Two-terminal network with negative resistance
Abstract Generated abstract
This paper examines the use of controllable negative resistance for compensating active losses in complex impedances, with attention to measuring applications such as electromagnetic defectoscopy and structuroscopy. It analyzes a two-terminal network based on an amplifier with deep positive feedback, showing that although such a circuit can raise the equivalent resistance and quality factor, its usefulness is limited by instability of the amplifier gain and output resistance. The authors propose a stabilization circuit that compares amplifier input and output voltages and automatically adjusts gain through a control signal. The derived relations indicate that this method can improve negative-resistance stability without reducing the depth of positive feedback, making the stabilized two-terminal network suitable for measurement circuits.
Full Text
Two-Terminal Network with Negative Resistance
V. K. Zhukov, I. G. Leshchenko, A. V. Emelyanenko
(Presented by the scientific seminar of the Department of Information and Measuring Technology)
An important feature of negative resistances, which makes it possible to use them in measuring technology, is the possibility of neutralizing, with their aid, the active component of a complex impedance and thereby increasing its equivalent quality factor. Especially effective in this respect is the case in which the active component of the complex impedance considerably exceeds the reactive component and the application of known methods for measuring complex impedances does not give the desired result.
Of interest is the use of controllable negative resistances in electromagnetic defectoscopy and structuroscopy for compensating active losses both in the transducer itself and in the shield, which makes it possible to increase substantially the sensitivity to the useful controlled parameter. Compensation of the active losses of two-terminal networks can be reduced to a simplified circuit in which a negative resistance is connected in parallel with the loss resistance. The resulting equivalent active resistance
\[ R_{\mathrm{e}}=\frac{RR_{-}}{R_{-}-R}, \]
turns out to be greater than each of the connected resistances and, in the limit, under complete compensation, is equal to \(\infty\).
Of all known elements [2, 3] with a falling section of the current–voltage characteristic, the most suitable, from the standpoint of energy, dynamic range, and controllability, is an amplifier encompassed by deep positive feedback (PFB). The input resistance of such an amplifier turns out to be negative (Fig. 1) and equal to
\[ R_{-}=\frac{R_{\mathrm{out}}+R_{\mathrm{p.f.}}}{1-k^{*}} . \tag{1} \]
The principal disadvantage of negative resistances based on amplifiers with PFB is their high instability
\[ \frac{\Delta R_{-}}{R_{-}}= \frac{\Delta R_{\mathrm{out}}}{R_{\mathrm{out}}} + \frac{\Delta k^{*}}{k^{*}}\cdot \frac{k^{*}}{1-k^{*}}, \tag{2} \]
Fig. 1
which limits the possibility of using them in measuring equipment. The instability of \(R_{-}\) has an especially strong effect on the value of \(R\):
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \frac{\Delta R_{-}}{R_{-}}(1-\eta), \tag{3} \]
where \(\eta=\dfrac{R_{\mathrm{e}}}{R_x}\) is the regeneration coefficient.
Substituting into (3) the value of \(\dfrac{\Delta R_{-}}{R_{-}}\) from (2) and taking \(k\gg 1\), we obtain the expression
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \left( \frac{\Delta R_{\text{out}}}{R_{\text{out}}} + \frac{\Delta k^*}{k^*} \right)(1-\eta), \tag{4} \]
from which it follows that the instability of the resistance formed by parallel-connected active positive and negative resistances is increased by a factor of \((1-\eta)\).
The use of negative feedback [1] does not make it possible to stabilize the value of the negative resistance at a given regeneration coefficient, since it reduces the depth of the positive feedback. It is therefore expedient to consider other stabilization methods.
Figure 2 shows one possible circuit of a stabilized two-terminal network with negative resistance.
Fig. 2
The operation of the circuit is based on comparison of the input and output voltages of the amplifier and automatic adjustment of the gain by a difference signal. In particular, if the gain of amplifier \(Y_1\) differs from its nominal gain \(k_0\), the input voltage \(U_1\) becomes unequal to the voltage \(U_{\mathrm{d}}\) taken from the calibrated divider KD. At the output of the comparison circuit CC there arises a voltage proportional to the difference between the voltages \(U_1\) and \(U_{\mathrm{d}}\); it is amplified by amplifier \(Y_2\) and applied to the phase-sensitive rectifier PSR, from the output of which a dc voltage \(U_{\mathrm{ctrl}}\) is taken, controlling the gain of amplifier \(Y_1\).
If the gain of the amplifier is represented in the form
\[ k=\frac{U_2}{U_1}=\frac{R_{\mathrm{oc}}k^{*}+R_{\mathrm{out}}}{R_{\mathrm{oc}}+R_{\mathrm{out}}}, \tag{5} \]
then the expressions for the instability of \(R_{-}\) and \(R_{\mathrm{e}}\) take the form:
\[ \frac{\Delta R_{-}}{R_{-}}=\frac{\Delta k}{1-k}; \tag{6} \]
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \frac{\Delta k}{1-k}\,(1-\eta), \tag{7} \]
To determine the instability of the gain \(k\), let us denote by
\(k_0\) — the gain of amplifier \(Y_1\) without stabilization;
\(k_{\mathrm{e}}=k_1 k_2 k_3\) — the equivalent transmission coefficient,
where
\(k_1\) — the transmission coefficient of the comparison circuit;
\(k_2\) — the gain of the amplifier;
\(k_3\) — the transmission coefficient of the PSR.
The dependence of the gain of amplifier \(Y_1\) on \(U_{\mathrm{ctrl}}\) can be expressed as follows:
\[ k=k_0(1+k_y U_{\mathrm{ctrl}}), \tag{8} \]
where
\[ U_{\mathrm{ctrl}}=(U_1-U_{\mathrm{d}})k_{\mathrm{e}}; \]
\[ U_{\mathrm{d}}=\frac{U_1 k}{k_{\mathrm{d}}}; \]
\(k_{\mathrm{d}}\) — the division coefficient of the calibrated divider;
\(k_y\) — the control coefficient.
Substituting its value for \(U_{\mathrm{ctrl}}\) in (8), we obtain:
\[ k=k_0\left[1+k_y k_{\mathrm{e}}U_1\left(1-\frac{k}{k_{\mathrm{d}}}\right)\right]. \tag{9} \]
Hence
\[ k=\frac{k_0+k_0 k_y k_{\mathrm{e}}U_1}{1+\dfrac{k_0 k_y k_{\mathrm{e}}}{k_{\mathrm{d}}}\cdot U_1}; \tag{10} \]
\[ \frac{\Delta k}{k}=\frac{\Delta k_0}{k_0}\left(\frac{1}{1+\frac{k_0 k_y k_{\mathrm{e}}}{k_{\mathrm{d}}}\cdot U_1}\right). \tag{11} \]
Assuming that \(k\gg 1\),
\[ \frac{\Delta R_{-}}{R_{-}}=\frac{\Delta k_0}{k_0}\left(\frac{1}{1+\frac{k_0 k_{\mathrm{e}} k_y}{k_{\mathrm{d}}}\cdot U_1}\right); \tag{12} \]
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \frac{\Delta k_0}{k_0} \left( \frac{1-\eta}{1+\frac{k_0 k_{\mathrm{e}} k_y}{k_{\mathrm{d}}}\cdot U_1} \right). \tag{13} \]
or
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \frac{\Delta k_0}{k_0}\frac{1-\eta}{k_{\mathrm{st}}}, \tag{14} \]
where \(k_{\mathrm{st}}=1+\dfrac{k_0 k_{\mathrm{e}} k_y}{k_{\mathrm{d}}}\,U_1\) is the stabilization coefficient equivalent to the depth of negative feedback \((1+k\beta)\).
When the divider division coefficient is chosen equal to the nominal gain of the amplifier, we obtain
\[ k=k_0;\qquad k_{\mathrm{st}}=1+k_{\mathrm{e}}k_y U_1, \tag{15} \]
from which it is clear that such stabilization does not reduce the depth of positive feedback and at the same time increases the stability of the negative resistance by a factor of \((1+k_{\mathrm{e}}k_y U_1)\).
If \(k_{\mathrm{e}}k_y U_1\gg 1\), then \(k_{\mathrm{st}}=k_{\mathrm{e}}k_y U_1\).
In this case the instability of the equivalent resistance will be equal to:
\[ \frac{\Delta R_{\mathrm{e}}}{R_{\mathrm{e}}} = \frac{\Delta R_x}{R_x}\cdot \eta + \frac{\Delta k_0}{k_0}\, \frac{1-\eta}{k_{\mathrm{e}}k_y\cdot U_1}. \tag{16} \]
The circuit (Fig. 2) can be simplified if \(k_{\mathrm{d}}\) is taken to be somewhat less or greater than the nominal amplifier gain \(k_0\), namely:
\[ k_{\mathrm{d}}=k_0\left(1\pm \frac{\Delta k_0}{k_0}\max\right), \tag{17} \]
where \(\dfrac{\Delta k_0}{k_0}\max\) is the maximum instability of the gain of amplifier \(U_1\).
This makes it possible to eliminate from the stabilization circuit the phase-sensitive rectifier and the reference-voltage source.
The analysis carried out shows that the introduction of a circuit for stabilizing the amplifier gain makes it possible to substantially increase the stability of the negative resistance.
The resulting stability of the negative resistance proves sufficient for the use of the two-terminal network in measuring circuits.
References
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G. Girnton. Stabilized negative resistances, Electronics, 1945, July.
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B. M. Polivoda, A. P. Nasonov. A meter of complex resistance with self-compensation. Measuring Technology, No. 7, 1969.
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V. V. Kogen-Dalin, B. A. Pashkov. Negative resistance for compensating losses in alternating-current circuits. Transactions of the Moscow Power Engineering Institute, issue LVII, 1964.