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1st stage ith stage nth stage

Fig. 4-26 Nomenclature for the stages in a multistage amplifier.

with a unit impulse, i.e., the time derivative of a unit step, so that V0(p) = 1.0 (which is the Laplace transform of the unit impulse).

Normalized over-all gain function

### Normalized gain function of the ith stage

We take next the general expression for the Laplace transform, expand t~vi in a power series, integrate term by term, and gather and identify terms with the definitions of Eqs. (4-39) and (4-42).

where TDN = delay time of n stages

Trn = rise time of n stages Note however that a similar relationship holds for the zth stage (one stage only). ^ ^

Now, since an{p) is the continued product of the a,(p), we take a continued product of series like Eq. (4-46) and gather terms in like powers of p, giving n an(p) = n ai(p) 1

Finally, we compare Eqs. (4-45) and (4-47) and equate the coefficients of the p and p2 terms, giving the following results:

The important result of Eq. (4-49) verifies the previous equation (4-37) in showing that the over-all rise time is the root mean square of the individual rise times of the stages. And Eq. (4-48) shows that the over-all delay time is the simple sum of the individual stage delay times.

Some of the by-products of Elmore's analysis are of interest. For instance, if one has a cascade of similar stages, should all stages be designed for the same gain and the same rise time, or might something be gained by having high gain and slow speed in one stage but counterbalanced by low gain and high speed in another stage? The answer apparently is that the minimum over-all rise time for a given gain is achieved by making all stages the same (see Sec. 4-10). The proof involves the use of Lagrange's method of undetermined multipliers, operating upon Eq. (4-49) and another equation which results from the fact that the rise time of each stage is proportional to the gain, since both are proportional to Rl-

For the condition of all stages identical, a simpler form of Eq. (4-49) is

Another interesting result is that there can be found an optimum gain per stage in order to achieve a given over-all gain with the minimum overall rise time. This gain per stage is \ft = 1.65 and is derived as follows:

A/Tr Aj

Wm/V2t C

where j? = efficiency compared with RC stage (as in Fig. 4-3).

Solve for dTRN/dn, and equate to zero to find TRN, min. The minimum value of Trn for a specified An occurs when n = 2 In An (4-52)

V2t ln An

Note that Eq. (4-54) gives the rise time according to Elmore's definition; for the 10 to 90 per cent value, replace \^2tt by 2.2.

The relationships given above have some practical limitations and require some judicious interpretation and application. First note that n has the same value, regardless of tube type or circuit type; for instance, if An is 105 (100 db), Eq. (4-52) says that 23 stages are called for, whether one uses 6AU6 or 6AH6 tubes. But it does not say that the same minimum rise time results in either case; from Table 4-1 and Eq. (4-54) it will be found that the 6AU6 would give 25y-{f)s, or 1.57, times as great a rise time as the 6AH6.

Also, the minimum of the rise-time function in Eq. (4-51) is a broad one, and one can violate the minimum conditions by quite a margin without serious detriment to the over-all rise time. Elmore provides an example of a 6AC7 amplifier (gm = 0.009 mho, C = 22 pf, 77 = 1.5) in which the 23 stages required for the 100-db gain give a rise time of 0.032 /¿sec; yet, with only 9 stages to give the same gain, the rise time is 0.044 /¿.sec. Thus, the rise time is impaired only 37.5 per cent for a 48 per cent reduction in the number of stages. Moreover, the larger load resistor in the latter case (400 ohms instead of 180 ohms) permits a larger output voltage to be realized from the amplifier (222 per cent greater).

4-10 Amplifiers with Nonidentical Stages. The conclusion that an amplifier should be made up of identical stages for minimum rise time is obtained by an analysis which is invalid if any of the stages has overshoot. Hence there would seem to be the possibility of designing an amplifier in which some stages have large overshoots (large m) coupled to other stages which tend to reduce the overshoot. Attempts to design such an amplifier on the basis of the steady-state amplitude response have usually resulted in amplifiers with excessive over-all overshoot.1'2 Designing the amplifier on the basis of a linear steady-state phase response gives an excellent transient response;3 however, the most straightforward data to interpret for the transient case are given by F. A. Muller.4 Either the linear-phase response or the Muller data give a faster amplifier than is obtained by using identical stages. Since the Muller data have the advantage of giving a specified overshoot, they will be presented here. Each stage of the amplifier is a shunt-peaked stage (Fig. 4-2a, where L may be zero in some stages). The data for each stage are given in terms of m = L/Rl2C, as before, and a normalized r which is defined as r< 4 7-"—^ (4-55)

In Eq. (4-56) Ct- is the total capacitance, and R, is the load resistor of the ith interstage. Table 4-3 gives the value of r,- and (where m; = Li/RfCi) for each stage. The rise time of the whole amplifier is given in normalized form as Tx. The actual rise time T^.y of the whole amplifier is given by

If the amplifier is to be designed for a given over-all rise time and the gm and Ci of each stage are known, the load resistor for an individual stage is

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