When the transient response of a linear network to an applied unit step function consists of a monotonic rise to a final constant value, it is found possible to define *delay time* and *rise time* in such a way that these quantities can be computed very simply from the Laplace system function of the network. The usefulness of the new definitions is illustrated by applications to low pass, multi‐stage wideband amplifiers for which a number of general theorems are proved. In addition, an investigation of a certain class of two‐terminal interstage networks is made in an endeavor to find the network giving the highest possible gain—rise time quotient consistent with a monotonic transient response to a step function.

## REFERENCES

1.

The notation and terminology adopted here is that found in M. F. Gardner and J. L. Barnes,

*Transients in Linear Systems*, (John Wiley and Sons, Inc., New York, 1942), Vol. 1.2.

The gain—rise time quotient is analogous to the more familiar gain‐band width product, but appears to be a more useful measure of amplifier performance in the case of amplifiers designed to amplify fast transients. The definition of rise time is considered in Section 2.

3.

For many applications it is important to avoid over‐compensation in an amplifier. This is particularly true for

*pulse amplifiers*used in nuclear physics (to amplify pulses obtained from an electrical detector of radiation) and for wideband amplifiers used in Studying fast electrical transients (such as amplifiers for cathode‐ray oscillographs). Video amplifiers used in television applications are not as critical in this respect since small oscillations resulting from over‐compensation do not impair the quality of television pictures.4.

The curve $e\u2032(t)$ may be considered to be the response of the amplifier to a

*unit impulse*applied at time $t\u2009=\u20090.$5.

6.

The individual stages in the amplifier must each have a monotonic transient response to the unit step function.

7.

This result appears to have been first noticed by Henry Wallman, and will be discussed in Chap. 7, Vol. 18 of the

*Radiation Laboratory Series*(McGraw‐Hill Book Company, Inc., New York, in press).8.

It has been called to the attention of the writer that this method of computing moments is closely related to methods used in mathematical statistics, the method of the moment generating function, and the method of the characteristic function. See, for instance, S. S. Wilks,

*Mathematical Statistics*(Princeton University Press, Princeton, New Jersey, 1943). The mathematical steps leading to Eq. (7) can be made more rigorous by noticing that the series for the integrand converges*uniformly*, thus permitting term‐wise integration between zero and a finite upper limit*T*. Since the integrated series, considered as a function of*T*, is also uniformly convergent when*s*is restricted to the region near the origin, the series must converge, as*T*approaches infinity, to the Laplace integral (6) from which it is derived.9.

By setting $C\u2009=\u20091,$ $R\u2009=\u20091,$ and expressing

*L*in units of $R2C,$ values of $TD$ and $TR$ are obtained in units of*RC*. This device enables the system function to be written immediately in a simple, normalized form.10.

See, for instance, H. W. Bode,

*Network Analysis and Feedback Amplifier Design*, (D. Van Nostrand Company, Inc., New York, 1945), pp. 408,*et seq.*11.

This statement is true provided that no coupling between stages exists except through the electron stream in the constituent amplifying tubes. This situation can be realized in practice if the tubes in the amplifier are pentodes.

12.

For instance, by using Lagrange’s method of undetermined multipliers, the differential of Eq. (14) must be zero, $\Sigma TRidTRi\u2009=\u20090,$ subject to the condition $(\Pi TRi)\u2009\Sigma dTRi/TRi\u2009=\u20090$ which is the differential of Eq. (15). After multiplying the latter equation by the undetermined multiplier α and adding it to the former, each coefficient of $dTRi$ must be identically zero, giving $TR1\u2009=\u2009TR2\u2009=\u2009\cdots .$ A proof that this condition leads to a

*minimum*rise time is scarcely needed.13.

It is assumed that $R\u226arp,$ the plate resistance of the tube.

14.

The pair of equations (20a) can be made the basis of a convenient nomograph to aid in the design of an amplifier of assigned rise time and total gain.

15.

Strictly speaking, two‐terminal and three‐terminal networks.

16.

See reference 10, pp. 177–181 for a discussion of Foster’s reactance theorem and of the various networks which can be used to realize an arbitrary reactance.

17.

It is of interest to note that if a switch is inserted in series with the capacitor

*C*in Fig. 3b, and the capacitor is initially given a unit charge, then Eq. (28) is the Laplace transform of the voltage developed across the network when the switch is closed at $t\u2009=\u20090.$ Since the voltage across the capacitor decreases linearly while it is being discharged into the remaining branch of the network, the current flowing through the resistor must have the form of a rectangular pulse (of amplitude $12$). The network can evidently be used (ideally, at any rate) to convert either a current impulse, or the sudden discharge of a capacitor, into a rectangular voltage pulse across a resistive load.18.

See, for instance, E. T. Whittaker and G. N. Watson,

*Modern Analysis*, (Cambridge University Press, Teddington, England, 1927), fourth edition, p. 136, example 7.
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© 1948 American Institute of Physics.

1948

American Institute of Physics

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