices themselves. The method to be described relies upon a large-signal equivalent circuit model of the transistors, an approximation that is dubious at best. The construction of the large-signal equivalent circuit requires averaging of nonlinear model elements such as transconductance and certain device capacitances over the appropriate signal voltage range. But the objective of the technique is not absolute accuracy. That is much less important than being able to identify the dominant contributors to the delay and risetime, since more accurate but less intuitive solutions are easily available through circuit simulation.

The propagation delay definition described above, the delay required to reach 50% of the logic swing, must be relaxed slightly to apply methods based on linear system analysis. It was first shown by Elmore [2] in 1948, and apparently rediscovered by Ashar [3] in 1964, that the delay time TD between an impulse function d(0) applied at t = 0 to the input of a network and the centroid or “center of mass” of the impulse response f(t) at the output is in many cases quite close to the 50% delay tP . The Elmore delay is called TD to avoid confusion with the propagation delay tP defined above. The Elmore delay is illustrated in Figure 74.1 and can be calculated by the normalized value of the first moment of the impulse response.

Two conditions must be satisfied to use this approach. First, the step response of the network is monotonic. This implies that the impulse response is purely a positive function. Monotonic step response is valid only when the circuit poles are all negative and real or the circuit is heavily damped. Owing to feedback through device and circuit elements and the underdamping effects caused by a source or emitter follower driving a common source or emitter stage, this condition is seldom completely correct. Complex poles often exist. But, strongly underdamped circuits are seldom useful for reliable logic circuits because their transient response will exhibit ringing, so efforts to compensate or damp such oscillations are needed in these cases anyway. Then, the circuit becomes heavily damped or at least dominated by a single pole and fits the above requirement more precisely. In any event, the method will yield a conservative estimate of delay and risetime.

Second, the correspondence between TD and tP is improved if the impulse response is symmetric in shape as in Figure 74.1. It is shown in Ref. [2] that cascaded stages with similar time constants have a tendency to approach a Gaussian-shaped distribution as the number of stages becomes large. Most logic systems require several cascaded stages, so this condition is often true as well.

Assuming that these conditions are approximately satisfied, the Laplace transform of the impulse response of a circuit is the same as the network function F(s) in the complex frequency s = s + jw. Then, the Elmore delay, TD, can be determined by

TD can be obtained directly from the network function F(s) as shown. But, the network function must be calculated from the large-signal equivalent circuit of the device including all important parasitics, driving impedances, and load impedances. This is notoriously difficult if the circuit includes a large number of capacitances or inductances. Tien [4] has described an approach for calculating the Elmore delay strictly from F(s) by breaking up the circuit into subnetworks. However, the algebraic labor is still considerable.

Risetime

The standard definition of risetime is the 10–90% time delay of the step response of a network. While convenient for measurement, this definition is analytically unpleasant to derive for anything except simple, first-order circuits. Elmore [2] demonstrated that the standard deviation of the impulse response could be used to estimate the risetime of a network. This definition provides estimates that are close to the usual risetime definition as indicated in Figure 74.1. The standard deviation of the impulse response can be calculated using

Since the impulse response frequently resembles the Gaussian function, the integral could be easily evaluated; however, the integration need not be performed. Lee [5] has pointed out that the transform techniques can also be used to obtain the Elmore risetime directly from the network function, F(s).

This result can also be used to show that the risetimes of cascaded networks add as the square of the individual risetimes. If two networks are characterized by risetimes TR1 and TR2, the total risetime, TR,total, is given by the root mean square (RMS) sum of the individual risetimes