UDC 621.372542.07

NULL CIRCUITS FREQUENCY-TUNABLE BY A SINGLE ELEMENT

E. G. EMEL’YANOV, N. M. STEPANENKO

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

Two null circuits are considered that are modifications of the TT-RC bridge. It is shown that, in terms of the uniformity of the frequency response, constancy of the quality factor, and a number of other properties, the TT-RC bridge with separated circuits is superior to the TT-RC bridge with two inputs.
Illustrations 2, references 6.

In the construction of selective amplifiers and oscillators, null circuits are widely used, including the double T-shaped RC bridge (TT-RC bridge). Tuning of an ordinary TT-RC bridge over a wide frequency range is possible by changing a minimum of two elements [1]; however, the TT-RC bridge with two inputs [2, 3, 4] and the TT-RC bridge with separated circuits [5, 6] make it possible to tune the frequency with a single element. This circumstance may prove decisive when choosing a tunable null circuit for a device whose adjustment must be carried out automatically, since it eliminates the need for precise tracking of the tuning elements.

Let us consider in more detail the basic properties of the TT-RC bridge with two inputs and of the TT-RC bridge with separated circuits.

The transfer coefficient of the TT-RC bridge with two inputs is described by the equation [3]

\[ \dot{T}= \frac{m\left(n^{2}-1\right)} {n^{2}-m-jn\sqrt{m}\cdot\frac{1}{q}}, \tag{1} \]

where \(m\) is the connection coefficient of the HF section;

\(q\) is the quality factor of the TT-RC bridge at \(m=1\);

\(n\) is the relative detuning;

\[ n=\frac{\omega}{\omega_{0}}; \]

\(\omega_{0}\) is the tuning frequency.

\[ q= \frac{\sqrt{C_{1}C_{2}R_{3}(R_{1}+R_{2})}} {C_{3}R_{1}+C_{2}(R_{1}+R_{2})}; \]

\[ \omega_{0}^{2}= \frac{1} {mC_{1}C_{2}(R_{1}+R_{2})R_{3}} = \frac{C_{1}+C_{2}} {k\,C_{1}C_{2}C_{3}R_{1}R_{2}}. \]

If

\[ \frac{R_1 R_2}{(R_1+R_2)R_3}=\frac{C_1+C_2}{C_3}, \]

then at the tuning frequency \(\omega_0\) the bridge transmission coefficient is equal to zero; moreover, with the bridge elements unchanged, the frequency \(\omega_0\) can be retuned by changing \(m\), i.e., by a single element.

However, first, retuning \(\omega_0\) by changing \(m\) leads to a substantial asymmetry of the amplitude-frequency characteristic (AFC) of the bridge. From (1) we have \(\dot T(n=0)=1\), \(\dot T(n=\infty)=m\). Consequently, already with a threefold change in \(\omega_0\) the null circuit practically degenerates either into a low-pass filter (when \(\omega_0\) is increased) or into a high-pass filter (when \(\omega_0\) is decreased).

Second, if at \(m=1\) the phase-frequency characteristic (PFC) of the bridge is symmetric with respect to \(\omega_0\), then changing \(m\) leads to a shift of the PFC, as is evident from the expression for the bridge PFC

\[ \varphi=\operatorname{arctg}\frac{n\sqrt{m}}{q(n^2-m)}. \tag{2} \]

Third, the selectivity of the TT-\(RC\) bridge with two inputs is maximal at \(m=1\) and decreases sharply as \(m\) is changed [3].

It should also be noted that the transmission coefficient of this bridge has a high sensitivity to changes in the element parameters, and that overall tuning is difficult.

The listed shortcomings, especially the shift of the PFC relative to \(\omega_0\) when \(m\) is changed, hinder the use of the TT-\(RC\) bridge with two inputs in active filters operating with a large frequency coverage.

Fig. 1. Schematic diagram of a TT-\(RC\) bridge with two inputs

Fig. 2. Schematic diagram of a TT-\(RC\) bridge with separated circuits

The transmission coefficient of a TT-\(RC\) bridge with separated circuits is described by the equation

\[ \dot T= \frac{K_1(1+j\omega\tau_2)-K_2\omega^2\tau_2\tau_3(1+j\omega\tau_1)} {(1+j\omega\tau_1)(1+j\omega\tau_2)(1+j\omega\tau_3)}, \tag{3} \]

where \(\tau_1=R_1C_1\), \(\tau_2=R_2C_2\), \(\tau_3=R_3C_3\). \(K_1\) and \(K_2\) are the transmission coefficients of the amplifiers, respectively, in the LF and HF branches of the bridge.

When the condition

\[ \tau_1=\tau_2 \tag{4} \]

is satisfied, (3) is rewritten in the form

\[ \dot T=K_1 \frac{ 1-\dfrac{K_2}{K_1}\omega^2\tau_1\tau_3 } { 1-\omega^2\tau_1\tau_3+j\omega(\tau_1+\tau_3) }. \tag{5} \]

It follows from (5) that at the frequency

\[ \omega_0=\sqrt{\frac{K_1}{K_2\tau_1\tau_3}} \tag{6} \]

the bridge transmission coefficient is equal to zero.

The tuning frequency \(\omega_0\) can be varied either by changing the ratio \(K_1/K_2\), or by a conjugate change of the time constants \(\tau_1\) and \(\tau_2\), or by changing only the time constant \(\tau_3\).

Of greatest interest is the possibility of retuning the frequency \(\omega_0\) by changing \(\tau_3\)—the time constant of the circuit consisting of resistor \(R_3\) and capacitor \(C_3\), i.e., by changing one element (either resistor \(R_3\) or capacitor \(C_3\)), without thereby violating the condition for obtaining a zero of the transfer coefficient at the frequency \(\omega_0\).

To avoid the undesirable effects inherent in circuits with two inputs, it is necessary to ensure equality of the gain coefficients of the amplifiers

\[ K_1 = K_2. \tag{7} \]

When condition (7) is satisfied, expression (5) can be written in the form

\[ T = \frac{K_1}{1 - j \frac{1}{qY}}, \tag{8} \]

where

\[ Y = \frac{\omega}{\omega_0} - \frac{\omega_0}{\omega} \]

is the generalized detuning;

\[ \omega_0 \]

is the tuning frequency of the bridge;

\[ \omega_0 = \frac{1}{\tau_1\sqrt{a}}; \tag{9} \]

\(a\) is the tuning parameter;

\(q\) is the quality factor of the bridge;

\[ q = \frac{\sqrt{a}}{1+a}; \tag{10} \]

\[ a = \frac{\tau_3}{\tau_1}. \]

The AFC and PFC of a TT-\(RC\) bridge with separated circuits are described by the equations:

\[ |H| = \frac{K_1}{\sqrt{1 + \frac{1}{(qY)^2}}}; \tag{11} \]

\[ \varphi = \operatorname{arctg}\frac{1}{qY}. \tag{12} \]

It follows from (11) and (12) that when a TT-\(RC\) bridge is retuned by changing \(\tau_3\) over a wide frequency band, symmetry of the AFC and PFC with respect to \(\omega_0\) is ensured.

The quality factor of the bridge is maximal at \(a=1\) (i.e., \(\tau_1=\tau_3\)) and is equal to \(q_m=0.5\). For \(a \ne 1\), always \(q<q_m\); however, the quality factor decreases considerably more slowly than in a TT-\(RC\) bridge with two inputs. The change in the quality factor within a specified range will be minimal if, at the center frequency of the range \(\omega_c=\sqrt{\omega_{\min}\cdot\omega_{\max}}\), the condition \(a=1\) is satisfied. In this case the quality factor is minimal at the boundaries of the range and, for example, with a twofold frequency coverage \(q_{\min}=0.47\).

It is simplest to ensure fulfillment of condition (7), and also to guarantee high stability of \(K_1\) and \(K_2\) and a minimum of introduced nonlinear distortions, if voltage followers are used as the amplifiers.

Let us consider the question of the sensitivity of a TT-\(RC\) bridge with separated circuits to the spread of the parameters of passive elements.

The spread of \(R_1, R_2, R_3, C_1, C_2, C_3\) leads to the appearance of a spread in the values of the time constants \(\tau_1, \tau_2, \tau_3\). Taking this spread into account, (3) is written in the form

\[ \dot{T}+\delta_T = \frac{ K_1[1+j\omega(\tau_2+\delta_2)] + K_2\omega^2(\tau_2+\delta_2)(\tau_3+\delta_3)[1+j\omega(\tau_1+\delta_1)] }{ [1+j\omega(\tau_1+\delta_1)][1+j\omega(\tau_2+\delta_2)][1+j\omega(\tau_3+\delta_3)] }. \tag{13} \]

It follows from (13) that a zero of the bridge transfer coefficient at the tuning frequency occurs when the conditions

\[ \tau_1=\tau_2 \quad \text{and} \quad \delta_1=\delta_2 \tag{14} \]

are simultaneously satisfied.

A possible shift of the tuning frequency relative to \(\omega_0\), determined by (6) or (9) without allowance for spread, can always be compensated by a small change in \(\tau_3\).

It should be noted:

a) the inclusion of amplifiers in the TT-\(RC\) bridge circuit makes it possible to carry out independent adjustment of the time constants of all circuits and, consequently, tuning of the TT-\(RC\) bridge with separated circuits as a whole, even if elements with large tolerances are used to construct the bridge;

b) to obtain a zero of the transfer coefficient at the tuning frequency, it is sufficient to satisfy conditions (14), which makes it possible to make effective use of parametric stabilization of the characteristics of the TT-\(RC\) bridge with separated circuits.

The analysis performed and the experimental verification of the results have shown the promising nature of using a TT-\(RC\) bridge with separated circuits, despite a certain complexity of the circuit, and its clear advantages over the TT-\(RC\) bridge with two inputs.

REFERENCES

1. Yu. A. Andreev, V. O. Kobak. Double T-shaped bridges in selective amplifiers. L., “Sudpromgiz,” 1962.

2. D. Roy. Null circuits with two inputs. TIIER, 1967, vol. 55, No. 2 (Russian translation).

3. Ganguli. Null circuits with two inputs. TIIER, 1966, vol. 54, No. 3 (Russian translation).

4. Swamy. Null circuit with two inputs. TIIER, 1966, vol. 54, No. 8 (Russian translation).

5. Frequency responsive Network. U.S. Patent No. 3296463.

6. Frequency responsive Network. U.S. Patent No. 3296464.