ACTIVE NOTCH FILTER WITH AN EXTREMELY LOW LEVEL OF INTRINSIC NONLINEAR DISTORTION

E. G. Emel’yanov, M. S. Roitman

(Presented by the scientific and technical seminar of the Department of Radio Engineering)

In the investigation and development of ultralinear measuring amplifiers and generators with a precision curve shape, it becomes necessary to measure very low levels of nonlinear distortion \((K_f\) less than \(0.05\%)\). Industrially produced nonlinear-distortion meters (S6-1, S6-1A), harmonic analyzers (S5-2, S5-3), and spectrum analyzers do not make it possible to measure such low levels of \(K\) because of the comparatively high level of their own distortion (of the order of \(0.05 \div 0.1\%\)) and their limited dynamic range \((\leq 60\ \text{dB})\) [1].

If, in the investigation of measuring amplifiers, a substantial increase in resolving power and expansion of the dynamic range of serial instruments can be achieved by the use of compensation measurement methods [2], then in the investigation of generators, selective amplifiers, etc., the application of the amplitude-compression method gives a positive effect. The method consists in incomplete suppression of the fundamental harmonic relative to the higher harmonics [3, 4, 5].

Filters used for amplitude compression must provide:

— an extremely low \((K_f < 0.001\%)\) level of intrinsic nonlinear distortion;

— the required \((\geq 20\ \text{dB})\) suppression of the signal at the fundamental frequency;

— a small (preferably less than \(+3\%\)) nonuniformity of the transmission coefficient for the higher (second, third, etc.) harmonics.

In works [3, 4, 5], preference was given to passive \(RC\) filters, despite the considerable \((5 \div 7\ \text{dB})\) nonuniformity of the transmission coefficient for the higher harmonics, and only because passive filters do not introduce their own nonlinear distortions into the signal under investigation.

However, for purposes of amplitude compression when measuring low levels of nonlinear distortion, active notch filters (ANF) can be successfully used, constructed on the basis of cathode (emitter) followers and passive null \(RC\) circuits, for example, double T-shaped \(RC\) bridges, or, for short, TT–\(RC\) bridges (Fig. 1).

It is known that the level of nonlinear distortion introduced into the signal spectrum by an active filter is determined by the nonlinear distortion of the basic amplifier (in this case, the follower).

The nonlinear distortions introduced by a simple cathode follower, with an output signal on the order of \(1 \div 1.5\) V, can be reduced to a level of \(0.002\%\). And a cathode follower with parallel control, at a certain depth of positive feedback, even with an output signal on the order of 5 V, makes it possible to achieve almost complete compensation of its own nonlinear distortions [6].

Fig. 1. Block diagram of an active notch filter (ANF)

The output signal of the ANF, for an input signal with small \(K_f\), is determined mainly by the magnitude of the first harmonic component. The latter, in turn, is at least an order of magnitude lower than the magnitude of the first harmonic component at the input of the ANF. Therefore, the determining factor is the nonlinear distortion introduced into the signal spectrum by the input device connected between the ANF and the signal source.

Let us consider the influence of the follower gain, the input and output resistance of the follower, and the output resistance of the signal source on the main characteristics of the ANF.

Fig. 2, a. Structural diagram of the ANF

Graph. ANF

In the circuit of Fig. 1, \(Z_g\) corresponds to the output resistance of the signal source; \(Z_n\), which is the load of the TT-\(RC\) bridge, corresponds to the resistance of the follower between points “1” and “2”; \(Z_0\) corresponds to the output resistance of the follower.

The block diagram of Fig. 1 corresponds to the structural diagram of Fig. 2a, which in turn corresponds to the graph of Fig. 2b.

Here:

\(K_1\) — gain of the follower;

\(T_1\) — transfer coefficient of the TT-\(RC\) bridge for the generator signal;

\(T_2\) — transfer coefficient of the TT-\(RC\) bridge for the feedback signal (see Fig. 3), taking into account the influence of the external circuits \((Z_g, Z_0, Z_n)\).

The transfer coefficient of the ANF is found from the formula

\[ K=\frac{K_1 \cdot T_1}{1-K_1 \cdot T_2}. \tag{1} \]

Under the condition \(Z_n=\infty\), the relation holds

\[ T_1+T_2=1, \tag{2} \]

and

\[ K=\frac{K_1\cdot T_1}{1-K_1(1-T_1)}. \tag{3} \]

Below we shall consider the case when the repeater has zero phase shift (\(K_1\) is a purely real quantity).

Fig. 3. Circuit of a TT—RC bridge taking external circuits into account

\[ T_1=\left.\frac{U_3}{U_1}\right|_{U_2=0},\qquad T_2=\left.\frac{U_3}{U_2}\right|_{U_1=0}. \]

Variant I. \((Z_{\mathrm{g}}=0,\ Z_0=0,\ Z_{\mathrm{n}}=\infty)\). From [7], when the conditions for zero balance of the TT—RC bridge are satisfied, we have

\[ T_1=\frac{1}{1-j\frac{D}{Y}}, \]

then

\[ K=\frac{K_1}{1-j\frac{D(1-K_1)}{Y}}, \tag{4} \]

and

\[ Q_{\mathrm{e}}=\frac{1}{D(1-K_1)}. \tag{5} \]

where:

\[ D \]
— attenuation of the TT—RC bridge;

\[ Y=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \]
— generalized detuning;

\[ \omega_0 \]
— tuning frequency of the TT—RC bridge;

\[ Q_{\mathrm{e}} \]
— equivalent quality factor of the active rejection filter.

The amplitude-frequency (AFC) and phase-frequency (PFC) characteristics can be calculated from the formulas

\[ |K|=K_1\left[1+D^2(1-K_1)^2Y^{-2}\right]^{-1/2}, \tag{6} \]

\[ \varphi=\operatorname{arc\,tg}\frac{D(1-K_1)}{Y}. \tag{7} \]

At the tuning frequency \((Y=0)\)

\[ |K||_{Y=0}=K_0=0. \]

For large detuning \((Y \to \pm\infty)\)

\[ |K|\big|_{Y\to\pm\infty}=K_m=K_1. \tag{9} \]

A necessary condition for realizability of an ANF according to the circuit of Fig. 1 is the condition

\[ K_1<1. \tag{10} \]

From formulas (4), (5), and (6) we have that, for a small difference \((1-K_1)\), the equivalent \(Q\) of the ANF can be very large (100 and more), the phase shift \(\varphi\) rapidly dies out with detuning, and the modulus of the transmission coefficient differs only slightly from \(K_1\) even at small detuning.

Example 1. If \(K_1=0.98\) and \(D=4\), then \(|K|=0.99K_1\) at \(Y=\pm0.56\), i.e., for a relative frequency detuning of \(\pm33\%\).

Variant 2. \((Z_0=R_0,\ Z_{\Gamma}=0,\ Z_{\mathrm{n}}=\infty)\)

\[ T_1=\frac{(1-T_0)Y-jDT_0}{(1-T_0)Y-jD}, \]

and, taking (2) into account, we have

\[ K=K_1\frac{(1-T_0)Y-jDT_0}{(1-T_0)Y-jD[1-K_1(1-T_0)]}. \tag{11} \]

For the case of a symmetrical TT–RC bridge \((R_1=R_2=2R_3),\ C_1=C_2=0.5C_3)\)

\[ T_0=\gamma/(1+\gamma), \tag{12} \]

\[ D=4/(1+4\gamma)^{1/2}, \tag{13} \]

\[ \gamma=R_0/R_1, \tag{14} \]

\[ \omega_0=1/R_1C_1(1+4\gamma)^{1/2}. \tag{15} \]

The amplitude-frequency and phase-frequency characteristics of the filter can be calculated from the formulas

\[ |K|=K_1\left\{\frac{(1-T_0)^2Y^2+D^2T_0^2}{(1-T_0)^2Y^2+D^2[1-K_1(1-T_0)]^2}\right\}^{1/2}, \tag{16} \]

\[ \varphi=\operatorname{arctg}\frac{D[1-K_1(1-T_0)]}{(1-T_0)Y} -\operatorname{arctg}\frac{DT_0}{(1-T_0)Y}. \tag{17} \]

At the tuning frequency \((Y=0)\)

\[ |K|\big|_{Y=0}=K_0=\frac{K_1T_0}{1-K_1(1-T_0)}, \tag{18} \]

\[ \varphi\big|_{Y=0}=\varphi_0=0. \tag{19} \]

For large detuning

\[ |K|\big|_{Y\to\pm\infty}=K_m=K_1, \tag{20} \]

\[ \varphi\big|_{Y\to\pm\infty}=0. \tag{21} \]

The equivalent \(Q\) of the ANF is

\[ Q_{\mathrm{e}}= \left. \frac{d\left|\dfrac{K}{K_m}\right|}{dY} \right|_{Y=0} \simeq \frac{1-T_0}{D[1-K_1(1-T_0)]}. \tag{22} \]

Analysis of formulas (13)–(22) shows that inclusion of the resistance \(R_0\) in the TT–RC bridge circuit leads to substantial quantitative and qualitative changes in the properties of the ANF:

1 — the transmission coefficient of the ANF at the tuning frequency \(\omega_0\) is not equal to zero \((K_0>0)\);

2 — the value \(K_0\) depends on the transmission coefficient of the TT-\(RC\) bridge at the tuning frequency and on the depth of negative feedback;

3 — there is no phase shift at the tuning frequency \((\varphi_0=0)\);

4 — there is a “downward” shift of the ANF tuning frequency relative to the tuning frequency of the TT-\(RC\) bridge.

The maximum phase shift \(\varphi_{\mathrm{cr}}\) is observed at a certain detuning \(Y_{\mathrm{cr}}\)

\[ Y_{\mathrm{cr}}=\pm \frac{DT_0}{1-T_0}\,[1-K_1(1-T_0)]^{1/2}. \]

Example 2.
\(R_1=20\ \text{k}\Omega,\ R_0=20\ \Omega\), permissible nonuniformity of the transmission coefficient over the harmonics not more than \(\pm 2\%\).

As follows from (16), the transmission coefficient of the follower must be not less than \(K_{1,\min}=0.92\).

In this case \(K_0=0.012\), i.e., the ANF provides suppression of the fundamental-frequency signal only by a factor of 80. If this is insufficient, then, while preserving the same nonuniformity of the transmission coefficient over the harmonics, reducing \(K_0\) is possible only by decreasing \(T_0\).

The above analysis of the ANF characteristics was carried out under the assumption that the resistance \(R_0\) is included in the TT-\(RC\) bridge circuit and the output signal is taken from point 3. In reality, however, point 2 is the output of the ANF. Such a simplification is legitimate and leads to an insignificant error if \(R_0 \ll R_1\). In the real case, a feedthrough signal is always present at the ANF output; taking it into account, the ANF transmission coefficient is characterized by the formula

\[ K_{\Sigma}=K_1 \frac{(1-T_0)Y-jDT_0} {(1-T_0)Y-jD[1-K_1(1-T_0)]} +\Delta . \tag{24} \]

For the case of a simple cathode follower, a TT-\(RC\) bridge symmetrical with respect to the bridge, and \(Z_g=0,\ Z_n=\infty\), we have

\[ \Delta= \frac{(1-K_1)\cdot DT_0(0.25D\sigma-j1)} {(1-T_0)Y-jD[1-K_1(1-T_0)]}, \tag{25} \]

where

\[ K_1=\frac{SR_{\kappa}}{1+SR_{\kappa}} \]

is the transmission coefficient of the cathode follower;

\[ T_0=\frac{R_0}{R_1+R_0}; \]

\[ R_0=\frac{R_{\kappa}}{1+SR_{\kappa}} \]

is the output resistance of the cathode follower;

\[ D=4(1+4R_0/R_1)^{1/2}; \]

\[ \sigma=\frac{\omega}{\omega_0} \]

is the relative detuning;

\[ \omega_0=1/R_1C_1(1+4R_0/R_1)^{1/2}; \]

\(S\) — tube transconductance;

\(R_{\kappa}\) — resistance in the cathode circuit.

The magnitude of the feedthrough signal depends both on the depth of negative feedback and on the ratio between \(R_1\) and \(R_0\); moreover, at the tuning frequency the feedthrough is determined only by the value \(T_0\).

The total transfer coefficient of the ANF at the tuning frequency is

\[ K_{\Sigma_0}=K_0+\Delta_0=T_0\frac{1+j(1-K_1)}{1-K_1(1-T_0)}, \tag{27} \]

i.e., the total signal at the output of the ANF has practically not changed in magnitude, but has a phase shift, although a small one, relative to the input signal. The output signal is in phase with the input signal at the frequency

\[ \omega'=1/R_1C_1\{1+4R_0/(K_1R_1-R_0)\}^{1/2}. \tag{28} \]

At this frequency, shifted “upward” relative to \(\omega_0\), the transfer coefficient of the ANF is

\[ K'=R_k/(R_1+R_k). \]

The direct-transmission signal and the effects associated with it can be eliminated by introducing one more follower into the ANF circuit (Fig. 4).

Fig. 4. Practical transistor circuit of an ANF using transistors

In addition, such a construction of the transistor version of the ANF makes it possible to provide the TT-\(RC\) bridge with a mode close to the no-load mode.

Variant 3.

\[ (Z_0=R_0,\; Z_r=R_r,\; Z_n=R_n). \]

Relation (2) is not satisfied.

In the general case, the AFC and PFC of the TT-\(RC\) bridge are asymmetric; however, when the condition for a symmetric TT-\(RC\) bridge is satisfied,

\[ R_1^2+4R_0=2R_rR_n \tag{29} \]

frequency symmetry takes place.

In this case

\[ T_1=T_m\frac{(T_m-T_0)(1-\sigma^2)+j\sigma DT_0}{(T_m-T_0)(1-\sigma^2)+j\sigma DT_m}, \tag{30} \]

\[ T_2=T_m\frac{-\sigma^2(T_m-T_0)\cdot B+j\sigma DT_0\cdot A}{(T_m-T_0)(1-\sigma^2)+j\sigma DT_m}, \tag{31)*} \]

\[ \text{*) Fulfillment of condition (29) guarantees the symmetry of } T_1;\ \text{the characteristic } T_2 \text{ is symmetric only when } R_r=0,\ R_n=\infty. \]

and the formula for the transfer coefficient of the ANF has the form

\[ K = K_1 \frac{(T_m - T_0)(1-\sigma^2)+j\sigma DT_0} {(T_m - T_0)[1-\sigma^2(1-K_1B)] + j\sigma DT_0(1-K_1A)}, \tag{32} \]

where

\[ T_m = [1+2m(m+2)(1+4\gamma)^{-1}]^{-1}, \]

\[ T_0 = \gamma[1+m+\gamma+2m(m+m\gamma+2\gamma)(1+4\gamma)^{-1}]^{-1}, \]

\[ D = 4\gamma(T_m - T_0)/T_0(1+4\gamma)^{1/2}, \]

\[ \sigma = \omega/\omega_0, \]

\[ \omega_0 = 1/R_1C_1(1+4\gamma)^{1/2}, \]

\[ A = (1+m)\cdot\gamma^{-1}, \]

\[ B = 4m\cdot(1+4\gamma)^{-1}, \]

\[ m = R_r/R_1,\quad \gamma = R_0/R_1. \]

For purposes of amplitude compression, it is advisable to design the ANF with an input follower (the level of nonlinear distortion of the input follower should be as low as possible). In this case the following ratios are realistically ensured:

\[ m \ll 0.01,\quad \gamma \ll 0.01, \]

and the calculation of the principal parameters of the ANF can be carried out using the formulas valid for variant 2.

In conclusion, it should be noted that at the tuning frequency of the ANF one can obtain practically complete suppression of the signal; for this it is sufficient only to slightly unbalance the TT-RC bridge, for example by changing the value of the resistance \(R_3\).

REFERENCES

1. Radio-Measuring Instruments. Catalog-prospectus, VNIITEIR, Moscow, 1968, 1969.

2. M. S. Roitman. Measurement of nonlinear distortion of an amplifier. Second All-Union Scientific and Technical Conference on Measuring Amplifier Technology (abstracts of papers). Leningrad, 1967.

3. M. S. Roitman. Generator of purely sinusoidal voltage. Izv. vuzov SSSR—Radioelektronika, Kiev, No. 8, 1967.

4. Yu. F. Khimerik, Z. F. Lozevskii, Yu. G. Bulyubash. Measurement of small values of the nonlinear-distortion coefficient. Izmeritel’naya Tekhnika, No. 9, Moscow, 1968.

5. N. B. Petrov. Measurement of small coefficients of nonlinear distortion. Izmeritel’naya Tekhnika, No. 7, Moscow, 1968.

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