ON THE LIMIT PROPERTIES OF A MULTISTAGE BANDPASS AMPLIFIER

E. I. FIALKO

Statement of the Problem

In some cases it is necessary to know not only the amplitude-frequency characteristic of a radio device, but also its phase-frequency characteristic (as, for example, in the analysis of distortions of pulse signals in radio filters). In a number of cases, in general, only the phase-frequency characteristic is of interest (for example, in certain phase-measuring devices).

There are several methods for constructing phase-frequency characteristics. As is known, between the amplitude-frequency and phase-frequency characteristics \((K(\omega)\) and \(\varphi(\omega))\) there exists a quite definite relation [1]:

\[ \int_{0}^{\infty} \frac{K(\omega)}{\omega}\cdot \cos[\omega t-\varphi(\omega)]\,dt = 0.{}^{1)} \tag{1} \]

Therefore, knowing \(K(\omega)\), we can, generally speaking, find \(\varphi(\omega)\). This method of determining the phase-frequency characteristic was implemented by A. P. Belousov [2] for the case when the amplitude-frequency characteristic of the system is described by a probability (Gaussian) curve; it was shown that the phase-frequency characteristic is a straight line perpendicular to the frequency axis. As is known, the probability curve is the limiting form of the amplitude-frequency characteristic of a multistage resonant amplifier [3].

In the present work another limiting case is considered—a multistage bandpass amplifier with an infinitely large number of stages.

We shall confine ourselves to two particular problems: we shall find the limiting form of the phase-frequency characteristic of a multistage bandpass amplifier built from identical two-circuit filters with very weak and optimal coupling between the filter circuits (where the circuits included in the filter are tuned to one frequency and have equal attenuations. In what follows such circuits will be called identical). Since solving the integral equation (1) is very cumbersome, we shall use a simpler method of constructing the phase-frequency characteristic, set forth below.

Phase-Frequency Characteristic of a Bandpass Stage of a Multistage Bandpass Amplifier

The phase-frequency characteristic of a bandpass stage with two coupled identical circuits is expressed by the relation [3;4]:

\[ \varphi = \operatorname{arc\,tg}\frac{1-x^{2}+\eta^{2}}{2x}, \tag{2} \]

\[ {}^{1)} \]
This integral equation is valid for the case of absence of a constant component of the current in the steady-state regime.

where

\(x\) — generalized detuning of the filter circuits,
\(\eta\) — relative coefficient of coupling between the filter circuits:

\[ x=\frac{\Delta f}{\Delta F_0}\left(1+\frac{f_0}{f}\right), \tag{3} \]

\(\Delta f=f-f_0\) — absolute detuning of the filter circuits,
\(f_0\) — resonant frequency of the circuits,
\(\Delta F_0\) — passband of a circuit, measured at the 0.7 level;
\(K_{cv}\) — coefficient of coupling between the filter circuits,
\(d\) — damping of the circuit.

For the case of optimum coupling between the filter circuits \(K_{cv}=d\), and formula (2) takes the form:

\[ \varphi=\operatorname{arc\,tg}\frac{1-\left(\dfrac{x}{\sqrt{2}}\right)^2}{x}. \tag{4} \]

In what follows we shall assume that

\[ \frac{f}{f_0}\simeq 1. \]

This assumption is valid if the resultant passband of the amplifier is small in comparison with the frequency \(f_0\) to which the circuits are tuned, and if the detunings under consideration are comparable with the resultant passband. (Let us note that of greatest interest is that part of the phase-frequency characteristic which lies in the frequency interval commensurate with the amplifier passband.) Therefore

\[ x\simeq \frac{2\Delta f}{\Delta F_0}. \tag{5} \]

Let us also note that, in the case of optimum coupling between the filter circuits, the passband of the filter is equal to [3]

\[ \Delta F_1=\sqrt{2}\,\Delta F_0. \tag{6} \]

Let us find the limiting form of the phase-frequency characteristic of a cascade as the number of cascades is increased and a specified (constant) resultant passband is maintained. To this end, we shall first express the passband of one cascade \((\Delta F_1)\) through the number of cascades \((n)\) and the passband of the entire amplifier \((\Delta F_n)\).

As is known, for sufficiently large \(n\),

\[ \Delta F_1=\Delta F_n\cdot \sqrt[4]{n}\cdot \frac{1}{\sqrt[4]{\ln 2}}. \tag{7} \]

Substituting (7) into (6), and then into (5), we obtain:

\[ x=\frac{2\sqrt{2}\sqrt[4]{\ln 2}}{\Delta F_n\cdot \sqrt[4]{n}}\,\Delta f. \tag{8} \]

If we are interested in the form of the phase-frequency characteristic within the passband (or at detunings comparable with the passband) of the amplifier, then for sufficiently large \(n\), \(x\ll 1\), and formula (4) takes the form:

\[ \varphi_1\simeq \operatorname{arc\,tg}\frac{1}{x}. \tag{9} \]

Expanding \(\arc \tg x\) in a series and neglecting the terms of the series containing \(x\) in the 3rd and higher powers, we obtain

\[ \varphi_1 \approx \frac{\pi}{2} - \Delta f \cdot \tau_1, \tag{10} \]

where

\[ \tau_1=\frac{[[unclear: numerator of fraction]]}{\Delta F_n\sqrt{n}}. \tag{11} \]

As we see, the phase-frequency characteristic of one bandpass stage, as \(n\) increases, straightens and, rotating about the point \(\left(\varphi_1=\frac{\pi}{2}; f=f_0\right)\), approaches a horizontal straight line raised by a constant phase angle equal to \(\frac{\pi}{2}\) (see Fig. 1).

Let us now consider the case of very weak coupling \((\eta \to 0)\). Theoretically such a case could occur with an unlimited increase in the number of stages, but while maintaining an unchanged resulting passband; in this case the attenuation \(d\) of the circuit increases as \(n\) increases, while the coupling coefficient cannot exceed \(K_{\mathrm{c}\,\max}=1\); therefore

\[ \eta=\frac{K_{\mathrm{c}}}{d}\to 0. \]

In this case (see formula (2)),

\[ \varphi=\arc \tg \frac{1-x^2}{2x}. \tag{12} \]

Fig. 1. Phase-frequency characteristic \(\varphi(f)\) of a stage of a multistage bandpass amplifier (in the region of small detunings).

\[ n_2>n_1\gg 1 \]

Let us express the reduced detuning \(x\) through the passband of the amplifier.

As is known, in the case of very weak coupling

\[ \Delta F_0=\Delta F_n\sqrt{n}\cdot \sqrt{\frac{2}{\ln 2}}. \tag{13} \]

Substituting (13) into (5), we have:

\[ x=\frac{\sqrt{2\ln 2}}{\Delta F_n\sqrt{n}}\cdot \Delta f. \tag{14} \]

If the detunings \(\Delta f\) are commensurate with \(\Delta F_n\), then for a very large number of stages \(x\ll 1\), and formula (12) takes the form

\[ \varphi \approx \arc \tg \frac{1}{2x}. \tag{15} \]

Expanding \(\arc \tg 2x\) in a series and neglecting the terms of the series beginning with those containing \(x\) in the third power, we obtain:

\[ \varphi_1 \approx \frac{\pi}{2} - \Delta f \cdot \tau_1, \tag{16} \]

where

\[ \tau_1=\frac{2\sqrt{2\ln 2}}{\Delta F_n\sqrt{n}}. \]

Thus, in this case as well, the phase-frequency characteristic of the stage, as \(n\) increases, straightens and, rotating about the point

\[ \left(\varphi_1=\frac{\pi}{2}; f=f_0\right), \]

tends to a horizontal position (Fig. 1).

Let us now proceed to the consideration of a multistage bandpass amplifier.

Phase-Frequency Characteristic of a Multistage Bandpass Amplifier

The Case of Very Weak Coupling

The phase-frequency characteristic of an \(n\)-stage amplifier \((\varphi_n)\) in the case of identical stages may be represented in the form

\[ \varphi_n=n\cdot\varphi_1. \tag{17} \]

Assuming \(n\) to be large, from (16) and (17) we obtain

\[ \varphi_n=n\frac{\pi}{2}-\tau_n\cdot\Delta f, \tag{18} \]

where, in the case of very weak coupling,

\[ \tau_n=\frac{2\sqrt{2\ln 2}\sqrt{n}}{\Delta F_n}. \tag{19} \]

A number of conclusions follow from formulas (18) and (19).

For sufficiently large \(n\), the phase-frequency characteristic is close to a straight line. More precisely, the section of the phase-frequency characteristic corresponding to generalized detunings \(x\ll 1\) is rectilinear.

The phase-frequency characteristic of a bandpass amplifier contains, along with the component \(\tau_n\cdot\Delta f\), also a constant component \(n\frac{\pi}{2}\). In this respect a bandpass amplifier differs substantially from a resonant amplifier, whose phase-frequency characteristic contains no constant component and, for \(n\gg 1\), is a straight line passing through the point \((\varphi=0; f=f_0)\).

As the number of stages increases, the phase-frequency characteristic of the bandpass amplifier becomes steeper and at the same time rises in jumps (see Fig. 2). In the limit, as \(n\to\infty\), the phase-frequency characteristic inclines to the frequency axis at an angle \(\frac{\pi}{2}\) and contains a constant term

\[ \frac{\pi}{2}, \]

which assumes an infinite set of values that are multiples of \(\frac{\pi}{2}\).

It can be shown that the amplitude-frequency characteristic of a bandpass amplifier, as \(n\to\infty\) and \(\eta\to 0\), tends to a quite definite limit:

\[ e^{-a\Delta f^2}. \tag{20} \]

Thus, the relationship between the limiting amplitude-frequency and phase-frequency characteristics of a multistage bandpass amplifier proves to be nonunique; moreover, one amplitude-frequency characteristic corresponds to an infinite set of phase-frequency characteristics.

However, in some cases the infinite set of phase-frequency characteristics can be reduced to a finite number of lines (for example, in

Fig. 2. Phase-frequency characteristics \(\varphi_n(f)\) of multistage bandpass amplifiers (in the region of small detunings).

\[ n_3>n_2>n_1 \]

In the figure, the vertical axis is labeled \(y\), and the horizontal axis is labeled \(\Delta f=f-f_0\). The plotted lines are marked \(n_1,n_2,n_3,n_4,n_5,n_6\), with levels \(n_1\frac{\pi}{2}, n_2\frac{\pi}{2}, \ldots, n_6\frac{\pi}{2}\).

the case where the phase angle enters the argument of a periodic function). Indeed, representing \(n\) (\(n \gg 1\)) in the form

\[ n=4m+K, \]

where \(m\) is an integer, and \(K\) takes four values

\[ K=0;1;2;3, \]

we write formula (18) in the following form:

\[ \varphi_n \simeq m\,2\pi+K\frac{\pi}{2}-\tau_n\Delta f . \tag{21} \]

If the phase angle \(\varphi_n\) enters the argument of some periodic function \(F\), then

\[ F(\varphi_n)=F(\varphi_n-m\cdot 2\pi), \]

and the result will be the same as if the phase-frequency characteristic had the form

\[ \varphi_n^{(*)}=K\frac{\pi}{2}-\tau_n\Delta f . \tag{22} \]

As the number of stages increases, the slope of the phase-frequency characteristic \(\varphi_n^{(*)}\) increases, while the constant phase shift takes the values

\[ 0;\ \frac{\pi}{2};\ \pi;\ \frac{3}{2}\pi, \tag{23} \]

then again \(0, \dfrac{\pi}{2}\), etc. (Fig. 3). Thus, the phase-frequency characteristics will split into 4 groups; in the limit, as \(n \to \infty\), the phase-frequency characteristic will assume 4 values:

\[ \varphi_n^{(\infty)} = \begin{cases} -\tau_n \Delta f,\\ \dfrac{\pi}{2} - \tau_n \Delta f,\\ \pi - \tau_n \Delta f,\\ \dfrac{3}{2}\pi - \tau_n \Delta f \end{cases} \qquad (\tau_n \to \infty) \tag{24} \]

Analyzing, for example, the passage of a pulse signal through a band-pass radio filter with amplitude-frequency characteristic

\[ y = e^{-a \Delta f^2}, \]

it is necessary to consider 4 particular cases, differing from one another by the different value of the constant component of the phase-frequency characteristic.

Let us recall that, in the analysis of the passage of a pulse signal through a multistage resonant amplifier, this complication is absent, owing to

Fig. 3. Phase-frequency characteristics \(\varphi_n^{(\infty)}(f)\) of multistage band-pass amplifiers (in the region of small detunings)

\[ n_6 > n_5 > \ldots > n_1 \]

a one-to-one dependence between the limiting amplitude-frequency and phase-frequency characteristics of a multistage resonant amplifier. Note that the magnitude of the constant phase shift will affect only the displacement in time of the high-frequency filling, and will not affect the shape of the signal envelope.

Case of optimal coupling

The phase-frequency characteristic of an \(n\)-stage band-pass amplifier in the case of identical stages and optimal coupling between the filter circuits can be represented in the form (see (17), (10), and (11)):

\[ \varphi_n = n \dfrac{\pi}{2} - \tau_n \Delta f, \tag{25} \]

where

\[ \tau_n = - \frac{2\sqrt{2}\,\sqrt[4]{\ln 2}}{\Delta F_n}\cdot n^{3/4}. \tag{26} \]

Consequently, as the number of stages is increased, the phase-frequency characteristic straightens, becomes steeper, and at the same time rises by jumps (Fig. 2).

This case ($\gamma=1$) differs from the preceding one only in that, for the same pass band and the same number of stages, the steepness of the phase-frequency characteristic in the case of optimum coupling will be greater than in the case of very weak coupling (see (26) and (19)).

Sometimes, for example when the phase angle enters into the argument of a periodic function, the family of characteristics (25) may be represented in the form of four groups (see (21)—(23)):

\[ \varphi_n^{(i)}= \begin{cases} -\tau_n \Delta f, & \text{for } n=4m,\\[4pt] \dfrac{\pi}{2}-\tau_n \Delta f, & \text{for } n=4m+1,\\[6pt] \pi-\tau_n \Delta f, & \text{for } n=4m+2,\\[4pt] \dfrac{3}{2}\pi-\tau_n \Delta f, & \text{for } n=4m+3, \end{cases} \tag{27} \]

where $m$ is an integer, and $\tau_n$ is determined by formula (26). Optimum coupling is theoretically feasible for $n<n_{\max}$, where $n_{\max}$ is the integer part of

\[ \left\{\left(\frac{f_0}{\Delta F_n}\right)^4 \cdot 4 \ln 2 \right\}. \]

Usually $n_{\max}\gg 1$; therefore the amplitude-frequency characteristic of the amplifier under consideration, for a number of stages equal to $n_{\max}$, will differ very little from the curve

\[ \nu=e^{-3\Delta f^4}. \tag{28} \]

Formula (28) is obtained in passing to $n\to\infty$.

Although the passage to $n>n_{\max}$ is not physically justified, the function (28) nevertheless gives a good approximation to the amplitude-frequency characteristics of multistage bandpass amplifiers of this type. But in approximating the amplitude-frequency characteristic of an amplifier by the function (28), it is necessary to find the phase-frequency characteristic corresponding to it. For this purpose one must likewise pass to $n\to\infty$ in (27).

Thus we arrive at the same conclusions as in the preceding case. In particular, the relation between the amplitude-frequency characteristic, represented in the form (28), and the corresponding phase-frequency characteristic proves to be non-unique: to one amplitude-frequency characteristic there corresponds an infinite set of phase-frequency characteristics; however, in some cases the infinite set of phase-frequency characteristics can be reduced to a finite number of lines.

Conclusions

1. The phase-frequency characteristic of a multistage bandpass amplifier is linear within the limits of small generalized detunings and, in addition to the component $-\tau_n\Delta f$, contains the constant component $\left(n\cdot\dfrac{\pi}{2}\right)$.

2. The limiting phase-frequency characteristic of the multistage bandpass amplifier considered is linear and is inclined at an angle $\dfrac{\pi}{2}$ to the frequency axis.

1. Comparing the result obtained with the data presented in the works of Siforov, Belousov, and other authors, one may conclude that the limiting amplitude-frequency characteristics correspond to linear phase-frequency characteristics inclined at an angle \(\frac{\pi}{2}\) to the frequency axis.

REFERENCES

1. Kharkevich A. A. Theory of Electroacoustic Apparatus. Svyazizdat, 1939.
2. Belousov A. P. Radio Engineering, No. 6, 1946.
3. Siforov V. I. Radio Receiving Devices. LKBVIA, 1947.
4. Chistyakov N. I. Resonant Amplifiers and Preliminary Selectors. Svyazizdat, 1939.