On the Limiting Form of the Amplitude-Frequency Response of a Multistage Band-Pass Amplifier

E. I. Fialko

1. Preliminary Remarks

The properties of band-pass amplifiers have been considered in sufficient detail in a number of works by Soviet researchers, in particular in the works of V. I. Siforov [1,2], N. I. Chistyakov [3], A. A. Kolosov [4], and others.

In the present article a particular question is considered, namely: the limiting properties of an amplifier assembled on band-pass filters with optimal coupling between the filter circuits (the circuits having equal damping and identical resonant frequencies). This problem is of interest not only because its solution determines the highest selectivity attainable in this type of amplifier, but also because the limiting properties corresponding to an infinite number of stages are sufficiently objective also for the case of a finite, but very large, number of stages.

From the point of view of real sensitivity, it is useful to clarify the relation between the noise band and the pass band of the amplifier.

In addition, a comparative evaluation of the limiting properties of multistage band-pass and resonant amplifiers is of interest.

To solve these problems it is necessary to find the equation of the amplitude-frequency response of a multistage band-pass amplifier with a fixed resultant pass band for an infinitely large number of stages.

We shall first consider a band-pass amplifier with a finite number of stages.

2. Multistage Band-Pass Amplifier

As is known [1—3], the amplitude-frequency response of a stage of a band-pass amplifier with two coupled identical¹ circuits is described by the equation

\[ y=\frac{1+\eta^2}{\sqrt{(1-x^2+\eta^2)^2+4x^2}}, \tag{1} \]

where \(x\) is the generalized detuning of the circuits of the band-pass filter,
\(\eta\) is the relative coefficient of coupling between the filter circuits;

\[ x=\frac{\Delta f}{\Delta F_0}\left(1+\frac{f_0}{f}\right), \tag{2} \]

\(\Delta f=f-f_0\) is the absolute detuning of the circuits,
\(f_0\) is the resonant frequency of the circuits,

\(\Delta F_0=d f_0\) is the pass band of the circuit, measured at the level 0.7;

\[ \eta=\frac{k_{\mathrm{sv}}}{d}, \]

\(k_{\mathrm{sv}}\) is the coefficient of coupling between the filter circuits,
\(d\) is the damping of the circuit.

With optimum coupling between the circuits of the band-pass filter \(k_{\mathrm{sv}}=d\) (\(\eta=1\)), and formula (1) takes the form

\[ y_1=\frac{2}{\sqrt{4+x^4}} . \tag{3} \]

The amplitude-frequency characteristic of a band-pass amplifier consisting of \(n\) identical stages is described by the equation

\[ y_n=y_1^n=\left(\frac{2}{\sqrt{4+x^4}}\right)^n . \tag{4} \]

We are interested in the change in the form of this characteristic as the number of stages is increased, in the case when the resulting pass band is kept constant. But in order for the resulting pass band to remain unchanged when a new stage is connected, it is necessary to increase the pass band of each circuit \(\Delta F_0\). Thus the generalized detuning \(x\) proves to depend also on \(n\). Let us find this dependence; for this, first of all, we shall express the magnitude of the resulting pass band of the amplifier \(\Delta F_n\) through the pass band of the circuit \(\Delta F_0\) and the number of stages \(n\).

Usually the pass band is measured at the level

\[ y=\frac{1}{\sqrt{2}}\simeq 0.7 . \]

Putting in formula (4) \(y_n=\frac{1}{\sqrt{2}}\), we find the generalized detuning corresponding to the level 0.7

\[ x_{0.7}=\sqrt{2}\sqrt[4]{\sqrt[n]{2}}-1 . \tag{5} \]

From (5) and (2) we find the positive and negative absolute detunings \((\Delta f_{0.7(+)},\ \Delta f_{0.7(-)})\), corresponding to \(y_n=\frac{1}{\sqrt{2}}\), and determine the resulting pass band

\[ \Delta F_{0.7}=\Delta f_{0.7(+)}+\left|\Delta f_{0.7(-)}\right| . \]

Owing to a certain asymmetry of the resonance characteristic of an oscillatory circuit, the resulting characteristic of the band-pass amplifier will also be asymmetric, and

\[ \Delta f_{0.7(+)}>\left|\Delta f_{0.7(-)}\right| . \]

In the case when the resulting pass band \(\Delta F_n\) is small in comparison with the resonant frequency \(f_0\), and the detunings \(\Delta f\) are commensurate with \(\Delta F_n\)

\[ \left(\text{i.e., when } \frac{f}{f_0}\simeq 1\right), \]

the resulting characteristic practically becomes symmetric, which makes it possible to simplify the calculations substantially.

For \(\dfrac{f}{f_0} \simeq 1\), the normalized detuning takes the form

\[ x=\frac{2\Delta f}{\Delta F_0}. \tag{6} \]

Taking into account that \(2\Delta f_{0.7}=\Delta F_n\), from (5) and (6) we find

\[ \Delta F_n=\Delta F_0\cdot \sqrt[4]{\,2\cdot \sqrt[n]{2}-1\,}. \tag{7} \]

Remark. Let us express the passband of the band-pass cascade \(\Delta F_1\) in terms of the passband of the circuit \(\Delta F_0\) (in formula (7) put \(n=1\)):

\[ \Delta F_1=\sqrt{2}\,\Delta F_0. \tag{8} \]

Substituting (8) into (7), we express the amplifier bandwidth in terms of the cascade bandwidth

\[ \Delta F_n=\Delta F_1\cdot \sqrt[4]{\,\sqrt[n]{2}-1\,}. \tag{9} \]

Representing \(\sqrt[n]{2}\) in the form of a power series and, for sufficiently large \(n\), neglecting terms of the series beginning with the 3rd, we obtain

\[ \Delta F_n=\Delta F_0\sqrt{2}\, \frac{\sqrt[4]{\ln 2}}{\sqrt[4]{n}}, \tag{10} \]

\[ \Delta F_n=\Delta F_1\, \frac{\sqrt[4]{\ln 2}}{\sqrt[4]{n}}, \tag{11} \]

or

\[ \Delta F_n=1.295\,\frac{\Delta F_0}{\sqrt[4]{n}}, \tag{10a} \]

\[ \Delta F_n=0.915\,\frac{\Delta F_1}{\sqrt[4]{n}}. \tag{11a} \]

For a sufficiently large number of cascades, the normalized detuning is equal to (see (6) and (10))

\[ x=\frac{\Delta f}{\Delta F_n}\, \frac{\sqrt[4]{\ln 2}}{\sqrt[4]{n}}\cdot 2\sqrt{2}. \tag{12} \]

After substituting (12) into (4) and after simple transformations, the amplitude-frequency response of a multistage band-pass amplifier in the case of a sufficiently large number of cascades takes the form

\[ y=\frac{1}{\left(\sqrt{\,1+\left(\dfrac{2\Delta f}{\Delta F_n}\right)^4 \dfrac{\ln 2}{n}\,}\right)^n}. \tag{13} \]

3. Limiting Form of the Amplitude-Frequency Response of a Multistage Band-Pass Amplifier

It is quite obvious that the connection of each subsequent stage, in the case of a sufficiently large number of stages and while maintaining an unchanged resultant passband, will change the form of the resultant amplitude-frequency response only slightly (let us recall that the case under consideration is that in which the stages have identical amplitude-frequency responses).

Let us find the limiting expression for the amplitude-frequency response of the amplifier for \(n \gg 1\).

We write equation (13) in the form

\[ y_n \simeq \left[1-\frac{1}{2}\left(\frac{2\Delta f}{\Delta F_n}\right)^4 \frac{\ln 2}{n}\right]^n . \tag{14} \]

Introducing

\[ \beta=-\frac{\ln 2}{2}\left(\frac{2\Delta f}{\Delta F_n}\right)^4 , \tag{15} \]

we obtain

\[ y_n \simeq \left(1+\frac{\beta}{n}\right)^n . \tag{16} \]

Making the formal transition to the limit and taking into account that, for finite detunings, \(\beta\) is a finite quantity, we obtain

\[ y=\lim_{n\to\infty} y_n=e^\beta \]

or

\[ y=e^{-\frac{\ln 2}{2}\left(\frac{2\Delta f}{\Delta F_n}\right)^4} \tag{17} \]

and finally

\[ y=e^{-\frac{5.55}{\Delta F_n^4}\,\Delta f^4}, \tag{17a} \]

where \(\Delta F_n\) is the resultant passband, measured at the level \(0.7\). Let us note that for \(n\to\infty\) the passband of the circuit \(\Delta F_0 \to \infty\); physically, of course, both of these conditions are unrealizable.

Moreover, as the number of stages \(n\) increases, the required attenuation of the circuit \(d\) also increases (for constant \(\Delta F_n\)).

Indeed, from (10a) it follows that

\[ d=\Delta F_n\cdot \sqrt[4]{n\cdot 0.77}\cdot \frac{1}{f_0}. \tag{18} \]

For very large \(n\), the required attenuation \(d>1\). It is clear that in this case the optimum coupling between the filter circuits is no longer realizable: as is known, the theoretical limit of the coupling coefficient is

\[ k_{\mathrm{c.max}}=1. \]

Thus, there exists some maximum number of stages \(n_{\max}\), above which the optimum coupling is unrealizable not only practically, but also theoretically.

Using formula (18), it is not difficult to show that \(n_{\max}\) is equal to the integer part of the quantity

\[ 4\ln 2\left(\frac{f_0}{\Delta F_n}\right)^4 \simeq 2.8\left(\frac{f_0}{\Delta F_n}\right)^4 . \tag{19} \]

Since usually the resonant frequency is considerably greater than the pass band of the amplifier, \(n_{\max}\) is very large and

\[ \left(1+\frac{\beta}{n_{\max}}\right)^{n_{\max}} \]

differs very little from \(e^\beta\).

In practical circuits the number of stages \(n\) is considerably smaller than \(n_{\max}\), but even in this case the amplitude-frequency characteristics of multistage amplifiers are well approximated by equation (17).

As an illustration, Fig. 1 shows:

Fig. 1

1. The amplitude-frequency characteristic of a five-stage band-pass amplifier, plotted from the formula

\[ y_5=\left(\frac{1}{\sqrt{1+\dfrac{x^4}{4}}}\right)^5, \]

where

\[ x=\frac{2\sqrt{2}\,\sqrt[4]{\ln 2}}{\Delta F_n\sqrt[4]{5}}\,\Delta f . \]

1. The curve

\[ y_\infty=e^{-\frac{\ln 2}{2}\left(\frac{2\Delta f}{\Delta F_n}\right)^4} \]

for the case \(\Delta F_n=10^6\) cycles/sec.

As can be seen, the discrepancy between these curves is negligible for \(|\Delta f|\leq \frac{1}{2}\Delta F_n\).

Thus, the limiting transition \((n\to\infty)\) is unlawful from the physical point of view, and the amplitude-frequency characteristics of real amplifiers cannot coincide exactly with the limiting curves. However, the deviation of the amplitude-frequency characteristics of real multistage amplifiers from the limiting characteristic is very small.

4. Rectangularity Coefficient of the Limiting Amplitude-Frequency Characteristic of a Multistage Band-Pass Amplifier

The shape of the characteristic is usually evaluated by the rectangularity coefficient, that is, by the ratio of the passbands at the levels 0.7 and 0.1

\[ k_{pr}=\frac{\Delta F_{0.7}}{\Delta F_{0.1}} . \tag{20} \]

Substituting \(y=0.1\) into (17) and taking into account that \(2\Delta f_{0.1}=\Delta F_{0.1}\), we obtain:

\[ 0.1=e^{-\frac{\ln 2}{2}\left(\frac{1}{k_{pr}}\right)^4}, \]

whence

\[ k_{pr}\simeq 0.62. \]

As is known, the rectangularity coefficient of the amplitude-frequency characteristic of a resonant amplifier does not exceed 0.39. Consequently, the selective properties of a multistage band-pass amplifier with optimal coupling between the circuits are considerably higher than those of a multistage resonant amplifier (for identical passbands):

\[ \frac{k_{pr.\,pol}}{k_{pr.\,res}}=1.6. \]

The shape of the amplitude-frequency characteristic determines not only the selectivity, but also the ratio between the noise bandwidth and the passband. This ratio is of interest in connection with the problem of real sensitivity.

5. Noise Band

By definition [2], the noise band \(\Delta F_{\mathrm{sh}}\) is equal to:

\[ \Delta F_{\mathrm{sh}}=\int_{0}^{\infty} y^2\,df, \tag{21} \]

In determining \(\Delta F_{\mathrm{sh}}\), one should use the expression for the amplitude-frequency characteristic for arbitrary detunings. However, since the branches of the amplitude-frequency characteristic fall off sufficiently rapidly as the detuning increases, in the case where the passband is small in comparison with the resonant frequency, equation (17) may be used. Substituting (17) into (21), we obtain

\[ \Delta F_{\mathrm{sh}}=\int_{0}^{\infty} e^{-\ln 2\left(\frac{2\Delta f}{\Delta F_n}\right)^4}\,df, \tag{22} \]

where \(\Delta f=f-f_0\).

Introducing the new variable

\[ t=\frac{\sqrt[4]{\ln 2}\cdot 2(f-f_0)}{\Delta F_n}, \]

we reduce (22) to the form:

\[ \Delta F_{\mathrm{sh}}=\Delta F_n\frac{1}{2\sqrt[4]{\ln 2}}\int_{0}^{\infty} e^{-t^4}\,dt,\quad \text{where } \xi=2\sqrt[4]{\ln 2}\,\frac{f_0}{\Delta F_n}, \tag{23} \]

For the case

\[ \frac{\Delta F_n}{f_0} \ll 1 \]

we may, without introducing appreciable error, replace the lower limit by taking it equal to “\(-\infty\)” (since the branches of the curve \(e^{-t^4}\) fall off very rapidly). Taking into account also the fact that the integrand is an even function, we obtain:

\[ \Delta F_{\mathrm{sh}} \simeq \Delta F_n \frac{1}{\sqrt[4]{\ln 2}} \int_0^\infty e^{-t^4}\,dt. \tag{24} \]

Using reference data, we find [5]:

\[ \int_0^\infty e^{-t^4}\,dt = \left(\frac{1}{4}\right)! = 0.9064. \]

Substituting the last value into (24), we obtain:

\[ \Delta F_{\mathrm{sh}} \simeq 0.99\,\Delta F_n, \tag{25} \]

that is, the noise bandwidth of the band-pass amplifier under consideration is practically equal to the passband at the level 0.7.

Let us compare the noise bandwidths of multistage band-pass and resonant amplifiers having identical passbands. Using the known limiting equation for the amplitude-frequency characteristic of a multistage resonant amplifier [2],

\[ y = e^{-\alpha \Delta f^2}, \tag{26} \]

it is easy to show that the passband measured at the level 0.7 and the noise bandwidth are related by the simple relation:

\[ \Delta F_{\mathrm{sh}} \simeq 1.06\,\Delta F_n. \tag{27} \]

Comparing formulas (25) and (27), we see that the noise bandwidth of a multistage band-pass amplifier is smaller than the noise bandwidth of a multistage resonant amplifier by several percent (with the passbands being the same). Consequently, use of the multistage band-pass amplifier under consideration will apparently increase (though only slightly) the real sensitivity of the receiving device in comparison with the case of using a multistage resonant amplifier. This question requires more detailed investigation.

Conclusions

The paper has considered a particular case of a multistage band-pass amplifier, namely: an amplifier all of whose stages have identical amplitude-frequency characteristics, each stage containing a two-circuit band-pass filter with optimally coupled circuits; the filter circuits have identical attenuations and are tuned to one and the same frequency.

As the number of stages is increased, the amplitude-frequency characteristic of such an amplifier is described sufficiently well by the function

\[ y = e^{-a \Delta f^4}, \tag{28} \]

where

\[ a = \frac{8\ln 2}{\Delta F^4}, \]

\(\Delta F\) is the resulting passband measured at the level

\[ y = 0.7; \]

\(\Delta f = f - f_0\) is the absolute detuning of the circuits.

The rectangularity coefficient of characteristic (28), which evaluates the limiting selective properties of the band-pass amplifier considered, is equal to:

\[ k_{\mathrm{pr}} \simeq 0.62 \]

and considerably exceeds the limiting value of the rectangularity coefficient of a multistage resonant amplifier \((k_{\mathrm{pr}} \simeq 0.39)\).

The noise bandwidth is practically equal to the passband measured at the level 0.7.

References

1. Siforov V. I. Band-Pass Amplifiers, ONTI, 1936.
2. Siforov V. I. Radio Receiving Devices, Publ. LKVVIA, 1947.
3. Chistyakov N. I. Resonant Amplifiers and Preliminary Selectors, Svyazizdat, 1939.
4. Kolosov A. A. Resonant Systems and Resonant Amplifiers, Svyazizdat, 1949.
5. Jahnke and Emde. Tables of Functions, OGIZ, 1948.

1. Formula (1) is valid in the case when the circuits entering into the band-pass filter have identical damping and identical frequencies. For brevity, such circuits are called identical here and below. ↩