Active notch filter
Abstract Generated abstract
This paper examines the stability of signal suppression in active notch filters built with follower stages and passive RC null networks. It proposes and analyzes a circuit in which a damping pole is formed with minimal adjustment, reducing the dependence of the rejection level on the follower output resistance and allowing the active filter rejection frequency to coincide with that of the passive network. Derived amplitude and phase relations indicate a frequency shift between the amplitude and phase characteristics, while nonlinear distortion remains small under typical feedback conditions. Experimental tests with tube and transistor followers and several null networks are reported to confirm the main theoretical conclusions.
Full Text
ACTIVE NOTCH FILTER
Yu. K. Rybin
(Submitted by the scientific seminar of the Department of Radio Engineering)
One of the important characteristics of an active notch filter (ANF) is the level of signal suppression at the notch frequency and its stability. Unfortunately, in ordinary ANFs implemented on the basis of a follower and a passive \(RC\)-network, the suppression level depends on the degree to which the follower characteristics approximate the ideal ones, namely on its input and output resistances and their time and temperature stability. In [1] an analysis was carried out of an ANF circuit in which the influence of these resistances was compensated by forming an additional channel for signal passage to the output. However, the stability of the suppression level in this circuit is determined by the accuracy with which the compensation conditions are fulfilled, which may vary when external conditions change.
A practically drawback-free ANF circuit is shown in Fig. 1. In this circuit a damping pole is formed, but practically without any additional adjustments. This circumstance is of substantial importance in the case of a strictly fixed tuning of the passive network (for example, when using notch filters based on distributed \(RC\)-lines).
Fig. 1
Let us analyze the circuit, assuming that the resistance of the grid–cathode section is equal to \(\infty\).
In this case the transmission coefficients from the input to the grid and cathode have the form
\[ \dot{T}_{12}=\frac{T_{0}+jQY'}{1+jQY'}\ldots, \tag{1} \]
\[ \dot{T}= \frac{ T_{0}+jQK_{\mathrm{p}}Y+jT_{0}\dfrac{\omega}{\omega_{0}} }{ 1+jQY' }\ldots, \tag{2} \]
where \(T_0\) is the transmission coefficient at \(Y=0\);
\[ T_0=\frac{\alpha}{1+\alpha-K_{\mathrm{p}}}, \]
\(\alpha\) is a coefficient equal to
\[ \frac{R_{\mathrm{out}}}{R}; \]
\(R_{\mathrm{out}}\) is the output resistance of the cathode follower,
\(K_{\mathrm{p}}\) is the transmission coefficient of the cathode follower;
\(Q\) is the equivalent quality factor of the active rejection filter:
\[ Q=\frac{q}{1+\alpha-K_{\mathrm{p}}}\cdot \sqrt{1+\frac{\alpha}{q}}, \]
\(q\) is the quality factor of the \(2T\)-\(RC\) bridge;
\(Y,\;Y'\) are the generalized detunings without and with allowance for \(R_{\mathrm{out}}\),
\[ Y=\left(\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\right),\qquad Y'=\left(\frac{\omega}{\omega_0}-\frac{\omega'_0}{\omega}\right). \]
\[ \omega'_0=\frac{\omega_0}{\sqrt{1+\frac{\alpha}{q}}}; \qquad \omega_0 \text{ is the rejection frequency.} \]
It is obvious that the transmission coefficient of the filter to the output is equal to
\[ \dot T_{16}=-(\dot T_{12}-\dot T_{15})S R_a \ldots , \tag{3} \]
where \(S\) is the tube transconductance.
Substituting (1) and (2) into (3), after simple transformations we obtain
\[ \dot T_{16}=-\frac{K}{1+K\beta}\cdot \frac{jQY}{1+jQY'} \ldots , \tag{4} \]
where \(K\) is the gain of the amplifier without feedback,
\(\beta\) is the feedback coefficient
\[ \beta=\frac{R_k}{R_a}. \]
Hence the amplitude-frequency and phase-frequency characteristics, respectively, are
\[ |\dot T_{16}|=\frac{K}{1+K\beta}\frac{QY}{\sqrt{1+Q^2Y'^2}}\ldots , \tag{5} \]
\[ \varphi=\operatorname{arctg}\frac{1}{QY'}\ldots . \tag{6} \]
Analyzing expressions (5) and (6), one may conclude that:
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The rejection frequency of the active filter and of the passive null network coincide \(\left(|\dot T_{16}|=0 \text{ at } Y=0\right)\).
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Between the amplitude-frequency and phase-frequency characteristics there is a frequency shift \(\left(|T_{16}|=0 \text{ at } Y=0,\ \varphi=\pm 90^\circ \text{ at } Y'=0\right)\).
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The rejection level is practically independent of the output resistance.
The coefficient of nonlinear distortion of the filter is small, since usually \(\beta \simeq 1\), i.e., it is approximately equal to the distortion of the cathode follower. Without carrying out an additional analysis, it can be shown that an analogous effect is also achieved with any other null network.
Experimental verification
Active notch filters (ANFs) with various null networks were investigated, both with a tube and with a transistor follower; in the latter case the follower was implemented using a composite transistor. The results of the experimental verification of the ANF according to the circuit shown in Fig. 2 are presented in Fig. 3.
Fig. 2
For comparison, amplitude–frequency characteristics are given for the passive null network (curve 1), at the emitter of \(T2\) (curve 2), and at the output (curve 3).
Fig. 3
As can be seen from Fig. 3, the main conclusions of the theoretical analysis are confirmed quite convincingly by experiment.
LITERATURE
- M. Z. Chapovskii, V. N. Loveiko. Active notch filter with improved characteristic. In: Problems in the Theory and Practice of Active Filters. Taganrog, 1970, issue 29.